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The comparative roles of connected and disconnected trajectories in the evaluation of the semiclassical coherent-state propagator

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The comparative roles of connected and disconnected trajectories in the evaluation of the semiclassical coherent-state propagator
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Rubin, Andrew E
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iv, 74 leaves : ill. ; 29 cm.

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Thesis (Ph.D.)--University of Florida, 1998.
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Includes bibliographical references (leaves 72-73).
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Typescript.
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Vita.
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by Andrew E. Rubin.

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THE COMPARATIVE ROLES OF CONNECTED AND DISCONNECTED TRAJECTORIES IN THE EVALUATION OF THE SEMICLASSICAL COHERENT-STATE PROPAGATOR












By
ANDREW E. RUBIN












A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FUFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1998














TABLE OF CONTENTS

page
A B ST R A C T ................................................................. iv

SECTIONS

INTRODUCTION AND REVIEW OF LITERATURE ................... 1

THE FEYNMAN PATH INTEGRAL ..................................... 4

LEBESGUE INTEGRATION ............................................ 8
STOCHASTIC VARIABLES AND WIENER MEASURE ................. 11

Random Variables in One Dimension ............................ 12
Stochastic Variables ............................................ 14
W iener Processes ............................................... 17
Stochastic Integrals ............................................. 23

REVIEW OF COHERENT STATES ..................................... 25

THE COHERENT-STATE PATH INTEGRAL ........................... 28

DERIVATION OF THE SEMICLASSICAL COHERENT-STATE
PRO PAG ATO R .................................................. 32
Boundary Conditions for Eqs. (7.7) ............................. 33
Sem iclassical Action, F ......................................... 34
A m plitude Factor, E ............................................ 37

FORMULATION OF THE SEMICLASSICAL COHERENT-STATE
PROPAGATOR IN SUMMARY ................................... 42

APPLICATION OF THE SEMICLASSICAL COHERENT-STATE
PRO PAG ATO R ................................................... 45
Origin of M ultiple Solutions ..................................... 45
Non-contributing Solutions ...................................... 47

NATURE OF THE TRAJECTORIES .................................... 55

Continuously Connected and Disconnected Trajectories ......... 57 Continuously Connected and Disconnected Regions ............. 60










NUMERICAL RESULTS ................................................ 63

SUMMARY AND CONCLUSION ........................................ 70

R E FE R E N C E S .............................................................. 72

BIOGRAPHICAL SKETCH ................................................. 74














Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

THE COMPARATIVE ROLES OF CONNECTED AND DISCONNECTED
TRAJECTORIES IN THE EVALUATION OF THE SEMICLASSICAL COHERENT-STATE PROPAGATOR


By

Andrew E. Rubin

December 1998

Chairman: John R. Klauder
Major Department: Physics

The Feynman path integral is discussed and why, strictly speaking, this is not an actual integral in the sense of Lebesgue. An exact path integral expression for the coherent-state propagator, developed by Daubechies and Klauder, is presented and discussed. A semiclassical approximation to this path integral obtained by Klauder is then derived from the exact path integral expression. This approximation which was subsequently modified by Adachi is applied to the quartic oscillator. The evaluation of this semiclassical expression involves classical trajectories which must satisfy complex boundary conditions. It is found that these complex classical trajectories fall into two broad categories basically characterized by the descriptive titles "continuously connected" and "disconnected" given to the two different types. The continuously connected type is found to always contribute in the evaluation of the semiclassical propagator while the disconnected type will only contribute under specific conditions.















INTRODUCTION AND REVIEW OF LITERATURE


The Feynman path integral representation of the x-to-x propagator involves taking the continuum limit of a time slicing formulation (see Sec. 2). Strictly speaking this formulation results in a linear functional but not an actual integral (in the sense of Lebesgue). The Feynman expression has motivated various attempts at providing well defined integral expressions for quantum mechanical propagators (not necessarily x-to-x) [1]. In the interest of providing such an expression Klauder and Daubechies [2] derived an exact path integral expression involving Wiener measure (Sec. 4) for the propagator,

K(p", q", T; p', q', 0) = (p", q"le- Tlp', q'), (1.1) where R- is the Hamiltonian operator corresponding to the classical Hamiltonian H, and Ip, q) is a coherent state (Sec. 5).

In Sec. 2 the Feynman path integral will be discussed, and in Sec. 3 the rigorous definition of an integral in the sense of Lebesgue will be presented along with why the Feynman integral does not qualify. The coherent-state path integral itself is presented in Sec. 6 and, in the interest of understanding this formulation, stochastic variables and Wiener measure are discussed in Sec. 4, followed by coherent states in Sec. 5.

Following the derivation of the exact path integral expression for the propagator, Klauder [3] obtained a semiclassical approximation for it,

Ksc(p", q", T; p', q', 0) = Eef/ (1.2) This semiclassical expression is evaluated along trajectories q(t) and p(t) satisfying










Hamilton's equations,

: OH _ OH
q- and p- (1.3) but subject to complex boundary conditions (CBC)

q' +ip' = q' + ip', (1.4)

Q4" - ip" = q" - ip"

with q' = q(0), i" =(T), p' = P(0), p" = P(T) and Q, q',p', q",p" E R. [Here a single trajectory refers to a single set (q(t),p(t)) satisfying (1.3) and (1.4).] As a result of the CBC the functions 1(t) and p(t) are generally complex valued which causes the "action" F also to be complex. Equation (1.2) along with the CBC (1.4) will be derived from the path integral expression for the propagator in Sec. 7. The results of Sec. 7 will be summarized and further discussed in Sec. 8.

In actual application of Eq. (1.2), it is found that for a given set of CBC there will in general exist more than one complex trajectory satisfying Hamilton's equations. It was originally expected [3] that only a particular one of these solutions (see Sec. 10) was needed. However, it was shown by Adachi [4] that the number of contributing trajectories [for given boundary conditions (1.4)] varies according to specific rules so that Eq. (1.2) should be replaced by

Ksc(p",q",T;p',q',0) = Ee'. (1.5)
contributing
trajectories
However, Adachi's scheme (see Sec. 9) for the general application of (1.5) is arrived at semi-empirically, and in the interest of studying quantum chaos, a somewhat special system was used, that is, the delta kicked rotator
1 20
H--yp +Kcosq E 6(t-n), KcR, (1.6)
2 =-o0o










so that with = 4(n-) and pn = p(n-), one obtains the discrete equations

4n+1 4n +'pn+l, (1.7)

Pn+1 = n +Ksin 4.

Here (1.2) will be applied to an integrable system, the quartic oscillator,
1_2 1_4
H=1p2 + q 4. (1.8)
2 4

It is found that all the features of the kicked rotator system which lead to Adachi's semi-empirical methods, such as interference effects near a "phase-space caustic" (see Sec. 9) are also present in this system. In addition we discover some interesting properties of complex classical trajectories, i.e. solutions of (1.3) subject to the CBC (1.4). These and related topics are discussed in Secs. 9 through 11 and summarized in Sec. 12.















THE FEYNMAN PATH INTEGRAL


The x-to-x quantum mechanical propagator is given as,

Q(x", T; x', O) = (x" I-e -Tlx)., (2.1) where 7 is the Hamiltonian operator corresponding to the classical Hamiltonian H. A lattice formulation for this propagator can be derived by first requiring that H = T+V, so that W = T+V [with T = p2/(2m) so that T= p2/(2m)], and then making use of the Trotter Product Formula [5]
a2
e-a(T+V)-N e -aT/Ne-aV/N + O( -a), (2.2) with a = iT/h; then for sufficiently large N, the term e-iT/h in (1.1) becomes,

eiTTh/h = e-aT+V) = [e-aT+V)/N]N '- [e-aT/Ne-a/N]N, (2.3) so that (2.1) becomes,

Q(x",T; x', 0) = lim (x"I[e -a/Ne-av/N]Nxi). (2.4) N-.. oo

A resolution of unity in terms of position eigenstates,

f dxijxi)(xil = i, (2.5) is then inserted between each of the N products in Eq. (2.4), so that I = 1, 2,.. N-I and (2.4) becomes,
N-1
Q(x", T;x',0) = lim I dX.' dxN-1 11 (Xk+1I[e-aT/Ne-aV/NulXk) (2.6) N--*oo f k










with x0 = x' and XN = x". Each integration in this expression is thought of as corresponding to the fixed and equally spaced times, t = k(T/N) = kE with k = 1, 2,... , N - 1; values of the xk along these time slices are thought of as lattice points. Assuming V is a function of position only,

e-v/Nx~I - Ixe(Xk)/e. (2.7) We may then evaluate the matrix elements, (xk+i Ie- aT/N Xk) = x+l -af,2/(2mN) jxk), occuring in Eq. (2.6) using the resolution of unity,

f dplp)(pl i, (2.8)

and the formula,
1 _� (2.9)
(px)- - 2rh
so that

(qle -aT/INIx) f dP(qle -af2/(2mN)1P)(PIX) (2.10) dpe-aP2/(2mN)(qlP)(Pjx) 2irh 1 dpe-ap2/(2mN eK ) 2(hfdp ex p[ a 2 e x)p]
-- dp e p a p2 + -(q- x)


This integral can then be evaluated using the general formula,

f e-a2+Oydy = e02/(4.), (2.11) so that (2.10) becomes

(qle-aT/NIx) = 2iah exp [-' (q - x)2 (2.12) Formula (2.6) then becomes

Q(x", T; x', 0) (2.13) = lim dxl... dxNN-1 mN l exp [-N (Xk�1 Xk)2 -V(Xk)
N-oo- 2ah2 k=O










which becomes,

Q(x", T; x', 0) (2.14)

pim fdx... dxN_1 ( [ 2 exp E mN (Xk+l Xk)2 - TV(xk)]
_ 00 f GiihT) hk=O I2

after combining exponentials and using a = iT/h. Now using E T/N, this expression takes on the more lattice-like form,

Q(x", T; x', 0) (2.15)

- Nlimo J dxl dxN l (2.. _N--) exp { 1 E E0 (Xk+l Xk) _ V(xk)1 }

The Feynman path integral may be obtained from (2.15) by first identifying the expression, E EN-o{(m/2)[(Xk+l - Xk)/E]2 - V(xk)}, with the classical action over the broken line path, x', xl,..., XN-1, X". Then interpreting this set of lattice points in the limit E --+ 0 as one possible "path" one obtains the formal expression,

F(x", T; x', 0) = AfJ e s[x(t)] I dx(t), (2.16) where K is an infinite normalization constant and,

Six(t)] = foTL[x(t),.b(t)]dt; (2.17) the integral being taken over all paths satisfying x(0) = x' and x(T) = x". We may denote the continuous Cartesian product of the values of position at each time between 0 and T as R(OT). The entire path space with x(0) = x' and x(T) = x" fixed, is then given by, SP {x'} x R(oT) x {x"}.
It should be noted that the paths contained in Sp need not be continuous since Xk+1 - Xk need not approach zero as E -+ 0; continuous paths also, need not be differentiable, since Xk+1 - Xk may approach zero, but not in a way such that (Xk+l - Xk)/6 remains finite; an average path in Sp, in fact, will suffer discontinuities everywhere,










although it is generally believed that (2.16) is supported on a subset of Sp containing much smoother paths.

A semiclassical approximation for the x-to-x propagator, (x" le-iT/H x'), can be derived from the lattice expression (2.15) or analogously from the path integral formula (2.16). This expression is given as

J(x", T; x', 0) A K S(X/",X), (2.18) where,

A= ( [ x j (2.19) and,

S(x", I') [P)(t) H(p, x)]dt. (2.20) The sum in (2.19) is taken over each set of x(t), p(t) satisfying the extremal equations
OH OH
OH and 9H '(2.21) subject to the boundary conditions,

x(0) = x' and x(T) = x". (2.22)

In n dimensions the expression for A above generalizes to include a determinant, known as the Van Vleck determinant [6], and is given as


A (2 {det [J (k,q = 1,...n). (2.23) The importance of this determinant in approximating the wave function was previously realized by Van Vleck while studying the correspondence between certain quantum formulas, in the "transformation theory" of Dirac [7] and Jordan [8], and their corresponding classical expressions.














LEBESGUE INTEGRATION


As was mentioned in the introduction the Feynman path integral is a linear functional not an actual integral, this being due to the lack of a measure. In this section measure and the Lebesgue integral will be discussed as well as why the Feynman integral fails to possess an actual measure.

The fundamental ingredient in the theory of Lebesgue integration (on the real line) is a function called the measure which takes as its argument any of a wide class of subsets of the real line known as measurable sets (see for example the first citation of Ref. 9). The measure must satisfy the properties of being both nonnegative, that is if S is some set of points (or the null set), its measure r(S) > 0, as well as that of countable additivity, that is if

A U An, with AinA = 0, for any i, k such that i $ k, then, (3.1)

m(A) = m(An).

Additional properties are sometimes specified.

The measure as originally defined by Lebesgue (Lebesgue measure) generalized the concept of length so that if a set S is some open interval, its measure m(S), is equal to the length of that interval, and if S is a single point then m(S) would be zero, [although many other measures can be specified which satisfy properties (3.1)]. From the countably additive property of measures it follows that the Lebesgue measure of a countable set of open intervals is the sum of the lengths of those intervals, and the Lebesgue measure of a discrete set of points such as the integers is zero. In fact the Lebesgue measure of the rational numbers (within some bounded or unbounded










interval) is zero (a demonstration of this can readily be found in the second citation of Ref. 9). The Lebesgue measure of the irrationals in an interval (a, b), a < b, would just be the length, b - a, of the interval, since the interval can be decomposed into both rational, R(a,b), and irrationals, I(a,b), so that

m[(a, b)] = b - a = m(R(a,b)) + m(I(,,b)) = 0 + m(I(a,b)) = m(I(a,b)) (3.2) by the additive property (3.1).

Using a general measure the Lebesgue integral of a nonnegative real-valued function, f(x), on an interval a < x < b, with a < b, is then defined by partitioning the y-axis (as opposed to the x-axis as in the Riemann scheme) over the range of f(x) on a < x < b. Then let Ei be the set of x values for which Yi-1 < f(x) < yi, with

1... N, then form the sum i=1 77m(Ei), where qi is any value within [yi-l, yi], and m(Ei) is the measure of the set E. The integral is then given by b +N


with each interval Ayi = Yi - Yi-1 of the partition approaching zero. This definition of the integral can be generalized to more complex functions in a straightforward manner.

Due to the additive property of the measure (3.1), the value of the integral (3.3), is insensitive to a redefinition in the values of the function at points x in the interval [a, b] constituting a set of measure zero. For example the integral (3.3) would have the same value for the function f(x) = x2, as for the function
x' for x irrational
0 for x rational
on any bounded interval.

In the case of a Lebesgue measure the differential, dm(x), is written as dx and it can be shown that whenever some function is Riemann integrable it is Lebesgue integrable to the same value. There are, however, certain functions which are Lebesgue










integrable which fail to be Riemann integrable, such as the Dirichlet function which has the value one when x is rational and zero when x is irrational (a demonstration of this involving the Dirichlet function can readily be found in the second citation of Ref. 9). The integral (3.3) has other significant advantages as well, such as the dominated convergence theorem [9].
This definition of the Lebesgue integral also carries over naturally to functions defined on any arbitrary space on which a measure can be defined. For example a function the domain of which consists of paths, x(t), from some appropriate path space on which a measure can be defined.

According to the Lebesgue dominated convergence theorem, ep[_(1 + a) ZFNj' X?]
lim C dxl ... dxN-1 (# 1 (3.4)
a-O o(V7r )N-1

- dxl ... dXN-1 lim exp [- a = i -) FO1_t0 (V#)N-1
with N defined as in Sec. 2 and a > 0. The left hand side of (3.4) is equal to lima-O 1/(1 + a)(N-1)/2 = 1 which is clearly equal to the right hand side. If we now replace a in (3.4) by E, with c = TIN as defined in Sec. 2, the left hand side of (3.4) becomes equal to limNg-+ 1/(1 +T/N)(N-1)/2 = e-T/2, while the right hand side becomes equal to

Iexp - x2)l 11 -- i=, (3.5)
t
with t denoting a continuous index such that t E (0, T); this shows that (for any T > 0) the dominated convergence theorem no longer holds as c = TIN -4 0, implying that limN-+o dxl ... dxN-1 - 1- dx(t) no longer provides us with a measure.














STOCHASTIC VARIABLES AND WIENER MEASURE


An actual measure satisfying the properties discussed in Sec. 3 can be constructed on the set of all paths, X(t) G {{x'} x R(o,T) x {x"}}, obtained from the limit of the lattice formulation as N -- co and E = T/N -+ 0 (see Sec. 2). It is first convenient to consider paths on the lattice such that the initial point x' is fixed at x' =0 while the final point x" XN is left free, i.e., sets of lattice points such that IX1,...,XN1,x"} {O, X1,...,XN-I,XN} with E = T/N, x' = 0 and XN free to vary.

A Gaussian weight factor,

1 1
exp -- (x,+l - X,)2 (4.1)
/2lrE 12E

with i = 1, 2,..., N is then associated with each step in the lattice, so that the weight or probability density associated with a given path becomes, N-1 1 [1]
PN(Xl,X2,...,XN) f i exp i (4.2) i=O 2lrE I2f

( lrE2)Nexp 1 N-1(xi+-iX,)2 E. The expression,

MN(S) fPN(Xl,X2,...,XN)dxl ...dxN, (4.3) then provides a measure on the N-dimensional path space with S being any arbitrary but fixed region of this space [10]; the expression PNdxl ... dxN can then be though of as the measure of the infinitesimal region dxl ... dxN about the path










{0, X1,... , XN1, XN}. Also note that (4.3) integrated over the entire path space is unity.

The limit of PN dxl ... dXN as N --+ co and E = T/N -+ 0 continues to provide us with a measure called the Wiener measure [10], the measure now being over the function space {0} x R('T]; denoting this limit of PN dxl ... dxN by diuw(X) we then obtain,

dliw(X) =Afe-2 fo x2(tdt fi dX(t), (4.4) where A/ is an infinite normalization factor and fI dX(t) represents an infinitesimal volume about X(t). The measure of some arbitrary but fixed region of path space, SP, can then be written as,

pw(Sp) = fs e- f 2(t)dt dX(t). (4.5) Note that since the value of (4.3) integrated over the entire path space is constant and equal to unity its value will remain the same in the limit as N --+ c so that (4.5) continues to be normalized to unity when Sp equals the entire path space.

An analogous measure, known as a pinned Wiener measure, can be derived on the function space {x'} x R(oT) x {x"} simply by using (4.2) with x0 = x' as well as XN = x" both arbitrary but fixed and excluding the integration over dxN from (4.3). This measure will no longer be normalized to unity but will have the value (1/v2rT)e[-(x" '-X)2/(2T)] over the entire space. Both these measures can also be derived from the more general theory of stochastic variables which will be subsequently discussed.

Random Variables in One Dimension In one dimension a random variable, x, is defined as a variable which can take on a real value from some specified set of values with a given probability. The moments










of the distribution are defined as

(x n), n =1,2, 3.... (46) where (F) = (F(-)) denotes the average of some arbitrary function F(x) over the entire ensemble of values normalized so that (1) = 1. Note that in the case where the set of values which x may assume is R, (F) is given by,

(F) = J F(x)P(x)dx, (4.7) where P(x) is the probability distribution of x on R.

It is also convenient to introduce the connected moments, (Xn)c, which are defined by the relation,
(eax) e(eax-)c = e[a(x)c + a(x�+!3(X3)o+j, (4.8)


with a E C. The first two connected moments are given by,

WxC = Wx, (4.9)

(X 2)c = X2)-_ (X)2,


note that (x2), is equivalent to the variance of the distribution. Another useful function is the characteristic function,

C(s) = (e"'), (4.10) with s E R, which contains information on the moments encoded within it. It should also be noted that in the case where C(s) is known (and x E R) it follows from (4.8) that,

(eZ. i eisxP(x)dx = C(s), (4.11)










and P(x) can be found by way of the inverse Fourier transform,

P(x) = J ef esC(s)ds. (4.12)

Various classes of probability distributions can now be described by specifying certain constraints on the values of the connected moments, for example a Gaussian distribution is defined as one for which all connected moments vanish except the first two. In this case we find, using (4.8), that the characteristic function has the form,

C(s) (eiSx) = e s (4.13) since (x), = (x). The probability distribution, P(x), for a Gaussian distribution can now be obtained from (4.13) and is found to be,

P(x) = 1 exp (X2 W , (4.14) the familiar form for a Gaussian distribution in one dimension. It should be noted that particular specifications of the values of the connected moments are generally equivalent to specifying a measure on the set of values of which the random variables may take on, although under certain circumstances this measure so determined is not unique.

Stochastic Variables
In the theory of stochastic variables a measure on some space of functions (such as {x'} x R(oT) x {x"} defined in Sec. 2) may be defined indirectly by specifying an appropriate complete set of correlation functions. The correlation functions being a generalization of the moments (4.6) (when x is a continuous variable), are defined as

(X(tl)X(t2) ... X(tk)), for k = 1, 2,3, ... , (4.15) here ((-)) = f(.)djt(X) with dpi(X) being the measure of the infinitesimal volume dX(t), about X(t), and (1) generally being normalized to unity, although this may










not always be the case. The X(ti), i = 1,2,...,k in (4.15) represent any given function (or stochastic variable), X(t), from the space of functions, evaluated at the arbitrary but fixed times t = t1, t2,. . . , tk, with t always taken to be finite so that 0 < ti < T with 0 < T < co or 0 < ti < T in the case where T is infinite. The correlation functions (4.15) are also chosen so that, when each t, equals some fixed value t, they will correspond to the moments of a one dimensional probability distribution,

(Xn (t)), n =1,2,3,., (4.16) that is X(t), for any fixed value of t, is equivalent to a random variable in one dimension. As stochastic variables are generally defined on a function space for which they can take on continuous values for fixed values of t, such as Sp = {x'} x R(oT) X { x"} for example, X(t) can be viewed as representing the corresponding value of x on the t'th time slice. In other words, for the space Sp above, for example, X(t) E R for any given X(t) with fixed t E (0, T). Considering this, and that X(t) is a well defined random variable for fixed t, it follows that when t is such that X(t) E R, (F{X(t)}) is equivalent to,

(F{ X(t)}) F{x}P(x,t)dx, (4.17) where P(x, t) is the probability distribution of X(t) on the t'th time slice and F{X(t)} is a general function of X(t).
It is again convenient to generalize (4.8) and define connected correlation functions by the relation,

_~') e 2 i (4.18)


where (s, X) =f6 s(t)X(t)dt with s(t) some arbitrary but fixed function for which the integral is convergent. The first two connected correlation functions are found










from (4.18) to be,

(X(t)), = (X(t)), (4.19)

X(tl)X(t2))C, = (X(t1)X(t2)) - (X(t))(X(t2)).

The characteristic functional, the generalization of Eq. (4.10), is defined as,

C{s} = (ei(S'x)). (4.20) In particular note that by setting s(t') = a6(t' - t) with a E R, C{s} becomes,

C(a) = (eiaX(t)), (4.21) the characteristic function for the stochastic variable X(t). Utilizing (4.17) we obtain

C(a) = (eiaX(t)) = J eiaxP(x, t)dx, (4.22) which can be used to find the distribution P(x, t) by way of the inverse Fourier transform as in the one dimensional case.

Various types of distributions or stochastic processes (the term process being implied by the time evolution of the stochastic variables along with their particular properties) can now be described by specifying certain constraints on the values of the connected moments. In particular a Gaussian stochastic process [10] is defined by the first two connected correlation functions with all higher-order connected correlation functions being equal to zero. Using (4.18) and (4.19) the characteristic functional for a Gaussian process is found to be,
C{s} = (e(Sx)) = e[i(sx)c- ((sx)2)c] (4.23)


-- etf s(t)(X(t))dt- L f s(tl)s(t2)(X(t1)X(t2)),dtldt2 In particular, by setting s(t') = a6(t' - t) with a E R in (4.23), one obtains the characteristic function for a Gaussian process,

(eiax(t)) = eia(x(t)) e-a2(x2(t))c, (4.24)










note that this has the same form as (4.13) with x replaced by X(t). This now allows us to use (4.22) to obtain,

J eicXP(x,t)dx= eia(X(t))e- 2Kx2(t))c, (4.25) which allows us to find the distribution P(x, t) by way of the inverse Fourier transform. Similarly to (4.14) we now obtain,

P(xt 1 exp [X - (X(t)]2 (4.26)
P ) 27r(X2(t))Cx 2(X(t)

showing that the paths X(t) have a Gaussian distribution over any given time slice.


Wiener Processes
The measure discussed at the beginning of this section follows from a particular type of Gaussian process known as a Wiener process or Brownian motion. A Wiener process is characterized by the conditions,

W(O) = 0, (W(t)) = 0, (4.27) where t > 0, with the connected two-point correlation function given by,

(W(t1)W(t2)) = min(tl,t2), (4.28) (W(tl)W(t2)), being equal to (W(tl)W(t2) since (W(t) = 0, with all higher-order connected correlation functions vanishing as is the case for all Gaussian processes. Note that the path space on which the measure is defined is {0} x R(oT]. This process is easily generalized to the case where W'(0) = x' with x' arbitrary. In this case (4.27) is replaced by, W'(0) = x' and (W'(t)) = x'; the variance given by the right hand side of (4.28) remains the same, however, we must write (W'(t1)W'(t2) , = min(t1,t2), since (W'(t)) no longer has a zero value; as the process is still Gaussian, again, all higher-order connected correlation functions must vanish. It is easily seen through










direct calculation that this process is entirely equivalent to W'(t) = W(t) + x'. It is sometimes useful to define a Wiener process with a different diffusion constant. This process, W,(t), is identical to W(t), with the exception that (4.28) is replaced by (Wv (tl) Wv (t2)) = v min(t1, t2). Again this process is simply related to W (t) through the equation W (t) = v-iW(t). The process W(t) itself is sometimes referred to as a standard Wiener process.

The nature of Brownian paths
In the case of the Wiener measure one finds that the measure is supported only on a special class of paths in {0} x R(,T], that is all sets of paths outside this class have measure zero. To show this first define X(t) to be a Gaussian variable with mean value (X(t)) equal to zero so that (4.26) becomes,

P(x,t) = 1 exp X (4.29)
2ijr-X2(t)) I-2(X2(t))J
Therefore, as (X2(t)) -- 0, equation (4.29) for P(x, t) tends towards a delta function for any value of t, hence the measure can only be supported (nonzero) on X(t) such that X(t) = 0. Also note that if (X2(t)) = 0, it follows that,

J x2p(x, t)dx = 0, (4.30) therefore since x2 > 0 the same conclusion applies. Now if (X2(t)) = co we see that P(x, t) 0 for any finite value of x, hence it can be inferred that in the limit as (X2(t)) - 0,
fbP(xt)dx= 0 (4.31)


for any a, b E R with a < b. On the other hand since f P(x, t)dx = 1, independent of (X2(t)), we have lim(x(t))_ ff_ P(x,t)dx = 1. Now since ff. Pdx = f - Pdx + f', Pdx+f: Pdx (with a > 0) it follows from (4.31) that in the limit as (X2(t)) -+ cc

-P(x,t)dx + P(x,t)dx=1 (4.32)










for any finite but arbitrarily large a. Hence, this heuristic argument demonstrates that the measure is not supported on any finite value of X(t) and so must only be supported on values of X(t) such that X(t) = �lo.

Next define X(t) to be X(t) = W(t + At) - W(t) with At > 0, since the sum or difference of any two Gaussian random variables is Gaussian, X is Gaussian with

(X) = (W(t + At)) - (W(t)) = 0 and

(X2) = (W2(t + At) + W2(t) - 2W(t + At)W(t)) (4.33)

= (t+At)+t-2t=At.

Hence as At -* 0 the variance, (X2) -+ 0 so that the measure becomes concentrated only on values where X = 0. Although X(t) represents the difference between values of the function W(t) at times t + At and t, Eq. (4.33) is not enough in itself to insure continuity of the W(t)'s. For example if a W(t) possesses an isolated jump discontinuity at t = td this will result in two jump discontinuities in the function X(t) with the middle segment of X ranging over the interval (td - At, td] or [td - At, td) [depending whether the jump in W(t) occurred at or after td respectively]. Now as At -+ 0 this middle segment will shrink to zero in length while the other two segments both converge to X(t) = 0 and rejoin at this point. We may therefore rewrite (4.33) as

([W(t + At) - W(t)]2) = At = a At + ad At, (4.34) where a, and ad are proportionality constants with a, related to the average slopes of the continuous segments of the W(t)'s within the At window [including the continuous segments of the W(t)'s possessing discontinuities within the window] and ad measuring the average contribution of the discontinuities within the window. In the case of higher moments of X(t) it is expected that the contribution of the discontinuities will still










be proportional to the first power of At, as this contribution has nothing to do with the average slopes of the W(t)'s within the At window, and so should be directly proportional to the width of the window. In particular for (X4(t)) we may write

([W(t + At) - W(t)]4) = C(At) + kd At, (4.35) where C(At) is some function related to the average slopes of the continuous segments of the W(t)'s within the At window and kd is a proportionality constant. Using the relation,

(S(t1)S(t2)S(t3)S(t4)) = (S(t1)S(t2)S(t3)S(t4)), + (S(t1)S(t2))(S(t3)S(t4)) + (S(t1)S(t3))(S(t2)S(t4)) + (S(t1)S(t4)) S(t2)S(t3)), for a general stochastic variable, S(t), with (S(t)) = 0, and recalling that

(S(t1)S(t2)S(t3)S(t4)), = 0

if S(t) is Gaussian as well, we may calculate,

([W(t + At) - W(t)]4) = 3(At)2. (4.36) Comparing (4.36) with (4.35) we find kd = 0, hence the Wiener measure is only supported on functions W(t) which are continuous. This result is closely related to a theorem from which the continuity of the Wiener paths also follows [11]. We may now define X = [W(t + At) - W(t)]/(At) with At > 0 which is still Gaussian for any given At. Again, (X) = [(W(t + At)) - (W(t))]/(At) = 0, while the variance is found to be (X2) = 1/(At). In this case as At -+ 0, the variance (X2) -* co, so that the measure is concentrated only on values where X = +oo. Now in this case as At -+ 0, X -4 dW(t)/dt = �oo, hence the Wiener measure is supported on functions W(t) which are everywhere continuous and nowhere differentiable.










Explicit form of Wiener measure

With X(t) = W(t+)-W(t), as earlier, this variable is Gaussian with (X(t)) = 0, the variance is found to be (X2(t)) = E, and it then follows from Equation (4.26) that,

1 [X2]
P(x t) exp - . (4.37)
v'2i re [_2E1

In this case x represents the difference in values of some continuous path, W(t), between the times t1 = t + c and t, = t; x may therefore be written, x = xf - xi, (4.37) then becomes,

1 F(Xf - j2
P(xf,tf;xi,ti) = :1 exp [ 2- (4.38) Recalling that (4.38) is the probability distribution of the random variable X(t) at time t, P(xf, tf; xi, ti) can then be understood as the probability density of some continuous function, W(t), in the function space, going from some arbitrary but fixed point xi at time ti, to some other arbitrary but fixed point xf at time tf. Note that f!0 P(xf, tf; xi, ti)dxf is equal to unity. Then dividing the time T into N intervals so that E = T/N, the probability density for a path, W(t), going from the fixed point x' = 0 at to = 0, and passing through the set of arbitrary but fixed points X1, x2,... , xN at the equally spaced times tj = E, t2 2E,..., tN = T becomes, N11 ]
PN (X1, X2 ,,N)-= ) 17 ,exp --(Xi+,- X)2 .(4.39) =0 2
Thus we recover equation (4.2) from which equation (4.4) for the explicit form of the Wiener measure follows.

Browian Bridge
In the evaluation of the Feynman path integral or the coherent-state path integral it is necessary to consider paths starting from some fixed value, x', at time t = 0 and ending at some fixed value, x" at t = T. It is therefore necessary to consider the










subset of Wiener paths which satisfy these conditions. This particular ensemble of paths, B(t), should be Gaussian with an average which varies linearly from x' to x", i.e., (B(t)) = x' + (x" - x')t/T. The variance (on some fixed time slice), (B2(t))c, must be zero at both t = 0 and t = T, therefore if

(B2(t))c = at2 + bt + c, (4.40) it follows that c = 0 and a = -b/T, so that (B2(t)), = bt(1-t/T). In the limit as T -+ co the paths will no longer be pinned at x", hence, limT.,,,(B2(t))C = (W'2(t))C = t so that b = 1 and

(B2(t)), = t(1 - t/T). (4.41) A Gaussian process having exactly these properties, and therefore entirely equivalent to extracting this subset of paths is given by,

B(t) =x+ (x" - x') + W(t) - -W(T). (4.42) Direct calculation now yields,

(B(t1)B(t2)), = min(tl, t2) W2 (4.43) T'

which clearly reduces to (4.41) in the case where tl = t2 = t.
The measure on the function space, {x'} x R(o,T) x {x"}, mentioned near the beginning of this section also follows from this process. However in this case we have,

(0) f(.)dAB(X) (4.44)
f dAB(X)
due to the fact that (1) = 1 for a Gaussian process and
dp (X) = exp (X1-- X (4.45)










Stochastic Integrals
Due to the nondifferentiability of the Wiener paths, W(t), special care must be taken in defining integrals involving the derivatives of such paths.
In the case of the integral, fT X2(t)dt, found in (4.4) [with X(t) = W(t)], the integral is defined on the lattice and is equal to infinity. It should be noted however that this integral, or the term e-2 fo r2(t)dt for example, should only appear in the form le- fo r X2(i)dt I dX(t), since this term is defined by

Are-! fo2 0 2(t)dt 11 dX (t) (4.46)

lim (- exp -2 ] dxl ... dxN which collectively represents a well defined mathematical expression, while terms such as Af or e-2fo X2(t)dt alone, are equal to infinity and zero respectively. An integral of the form foT s(t)W(t)dt, as found in Eq. (4.20) [also with X(t) = W(t)], poses no problem since both s(t) and W(t) are both finite and continuous so that s(t)W(t) is finite and continuous and hence falls within the scope of both the Reimann and Lebesgue definitions of integration.
The coherent-state path integral (to be discussed in Sec. 6), however, involves integrals of the form
fo Y(t)X(t)dt. (4.47)


There are two standard prescriptions for defining this integral,
f T N-1
Y(t)X(t)dt = lim E Y(ti) [Z(ti+) - X(ti)], (4.48) N---oo i=

and,
ytXt)dt = lim [Y(ti+i) + Y(ti)] [X(ti+l) - X(ti)], (4.49)
fo N-+ooi=O2










again where tk = ke with k = 0, 1,... , N and E = T/N, the first being due to It6 and the second, sometimes called the midpoint rule, due to Stratonovitch; these two prescriptions may or may not give the same answer depending upon circumstances. In the case where Y(t) and X(t) are both continuous and differentiable functions, or in the case where Y(t) and X(t) are independent Wiener paths (as is the case for the coherent-state path integral in suitable coordinates) both (4.48) and (4.49) yield the same values, however, if Y(t) = X(t) = W(t) we will see that these prescriptions are not equivalent. Subtracting (4.48) from (4.49) we obtain,
N-1 1 1
lim 1: [W(ti+i) - W(ti)]2 -T : 0, (4.50)
N-+oo i=O 2

where we have used the fact (which may be found in Ref. 12) that [W(t+E)-W(t)]2 = E for t + E < T and e sufficiently small. The coherent-state path integral involves an integral of the form,

Jf. [p(t)4(t) - q(t)15(t)]}I dt (4.51) where p(t) and q(t) are independent Wiener paths; this integral will be defined according to the Stratonovitch prescription.














REVIEW OF COHERENT STATES


Conventional canonical coherent states are defined as the normalized eigenstates of the annihilation operator a = (1/v/-2-h)(Qq + i3), so that

alz) = zlz), (zlz) =1, with z E C and [4,P] =ih. (5.1) It can be seen that Iz) can be written as
11.1 zat1z o Zn Iz) = e1zIeza 10) or Iz) = e-1z2 y n In) (5.2) nOv
where 10) and In) are the zeroth and nth energy eigenstates of the harmonic oscillator. Therefore

(zlz') = +e-2 +Iz'1 )ezz (5.3) while in the position representation

(qlz') = exp (IzI2 Q 2 1 12 20 (5.4)
-x -q _ -z + V -fqz') 54


If we define z = (1/ V2h)(Qq + ip) (q,p E R) then (5.3) takes the form,

(p, qlp', q') = exp {_(qpi - p - [Q(q q,)2 , while (5.4) becomes

(qjP', q') = exp - (q - + - (q - q'/2)p'] , (5.6) where Iz) = Ip, q).
The coherent states also satisfy the resolution of unity
dpdq (7
f fIP, q)(Pq ' = (5.7) 27rh










However they are not orthogonal and exhibit a linear dependency in the form

Ip',q') dpdq (p, qlp', q')1p, q) (5.8) dpq exp I-i(qP - Pq') - [1(q + q-1(p-- pl)2]} p, q)


Therefore there are more coherent states then are necessary to form a basis for the Hilbert space and the resolution of unity is also referred to as an overcompleteness relation.
It follows from (5.7) that,

IS) = f f Ip, q)(p, qS) dpdq (5.9) 27rh
where IS) is an arbitrary state vector, hence the coherent states give us a useful representation of the Hilbert space, i.e., O(p, q) =_ (p, q1S). It can be shown that the functions O(p, q) are square-integrable, bounded, continuous [13], and constitute a subspace, H0, of the vector space L2(R2). From Eq. (5.9) it follows that,

(p, qIS) = (p,qIp',q')(p',q'IS) 2hdpdq (5.10) hence (5.5) constitutes a reproducing kernel for the space H0; more generally Eq. (5.5) constitutes the integral kernel of a projection operator, P, on L2(R2) onto the space HO; note that for a general function, f(p, q) L2(R2), f(p, q) = fo(p, q) + f�(p, q) where fo E H0 and fi I Ho, therefore Pof(p, q) = fo(p, q), the integral analog of this equation being,

f f (P, l', q') f (p', q')ddq fo (p, q). (5.11)

It can also be shown that a square-integrable function g(p, q) is contained in H0 if and only if Ag(p, q) = 0, where

A = 1 [(iOp - q/2)2 + (iOq + p/2)2 - 1] . (5.12)
2=










An integral transform exists allowing us to obtain the coherent state representation, 0 (p, q) (p, qIS), of an arbitrary state vector IS), from its position representation, O(x) = (xIS). Using the resolution of unity and Eq. (5.6), we obtain the coherent-state transformation,

0(p, q) = (p, q1S) = C ) e-K f e )2h e-KxP(x)dx. (5.13) And by integrating the coherent-state transformation we obtain an inverse transformation,
1 ( ) f
O-W J e=(xP/Xli(P K)dP (5.14) 2Q \irhl













THE COHERENT-STATE PATH INTEGRAL


The coherent-state path integral is given as,

P,(h; p", q", T; p', q', 0) (6.1)

2reT/2 fexp {i f ( - qP) - h(p, q)] dt-diV((p, q),

where /4t is the product of two independent Wiener measures [i.e. dj4 (p, q) dp1v (p)dpv (q)] pinned at p', q' at t = 0 and p", q" at t = T > 0 with diffusion constant v [here, with the exception of (6.4), h and Q will both equal unity]. Note that in this case the explicit form of the Wiener measure given by (4.4) is modified to,

dpw(X) = A/"e--L 1oI2(~t dX (t), (6.2)


due to the presence of the diffusion constant v [as can be seen from (4.35) with the variance e replaced by vc], with the normalization now given by
f ,1 (p,,_)+,_ q,)_ ]
dpw (P, q) 2vT exp . p)2 + (q" - (6.3) 2vT

The function h(p, q) in the integral is related to the Hamilton operator 7t by,

h(p, q) = exp h (,2 + aq) (p, qI7 jp, q). (6.4) It is show in Ref. 2 that the relation,

(V", q"e- KT IP', q') (6.5)

= lim 27re'T/2 exp {i (p - qP) - h(p, q)] dt}dpv (p, q)










holds for a wide class of Hamiltonians, 7, including all those that are polynomials in P5 and 4 (P and 4 being operators corresponding to the Cartesian momentum and position coordinates). It should be noted that an analogous path integral representation for the configuration space propagator, (x"[e-KT Ix'), involving Wiener measure cannot be constructed due to the presence of the factor exp[i f e12dt] resulting from the kinetic energy term in the action [14].
In the case h = 0 Eq. (6.1) can be solved explicitly, the result being

P,,(h = O; p", q", T; p', q', 0) (6.6) e(p -T21p!)2 q)2] exp j(p q" - p q) - - coth(vT/2) [P+ (q" -q 2sinh(vT/2)
For T > 0 we find,

lim P, (0; p", q", T; p', q', 0) (6.7) exp i (p'q" - p"q')- 1 [(P - p')2 + (q" - q)] in agreement with (5.5), as expected for the case of no time evolution; for T -+ 0 for finite v, however, we find P,(0; p", q", T; p', q', 0) approaches 21rJ(p"-p')J(q"-q'). This reflects the fact (shown in the second paper listed in Ref. 2) that P,(0; p", q", T; p', q', 0) is the integral kernel of the operator E�(v, T) = e-,,AT on L2(R2) with A as given in Sec. 5 [note that if B is an operator its integral kernel is, B(p",p';q",q') = (p", q"jB~p', q')]. It is shown in the second paper listed in Ref. 2 that as v -+ co with T > 0, the operator E�(v, T) -+ Po, the projection operator on L2(R2) discussed in Sec. 5 whose integral kernel is (6.7), whereas if T = 0, clearly E0(v, 0) = I. More generally, for h(p, q) satisfying

f dpdqIh(p, q)12 exp [-a(p2 + q2)] < cc (6.8) for all a > 0, it is shown that Pv (h; p", q", T; p', q', 0) is the integral kernel of a strongly continuous contraction semigroup, E(v, h; T), on L2(R2). In particular for uniformly









bounded h (i.e., Ih(p, q) 0) of the h = 0 result, lim,_, E�(v, T) = P, is shown to be,

lim E(v, h; T) = Poe-WphPoT P (6.9) the integral kernel of the operator, PoeiPohPoTPo, being the propagator itself, i.e.,

[PoeiP'hp�TP0](p'', q", T; p', q', 0) = (p", q"e- KiTnp', q'). (6.10) The integral kernel of E(v,, h; T), that is P. (h; p", q", T;p', q', 0), is likewise shown to converge to,

[PoeipohPo] (p", q", T; p', q', 0) = (p", q"te- Tlp', q') (6.11) in the sense of Schwartz distributions [15] for T > 0.
Following this, pointwise convergence of 'P,(h;p", q", T;p', q', 0) to (p", q" e- kTI lp', q') for T > 0 is shown assuming h satisfies the condition,

JdpdqjIh (p, q) I exp [1 OI(p2 + q')J < 00 (6.12) for some 0 < /3 < 1 in addition to (6.8). In order to do this use is made of the expansion,

P,(h; p", q", T; p', q', 0) (6.13)

P ,(0; p", q", T; p', q', 0) - i f/ dt(Op,,,q",,,T-t, h(p, q)Op,q,,v,t)
fT dt ftl dt2(h(p, q) p, qvTtE(v, h; tl - t2)h(p, q) Op,,q,,,,t2),


with Op,q,v,t(P, q) - P(0; p, q, t; Pi, ql, 0), its explicit form being obtained from (6.6), and inner products being defined by,

(f g) = f pf,( q)g(p, q), (6.14)







31

and the expansion,

(P" q le- TnlP', q') = (p", q"lP', q') - i f, dt (p", q" I W p', q') 6.5

- fo dt, o dt2(p", q" 1He-ITNIQ1 lp', q').


Eq. (6.15) is first generalized to L2(R2) then subtracted from (6.13). It is then shown that as v -* oo the difference IP,(h; p", q", T; p', q', O) - (p", q"Ie -kT1lp, q'I tends to zero.














DERIVATION OF THE SEMICLASSICAL COHERENT-STATE PROPAGATOR


Using the explicit form of the Wiener measure (6.2), Eq. (6.5) now becomes,

K(p", q", T; p', q', 0) (7.1)

lim 27rehvT/2 exp { ['(p - qP) - h(p, q)] dt

2 1 f (p2 + 42) dt}J dp(t)dq(t),

where hi is now the diffusion constant (and 0 = 1). We seek a semiclassical approximation to this integral of the form

Ksc(p", q", T; p', q', O) = Eeif/, (7.2) where F represents the "action",

F lim fT [ -(p4 - qvv) - H(pv, qv) + (P2 + 42)dt, (7.3)
V400o 12 2v I/
evaluated along extremal trajectories, q , and p , with E representing the contribution of quadratic deviations from these rays. Note that h(p , q,) has been replaced by the classical Hamiltonian, H(pv, qv), in (7.3) since h(p, q) --+ H(p, q) as h -* 0 (as can be seen from (6.4) and Ref. 16). The extremal equations following from (7.3) are easily shown to be,

tM -OH = iiW(t) (7.4)
~-; '
O H .__ _ ( t
pM(t) + Oq(t) - -,t (7.5) subject to the boundary conditions qv(0) = q', pv(O) = p', qv(T) = q" and pv(T) = p". These equations are analogous to the Navier-Stokes equations [17] for an incompressible fluid with a small viscosity and approximate solutions for large v (exact in the









V -+ co limit) are given by [3],

qV(t) = (q' - i')e- t + q(t) + (q" - if)e -v(T-t) (7.6)

PVT) = (P' -V)e-vt + (t) +(p"- P")e -v(T-t)

where 4' = 4(0), 4" = ((T), p" =(T) with q(t) and p(t) satisfying the classical equations,
- OH OH 77
q= - and = '(7.7) subject to boundary conditions which will be determined shortly.

Boundary Conditions for Eqs. (7.7)
Substituting equations (7.6) into (7.4), yields the result,
1.(-t _q 1 (9H
- q')e-evt + -q + ( / - j")V(T - ! OH (7.8)

1 ..
i[(p' - p')e-vt + -2 P+ (p" - ")e-'. Now setting t = 0 yields,
1., (q, _ 1 aH
-(q'H- ') + + - (7.9) Si(p') + -2-P + - ,p"eVT] with q'= 4(0) and - = (0), which in the limit v -+ co, imposes the initial boundary condition,

q' + il' = q' + ip', (7.10) on q(t) and p(t). Similarly, setting t = T in (7.8) yields,

vT 1 - it . ,I 1OaH
-('Ye"+- + W - (7.11) ii V ~pV It=T
=i[(p' - P )e- v + 1 "] 12P+ P










with q" 4(T) and p = P(T), which in the limit v -+ oo, now imposes the final boundary condition,

- " - ip" = q11 - ip", (7.12)

on q(t) and p(t). The same process applied to (7.5) will produce identical results. It is seen from these mixed boundary conditions that the 1(t) and p(t) satisfying (7.7) and utilized in (7.6) are generally complex valued, hence qv(t) and pv(t) are complex as well. Note that (7.10) and (7.12) may appear to over specify the solutions of the classical equations (7.7), however if we set ' q' + w and p' = p' + w, where w C, the initial boundary condition ' + ip' = q' + ip' is automatically satisfied while the combination ' - ip' = q'- ip'+ 2w remains completely free. Hence w may be chosen so that the combination q - ip' evolves in time to the specified final boundary condition (7.12).

Semiclassical Action, F
Using Eqs. (7.6) we are able to find a semiclassical expression for F, defined by Eq. (7.3), entirely in terms of the position, q(t), and the momentum, p(t), satisfying the complex boundary conditions (CBC) given by (7.10) and (7.12). First note that for arbitrary differentable functions, p(t) and q(t), (p4 - q3) = pq - [d(pq) - p4], so that
fT -(p - qj3)dt -[p(T)q(T) - p(O)q(0)] + fp4 dt, (7.13)


so that for the extremals, p,(t) and q,(t),
(p4. - q j.,)dt =-(p q - p'q') + fT p,4, dt, (7.14)



hence F becomes,

F =-(pq - p'q') + lim j , [P, - H(p.,, q.,) + 2_-(j2 + 42) dt. (7.15)









Using (7.6) we find,

P'4' [(q" - -4")(p' -)- (q' _ -q')(p"l _ p1)]Ve-vT (7.16)

-(q' - i')(p' - 2)]ve-2t + (q" - -")(p" - p")] e-2&(T-t)

-(q, -i)pve-" + (q"- V")pve -(T-t)
(- p') - + (p" - P") -e-(T-t) + .


The first term on the right hand side of (7.16) clearly makes no contribution to F in the v -4 oo limit, while the second and third terms are readily integrated to yield,

lim [ (q' - ')(p' - P)(e-2T- 1)+ - ")(p" - - e2T) (7.17)

2(q' V) ~'(P'- + 2-(q" -7)(p"In order to evaluate the fourth and fifth terms consider the following relation where f(t) is an arbitrary differentable function on [0, T], with T > 0,
f (t) e-vtdt [f(T)e -vT f(0)] + fj T(t)e-vtdt. (7.18) Clearly, as v -+ oo, the first term on the right hand side approaches f(0); now consider the term,
1fOT f(t)e- tdt K j

e- tdt = j mrax [1- e

{where Ifmaxj is the maximum value of Ii(t)j on [0,T] }, this term clearly tends to zero as v -4 oo so that,

lim f f(t)ve-'tdt = f(0). (7.20) In an analogous manner it can be shown,

lim fT(t)ue-"(T-)dt = f(T). (7.21)










From Eq. (7.19), and its analog for the integral f0T f(t)e-(T-t)dt, the sixth and seventh terms are seen to make zero contribution to F. Hence for q(t) and P(t) differentable on [0, T] with T > 0 it is found that,

lm p'Il dt - (q' - -)(p' - V) + 1(q" - Y')(p" - V') (7.22)

-(q' - q')V + (q" - Y')V' +o p- dt,

which after rearranging terms can also be written,
lim pf~ vdt -(p Y" - q"P" + q'V- p'-') (7.23) jT~
+(pq" - p'q') - 2 W - VV + fo P A
2

and since (7.13) is true for arbitrary differentable functions q(t) and p(t) this can once again be written,

lim-pjvvdt (pl11 - q"V"+ q1 - p'') (7.24)
v-oo J0 2
+ -( P' q') � jT o ( - )]dt.


Now since the functions q' and pv satisfy the extremal equations (7.4) and (7.5) in the limit v -- oo, given the CBC (7.10) and (7.12), qv = q. and p' = pv in this limit. Therefore since,
T1 0 0 TT
lim p,4, dt = lim p,4v dt - lim p%' dt - lim dt, (7.25) Eq. (7.15) for F may be written,
F = - 2p",_jq ""+ &' _ P'') +_tr(1 . - q- -qp)- ( )d 2726
2 0o 2



Note that we may replace H(p,,q,) by H(P,q) in (7.24) since qv(t) -* q(t) and pv(t) -> P(t) for 0 < t < T as v -+ oo and in this limit, q' = q, and pv = pv, hence










in this limit the corresponding functions q,, q and p,, F differ from each other only at t = 0 and t = T (a set of measure zero) and will not effect the value of (7.26). Now in a manner similar to that of the derivation of (7.22), and utilizing the fact that q" = q, and p' = p. in the v -+ oo limit, it can be shown that,
lim (05 V V)d (7.27) = ![(p1 _,)2 + (q' _ 4')2 + (P" _ jyI)2 + (q" _ q/,)2].

We may now rearrange the CBC so that, q'-q' = -i(p'-P') and q"-" = i(p" showing that
--m [ (p + 42 )dt = 0. (7.28)
V-- cl T

The semiclassical expression for F is now found to be,
F - / (pq- � q I - P'P') -[- )-HP, ()]A1 (7.29)


evaluated along trajectories satisfying Hamilton's equations,
OH OH
4= - and P = -- (7.30) but subject to the CBC,

-+ ip' q' + ip', (7.31)
=p q" -ip".


Amplitude Factor, E
In order to evaluate that part of (7.2) representing quadratic deviations from the extremal trajectories we may rewrite (7.1) using (7.13) so that,

K(p", q", T; p', q', 0) (7.32)

lim 27re hvT/2 f exp {J [1p4 - h(p,q) dt

2hv f (52 � q2) dt} f1 dp(t)dq(t)exp -(P q -p'q')










Letting q(t) = q.(t) + v(t), p(t) = p,(t) + u(t) and replacing h(p,q) in (7.32) by H(p, q), we may expand H(p, q) about an extremal path to obtain,

H(p, q) = H(p,, q.,) + [,OpH(pl,, q,)]u + [Oq,,H(p,, q,)]v (7.33)
1 1
+1a(pv, qv) + O(pvqv) + 1Y(pv, qv) +'", where,

a(pv,q.) = O H(pv,,q.), /3(pv,q) = Op Oq H(p.,q.,) and (7.34)
'Y(Pv, qv.) = 0aq .~ v, qv),


E then takes the form,
f in [ri_ 1 1 q)]d lim 27rel'T/2] exp { [u - f(pv, qv) - qv) - -Y(pv, q dt (7.35)
V_+00 fp W [2p 2 (735

_h J (i,2 + 2 ) dt}J du (t) dv (t),

subject to the boundary conditions u(0) = v(0) = u(T) = v(T) = 0. Note that the factor I q" - p'q') has been incorporated into F. The amplitude factor, E, now has the form of the coherent-state path integral representation of a propagator with time dependent Hamiltonian,
1 1 1
H = -1aP2 + /(PQ + QP) + -YQ2, (7.36)
2 2 2
where [Q, P] = i (with h = 1), so that in Dirac notation

E = (O,OITexp -i fT (1aP2 + f3P.Q+ 17Q2 dt 0,0), (7.37) where P - Q = i(PQ + QP). It should be noted that since (7.35) is given in terms of the extremal rays p,(t) and qv(t) in the v -+ co limit the a, 0 and y in (7.36), used in (7.37), should be given in terms of poo(t) and q..(t), hence we are at liberty to replace these by p(t) and 4(t), since the corresponding functions differ only on a










set of measure zero (as mentioned earlier). Hence from (7.36) onward, a, 0 and 'Y will be given by,

a = ( ,X) (,H(,q), 3 =3(,) = apA6'qH(p,q) and (7.38)

-y =(Pq) = H(, ).

Also note that the operators 1P2, P. Q and !Q' form a basis of Lie algebra such that,

[!Q2' p2] i iP Q [!Q2, p. Q Q2, [1p2 p. Q -ip2, (7.39) and can be faithfully represented by the 2-dimensional matrices,
1p2 0 1 1Q2[ 0 0, P.Q= il. (7.40)
2 0= 1 0 ' 0 -i In this representation H takes the form,
H = [i -i0 a ] , (7.41)


so that in this representation the operator,

T xp[_fT/ 1 \p p 1_Q (.2 may be written

T expif iI dtj A= (7.43)
0 -i-Y -0 C D "

E can now be evaluated by seeking an alternative expression for (7.43) in the form,

exp [il(P2 + Q2)] exp [ii7P. Q] exp [i 1(P2 + Q2)1. (7.44) Noting that (P2 +Q2) = al and that P.Q = ia3, where a, and U3 are Pauli matrices, the matrix representation of (7.44) is easily found to be,

exp [i 1 (P2 + Q2)] exp [irIP . Q] exp [ic 1(P2 + Q2)j (7.45) [ cos isin I] [ e-" 0 1 [ cos C isinci isin cos J [ 0 e7 isin( cos J










From (5.2) it is seen that 10, 0) = 10), where 10) is the zeroth energy eigenstate of the harmonic oscillator (with h = Q = 1), so that 1(p2 + Q2)j1) = 10) and we obtain,

E= (01 exp [i 1 (P2 + Q2)] exp [i1P. Q] exp [iC1(P2 + Q2)] 10) (7.46)

exp [il ( + )] (01 exp [iiP . Q] 10),

so that using [Q, P] = i and (xJO) = exp(-x2/2)/7r1/4 this becomes,

ei( +()/2 e 7/2 e-X2/2e?/x8 e-x2/2 dx. (7.47) In order to evaluate this integral note that,

eoX8e X2/2= 1 (xax)e:v2/2 q (7.48) k=O k
but,

(x)ke -212 = [(xa)ke(X2e2)/2 k e-(2)/2 (7.49)
[ 17--0 = J '


since, (xOx)ke- (X2e2fl)/2 - ae -(X2e2)/2, so that 1[ = 0 [ k-(x 2,22q)]/ (7.50) k=0 ! q r~ o-X , / since the middle term of (7.50) is just the Taylor series expansion of e-( z 2e)/2. Eq. (7.47) then becomes,

E = ei( +()/2 e,'/2 / -(l+e21)x2/2dx - ei(C+()/2 (7.51)
lr Vc-osh 71 From (7.45) it is seen that,

(A+ D)/2 = cosh(77) cos( +C), (7.52)

(B+C)/2 = icosh(7/)sin(+C), from which it follows,

E = 1/V/[A + D - (B + C)]/2 (7.53)










This result may be reinterpreted in terms of the linearized Hamiltonian equations of motion obtained by taking differentials of (7.30), this leads to,

t = (t) (t) + a(t)f(t), (7.54)
P(t) =-7)(t00() - (t)M(t,


where again a(t), 03(t) and -y(t) are given in terms of (7.38). Since p and 4 are subject to the fixed initial boundary condition of (7.31) we find J(q' + ip') = + ip = 0 so that jY = ii. Eqs. (7.54) may be rewritten in matrix form,




note that this is in the form of Schr6dinger's equation, iv= Hfi, with H given by (7.41), thus it follows that a solution is given by,
[ 1 - A B] [] (7.56)


From (7.56) and the relation j= ij it follows that, P' + iq" [A + D - (B + C)]iT, thus it is convenient to choose i = -i/2 so that,

E = , 1 (7.57) Hence E is given in terms of solutions of (7.54), evaluated at t = T, and subject to initial conditions,


= -i/2, 9 = 1/2.


(758)













FORMULATION OF THE SEMICLASSICAL
COHERENT-STATE PROPAGATOR IN SUMMARY


The semiclassical coherent-state propagator (SCSP) as derived in Sec. 7 (and previously derived in Ref. 3) is given as,

Ksc(p", q", T; p', q', O) = Eef/. (8.1) The action F is given as

(p q"+ q'P' - P'i) + - [-q( )-H(,q)]dt (8.2) with 4(t) and P(t) satisfying the extremal equations
_ OH OH
4=- and =-- (8.3) Each extremal solution (4(t), p(t)) is subject to CBC

Qq'+ip' = Qq'+ip', (8.4)

-q' - ip" = Qq" - ip"

with 4' = q(0), q" = q(T), 1' = p(O), p" = p(T) and Q, q',p', q",p" E R.
The amplitude factor E, is calculated from quadratic deviations about the extremal (complex classical) trajectory in the exact path integral expression for the propagator [see Eq. (6.5) and (7.1)]. It is given as

E T1 (8.5)
V (T) + ifl(T)
where (t) and ji(t) are solutions of the linearized Hamiltonian equations of motion,

q2(t) (8.6)



42










[HF(p, q) = O(49H(p, 4),etc.] subject to the initial conditions

4(0) =-i/(20) and f(O)= 1/2. (8.7)

The need for CBC. In order to evaluate the propagator (8.1) we must use solutions to (8.3) which in some sense connect the entrance label (q', p') with the exit label (q", p") in a time interval T. If we were to make an obvious choice and set 4' = q' and 1Y = p' we would arrive at the fixed real value of position and momentum " = qT and " = PT after a time T. This would be acceptable only if q" = qT and p" = PT; clearly choosing i' and 1' to be real is too strict a requirement for arbitrary q" and p". Therefore in order to have extremal trajectories connecting some given entrance label (q',p') to an arbitrary exit label (q",p") the functions q(t) and p(t) must be complexified according to some set of CBC.

Why CBC (8.4)? There are many things in the formalism of the coherent-state propagator which are suggestive of the CBC (8.4) (also derived in Sec. 7), for example the quantum mechanical coherent-state propagator

(p", q"le- kT lp', q') = (z"Ie- ATHnlz) = e-( z"2+Iz'12)(Oleae e 10) (8.8) with z = (1/v/2-h)(Qq + ip). Furthermore, exact analytic expressions for the quantum mechanical coherent-state propagator exist for the three cases H = 0, H = p2/2 and H =(p2 + q2) [18,3] and when one applies the CBC (8.4) to Eqs. (8.1), (8.2) and (8.3) for these three H's one re-obtains these exact analytic expressions.

Understanding CBC (8.4). Consider the initial boundary condition 04' + i'=

Qq'+ip'. If we set q' = q'+w, p' = p'+ifw (w E C), as remarked in Sec. 7, it is easily seen that this initial condition is satisfied for any complex number w. Therefore for any fixed entrance label (q', p') we are free to choose any w so that the final boundary condition -q" + iV" = �Qq" + ip" is satisfied. The situation is easily understood if









we define two new variables P(t) = q(t) + ip(t), Q(t) = Q(t) - ip(t), which up to an unimportant multiplicative constant are new canonical variables [19]. A given entrance label (q', p') will then fix our initial value of momentum P'. Then using q' = q' + w and p' =p' + if2w our initial position Q' can be written as Q' = Qq' - ip' + 2Qw. Since w is arbitrary we are free to choose any value of Q' that will satisfy the final boundary condition Q" = Q4" - ip" = �Qq" - ip". Let us also note that when w = 0, q' = q' and ' = p' and we obtain a unique real classical ray. If w =A 0, on the other hand, the classical rays are complex.
An extremal trajectory can be viewed now as either a (1(t), P(t)) pair or its corresponding (Q(t), P(t)) pair. Now, since we fix P' [or equivalently our entrance label (q', p')] and choose Q' to yield a specific value of Q", Q" = fq" - ip", it will be convenient to alternatively view our trajectories as those functions Q(t) which yield the mapping Q' -* Q". Also, a trajectory will be fixed by the choice of initial phase space coordinates, Q' and P', but since our initial momentum P' is strictly fixed by our choice of entrance label (q', p'), each of the trajectories used in the evaluation of the SCSP will be determined only by its initial position Q'. We may thus associate each trajectory with its unique initial value Q'. Therefore relevant quantities such as F and Ksc may be viewed as functions of Q'. Real valued quantities such as FR = Re(F), F = Im(F) and IKsc may now be pictured as functions over the complex Q'-plane.














APPLICATION OF THE
SEMICLASSICAL COHERENT-STATE PROPAGATOR

Origin of Multiple Solutions
In actual practice there will generally be more than one trajectory satisfying a set of CBC. This comes about due to the mapping Q' -+ Q" (where Q' -4 Q" always implies fixed P' and T) being analytic. We will first discuss how multiple solutions arise and then demonstrate the analyticity of the map Q' -+ Q" and discuss its critical points.
Multiple solutions of mapping Q' -4 Q". In general if an analytic map f(z) contains critical points, that is z0 for which f'(zo) = 0, the mapping will be many-to-one in the neighborhood of z0. Specifically, the mapping will be (n+l)-to-one for an nth order critical point, i.e., if f(n+l)(zo) 0 0 is the lowest order non-vanishing derivative of f(z) at z = z0. For example, if the critical point is 1st order, f'(zo) = 0 and f"(zo) :$ 0, the mapping will be two-to-one in the neighborhood of z0 as can be seen from the equation f(z) - f(zo) � if"(zo)(Z _ Zo)2 which holds in the neighborhood of z0; at the point z0 itself we also see from this equation that the mapping is one-toone. The many-to-one property of the analytic map containing critical points will hold globally (except at images of critical points themselves) due to continuity, the exact number of points mapping to a given image point depending on the number of critical points and their respective order. For example, the mapping f(z) = z4/4 + z3/3 contains two critical points, one 1st order at z = -1 and one 2nd order at z - 0, and is globally four-to-one.

Analyticity of mapping Q' -+ Q". To demonstrate the analyticity of the mapping Q'-+ Q" we first recall that a solution to (8.6) for arbitrary T is given by (7.56) and









can be written,

-i ' = -iAr+Bp (9.1)
P' = -iC + D '

(0), ' = q(T), etc.] where A, B, C and D are complex constants depending upon the given arbitrary but fixed trajectory with P' and T fixed as always. Now aQ"/aQ' = JQ"/SQ' where 6Q' - Q - i;' and 6Q" - Qi7- iY' is the corresponding change in Q". Since this derivative will always be evaluated holding T and P' constant we get 5P' - + ip = 0 so that ' = i and JQ' = 2f with T arbitrary. Therefore OQ"/OQ' = (f" - ip")/(2), but using (9.1) and f' iy' we find QT'- ig' [f(A + D) - (C + Q2B)]'. And so it is found that
OQ" _ '- ii' _ Q(A + D)- (C + Q 2B) (9.2)
OQ' 2Q' 2(
independent of the arbitrary initial change in Q', 6Q' = 2QT.
Critical points of mapping Q' -* Q". The critical points of this mapping are the values of Q' for which i9Q"/Q' = 0. Notice that since i9Q"/aQ' = (i'- i")/(2f2') we can write
1 = 2f' (93)
WQ "/-a Q, V 7"- iP'7(.3

Now if we use the value of ' given in Eq. (8.7), 4' = -i/(2M), (used for the evaluation of the amplitude factor E) in the above equation we find
1 _ 2f' 1 E.
/Q4,/'Q, Q4"- i1 1 (T) + iff(T) (9.4) Therefore at the critical points of the mapping Q' -+ Q" the amplitude factor E diverges; such a critical point is referred to as a phase space caustic (PSC)[4].
It is worth mentioning in conclusion that for the three cases H = 0, H = P2/2 and H = 1 + q2), PSCs of Q' -4 Q" do not exist and consequently the map is










one-to-one. For example, for the harmonic oscillator, H = !(p2 + 42), Q1 -* Q" (with Q = 1) takes the form Q" = QteiT so that OQ"/OQ' = eiT which is never zero. Clearly in this case Q' -4 Q" is one-to-one.

Non-contributing Solutions

In this subsection we will first discuss how non-contributing solutions arise. Following this we will explain Adachi's method of drawing "Stokes lines" in the Q'-plane which separate regions of non-contributing solutions from those of contributing solutions.
In order to satisfy the initial boundary condition it was remarked that we can set = q' + w and p' = p' + if~w where w is an arbitrary complex number. Therefore, excepting the case where w = 0 (which leads to the only real classical ray) q' and 1' will be complex and so our extremal rays, (4(t),p(t)), will be complex and, in general, as a consequence so will F. This has important consequences for the SCSP. Note that since F is complex it can be written F = FR + iF, so the SCSP can be written Ksc = EeiFR/he-FI/'; from this expression we can see the importance of F in determining the amplitude of the SCSP. It follows from the Schwarz inequality that for the quantum mechanical coherent-state propagator IKI < 1, and as h decreases a valid semiclassical approximation Ksc should more nearly approach K and therefore the bound IKscI < 1 should hold for h sufficiently small. Clearly if F, < 0 the SCSP would diverge as h decreased and the bound IKscj < 1 would be violated. The amplitude factor E cannot prevent this divergence since it is independent of h [see (8.5)] and so remains constant as h is decreased. We can conclude from this that if there are in fact trajectories satisfying the CBC for which F1 < 0 they would have to be excluded from the evaluation of the SCSP.









The glaring question which now confronts us is, are there trajectories (or equivalently values of Q') for which F, < 0? For the three cases H = 0, H = p2/2 and H = 1(p + 2) for which no PSC are present and there is only one solution satisfying a given set of CBC the answer is no. In fact, for the H = 0 (with Q arbitrary) and H =(p2 + 42) (with [ = 1) cases, F1 = JIw2 and is independent of time. If there did in fact exist an H leading to Q' - Q" being one-to-one for which F < 0 for any rays it would actually be a catastrophe. Note that Q' -* Q" being one-to-one tells us that if we wish to evaluate the SCSP at a particular exit label (q", p"), so that Q" = [2q" - ip", there will be only one ray Q(t) with which we can do it. We also know that if F, < 0 for a given ray Q(t) it must be excluded from the evaluation of the SCSP. Therefore, for such rays evaluation of the SCSP would be impossible at the unique exit labels corresponding to Q", (q",p") = (Re(Q")/2, -Im(Q")).
It is also necessary to mention that when T = 0, independently of H, Ksc will reduce to the overlap of two coherent states (Eq. 5.5), as in the H = 0 case, so that F, = 4[Q(q - q')2 + [-1(p _ p,)2] = lwl2, which is always larger than or equal to zero. Again Q' -+ Q" is one-to-one and possesses no PSCs. Note that when T = 0, q' = q" and 1Y = iY' so that Q' = Qij - i 2 =4" - ig" = Q", so that Q' -+ Q" takes the simple form Q' = Q". This is obviously one-to-one and possesses no PSCs since OQ"/OQ' = 1.
For T > 0, and for more complicated systems for which PSCs exist and consequently there is more than one solution for a given set of CBC (Q' -+ Q" many-to-one), we find the answer to our question is yes. In fact, viewing F as a function of Q' we will find large regions of the Q'-plane over which F < 0 and thus the corresponding rays Q(t) cannot contribute to the evaluation of the SCSP. Figs. la and lb show F over the Q'-plane for the quartic oscillator (Eq. 1.8) with the parameter values n = 30 (30 time steps, each step being of time At = .001), Q = 1 and (q',p') = (0,0) for













flu (F) "rm(F)







-25 10 50 10
-500
100 Im(Q' ) -10 0 1m(Q" )

Re(Ql) 10 -10 Re(Q') -10




Figure 1. F, over the Q'-plane. Parameter values are n=30 (T = n At, At = .001), = 1 and (q',p') = (0, 0). (a) Clipped above at F1 = 40. (b) Clipped above at F, = 175 and below at F, = 0.


the fixed entrance label (Sec. 8). Fig. la is clipped above at F 40 and the regions of F < 0 can be clearly seen. Fig. lb is clipped above at F1 = 175 and below at F, = 0; the regions of F < 0 are clearly seen as the two flat regions in the lower left and upper right hand corners of the Q'-plane. In both figures the point Q' = (0, 0) corresponds to the real classical ray (w = 0), hence F is purely real along this ray and F = 0 at this point.

We have seen that a relationship exists between the many-to-one property of Q' -+ Q", the presence of PSCs and F < 0; it has in fact been argued above that F1 < 0 can only occur in the presense of PSCs. It was first observed by Adachi [4], for the case of the delta kicked rotator, that near each PSC there formed a region for which F < 0; the same phenomenon is observed for the quartic oscillator. This











15

10 10 5 5 00



-10 -10

-15 -15
-15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15

Figure 2. Parameter values are n = 30 (T = nzAt, At = .001), Q = 1 and (q',p') = (0,0). The horizontal axis represents Re(Q') and the vertical axis Im(Q') in each figure. (a) Contour plot of Fig. lb. The two dots represent PSCs. Near each dot is formed a region of F < 0, shown here as darkened regions. (b) Superimposed contour maps of Re(Q") and Im(Q"). Q' -- Q" is conformal except at PSCs (two dots) therefore PSCs are seen as "defects" in the mesh pattern.


can be seen from Figs. 2a and 2b. For both these figures, as in Figs. la and 1b, n = 30 [T = n At = 30(.001) --= .03], Q = 1 and (q',p') = (0,0). Fig. 2a is the contour plot of Fig. lb and the two points drawn mark the two PSCs occurring at Qp SC = (-6.76, -6.57) and Q's = (6.76, 6.57). Near each of these PSCs we find there is formed a region of F, < 0, represented by the two darkened regions of Fig. 2a. At both of the PSCs shown F = 10.0 and at the point Q' = (0, 0), corresponding to w = 0, F = 0. Fig. 2b shows the contour maps of both Re(Q") and Im(Q") superimposed over the same region of the Q'-plane as in Fig. 2a. Since Q' -+ Q" is analytic everywhere and 9Q"/Q' 0 0 everywhere except at the PSCs (see end of previous subsection), our mapping Q' -+ Q" is conformal everywhere except at the PSCs. As a consequence of conformality the contour lines of Re(Q") and Im(Q") in










Fig. 2b are seen to intersect one another at right angles. The two dots in Fig. 2b mark the position of the PSCs. We see clearly from Fig. 2b that the PSCs appear as "defects" in the mesh pattern of the contour lines. Stokes lines

In order to find the exact boundary line between the contributing and noncontributing regions of the Q'-plane, Adachi introduced the construct of "Stokes lines." We know that in the neighborhood of a PSC the mapping Q' -+ Q" is manyto-one (beginning of present section); let us assume for the sake of illustration that it is two-to-one. Therefore for every Q' in the neighborhood of a PSC there will be a Q' (p for pair) which maps to the same Q". Adachi then instructs us to form the quantity AF1(Q') = FI(Q') - F1(Q,') and consider its steepest ascent lines in the Q'-plane originating from the PSC. Note that at the PSC itself, AFI(Qpsc) = 0, since at this point Q' = Q,. We then choose two of these steepest ascent lines which surround the smallest region of the Q'-plane containing the F < 0 region (see Fig. 3a). This will be our non-contributing region. In practice it is found that the region for which F1 < 0 is formed near the PSC (Fig. 2a), but at the PSC itself FI is always found to be greater than zero. Therefore, contained in the non-contributing region, between the Stokes lines themselves and the F < 0 region, there will also be a region for which F, > 0 (again see Fig. 3a).

In order to illustrate this let us assume that Q' -+ Q" is given by Q" = Q2. Then aQ/aQ' = 2Q' so that there is a PSC at Q'PSC = 0. Let us further assume that F is such that the steepest ascent lines of the quantity AFI(Q') = F1(Q') - FI(Q,') which emanate from the PSC are given as in Fig. 3a. The two steepest ascent lines A and B are then chosen to remove the F < 0 region as shown.



























Figure 3. A hypothetical illustration of the use of Stokes lines in the evaluation of the SCSP. The mapping Q' --+ Q" is chosen to be Q" = Q2, which possesses a PSC at QP 0 (Sec. 9). The "action" F is further assumed to be such that the quantity AF1Q') F, F(Ql) - F1(Q') (Sec. 9) has steepest ascent lines A,B and C. (a)AF has steepest ascent lines A,B and C. The lines A and B surround the F, < 0 region and are therefore chosen as Stokes lines. Stokes lines in general (A and B in this example) define the boundary of the non-contributing region (which always includes the F < 0 region). (b) The Stokes lines A' and B' in the Q"-plane are images of the Stokes lines A and B in the Q'-plane. The number of contributing trajectories in Eq. (1.5) therefore changes from two to one when the exit label, Q", moves across A' or B from the right-half Q"-plane to the left-half Q"-plane.


The corresponding situation in the Q"-plane is shown in Fig. 3b. Recall that each

point in the Q"-plane corresponds to an exit label, (q",p"), at which we evaluate the
SCSP (where a fixed entrance label is understood), from Q" = Q-" - il 2 = q" - ip"
or alternatively (q",p") = (Re(Q")/Q, -Im(Q")) (Sec. 8); we will therefore use the

term "exit label" to refer to either the pair (q", p") or a point Q" in the Q"-plane.
Let us also recall that each point, Q', in the Q'-plane also represents a trajectory,

(q(t), p(t)), along which the SCSP is evaluated (Sec. 8). The non-contributing region
represented by region I in Fig. 3c would map to the left half Q"-plane (LHQ"P) under

- Q=2, however we are abandoning this region in our evaluation of the SCSP. This

























Q -plane Q plane

Figure 3. Continued. (c) Each exit label Q" in the Q"-plane has two pre-images, Q' and QP,, in the Q'-plane. As the exit label Q" is varied along the curve connecting Q"1 and Q", it crosses the Stokes line B'. As this happens the pre-image Q' crosses the Stokes line B. When this occurs its corresponding trajectory suddenly becomes noncontributing causing the SCSP to discontinuously change from Ksc = EeiF/'IQ, + Ee iF/hIQ,P to KsC = EeiF/hIQ, .


is not a disaster when it comes to the evaluation of the SCSP at these exit labels, since region I which is contributing also maps to the LHQ"P. In other words, each point, Q", in the LHQ"P will have a pre-image in both region I and l, so that if we wish to evaluate the SCSP at one of these exit labels we will use its pre-image from region I but discard the one from the non-contribution region (region I). Regions I and IV are both contributing and both map to the right half Q"-plane (RHQ"P), therefore points in the RHQ"P will correspond to two contributing rays. In other words, if we wish to evaluate the SCSP at an exit label in the RHQ"P we must use the rays corresponding to each Q' pre-image (unless either of their contributions is negligibly small). Therefore, for these exit labels (in the RHQ"P), Eq. (1.5) becomes a sum over two contributing rays, i.e., Ksc(p", q", T; p', q', 0) = Ee/IQ, + Eef/iQ;.










Fig. 3c shows an exit label Q" in the Q"-plane and its two pre-images Q' and Q', (p for pair) in the Q'-plane. The two pre-images Q' and Q'~ both originate in the contributing region (regions 11, 11, and IV in Fig. 3c) of the Q'-plane and should be used in the evaluation of the SCSP. However, if we continuously vary our exit label, Q", so that it moves from Q"' to Q", crossing the Stokes line B' as shown in Fig. 3c, the two corresponding pre-images will continuously vary from Q' and Q' to Q and Q' , respectively. During this process, as our exit label crosses B', the pre-image Q' will cross B (from region I to region I) and abruptly change from contributing to non-contributing. Now as this happens the number of rays used in the evaluation of the SCSP abruptly changes from two to one, i.e. the SCSP changes from Ksc = Ee kQ'+Eeif/a I to Ksc = Eeif/h IQ,. Clearly this leads to a discontinuity in the propagator along the lines A' and B' in the Q"-plane. Now it must be remarked here that the method used for obtaining the Stokes lines (using the steepest ascent lines of AFI) was constructed as a general method of minimizing this discontinuity. Since the steepest ascent lines are chosen so that AFI(Q') = FI(Q') - FI(Q,) > 0 holds to as great an extent as possible, FI(Q') > FI(Q,) also holds to as great an extent as possible. It follows that e- f/?tQ, < e -F/ Q, by as much as possible. Now returning to the example of Fig. 3a, when the Stokes line B' is crossed the term Eeif/h Q, is dropped from KsC. But since the magnitude of a term Eeif/h in Ksc is IEeif/ill = IEeiFR/he - /hI IEle-FI/, and if the amplitude factors of both terms in Ksc (before crossing B') are approximately equal, then the magnitude of EeiF/ Q, will be smaller than that of Eeif/ IQ, by as much as possible. This situation (of minimum discontinuity) will hold in general when crossing a Stokes line, given the approximate equality of the magnitude of the amplitudes and especially so when F1>>h.














NATURE OF THE TRAJECTORIES


When written in the form Q = [1/(Qv2Y)](f4 - ip), P = (1/v"2)(Q4 + ip), the coordinates, Q and P, become canonical (see Sec. 8 and Ref. 19). We can therefore invert these to obtain 4 and P in terms of Q and P and so rewrite (1.8) as H = (P-QQ)2/(4i)-(QQ+P)4/(2Q)4. Now using Q = H/P we can approximate Q" to first order in time as Q" -- Q' +Qjt=o T = Q'� [(P'- 2Q')/(2i) - (QQ'+P')3/(44)] T. This expression can now be used to obtain a rough approximation of the position of the PSCs by forming the quantity (Sec. 9) i9Q"/.9Q' = 1 - [Q/(2i) + 3(QQ' + P')2/(4f23)] T = 0 and solving for Q' to obtain Q' = -P'/1�(2 f/3)[1/T+iQ/2f1/2; we say that these two PSCs are 1st order in T. We know that at T = 0 no PSCs exist (Sec. 9); therefore this expression suggests that two PSCs are formed at infinity in the Q'-plane and quickly move in towards the fixed point -P'/I = -(Qq' + ip')/Q. This situation is what is observed numerically, and can be seen from Figs. 4a, 2b and 4b. Figs. 4a and 4b are both contour plots of Re(Q") and Im(Q") superimposed over a region of the Q'-plane as was Fig. 2b; each has the parameter values Q = 1 and (q',p') = (0,0). Fig. 4a corresponds to n = 10 (T = nAt, At = .001), the two dots representing the PSCs are located at Q'psc=(11.6,11.5) and (-11.6,-11.5). In Fig. 2b (n = 30) the PSCs have moved in towards -P'/IQ = -(fq',p')/Q = (0,0) to the points Q'psc=(6.8,6.6) and (-6.8,-6.6). In Fig. 4b, for which n = 100, the PSCs have continued to move in and are now located at Q psc=(3.8,3.5) and (-3.8,-3.5). As time continues to increase additional PSCs are seen to move in from infinity; in Fig. 4c, n = 250, and four additional PSCs (seen as "defects" in the mesh pattern) have









56


15 15 10 10 5 5 0 0


-5 -5


-10 -10


-15 -15
-15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15

15


10


5


0


-5


-10



-15 -10 -5 0 5 10 15

Figure 4. Superimposed contour maps of Re(Q") and Im(Q"). The parameter values are 0 = 1, (q',p') = (0,0) and At = .001 with T = nAt. For parts a and b, as time T is increased the two PSCs move in from infinity towards the point -P' =
-(q', p') = (0, 0). The horizontal axis represents Re(Q') and the vertical axis Im(Q') in each figure. (a) n = 10 and two PSCs (two dots) are seen at Q'psc=(11.6,11.5) and (-11.6,-11.5). (b) n = 100. The two PSCs have moved to Q'sc=(3.8,3.5) and (-3.8,-3.5). (c) n = 250. Four additional PSCs (seen as "defects" in the mesh pattern) have appeared. The white flower-like regions the on top and bottom of this figure result from Re(Q") and Im(Q") being sharply peaked in these regions. appeared. The white flower-like regions on the top and bottom of Fig. 4c result from Re(Q") and Im(Q") being sharply peaked in these regions and should be ignored.










Continuously Connected and Disconnected Trajectories
At time T = 0 the mapping Q' - Q" takes the simple form Q" = Q' (Sec. 9) and is clearly one-to-one. Furthermore, for a general dynamical system with a fixed initial value of "momentum" and a fixed final value of "position", it is generally possible to find an initial value of position for smaller and smaller time intervals T, which gets closer and closer to the final value of position. We have also seen that PSCs, which are the source of multiple solutions (Sec. 9) for a fixed set of CBC (Sec. 8) are formed at infinity at T = 0+. We may therefore conjecture that at T = 0+ only one of the solutions satisfying a fixed set of CBC will be such that Q' is arbitrarily close to Q" and for each of the others the initial position Q' will appear at infinity in the Q'-plane.
In order to view this conjecture in another way let us consider the trajectories as the functions Q(t) [recalling that each P(O) is always fixed at P' = �2q' + ip' as was discussed in Sec. 8]. First let us suppose we have two trajectories, Q(t), such that both reach the same final point Q(T) = Q" in some sufficiently small but finite time interval T; thus both rays satisfy the same set of CBC, P(0) = P' = q' + ip' and Q(T) = Q" = fq" - ip". Given this, our conjecture will demand that if we decrease our time interval T toward zero, while keeping the final point of both rays fixed at Q" (CBC fixed while T decreases), one ray will continuously shrink in length as its initial point Q' converges to Q", while the other ray will continuously increase in length as its initial point Q' tends to infinity. Re-expressing the conjecture in this form also allows us to see that both trajectories cannot continuously shrink in length so that both their initial points converge to the point Q". First recall that the mapping Q' -4 Q" is analytic (Sec. 9) and an analytic map f(z) is always one-to-one in a sufficiently small neighborhood of a point zo if f'(zo) : 0. With this in mind let us view Q' -+ Q" as a mapping from the Q'-plane to the Q"-plane. Now if both trajectories, Q(t), discussed










above were to continuously shrink in length their initial points, Q', would converge to the fixed point Q' = Q" in the Q'-plane making the mapping Q' -* Q" two-to-one in the neighborhood of Q' = Q". But this could only happen if e9Q"/OQ' (P' fixed) equaled zero at the point Q' = Q", but this would make the point Q' = Q" a PSC and we know that the PSCs don't stay fixed but tend towards infinity in the Q'-plane as T is decreased. Hence both rays cannot continuously shrink to zero in length as T shrinks to zero.

This conjecture has in fact been shown to hold true numerically; we will therefore call the ray which shrinks to zero in length "continuously connected" (CC), since it is CC with the ray Q' = Q" at T = 0, and the one whose initial point Q' tends to infinity "disconnected" (DC). Figs. 5a,b,c and d display this phenomenon for T ranging from T = .06 down to T = .0001 with the fixed value of Q" = (-12.6, 24.6). In these figures the horizontal axis represents Re(Q') and the vertical axis vertical represents Im(Q'); the DC ray is portrayed as the dotted curve and the CC ray as the solid curve. In Fig. 5a, T = .06, and the length of the disconnected ray (LDc) is 30.5 and the length of the connected ray (Lcc) is 20.9. In Figs. 5b, T = .004, LDC = 36.4 and LcC 13.1; in Fig. 5c, T = .0004, LDC = 57.6 and LCC 4.9; finally, in Fig. 5d, T .0001 and the DC ray has increased in length to LDC 98.6 while the CC one has shrunk down to LCC = 1.8. We therefore see that there will always be one and only one CC ray for any given exit label Q". We may then in some sense think of the DC trajectories as "extra" trajectories. It is also worth mentioning here a simple algebraic analog to this continuously connected, disconnected ray situation. Consider the roots of the quadratic equation y = ax2+ bx + c which are given by x� = (b/2a) (-1 � V/1 - 4ac/b2 ). Letting a play the role of T, we find that as a -+ 0, the root x+ approaches the fixed value x+ = -c/b, while the root x- Z_ (-b/a)(1 - ac/b2) extinguishes itself by tending to infinity. Thus as a -+ 0, x+ plays the role of the CC















Q ' -plane
5

0


0

5

0

5
-75 -50 -25 0 25 50 75



Q -plane
5 0.

5

0

5
0

5


-75 -50


-25 0


Q' -plane


0




5
0

5
5 ... . . .._ __ __ _-75 -50 -25 0 25 50 75



Q' -plane
5

0
5

0

5

0

5


-25 0


Figure 5. A "continuously connected" ray (solid curve) and a "disconnected" ray (dotted curve) both reaching the same exit label Q" = (-12.6, 24.6) in various time intervals, T. As T is decreased the initial point Q' of the DC ray tends to infinity as the initial point of the DC ray tends to Q". The horizontal axis represents Re(Q') and the vertical axis Im(Q') in each figure. (a) T = .06. (b) T = .004. (c) T = .0004.
(d) T = .0001.




trajectory and x- the role of the DC trajectory.

It was mentioned in the Introduction that in order to evaluate the SCSP, Ksc, for a fixed set of CBC it was originally expected [3] that only one particular trajectory was needed. This is the CC ray just discussed. As was mentioned above, for a given time interval T, each exit label Q" will have one and only one CC ray, Q(t), such


25 50 75 -75 -50


25 50 75










that Q(T) = Q". In the evaluation of the SCSP at a fixed exit label Q" = q" - ip" (for some fixed entrance label P' = Qq' + ip') it is found that the CC ray is always contributing (see Secs. 1 and 9). It is also found that this contribution is generally the dominant one. The DC rays represent quantum mechanical interference and are found to play an important role in the region of PSCs, the PSCs themselves signaling interference.

Continuously Connected and Disconnected Regions

As was discussed above for the case of the quartic oscillator (1.8), PSCs are formed at infinity in the Q'-plane when T = 0+ and quickly move in towards the point Q' = -P'; these PSCs in turn give rise to DC rays whose initial points, Q', (for fixed Q") are also formed at infinity, and for sufficiently small time, T, move in towards Q". In addition to this it was discussed in Sec. 9, for the case of the quartic oscillator (and delta kicked rotator) that near each PSC there formed a region in the Q'-plane for which F, < 0. This is highly suggestive of a relationship between the DC rays and the F, < 0 regions of the Q'-plane. It is actually found that every trajectory, Q(t), whose initial point Q' lies in a F, < 0 region is a DC one. It is further found, after a short time T (T=.03, for example, with parameter values q' = p' = 0 and Q = .5), that the DC rays themselves form a region with well defined boundaries surrounding the F < 0 region with which they are associated. This boundary which will be called the "connected-disconnected line" (CDL) must pass through the PSC associated with the F < 0 region which it encompasses. To see this assume for a fixed time T, that a ray, Q(t), whose initial point is a PSC, Q'PSC, goes to the final point Q . Now let us assume that in the neighborhood of the PSC the mapping Q'--4 Q" is two-to-one; this assumption is unnecessary but simplifies the argument and is the actual case for the quartic oscillator. Now an exit label Q" arbitrarily close










to " will have two pre-images (since Q' - Q" is two-to-one near Q'Psc) which by continuity will be arbitrarily close to Q' Now since there is only one CC ray one of the pre-images must correspond to a CC ray, (Qcc), and the other to a DC ray, (Q D). Since at Q'PsC itself the mapping is necessarily one-to-one (Sec. 9), as approaches Q"SC, Q' and Q' must converge to Q Now the CDL separates regions in the Q'-plane for which the corresponding trajectories are either CC or DC, therefore the CDL must pass between Qc' and Q'DC and will be forced to pass through Q'Psc as Q" -+ Q"

We should also remark here that since an arbitrary exit label Q" (for fixed time T) will have a unique CC ray associated with it, the CC region of the Q'-plane will map to the Q"-plane in a one-to-one manner. It was discussed in Sec. 9 that the Stokes lines (which separate contributing and non-contributing regions of the Q'plane) themselves emanate from each PSC. The Stokes lines then must necessarily fall within the disconnected region, that is between the CDL and the F, < 0 region. If this were not the case the Stokes lines would fall within the CC region and we would be throwing away each entire DC region as well as parts of the CC region, thus only CC contributions to the SCSP would remain. And as was remarked above, the CC rays map in a one-to-one manner onto the Q"-plane, so that the exit labels Q" corresponding to the CC rays thrown away would receive no contribution at all. This general situation is depicted in Fig. 6. From this figure we can see that only DC rays whose initial points lie in the regions between the CDL and the Stokes lines will play a role in the evaluation of the SCSP.








































Figure 6. The connected-disconnected lines separate the DC regions of the Q'-plane from the CC region. The Stokes lines must always fall within the DC regions.















NUMERICAL RESULTS


It was seen in the previous section that as a result of the formation of DC regions of the Q'-plane the Stokes lines must fall within these regions. As a result of this only the regions of the Q'-plane between the CDLs and the Stokes lines (see Fig. 6) can give rise to contributing DC rays. It was also seen in the previous section that the CC rays map in a one-to-one manner onto the Q"-plane. Therefore since the entire CC region is contributing and only the cusps of the DC regions are contributing each exit label Q" will have a CC contribution, but only some will have DC contributions.

In comparing Ksc with the actual quantum mechanical coherent state propagator (1.1) it is of interest to study the relative contributions of both the DC and CC rays to KSc [Ksc = KDC + Kcc according to (1.5)] as a Stokes line is approached and passed through. In order to do this we may evaluate the SCSP along a DC trajectory, Q(t), whose initial point, Q'C, lies just within the DC region near the CDL. We may then evaluate the SCSP along the CC trajectory whose exit label Q" is the same as that of the DC ray. We may continue this process choosing values of QD which lie further and further into the DC region and so closer to the F, < 0 region. When a Stokes line is reached the DC ray becomes non-contributing; we should expect this to be signaled by the CC contribution alone becoming a good approximation to the full propagator K, i.e. K _ Ksc = Kcc at and past a Stokes line. We may also note that since the Stokes lines are steepest ascent lines of the quantity AFI(Q') = FI(Q') - FI(Q,') (Sec. 9) and occur within DC regions of the Q'-plane, that on the Stokes lines themselves AF, > 0, Q' = Q' and = Q so that










FI(Q'DC) > FI(Q'cc). In other words at a point Q" on the image of a Stokes line in the Q"-plane FIDc will be larger than Fi0,. It should also be mentioned that if only the CC rays were contributing (see Secs. 1 and 10), as was originally expected [3], the Stokes lines would coincide with the CDLs so that K - Ksc = Kcc at the first Q'C point chosen in the above procedure.
In studying the behavior of Ksc in approaching a Stokes line the relevant data will be arranged in two tables. Each row of the first table will give the initial points of the DC and CC rays, Q'DC and Q'Cc (reaching a common exit label Q"), the corresponding values of E and F evaluated along these rays, and the exit label Q" itself. Each corresponding row of the second table gives the values KDC and Kcc corresponding to these rays, their sum Ksc, and the value of the full quantum mechanical propagator K corresponding to the particular exit label Q" as found by numerical solution of the Schr6dinger equation. The modulus of KDC, Kcc, Ksc and K is given directly below each value. The quantity .TC - [Kcc[/IKscl is also given; when Q" moves past a Stokes line a value of Tcc - 1 is expected.
Before studying the approach to a Stokes line let us first look at the value of Ksc near its peak modulus (recall IKI < 1, Sec. 9). This may be done by evaluating Ksc at the exit label Q" corresponding to the unique real classical ray, the initial point of this ray always being Q' = 2q'-ip' (w = 0, Sec. 8). Let us choose parameter values h = .1, 2 = .5, q' = 1, p' = 0 and n = 300 (where T = nAt with At = .001). This choice gives Q' = (.5, 0), which lies in the CC region of the Q'-plane, and its corresponding exit label is Q" = Q(T) = (.48,.29). Along this path ECc = (.82, -.35), FCC = (7.07x10-2, 0), Kcc = (.85, .27), and IKccI = .89. This compares well with the actual value of the propagator K = (.84, .28) with IK = .89. With these parameter values we find PSCs formed at the points Q'Psc = (.30, .77) and QPsc = (-1.29,-.62). Each of these PSCs will have a DC region associated with it, and each of these DC









regions will have a DC ray with the desired exit label Q" = (.48, .29). The DC ray originating in the region associated with Q sc= (.30, .77) has QDC = (.23,1.43); evaluating the SCSP along this ray gives EDC (.40, .65), FDC = (.56, -.11), and KDC = (2.23,.84) so that IKDcI = 2.38. Note that FIDc < 0 along this ray so that it is clearly non-contributing (Sec. 9). The DC ray associated with Q'Psc = (-1.29, -.62) is found to have Q'D (-2.19, -1.35); the SCSP evaluated along this ray gives EDC = (.09, -.55), FDC (-2.69,1.65), and KDc = (-3.79,.09)xlO-8 so that IKDcI= 3.79x10-8. Therefore even if this value of QDC falls within the contributing part of the DC region associated with this PSC, its contribution, KDC, to Ksc, is negligible. For this set of parameter values it is found that at exit labels where K is large (K > . 1) the DC rays are either non-contributing or make a negligible contribution to Ksc and Ksc = Kcc provides a good approximation to K.
In Table Ia the initial points QDC were chosen within the DC region corresponding to Q'PSC = (.30, .77). The points were chosen moving up along the line Re(Q c)
1.15 starting at the point Q' c = (1.15, .4) which lies about .2 units above the CDL. For exit labels which correspond to points QDC along this line starting at the CDL [about Q'=(1.15,.2)] and moving up to the second point in Table Ia, Q'DC = (1.15, .5), this DC ray alone provides a good approximation to K, the value of Kcc being either negligible or small for these exit labels. For each exit label in Table Ia the additional DC ray originating in the DC region associated with Q'Psc = (-1.29, -.62) is found to be non-contributing (since for each FIDc < 0). For example, the DC ray originating in the DC region associated with Q'psc = (-1.29, -.62) whose exit label is Q" = (.17,1.31) has initial point Q' c = (-2.05, -1.62) and along this ray EDC = (.01, -.50), FDC = (-4.08, -.63) so that KDC = (-.24, 2.75)x102 and IKDcI = 276.
Clearly this ray is non-contributing since FIDc < 0. As we proceed upward, we see from Table Ib, that the CC ray quickly becomes the dominant contributor so that at










Table Ia
DC EDC FDC c Ecc Fcc Q
(1.15,0.4) (0.49,-0.55) (1.13,0.42) (-0.40,1.34) (0.64,0.31) (0.38,1.12) (0.17,1.31)
(1.15,0.5) (0.46,-0.58) (1.25,0.64) (-0.45,1.26) (0.68,0.28) (0.39,1.03) (-0.01,1.28)
(1.15,0.6) (0.43,-0.61) (1.31,0.90) (-0.50,1.17) (0.72,0.25) (0.42,0.97) (-0.18,1.23)
(1.15,0.7) (0.39,-0.63) (1.30,1.19) (-0.56,1.08) (0.75,0.20) (0.44,0.95) (-0.35,1.16)
(1.15,0.8) (0.34,-0.65) (1.20,1.50) (-0.62,0.99) (0.78,0.16) (0.46,0.96) (-0.51,1.06)
(1.15,0.9) (0.29,-0.65) (1.00,1.82) (-0.69,0.88) (0.80,0.10) (0.46,1.00) (-0.66,0.95)


Table lb
K KDC Kcc Ksc Jcc
(-5.77,-9.84)x10-' (-5.57,-9.85)x10-a7' (-4.56,-8.92)x10-6 (-5.57,-9.86)X10-3
1.14x10-2 1.13x10-2 .001x10-2 1.13x10-2 .001
(0.74,-1.06)x10-3 (0.77,-1.03)x10-3 (-0.96,-2.33)x10-5 (0.76,-1.05)X10-3
1.30x10-3 1.28x10-3 0.03x10-3 1.30x10 - .023
(7.67,-8. 19)X10-5 (8.67,-3.74)X10-5 (-0.98,-4.35)x10-I' (7.69,-8.09)x10-5
1.12x10-4 0.94x10-4 0.45x10-4 1.12x10-4 .402 (0.18,-5.96)x10-5 (4.10,-2.97)xlO-6 (-0.30,-5.68)xlO-5 (0.11,-5.98)x10-5
5.96x10-5 .51x10-5 5.69x10 -5 5.98x10-5 .952
(0.27,-5.38)x10-5 (-0.27,-2.20)X10-7 (0.21,-5.36)x10-5 (0.21,-5.38)x10-5
5.38x10 - 0.02x10-5 5.36x10-5 5.38x10-5 .996 (-0.04,-3.84)x10-5 (-7.53,4.75)x10-9 (-0.07,-3.82)x10- (-0.07,-3.82)x10- 1
3.84x10-5 .0009X10-5 3.82x10-5 3.82x10-5


Q" = (-.66, .95) the value of Fcc is very close to unity indicating that we are near a Stokes line. From this point and upwards, the CC ray alone will provide a good approximation to K. We should also note that at Q" = (-.66, .95), FIDc is larger than Fcc as should be the case near a Stokes line as was discussed earlier in this section. As points above Q'Dc = (1.15, .9) are chosen FIDc will continue to increase, reach a peak, decrease, and then cross into the F < 0 region at Q'DC = (1.15, 2.17).

We will now look at the SCSP for the parameter values h = .1, 0 = .5, q' = 1, p' = .5 and n = 400 (where T = n At and At = .001). As before let us first look at the propagator near its peak value by evaluating it at the exit label Q"









corresponding to the unique real classical ray. For these parameter values this ray will have initial point Q' = q - ip' = .5 - i.5, which again lies in the CC region, and corresponding exit label Q" = (.55, -.01). Along this ray Ecc = (.69, -.41), Fcc = (.13, 0), and Kcc = (.57, .57) with modulus IKccI = .81. Again this compares well with the value of the quantum mechanical propagator evaluated at this exit label, K = (.57, .59) with modulus IKI = .82. With these parameter values PSCs are found at the points Q'Psc = (.11, .19) and Q'p = (-1.30, -.98). The DC ray originating in the DC region associated with Q'Psc = (.11, .19) is found to have initial point Q'DC = (-.10, .82), and along this ray EDC = (.45, .52), FDC = (.65, .30), and KDC = (.015, .030) with modulus IKDc = .03; it can be inferred that this ray originates in the non-contributing part of the DC region since the addition of this value of KDC to Kcc = (.57, .57) worsens the approximation. The additional DC ray originating in the DC region associated with Q'sc = (-1.30, -.98) is found to have initial point Q'I =(-2.11,-1.73), and along this ray EDC = (.09,-.52), FDC = (-2.77, 1.49) and KDC = (-1.28, 1.26)x10- with modulus IKDcI = 1.80x10-7. This value of KDC is negligible in comparison to Kcc so in this case we can infer that QC is either close to or past the Stokes line (coming from a contributing region of the Q'-plane). For this set of parameter values it is found that at exit labels where K > .2 the DC rays are either non-contributing or make a negligible contribution to Ksc so that Kcc alone provides a good approximation to K.
In Table lla the initial points Q' c were chosen within the DC region corresponding toQs = (.11, .19). The points were chosen moving up along the line Re(Q' c) = 1 starting at the point QD = (1, -.3) which lies about .2 units above the CDL. For exit labels which correspond to points QDc along this line starting at the CDL [about Q'=(1,-.5)] and moving up to the third point in Table lfa, Q'DC = (1, -.1), this DC ray alone provides a good approximation to K, the value of Kcc being either neg-










Table Ila
Q'DC EDC FDC Q1'_ Ecc Fcc Q"1 DC CC
(1,-0.3) (0.48,-0.48) (0.94,0.17) (-0.50,0.86) (0.53,0.27) (0.25,1.44) (0.39,0.95)
(1,-0.2) (0.46,-0.51) (1.09,0.35) (-0.55,0.80) (0.57,0.25) (0.18,1.28) (0.17,0.94)
(1,-0.1) (0.44,-0.53) (1.19,0.60) (-0.60,0.73) (0.60,0.23) (0.13,1.18) (-.04,0.91)
(1,0.0) (0.41,-0.56) (1.22,0.93) (-0.66,0.66) (0.64,0.20) (0.09,1.14) (-0.24,0.85)
(1,0.1) (0.38,-0.58) (1.18,1.31) (-0.72,0.59) (0.67,0.16) (0.06,1.15) (-0.43,0.78)
(1,0.2) (0.34,-0.59) (1.05,1.75) (-0.79,0.51) (0.70,0.11) (0.01,1.21) (-0.63,0.69)
(1,0.3) (0.30,-0.59) (0.82,2.23) (-0.86,0.43) (0.72,0.06) (-0.05,1.32) (-0.81,0.57)



Table lib
K KDC KCC Ksc .Tcc
(-9.19,8.92)x10-2 (-9.20,8.77)x1O-2 (-3.43,0.45)x10-' (-9.20.8.77)x10-2
0.128 0.127 .0000003 .127 0 (-1.74,-1.31)xlO-2 (-1.72,-1.32)x1O-2 (-1.04,1.38)xlo-6 (-1.72,-1.32)x10-2
2.18x10-2 2.16x10-2 .0002x10-2 2.16x10-2 0 (0.00,-1.69)x10-3 (0.02,-1.69)x10-3 (-0.43,4.76)x106 (0.02,-1.69)x10-3
1.69x10-3 1.69x10-3 .005x10-3 1.69x10-3 .003 (2.19,-5.56)x10-5 (2.02,-6.26)x10-5 (2.59,7.11)x10-6 (2.28,-5.55)x10-5
5.98x10-5 6.57x10 - .76x10-5 6.00x10-5 .127 (4.64,3.72)x10- (-0.20,-1.38)x10-6 (4.93,5.08)x10-6 (4.73,3.70)x10-6
5.95x10-6 1.40x10-6 7.08x10-6 6.01x10-6 1.178 (3.78,1.20)x10-6 (-1.76,-0.13)x10- (3.82,1.17)xlO-6 (3.80,1.17)xlO-6
3.97x10-6 .02x10-6 4.00xlO-6 3.98x10-6 1.005 (1.29,-0.51)x10-6 (1.00,1.04)x10-1� (1.28,-0.52)x10-6 (1.28,-0.52)x10-6
1.38x10j6 .0001x10-6 1.38x10-6 1.38x10-6 1


ligible or small for these exit labels. For each exit label in Table la the additional DC ray originating in the DC region associated with Q'Psc = (-1.30, -.98) is found to have Fve < 0 and is therefore clearly non-contributing. For example, the additional DC ray originating in this region associated with the exit label Q" = (.39, .95) has initial point QDC = (-2.01, -1.94), and along this ray EDC = (.03,-47), FDC = (-4.37, -.20) so that KDc = (1.07,-3.33) and IKDcI = 3.50. As we proceed upward, we see from Table ib, that the CC ray again quickly becomes the







69

dominant contribution so that at Q" = (-.63, .69) the value of Fcc is very close to unity indicating that we are near a Stokes line. From this point and upwards, the CC ray alone will provide a good approximation to K. Again let us note that at Q"f = (-.63, .69), FIDC is larger than Fcc as should be the case near a Stokes line as was discussed earlier in this section. As points above Q'0o = (1, .3) are chosen, FIDc will continue to increase till it reaches a peak, then decrease, entering the F, < 0 region at Q'C = (1, 4.75).














SUMMARY AND CONCLUSION


It has been found that the complex classical trajectories Q(t) = Q(t)- i(t), P(t) = Q (t) + ip(t) satisfying the complex boundary conditions (CBC) (1.4), along which the semiclassical coherent-state propagator (SCSP) (1.5) is evaluated, are of two types: continuously connected (CC) or disconnected (DC). For a fixed set of CBC there will always be one unique CC ray Q(t) which is always contributing in the evaluation of Eq. (1.5) and for each phase space caustic (PSC) formed in the Q'-plane there will be an additional DC ray satisfying the set of CBC which may or may not be contributing. These two types of rays themselves are found to lie in associated regions in the Q'-plane whose boundary line [referred to as a connected-disconnected line (CDL)] will pass through a PSC; therefore for each PSC there will be a DC region of the Q'-plane. Along with each PSC there is also formed a non-contributing region (separated from contributing regions by Stokes lines) of the Q'-plane containing an F1 < 0 region [4]. Each of these non-contributing regions has been found to fall within a DC region. Therefore the only exit labels Q" = Qq" - ip" at which DC rays contribute in the evaluation of (1.5) will be those corresponding to the initial points Q'DC found between the CDL and the Stokes lines (see Fig. 6).
The comparative roles of the continuously connected and disconnected trajectories in the correct evaluation of the semiclassical coherent-state propagator can be understood through a comparison of the original [3] and modified [4] form of the theory. In the original form of the theory of the semiclassical coherent-state propagator presented by Klauder [3] only the continuously connected rays were thought to con-










tribute so that Ksc was given by Eq. (1.2). Our work has shown that the continuously connected rays always contribute but at certain exit labels the evaluation of the semiclassical coherent-state propagator must be supplemented by a disconnected ray. In Adachi's treatment of the delta kicked rotator [4] it was necessary to replace Eq. (1.2) by Eq. (1.5) and introduce the use of Stokes lines; however, in Adachi's treatment no distinction was made between continuously connected and disconnected rays. The results of this paper support Adachi's modified form of the theory although it has been found that the non-contributing regions of the Q'-plane fall within disconnected regions so that all non-contributing rays are disconnected.

Related work can also be found in Ref. 20.















REFERENCES


1. A representative sample of references is the following: I. M. Gel'fand and A. M.
Yaglom, J. Math. Phys. 1, 48 (1960); D. G. Babbitt, J. Math. Phys. 4, 36 (1963); E. Nelson, J. Math. Phys. 5, 332 (1964); K. t6, Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability (California U. P., Berkeley, 1967), Vol. 2, Part 1, pp. 145-161; J. Tarski, Ann. Inst. H. Poincar6 17, 313 (1972); K. Gawedzki, Rep. Math. Phys. 6, 327 (1974); A. Truman, J. Math.
Phys. 17, 1852 (1976); S. A. Albeverio and R. J. Hoegh-Krohn, Mathematical Theory of Feynman Path Integrals (Springer, Berlin, 1976); V. P. Maslov and A.
M. Chebotarev, Theor. Math. Phys. 28, 793 (1976); C. DeWitt-Morette, A. Maheshwari, and B. Nelson, Phys. Rep. 50, 255 (1979); D. Fujiwara, Duke Math.
J. 47, 559 (1980); P. Combe, R. Hoegh-Krohn, R. Rodriguez, and M. Sirugue, Commun. Math. Phys. 77, 269 (1980); F. A. Berezin, Soy. Phys. Usp. 23, 763 (1980); T. Ichinose, Proc. Jpn. Acad. Ser. A 58, 290 (1982); I. Daubechies and J.
R. Klauder, J. Math. Phys. 23, 1806 (1982); R. H. Cameron and D. A. Storvick, Mem. Amer. Math. Soc. 46, No. 288 (1983); T. Hida and L. Streit, Stoch. Proc.
Appl. 16, 55 (1983); N. K. Pak and I. Sokman, Phys. Rev. A 30, 1629 (1984); F.
Steiner, Phys. Lett. 106 A, 363 (1984); C. DeWitt-Morette, Acta Phys. Austr.
Suppl. 26, 101 (1984); A. Young and C. DeWitt-Morette, Ann. Phys. 169 140 (1986); M. Bohm and G. Junker, J. Math. Phys. 28, 1978 (1987); D. Castrigiano and F. Staerk, J. Math. Phys. 30, 2785 (1989); M. de Faria, J. Potthoff, and L.
Streit, J. Math. Phys. 32, 2123 (1991); W. Fischer, H. Leschke, and P. Miiller, J. Phys. A: Math. Gen. 25, 3835 (1992); L. S. Schulman, J. Math. Phys. 36,
2546 (1995); C. Grosche and F. Steiner, J. Math. Phys. 36, 2354 (1995).
2. J. R. Klauder and I. Daubechies, Phys. Rev. Letters 52, 1161-1164 (1984); I.
Daubechies and J. R. Klauder, J. Math. Phys. 26, 2239-2256 (1985).
3. J. R. Klauder, in Random Media, Ed. G. Papanicolaou, Vol. 7, IMA Series in
Mathematics and its Applications, Springer-Verlag, New York, 1987, pp. 163182.
4. S. Adachi, Ann. Phys. (NY) 195, 45-93 (1989).
5. See, e.g., L. Schulman, Techniques and Applications of Path Integration (Wiley,
New York, 1981).
6. J. H. Van Vleck, Proc. Nat. Acad. U. S. Sci. 14, 178 (1928).
7. P. A. M. Dirac, Proc. Roy. Soc. London, 113A, 621 (1927).
8. P. Jordan, Zeits. Physik, 40, 809 and 44, 1 (1927).










9. N. Dunford and J. Schwartz, Linear Operators, Vol. 1, (Interscience Publishers,
New York, 1966); a particularly readable elementary account is given in, I.
Stakgold, Green's Functions and Boundary Value Problems (Wiley, New York,
1979).
10. See, e.g., T. Hida, Brownian Motion (Springer-Verlag, New York, 1980), or T.
Hida and M. Hitsuda, Gaussian Processes (American Mathematical Society,
Providence, RI, 1993).
11. See, e.g., A. V. Skorokhod, Studies in the Theory of Random Processes (AddisonWesley, Reading, MA, 1965), pg. 3.
12. K. t6, Applied Mathematics and Optimizatiom 1, 347-381 (1975).
13. See, e.g., Coherent States, Eds. J. R. Klauder and B.-S. Skagerstam, World
Scientific, Singapore, 1985, pp. 169-184.
14. R. H. Cameron, J. Anal. Math. 10, 287 (1962/63).
15. L. Schwartz, Thdorie des Distributions (Hermann, Paris, 1966) or, I. M. Gel'fand
and N. Ya. Vilenkin, Generalized Functions, Vol. 1, (Academic Press, New York
and London, 1964).
16. J. R. Klauder, J. Math. Phys. 4, 1058-1073 (1963); reprinted in Ref. 13, pp.
169-184.
17. See, e. g., R. S. Brodkey, The Phenomena of Fluid Motions (Addison-Wesley,
Reading, MA, 1967), Chap. 9.
18. J. R. Klauder, in Path Integrals, Eds. G. J. Papadopoulos and J. T. Devreese,
Plenum, New York, 1978, pp. 5-38; Phys. Rev. D 19, 2349-2356 (1979).
19. Y. Weissman, J. Chem. Phys. 76, 4067-4079 (1982); J. Phys. A 16, 2593-2701
(1983).
20. M. KuA, F. Haake, and Bruno Eckhardt, Z. Phys. B 92, 221-233 (1993); M.
Kug, F. Haake, and D. Delande, Phys. Rev. Letters 71, 2167-2171 (1993); R.
Scharf and B. Sundaram, Phys. Rev. E 49, 4767-4770 (1994).














BIOGRAPHICAL SKETCH


The author is originally from New York City and obtained his B.S. in physics from the Polytechnic University of New York.











I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of PhilosopAY.

JohW . Klauder, Chair man Protssor of Physics

I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Ph* sophy

Henr&NVan Rinsvelt
Professor of Physics

I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of P)1phy.

J. Robert Buchler
Professor of Physics

I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy.

Khandker A. Muttalib
Associate Professor of Physics

I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Pmosophy.

Paul Robinson
Professor of Mathematics

This dissertation was submitted to the Graduate Faculty of the Department of Physics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy.

December 1998
Dean, Graduate School




Full Text

PAGE 1

THE COMPARATIVE ROLES OF CONNECTED AND DISCONNECTED TRAJECTORIES IN THE EVALUATION OF THE SEMICLASSICAL COHERENT-STATE PROPAGATOR By ANDREW E. RUBIN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FUFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1998

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TABLE OF CONTENTS page ABSTRACT iv SECTIONS INTRODUCTION AND REVIEW OF LITERATURE 1 THE FEYNMAN PATH INTEGRAL 4 LEBESGUE INTEGRATION 8 STOGHASTIG VARIABLES AND WIENER MEASURE 11 Random Variables in One Dimension 12 Stochastic Variables 14 Wiener Processes 17 Stochastic Integrals 23 REVIEW OF COHERENT STATES 25 THE COHERENT-STATE PATH INTEGRAL 28 DERIVATION OF THE SEMIGLASSIGAL COHERENT-STATE PROPAGATOR 32 Boundary Gonditions for Eqs. (7.7) 33 Semiclassical Action, F 34 Amplitude Factor, E 37 FORMULATION OF THE SEMIGLASSIGAL COHERENT-STATE PROPAGATOR IN SUMMARY 42 APPLICATION OF THE SEMICLASSICAL COHERENT-STATE PROPAGATOR 45 Origin of Multiple Solutions 45 Non-contributing Solutions 47 NATURE OF THE TRAJECTORIES 55 Continuously Connected and Disconnected Trajectories 57 Continuously Connected and Disconnected Regions 60 ii

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NUMERICAL RESULTS 63 SUMMARY AND CONCLUSION 70 REFERENCES 72 BIOGRAPHICAL SKETCH 74 iii

PAGE 4

Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THE COMPARATIVE ROLES OF CONNECTED AND DISCONNECTED TRAJECTORIES IN THE EVALUATION OF THE SEMICLASSICAL COHERENT-STATE PROPAGATOR By Andrew E. Rubin December 1998 Chairman: John R. Klauder Major Department: Physics The Feynman path integral is discussed and why, strictly speaking, this is not an actual integral in the sense of Lebesgue. An exact path integral expression for the coherent-state propagator, developed by Daubechies and Klauder, is presented and discussed. A semiclassical approximation to this path integral obtained by Klauder is then derived from the exact path integral expression. This approximation which was subsequently modified by Adachi is applied to the quartic oscillator. The evaluation of this semiclassical expression involves classical trajectories which must satisfy complex boundary conditions. It is found that these complex classical trajectories fall into two broad categories basically characterized by the descriptive titles “continuously connected” and “disconnected” given to the two different types. The continuously connected type is found to always contribute in the evaluation of the semiclassical propagator while the disconnected type will only contribute under specific conditions. IV

PAGE 5

INTRODUCTION AND REVIEW OF LITERATURE The Feynman path integral representation of the x-to-x propagator involves taking the continuum limit of a time slicing formulation (see Sec. 2). Strictly speaking this formulation results in a linear functional but not an actual integral (in the sense of Lebesgue). The Feynman expression has motivated various attempts at providing well defined integral expressions for quantum mechanical propagators (not necessarily x-to-a:) [1]. In the interest of providing such an expression Klauder and Daubechies [2] derived an exact path integral expression involving Wiener measure (Sec. 4) for the propagator, K{p",q",T-,p',q',0) = {p",q"\e , q') , ( 1 . 1 ) where H is the Hamiltonian operator corresponding to the classical Hamiltonian H, and Ip, g) is a coherent state (Sec. 5). In Sec. 2 the Feynman path integral will be discussed, and in Sec. 3 the rigorous definition of an integral in the sense of Lebesgue will be presented along with why the Feynman integral does not qualify. The coherent-state path integral itself is presented in Sec. 6 and, in the interest of understanding this formulation, stochastic variables and Wiener measure are discussed in Sec. 4, followed by coherent states in Sec. 5. Following the derivation of the exact path integral expression for the propagator, Klauder [3] obtained a semiclassical approximation for it, Ksc{p",q",T-,p',q',0)^Ee^^/\ (1.2) This semiclassical expression is evaluated along trajectories q{t) and p{t) satisfying 1

PAGE 6

2 Hamilton’s equations, dH . dH — and p = — , op oq but subject to complex boundary conditions (CBC) (1.3) Q.q' + ip = Vlq' + ip, (1.4) Uq"-ip" = Qq"-ip" with q' = g(0), q" = q{T), p' = p(0), p" = p{T) and fl, q',p', q",p'' E R. [Here a single trajectory refers to a single set {q{t),p{t)) satisfying (1.3) and (1.4).] As a result of the CBC the functions q{t) and p{t) are generally complex valued which causes the “action” F also to be complex. Equation (1.2) along with the CBC (1.4) will be derived from the path integral expression for the propagator in Sec. 7. The results of Sec. 7 will be summarized and further discussed in Sec. 8. In actual application of Eq. (1.2), it is found that for a given set of CBC there will in general exist more than one complex trajectory satisfying Hamilton’s equations. It was originally expected [3] that only a particular one of these solutions (see Sec. 10) was needed. However, it was shown by Adachi [4] that the number of contributing trajectories [for given boundary conditions (1.4)] varies according to specific rules so that Eq. (1.2) should be replaced by Ksc{p",q",T-p',q',0)^ ^ (1.5) contributing trajectories However, Adachi’s scheme (see Sec. 9) for the general application of (1.5) is arrived at semi-empirically, and in the interest of studying quantum chaos, a somewhat special system was used, that is, the delta kicked rotator H = -p'^ + K cos q ^ 6{t-n), KeK, n=— (X) ( 1 . 6 )

PAGE 7

3 so that with = q{n ) and = p{n ), one obtains the discrete equations 9n+l 9n”^Pn+l> (1-'^) Pn+i = Pn + Ksinq^. Here (1.2) will be applied to an integrable system, the quartic oscillator, ( 1 . 8 ) It is found that all the features of the kicked rotator system which lead to Adachi’s semi-empirical methods, such as interference effects near a “phase-space caustic” (see Sec. 9) are also present in this system. In addition we discover some interesting properties of complex classical trajectories, i.e. solutions of (1.3) subject to the CBC (1.4). These and related topics are discussed in Secs. 9 through 11 and summarized in Sec. 12.

PAGE 8

THE FEYNMAN PATH INTEGRAL The a:-to-a: quantum mechanical propagator is given as, Q{x",T-,x',0) = {x"\e~^'^'^\x'), (2.1) where H is the Hamiltonian operator corresponding to the classical Hamiltonian H. A lattice formulation for this propagator can be derived by first requiring that H = T + T, so that Ti = T+V [with T = /{2m) so that T = p^/(2m)], and then making use of the Trotter Product Formula [5] g-a{r+V)/V ^ ^-aTlN^-aV/N ( 2 . 2 ) with a = iT/h] then for sufficiently large N, the term e in (1.1) becomes, ^ g-a(T+V) ^ jg-a(r+V)/7VjiV ^ jg-aT/iVg-aV/iVjiV^ (2.3) so that (2.1) becomes. g(x",T;x',0) = lim (2.4) N—^oo A resolution of unity in terms of position eigenstates, j dxt\xi){xi\ = i, (2.5) is then inserted between each of the N products in Eq. (2.4), so that I = 1, 2, . . . , A’ — 1 and (2.4) becomes, dxi--dxN-i n (2.6) fc =0 Q{x",T-,x',0) = lim [ N^oo J 4

PAGE 9

5 with Xo = x' and xjv = x” . Each integration in this expression is thought of as corresponding to the fixed and equally spaced times, t = k{T/N) = ke with k = — 1; values of the Xk along these time slices are thought of as lattice points. Assuming y is a function of position only, . (2.7) We may then evaluate the matrix elements, {xk+i\e~‘^'^^^\xk) = {xk-\ri\e~°'^^ , occuring in Eq. (2.6) using the resolution of unity, J dp\p){p\ = i, (2.8) and the formula, so that (9|e -aT/JV X) = j dp e-V/(2m«) (,|p) {p\x) = ~j dp, (2.10) -ap^ /{2mN) ^^p{q-x) 1 27rh I dp exp Q, 2 ^ / This integral can then be evaluated using the general formula, ./?V(4a) / = £ SO that (2.10) becomes mN y 2TTah^ Formula (2.6) then becomes exp mN 2aN {q xf ( 2 , 11 ) ( 2 . 12 ) Q(i",T;x',0) = Jim/*,...dx„_.(^)“ nexp (2.13) k=0 fnN 2 “ T^/ n' -j^r(x*)

PAGE 10

6 which becomes, (2,14) = fc/ / mN \ 2 N-l r ISnSrj “P U £ ilT mA^ T TV after combining exponentials and using a = iT/h. Now using e = T/N, this expression takes on the more lattice-like form, Q{x",T;x',0) (2.15) = lim N—^oo /*' (sSs) ’ “p {r £ [y v{x,) The Feynman path integral may be obtained from (2.15) by first identifying the expression, e £)^o^{(^/2)[(a;fc+i — Xjt)/e]^ — V^(a;*;)}, with the classical action over the broken line path, x',xi, . . . ,xn-i,x". Then interpreting this set of lattice points in the limit e ^ 0 as one possible “path” one obtains the formal expression. F{x", T; x', 0)=M I H (2.16) where J\f is an infinite normalization constant and, 5[x(t)] = f L[x{t),x{t)]dt\ (2.17) J 0 the integral being taken over all paths satisfying a;(0) = x' and x{T) = x". We may denote the continuous Cartesian product of the values of position at each time between 0 and T as The entire path space with a:(0) = x' and x{T) = x" fixed, is then given by, Sp = {x'} x R(°T) x {x"}. It should be noted that the paths contained in Sp need not be continuous since acfc+i — Xk need not approach zero as e — > 0; continuous paths also, need not be differentiable, since Xk+i — x^ may approach zero, but not in a way such that (a^^+i — X)t)/e remains finite; an average path in Sp, in fact, will suffer discontinuities everywhere.

PAGE 11

7 although it is generally believed that (2.16) is supported on a subset of Sp containing much smoother paths. A semiclassical approximation for the a;-to-a: propagator, can be derived from the lattice expression (2.15) or analogously from the path integral formula (2.16). This expression is given as J{x\T;x',0) = (2.18) where, A ( ' i" 'd‘^S{x\x') \2'xh) dx"dx' (2.19) and. S(x'',x) = f \p{t)x{t) H{p,x)]dt. (2.20) J 0 The sum in (2.19) is taken over each set of x{t),p(t) satisfying the extremal equations ( 2 . 21 ) subject to the boundary conditions. . dH , OH x=— and p=— , op Ox x(0) = x' and x{T) = x" . ( 2 . 22 ) In n dimensions the expression for A above generalizes to include a determinant, known as the Van Vleck determinant [6], and is given as ,x') The importance of this determinant in approximating the wave function was previously realized by Van Vleck while studying the correspondence between certain quantum formulas, in the “transformation theory” of Dirac [7] and Jordan [8], and their corresponding classical expressions. {k,q = 1,. . .n). (2.23) A = 27rh det 'd^S(x" dxld:

PAGE 12

LEBESGUE INTEGRATION As was mentioned in the introduction the Feynman path integral is a linear functional not an actual integral, this being due to the lack of a measure. In this section measure and the Lebesgue integral will be discussed as well as why the Feynman integral fails to possess an actual measure. The fundamental ingredient in the theory of Lebesgue integration (on the real line) is a function called the measure which takes as its argument any of a wide class of subsets of the real line known as measurable sets (see for example the first citation of Ref. 9). The measure must satisfy the properties of being both nonnegative, that is if S is some set of points (or the null set), its measure m{S) > 0, as well as that of countable additivity, that is if A = with AidAk = 0, for any i, k such that i ^ k, then, (3.1) OO "^(A) = "^(An). n=l Additional properties are sometimes specified. The measure as originally defined by Lebesgue (Lebesgue measure) generalized the concept of length so that if a set S is some open interval, its measure m{S), is equal to the length of that interval, and if 5 is a single point then m{S) would be zero, [although many other measures can be specified which satisfy properties (3.1)]. From the countably additive property of measures it follows that the Lebesgue measure of a countable set of open intervals is the sum of the lengths of those intervals, and the Lebesgue measure of a discrete set of points such as the integers is zero. In fact the Lebesgue measure of the rational numbers (within some bounded or unbounded 8

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9 interval) is zero (a demonstration of this can readily be found in the second citation of Ref. 9). The Lebesgue measure of the irrationals in an interval (a, b), a
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10 integrable which fail to be Riemann integrable, such as the Dirichlet function which has the value one when x is rational and zero when x is irrational (a demonstration of this involving the Dirichlet function can readily be found in the second citation of Ref. 9). The integral (3.3) has other significant advantages as well, such as the dominated convergence theorem [9]. This definition of the Lebesgue integral also carries over naturally to functions defined on any arbitrary space on which a measure can be defined. For example a function the domain of which consists of paths, x{t), from some appropriate path space on which a measure can be defined. According to the Lebesgue dominated convergence theorem, exp f-(l + a) xf lim / a— >0 J —Q dxi • dxN-1(0r) N-l (3.4) poo exp J — oo LLdu XlXll a— >0 (0F)^-1 with N defined as in Sec. 2 and a > 0. The left hand side of (3.4) is equal to limQ_^o 1/(1 + = 1 which is clearly equal to the right hand side. If we now replace a in (3.4) by e, with e = T/N as defined in Sec. 2, the left hand side of (3.4) becomes equal to lim^v-^-oo 1/(1 + while the right hand side becomes equal to /-p(-E-?)n^=ni=i. (3.5) with t denoting a continuous index such that t € (0,T); this shows that (for any T > 0) the dominated convergence theorem no longer holds as e = T/N 0, implying that limAr_,.oo dxi dx^-i — FI dx{t) no longer provides us with a measure.

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STOCHASTIC VARIABLES AND WIENER MEASURE An actual measure satisfying the properties discussed in Sec. 3 can be constructed on the set of all paths, X{t) G {{a:'} x R(°T) y {x"}}, obtained from the limit of the lattice formulation as V -> oo and e = T/N 0 (see Sec. 2). It is first convenient to consider paths on the lattice such that the initial point x' is fixed at x' = 0 while the final point x" = xn is left free, i.e., sets of lattice points such that {x',xi, . . . = {0,Xi, . . . , iCjv} with e = T/N, x' = 0 and x^ free to vary. A Gaussian weight factor. with z = 1, 2, . . . , V is then associated with each step in the lattice, so that the weight or probability density associated with a given path becomes. then provides a measure on the A^-dimensional path space with S being any arbitrary but fixed region of this space [10]; the expression P^dxi dx^ can then be though of as the measure of the infinitesimal region dxi dx^ about the path (4.1) (4.2) The expression, TTIn{S) = / Pn{xi,X 2, . . . ,x^)dxi dxN, Js (4.3) 11

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12 {0, Xi, . . . , xjv_i, Also note that (4.3) integrated over the entire path space is unity. The limit of dxi -dx^ as N oo and e = T/N — > 0 continues to provide us with a measure called the Wiener measure [10], the measure now being over the function space {0} x R(°T]. denoting this limit of Pjv dx\ dx^r by dfj,w{X) we then obtain, d^xw{X) = Me--^ So (4.4) where A/” is an infinite normalization factor and H dX (t) represents an infinitesimal volume about X{t). The measure of some arbitrary but fixed region of path space, Sp, can then be written as, l^w{Sp) = W / e 2 /o ^ dX (t) . (4.5) Sp Note that since the value of (4.3) integrated over the entire path space is constant and equal to unity its value will remain the same in the limit as AT — ^ oo so that (4.5) continues to be normalized to unity when Sp equals the entire path space. An analogous measure, known as a pinned Wiener measure, can be derived on the function space {x'} x x {x"} simply by using (4.2) with Xq = x' as well as xat = x" both arbitrary but fixed and excluding the integration over dx^r from (4.3). This measure will no longer be normalized to unity but will have the value (l/-\/27rT)e[“(^"“®’)^/(^^)] over the entire space. Both these measures can also be derived from the more general theory of stochastic variables which will be subsequently discussed. Random Variables in One Dimension In one dimension a random variable, x, is defined as a variable which can take on a real value from some specified set of values with a given probability. The moments

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13 of the distribution are defined as (a;"), n = 1,2,3,... (4.6) where (F) = (F(-)) denotes the average of some arbitrary function F{x) over the entire ensemble of values normalized so that (1) = 1. Note that in the case where the set of values which x may assume is R, (F) is given by, (F) = j F{x)P{x)dx, (4.7) where P{x) is the probability distribution of x on R. It is also convenient to introduce the connected moments, {x")^, which are defined by the relation, (e“^) = (4 3) with a e C. The first two connected moments are given by, {x)c = (x), (4.9) note that (x^)^ is equivalent to the variance of the distribution. Another useful function is the characteristic function, CM = {e"’), ( 4 . 10 ) with s G R, which contains information on the moments encoded within it. It should also be noted that in the case where C{s) is known (and a: G R) it follows from (4.8) that. (e‘") = j e"‘P(x)dx = C(s), ( 4 . 11 )

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14 and P{x) can be found by way of the inverse Fourier transform, P{x) = ^ / e-“"C(s)ds. (4.12) Various classes of probability distributions can now be described by specifying certain constraints on the values of the connected moments, for example a Gaussian distribution is defined as one for which all connected moments vanish except the first two. In this case we find, using (4.8), that the characteristic function has the form, C{s) = (e”^) = (4.13) since {x)^ = {x). The probability distribution, P{x), for a Gaussian distribution can now be obtained from (4.13) and is found to be. P{x) 1 \/27r^ exp (^))' 2(x^)e (4.14) the familiar form for a Gaussian distribution in one dimension. It should be noted that particular specifications of the values of the connected moments are generally equivalent to specifying a measure on the set of values of which the random variables may take on, although under certain circumstances this measure so determined is not unique. Stochastic Variables In the theory of stochastic variables a measure on some space of functions (such as {a;'} x R(0,T) X {x"} defined in Sec. 2) may be defined indirectly by specifying an appropriate complete set of correlation functions. The correlation functions being a generalization of the moments (4.6) (when x is a continuous variable), are defined as {X{h)X{t 2 )---X{tk)), for /c = 1,2,3,... , (4.15) here ((•)) = J{-)dfi{X) with dii{X) being the measure of the infinitesimal volume dX{t), about X{t), and (1) generally being normalized to unity, although this may

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15 not always be the case. The X{ti), i = 1,2, in (4.15) represent any given function (or stochastic variable), X{t), from the space of functions, evaluated at the arbitrary but fixed times t = ti, ^ 2 , • • • , 4, with t always taken to be finite so that 0 < ti < T with 0 < T < oo or 0 < t,< T in the case where T is infinite. The correlation functions (4.15) are also chosen so that, when each ti equals some fixed value t, they will correspond to the moments of a one dimensional probability distribution, that is X(t), for any fixed value of t, is equivalent to a random variable in one dimension. As stochastic variables are generally defined on a function space for which they can take on continuous values for fixed values of t, such as Sp = {a:'} x x the fth time slice. In other words, for the space Sp above, for example, X(t) € R for any given X{t) with fixed t G (0, T). Considering this, and that X{t) is a well defined random variable for fixed t, it follows that when t is such that X{t) e R, (F{A(t)}) is equivalent to, where P{x, t) is the probability distribution of A(t) on the t’th time slice and F{X{t)} is a general function of X{t). It is again convenient to generalize (4.8) and define connected correlation functions by the relation. (4.16) {a:"} for example, X(t) can be viewed as representing the corresponding value of x on inm}) = J F{x}P{x,t)dx, (4.17) (4.18) where {s,X) — s(t)X{t)dt with s{t) some arbitrary but fixed function for which the integral is convergent. The first two connected correlation functions are found

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16 from (4.18) to be, = (X(i)>. (4,19) {X{h)X(h))^ = (X(l,)X(t,)) (Jf((0)(X(t,)), The characteristic functional, the generalization of Eq. (4.10), is defined as, C{s} = (4.20) In particular note that by setting s{t') = aS{t' t) with a G R, C{s} becomes, C{a) = (4.21) the characteristic function for the stochastic variable X{t). Utilizing (4.17) we obtain C{a) = = I e‘“P(x, t)dx, (4.22) which can be used to find the distribution P{x, t) by way of the inverse Fourier transform as in the one dimensional case. Various types of distributions or stochastic processes (the term process being implied by the time evolution of the stochastic variables along with their particular properties) can now be described by specifying certain constraints on the values of the connected moments. In particular a Gaussian stochastic process [10] is defined by the first two connected correlation functions with all higher-order connected correlation functions being equal to zero. Using (4.18) and (4.19) the characteristic functional for a Gaussian process is found to be, C{s} = (4-23) = eif s{t){X{t))dt-^^ J s{h)s{t2){X{tl)X{t2))^dtidt2 In particular, by setting s{t') = a5{t' t) with a G R in (4.23), one obtains the characteristic function for a Gaussian process. (4.24)

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17 note that this has the same form as (4.13) with x replaced by X{t). This now allows us to use (4.22) to obtain, I e‘“’P(x, t)dx = (4.25) which allows us to find the distribution P{x, t) by way of the inverse Fourier transform. Similarly to (4.14) we now obtain, P{x,t) = exp < -IlzW])]; (4.26) showing that the paths X{t) have a Gaussian distribution over any given time slice. Wiener Processes The measure discussed at the beginning of this section follows from a particular type of Gaussian process known as a Wiener process or Brownian motion. A Wiener process is characterized by the conditions, W(0) = 0, {W{t)) = 0, (4.27) where t > 0, with the connected two-point correlation function given by, {W {ti)W{t 2 )) = min(ti, fs), (4.28) {W {ti)W (t 2 )) ^ being equal to {W {ti)W {t 2 )) since (VF(t)) = 0, with all higher-order connected correlation functions vanishing as is the case for all Gaussian processes. Note that the path space on which the measure is defined is {0} x This process is easily generalized to the case where W'(0) = x' with x' arbitrary. In this case (4.27) is replaced by, W'(0) = x' and {W{t)) = x'\ the variance given by the right hand side of (4.28) remains the same, however, we must write {W {ti)W [ 12 ]) c = min(ti,f 2 ), since {W'{t)) no longer has a zero value; as the process is still Gaussian, again, all higher-order connected correlation functions must vanish. It is easily seen through

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18 direct calculation that this process is entirely equivalent to W'{t) = W{t) x'. It is sometimes useful to define a Wiener process with a different diffusion constant. This process, Wi,{t), is identical to W{t), with the exception that (4.28) is replaced by (W'u{ti)W^{t 2 )) = i^min(fi, ^ 2 )Again this process is simply related to W{t) through the equation W„{t) = ^/uW{t). The process W{t) itself is sometimes referred to as a standard Wiener process. The nature of Brownian paths In the case of the Wiener measure one finds that the measure is supported only on a special class of paths in {0} x that is all sets of paths outside this class have measure zero. To show this first define X{t) to be a Gaussian variable with mean value {X{t)) equal to zero so that (4.26) becomes, P{x,t) = 1 exp (4.29) Therefore, as {X‘^{t)) — ) 0, equation (4.29) for P{x,t) tends towards a delta function for any value of t, hence the measure can only be supported (nonzero) on X{t) such that X{t) = 0. Also note that if (A^(t)) = 0, it follows that, J x‘^P{x,t)dx = 0, (4.30) therefore since x"^ > 0 the same conclusion applies. Now if (X^(t)) = 00 we see that P{x, t) = 0 for any finite value of x, hence it can be inferred that in the limit as {X^{t)) 00 , pb / P{x,t)dx = 0 (4.31) J a for any a, 6 G R with a < b. On the other hand since / P{x, t)dx = 1, independent of {X^{t)), we have lim(x 2 p))^oo /.!^ -P(a;, t)dx = 1. Now since f^^Pdx = fl^Pdx + f-a Pdx + fff Pdx (with a > 0) it follows from (4.31) that in the limit as (X^(f)) — >• 00 / —a poo P{x,t)dx+ / P{x,t)dx = l (4.32) -00 Ja

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19 for any finite but arbitrarily large a. Hence, this heuristic argument demonstrates that the measure is not supported on any finite value of X{t) and so must only be supported on values of X{t) such that X{t) = ±oo. Next define X{t) to be X{t) = W{t + At) — W{t) with At > 0, since the sum or difference of any two Gaussian random variables is Gaussian, X is Gaussian with (A) = {W{t + At)) {W{t)) = 0 and (X^) = {W^{t + At) + W\t)-2W{t + At)W{t)) (4.33) — (t "h At) + 1 — 2t — At. Hence as At — )• 0 the variance, {X'^) — ) 0 so that the measure becomes concentrated only on values where A = 0. Although X{t) represents the difference between values of the function W{t) at times t + At and t, Eq. (4.33) is not enough in itself to insure continuity of the VE(t)’s. For example if a W{t) possesses an isolated jump discontinuity at t = tj this will result in two jump discontinuities in the function X{t) with the middle segment of A ranging over the interval (t^ At, td] or [td At, t^) [depending whether the jump in W{t) occurred at or after td respectively]. Now as At — > 0 this middle segment will shrink to zero in length while the other two segments both converge to A(t) = 0 and rejoin at this point. We may therefore rewrite (4.33) as ([W(t + At) W(t)j^) = At = Uc At + OdAt, (4.34) where Gc and are proportionality constants with Oc related to the average slopes of the continuous segments of the W (t)’s within the At window [including the continuous segments of the W (t)’s possessing discontinuities within the window] and measuring the average contribution of the discontinuities within the window. In the case of higher moments of X{t) it is expected that the contribution of the discontinuities will still

PAGE 24

20 be proportional to the first power of At, as this contribution has nothing to do with the average slopes of the l^(t)’s within the At window, and so should be directly proportional to the width of the window. In particular for {X‘^{t)) we may write {[W{t + At) W{t)Y) = C{At) + k. At, (4.35) where C{At) is some function related to the average slopes of the continuous segments of the iy(t)’s within the At window and kj is a proportionality constant. Using the relation, {S{t,)S{t2)S{h)S{U)) = {S{U)S{t2)S{t,)S{U)), + {S{ti)S{t2)){S{h)S{U)) + {S{t,)S{h)){S{t2)S{U)) + {S{h)S{U)){S{t2)S{ts)), for a general stochastic variable, ^(t), with {S{t)) = 0, and recalling that {S{h)S{t2)S{h)S{U))^ = 0 if S{t) is Gaussian as well, we may calculate, {[W{t + At) W{t)]‘^) = 3{At)\ (4.36) Comparing (4.36) with (4.35) we find kd = 0, hence the Wiener measure is only supported on functions W{t) which are continuous. This result is closely related to a theorem from which the continuity of the Wiener paths also follows [11]. We may now define X = [IU(t + At) — W{t)]/ [At) with At > 0 which is still Gaussian for any given At. Again, {X) = [(W(t + At)) — (W(t))]/(At) = 0, while the variance is found to be (A^) = l/(At). In this case as At 0, the variance (A^) oo, so that the measure is concentrated only on values where A = ± 00 . Now in this case as At -4 0, A dW[t)/dt = ± 00 , hence the Wiener measure is supported on functions W (t) which are everywhere continuous and nowhere differentiable.

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21 Explicit form of Wiener measure WithX(t) = W{t + e) — W{t), as earlier, this variable is Gaussian with (X{t)) = 0, the variance is found to be {X^{t)) = e, and it then follows from Equation (4.26) that, P{x,t) = 1 V2 7re exp X {4.37) In this case x represents the difference in values of some continuous path, W{t), between the times tf = t + e and ti = t\ x may therefore be written, x = Xf — Xi, (4.37) then becomes. P{Xf,tf,Xi,ti) =: V2 ne exp {Xf Xjf 2e (4.38) Recalling that (4.38) is the probability distribution of the random variable X{t) at time t, P[xf,tf]Xi,ti) can then be understood as the probability density of some continuous function, W{t), in the function space, going from some arbitrary but fixed point Xi at time tj, to some other arbitrary but fixed point x/ at time tf. Note that ^T"ooP[xf,tf\,Xi,ti)dxf is equal to unity. Then dividing the time T into N intervals so that e = T/N, the probability density for a path, W{t), going from the fixed point x' = 0 at to = 0, and passing through the set of arbitrary but fixed points Xi,X 2 , . . . ,xn at the equally spaced times ti = e,t 2 = 2e, . . . ,1^ = T becomes. N-l Pn{xi,X2,...,Xn)n /K — =0 TTC (4,39) Thus we recover equation (4.2) from which equation (4.4) for the explicit form of the Wiener measure follows. Browian Bridge In the evaluation of the Feynman path integral or the coherent-state path integral it is necessary to consider paths starting from some fixed value, x', at time t = 0 and ending at some fixed value, x" at t = T. It is therefore necessary to consider the

PAGE 26

22 subset of Wiener paths which satisfy these conditions. This particular ensemble of paths, B{t), should be Gaussian with an average which varies linearly from x' to x", i.e., = x' + (x" — x')t/T. The variance (on some fixed time slice), {B'^{t))c, must be zero at both t = 0 and t — T, therefore if {B'^(t))c = at^ + bt + c, (4.40) it follows that c = 0 and a = -b/T, so that {B‘^{t))c = bt{l-t/T). In the limit as T ^ oo the paths will no longer be pinned at x", hence, limr_>oo(-B^(0)c = (W^'^(O)c =" t so that 6=1 and {B\t)), = t{l-t/T). (4.41) A Gaussian process having exactly these properties, and therefore entirely equivalent to extracting this subset of paths is given by, B{t) = x' + (x" x')^ + W{t) ^W{T). (4.42) Direct calculation now yields, {B{ti)B{t2))c = min(ti,t2) (4.43) which clearly reduces to (4.41) in the case where ti — t 2 = t. The measure on the function space, {x'} x R(°T) x {x"}, mentioned near the beginning of this section also follows from this process. However in this case we have. = f(-)dMB(X) JdpBiX) (4.44) due to the fact that (1) = 1 for a Gaussian process and / dfi(X) exp (x" x') 2T (4.45)

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23 Stochastic Integrals Due to the nondifferentiability of the Wiener paths, W{t), special care must be taken in defining integrals involving the derivatives of such paths. In the case of the integral, X‘^{t)dt, found in (4.4) [with X{t) = W{t)], the integral is defined on the lattice and is equal to infinity. It should be noted however that this integral, or the term e~^^o ^ for example, should only appear in the form A/’e~ 2 /o ^ dX{t), since this term is defined by (4.46) N lim exp ^ ^ (Xi+i Xi'' 2 -^ 2 ^ ^ t=0 dx\ dx N which collectively represents a well defined mathematical expression, while terms such as A/" or e“ 2-^0 ^ alone, are equal to infinity and zero respectively. An integral of the form s{t)W{t)dt, as found in Eq. (4.20) [also with X{t) = W{t)], poses no problem since both s{t) and W{t) are both finite and continuous so that s{t)W{t) is finite and continuous and hence falls within the scope of both the Reimann and Lebesgue definitions of integration. The coherent-state path integral (to be discussed in Sec. 6), however, involves integrals of the form r Y{t)X{t)dt. There are two standard prescriptions for defining this integral, rT lo (4,47) r Y(t)X(t)dt = Jim Y(U) (Jf (ti„) X((,)] , (4.48) and. i=0 I Y{t)X{t)dt = Jin^ XI \ , (4.49)

PAGE 28

24 again where tk = ke with k = 0,1, ... ,N and e = T/N, the first being due to Ito and the second, sometimes called the midpoint rule, due to Stratonovitch; these two prescriptions may or may not give the same answer depending upon circumstances. In the case where Y{t) and X{t) are both continuous and differentiable functions, or in the case where Y{t) and X{t) are independent Wiener paths (as is the case for the coherent-state path integral in suitable coordinates) both (4.48) and (4.49) yield the same values, however, if Y{t) = X{t) = W{t) we will see that these prescriptions are not equivalent. Subtracting (4.48) from (4.49) we obtain, Jto s' 5 (W'(*i+.) mtdf = \t^0. (4,50) where we have used the fact (which may be found in Ref. 12) that [W{t+e)-W{t)Y = e for t + e < T and e sufficiently small. The coherent-state path integral involves an integral of the form, / \p{i)Q{t) g(^)p(f)]| dt (4.51) where p{t) and q{t) are independent Wiener paths; this integral will be defined according to the Stratonovitch prescription.

PAGE 29

REVIEW OF COHERENT STATES Conventional canonical coherent states are defined as the normalized eigenstates of the annihilation operator a = {l/\/2flh){Q,q + ip), so that a\z) = z\z), {z\z) = 1, with z e C and [q,p] = ih. (5.1) It can be seen that I 2 ) can be written as |z) = or \z) = 'V' -^j=\n) (5.2) n=o vn! where |0) and |n) are the zeroth and nth energy eigenstates of the harmonic oscillator. Therefore while in the position representation If we define z = (l/\/2fIfi)(ng + ip) [q,p G R) then (5.3) takes the form, (p, q\p', q') ^ ^ (?p' P^') “ ^ (P “ } > while (5.4) becomes {qW, 5') = I ^) exp where \z) = \p,q). The coherent states also satisfy the resolution of unity dpdq I j \p,q){p.q\ 2nh = 1 . (5.3) (5.4) (5.5) (5.6) (5.7) 25

PAGE 30

26 However they are not orthogonal and exhibit a linear dependency in the form \p\q') = j J ^^{P,q\p',Q')\p,Q) (5.8) = / / ^ ^ I"' Therefore there are more coherent states then are necessary to form a basis for the Hilbert space and the resolution of unity is also referred to as an overcompleteness relation. It follows from (5.7) that, where |5) is an arbitrary state vector, hence the coherent states give us a useful representation of the Hilbert space, i.e., ’ip{p,q) = (p, g|5). It can be shown that the functions 'ip{p,q) are square-integrable, bounded, continuous [13], and constitute a subspace. Ho, of the vector space L^(R^). From Eq. (5.9) it follows that, {p,Q\S) = j j{p,q\p',q'){p',q'\S)^^^, (5.10) hence (5.5) constitutes a reproducing kernel ior the space Hoi more generally Eq. (5.5) constitutes the integral kernel of a projection operator, Pq, on L^(R^) onto the space Hoi note that for a general function, f{p,q) e f{p,q) = fo{p,q) + f±(p,q) where /o £ Hq and f± ± Hq, therefore Pof{p, q) = /o(p, q), the integral analog of this equation being, f j {P, q\p', Q')f{p', = /o(Pq)(5-11) It can also be shown that a square-integrable function g{p, q) is contained in Hq if and only if Ag{p, q) = 0, where ^ ^ [('®p «/2)^ + (i£>, + pft? 1 ( 5 . 12 )

PAGE 31

27 An integral transform exists allowing us to obtain the coherent state representation, ip{p,q) = (p,q\S), of an arbitrary state vector 15), from its position representation, (l)[x) = (a:|5). Using the resolution of unity and Eq. (5.6), we obtain the coherent-state transformation, '^{p,q) = {p,q\S) = J (5.13) And by integrating the coherent-state transformation we obtain an inverse transformation, 3 0(a:) = (xi5) = ^ J e^^^P^^^^p{p,x)dp. (5.14)

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THE COHERENT-STATE PATH INTEGRAL The coherent-state path integral is given as, V^{h-,p",q",T-p',q',0) (6.1) = 27re'^^/2 J exp J ^^{pq qp) h{p, q) dt^dij!'^{p, q), where is the product of two independent Wiener measures [i.e. d/j,‘^{p,q) = dfJ‘wip)dfJ-wiQ)] pinned at p',q' at t = 0 and p”,q" at t = T > 0 with diffusion constant u [here, with the exception of (6.4), h and fl will both equal unity]. Note that in this case the explicit form of the Wiener measure given by (4.4) is modified to. ( 6 . 2 ) due to the presence of the diffusion constant v> [as can be seen from (4.35) with the variance e replaced by i^e], with the normalization now given by /
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29 holds for a wide class of Hamiltonians, including all those that are polynomials in p and q {p and q being operators corresponding to the Cartesian momentum and position coordinates). It should be noted that an analogous path integral representation for the configuration space propagator, involving Wiener measure cannot be constructed due to the presence of the factor exp[f| f q^dt\ resulting from the kinetic energy term in the action [14]. In the case = 0 Eq. (6.1) can be solved explicitly, the result being = 0;p",g",T;p',5',0) (6.6) ( i 1 r n 'I = 2itahpV2) 4 [(?" r'f + («" }. For T > 0 we find. P"' (6 '^) = exp {^(P'?" P''q') \ [(/ p'f + {q" q'f] | , in agreement with (5.5), as expected for the case of no time evolution; for T — ) 0 for finite V, however, we find P„(0;p", g", T;p', 5', 0) approaches 2ttS{p" p')6{q" -q'). This refiects the fact (shown in the second paper listed in Ref. 2) that P„(0; p”, q", T; p', q', 0) is the integral kernel of the operator E°{u,T) = on L^(R^) with A as given in Sec. 5 [note that if B is an operator its integral kernel is, B{p",p'-,q",q') — {p",q"\B\p',q% It is shown in the second paper listed in Ref. 2 that as u ^ 00 with T > 0, the operator E^{u,T) — > Pq, the projection operator on Z/^(R^) discussed in Sec. 5 whose integral kernel is (6.7), whereas if T = 0, clearly P°(i^, 0) = i. More generally, for /i(p, q) satisfying I dpdq\h{p,q)\\xp [-a{p^ + q^) < 00 (6.8) for all Q! > 0, it is shown that Vu{h\ p", q", T ; p', q', 0) is the integral kernel of a strongly continuous contraction semigroup, E{u,h]T), on L^(R^). In particular for uniformly

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30 bounded h (i.e., \h{p,q)\ < k with 0 < k < oo), E{u,h\T) has the form In this case the analog (for T > 0) of the h = 0 result, lim^^.oo T) = Pq, is shown to be, h;T) = Poe (6.9) the integral kernel of the operator, Po, being the propagator itself, i.e., lPoe-“’<‘’'^"^Po]{p",q".T-p'.q'.0) = {p" ,g"\e~i™\p' ,g'). (6.10) The integral kernel of E{u,h]T), that is Vi,{h;p'',q",T-,p',q',0), is likewise shown to converge to, 9", T; p', q' . 0) = 0. Following this, pointwise convergence of'P^{h]p",q",T]p',q',0) to {p",q"\e~^'^'^\p',q') for T > 0 is shown assuming h satisfies the condition. J dpdq\h{p, q)\^ exp -^P{P^ + Q^) < oo (6.12) for some 0 < ,0 < 1 in addition to (6.8). In order to do this use is made of the expansion. V^{h-,p",q",T-p',q',0) (6.13) = Vt.i0]p",q",T;p',q',0) i f df T-i, h(p, J 0 rT rti / dti dt2{h{p,q)(t)pn^gn „T_t,E{u,h]ti-t2)h{p,q)(i)pi^g,^^^t2), with 4>puqi,‘',t{P^ Q) ~ ^1^(0; P> 9) 9i) 0), its explicit form being obtained from (6.6), and inner products being defined by. if, 9) = J ^^r{p,9)g(p, 9). (6.14)

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31 and the expansion, (/, q”\e-^'^'^\p', q') = {p", q"\p', q') i f dt{p", q"\'H\p', q') (6.15) J 0 -[ dti f dt2{p”,q"\'He^~^'^'^^'H\p',q'). Jo Jo Eq. (6.15) is first generalized to L^(R2) then subtracted from (6.13). It is then shown that as -> oo the difference \V^{h-,p",q",T-,p',q',0) {p”,q"\e-^'^'^\p',q')\ tends to zero.

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DERIVATION OF THE SEMICLASSICAL COHERENT-STATE PROPAGATOR Using the explicit form of the Wiener measure (6.2), Eq. (6.5) now becomes, K{p",q\T-p\q',0) (7.1) = y exp I ^ y ^{pq ~ qp) ~ h{p, q) dt where hu is now the diffusion constant (and D = 1). We seek a semiclassical approximation to this integral of the form Ksc{p",q",T-p',q',0) = Ee^^/\ (7.2) (pI + e.) 2u dt, (7.3) where F represents the “action”, ^ y Q‘'P>') ^(pi'^ q^) evaluated along extremal trajectories, q^ and with E representing the contribution of quadratic deviations from these rays. Note that h{p,,,q,,) has been replaced by the classical Hamiltonian, H{p„,q,,), in (7.3) since h{p,q) H{p,q) as /i 0 (as can be seen from (6.4) and Ref. 16). The extremal equations following from (7.3) are easily shown to be, qu{t) Pu{t) + dH dpu{t) dH dqu{t) = iPvit) —I V qu{t) u (7.4) (7.5) subject to the boundary conditions g^(0) = q' , p^(0) = p' , q,,{T) = q" and p^(T) = p". These equations are analogous to the Navier-Stokes equations [17] for an incompressible fluid with a small viscosity and approximate solutions for large u (exact in the 32

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33 V oo limit) are given by [3], 9"(*) = (?' s')*:-" + 5(0 + («" (7.6) p-(t) = (p'-p')e->'‘+p(() + where q' = q{0), q" = q{T), p' = p(0), p" = p{T) with q{t) and p{t) satisfying the classical equations, ^ dH . dH 9=^ (7.7) subject to boundary conditions which will be determined shortly. Boundary Conditions for Eqs. (7.7) Substituting equations (7.6) into (7.4), yields the result, ~(q' ~ 5')e+ i? + (," 5")e-'7-<) u 1/ dp‘' (7.8) 1 .. = i|(p' p')e-‘" + -^p+ {p" Now setting t = 0 yields, 1 {q' q') + ^q + {q" 9")e-^^ 1 dH u dp'' (7.9) «=0 = i[{p' p') + ^P + (/ p”)e with q = q{0) and p = p(0), which in the limit u oo, imposes the initial boundary condition. q' + ip' = q' ip', (7.10) on q{t) and p{t). Similarly, setting t = T in (7.8) yields. y-q')e-''^ + -q'^-{q"-q")-t V 1/ dp'^ (7.11) t=T 1 i[(p' p')e +^p + {p" p")],

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34 with q = q{T) and p = p(T), which in the limit i/ — )• oo, now imposes the final boundary condition, (7.12) on q(t) and p{t). The same process applied to (7.5) will produce identical results. It is seen from these mixed boundary conditions that the q{t) and p{t) satisfying (7.7) and utilized in (7.6) are generally complex valued, hence q''{t) and p‘'{t) are complex as well. Note that (7.10) and (7.12) may appear to over specify the solutions of the classical equations (7.7), however if we set '^ = q' -\-w and p' = p' + w, where w e C, the initial boundary condition q' + ip' = q' + ip' is automatically satisfied while the combination q' — ip' = q' — ip' + 2w remains completely free. Hence w may be chosen so that the combination q' — ip' evolves in time to the specified final boundary condition Using Eqs. (7.6) we are able to find a semiclassical expression for F, defined by Eq. (7.3), entirely in terms of the position^ 9(0) the momentum, p{t), satisfying the complex boundary conditions (CBC) given by (7.10) and (7.12). First note that for arbitrary differentable functions, p(t) and q{t), {pq qp) = pq [J^(p?) pq], so that (7.12). Semiclassical Action, F ^(p?-9P)*= -^[p(r)?(T)-p(0)?(0)] + ^ pqdt, (7.13) SO that for the extremals, Pu{t) and q^it), (7.14) hence F becomes. F P'^') + ^ P>'^'' H{Pu, qu) + ^{pI + ql) dt. (7.15)

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35 Using (7.6) we find, p-q= [(," rw P') (q' i)(p" p'OI-'e-'’’ (7.16) -( oo limit, while the second and third terms are readily integrated to yield. lim U^OO \( 0, r f{t)ue-‘'^dt = -[f{T)e-'^ /(O)] + T /(t)e-‘^‘dt. ^0 Jo (7.18) Clearly, as — ) oo, the first term on the right hand side approaches /(O); now consider the term, /o , pT 1 r f{t)e-‘'^dt < e-'^dt Jo Jo ' (7.19) 1 r , 1 < /max / /max 1 Jo V (where |/max| is the maximum value of \f{t)\ on [0,T] }, this term clearly tends to zero as z/ — ) oo so that, rT lim [ f{t)ve ''^dt = /(O). Jo In an analogous manner it can be shown, = /(r). (7.20) (7.21)

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36 From Eq. (7.19), and its analog for the integral the sixth and seventh terms are seen to make zero contribution to F. Hence for q{t) and p{t) differentable on [0,T] with T > 0 it is found that, Ji™ / dt = -^(q'q){p' p') + ^{q" q"){p" f) (7.22) -( Jq Jq i/->oo Jq i/->-oo u-^ooJq Eq. (7.15) for F may be written, F = ~2{p"q" q”p" + q'p' p'i) + ~ ~ p^'q^'dt, (7.25) dt (7.26) Note that we may replace H{p^,q^) by H{p,q) in (7.24) since q''{t) q{t) and p''{t) p(t) for 0 < t < T as ^ oo and in this limit, q'' = q^ and p‘' = p^, hence

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37 in this limit the corresponding functions q^, q and p^, p differ from each other only at t = 0 and t = T {a set of measure zero) and will not effect the value of (7.26). Now in a manner similar to that of the derivation of (7.22), and utilizing the fact that q‘' = q„ and p" = p^ in the v oo limit, it can be shown that, 1 (7.27) = ^l(p' f? + W + ip" f? + W ?")"]. We may now rearrange the CBC so that, q' — q' = -i{p' —p') and q" — q” = i{p" —p"), showing that The semiclassical expression for F is now found to be, F = 2 (t’V q"p” + q'p' p'q') + ^ ~ ^^P’ evaluated along trajectories satisfying Hamilton’s equations, ^ dH . OH 5=^ and P=-^, but subject to the CBC, (7.28) (7.29) (7.30) q' + ip' = q' + ip', (7.31) q" — ip" = q" — ip" , Amplitude Factor, E In order to evaluate that part of (7.2) representing quadratic deviations from the extremal trajectories we may rewrite (7.1) using (7.13) so that, K{p",q!',T-p',q',Q) (7.32) = lim 27re^‘'^/^ I/— VOO 1 -pq h{p, q) 2hu / if + e) dt} n dp{t)dq{t) exp -^(pV-pV)

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38 Letting q{t) = q^{t) + p{t) = Pv{^) + u{t) and replacing h[p,q) in (7.32) by H{p,q), we may expand H{p,q) about an extremal path to obtain, H{p,q) = H{p„,q^) + [dp^H{p^,q^)]u + [dg^H{p^,q^)]v + Qi^) + P{Pu, qu) + \^{Pu, qv) + where, (7.33) Oi{Pu, qv) = dl^H{p^, q^), P{p^, q^) = dp^dq^H{p^, q^) and 7{Pv>qv) = dg H{p^,q^), (7.34) E then takes the form. ^ ^ J ~ \7{Pv,qv) dt (7.35) subject to the boundary conditions u(0) = u(0) = u{T) = v{T) = 0. Note that the factor \{p''q" —p'q') has been incorporated into F. The amplitude factor, E, now has the form of the coherent-state path integral representation of a propagator with time dependent Hamiltonian, H = \aP^ + \l}(PQ + QP) + \lQ'‘, (7.36) where [Q, P] = i (with h = 1), so that in Dirac notation rT n E = (0, 0|T exp | 0 , 0 ), (7.37) where P Q = \{PQ + QP). It should be noted that since (7.35) is given in terms of the extremal rays p„{t) and q^{t) in the u oo limit the a, /? and 7 in (7.36), used in (7.37), should be given in terms ofpoo(0 qooit), hence we are at liberty to replace these by p{t) and q{t), since the corresponding functions differ only on a

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39 set of measure zero (as mentioned earlier). Hence from (7.36) onward, a, P and 7 will be given by, a = a{p, q) = d^H{p, q), j3 = ^{p, q) = dpdgH{p, q) and (7.38) 7 = 1{P,0) = d^H{p,q). Also note that the operators P Q and form a basis of Lie algebra such that. [ 2 ^ 2 . iP • Q, Q\PQ -iP\ and can be faithfully represented by the 2-dimensional matrices, 1 lp2 2 0 1 0 0 :<3 = 0 0 1 0 PQ i 0 0 -i In this representation H takes the form, H -ij3 a 7 (7.39) (7.40) (7.41) (7.42) (7.43) E can now be evaluated by seeking an alternative expression for (7.43) in the form. so that in this representation the operator, rT n T exp may be written T exp i-r —ia = A B ' \ Jo —ip . C D exp i^^{P^ + Q^) exp [irjP Q] exp i(l(P^ + < 3 ") (7.44) Noting that |(P^-f Q^) = cti and that P-Q = ias, where
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40 From (5.2) it is seen that [0, 0) = |0), where |0) is the zeroth energy eigenstate of the harmonic oscillator (with h = Q, = 1), so that |(P^ + Q^)|0) = ||0) and we obtain, E = (0| exp = exp exp [ir)P Q] exp (0|exp [irjP Q] |0), + Q^) |o> (7.46) SO that using [Q, P] = i and (a:|0) = exp(— a:^/2)/7r^/^ this becomes, =i«+C)/2 \/7r J (7.47) In order to evaluate this integral note that. OO 1 fc =0 (7.48) but. r )=0 dk dr]^ ,-{ PP ^)!2 (7.49) 77=0 since, (xdx)^e (x^e^’^)i 2 ^ ^rixdx^-x^/2 E^ A:! k=0 dk dr}^ ,-{x^e^P)l2 T]k = (7.50) 77=0 since the middle term of (7.50) is just the Taylor series expansion of e Eq. (7.47) then becomes, ,i(4+0/2 ^ __ ^i(C+0/2 E = 7T v^cosh rj (7.51) From (7.45) it is seen that, {A + D)/2 = cosh ( 77 ) cos (^ + C), {B + C)/2 = 7 cosh ( 77 ) sin (^ + C), (7.52) from which it follows. E=l/^[A + D-(B + C)]/2 . (7.53)

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41 This result may be reinterpreted in terms of the linearized Hamiltonian equations of motion obtained by taking differentials of (7.30), this leads to, q{t) = p{t)q{t) + a{t)p{t), (7.54) P{i) = ~j{t)q{t) /3{t)p{t), where again a(t), P{t) and -y{t) are given in terms of (7.38). Since p and q are subject to the fixed initial boundary condition of (7.31) we find 6{q' + ip') = ^ + ip' = 0 so that ^ = i^. Eqs. (7.54) may be rewritten in matrix form, (7.55) -iq ip Q 1 1 . P . ^ -ip note that this is in the form of Schrodinger’s equation, iv = Hv, with H given by (7.41), thus it follows that a solution is given by. ' -i^' A B ' 1 1 «s>. 1 C D L ^ J (7.56) From (7.56) and the relation ^ = i^ it follows that, p' + i^' = [A + D — (B + C)]i^, thus it is convenient to choose ^ = —i/2 so that. E = 1 (7.57) Hence E is given in terms of solutions of (7.54), evaluated at t = T, and subject to initial conditions. ^ = -i/2, ^ = 1/2. (7.58)

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FORMULATION OF THE SEMICLASSICAL COHERENT-STATE PROPAGATOR IN SUMMARY The semiclassical coherent-state propagator (SCSP) as derived in Sec. 7 (and previously derived in Ref. 3) is given as, Ksc{p",q",T-,p',q',0) = Ee“'/’‘. (8.1) The action F is given as F = -^(pV q''p" + q'p' pV) + [^(P t QP) H{p, q)]dt with q{t) and p{t) satisfying the extremal equations ^ dH . dH and p= — . dp dfq Each extremal solution {q(t),p{t)) is subject to CBC ( 8 . 2 ) (8.3) Q,q' + ip' = ilq' -Iip', (8.4) nq"-ip" = Qq"-ip" with q' = q{0), q" = q{T), p' = p(0), p" = p{T) and fl,q' ,p' ,q" ,p" G R. The amplitude factor E, is calculated from quadratic deviations about the extremal (complex classical) trajectory in the exact path integral expression for the propagator [see Eq. (6.5) and (7.1)]. It is given as E= 7 = ^ ( 8 . 5 ) ^p(T) + mq(T) where q{t) and p{t) are solutions of the linearized Hamiltonian equations of motion, q{t) = Hpg{p,q)q{t) + Hpp{p,q)p{t) (8.6) p{t) = -Hgg{p, q) q{t) Hgp{p, q) p{t) 42

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43 [Hpg{p,q) = d qd pH {p,q), etc.] subject to the initial conditions g(0) = — ^'/(2n) and p(0) = 1/2. (8.7) The need for CBC. In order to evaluate the propagator (8.1) we must use solutions to (8.3) which in some sense connect the entrance label {q',p') with the exit label {q”,p") in a time interval T. If we were to make an obvious choice and set q' = q' and p' = p' we would arrive at the fixed real value of position and momentum q" — qT and p" = pt after a time T. This would be acceptable only if q" — qx and p" = px\ clearly choosing q' and p' to be real is too strict a requirement for arbitrary q" and p" . Therefore in order to have extremal trajectories connecting some given entrance label {q',p') to an arbitrary exit label {q",p") the functions q{t) and p{t) must be complexified according to some set of CBC. Why CBC (8.4)? There are many things in the formalism of the coherent-state propagator which are suggestive of the CBC (8.4) (also derived in Sec. 7), for example the quantum mechanical coherent-state propagator (P .9 I -^T-H \p\q') = {z"\e T(b"P+bT) (0|e^ | 0 ) ( 8 . 8 ) with z = (l/\/2Vlh){Q.q + ip). Furthermore, exact analytic expressions for the quantum mechanical coherent-state propagator exist for the three cases H = 0, H = p‘^/2 and H = |(p^ + q^) [18,3] and when one applies the CBC (8.4) to Eqs. (8.1), (8.2) and (8.3) for these three H's one re-obtains these exact analytic expressions. Understanding CBC (8.4), Consider the initial boundary condition Up' 4ip' = Q,q' + ip' . If we set q' = q' + w, p' = p' + iflw {w G C), as remarked in Sec. 7, it is easily seen that this initial condition is satisfied for any complex number w. Therefore for any fixed entrance label {q',p') we are free to choose any w so that the final boundary condition f2p" 4ip" = Q,q" + ip" is satisfied. The situation is easily understood if

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44 we define two new variables P{t) = flq{t) + ip{t), Q{t) = — ip{t), which up to an unimportant multiplicative constant are new canonical variables [19]. A given entrance label {q',p') will then fix our initial value of momentum P' . Then using q' = q' + w and p' = p' -\iflw our initial position Q' can be written as Q' = Q,q' — ip' + 2Qw. Since w is arbitrary we are free to choose any value of Q' that will satisfy the final boundary condition Q" = Qq" — ip" = Vtq" — ip" . Let us also note that when u; = 0, q' = q' and p' = p' and we obtain a unique real classical ray. If w ^ 0, on the other hand, the classical rays are complex. An extremal trajectory can be viewed now as either a {q{t),p{t)) pair or its corresponding {Q{t), P{t)) pair. Now, since we fix P' [or equivalently our entrance label {q',p')] and choose Q' to yield a specific value of Q", Q" = flq" — ip", it will be convenient to alternatively view our trajectories as those functions Q{t) which yield the mapping Q' Q" . Also, a trajectory will be fixed by the choice of initial phase space coordinates, Q' and P' , but since our initial momentum P' is strictly fixed by our choice of entrance label {q',p'), each of the trajectories used in the evaluation of the SCSP will be determined only by its initial position Q' . We may thus associate each trajectory with its unique initial value Q' . Therefore relevant quantities such as F and Ksc may be viewed as functions of Q'. Real valued quantities such as Fr = Re(F), Fj = Im(F) and \Ksc\ may now be pictured as functions over the complex Q'-plane.

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APPLICATION OF THE SEMICLASSICAL COHERENT-STATE PROPAGATOR Origin of Multiple Solutions In actual practice there will generally be more than one trajectory satisfying a set of CBC. This comes about due to the mapping Q' Q" (where Q' -> Q” always implies fixed P' and T) being analytic. We will first discuss how multiple solutions arise and then demonstrate the analyticity of the map Q' — )• Q" and discuss its critical points. Multiple solutions of mapping Q' Q". In general if an analytic map /(z) contains critical points, that is zq for which f'{zo) = 0, the mapping will be many-to-one in the neighborhood of Zq. Specifically, the mapping will be (n-l-l)-to-one for an nth order critical point, i.e., if 0 is the lowest order nonvanishing derivative of /(z) at z = Zq. For example, if the critical point is 1st order, /'(zq) = 0 and /"(zo) 7 ^ 0, the mapping will be two-to-one in the neighborhood of Zq as can be seen from the equation /(z) ~ f{zo) + 2 f"(^o){z — which holds in the neighborhood of Zq; at the point zq itself we also see from this equation that the mapping is one-toone. The many-to-one property of the analytic map containing critical points will hold globally (except at images of critical points themselves) due to continuity, the exact number of points mapping to a given image point depending on the number of critical points and their respective order. For example, the mapping /(z) = z^/4 -|z^/3 contains two critical points, one 1st order at z = — 1 and one 2nd order at z = 0, and is globally four-to-one. Analyticity of mapping Q' — )• Q” . To demonstrate the analyticity of the mapping Q' -> Q" we first recall that a solution to (8.6) for arbitrary ^ is given by (7.56) and 45

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can be written, 46 -i^' = -iA^ + Bp' (9.1) f = + = 9 ( 0 )) = q{T), etc.] where A,B,C and D are complex constants depending upon the given arbitrary but fixed trajectory with P' and T fixed as always. Now dQ" /dQ' = 5Q" /SQ' where 5Q' = — ip and 5Q" = flp' — ip' is the corresponding change in Q". Since this derivative will always be evaluated holding T and P' constant we get 5P' = + ip = 0 so that p = iftp and SQ' = 2Q,p with p arbitrary. Therefore dQ" /dQ' = {Q.p' — ip')/{2Vtp), but using (9.1) and p = iVlp we find Q.p' — ip' = [Q(>1 + D) — (C + VPB)]P . And so it is found that dQ" _ np' ip' __ n{A + D) {C + Q^B) dQ' 2Q,p 2f] independent of the arbitrary initial change in Q' , 6Q' — 2Q,p . Critical points of mapping Q' Q" . The critical points of this mapping are the values of Q' for which dQ" /dQ' = 0. Notice that since dQ" /dQ' = {Q,p' — ip') / {2Q,p) we can write 1 pQ“IBQ' Now if we use the value of p given in Eq. (8.7), p = —i/{2Q), (used for the evaluation of the amplitude factor E) in the above equation we find 1 pQ"/BQ' Therefore at the critical points of the mapping Q' — >• Q" the amplitude factor E diverges; such a critical point is referred to as a phase space caustic (PSC)[4]. It is worth mentioning in conclusion that for the three cases H = 0, H = p^/2 and H = |(p^ + q^), PSCs of Q' — ) Q" do not exist and consequently the map is 2Qp ^q" •' ^p{T) + inq{T) E. (9.4) 2flp Qq" — ip' (9.3)

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47 one-to-one. For example, for the harmonic oscillator, H — + Q^), Q' Q" (with n = 1) takes the form Q" — Q'e'^ so that dQ" /dQ' = which is never zero. Clearly in this case Q' — > Q" is one-to-one. Non-contributing Solutions In this subsection we will first discuss how non-contributing solutions arise. Following this we will explain Adachi’s method of drawing “Stokes lines” in the Q'-plane which separate regions of non-contributing solutions from those of contributing solutions. In order to satisfy the initial boundary condition it was remarked that we can set q' = q' + w and p' = p' + iflw where w is an arbitrary complex number. Therefore, excepting the case where w = 0 (which leads to the only real classical ray) q' and p' will be complex and so our extremal rays, {q{t),p{t)), will be complex and, in general, as a consequence so will F. This has important consequences for the SCSP. Note that since F is complex it can be written F = Fr + iFj, so the SCSP can be written Ksc = from this expression we can see the importance of F/ in determining the amplitude of the SCSP. It follows from the Schwarz inequality that for the quantum mechanical coherent-state propagator \K\ < 1, and as h decreases a valid semiclassical approximation Ksc should more nearly approach K and therefore the bound \Ksc\ < 1 should hold for h sufficiently small. Clearly if F/ < 0 the SCSP would diverge as fi decreased and the bound \Ksc\ < 1 would be violated. The amplitude factor E cannot prevent this divergence since it is independent of h [see (8.5)] and so remains constant as h is decreased. We can conclude from this that if there are in fact trajectories satisfying the CBC for which F/ < 0 they would have to be excluded from the evaluation of the SCSP.

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48 The glaring question which now confronts us is, are there trajectories (or equivalently values of Q') for which Fj < 0? For the three cases H = 0, H = p^/2 and H = + 9^) for which no PSC are present and there is only one solution satisfying a given set of CBC the answer is no. In fact, for the H = 0 (with Q arbitrary) and H = |(p^ + q^) (with n = 1) cases, Fj = \w\'^ and is independent of time. If there did in fact exist an H leading to Q' — > Q” being one-to-one for which Fj < 0 for any rays it would actually be a catastrophe. Note that Q' Q" being one-to-one tells us that if we wish to evaluate the SCSP at a particular exit label {q",p"), so that Q" = Q,q" — ip", there will be only one ray Q{t) with which we can do it. We also know that if F/ < 0 for a given ray Q{t) it must be excluded from the evaluation of the SCSP. Therefore, for such rays evaluation of the SCSP would be impossible at the unique exit labels corresponding to Q" , {q" ,p") = (Re(Q")/fi, — Im(Q")). It is also necessary to mention that when T = 0, independently of H, Kgc will reduce to the overlap of two coherent states (Eq. 5.5), as in the H = 0 case, so that Fj — — q')^ + — p')'^] = I'fi’P, which is always larger than or equal to zero. Again Q' — ) Q" is one-to-one and possesses no PSCs. Note that when T = 0, q' = q" and p' = p" so that Q' = flq' — ip' = Qq" — ip" = Q", so that Q' — ) Q" takes the simple form Q' = Q". This is obviously one-to-one and possesses no PSCs since dQ"/dQ' = 1. For T > 0, and for more complicated systems for which PSCs exist and consequently there is more than one solution for a given set of CBC {Q' Q" many-to-one), we find the answer to our question is yes. In fact, viewing F as a function of Q' we will find large regions of the Q'-plane over which F/ < 0 and thus the corresponding rays Q{t) cannot contribute to the evaluation of the SCSP. Figs, la and lb show Fj over the Q'-plane for the quartic oscillator (Eq. 1.8) with the parameter values n = 30 (30 time steps, each step being of time At = .001), ft = 1 and {q',p') = (0,0) for

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49 Im(F) Im(F) -10 Re(Q') Im{Q') -10 Figure 1. Fj over the Q'-plane. Parameter values are n=30 (T = nAt, At = .001), Q = 1 and {q',p') = (0,0). (a) Clipped above at Fj = 40. (b) Clipped above at Fj = 175 and below at Fj = 0. the fixed entrance label (Sec. 8). Fig. la is clipped above at F/ = 40 and the regions of F/ < 0 can be clearly seen. Fig. lb is clipped above at F/ = 175 and below at F/ = 0; the regions of F/ < 0 are clearly seen as the two flat regions in the lower left and upper right hand corners of the Q'-plane. In both figures the point Q' = (0, 0) corresponds to the real classical ray {w = 0), hence F is purely real along this ray and F/ = 0 at this point. We have seen that a relationship exists between the many-to-one property of Q' — > Q", the presence of PSCs and F/ < 0; it has in fact been argued above that F/ < 0 can only occur in the presense of PSCs. It was first observed by Adachi [4], for the case of the delta kicked rotator, that near each PSC there formed a region for which F/ < 0; the same phenomenon is observed for the quartic oscillator. This

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50 Figure 2. Parameter values are n = 30 (T = n At, At = .001), Q, = 1 and {q',p') = (0,0). The horizontal axis represents Re(Q') and the vertical axis Im(Q') in each figure, (a) Contour plot of Fig. lb. The two dots represent PSCs. Near each dot is formed a region of Fj < 0, shown here as darkened regions, (b) Superimposed contour maps of Re(Q") and Im((5"). Q' — )• Q" is conformal except at PSCs (two dots) therefore PSCs are seen as “defects” in the mesh pattern. can be seen from Figs. 2a and 2b. For both these figures, as in Figs, la and lb, n = 30 [T = nAt = 30(.001) = .03], ft = 1 and {q',p') = (0,0). Fig. 2a is the contour plot of Fig. lb and the two points drawn mark the two PSCs occurring at Q'psc — (-6.76,-6.57) and Q'psc — (6.76,6.57). Near each of these PSCs we find there is formed a region of F/ < 0, represented by the two darkened regions of Fig. 2a. At both of the PSCs shown F/ = 10.0 and at the point Q' — (0,0), corresponding to w ~ 0, Fi = 0. Fig. 2b shows the contour maps of both Re(Q”) and Im(<5”) superimposed over the same region of the Q'-plane as in Fig. 2a. Since Q' Q" is analytic everywhere and dQ" /dQ' 7^ 0 everywhere except at the PSCs (see end of previous subsection), our mapping Q' — )• Q" is conformal everywhere except at the PSCs. As a consequence of conformality the contour lines of Re(<5") and lm{Q") in

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51 Fig. 2b are seen to intersect one another at right angles. The two dots in Fig. 2b mark the position of the PSCs. We see clearly from Fig. 2b that the PSCs appear as “defects” in the mesh pattern of the contour lines. Stokes lines In order to find the exact boundary line between the contributing and noncontributing regions of the Q'-plane, Adachi introduced the construct of “Stokes lines.” We know that in the neighborhood of a PSC the mapping Q' Q" is manyto-one (beginning of present section); let us assume for the sake of illustration that it is two-to-one. Therefore for every Q' in the neighborhood of a PSC there will be a Q'p {p for pair) which maps to the same Q" . Adachi then instructs us to form the quantity AFj(Q') = Fj{Q') — Fi{Q'p) and consider its steepest ascent lines in the Q'-plane originating from the PSC. Note that at the PSC itself, AFj{Qpsc) = 0, since at this point Q' = Q'p. We then choose two of these steepest ascent lines which surround the smallest region of the Q'-plane containing the Fj < 0 region (see Fig. 3a). This will be our non-contributing region. In practice it is found that the region for which F/ < 0 is formed near the PSC (Fig. 2a), but at the PSC itself Fj is always found to be greater than zero. Therefore, contained in the non-contributing region, between the Stokes lines themselves and the Fj < 0 region, there will also be a region for which F/ > 0 (again see Fig. 3a). In order to illustrate this let us assume that Q' — ) Q" is given by Q" = Q'^. Then dQ" /dQ' = 2Q' so that there is a PSC at Q'psc = 0Let us further assume that Fj is such that the steepest ascent lines of the quantity AFj{Q') = Fj{Q') — Fj{Q'p) which emanate from the PSC are given as in Fig. 3a. The two steepest ascent lines A and B are then chosen to remove the F/ < 0 region as shown.

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52 Figure 3. A hypothetical illustration of the use of Stokes lines in the evaluation of the SCSP. The mapping Q' Q" is chosen to be Q” — which possesses a PSC at Q'psc — 0 (Sec. 9). The “action” F is further assumed to be such that the quantity AF/IQ') = Fi{Q ) — Fi{Q'p) (Sec. 9) has steepest ascent lines A,B and C. (a)AF/ has steepest ascent lines A,B and C. The lines A and B surround the Fj < 0 region and are therefore chosen as Stokes lines. Stokes lines in general (A and B in this example) define the boundary of the non-contributing region (which always includes the Fj < 0 region), (b) The Stokes lines A' and B' in the (5”-plane are images of the Stokes lines A and B in the Q^-plane. The number of contributing trajectories in Eq. (1.5) therefore changes from two to one when the exit label, Q", moves across A' or B' from the right-half (5"-plane to the left-half Q”-plane. The corresponding situation in the Q"-plane is shown in Fig. 3b. Recall that each point in the (5"-plane corresponds to an exit label, {q",p''), at which we evaluate the SCSP (where a fixed entrance label is understood), from Q” = Qq" — ip" = flq" — ip" or alternatively {q",p") = (Re(Q")/fl, -Im((5")) (Sec. 8); we will therefore use the term “exit label” to refer to either the pair {q",p") or a point Q" in the Q"-plane. Let us also recall that each point, Q', in the also represents a trajectory, (Qi't) ,p{t)) , along which the SCSP is evaluated (Sec. 8). The non-contributing region represented by region I in Fig. 3c would map to the left half Q"-plane (LHQ"P) under Q" = Q'^, however we are abandoning this region in our evaluation of the SCSP. This

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53 Figure 3. Continued, (c) Each exit label Q” in the Q^-plane has two pre-images, Q' and Qp, in the Q'-plane. As the exit label Q" is varied along the curve connecting Q'( and Q' 2 , it crosses the Stokes line B'. As this happens the pre-image Q' crosses the Stokes line B. When this occurs its corresponding trajectory suddenly becomes noncontributing causing the SCSP to discontinuously change from Ksc = Ee^^/^\qi -Ito Ksc = Ee^^'%,^. is not a disaster when it comes to the evaluation of the SCSP at these exit labels, since region HI which is contributing also maps to the LHQ"P. In other words, each point, Q" , in the LHQ"P will have a pre-image in both region I and HI, so that if we wish to evaluate the SCSP at one of these exit labels we will use its pre-image from region HI but discard the one from the non-contribution region (region I). Regions II and IV are both contributing and both map to the right half Q"-plane (RHQ"P), therefore points in the RHQ"P will correspond to two contributing rays. In other words, if we wish to evaluate the SCSP at an exit label in the RHQ"P we must use the rays corresponding to each Q' pre-image (unless either of their contributions is negligibly small). Therefore, for these exit labels (in the RH<5"P), Eq. (1.5) becomes a sum over two contributing rays, i.e., Ksc{p", q", T\p', q', 0) = .

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54 Fig. 3c shows an exit label Q” in the Q"-plane and its two pre-images Q[ and Q[p (p for pair) in the Q'-plane. The two pre-images Q[ and Q[p both originate in the contributing region (regions H, M, and IV in Fig. 3c) of the Q'-plane and should be used in the evaluation of the SCSP. However, if we continuously vary our exit label, Q", so that it moves from Q'( to Q 2 , crossing the Stokes line B' as shown in Fig. 3c, the two corresponding pre-images will continuously vary from Q[ and Q'lp to Q 2 and Q' 2 p, respectively. During this process, as our exit label crosses B', the pre-image Q[ will cross B (from region H to region I) and abruptly change from contributing to non-contributing. Now as this happens the number of rays used in the evaluation of the SCSP abruptly changes from two to one, i.e. the SCSP changes from Ksc = to Kgc = Clearly this leads to a discontinuity in the propagator along the lines A' and B' in the Q"-plane. Now it must be remarked here that the method used for obtaining the Stokes lines (using the steepest ascent lines of AFj) was constructed as a general method of minimizing this discontinuity. Since the steepest ascent lines are chosen so that AFj{Q') = Fj{Q') — Fj{Q'p) > 0 holds to as great an extent as possible, Fj{Q') > Fj{Q'p) also holds to as great an extent as possible. It follows that much as possible. Now returning to the example of Fig. 3a, when the Stokes line B' is crossed the term is dropped from KscBut since the magnitude of a term Ee^^!^ in Ksc is and if the amplitude factors of both terms in Ksc (before crossing B') are approximately equal, then the magnitude of will be smaller than that of Ee'’^l^\Qi^ by as much as possible. This situation (of minimum discontinuity) will hold in general when crossing a Stokes line, given the approximate equality of the magnitude of the amplitudes and especially so when Fi » h.

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NATURE OF THE TRAJECTORIES When written in the form Q = [l/(Q\/^)](f2g — ip), P = {l/\/^){Q,q + ip), the coordinates, Q and P, become canonical (see Sec. 8 and Ref. 19). We can therefore invert these to obtain q and p in terms of Q and P and so rewrite (1.8) as H = {P—QQ)‘^/{Ai) — {Q,Q-\-P)^/(2Q,)'^. Now using Q = dH /dP we can approximate Q" to first order in time as Q" ~ Q'+Q\t=oT = Q' + [{P' — Q.Q') / {2i) — {Q.Q' + P')^ / {40,'^)] T. This expression can now be used to obtain a rough approximation of the position of the PSCs by forming the quantity (Sec. 9) dQ"/dQ' = 1 — [Vt/{2i) + Z{VtQ' + P')^/(4fl^)] T = 0 and solving for Q' to obtain Q' = —P' /^±{2^JQ./'2>)[l|T+iU/2Yl'^^, we say that these two PSCs are 1st order in T. We know that at T = 0 no PSCs exist (Sec. 9); therefore this expression suggests that two PSCs are formed at infinity in the Q'-plane and quickly move in towards the fixed point —P'/Q — —{^q' + ip') /ft. This situation is what is observed numerically, and can be seen from Figs. 4a, 2b and 4b. Figs. 4a and 4b are both contour plots of Re((5”) and Im((5”) superimposed over a region of the as was Fig. 2b; each has the parameter values = 1 and {q'^p') = (0,0). Fig. 4a corresponds to n = 10 (T = nAt, At — .001), the two dots representing the PSCs are located at Qp5(^=(11.6,11.5) and (-11.6,-11.5). In Fig. 2b (n = 30) the PSCs have moved in towards —P'/Q — —{filq',p')/Vt = (0,0) to the points Q'pgQ={6.8,6.6) and (-6. 8,-6. 6). In Fig. 4b, for which n = 100, the PSCs have continued to move in and are now located at Q'pg(^={3.8,3.5) and (-3. 8,-3. 5). As time continues to increase additional PSCs are seen to move in from infinity; in Fig. 4c, n = 250, and four additional PSCs (seen as “defects” in the mesh pattern) have 55

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56 Figure 4. Superimposed contour maps of Re{Q") and Im(Q"). The parameter values are = 1, {q',p') = (0,0) and At = .001 with T = nAt. For parts a and b, as time T is increased the two PSCs move in from infinity towards the point —P' = = (0,0). The horizontal axis represents Re(Q') and the vertical axis Im(Q') in each figure, (a) n = 10 and two PSCs (two dots) are seen at Qp5c.=(11.6,11.5) and (-11.6,-11.5). (b) n = 100. The two PSCs have moved to Qp5<^=(3.8,3.5) and (-3. 8,-3. 5). (c) n = 250. Four additional PSCs (seen as “defects” in the mesh pattern) have appeared. The white flower-like regions the on top and bottom of this figure result from Re((5") and Im((5") being sharply peaked in these regions. appeared. The white flower-like regions on the top and bottom of Fig. 4c result from Re(Q") and Im(Q") being sharply peaked in these regions and should be ignored.

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57 Continuously Connected and Disconnected Trajectories At time T = 0 the mapping Q' — > Q" takes the simple form Q” = Q' (Sec. 9) and is clearly one-to-one. Furthermore, for a general dynamical system with a fixed initial value of “momentum” and a fixed final value of “position”, it is generally possible to find an initial value of position for smaller and smaller time intervals T, which gets closer and closer to the final value of position. We have also seen that PSCs, which are the source of multiple solutions (Sec. 9) for a fixed set of CBC (Sec. 8) are formed at infinity at T = O"*". We may therefore conjecture that at T = O"*" only one of the solutions satisfying a fixed set of CBC will be such that Q' is arbitrarily close to Q" and for each of the others the initial position Q' will appear at infinity in the Q'-plane. In order to view this conjecture in another way let us consider the trajectories as the functions Q{t) [recalling that each P(0) is always fixed at P' = flq' + ip' as was discussed in Sec. 8]. First let us suppose we have two trajectories, Q{t), such that both reach the same final point Q{T) = Q" in some sufficiently small but finite time interval T; thus both rays satisfy the same set of CBC, P{0) = P' = Qq' -|ip' and Q{T) = Q" = itq" — ip". Given this, our conjecture will demand that if we decrease our time interval T toward zero, while keeping the final point of both rays fixed at Q" (CBC fixed while T decreases), one ray will continuously shrink in length as its initial point Q' converges to Q", while the other ray will continuously increase in length as its initial point Q' tends to infinity. Re-expressing the conjecture in this form also allows us to see that both trajectories cannot continuously shrink in length so that both their initial points converge to the point Q" . First recall that the mapping Q' — ) Q" is analytic (Sec. 9) and an analytic map f{z) is always one-to-one in a sufficiently small neighborhood of a point zq if /'{zq) ^ 0. With this in mind let us view Q' Q" as a mapping from the Q'-plane to the Q”-plane. Now if both trajectories, Q{t), discussed

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58 above were to continuously shrink in length their initial points, Q', would converge to the fixed point Q' = Q" in the Q'-plane making the mapping Q' Q" two-to-one in the neighborhood of Q' = Q" . But this could only happen if dQ" jdQ' {P' fixed) equaled zero at the point Q' = Q", but this would make the point Q' = Q" a PSC and we know that the PSCs don’t stay fixed but tend towards infinity in the Q'-plane as T is decreased. Hence both rays cannot continuously shrink to zero in length as T shrinks to zero. This conjecture has in fact been shown to hold true numerically; we will therefore call the ray which shrinks to zero in length “continuously connected” (CC), since it is CC with the ray Q' = Q" at T = 0, and the one whose initial point Q' tends to infinity “disconnected” (DC). Figs. 5a,b,c and d display this phenomenon for T ranging from T = .06 down to T == .0001 with the fixed value of Q" = (—12.6,24.6). In these figures the horizontal axis represents Re(Q') and the vertical axis vertical represents Im((5'); the DC ray is portrayed as the dotted curve and the CC ray as the solid curve. In Fig. 5a, T = .06, and the length of the disconnected ray (Ldc) is 30.5 and the length of the connected ray (Lee) is 20.9 . In Figs. 5b, T = .004, Ldc = 36.4 and Lee = 13.1; in Fig. 5c, T = .0004, Lee = 57.6 and Lee = 4.9; finally, in Fig. 5d, T = .0001 and the DC ray has increased in length to Loe = 98.6 while the CC one has shrunk down to Lee — 1-8. We therefore see that there will always be one and only one CC ray for any given exit label Q” . We may then in some sense think of the DC trajectories as “extra” trajectories. It is also worth mentioning here a simple algebraic analog to this continuously connected, disconnected ray situation. Consider the roots of the quadratic equation y — ax^ ^hx-\-c which are given by x± = {b/2a) (— 1 i Y^l — Aac/b^ j. Letting a play the role of T, we find that as a ^ 0, the root x+ approaches the fixed value x^ = ~c/b, while the root ~ {—b/a){l — aefb^) extinguishes itself by tending to infinity. Thus as a -> 0, X-^. plays the role of the CC

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59 -75 -50 -25 0 25 50 75 75 50 25 0 -25 -50 -75 -75 -50 -25 0 25 50 75 Q' -plane V" 75 50 25 0 -25 -50 -75 -75 -50 -25 0 25 50 75 -75 -50 -25 0 25 50 75 Q' -plane Figure 5. A “continuously connected” ray (solid curve! and a “disconnected” ray (dotted curve) both reaching the same exit label Q" = (—12.6,24.6) in various time intervals, T. As T is decreased the initial point Q' of the DC ray tends to infinity as the initial point of the DC ray tends to Q” . The horizontal axis represents Re(Q') and the vertical axis Im((5') in each figure, (a) T = .06 . (b) T = .004 . (c) T = .0004 . (d) T = .0001 . trajectory and the role of the DC trajectory. It was mentioned in the Introduction that in order to evaluate the SCSP, Ksc, for a fixed set of CBC it was originally expected [3] that only one particular trajectory was needed. This is the CC ray just discussed. As was mentioned above, for a given time interval T, each exit label Q" will have one and only one CC ray, Q{i)^ such

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60 that Q{T) = Q” . In the evaluation of the SCSP at a fixed exit label Q” = Q,q" — ip" (for some fixed entrance label P' = Qq' + ip') it is found that the CC ray is always contributing (see Secs. 1 and 9). It is also found that this contribution is generally the dominant one. The DC rays represent quantum mechanical interference and are found to play an important role in the region of PSCs, the PSCs themselves signaling interference. Continuously Connected and Disconnected Regions As was discussed above for the case of the quartic oscillator (1.8), PSCs are formed at infinity in the Q'-plane when T = 0+ and quickly move in towards the point Q' = — P'; these PSCs in turn give rise to DC rays whose initial points, Q', (for fixed Q") are also formed at infinity, and for sufficiently small time, T, move in towards Q". In addition to this it was discussed in Sec. 9, for the case of the quartic oscillator (and delta kicked rotator) that near each PSC there formed a region in the Q'-plane for which Fj < 0. This is highly suggestive of a relationship between the DC rays and the Fj < 0 regions of the Q'-plane. It is actually found that every trajectory, Q{t), whose initial point Q' lies in a P/ < 0 region is a DC one. It is further found, after a short time T (T=.03, for example, with parameter values q' = p' = 0 and fl — .5), that the DC rays themselves form a region with well defined boundaries surrounding the Fj < 0 region with which they are associated. This boundary which will be called the “connected-disconnected line” (CDL) must pass through the PSC associated with the P/ < 0 region which it encompasses. To see this assume for a fixed time T, that a ray, Q{t), whose initial point is a PSC, Q'psci goes to the final point Q'pscNow let us assume that in the neighborhood of the PSC the mapping Q' — > Q" is two-to-one; this assumption is unnecessary but simplifies the argument and is the actual case for the quartic oscillator. Now an exit label Q" arbitrarily close

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61 to Qpsc have two pre-images (since Q' — ) Q" is two-to-one near Q'psc) which by continuity will be arbitrarily close to Q'pscNow since there is only one CC ray one of the pre-images must correspond to a CC ray, {Q'cc)^ the other to a DC ray, [Q'dc)Since at Q'psc itself the mapping is necessarily one-to-one (Sec. 9), as Q" approaches Q'psci Q'cc Q'^q must converge to Q'pscNow the CDL separates regions in the Q'-plane for which the corresponding trajectories are either CC or DC, therefore the CDL must pass between Q'^(^ and Q'p,c and will be forced to pass through Q'psc as Q" — > Q'pscWe should also remark here that since an arbitrary exit label Q" (for fixed time T) will have a unique CC ray associated with it, the CC region of the Q'-plane will map to the (5"-plane in a one-to-one manner. It was discussed in Sec. 9 that the Stokes lines (which separate contributing and non-contributing regions of the Q'plane) themselves emanate from each PSC. The Stokes lines then must necessarily fall within the disconnected region, that is between the CDL and the F/ < 0 region. If this were not the case the Stokes lines would fall within the CC region and we would be throwing away each entire DC region as well as parts of the CC region, thus only CC contributions to the SCSP would remain. And as was remarked above, the CC rays map in a one-to-one manner onto the (5”-plane, so that the exit labels Q" corresponding to the CC rays thrown away would receive no contribution at all. This general situation is depicted in Fig. 6. From this figure we can see that only DC rays whose initial points lie in the regions between the CDL and the Stokes lines will play a role in the evaluation of the SCSP.

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62 Figure 6. The connected-disconnected lines separate the DC regions of the Q'-plane from the CC region. The Stokes lines must always fall within the DC regions.

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NUMERICAL RESULTS It was seen in the previous section that as a result of the formation of DC regions of the Q'-plane the Stokes lines must fall within these regions. As a result of this only the regions of the Q'-plane between the CDLs and the Stokes lines (see Fig. 6) can give rise to contributing DC rays. It was also seen in the previous section that the CC rays map in a one-to-one manner onto the (5”-plane. Therefore since the entire CC region is contributing and only the cusps of the DC regions are contributing each exit label Q" will have a CC contribution, but only some will have DC contributions. In comparing Ksc with the actual quantum mechanical coherent state propagator (1.1) it is of interest to study the relative contributions of both the DC and CC rays to Ksc [Ksc = Kdc + Kcc according to (1.5)] as a Stokes line is approached and passed through. In order to do this we may evaluate the SCSP along a DC trajectory, Q{t), whose initial point, lies just within the DC region near the CDL. We may then evaluate the SCSP along the CC trajectory whose exit label Q” is the same as that of the DC ray. We may continue this process choosing values of Q'^/c which lie further and further into the DC region and so closer to the F/ < 0 region. When a Stokes line is reached the DC ray becomes non-contributing; we should expect this to be signaled by the CC contribution alone becoming a good approximation to the full propagator F, i.e. K ~ Ksc = Kcc at and past a Stokes line. We may also note that since the Stokes lines are steepest ascent lines of the quantity AFi{Q') = Fi(Q') — Fi{Q'p) (Sec. 9) and occur within DC regions of the Q'-plane, that on the Stokes lines themselves AFj > 0, Q' = Q'j^^ and Q'^ — Q'^q so that 63

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64 F i{Q'dc) > Fi{Q'cq). In other words at a point Q" on the image of a Stokes line in the Q"-plane ^Idc will be larger than PiccIt should also be mentioned that if only the CC rays were contributing (see Secs. 1 and 10), as was originally expected [3], the Stokes lines would coincide with the CDLs so that K ~ Ksc — Kcc at the first Q'dc poiiif chosen in the above procedure. In studying the behavior of Ksc in approaching a Stokes line the relevant data will be arranged in two tables. Each row of the first table will give the initial points of the DC and CC rays, Q'^q and Q'qq (reaching a common exit label Q"), the corresponding values of E and F evaluated along these rays, and the exit label Q" itself. Each corresponding row of the second table gives the values K£)c and Kcc corresponding to these rays, their sum Ksc, and the value of the full quantum mechanical propagator K corresponding to the particular exit label Q” as found by numerical solution of the Schrodinger equation. The modulus oi Kjjc, Kcc, Ksc and K is given directly below each value. The quantity Fee = \Kcc\/\Ksc\ is also given; when Q" moves past a Stokes line a value of Fee — 1 is expected. Before studying the approach to a Stokes line let us first look at the value of Ksc near its peak modulus (recall \K\ < 1, Sec. 9). This may be done by evaluating Ksc at the exit label Q” corresponding to the unique real classical ray, the initial point of this ray always being Q' = ilq'—ip' {w = 0, Sec. 8). Let us choose parameter values ^ = .1, Q = .5, q' = 1, p' = 0 and n = 300 (where T = n At with At = .001). This choice gives Q' = (.5,0), which lies in the CC region of the Q'-plane, and its corresponding exit label is Q” = Q{T) — (.48, .29). Along this path Ecc = (.82,— .35), Fee — (7.07x10“^, 0), Kcc = (-85, .27), and \Kcc\ = -89 . This compares well with the actual value of the propagator K = (.84, .28) with \K\ = .89. With these parameter values we find PSCs formed at the points Q'psc = (.30, .77) and Q'psc — (—1-29, —.62). Each of these PSCs will have a DC region associated with it, and each of these DC

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65 regions will have a DC ray with the desired exit label Q" = (.48, .29). The DC ray originating in the region associated with Q'psc — (-30, .77) has Q'^c — (-23,1.43); evaluating the SCSP along this ray gives Epc = (.40, .65), Fpc = (.56, —.11), and Kdc = (2.23, .84) so that \Kdc\ = 2.38. Note that < 0 along this ray so that it is clearly non-contributing (Sec. 9). The DC ray associated with Q'psc = (—1.29, —.62) is found to have Q'pc — (~2.19, —1.35); the SCSP evaluated along this ray gives Fpc = (.09,— .55), Fpc = (—2.69,1.65), and Kpc = (— 3.79, .09)xl0~® so that \Kp)c\ = 3.79x10“®. Therefore even if this value of Q'pc falls within the contributing part of the DC region associated with this PSC, its contribution, Kpc, to Ksc, is negligible. For this set of parameter values it is found that at exit labels where K is large {K > .1) the DC rays are either non-contributing or make a negligible contribution to Kgc and Ksc — Kcc provides a good approximation to K. In Table la the initial points Q'^q were chosen within the DC region corresponding to Q'psc = (-30, .77). The points were chosen moving up along the line Re(Q'£)c) = 1.15 starting at the point Q'pc — (113, .4) which lies about .2 units above the CDL. For exit labels which correspond to points Q'pc along this line starting at the CDL [about Q'={1.15,.2)] and moving up to the second point in Table la, Q'pc = (1-13, -5), this DC ray alone provides a good approximation to K, the value of Kcc being either negligible or small for these exit labels. For each exit label in Table la the additional DC ray originating in the DC region associated with Q'psc — (~l-29, —.62) is found to be non-contributing (since for each < 0). For example, the DC ray originating in the DC region associated with Q'psc — (~l-29, —.62) whose exit label is Q" = (.17, 1.31) has initial point Q'pc = (—2.05, —1.62) and along this ray Fpc = (.01,— .50), Fdc — (—4.08, —.63) so that K^c — ( — .24, 2.75)xl0^ and \Kdc\ = 276. Clearly this ray is non-contributing since < 0. As we proceed upward, we see from Table Ib, that the CC ray quickly becomes the dominant contributor so that at

PAGE 70

66 Table la Q'dc Edc Fdc Q'ce Fee Fee Q" (1.15,0.4) (0.49,-0.55) (1.13,0.42) (-0.40,1.34) (0.64,0.31) (0.38,1.12) (0.17,1.31) (1.15,0.5) (0.46,-0.58) (1.25,0.64) (-0.45,1.26) (0.68,0.28) (0.39,1.03) (-0.01,1.28) (1.15,0.6) (0.43,-0.61) (1.31,0.90) (-0.50,1.17) (0.72,0.25) (0.42,0.97) (-0.18,1.23) (1.15,0.7) (0.39,-0.63) (1.30,1.19) (-0.56,1.08) (0.75,0.20) (0.44,0.95) (-0.35,1.16) (1.15,0.8) (0.34,-0.65) (1.20,1.50) (-0.62,0.99) (0.78,0.16) (0.46,0.96) (-0.51,1.06) (1.15,0.9) (0.29,-0.65) (1.00,1.82) (-0.69,0.88) (0.80,0.10) (0.46,1.00) (-0.66,0.95) Table Ib K Kne Fee Kse Fee (-5.77,-9.84)xl0-^ 1.14x10-2 (-5.57,-9.85)x10-3 1.13x10-2 (-4.56,-8.92)xl0-3 .001x10-2 (-5.57,-9.86)x10-3 1.13x10-2 .001 (0.74,-1.06)xl0-^ 1.30x10-3 (0.77,-1.03)xl0-3 1.28x10-3 (-0.96,-2.33)x10-3 0.03x10-3 (0.76,-1.05)xl0-3 1.30x10-3 .023 (7.67,-8.19)xl0-^ 1.12x10-'^ (8.67,-3.74)xl0-3 0.94x10-^ (-0. 98,-4. 35)x 10-3 0.45x10-'^ (7.69,-8.09)x10-3 1.12xl0-'‘ .402 (0.18,-5.96)x10-3 5.96x10-3 (4.10,-2.97)x10-3 .51x10-3 (-0.30,-5.68)xl0-3 5.69x10-3 (0.11,-5.98)x10-3 5.98x10-3 .952 (0.27,-5.38)xl0-3 5.38x10-3 (-0.27,-2.20)xl0-^ 0.02x10-3 (0.21,-5.36)x10-3 5.36x10-3 (0.21,-5.38)x10-3 5.38x10-3 .996 (-0.04,-3.84)xl0-3 3.84x10-3 (-7.53,4.75)xl0-^ .0009x10-3 (-0.07,-3.82)xl0-3 3.82x10-3 (-0.07,-3.82)x10-3 3.82x10-3 1 Q" = (—.66, .95) the value of Tec is very close to unity indicating that we are near a Stokes line. From this point and upwards, the CC ray alone will provide a good approximation to K. We should also note that at Q" = (—.66, .95), is larger than as should be the case near a Stokes line as was discussed earlier in this section. As points above Q'^c = (115, .9) are chosen ^Idc will continue to increase, reach a peak, decrease, and then cross into the Fj < 0 region at Q'j^q = (1.15,2.17). We will now look at the SCSP for the parameter values h = .1, 0, = .5, q' = 1, p' = .5 and n = 400 (where T = nAt and At = .001). As before let us first look at the propagator near its peak value by evaluating it at the exit label Q"

PAGE 71

67 corresponding to the unique real classical ray. For these parameter values this ray will have initial point Q' = Qq' — ip' = .5 — i.5, which again lies in the CC region, and corresponding exit label Q" = (.55, —.01). Along this ray Ecc = (-69,— .41), Fee = (13,0), and Kec = (-57, .57) with modulus \Kec\ = -81 Again this compares well with the value of the quantum mechanical propagator evaluated at this exit label, K = (.57, .59) with modulus \K\ = .82 . With these parameter values PSCs are found at the points Q'pse — ( Hi -19) Q'psc — (~l-30, —.98). The DC ray originating in the DC region associated with Q'pse — (•H,-19) is found to have initial point Q'dc — (~-10i -82), and along this ray Ep,c = (-45, .52), Fee = (-65, .30), and Kpc = (.015, .030) with modulus \Kec\ = -03 ; it can be inferred that this ray originates in the non-contributing part of the DC region since the addition of this value of Kpc to Kec = (-57, .57) worsens the approximation. The additional DC ray originating in the DC region associated with Q'pse = (-1.30, —.98) is found to have initial point Q'dc = (-2.11,-1.73), and along this ray Epc — (-09, —.52), Fee = (—2.77,1.49) and Kec = (“1-28, 1.26)xl0“^ with modulus \Kec\ = 1.80x10“’^. This value of Kec is negligible in comparison to Kec so in this case we can infer that Q'ec is either close to or past the Stokes line (coming from a contributing region of the Q'-plane). For this set of parameter values it is found that at exit labels where K > .2 the DC rays are either non-contributing or make a negligible contribution to Ksc so that Kec alone provides a good approximation to K. In Table la the initial points Q'ec were chosen within the DC region corresponding to Q'psc = (-Hi -19)The points were chosen moving up along the line Re{Q'ec) — 1 starting at the point Q'ec = (li “-3) which lies about .2 units above the CDL. For exit labels which correspond to points Q'ec along this line starting at the CDL [about (5'=(l,-.5)] and moving up to the third point in Table Ha, Q'p,e = (1, — -1), this DC ray alone provides a good approximation to AT, the value of Kec being either neg-

PAGE 72

68 Table Ea Q'dc Edc Fdc Q'cc Fee Fee Q" (l,-0.3) (0.48,-0.48) (0.94,0.17) (-0.50,0.86) (0.53,0.27) (0.25,1.44) (0.39,0.95) (1.-0.2) (0.46,-0.51) (1.09,0.35) (-0.55,0.80) (0.57,0.25) (0.18,1.28) (0.17,0.94) (l.-O.l) (0.44,-0.53) (1.19,0.60) (-0.60,0.73) (0.60,0.23) (0.13,1.18) (-.04,0.91) (1,0.0) (0.41,-0.56) (1.22,0.93) (-0.66,0.66) (0.64,0.20) (0.09,1.14) (-0.24,0.85) (1.0.1) (0.38,-0.58) (1.18,1.31) (-0.72,0.59) (0.67,0.16) (0.06,1.15) (-0.43,0.78) (1,0.2) (0.34,-0.59) (1.05,1.75) (-0.79,0.51) (0.70,0.11) (0.01,1.21) (-0.63,0.69) (1,0.3) (0.30,-0.59) (0.82,2.23) (-0.86,0.43) (0.72,0.06) (-0.05,1.32) (-0.81,0.57) Table Eb K Kdc Kec Ksc Fee (-9.19,8.92)x10-2 0.128 (-9.20,8.77)xl0-2 0.127 (-3.43,0.45)xl0-'^ .0000003 (-9.20.8.77)x10-2 .127 0 (-1. 74,-1. 31)xl0-=^ 2.18x10-2 (-1.72,-1.32)x10-2 2.16x10-2 (-1.04,1.38)x10-3 .0002x10-2 (-1.72,-1.32)x10-2 2.16x10-2 0 (0.00,-1.69)x10-3 1.69x10-3 (0.02,-1.69)x10-3 1.69x10-3 (-0.43,4.76)x10-3 .005x10-3 (0.02,-1.69)x10-3 1.69x10-3 .003 (2.19,-5.56)x10-3 5.98x10-3 (2.02,-6.26)x10-3 6.57x10-3 (2.59,7.11)x10-3 .76x10-3 (2.28,-5.55)x10-3 6.00x10-3 .127 (4.64,3. 72)xl0-® 5.95x10-3 (-0.20,-1.38)xl0-3 1.40x10-3 (4.93,5.08)x10-3 7.08x10-3 (4.73,3.70)x10-3 6.01x10-3 1.178 (3.78,1. 20)xl0-« 3.97x10-3 (-1.76,-0.13)x10-3 .02x10-3 (3.82,1. 17)x 10-3 4.00x10-3 (3.80,1. 17)xl0-3 3.98x10-3 1.005 (1. 29,-0. 51)xl0-3 1.38x10-3 (1.00,1.04)xl0-^3 .0001x10-3 (1.28,-0.52)x10-3 1.38x10-3 (1.28,-0.52)xl0-3 1.38x10-3 1 ligible or small for these exit labels. For each exit label in Table Ea the additional DC ray originating in the DC region associated with Q'pgc = (—1.30, —.98) is found to have < 0 and is therefore clearly non-contributing. For example, the additional DC ray originating in this region associated with the exit label Q” = (.39, .95) has initial point Q'j^c = (—2.01, —1.94), and along this ray Edc = ( 03, —47), Fdc = (—4.37, —.20) so that Kdc = (1.07,-3.33) and |F£)c| = 3.50. As we proceed upward, we see from Table Eb, that the CC ray again quickly becomes the

PAGE 73

69 dominant contribution so that at Q" — (—.63, .69) the value of Tec is very close to unity indicating that we are near a Stokes line. From this point and upwards, the CC ray alone will provide a good approximation to K. Again let us note that at Q" — (—.63, .69), F/pj, is larger than as should be the case near a Stokes line as was discussed earlier in this section. As points above Q'qq — (1, .3) are chosen, F/^^ will continue to increase till it reaches a peak, then decrease, entering the F/ < 0 region at = (1,4.75).

PAGE 74

SUMMARY AND CONCLUSION It has been found that the complex classical trajectories Q{t) = Q,q{t) — ip{t), P{t) = riq{t) + ip{t) satisfying the complex boundary conditions (CBC) (1-4), along which the semiclassical coherent-state propagator (SCSP) (1.5) is evaluated, are of two types: continuously connected (CC) or disconnected (DC). For a fixed set of CBC there will always be one unique CC ray Q{t) which is always contributing in the evaluation of Eq. (1.5) and for each phase space caustic (PSC) formed in the Q'-plane there will be an additional DC ray satisfying the set of CBC which may or may not be contributing. These two types of rays themselves are found to lie in associated regions in the Q'-plane whose boundary line [referred to as a connected-disconnected line (CDL)] will pass through a PSC; therefore for each PSC there will be a DC region of the Q'-plane. Along with each PSC there is also formed a non-contributing region (separated from contributing regions by Stokes lines) of the Q'-plane containing an Fj < 0 region [4]. Each of these non-contributing regions has been found to fall within a DC region. Therefore the only exit labels Q" = Clq" — ip" at which DC rays contribute in the evaluation of (1.5) will be those corresponding to the initial points Q'dc found between the CDL and the Stokes lines (see Fig. 6). The comparative roles of the continuously connected and disconnected trajectories in the correct evaluation of the semiclassical coherent-state propagator can be understood through a comparison of the original [3] and modified [4] form of the theory. In the original form of the theory of the semiclassical coherent-state propagator presented by Klauder [3] only the continuously connected rays were thought to con70

PAGE 75

71 tribute so that Ksc was given by Eq. (1.2). Our work has shown that the continuously connected rays always contribute but at certain exit labels the evaluation of the semiclassical coherent-state propagator must be supplemented by a disconnected ray. In AdachiÂ’s treatment of the delta kicked rotator [4] it was necessary to replace Eq. (1.2) by Eq. (1.5) and introduce the use of Stokes lines; however, in AdachiÂ’s treatment no distinction was made between continuously connected and disconnected rays. The results of this paper support AdachiÂ’s modified form of the theory although it has been found that the non-contributing regions of the Q'-plane fall within disconnected regions so that all non-contributing rays are disconnected. Related work can also be found in Ref. 20.

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REFERENCES 1. A representative sample of references is the following; I. M. GelÂ’fand and A. M. Yaglom, J. Math. Phys. 1, 48 (1960); D. G. Babbitt, J. Math. Phys. 4, 36 (1963); E. Nelson, J. Math. Phys. 5, 332 (1964); K. Ito, Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability (California U. P., Berkeley, 1967), Vol. 2, Part 1, pp. 145-161; J. Tarski, Ann. Inst. H. Poincare 17, 313 (1972); K. Gawedzki, Rep. Math. Phys. 6, 327 (1974); A. Truman, J. Math. Phys. 17, 1852 (1976); S. A. Albeverio and R. J. Hoegh-Krohn, Mathematical Theory of Feynman Path Integrals (Springer, Berlin, 1976); V. P. Maslov and A. M. Chebotarev, Theor. Math. Phys. 28, 793 (1976); C. DeWitt-Morette, A. Maheshwari, and B. Nelson, Phys. Rep. 50, 255 (1979); D. Fujiwara, Duke Math. J. 47, 559 (1980); P. Combe, R. Hoegh-Krohn, R. Rodriguez, and M. Sirugue, Commun. Math. Phys. 77, 269 (1980); F. A. Berezin, Sov. Phys. Usp. 23, 763 (1980); T. Ichinose, Proc. Jpn. Acad. Ser. A 58, 290 (1982); I. Daubechies and J. R. Klauder, J. Math. Phys. 23, 1806 (1982); R. H. Cameron and D. A. Storvick, Mem. Amer. Math. Soc. 46, No. 288 (1983); T. Hida and L. Streit, Stoch. Proc. Appl. 16, 55 (1983); N. K. Pak and I. Sokman, Phys. Rev. A 30, 1629 (1984); F. Steiner, Phys. Lett. 106 A, 363 (1984); C. DeWitt-Morette, Acta Phys. Austr. Suppl. 26, 101 (1984); A. Young and C. DeWitt-Morette, Ann. Phys. 169 140 (1986); M. Bohm and G. Junker, J. Math. Phys. 28, 1978 (1987); D. Castrigiano and F. Staerk, J. Math. Phys. 30, 2785 (1989); M. de Faria, J. Potthoff, and L. Streit, J. Math. Phys. 32, 2123 (1991); W. Fischer, H. Leschke, and P. Muller, J. Phys. A: Math. Gen. 25, 3835 (1992); L. S. Schulman, J. Math. Phys. 36, 2546 (1995); C. Grosche and F. Steiner, J. Math. Phys. 36, 2354 (1995). 2. J. R. Klauder and I. Daubechies, Phys. Rev. Letters 52, 1161-1164 (1984); I. Daubechies and J. R. Klauder, J. Math. Phys. 26, 2239-2256 (1985). 3. J. R. Klauder, in Random Media, Ed. G. Papanicolaou, Vol. 7, IMA Series in Mathematics and its Applications, SpringerVerlag, New York, 1987, pp. 163182. 4. S. Adachi, Ann. Phys. (NY) 195, 45-93 (1989). 5. See, e.g., L. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981). 6. J. H. Van Vleck, Proc. Nat. Acad. U. S. Sci. 14, 178 (1928). 7. P. A. M. Dirac, Proc. Roy. Soc. London, 113A, 621 (1927). 8. P. Jordan, Zeits. Physik, 40, 809 and 44, 1 (1927). 72

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73 9. N. Dunford and J. Schwartz, Linear Operators, Vol. 1, (Interscience Publishers, New York, 1966); a particularly readable elementary account is given in, I. Stakgold, GreenÂ’s Functions and Boundary Value Problems (Wiley, New York, 1979). 10. See, e.g., T. Hida, Brownian Motion (SpringerVerlag, New York, 1980), or T. Hida and M. Hitsuda, Gaussian Processes (American Mathematical Society, Providence, RI, 1993). 11. See, e.g., A. V. Skorokhod, Studies in the Theory of Random Processes (AddisonWesley, Reading, MA, 1965), pg. 3. 12. K. Ito, Applied Mathematics and Optimizatiom 1 , 347-381 (1975). 13. See, e.g., Goherent States, Eds. J. R. Klauder and B.-S. Skagerstam, World Scientific, Singapore, 1985, pp. 169-184. 14. R. H. Cameron, J. Anal. Math. 10, 287 (1962/63). 15. L. Schwartz, Theorie des Distributions (Hermann, Paris, 1966) or, I. M. GelÂ’fand and N. Ya. Vilenkin, Generalized Functions, Vol. 1, (Academic Press, New York and London, 1964). 16. J. R. Klauder, J. Math. Phys. 4, 1058-1073 (1963); reprinted in Ref. 13, pp. 169-184. 17. See, e. g., R. S. Brodkey, The Phenomena of Fluid Motions (AddisonWesley, Reading, MA, 1967), Chap. 9. 18. J. R. Klauder, in Path Integrals, Eds. G. J. Papadopoulos and J. T. Devreese, Plenum, New York, 1978, pp. 5-38; Phys. Rev. D 19, 2349-2356 (1979). 19. Y. Weissman, J. Chem. Phys. 76, 4067-4079 (1982); J. Phys. A 16, 2593-2701 (1983). 20. M. Kus, F. Haake, and Bruno Eckhardt, Z. Phys. B 92, 221-233 (1993); M. Kus, F. Haake, and D. Delande, Phys. Rev. Letters 71, 2167-2171 (1993); R. Scharf and B. Sundaram, Phys. Rev. E 49, 4767-4770 (1994).

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BIOGRAPHICAL SKETCH The author is originally from New York City and obtained his B.S. in physics from the Polytechnic University of New York. 74

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality as a dissertation for the degree of Doctor of Philosophy. ' A John^. Klauder, Chair ProSP^sor of Physics lirman I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of f^jpsophyv— v Henrft'Van Rinsvelt Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor c^^^hiJ^opl^ J. Robert Buchler Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. t C. L Khandker A Muttalib Associate Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of PMosophy. Paul Robinson Professor of Mathematics This dissertation was submitted to the Graduate Faculty of the Department of Physics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December 1998 Dean, Graduate School