Singular dynamics in quantum mechanics and quantum field theory

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Singular dynamics in quantum mechanics and quantum field theory
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Thesis (Ph. D.)--University of Florida, 1994.
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Includes bibliographical references (leaves 101-102).
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by Chengjun Zhu.
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SINGULAR DYNAMICS
IN
QUANTUM MECHANICS AND QUANTUM FIELD THEORY




By


CHENGJUN ZHU


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


UNIVERSITY OF FLORIDA


1994
























Dedicated to

My Parents and Friends




















ACKNOWLEDGMENTS


I wish to thank my supervisor, Prof. J. R. Klauder, for his efforts, patience

and encouragement throughout the time I have been at the University of Florida.
He introduced me to most of the topics involved in this dissertation and always

was ready for a discussion if needed. Over the years, his enthusiasm and
encouragement have been constant sources of inspiration to push the project
forward.
I am grateful to the other members in my supervisory committee, Prof.
Pierre Ramond for his valuable comments and suggestions, Prof. Charles

Hooper, Prof. Robert Buchler, Prof. David Tanner and Prof. Christopher Stark,
for patiently sitting through my oral exam and thesis defense. Special thanks

should go to Prof. Buchler for his kindness, concern and heuristic suggestions

when I was very depressed in job-hunting.
Also I want to express my deep thanks to my fellow graduate students
and colleagues, for their help and encouragement throughout the years.









TABLE OF CONTENTS


pages

ACKN OW LEDGM ENTS ......................................................................................... iii

A BSTR A CT ................................................................................................................ vi

PART I CLASSICAL SYMPTOMS OF QUANTUM ILLNESSES

CHAPTERS

1 INTRO DUCTION .................................................................................. 2

1.1 Basic Concepts of Operator Theory ...................................... 3
1.2 Symmetric Operators and Extensions ................................ 4
1.3 The Construction of Self-Adjoint Extensions .................... 8

2 CLASSICAL SYMPTOMS OF QUANTUM ILLNESSES ............. 10

2.1 Principal Assertions ............................................................. 10
2.2 Classical Symptoms of Quantum Illnesses .......................... 12
2.3 More General Examples ...................................................... 16
2.3.1 Examples of One-Dimensional Hamiltonians ........ 16
2.3.2 Examples of Three-Dimensional Hamiltonians ....... 20

3 SELF-ADJOINTNESS OF HERMITIAN HAMILTONIANS .......... 22

3.1 Three Different Cases of Hermitian Hamiltonians.............. 22
3.2 The Application of Boundary Conditions and Self-Adjoint
Extensions at Regular Points ............................................ 32
3.3 The Application of Boundary Conditions and Self-Adjoint
Extensions at Singular Points ............................................ 34
3.4 Conclusion .............................................................................. 38

PART II OPERATOR ANALYSIS AND FUNCTIONAL INTEGRAL
REPRESENTATION OF NONRENORMALIZABLE MULTI
COMPONENT ULTRALOCAL MODELS

4 INTRO DUCTION .............................................................................. 44

5 CLASSICAL ULTRALOCAL MODEL AND THE STANDARD
LATTICE APPROACH ....................................................................... 48

6 OPERATOR ANALYSIS OF MULTI-COMPONENT ULTRALOCAL
M O D ELS .............................................................................................. 53








6.1 Operator Analysis of Single-Component Ultralocal Models
......................................................................................... 54
6.2 Operator Analysis of Finite-Component Ultralocal Fields
......................................................................................... 60
6.3 Operator Analysis of Infinite-component Ultralocal Fields
......................................................................................... 65

7 THE CLASSICAL LIMIT OF ULTRALOCAL MODELS ................. 73

7.1 Ultralocal Fields and Its Associated Coherent States ......... 74
7.2 Selection of the Coherent States for Ultralocal Fields
and the Classical Limit .......................................................... 77

8 PATH-INTEGRAL FORMULATION OF ULTRALOCAL MODELS
........................................................................... 86
8.1 Operator Solutions of Ultralocal Models ........................ 87
8.2 Euclidean-Space Path-Integral Formulation of
Single-Component Ultralocal Fields .................................. 92
8.3 Euclidean-Space Path-Integral Formulation of
Multi-Component Ultralocal Fields ................................... 96

REFERENCES ......................................................................................... 101

BIOGRAPHICAL SKETCH ...................................................................... 103











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


SINGULAR DYNAMICS
IN
QUANTUM MECHANICS AND QUANTUM FIELD THEORY


By

Chengjun Zhu

April, 1994


Chairman: Dr. John R. Klauder
Major Department: Physics


The classical viewpoint that the pathologies of quantum Hamiltonians are
reflected in their classical counterparts and in the solution of the associated
classical dynamics is illustrated through the properties of classical equations of
motion. Rigorous mathematical proofs are provided by calculating the
deficiency indices, which further help to analyze the possibility of extending the
Hermitian Hamiltonian to a self-adjoint one. A few examples of how to add
boundary conditions on wave functions in order to have self-adjoint extensions,
as well as to cure the illness of certain Hamiltonians, are discussed.
The nontrivial (non-Gaussian), nonperturbative quantum field theory of
O(N)-invariant multi-component, nonrenormalizable ultralocal models is
presented. An indefinite nonclassical and singular potential has replaced the
nonvanishing, positive-definite nonclassical and singular potential appeared in
the single-component case. The operator theory of multi-component ultralocal








fields remains noncanonical even with the disappearance of the singular
potential. The validity of this nontrivial solution is supported by the fact that the
nontrivial quantum solution reduces to the correct classical theory in a suitable
limit as h -- 0. The nontrivial (non-Gaussian) path integral formulation
constructed by the nonperturbative operator solution, involving the nonclassical
and singular potential, replaces the standard lattice approach, which invariably
leads to a Gaussian theory regardless of any nonlinear interactions. The
appearance of the nonclasssical, singular potential suggests that we can not
always place the classical Lagrangian or classical Hamiltonian directly into the
path-integral formulation, or in other words, a straightforward canonical
quantization of fields with infinite degrees of freedom does not always apply.
These differences suggest plausible modifications in the lattice-space formulation
of relativistic nonrenormalizable models that may lead to nontriviality for N-
4
component models such as (%, n > 5 (and possibly n = 4), for N >! 1.















PART I


CLASSICAL SYMPTOMS
OF
QUANTUM ILLNESSES














CHAPTER 1
INTRODUCTION


The rules of quantization, as laid down in 1926 by Schrodinger, have stood
the test of time and have provided the basis for applying (and teaching)
nonrelativistic quantum mechanics1. Because of the overwhelming success of the
theory, it is not surprising that physicists have little use for several mathematical
niceties that go by the largely unfamiliar code words of "self-adjoint extension,"
"deficiency indices," operator domains," etc. (which we will briefly define in
subsequent sections). Z3
These concepts typically become important when the quantization
prescriptions are ambiguous or otherwise incomplete all by themselves4. Our
goal is to show that such quantum technicalities are actually reflected in the
classical theory by corresponding difficulties that are easy to see and understand.
It will be clear that such an evident classical difficulty must lead to some kind of
quantum difficulty, and thus the naturalness of the corresponding quantum
technicality will become apparent. We may go so far as to assert that there are
sufficient classical symptoms of any quantum illness that a complete diagnosis is
possible already at the classical level (Chapter 2). Of course, resolving the
problems and effecting a full cure, when one exists, can only take place at the
quantum level (Chapter 3).
This chapter contains basically a summary of the mathematical concepts
and theorems prepared for later chapters. In Chapter 2, we will discuss the
properties of classical solutions of given examples, whose "symptoms" can be







used to diagnose any potential quantum "illnesses". In Chapter 3, we will
provide the mathematical proof for all the assertions made in Chapter 2 and will
also give a few examples of how to add boundary conditions on the wave
functions in order to extend the Hamiltonians to be self-adjoint and so to cure the
illness of certain Hamiltonians.


1.1 Basic Concepts of Operator Theory


In this section we follow Ref. 3 closely. The operators we deal with in
quantum mechanics are frequently unbounded operators which need to be
defined on a domain D(A) to ensure that Aye r9 for V e D(A), where His
Hilbert space.
Generally, if A and B are two unbounded linear operators,


D (aA + PB) = D(A) n D(B), a, 3 e C,
(aA + PB)y = a (A y) + f(B y), Vye D (aA + +PB).


For the product put


D (AB) ={ V: ye D(B), By V D(A)),
(AB)y= A (By), iye D (AB).


In general D(AB) D(BA), and hence AB BA, and even A A'1 A'A,
because their domains may not coincide. So extra care is needed in treating these
operators.







We recall that A is densely defined in Hif (and only if) His the closure of
D(A), i. e., for y e H, there is a sequence of elements of D(A) converting to ig,
[on -V, n E D(A)].
For a densely defined operator A for H9 into ,4 there exists a unique
adjoint operator At which satisfies


(A W, 0) = ( Vp, A t),


where Vy e D(A) and # e D(A t).
If an adjoint operator At = A on D(A), and D(A) = D(At), then A is called
a self-adjoint operator.


1.2 Symmetric Operators and Extensions


If At = A on D (A), then D(A) a D (At) always holds (like Hermitian
Hamiltonian operators). Now,
(a) A is called a self-adjoint operator when D(A) = D(At), and one writes A = At.
(b) A is called a symmetric operator when D(A) c D(At), and one writes A c At.
Further, if there exists a symmetric extension As of A, such that A c As =
Ast c At, then we say that A has the self-adjoint extension As (by enlarging D(A)
to D(As) and contracting D(At) to D(Ast) = D(As)).
Let At 0= i 4, with solutions 0 e N which are closed subspaces of 94
known as the deficiency spaces of A; and their dimensions, n respectively, are

called the deficiency indices. We shall also say for brevity that A has deficiency
indices (n+, n-).
Lemma 1. If A is symmetric, then D(At) = D(]A )N+@N.-, A denotes the
closure of A.





5

Lemma 2 A symmetric operator with finite deficiency indices has a self-
adjoint extension if and only if its deficiency indices are equal.
With the help of the two above Lemmas, the following theorem regarding
deficiency indices is straightforward:
THEOREM 1: Suppose the Hermitian operator A has deficiency indices
(n+, n-), then if
(i) n, = n- = 0, A is self-adjoint when D(A) is closed and the closure of A is self-

adjoint when D(A) is not closed.
(ii) n. = n- 0, A has self-adjoint extensions.
(iii) n, n-, A is not self-adjoint and has no self-adjoint extension.

This theorem is used to classify Hermitian operators, and we see that not all
Hermitian operators will generate full-time quantum mechanical solutions.


Example: A = i d/dx on L2(Q)


A t= id/dx, At = fi


id-j <= +i k, 04(x) ~ exp(T x).



(a) Q = (- oo, + ): Neither 0+ nor 0- is in L2(- oo, + oo), so n+ = n- = 0 and A is

self-adjoint on its natural domain. This fact is consistent with its classical
behavior (see Fig. 1(a)), where there is a full-time evolution.
(b) Q = [ 0, + *o): 0. = exp(x) is not in L2([ 0, + -o)), so (n+, n-) = (1, 0), A is

not self-adjoint and possesses no self-adjoint extension. This fact is consistent
with its classical behavior (see Fig. 1(b)), where there is no full-time evolution.
(c) 0 = [ 0, 1]: Both & are in 2([ 0, 1 ]), (n+, n-) = (1, 1), A has self-adjoint
extensions by adding the boundary condition Vy(1) = exp(i0) y(0), where 0 et.








The spectrum of A = 0 + 2n ir, n = 0, 1, 2, .... For different 0, we get a different
self-adjoint extension. The case of 0 = 0 is the usual case used in quantum
mechanics and is called the periodic boundary condition. See Fig. 1(c) for its
consistent classical behavior.
The deficiency indices are clearly of central importance in the classification
of Hermitian operators and in the construction of self-adjoint extensions. These
may be found easily if explicit solutions of A tg = i y are known, but unless

the coefficients of A are simple, the calculations will involve difficult special
functions. In order to avoid those tedious, sometimes even impossible
calculations, we will use a modified theorem of deficiency indices, especially
when both end points of Q are singular. First, let us look at an example with
quantum Hamiltonian H, where


H y= (p2 + x3)y= (. -+ x3)V
dx2


acting on functions defined on an interval S2 = (- oo, + oo). Both end points x =
are singular. A direct calculation of the deficiency indices seems incredibly
tedious since the behavior of a solution of Hy/ = i V at both end points must

be known. To simplify the difficulties, we separate 2 = (a, b), where a and b are
both singular, to two intervals with i2i = (a, c ] and Q2 = [ c, b), where c is a
regular point. Here comes the second theorem regarding deficiency indices:
THEOREM 2 Let Aa and Ab be symmetric operators associated with A on
121 and 2 respectively. If n, na, nb indicate the deficiency indices of A, Aa
and Ab respectively, then n= na + nib 2, respectively (see Chapter 2 of Ref.

3).


To apply this result to the above example, we have








HtV d 2 + x3) =i
dx2


Since H is a real operator, i. e., H = H, we have /. = y/+. So n+= n. and we only
need to consider V+ and n+. Let i.+ = V,


(-d 2 +X3)W=iyl.
dx2


The asymptotic behaviors at x --+ is given by


f'- x3y,= 0


lx ~ K/i5(2x5/2) ~ x- 3/4exp( x5/2)
) 5 5

(. Ih/5(-)- f 23/4cos ( 2/2. 2-r)


S(2 52) X /2 3/- si( 2), /2. )
"5 5 20


where Jv, Nv and Kv are the Bessel functions of the first and second kinds,
respectively. Obviously, the integration


l4(x) 2dx


converges at x = + oo.


Thus, n+, +.= 1. In the same way,


(1, 2) 3/4
Wx ___ /







so that both solutions of (1, satisfy


JyO(x)2dx converges at x = oo.


Therefore, n+, .= 2, and


n+=n+, + + n+, -2 =1 + 2-2= 1


(n+, n.) = (1, 1).


One boundary condition* is needed to extend H to be self-adjoint. The existence
of bound states v. demonstrates that the energy spectrum is discrete. This
result is exactly what is claimed in Chapter 2.
Later in Chapter 3, we will see examples for three different cases of the
deficiency indices: (i) n+= n. = 0, (ii) n+= n. 0, and (iii) n+& n..


1.3 The Construction of Self-Adjoint Extensions


Before we construct the self-adjoint extensions, we introduce some
notation (see Chapter 10 of Ref. 3):
(i) (, = (A t+, V) (b, At), where Ve D(A );


(ii) fh, f2 .. fn e D(At)= D(A) N+N. where fi = hi + i, i= 1,2,...,
hie D(A), i -eN+N..


If n+= n- the number of boundary conditions = n+; see page 260 of Ref 3.








We call fl, f2 *** fn linearly independent relative to D(A) if <, ..., are
linearly independent.
With the help of the preceding definitions, we introduce the following
theorem:
THEOREM 3 Let A be a symmetric operator with finite nonzero deficiency
indices n+ = n. = n. Suppose that fl, f f2, e D(At) are linearly independent
relative to D(A) and satisfy
fj) = 0 (i, j = 1, 2,...,n).
Then the subspace Mfof Hconsisting of all ye D(A ) such that (v, fi) = 0 ( i = 1, 2,
...., n) is the domain of a self-adjoint extension M of A, given by M iy = Atp for


(V, fi) = 0, i = 1, 2, ...., n, are called the number of n boundary conditions on
the wave function p.
In chapter 3, we will give a few examples of how to use theorem 3 to
obtain some of these self-adjoint extensions and thereby cure the illness of certain
Hamiltonians.














CHAPTER 2
CLASSICAL SYMPTOMS OF QUANTUM ILLNESSES


Without using the precise definitions and techniques of operator theory,
presented in chapter 1, we can simply show that the pathologies of quantum
Hamiltonians are reflected in their classical counterparts and in the solutions of
the associated classical equations of motion. This property permits one, on the
one hand, to appreciate the role of certain technical requirements acceptable
quantum Hamiltonian must satisfy, and, on the other hand, enables one to
recognize potentially troublesome quantum systems merely by examining the
classical systems.
In 2.1, we will make the principal assertions that three categories of
classical solutions correspond to three categories of the self-adjointness of
quantum Hamiltonian operators. Examples are given in 2.2 and 2.3, while the
mathematical proof will be presented in chapter 3.


2.1 Principal Assertions


Specifically, for the classical system with a real Hamiltonian function
H = 2m + V(q), the classical equation of motion
2m
mq'(t) = V'[q(t)]
admits two kinds of solution:
(i) q(t) is global (which is defined through -- < t < +0) and unique, e. g.,
H=p2+q4 ;







(ii) q(t) is locally unique, but globally possibly nonexistent (escapes to infinity
in finite time). This case may be also divided into two sub-cases:
extendible (q(t) can be extended to a global solution) or nonextendible.
For example, H = p2. q4, H = p2 q3 are extendible, but H = pq3 belongs
to nonextendible situation.
The corresponding quantum mechanical problem


i- ax, t) = HV(x, t) = [ d 2 + V(x) ] W(x, t)
at 2m dx2


has similar cases. To conserve probability, the time evolution of the wave
function must be effected by a unitary transformation. A unitary transformation
gives a prescription for wave function evolution for all times t, < t < -, or as
we shall sometimes also say, for full time. The generator of such a unitary
transformation times VT, which we identify with the Hamiltonian operator (up
to constants), must satisfy one fundamental property, namely, that of being "self-
adjoint." Being self-adjoint is stronger, i.e., more restrictive, than being
Hermitian, which is the generally accepted sufficient criterion. According to
Theorem 1 on page 5, typically, there are three qualitatively different outcomes
that may arise (i) no additional input is needed to make the Hamiltonian self-
adjoint; (ii) some additional input (i.e., the boundary conditions) is required to
make the Hamiltonian self-adjoint, and qualitatively different Hamiltonians may
thereby emerge depending on what choice is make for the needed input; and (iii)
no amount of additional input can ever make the Hermitian Hamiltonian into a
self-adjoint operator. Remember it is only self-adjoint Hamiltonians that have
fully consistent dynamical solutions for all time. Thus in case (iii) no acceptable
quantum dynamical solution exists in any conventional sense.







Therefore we make the general assertions that there are three categories of
classical solutions, which correspond to three categories of the self-adjointness of
quantum Hamiltonian operators:
q(t) Hamiltonian operator

(i) global & unique self-adjoint & unique
(ii) nonglobal but extendible self-adjoint extensions exist
(iii) nonglobal & nonextendible no self-adjoint H exists


2.2 Classical Symptoms of Quantum Illnesses


To be sure, the majority of classical systems one normally encounters--as
represented by their classical Hamiltonians, their Hamiltonian equations of
motion, and the solution to these equations of motion--are trouble free. But there
are exceptions, classical systems which exhibit one or another kind of singularity
in their solutions, and it is in these problematic examples that we will find the
anomalous behavior we seek to illustrate.
Let us consider three model classical Hamiltonians that will exhibit the
three kinds of behavior alluded to above.
Example a:


H=lp2+iq4



The equations of motion lead to


4=2 2E 44
4







where E __ 0 is the conserved energy. Nontrivial motion requires E > 0 and E 2
q4/4 in which case


qdq)
+ dq -= t + c




leads to a well-defined, unique solution valid for all time. This solution exhibits

a periodic behavior with an energy dependent period T(E). See example a in
Table I.
Example b:


H = 2 1 3.



The equations of motion lead to

42 = 2E .q
3
and E can assume any real value. Since q3 < 3E, we find that



+ dq -t+c.
2E 7- 12, 3



This solution has the property that q diverges, q -+ for finite times, for
any nonzero E; see the solid line of example b in the table. The indicator of this
behavior is the observation that the integral converges as the upper limit goes to
- oo leading to a finite value for the right side, namely, a finite value for the time.

Actually, this same trajectory diverges at an earlier time as well. As b illustrates,







the particle comes in from q = oo at some time, say t = to, and returns to q = 0 at
a later time, say T + to, where

.3E)1/3
dq
T=2 j -.q
2- v3


For any nonzero E, T < o0; in particular, by a change of variables, we learn that


T= 1/2 1/3f dx
11/6 sL X3'


where S(E) = E/ I E I, the sign of the energy. As I E I -+ 0 the particle spends more
and more time near q = 0, until, at E = 0, it takes an infinite amount of time to
reach (or leave) the origin.
Example b in the table illustrates the generic situation for E 0 and finite
T. How could one possibly expect the quantum theory to persist for all time
when the generic classical solution diverges at finite times? The only possible
way for this example to have a genuine quantum mechanics is for the particle to
enjoy a full-time classical solution. And to achieve that-and this is the important
point--whenever the particle reaches q = oo, we must launch the particle back
toward the origin with the same energy, once again follow the trajectory inward,
and then outward, until the particle again reaches q = oo, when we must again
launch the particle back toward the origin with the same energy, ..., and so on,
both forward and backward in time ad infinitum. In brief, to get full-time classical
solutions (as needed to parallel full time quantum solutions) we must recycle the







same divergent trajectory over and over again in an endless periodic fashion
(with period T(E) ); see b in the table with the solid and the dashed lines.


Example c:


H=pq3.


The equations of motion, in this case, lead to


4 =q3,


with the solution


q(t) 1=1/--2t.


The solution is valid for t < c/2, diverges at t = c/2, and becomes imaginary for t >
c/2. See example c in the table, where we chose the constant c = 0. In no way can
an imaginary solution be acceptable as a classical trajectory. We cannot relaunch
the particle once it has reached q = -o at t = c/2 following the kind of solution
we have found, the only solution there is, as we were able to do in case b. In
short there is no possibility to have a full-time classical solution in the present
example.
If there is no full-time classical solution, then there should be no full-time
quantum solution, and that is exactly what happens. The corresponding
quantum Hamiltonian may be chosen as Hermitian, but there are no technical
tricks that can ever make it self-adjoint. Once again, the signal of this quantum
behavior can be seen in the classical theory, i.e., a divergence of the classical







solution followed by a change of that classical solution from real to imaginary.
With such a classical symptom, it is no wonder that there is an incurable

quantum illness.


2.3 More General Examples


In what follows we discuss a number of hypothetical classical
Hamiltonians, examine qualitatively the nature of their classical solutions, and
address, based on the thesis illustrated in Sec. 2.1, the problem of making an
Hermitian Hamiltonian into a self-adjoint one, if that is indeed possible.

Let us dispense with the "healthy" cases at the outset. Whenever the
classical equations of motion admit global solution for arbitrary initial conditions,
then the quantization procedure is unambiguous (apart from classically
unresolvable factor-ordering ambiguities). Observe that the existence of such
global solutions is an intrinsic property of a Hamiltonian independent of any
particular set of canonical coordinates.


2.3.1. Examples of One-Dimensional Hamiltonians


Example d:


H=-2 2q4
2






qf 2E = -t + c.
f V2/ +q4









E can be any real value.
For E > 0, we notice that f dq / f2 E+q4 converges to, say T (E), which

means when the particle travels from to + -, the time interval is T (E). For
nonzero E, T is finite; see the solid line of example d in the table. In order to get a
full-time classical behavior, we must send the particle back when it gets q = + -;
see the two distinct dashed lines of d in the table. In quantum theory, this
situation corresponds to adding two boundary conditions (at q = c), which are
required to extend H to be self-adjoint.
For E < 0, the paths do not cross the origin, but there is no essential
difference in the quantum behavior from the case E > 0. That means we still need
to add two boundary conditions to extend H to be self-adjoint.


Example e:


H = 2 -q2
2






J d t + c,




(1/f2)ln q+9T+ E= t +c.


When the particle travels from to + o-, the time spent by the particle spans the
whole t-axis (see example e in the table), so there is a global solution for an







arbitrary initial condition. Thus the quantum Hamiltonian is unique and self-
adjoint.
Since there is no periodic behavior, the spectrum of the quantum
Hamiltonian is continuous.


Example ft


H=2pq2,


4 = 2q2,


2q = 1/(- t + c).


Global solutions exist (see example f in the table) and the quantum Hamiltonian
is self-adjoint. Since there is no periodic behavior the spectrum is continuous.


Example g-


H =2 p (1/q),


consider 0 = (- o, 0) or (0, + o),


4 = 2(1/q),


q2=4t+c.







similar to Example c, q becomes imaginary for t < c /4, and there is no
possibility of having a full-time classical solution, just as in the case of example c.
Therefore there is no way to extend the Hamiltonian to be self-adjoint.


Example h:


H=p2+2pq3,


4 =2 (p +q 3) =2 + E.


Its classical solution is similar to example d, and is identical to H = p2 q6, to
which it is canonically equivalent.


Example i:


H=p2+2p/q


q 0, so Q = (- c, 0) or (0, + -),


= 2(p + 1/q) = 2 E + 1/q2,


1 + Eq 2 = 2 E( t +c).


The solution to the motion reads


(2E)2(t + c)2 E q2 = 1.








See example i in the table, where we chose E > 0. Since q = 0 is a singular point,

we divide 9S into Q = (- -, 0) and (0, + cc). For example, when the particle reaches
q -> 0+, we have to send it back or cross over q = 0 to get a full-time classical

behavior. By choosing c, the full-time classical picture may be like the solid and
dashed lines of i in the table. In quantum theory, one boundary condition is
needed to extend H to be self-adjoint.

Two additional examples (I and m ) appear in the table without discussion
in the text.


2.3.2. Examples of Three-Dimensional Hamiltonians

To simplify the problem, we consider here only central potentials V(r). In
suitable units, the quantum Hamiltonian becomes

^= 2
H= V + V(r),


V _-i r2 UL
r2dr dr r2


Set
u(r)
=-) Ylm,


then
Hr =. + + V(r) for u(r).
dr2 r2


The condition that I y 12 d3r converges now becomes the condition that

folu(r) dr converges.
For V(r) = A /rn, n = 1,2,3,..., A >0,









Hr = pr2 (A /rn) + (1 ( +1)/r 2)


and the classical equation of motion is


t = 2 VE + (A /rn) (1 (I + 1)/r2).


Under both cases of E < 0 and E > 0, we see the difference between n = 1 and
n > 2. For n > 2, near the origin, i.e., r ~ 0, i= 2 VE + A /rn, so r could go as near
as r ~ 0. See example k in the table. In quantum theory, one condition is
required to extend the Hamiltonians to be self-adjoint. It corresponds to one
parameter (e.g., see parameter B in Ref. 4), which is needed to specify quantum
solutions. But for n=1, i = 2 -E + A/r- (/ + 1)/r2; with E < 0 and I > 0, the
motion oscillates between the two roots of the expression under the radical,
namely,


+ A 2 1(1+1) r+ + 2(+1)
21EI 4E2 2 E 2E| 4E2 IE


So it avoids the singular point r = 0, and therefore it has a full-time classical
evolution (see example j in the table), thus its quantum Hamiltonian is a self-
adjoint operator. That is why the hydrogen atom does not have any problems.














CHAPTER 3
SELF-ADJOINTNESS OF HERMITIAN HAMILTONIANS


Having recalled that simple and physically natural classical "symptoms"
are available to diagnose any potential quantum "illness," we now discuss some
of the standard techniques in Chapter 1 used to analyze Hamiltonian operators
and confirm that the classical viewpoint presented in Chapter 2 is fully in
agreement with conventional analyses.
In particular, we analyze in this chapter the self-adjointness of several
Hamiltonians which appeared in Chapter 2 by calculating their deficiency
indices directly. Therefore, we can verify the consistency of solutions of classical
and quantum Hamiltonians. According to Theorem 1 on page 5, we have three
different cases of the deficiency indices: (i) n+= n- = 0, (ii) n,= n-. 0, and (iii)
nl+ n.. Cases (i) and (iii) are simple, where either a densely defined operator is
or is not self-adjoint, but case (ii) is much more complicated. Discussions such as
how to add boundary conditions on the wave functions in order to extend the
Hamiltonians to be self-adjoint, and what kind of physical interpretation those
extensions imply, etc., will be presented in 3.2 and 3.3.


3.1 Three Different Cases of Hermitian Hamiltonians


In this section we will apply Theorem sl & 2 to four general types of
Hamiltonian, just as we did in 1.2 for H = p 2 + X3.







(1) H = p2 bxm,
(2) H = p xm + xm p,

(3) H = p2+pxm +xmp,
(4) H=p2+xpm +pmnx
where m is a positive integer and b is a real constant. Those four types of H cover
all the Hamiltonians discussed in chapter 2.


3.1.1 H=p2-bxm

Obviously, H is Hermitian. Similarly to what we did for H = p2 + x3 in
1.2, let






with p = -id /dx,


-j+ bx m)=Tiv.



As both b and m are real, V. = Thus, n+ = n. = n. So consider Vy+ only; we
write



The asymptotic behavior of this equation as x is given by


The asymptotic behavior of this equation as x > is given by


since we require v (oo) = 0. The solutions are given by Bessel functions:









(X) ~ Z1/(m + 2)( 2 f m+2)
m+2


where Zv ~ Jv, Nv, Iv, or Kv, which are Bessel functions of the first and second
kinds, respectively. To examine whether or not J I Vy(x) I 2dx converges at x oo,
we need to use the following asymptotic forms of the Bessel functions:
lyl -4+oo,


v(y) ~ cos [y- (v +1]


Nv(y) -2-sin [y- (v+


Iv(y) 1 exp(y)



Kv(y) --exp(-y)



If J I yix) 12dx converges, then V(x) e D(Ht) c A9.


Examples:
(a) H=p2 +x4 :

near x =-ithus,


near I xl ~ -+ 0; thus,


V -x4y= O.








Then b = 1 and m = 4. We should choose Z -~ I, K (with y = 1/3 1x 13), the
Bessel functions of the second kind.


For x ~ o,


p(x) ~ K1/6(-a),
3


so n+,L.0 = 1


n+ =n+,+. + n+, -2 =1 +1-2=0= n.,


i. e., (n+, n.) = (0, 0).


Thus, H is self-adjoint in L2(( oo, + oo)), which makes possible the global solution
of the Schrodinger evolution equation (see Sections 9.6 and 9.7 of Ref. 3); the

existence of the bound states Vp and Vy. implies a discrete spectrum of energy.


(b) H=p2 -x4 :

y" +x4, = iW


Here b = +1 and m = 4; thus

2)(X) ~ -xJ1/6 (3) and -AN1 6(1x3) ~ (1/x)cos or sin (1x3 r).
3 3 3 3


It is easy to check that both of the solutions are in L2, so n+, + = 2 = n+, ., as is

symmetric on x.


n+= 2 + 2-2 = 2 = n.,


i.e., (n+, n.) = (2, 2).


Two boundary conditions are required to extend H to be self-adjoint, and a
discrete energy spectrum is expected.

Indeed, even a WKB analysis leads to a discrete spectrum.


Iv will make the integral diverge, and therefore is not allowed.









(c) H = p2 x2 (p2 + x2 is similar to p2 + x4 )


v/ +x2 =- i


where b = 1 and m = 2.

2)-X) ~ 1/4(-2) and INi,4(ix2) ~ l-12os or sin (4- )
IYo 4 4 4 8


but neither of them are in 2(0f); therefore,


(n+, n.) = (0, 0)


and H is self-adjoint, but its spectrum is continuous because no bound states
exist. Again, this conclusion follows from WKB arguments as well.


(d) H = p2 + x (p2 x is basically the same )


V+ xV+ =-i+


v+ (x- i))v+ = 0.


For x > 0, b = -1 and m = 1,


IY+(x) -~ -iTK1/3(2-x -i)3/2),


which is in L2.







But for x <0, b = + 1, and m = 1,


V1," 2(x) 1~ [1TTi/3(34W +i)32)3, Ixr1 +Ni/3(2jWx +i)3/2) 1/W1/4


where we have ignored the unimportant phase term. Both of the solutions are
not in L2 ( the square integral of i.+ diverges at x ~ o). That means there is no
solution for x < 0, and no solution could be connected to iV+, x >0 either.
Therefore, we have

(i) (n, n.) = (0,0) for = (- 0),
(ii) (n+, n) = (0, 0) for = (-oo, +oo),
(iii) (n+, n.) = (1, 1) for 12 = [ 0, + -).
The spectrum is expected to be continuous in (i) and (ii), but discrete in (iii).
This is an interesting system. In section 3.2, we apply it to a particle in a
gravitational field and a uniform electrical field.


3.1.2 H=pxrm +x mp

By substituting p ~ i d/dx and using [ p, xm ] = imx m -1, we get


(p xm + x mp) /p = (2p xm imx m-1 )I/= V


2xm V = (T1 mx m -1)


(i) m = 1, 2x Vf = ( T1 1)I+,


x .=0








with Vi 1/x., which is not in 2 since it diverges too rapidly at x ~ 0, and Y- =
const., which is not in L2 either. Therefore,


(n., n.) = (0, 0)


Thus, H = p x + x p is self-adjoint and its spectrum is continuous.


(ii) m > 1,
2// v/ = (T1 mx m -)/xm


~t- m/2exp ( 1 xl -m)



As x ~ o, ~ 1/x-" m/2, which are square integrable, so o are regular points.
But now x = 0 is a singular point. Checking V near x = 0, we find

m = odd, n. = 1, n+ = 0, which has no self-adjoint extension. H is not self-
adjoint and has no complete evolution, just as was the case for H = p x3 + x3 p in
Chapter 2.
m = even, n. = n+ = 0, H is self-adjoint with continuous spectrum. See the
example of H = p x2 + x2 p in the previous chapter.
It is interesting to point out that for 0 = ( -, 0), (n+, n.) = (0, 1) if m = odd
and (1, 0) if m = even; for 1 = ( 0, + c), (n+, n.) = (0, 1) if m = odd and (0, 1) if m =
even. Therefore, H is not self-adjoint in both 2 = ( o, 0) and 1 = ( 0, + -), no
matter whether m is even or odd. To explain that by the classical picture (see
example c in the table), we say that neither of those intervals could contain a
complete temporal evolution.









3.1.3H=p2+ pxm +xmp


In the above section, we found that p xm + xm p is not self-adjoint if m =
odd. Here we will show that by adding p2 to it, the situation will be changed
dramatically. First let us examine the classical solutions:*


H=p2+pqm +q mp=p 2+2pqm


4=2p+2qm=2 VE+q2m


(i) If m = 1, the solution is like that for H = p 2 q2, where H is self-adjoint.
(ii) If m > 1, the solution is similar to that for H = p2 q4, where boundary
conditions must be added to make H self-adjoint.
Second, let us see that quantum features of H:


H=p2+pxm +xmp=p 2+2pxm-imx m-1


(p2+ 2p x m imx m -1)V = i


+ 2i x m + imx m -1 / = T iy


The asymptotic for near x ~ is


V + 2i xmy/+ imx m-l1 i = 0.


* H =p2+2pqm,2p=-2qm1+2 7E-+q2m 2p+2pm=+ 2VE+q2m







We introduce an equation with the solution given by the Bessel functions, i.e.,


0 +[(1 2a)/x T 2ifyxry-1]0 + [(a2- vy2)/x2. ij(

0 = xaexp (ipx 7) Zv(Px x).


Note that V just fits this equation by choosing


y=m+1, =l1/7,


v = 1/2(m + 1),


~ ~VWexp(-i--1xm +1) Zv( 1 (- xm+ 1)~ 1/
m +1 m +1


which is square integrable when m > 1*. So n+, = n., ,. = 2, and


(n+, n-) = (2, 2),


when m > 1.


So by adding p2, H always has self-adjoint extensions no matter whether m is
even or odd. This is exactly consistent with what we have just discussed from

the classical point of view.
As for the case m = 1, i+ ~ 1/if, which is not square integrable, so

(n+, n.) = (0, 0), which again agrees with what follows from the classical point of
view.


3.1.4H=p2+ xpm +p mx


Classical method:


* Here Zv Jn, Nv.


a = 1/2,








p = aH/q = 2 p m


which is similar to 4 = 2 q m based on H = 2 p qm; therefore,
when m = even, H is self-adjoint;
when m = odd, H is not self-adjoint.


Quantum method:
x -4 i d/dp, [x, pm] = imp m-1


(p2 + x p m + p mx )(p) = i /(p)


2ip my, = (i p2 imp m 1)y+


~ (1/V ) plm)xexp (--m 2(m 3) p 3-m (3.1)



For m > 1: (n+, n.) = (0, 0), if m = even,
(n+, n.) = (1, 0), if m = odd.


For m = 1, (n+, n.) = (0, 0) which has already been discussed in Section 3.1.3. We
see again that the classical picture is consistent with the operator analysis.
It is heuristic to observe that the last factor in Eq. (3.1) comes from the part
p2 in H, and it only induces a phase in the solution, therefore it would not change
the convergence or divergence of the square integral, i. e., the self-adjointness of
p 2 + x pm + pmx is exactly the same as x pm + pmx, and the latter one turns out
to be the same as p xm + xmp.
In other words, unlike the p2 in p2+ p xm +xmp, the p2 in
p2 + x p m + p mx will not change the properties of self-adjointness.








3.2 The Application of Boundary Conditions and Self-Adjoint Extensions at
Regular Points


Examples:


(a) A = p = i d/dx, with 2 = [b, a]


p+= iV V = exp (T x).


Thus, n. = n- = 1, apply theorem 3 to this case, we have


fl= C1i+ + C2V-,


C1, C2 C C


0 = (f, f = ( dfi l/dx, fl) (f, i dfl/dx) = i [ fifl* ]I


= i [ f -(b) f(b)- f (a) fi(a)].


Set fi(b)/fi(a) = z; we have z z* = 1, so z = exp(i0), 0 R. For self-adjoint D(M) =
[Vx): ( V, f) = 0 ],i.e.,


Wp(b) fi(b) W(a) fi(a) = 0


(b) / (a) = exp (iO)


(3.2)


let us look at the eigenfunctions of p:


- ~ exp (iAx)









Vi;(b)/ V(a) = exp (iA( b a))


Vf, has to satisfy (3.2) to be in D(M); so we get


A(b-a)=0+22rn n = 0, 1, 2, ...


,A=(0+2rnn )/(b-a).


The case of 0 = 0 is what is usually considered in quantum mechanics, and is
called the periodic boundary condition.


(b) H = p2 + x, with S2 = [0, + o)
As we already discussed in (d) of Section 3.1.1, n+ = n- = 1 for 2 = [0, + c).
Therefore*


0 = (fl, f) = ( p2f +xf, fl) (fl, p 2fl +xf) = (p2f, fl1) (f, p2f1)


= (ft f- fI f (0)f(0) fl(0)fl (0).


fi(0)/f(0) = [h(0)//f(0)] = tan 0


0 E 9t,


D(M) = [*K(x): ( V, f)= 0 ], so the boundary condition at x = 0 is


V(0)cosO V'(0)sin0 = 0

limx [fif/ -flfl* = 0, since when x + -, v/+ = f- K1 /3(2-(x -)3/2),
3
h= c pV+ + c2Vf-


(3.3)









In fact, at any regular point, the form is the same.
Consider a particle with 1/2 unit mass moving vertically in the earth's
gravitational field and set x = 0 at the surface of the earth, D = [0, + -). Then its
Hamiltonian is given by


H = p2 + mgx =p2 +x, by setting g = 2.


It is interesting to note that this system has a natural self-adjoint extension
set by y(0) = 0. That is the case of q = 0 in the above boundary condition (3.3).
Additionally consider a physical system of an electron in a uniform
electric field E applied in the + x direction. The form of the Hamiltonian is then
H = p2 + x with S2 = ( o, + oo). For such a system, no special boundary

condition is needed since n+ = n. = 0.

3.3 The Application of Boundary Conditions and Self-Adjoint Extensions at
Singular Points


In this section, we will look at the effect of an attractive central potential.
We discussed in the classical point of view that there is no trouble for systems
with nonrelativistic Coulomb potentials, since they have a full time evolution.
Here we will see that this holds because they are self-adjoint. We will also find
self-adjoint extensions for the super-attractive potentials [V(r) = -A/r n, n > 2].
After separating the angular part from the radial part, we have


Hr = d 2/dr2 +1 (I + 1)/r 2 + V(r)


which acts on u(r) with the requirement that Jo u(r)r dr must converge.









(i) V(r) = -A/r, A > 0


Htr = iu ,


becomes, at small r,


S+ /r (I + 1)/r2) =0



u ~ Z (2+ 1)(2 r) ~ (21f-) (2+ 1).


Thus, with I > 0,


nt0o = 1.


At large r,


u iu = 0


and its is not hard to find that

n+,+o= 1.


Therefore, (n+, n.) = (0, 0) for H with a Coulomb potential.


(ii) V(r) = -A/r2


u + ( +(i+A/r2- 1 (1 + 1)/r2)u = 0.



At small r, the eigenfunction UE and u+ have the same form:


u +(i+/r -(1)/r2) =0
u+ + ( i + I /r I (I + 1)/r2)u+ = 0








SA- l(l+1)
u + r2 u= 0
72


u, uE-rP,


p=1/2 1/4-(A-(l +1))


For an s-wave (I = 0) with A > 1/4 or A 1(I + 1) > 1/4, we have


u, UE r 1/2sin ('ln r ),


r1/2cos(A'ln r )


where A' = [A (I + 1) 1/4]1/2.


Then n, 0.= 2.


With n,. = 1, we get


(n+, n-) = (1, 1).
Now let us construct self-adjoint extensions: at small r,


fh = cl rl/2cos(A'ln r ) + c2r 1/2sin(A.'n r).


Using ( ,yr = limr -0 o [ ''* V'], and noting that ( ) is a bilinear form on

D(Ht) xD(Ht), and(g,, h )= -(h,g ), we have


(rl/2sin('ln r), r 1/2sin(A'ln r) y = 0


(r1/2cos(A'ln r ), r 1/2cos(A'ln r) )b = 0


(r 1/2os(A'ln r), r 1/2sin(A'ln r) y = A'
Thus,


0 = (fl, f )= (c2- ClC2)',


C1C2= real.


Consequently c2/c1 is also real. Set c2/cl = tan 0, where 0 e 9t; then









fi ~ rl/2cos(;'ln r 0)


The eigenfunctions have the same form at small r, i. e.,


UE = CErl/2os(9'ln r 0)


in order to satisfy (uE, fl )= 0. In Ref. 4, the author found the dependence of
eigenenergy on the choice of 0. Because of 0, we have a one-parameter family of
self-adjoint extensions.
Further, D(M) = [ u(r): (u, fl) = 0 ], and therefore the boundary condition
at r = 0 is given by


limr --o u'(r)r 1/2cos('ln r 0) u(r) -(r/2cos(ln r 0)) =0
L dr J


(iii) V(r) =-A/r", n > 2,


uE +(E +A/rn (1 + 1)/r2)UE = 0.
at small r,
UE +(a/rn)UE = 0


uE~ #Z.1/(n+1 i (-2)/2


Using the Bessel asymptotic forms, we get


UE rn/4sin(yr- (n -2)/2)








rn/4 cos(yr-(n-2)/2)


where y= 2- /(n -2). u have the same form, so nt 0o. = 2, (n+, n.) = (1, 1).

The boundary condition is similar to (ii)


fi = ci rn/4 sin(yr- (n -2)/2 ) + c2rn/4 cos(yr- (n -2)/2)


using


Thus,


( rn/4 sin(yr-("n-2)/2 ), rn/4 cos(yr- ("-2)/2). = A1/2


(f, f, )=0


gives C2/C1= tan 8,


f- ~ rn/4 sin(yr- (n -2)/2 + 0)


uE = CE rn/4 sin(yr-(n-2)/2 + 0)


D(M)=[u(r): (u, f)= 0 ],


gives the requirement on u(r) at r -+ 0 :


limr -+o u'(r)rn/4 sin(yr-(n"-2)/2 + 0)) u(r) -d-r n/4 sin(yr-(n-2)/2 + 0))] = 0
L dr J




3.4. Conclusion


By calculating the deficiency indices for several representative examples of
Hermitian Hamiltonians, we have analyzed the possibilities in each case of
extending the operator to a self-adjoint one. In so doing we have confirmed the


6e91







connection proposed in the previous chapter that relates possible extensions to
associated characteristics in the solutions of the classical equations of motion.
The purpose of this chapter has been twofold: on the one hand, by demonstrating
that anomalies in the classical solutions to a dynamical system are reflected in
anomalies of the quantum Hamiltonian, certain technical issues ( such as
deficiency indices, etc. ) are brought into the realm of everyday experience. On
the other hand, the intimate connection between classical and quantum
properties should enable one to examine a given system at a classical level in
order to assess what problems, if any, are expected to arise at the quantum level.







40







Table I, Summary of classical and quantum highlights associated with several model problems, Each model has a label, a classical li miltonian, typical
solution trajectories, nature of those solutions, character of quantum Ilamlltonlan. spectral properties, and. In some cases. related examples as well, 1 he
solid curve in the figure portion represents a typical trajectory or part of a trajectory In the case of a periodic orbit The dotted curve denotes an
alternative typical trajectory, and the dashed curve denotes a periodic extension of the basic orbit. In d. the figure illustrates two possibly distinct periodic
extensions. h is omitted because it is a related example of d.


Example Classical Qualitative Graph Nature of Self-adjoint Spectral Related Examples
Label Hamiltonian t(q) of equations Classical Quantum Properties m = 1,2,3,...
H of motion solutions Hamiltonian

t


a P' + 4 global unique discrete p2 + 9 "*





t

) one parameter
I6 p' -. -- periodic family of discrete p9 qs

s. 1 solution (one p' 92'"+
) boundary
condition)
t


c 2pq' =- partially nonexistent none pq" ,m > 1 ,m = odd

complex



t two parameter
family of
d p' -- periodic solution (two discrete q'" m>2

-' boundary p2 + 2pq," m > 2

conditions)





pe -2 globall unique continuous p9 + 2pq


___^__






















Clatonian

Hamiltonian


21,/q"' -' = 0,I4


p' + 2p/q'"


I x .,ip 1r
thrnole


1 nhie I (Co'nutied }

















Table I. (Continued.)


- IA





^-1


Fig. 1. Classical trajectories for the simple Hamiltonian H=p in three separate coordinate domains: (a) a (b) O














PART II


OPERATOR ANALYSIS
AND
FUNCTIONAL INTEGRAL REPRESENTATION
OF
NONRENORMALIZABLE MULTI-COMPONENT ULTRALOCAL MODELS














CHAPTER 4
INTRODUCTION


As we know, the features of infinitely many degrees of freedom and
noncompact invariance groups have made the structure of quantum field
theory very complex. So seeking for the proper formulation of problems in
quantum field theory has been a difficult yet exciting endeavor in our
research. Even though conventional perturbation theoryS'6, which is based on
a free field formulation, has proved successful in solving some problems, e.g.,
quantum electrodynamics (QED), we must ascertain whether the free field can
be generally used as the basis for a perturbation expansion7. One of the results
that arose from the traditional perturbation analysis based on free fields is the
appearance of nonrenormalizable interactions which some physicists regard
as hopeless. Therefore the structure of nonrenormalizable models has been
largely ignored over the years. Twenty years ago, the pseudo-free theory was
proposed by Klauder7'8 in the study of ultralocal field models to extend the
usefulness of perturbation theory. An argument based on asymptotic
convergence suggests that the free theory is connected with continuous
perturbations while the pseudo-free theory is related to discontinuous
perturbations, which have much to do with nonrenormalizable interactions.
At about the same time, a functional and operator approach to single-
component ultralocal field models was developed through general
nonperturbative and cut-off free arguments9,10. There Klauder gave an







alternative quantum theory of such models, which does not fit into the
canonical framework. He also showed that as any interaction is turned off.
after it was once introduced, the theory will not pass continuously to the free
theory, but rather to a distinctly different pseudo-free theory. Clearly,
conventional perturbation theory could not be applied to such a model.
The ultralocal model is obtained from covariant model by dropping the
space-gradient term. Since distinct spatial points there characterize
independent fields for all times, it unavoidably results in a
nonrenormalizable situation. We are interested in this model because it is
solvable by nonperturbative techniques and may give us some insight into
the structure of nonrenormalizable fields.
In chapter 5, we will show that just like nonrenormalizable relativistic
quantum field theories, on one hand they exhibit an infinite number of
distinct divergences when treated perturbatively, while on the other hand
they frequently reduce to (generalized) free fields when defined as the
continuum limit of conventional lattice-space formulation11, ultralocal
models are specialized nonrenormalizable theories that also exhibit infinitely
many perturbative divergences and an analogous (generalized) free-field
behavior when defined through a conventional lattice limit12. However, the
characterization of infinitely divisible distributions13 allows ultralocal models
alternative operator solutions9,10 (also Chapter 6), which lead to a nontrivial
(non-Gaussian) solution on the basis of operator methods. The validity of
this nontrivial solution is supported by the fact that the nontrivial quantum
solution reduces to the correct classical theory in a suitable limit as h -> 0; the
trivial (Gaussian) solution has no such correct classical limit (Chapter 7)12.
Recently, it has been found how to obtain the same nontrivial results
offered by operator techniques through the continuum limit of a







nonconventional lattice-space formulation14. The key ingredient in the
lattice-space formulation that leads to the correct behavior is the presence of
an additional nonclasssical, local potential besides the normally expected
terms.
In chapter 6, we will construct the operator theory of O(N)-invariant
multi-component nonrenormalizable ultralocal models, where N <- It has
been surprisingly found that the singular, nonclassical term in the
Hamiltonian which showed up in the one-component ultralocal fields can be

made to disappear15, when N, the number of field components, satisfies N>4.
Nevertheless, a similar singular, nonclassical term still appears in the
regularized path-integral formulation16 that leads to the same nontrivial
quantum results for O(N) invariant multi-component scalar fields (Chapter
8). Thus, for any N, the number of field components, the new path integral
formulation involving the singular, nonclassical term replaces the standard
lattice approach which invariably leads to a Gaussian theory regardless of any
nonlinear interactions, and supports the concept of a pseudo-free theory.
One advantage of deriving the operator solutions by a lattice-space
formulation is the clear focus the latter approach places on the differences
from traditional approaches needed to lead to nontriviality. These differences
for ultralocal models suggest plausible modifications in the lattice-space
formulation of relativistic nonrenormalizable modes that may lead to
nontriviality for N-component models such as 04, n > 5 (and possibly n = 4),
for N 2 1.

In my future research, I hope to extend analogous reformulations to
covariant quantum fields, the possible relevance of which has already been
noted23. The general argument is partially based on the realization that the





47

classical theory of a typical nonrenormalizable interaction exists and is

nontrivial24.














CHAPTER 5
CLASSICAL ULTRALOCAL MODEL AND THE STANDARD
LATTICE APPROACH


The classical Hamiltonian of an ultralocal scalar field is expressed as


Hci = ( 2(x) + Im2p2(x) + V[(p(x)] dx, (5.1)


where n, p denote the classical momentum and field respectively, and VI[(p]
(= Vi[ -p], for simplicity) the interaction potential. Here x is a point in
configuration space of arbitrary dimension, x 9n-1. This model evidently
differs from a conventional relativistic field theory by the absence of the term
-[Vqx)]2 The classical canonical equations of motion appropriate to (5.1)
become


(xt) = H (xt),
81(x,t)

,r(x,t) = -CHl= m2p(x,t) V1[(p(x,t)], (5.2)
8p(x,t)

p(x,t) = m29(x,t) Vi[(p(x,t)].


Unlike conventional canonical field quantization, the quantum theory
of ultralocal fields does not follow from standard canonical commutation
relations, etc., whenever a nonlinear interaction exists. In particular, we do
not have an equation such as (5.2) for local quantum operators (see Chap. 6).






Nevertheless, it is instructive to first examine these models from a
conventional lattice limit viewpoint.
Let us take a look at the result from a standard functional integral
approach (which is based on the standard canonical quantization) where the
vacuum-to-vacuum transition amplitude is formally given by 6:


Z[J] = N ( D exp if d44j(x,t)O(x,t) + 142(x,t) 1 (m2 ie) D2(x,t) g04(x,t)]I


where we have chosen VI () = gD4 as an example. Using a lattice-space
regularization in the space direction (but not in the time direction) we obtain


Z[J] = w lima--)s (1n lOD) exp(if dt Ik a Jk(t) f k (f ) 2 2 (J)

= lima-o 11 D exp if dt a[J k(t)0 (t) + _'2(t) (m2 ie) 2(t) gz4(t)]


where k labels points in the spatial lattice and a is the volume of the unit cell
in the spatial lattice. As we now show, the result for such an interacting
model leads in the continuum limit to the analog of a generalized free field.
From the preceding equation, using u = ( a, go =g/a, and < > as an average
in the complex distribution, we have


Z[J] = 9 limao n f Du exp if dt [J k(t)u(t) + 1i2(t) -1 (m2 ie) u2(t) -gou4(t)]}








= lima--o l < exp{i( dt Jk(ut)u(t)} >
k f


=lima-+oll { I- < [
k 2


J k(t)u(t) dt ]2> + <[
4!


Jk(t)u(t) dt]4 > +-.}


= exp {-1/2


d3 x dt dt' J(x, t) J(x, t') < u(t) u(t') > }.


Evidently (5.3) is a Gaussian result in which (assuming 0

< u(t) u(t') >


= fn fjDu u(t) u(t') exp i dt [2(t) -(m2 e) u2(t) -gu4(t)]


=<0 IQexp(-ilt-t'IH)Q I 0>


= < 0 I Q I n> e-it-t'lmn


(5.4)


= P il t-t'l mn
n 2mn


where mn, n=1, 2, 3,... denote the eigenvalues of H, Pn = I< 0 I Q I n > 12.
2mn
H = 1/2 (P2 + m2Q2 ) +gQ4 const., and the constant is chosen so that H 10 >=0.
It is easy to verify that I pn = 1, and since < 0 1 QH4Q 10 > < oo it follows that
n

X pnm3 < oo. Thus we have
n


Z[J] = exp {-1/4f d3 x dt dt' J(x, t) J(x, t') I (pn/mn) e -il t-t' mn }
J n


(5.3)









= [ exp {- PnJ d3x dt dt' J(x, t) J(x, t) e -it-t'm" }.
S 4mnJ


The ultralocal free field corresponds to g=g=0, in which case mn = mn,

Pi = 1, Pn = 0 (n*1) in (5.4), so that


< u(t) u(t') > = -Le -ilt-t'lm
2m
Then (5.3) gives



ZF[J] = exp {- d3x dt dt' J(x, t) J(x, t') e -i l t-t' Im
4Tm


= <0 I Texp { i d4x J(x,t)eF (x,t)} I 0 >.


(5.6)


Observe that the ultralocal free field operator O (x,t) satisfies
..m m
4OF (x,t) + m2(DF (x,t) = 0.


Comparing (5.5) and Eq. (5.6), we get


ZUJ]= l < 0 I Texp { i
n


d4x pJ(x,t)o (x,t)} I 0 >


d4x J(x,t) I fPnFr'(x,t)} I 0 >
n


(5.5)


= < 0 I Texp {i








= < 0 I Texp I i d4x J(x,t) (D(x,t)} I 0 >, (5.7)



where ((x,t) =J p-;*T'(x,t). cl}r(x,t) are ultralocal free field operators with
n
mass mn n = 1, 2,3... ,which satisfy OIF (x,t) + rn OP(x,t) = 0 for each n.

From (5.7) we see that our regularized interacting model has led in the
continuum limit to an ultralocal generalized free field, a natural analog of the
relativistic generalized free field17. As a consequence of
[(I(x,t), [c(y,t), c)(z,t)] ] = 0, which holds for a generalized free field under our
present conditions, it is clear that the field operator I) of an interaction like
VI (I) = go)4 with g>0 can not satisfy this commutation equation. Obviously
this (generalized) free-field behavior is limited neither to ge4 (which could be
replaced by other local powers), or to the space-time dimension n=4. This fact
indicates some sort of failure of the conventional formulation of canonical
field quantization for these models. Moreover, with the help of coherent state
techniques18,19 we find that the classical limit of an ultralocal generalized free
field does not limit as (h -- 0) to a classical field that satisfies the classical
nonlinear canonical Eq.(1.2); instead, the classical limit of
Il)(x,t) =1 p nI(x,t)is given, not surprisingly, by (p(x,t) =T n n
rn(x,t) + mn qTmn(xt) = 0 for each n.















CHAPTER 6
OPERATOR ANALYSIS OF MULTI-COMPONENT
ULTRALOCAL MODELS




The quantum theory of ultralocal scalar fields had been discussed
extensively by Klauder9'10 over two decades ago. The employment of probability
theory and Hilbert space methods with an emphasis on infinitely divisible
distributions and coherent states techniques, respectively, enables us to give a
proper quantization of these models, which otherwise are meaningless within
the conventional canonical formulation of quantum field theory (see chap. 5).
An operator solution of multi-component, nonrenormalizable,
ultralocal quantum field models is developed here along lines presented

earlier for single-component models. In 6.2, we will show that the
additional, nonclassical, repulsive potential that is always present in the
solution of the single-component case becomes indefinite and may even
vanish in the multi-component case. The disappearance of that nonclassical
and singular potential does not mean a return to standard field theory. The
operator solution of multi-component ultralocal fields remains
noncanonical. In 6.3, we will show that nontrivial, i.e., nongaussian, results
hold for any number N of components, and suitable nontrivial behavior
persists even in the infinite-component ( N=oo ) case as well.









6.1 Operator Analysis of Single-Component Ultralocal Models


Let us briefly summarize the operator analysis of single-component
ultralocal models9',10. Assume that the field operator V(x) becomes self-
adjoint after smearing with a real test function f(x) at sharp time. Since
distinct spatial points characterize independent fields for all time, we may
write the expectation functional as


E[f] < 0 exp {i dx (x) f(x)} I 0 > = e J dx L[f(x)]



= exp {- f dxf d [ 1 ei)f(x)] c2() }. (6.1)



Where in the last step, we have used the result of Levy's canonical
representation theorem for the infinitely divisible characteristic functions13.
and have eliminated a possible contribution of the Gaussian component,
which applies to the free field. The real, even function c(k) is called the
"model function".
An operator realization for the field Q(x) is straightforward. Let
At(x, k) and A(x, X) denote conventional, irreducible Fock representation
operators for which 10 > is the unique vacuum, A(x, X) 10 > =0, for all
xe 9,n1, ,e 9t. The only nonvanishing commutator is given by


[A(x, ?) At (x', ')] = 8(x-x') (1-X'). (6.2)


Introduce the translated Fock operators








B(x, X) = A(x, X) + c(X), Bt(x, X) = At(x, X) + c(7), (6.3)


Obviously, the operators Bt(x, X) and B(x, X) follow the same commutation
relation (6.2). Then the operator realization for the field c(x) is given by


CD(x) = f dX Bt(x, X) X B(x, X). (6.4)


The correctness of this expression relies on the fact that


< 0 1 exp{ if dxcD(x)f(x)} I 0 >


=< 0 I exp {if dxj dX Bt(x, X) f(x)B(x, X) I1 0 >


=< 0 I: exp {f dxj X Bt(x, X) (eiXf(x)-1)B(x, X) ): 1 0 >


= exp {- df dk [ 1 eidf(x)] c2(X) }


as required by (6.1).
The Hamiltonian operator is constructed from the creation and
annihilation operators and is given by


H = f dx dX Bt(x, X) h(O/aX, X) B(x, X)








= f dxf dX At(x, X) h(a/a., X) A(x, X). (6.5)


Where h(a/3X, X) = 1h2 /2+ V(X), which is a self-adjoint operator in the X
2
variable alone. It is necessary that h(a/3?, X) > 0 in order that H > 0 and that
10 > be a unique ground state. Equality of the two expressions in (6.5) requires
h(Oa/a, ,) c(X) = 0, implying that c(X) e L2 in order for 10 > to be unique. This
relation also determines V(X) as V(?) = l2 c"(X)/c(k). Assuming that
2
Sd 2 c2() < oo, so that < 0 2(f) 10 >
together with the condition J dX c2(.) = oo, we are led to choose


c(k) = I exp --Lm2 Y()' (6.6)



where y, called the "singularity parameter", satisfies 1/2 have (px- ih 3/3)


h(a/3a, X) = 1 +-- 2 + 7)h2 + m(y- 1/2)h + I22 + Vi()
2 ax2 2X2 2


=p12 Y+ )h2 + m(y- 1/2)h + im2X2 + VI(X), (6.7)
2 22 2


VI(X) = ( y" + y'2) + ( mhX + yh2/ ) y'. (6.7b)


This equation determines c(k) for any given interaction potential VI(.), at
least in principle. Notice that in addition to the free term lp2 + m22 and
2 2







the interaction term VI(X), there appears a nonvanishing, positive, singular
and nonclassical potential y(y+l)h2/2X2. It is important to note that this
additional potential makes the path-integral formulation of ultralocal fields
totally different than that of standard quantum field theory14.
The definition of renormalized local powers of the field follows from
the operator product expansion


4(x) 4)(y) = S(x-y) f dX Bt (x, .) X2 B(x, X) + : D(x) D(y)



which suggests the definition


2(x) = d Bt(x, ) 2 B(x,) = Z (x), Z-1 = 8 (0) (6.8)



or more generally,


k(x)= d Bt(x,X) B(x,) [Z(x)]k, k= 1,2,3,... (6.9)



For k > 1, these expressions are local operators, i.e., become operators when

smeared by a test function. For k 5 0, let 1 (x) = Z2 dX Bt(x, X) 1 B(x, X), and

f d (x, )- I exten for
dx Bt(x, X) 1B(x, X) (x), then we can extend (6.9) for renormalized
J x
negative powers of the field. But we should note that for k < 0, 4k(x) are not
local operators.







Corresponding to (6.7), the Hamiltonian may be expressed by
renormalized fields as


H (fnr(x) + Im2(D(x) + VI[ Dr(x)]ldx, (6.10)



where I = r+ y(y+ 1)h2 r2+ m(2y- 1)h Obviously it means that Fir
neither fits into the canonical framework( FIr = Or ) nor fulfills the standard
canonical commutation relation [Or(x), fir(y)] = i hS(x y). We should also
notice that although Or (= D = -i [ D, H ] /h = -ih f dX Bt(x, X) 3/3a B(x, X) ) is

not a local operator due to the assumption regarding c(,), neither are
S2 0 2
O2r, Or ,r local operators by themselves alone; only the combination FIr is a
well-defined local operator. These are major differences from standard
quantum field theory.
The Heisenberg field operator is given by


4(x,t) = eiHt/h O(x) e-iHt/h


=f dk Bt(x, X) eiht/ ke-iht/h B(x, )



=fdX Bt(x, X) X(t) B(x, .) (6.11)



which is well defined. The time-ordered truncated n-point vacuum
expectation values are given by


< 0 I T [O(xl,tl) D(X2,t2) *. (Xn,tn)] I 0 >T








= 8(XI-X2) 8(X2-X3) ... (Xn-l-Xn). J d c()T [ X(t) X(t2)... (tn)] c(). (6.12)



For ultralocal pseudo-free fields ( defined when the interaction
potential VI(X) vanishes ), we have the model function
c() = I exp {- mX2/2h }, and the expectation functional


E[f]= exp {- fdx dX 1 -tos [ ) exp m,2} (6.13)



which is obviously not a Gaussian solution such as for ultralocal free fields.
In 6.2, the operator analysis of finite-component ultralocal models is
presented. There we will see how an indefinite (positive, negative or
vanishing), singular, and nonclassical potential affects the operator solutions.
In 6.3, the expectation functional of infinite-component ultralocal
fields is discussed. Even in this case the solution of any interacting theory
does not reduce to that of the free theory in the limit of vanishing nonlinear
interaction and again supports the concept of a pseudo-free theory7'8.








6.2 Operator Analysis of Finite-Component Ultralocal Fields


Following the pattern of the single-component case, the field operator
and the Hamiltonian of N -component ultralocal fields are defined by


QD(x) = J dX Bt(x, X) X B(x, X) (6.14)



and

H = dx dX Bt(x, ) h(V, X) B(x,X) (6.15)
I' (6.15)
= d dX A(x, 1) h(V., h) A(x, h)


2
where h(Vx, ) = 2Vj + V(), X= Notice that here the N-component

ultralocal models under consideration have O(N) symmetry. The
commutation relation becomes


A(x, X), A(x', = 5(x-x') S(-,') (6.16)


and the unique ground state satisfies A(x, X) 10 > = 0. The operator B(x, X) is
related to A(x, X) by


B(x, X) = A(x, X) + c(). (6.17)


The renormalized local powers of the field are given by







rki,.. ik(x) = dX Bt(x, X) 1"".. "k B(x, X), ,... ik = 1, 2, 3,. N


Occasionally, we need to use the renormalized negative powers of the field
even though they are not local operators. Take a single-component case for an
example:


(kD(x) dBt(x,)ikB(x,X) = -[Z (i(x)]k, i = 1, 2... N, (6.18)


which can be extended to k<0, as long as we define
i (x) -=Z2 dXBt(x,X)lB(x,X) and dXBt(x,)-B(x,X) = Dn(x).


From the requirement that h(Vx, X) c(X) = 0, we have
V(?) = l-V2 hc(X)/c(k). The real, even model function c(X) becomes
2

c(%) = I- exp 1-2mX2 y()) (6.19)
X r 2 h


with N/2 F < f d X c2() = oo and f d 2 C2() < oo.


Next let us find out how the operator h(Vx, X) appears. From (6.19), it
follows that


V(X) =l~ V4 ) 2- c) h C(X)
2 c(X) 2 c() i=1 82









= F(f+2 + mh (r N/2) + -m2X2 + VI(%), (6.20)
2r2

where VI(1) = h2 (- y" + y'2 ) + y' { mhl + [F- (N-1)/2 ] h2/ }, which
2
corresponds to the interaction. Therefore


h(VL, ,) = 1-hV2 + V()
2


=- l/2V2 + F(F+2 N)2 + mh ( r N/2) + im2?2 + Vi(X). (6.21)



This base Hamiltonian of multi-component ultralocal fields is similar to that
of single-component ultralocal fields (6.7); but a surprising difference between
the two equations is that unlike the non-vanishing, positive, singular and
nonclassical potential 7(y+l)h2/2X2 appearing in (6.7), F(F+2 N)h2/2%2 is not
always positive and moreover it may vanish. These properties follow since
N/2 rF < N/2 + 1, i.e., 0 F N/2 < 1 and therefore


2 N/2 < r+2 N = 2 + (r N/2) N/2 < 3 N/2.


Summarizing, F+2 N>0 for N<4; F+2 N<0 for N>6; and F+2 N is
indefinite, when 45N<6, i.e., it may be larger than zero, smaller than zero, or
even equal to zero. As examples of vanishing F(F+2 N)h2/2X2 we have N=4
and F=2, or N=5 and r=3.


It is worth mentioning that the disappearance of F(F+2 N)h2/2X2 does
not mean a return to standard (canonical) field theory or standard path-







integral formulation; even in this case it can be shown that a similar non-

vanishing, singular, nonclassical potential arises in the regularized path-

integral formulation (Chapter 8).


It is worthwhile to discuss further the Hamiltonian which according to
(6.21), is given by


H = I 1Hir(x) + im2~r(x) + Vi[cr(x)] dx. (6.22)


-2 -.2 -2 0
Here nr (x) = Or (x) + F(F+2 N)h2 I r(x) I + mh (2F N) I Or(x) I Notice that
2 (x) 1 2oear
S(x) -=f ddX Bt(x, x) (-h2 2/2 )B(x, X) i=1, 2, -N, are not local operators,
0 -2
neither are Dri(x), Ir(x)I or I r(x)I but somewhat surprisingly
N 2 :.2
n W (x)a r(x) is a well-defined operator for N=4 and F=2 and
i=l
N 2 0 .-2 0
1 i (x) + mh I rl I Er(x) + mh IlO. I for N=5 and F=3. In the case of N=4
i=l
-*2 -.2
and F=2, we have Ur (x) = Or (x), which in the present case is a local operator,

but fir(x) = r(x) involves local forms rather than local operators due to the ill

defined 4ri(x).

Summarizing, the canonical equation r,(x) = Ir(x) and the standard
canonical commutation relation [4n(x),rIrj(y)] = ih8ij8(x-y) do not hold for

the multi-component ultralocal fields, just as for the single-component
ultralocal fields, irrespect of whether the singular potential F(F+2 N)h2/22
-2
[~ I r(x) I ] is present or not.

Analogous to the single-component case, the Heisenberg field operator
is given by










= dX Bt(x, X) eiht/hke-iht/h B(x, X)



= dX Bt(x, X) X(t) B(x, X) (6.23)


which is well defined. It follows that the time-ordered, truncated n-point
function reads


< 0 I T [(i,(xl,tl) i,2(X2,t2)... )i(Xntn)] I0 >


= S(xl-X2) 8(X2-X3)... 58(Xn-l-xn) f dX c(X) T[Ri,(tl) X2(t2) i (tn)] c(X). (6.24)









6.3 Operator Analysis of Infinite-Component Ultralocal Fields


The expectation functional of finite-component fields is


Elf]=<0 1 exp i dx D(x).f(x) 1 0>



= exp { g dx dX [ 1 eiX-f(x)] c2() }. (6.25)



Here we have introduced g, a scale factor which cannot be determined on
general grounds but rather represents the only arbitrary renormalization scale
involved in the operator construction. The possibility of its existence lies in
the fact that two model functions differing by a constant factor lead to the
same differential operator h and thus to the same Hamiltonian H.


An operator realization of the infinite-component field requires us to
give a proper measure which is well defined. In doing so, notice that we can
absorb the model function c(X) into the measure so that we have the new
measure dp(X) = g c2(X) fIdki. Let us redefine the commutator


[A(x, X) At(x', X') = 8(x-x') 8.(0; V'), (6.26)


where Sp(X; X') is related to the measure p by J dp(X') f(X')8p(X; V') = f(X). The

operator B and the infinite-component field are redefined as







B(x, X) = A(x, X) + 1, (6.27)


QD(x) = dp() Bt(x, X) X B(x, 1). (6.28)


Then we immediately recognize that (6.25) stays the same. The renormalized
local powers of the field and the Hamiltonian are given analogously by


rkil... ik(x) = dp(X) Bt(x, X) X.1. -ik B(x, X) il,"" ,ik = 1, 2, 3, N, (6.29)



and H = f dx dp(X) Bt(x, X) h(Vx, X) B(x, X)



= dx) dp(X) At(x, X) h(Vk, Z) A(x, X), (6.30)


with the requirement of h(VX, M)-1 = 0. The Heisenberg field operator is
given by


QD(x,t) = eiHt/h (0(x) e-iHt/h


= dp(X) Bt(x, ?) eiht/he-iht/h B(x, X)



= dp(X) Bt(x, X) X(t) B(x, X). (6.31)


The time-ordered, truncated n-point function reads








< 0 I T [di,(xi,ti) Oi2(X2,t2) in( Xntn)] I 0 >


= 8(XI-X2) 5(X2-X3)... 8(Xn-l-Xn) J dp(%) T[iI(ti) Xi2(t2)... i(tn)]-1. (6.32)


It is worth mentioning that the above formulation for the infinite-
component case is suitable for the finite component case as well.


Next we try to obtain the characteristic functional of infinite-
component fields as the limit of finite-component ones. Take the general
form of c(X) = Krexp {1- m2 y(X)\ in the finite case, then we have
2f -
E[f = exp g dx dX [ 1 ei-fx)] 1- exp [ -m 2y(?) } (6.33)


In order for this limit to exist as N-+o, we need to scale several parameters,
namely y=yN, m=mN and to choose a suitable factor g=gN. Following the
Ref. 22, suppressing the integration over x, let us consider

L[flimN ,gN dX[1-ei- l-1 exp lmNX2 2yN(k) (6.34)



= limN.-. o.4 j dr rr-1 d [ 1 eiXf- ] exp (mN + r) h2 2yN(X) .







where dr rn e- = -n- has been used. If we introduce the Fourier transform
f f+1
pair: e- 2yN() = J d( hN(() ei2, hN() =- dl2 e- 2yN() e- i12, L[f] can be

expressed as


limN.__ dr rF-1 d[ 1 eif] exp (mN + r) X2 + ia2 )hN() d4
(F-l)! I ex


The integration over X now involves simply Gaussian integration
Sdx e x2 +ifx* = /2 e-?/4a and leads to

f c
limN-- gN dr rF-I ( N/2 [ 1 e-?/ 4(mN/h + r it)] hN() d4,
(-'-1)!J mN/h + r idt


= limN -N/2NJ dr re(mN/rh i/r + )-N/2[ 1 e-?/ 4(mN/h + r it)] hN(V) d4,


where -1 < -- F-1-N/2 < 0. Now we are ready to make the choice of yN, mN
and gN. Assume that gN= glcN/2(F-1)!, mN= m/N and yNW) = y (X/fN) which
suggests that hN(() = N h(N4). Following a change of variables of 4, we take
the limit N---=, by using limN-o( 1 + m/rh it/r =exp- (m/h i
N 2r
and we get


L[f = g dr r exp- 2--(m/h- i) [ 1- e-2/ 4r] h() d4
if0








=g dr r exp- (m/h) 2y( )L [1-e-?/4r]
Jo



= g do 2 exp(- (m/h) a 2y(a)) [ 1 e-o&/2], (6.35)
=fo (2C)6+2


where -1 < 0 < 0, and in the last step, we have substituted a = 1/(2r).


We can also try in the following way to obtain the characteristic
functional of infinite-component fields as the limit of finite-component
ones. Take the general form of c(X) = K exp -1 m?2 y(?)I in the finite
2 h
case, then we have


E[f] = exp dx dp(X) [ 1 eif(x)]}



exp dx d [ 1 ei-f-x)] 1 exp[ 1mX 2y(?) ] (6.36)
\X ^2 h


In order for this limit to exist as N-oo, we need to scale several parameters,
namely y=yN, m=mN and to choose a suitable factor g=gN. Suppressing the
integration over x, let us consider


L[f] dpN(I) [ 1 ei-f(x)] (6.37)



= limNoo, gN dX [ 1 eik- f I exp [mN2 2yN(X)
f X 21-







= limN--. gN d~-- exp {- mN 2- 2yN(X) df [ 1 eiXf cos 0]. (6.38)


Where dQ= sinN-2 0 dO dK'. By changing the variable 0 --0/N + n/2 and
taking N large, we have cos 0= sin (0/N)= /N and therefore
~ ~ 2 N-2 2
sinN-2 0= cosN-2 (O/N) = [ 1 EO/N) ] = exp[ IO/N) (N 2) ]. Hence we
2 2
may write


( d [1- eif cos ]= dE exp[ 1- 2(N-2)][1-e-11f]f df'


= 2c ]N/2f dQ' { 1- exp [-1f2 2/(N-2)]}


= S-{ 1 exp [1- f22 /(N -2)] }. (6.39)
2

Where S denotes the spherical surface area in N-dimensions, N >> 1, namely
S N -N/2 dK'. Substituting (6.39) into (6.38), we have



L[f] =limN_4, gN d- 1 exp imN2- 2yN(X) S-{1 exp[ if2/(N-2)])
f 2l- N +1 h 2

=iNg( d -1)/2S X 1exp- fmN-2y( ) {1-expl N
= limN.ogN J dXNsp h eScxp a mNNge 2yN(CfN exp[-f2,2N-N ] }.


In the last step we have made a change of variables of x-VN- X', = 2T- N + 1.







Now we must choose yN, mN and gN. Assume that mN= m/N,
yN() = y (/y/N-) and gN= N' 1)/2g/S, where g is proportional to g in (6.35).
By taking the limit N- oo, we obtain


L[f] = g I d- exp (- lm2- 2y(?) [ 1 e- f22/2], (6.40)



where 1< p< 3. So the characteristic functional of infinite-component fields
becomes


E[f = exp {- dx dp(X) [ 1 eif(x)] }



= exp -[ f dx J dX- exp (- mX2- 2y() [1 e- f2X2 /2] (6.41)



From (6.41), we see that a nontrivial, i.e., non-Gaussian, result holds for
infinite-component fields, just as it does for the finite-component ones.


Notice that the scalings we have used here, such as mN= m/N and
yN(O) = y (1/-N-), are different from the standard ones. For example: for
-4 -4
V1 = D the standard scaling assumes (Vi)N = ( /N, while the non-standard
scaling employs mN= m/N and yN(X) = y (1/-N) which gives
(VI)N[)] = VI[/VN] = )4/N2 or equivalently (VI)N[c] = VI[I/ N] = 4/N2.
The nonstandard scaling admits a 1/N expansion relevant to the Poisson-
distributed finite-N solution, while the standard scaling does not. Another
similar non-standard scaling example was shown to hold in the independent-
-2 -2
value models where mr = m2/N, (VI)N[D ] = VI[ /N] are also used22.





72

Judging from these examples it would appear that the standard scaling may

fail for certain nonrenormalized fields.














CHAPTER 7
THE CLASSICAL LIMIT OF ULTRALOCAL MODELS




As given in chapter 5, the classical Hamiltonian of an ultralocal scalar
field may expressed as


Hc = J {J2(x) + 2m2(p2(x) + Vi[p(x)] d3x, (7.1)



where xe 9n-1', r, p denote classical fields, and Vi[q] (= VI[ -p], for simplicity)
the interaction potential. The classical canonical equations of motion
appropriate to (7.1) are


q(x,t) = 8 = (x,t),
5&(x,t)

7r(x,t) = H- cl m2qp(x,t) VJ(p(x,t)], (7.2)
5(p(x,t)


p(x,t) = m2(p(x,t) V[(p(x,t)].


In fact, these formulas do not take into account a vestige of the quantum
theory that really is part of the classical action. Since the additional,
nonclassical, repulsive potential proportional to (field )-2 in (6.10) belongs in








the quantum Hamiltonian density, as h -* 0 the coefficient of this term
vanishes save when 9(x) = 0. To account for this term we formally write
0.p-2(x,t) in the classical Hamiltonian density, and to respect this potential we
need to derive the equations of motion for p(x,t) by means of a scale
transformation20, namely, using 5p(x,t) = SS(x,t)-p(x,t). This leads to a related
but alternative set of classical equations of motion (see Examples 1 and 2
below).
In the following, we will show that the alternative quantum theory of
ultralocal scalar fields described in Chapter 6 does indeed lead to the required
classical limit as h -> 0, thereby giving additional support to such an
alternative, non-Gaussian solution.




7.1 Ultralocal Fields and Associated Coherent States


The proper quantum Hamiltonian of ultralocal fields given by (6.10)
has an evident connection with the classical Hamiltonian. In (6.10), the
subscript 'r' means the fields are renormalized, and instead of fl = (D, we have
n2 = 2r + O(h) r + O(h2) D-2


According to (6.4) and (6.5), the field operator ( and Hamiltonian
operator are given by


D(x) = J dX B (x, X) X B(x, X) (7.3)


B(x, X) = A(x, X) + c(X),








H d3xf dX At(x, X) h(-, X) A(x, X),
= f Xfa


H=J d3xf


with h(1, ) = -
AX


12
2


dX Bt(x, X) h( -, ) B(x, X),
ax


- + + m +VI(X)
2 2X2 2


= p2X2 + (+)h+M 2 + V1(), (7.
2 2 2


( p ih )


Here we write the interaction potential as Vi(X) instead of VI(X), since we
have used VI as the classical interaction potential in (7.1). By using


exp(-Ht) B(x, X) exp( Ht) = exp[ -4 h(--, ,) ] B(x, X) (7.


we can show that the Heisenberg field operator for any renormalized field
power is given by


5)


d,(x, t)=


dX B(x, X)exp[ -t h(--, ) ] X)exp[- it h(--, X) ] B(x, X)
h ax h ax


(7.4)


(7.7)


6)


= dk Bt(x, X) X(t) B(x, X),







where kX(t) = exp[ it h(2-, X) ] Xexp[ it h( 3-, ) ] Even though
h a h ax
[4Dr(x), nr(y)] # ih S(x y), the variables X and px satisfy the standard one-
dimensional canonical commutation relations. Therefore


X= R, h]=p,,


(7.8)


p=- [px, h ] = m2 V('
ih 3


We now introduce canonical coherent states for the ultralocal fields


(7.9)


where, for each ye L2, the unitary transformation operator


U[v] = exp f


dX d3x [V(x, X) A (x, X) V*(x, X)A(x, X)]


(7.10)


It is straightforward to show that


Ut [] A(x, X) U[V] = A(x, X) + V(x, X),


(7.11)


Ut [v] At(x, X) U[V] = At(x, X) + V'(x, X);


evidently the same relations hold for B(x, X) and Bt(x, X) as well.


lv >= u[v] I o>,







7.2. Selection of the Coherent States for Ultralocal Fields and the Classical
Limit


Let us employ a Dirac-like (first quantized) formulation for functions of
X, and in particular let us set


(7.12)


where


( I/2 a(X) ) = -exp [- ((x)]2 + (x)+ i


(x) 1 [ p(x) + i(x) ].
f 2


(7.13)


(7.14)


The expression (7.13) has the form of a conventional canonical coherent state,
g. is an arbitrary phase, and I a(x) ) = exp[a(x) at a*(x) a] I 0 ) = U[a(x)] I 0 ), is
similar to the ordinary coherent state I a) 18,19 except that in the present case
a is a function of x. a, at are the usual annihilation and creation operators,
which are related to X and pX by the relation


a=1 (X+ipx),


.= 2( a + at),


at 1 (X-ip.),


pk= i (at-a).
A/


Note well the appearance of h in these various expressions.
In terms of these expressions we have


(7.15)


y/(X, ) = ( h 1/2 a(X) ),


= < 0 I Ut[V] or(x) U[y] I0 >








= f dX < 01 [Bl(x, %) + v'(x, X)] I [B(x, X) + v(x, X)] I 0 >


= f dX [c(.) + v'*(x, X)] X [c(X) + V(x, X)]


Since c(k) always takes the form c(X) = exp --mx2 yl() it follows that
JXf 2f h


dX c (X) X y(x, X) = 0. Consequently


dX y*(x, X) X xV(x, X)


= lim o ( h"- 1/2 c() X ) I ( hI h 1/2(X)


= limh _o( h-1/2 (X(X) I I h-1/2 (X(x))


= q(x).


(7.16)


Before we calculate , we need to prove several useful
identities. By using


I [h- 1/2 (x)] X [h12 a(x)] = + (x),


UI[h-1/2 a(x)] p U[h" 1/2 a(X)] = p; + 7c(x),


we can calculate any arbitrary monomial in X's and p. 's:


lim ,r o


lim o< WVI r(X) I V > = limh o








( h-1/2 Oa(x) pOn -1/2 (x)


=(0 I(+ q(x))e'... (pX+ (x))en 0)


Because of the prefactor ifh in the definition of X [= (a + at) ] and px
[= i /( at a )], all the terms involving X and px will go to zero when we

take the limit h -4 0. Thus we have


limh 0 ( h- 1/2 a(x) I X01... p I h- 1/2 (x) ) = (pO(X). .*on(x).


Now with h(--, ) = h = 1-- + V(W) we calculate
83 2 4a2


X(t) = exp(.t h) Xexp( it h)
h

= X + Lt [h, X] + ,t)2 h, [h, ,] + -t[h, [h, [h, h1]]]] + i ...


= X + tp. -t2V'(X) [V"() p. + pxV"()] +-..
2 3! 2


(7.17)


(7.18)


After an expectation in the coherent states I h-1/2 Ca(x)) and the limit h -4 0,
we obtain


limh- o ( -1/2 (/(x) I X(t) I h1/2 a(x)


= q(x) + t x(x) -t2Vo[q(x)] -t3 Vo[q(x)] i(x) +...,


(7.19)







where we have used (7.17), and introduced


Vo[(p(x, t)] = limh o V [(p(x, t)].


(7.20)


Notice that Vo((p) no longer depends on h but it does contain a vestige of h
in the term 0-o-2(x,t).


We now prove that the result of (7.19) is just the solution q(x, t) of the
classical canonical equations with the Hamiltonian


Hci = d3x [2(x,t) + Vo[p(x, t)]],



q(x, t) = 7(x, t), ic(x, t) = VOp(x, t)].


(7.21)



(7.22)


With (p(x, 0) = (p(x), i(x, 0) = 7(x), we can use a Taylor series expansion:


p(x, t) = p(x, 0) +t qp(x, 0) +1 2 p(x, 0) + 13 3p(x. 0)
2 3! 43


= ((x) + tn(x, 0) + i(x, 0) + L(x, 0) + ...
2! 3!


=p(x) + tz(x) QVokp(x)] t dVo(p(x,t))It=
2! 3!dt


= q(x) + t 7(x) -Vo0[((x)] 3-Vo[q0(x)] (x) +....
2! 0 3!


(7.23)


In the same way, we can prove that







limh 0 ( h" 1/2 a(x) I pX(t) I h1/2 a(x)) = r(x, t).


Thus


= lim -4o<1 I


dX Bt(x, X) X(t) B(x, X) I V >


= lima -o( h- 1/2 a(x) I (t) I h1/2 a(x)) = (p(x, t);


(7.24)


and evidently
lim _> O0< V I r(x, t) I W > = x(x, t).


(7.25)


According to (7.22), q(x, t) satisfies the classical equation of motion, namely


p(x, t) = Vo[p(x, t)].


(7.26)


Let us present two examples:


Example 1. Ultralocal Pseudo-Free Theory


The quantum Hamiltonian of the ultralocal pseudo-free field is given


H = I d3x dk Bt(x, X) h(3--, ) B(x, X)


limh o< V I r(x, t) I w >







where
2 2
h(,)= p +4 + 2 2+ m(y 1/2)h.



The model function c(X) for the system is determined through (6.7b), and the
result is


c() 1L exp -_1m?2. (7.27)



Expressing the quantum Hamiltonian in terms of the renormalized
field operators, we have


H = f(12(x) + m24(x)d3x



where we have defined the local operator

2 *2 2 r 0
S= (Dr+ y(Y+l)2 (r + my-I .


Obviously this definition means that FIr neither fits into the canonical
framework nor fulfills the commutation relation [4Dr(x), nr(y)] = i hS(x y).


From the above analysis, the classical limit of the field operators in the
coherent states denoted by p(x, t) and 7(x,t) fits into the classical canonical
equation (7.22), with Vi((p) = 0, and q(x, t) is the solution of the classical
pseudo-free field equation of motion (7.26)20:







p(x, t) = 0.p-3(x,t) m2(p(x,t),


where the role of 0.-q3(x,t) is to assure that (p(x, t) > 0 or (p(x, t) < 0, namely,
that (p(x, t) is not allowed to cross through (p(x, t) = 0, This kind of solution can
be secured if we replace the above equation by the scale-covariant equation of
motion3


p(x,t) [p(x, t) + m2p(x, t)] = 0.


For readers interested in the equation of motion of the pseudo-free field
operators, Ref. 21 is recommended.




Example 2. Ultralocal Field with Interaction Potential g(p4 + (p6
2m 2


The quantum Hamiltonian is H = ( d3x dX Bt(x, X) h(, ) B(x, X),
I I
and the expressions of h(-, X) and c(X) for such a model are
ax


h(a ) p ,2 6 + m (- 1/2)+ 3/2) 2
S 2 2 2 2m2 m


c() = exp 1mX2 4} (7.28)
c J xp 2 h 4m "


The Hamiltonian expressed in terms of the renormalized field operators is








H = ((x) +1 [m2 + m3/2) (x) + gf(x) + (x dx,
f 2 In 2 m2


t I 2 1)h 2 1 0
with = Or+ 'yy+l)h2Or+ m(y- 1) Or Since the last two terms in
2
h(-, X) vanish when we take the classical limit, according to (7.20), the

corresponding classical Hamiltonian when h -* 0 is


HcI = 2(x) + 0"p-2(x) + Im22(x) + gp4(x) + 6(x) d3x,
S2 2 2m2 M


The canonical equations are


p(x,t) = 7r(x, t)


ic(x, t) = 0-p-3(x,t) m29(x,t) 4g(p3(x,t) g 5(x,t) (7.29)
m2
and the equation of motion is


q(x, t) = 0.9-3(x,t) m2(p(x,t) 4gp3(x,t) 5(x,t) ,
m2


according to (7.26).


Again the role of 0.p-3(x,t) is to assure that 9(x, t) > 0 or (p(x, t) < 0,
namely, that p(x, t) is not allowed to cross through (p(x, t) = 0, This kind of
solution can be secured if we replace (7.29) by the scale-covariant equation of
motion3








(p(x,t)p(x, t) = m2(p2(x,t) 4g4(x,t) 32 t)


which is satisfied by the classical limit of the
limh 0< V I r(x, t) 1V >, according to Eq. (7.24).


ultralocal field














CHAPTER 8
PATH-INTEGRAL FORMULATION OF ULTRALOCAL
MODELS


In an earlier chapter, we have pointed out that the path-integral of the
quartic self-coupled ultralocal scalar field leads to an trivial (Gaussian) result
when approached by the conventional lattice limit (Chap. 5). However, the
characterization of infinitely divisible distributions allows ultralocal models
alternative operator solutions (Chap. 6) which suggest a new expression of
path-integral with a nonvanishing, nonclassical potential. In this chapter, we
seek to replace the standard lattice formulation by a nonstandard one which
admits nontrivial results. Specifically, we will use these alternative,
nonperturbative operator solutions to construct nontrivial lattice-space path
integrals for nonrenormalizable ultralocal models. As we will see, the
indefinite, nonclassical, singular potential required for the nontriviality in
multi-component case has different effects on distributions compared to the
single-component case, however the essential property of reweighting the

distribution at the origin is similar. The appearance of additional
nonclasssical, singular potentials suggests that we can not always place the
classical Lagrangian or classical Hamiltonian directly into the path-integral
formulation, or in other words, a straightforward canonical quantization of
fields with infinite degrees of freedom does not always apply.







8.1. Operator Solutions of Ultralocal Models


In this section, we will select certain results, which will be used here,
from Chapter 6 on the operator analysis of ultralocal models15, and put them
into completely proper dimensional forms. Following Chapter 6, the N
-component ultralocal field operator can be expressed as


Q(x) = dX Bt(x, ?1) X B(x, 1) (8.1)



Where the operators Bt(x, ,) and B(x, X) are translated Fock operators, defined
through B(x, X) = A(x, X) + c(X), X =X/_ while At(x, X) and A(x, X) denote
conventional irreducible Fock representation operators for which 10 > is the
unique vacuum, A(x,) 10 > = 0. for all xe9Rn-1, X9N. The only
nonvanishing commutator is given by


A(x, X) At(x', = 8(x-x') 8(-X'). (8.2)


Obviously, the operators Bt(x, X) and B(x, X) follow the same commutation
relation (8.2). For the single-component case, (8.1) becomes


Q(x) = J dX Bt(x, X) X B(x, X). (8.3)



Renormalized products follow from the operator product expansion:











which suggests the definition of the renormalized square as


Z = b/8 (0).


(8.4)


QDij(x) = b dX Bt(x, ?1) i, B(x, X) = Z Ii(x) Dj(y),



Here we have introduced an auxiliary constant b
dimensionless. (In the previous chapters, we have
With [.] = dimension (*), we have [b]= [x]'".
renormalized local powers of the field are given by


designed to keep Z
generally chosen b=l).
More generally, the


ri,.., ik(x) = bkj dk Bt(x, ) ,i,.. Xlk B(x, X) =Zk-ii,.. .Di


(8.5)


il,---,ik= 1,2,3,... N and k = 1,2,3,...


Occasionally, we need to use the renormalized negative powers of the field
even though they are not local operators. Take the one-component case for
example:


IdS(x) = d.Bt(x,) (b)kB(x,) = [Z DOi(x)]k, i = 1, 2... N;


(8.6)


it can formally be extended to k < 0, as long as we define


,i (x) =Z2 d B (x,BX-B(x,X) and dBt(x,b)- B(x,1 ) = D(x).
J b2 i b2(







The Hamiltonian of O(N) invariant, N-component ultralocal fields can
be expressed as


H = dx= dX Bt(x, X) h(VX, ?) B(x, X)



= dx) dX At(x, X) h(VX, X) A(x, X). (8.7)



Where h(V;, X) = --h2V2, + V(X) is a self-adjoint operator in the X variable
2b
alone. The connection between these two relations holds when we insist that
h(VX, X) c(X) = 0, and in order for 10 > to be the unique ground state of H, it
follows that c() e L2(9tN). Since [C]2= [x]"n-R N and [X]2= [x]n [h], we have
[c]2 = [x]l-n(+N/2)[h]-N/2. Therefore, by taking into account of the dimensions,,
the model function may be expressed as


c(X) = h(2r-N)/4baJ exp (mb)-2 y(., b, h)4, (8.8)
Xr \ 2h


where a = [ n(1-r+N/2)-1 ] /2(n-1). With a proper account of dimensions, the
self-adjoint base Hamiltonian obeys


h(V, X) = h2V + +2-N)2 + mh(F-N/2) + Im2b%2+ IVi(bX).
2b 2bX2 2 b

(8.9)
Obviously, For the single-component case, (8.8) and (8.9) become


C(X) = h(27-1)/4b[n(3/2"y)-1]/2(n-l) 1 exp b rn2 y(X, b, h)}. (8.10)
I4Y 2 h









h(/, X) = l 2 2 + (Y+1)h2 + mh(y- 1/2) + 1m2bk2 + VI(b) .
2b a2 2bX2 2 b


(8.11)


The Hamiltonian may also be expressed in terms of renormalized
fields as


H= f


-2 ( 2 -
R1H (x) + Im24r(x) + Vi[Dr(x)] dx,
2 2


-2 -2 -2 0
where ir (x) = 0r (x) + F(F+2 N)h2b21 Dr(X) I + mh (2F N)b I dy(x) I ,


(8.12)


and in


terms of bare fields as


[ L 2 2 -
H= n ZH(x) + 1m2ZD (x) + 1-Vi[Z4(x)] dx
2 2 Z I


(8.13)


-2 -2 -- -2 0 -2
where ZH (x) = Zc (x) + F(F+2-N)h2b2Z I (D(x) I + mh (2F-N)bZ1 I Q1(x) I = Fir (x).


For the single-component case, the Hamiltonian becomes


H =



=f


11j2(x) +


m2r2(x) + VI[Or(X)] dx,


(8.14)



(8.15)


(Wz2(x) + Im2ZI2(x) + IVi[ZDr(x)]l dx


where f2 = r+ y(+l)h2b2 + mh(2y- 1)b D? and ZH2 = + 2 y+)h2b2-2Z3
+ mh(2y- 1)b 90Z-1 2.







The Hamiltonian operator (8.7) readily leads to the Heisenberg field
operator which is given by


D(x,t) = eiHt/h D(x) e-iHt/h=f dk Bt(x, X) eiht/he-iht/h B(x, X)



=dX Bt(x, X) X(t) B(x, X) (8.16)



After this partial review, we turn our attention to the path-integral
formulation of ultralocal models.






8.2 Euclidean-Space Path-Integral Formulation of Single-Component
Ultralocal Fields

In this section, we will start from the operator solution presented in
8.1 to derive the Euclidean time lattice-space formulation. Attention is
focused on the time-ordered vacuum to vacuum expectation functional

C[f] = < 0 I Tex dx dt QD(x,t) f(x,t) I 0 >. (8.17)

Insert (8.3), then we have

C[f] = < 0 I Texp dx dt f dX Bt(x, X) f(x,t)X(t) B(x, X) I 0 >,

By using the normal ordering technique, C[f] becomes

< 0 IT: exfp dx f dX Bt(x, X) {I exp[if dt f(x,t).(t)] 1 B(x, X)1 :1 0 >.

Notice that only the part inside { ) depends on time t Therefore we can
move the time-ordering sign T into the bracket, so we find that

C[fM = < 0 I: exp dx dX Bt(x, X) { Texp[4 dt f(x,t)X(t)] -1) B(x, ) : I 0 >

= exp dx dX c(X) I Texp[i dt f(x,t)X(t)] 1} c(X)







= exp dxf dX c(X) { Texp[f dt ( f(x,t)X h(W/3A, )) ] 1} c()]


= lima-.o exp{ af dX c8(X) { Texp[l


dt (j fk(x,t)X h(a/3k, X) ) ] 1} CS(1 )


Here in the last step we have employed a lattice-space regularization in which
a is the space volume. Let

c8(X) =h(2y'1)/4 ba (k2+ 2)-/2exp (_- 2m(X2+ 2) y(X, 8),


where 8 is the function of a, ac = [n(3/2-y)-1]/2(n-1), and choose 8 so that
af dX c (X) = 1. Then we find out that 8 = -h (a b2aF)1/(2t'), F = f dx (x2+1)-
J LI


for y> 1/2; and 8 f bn/[2(1-n)] exp[-1/(2ab)] for y= 1/2.


Therefore 8 -> 0, as


a--0, lima.o cS(.) = c(k), and


C[f] = lima-olk exp af dX c8(?) { Texp[
k -


dt ( ifk(x,t). hs(a/3A, %) ) ] 1) C(.)1


= lima-o I af
k f


d c8g() { Texp[


dt -fk(t). dt hs(a/3D, X) ] } c8(k)


where al/2c8(X) represents the normalized ground state of h8s(/ X, X), i.e.,
hs(/3A,, X) C8(?) = 0. To the required accuracy in 8, we have


hs(/3a,, X) = +-l--2 + Y+52 h2+ m(y-1 /2)h + 1m2bX2+ -Vi(bX)
2b a2 2b( 2)2 2 b
ax2 2b(X +68)