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BICOHERENT STATES, PATH INTEGRALS, AND SYSTEMS WITH CONSTRAINTS
By GAJENDRA TULSIAN 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 1995 To the dawn of the day when man on earth, as in heaven, will be judged only by the nobility of his deeds and nothing else. ACKNOWLEDGMENTS I would like to thank everybody who has helped me reach this point in my life. Foremost, I would like to thank Professor John R. Klauder, my advisor and mentor for the past three years. I thank him for his extensive help during my thesis work. He always had answers to all my questions and was extremely fair in professional matters. My only regret is that I am not enough of a scholar to have taken advantage of his vast knowledge in all areas of physics. I am grateful for the courses in physics and mathematics that he taught in the most lucid manner. As a teacher, I find myself copying his teaching style. I would also like to thank Professor Bernard F. Whiting for many enlightening discussions on the subject of dynamical constraints. Thanks are due to Professor S. V. Shabanov for introducing me to two of the models studied in this thesis. I thank Professors P. Ramond, J. Dufty and C. Stark for serving on my committee. On a personal note my greatest debt is to Jocelyn, my dear wife, for her incredible patience in the past six months. She took on more than her share of chores and put up with a unreasonably difficult man at the same time. I would like to thank my Mom and Dad for every thing they have done for me ... "Why did you spend all this time getting a Ph.D. if you can't find a job Thanks to all members of my family in India. I missed them the most when I was blue. Thanks go to all who were my friends in Gainesville: Thanu, Raju, P.P., Gary, Phil(l) (who shared with me many a stories, sad and happy, over pitchers of beer), George and Lori, Mike, Erica, Bill Shi, Thor (for all those scrabble games when I should really have been studying), Jose (even though he cheated at golf; how else could he have beaten me so often ?), Mark (who aspires to be as good a golfer as I), Jim, Sean, Andy, Jae Wan, Don, Sam, Beth, and Steve. Thanks to Dave, an unsolicited critic and a special friend. i i in Finally, I would like to add that even though I am extremely glad this 'leg of my journey' is coming to an end, if I had the choice, I would do it all over again. I have enjoyed the endeavour immensely. iv TABLE OF CONTENTS ACKNOWLEDGMENTS.................................iii ABSTRACT.........................................vii CHAPTERS 1. INTRODUCTION.................................... 1 2. CONSTRAINTS AND PATH INTEGRALS.....................12 2.1 Systems with Firstclass Constraint ......................12 2.1.1 Propagator.................................14 2.2 Systems with Secondclass Constraints....................15 2.3 Extended Coherent States............................17 2.3.1 Basic Example of ECS..........................17 2.3.2 Classical Limit..............................19 3. BICOHERENT STATES AND FIRSTCLASS CONSTRAINTS.........22 3.1 Toy Model.....................................22 3.2 The Path Integral.................................26 3.2.1 Projection Operator............................27 3.2.2 Propagator.................................29 3.2.3 Classical Limit..............................32 3.2.4 Constraint Hypersurface.........................36 3.3 The Quartic Potential ..............................37 3.4 The Measure...................................39 4. BICOHERENT STATES AND SECONDCLASS CONSTRAINTS.......42 4.1 Toy Model 1 ...................................42 4.1.1 Path Integral................................44 4.1.2 Classical Limit..............................47 4.1.3 Constraint Hypersurface.........................49 4.2 Toy Model 2 ...................................49 4.2.1 Projection Operator............................51 4.2.2 Path Integral................................52 4.2.3 Classical Limit..............................53 4.2.4 Constraint Hypersurface.........................54 5. E(2) COHERENT STATES...............................56 5.1 E(2) Coherent States and their Propagators..................59 5.1.1 E(2) Coherent State Representative of Functions on L2(S1,d6) 59 5.1.2 Surface Constant Fiducial Vectors...................62 5.1.3 Propagators................................64 5.2 The Universal Propagator............................65 5.2.1 Vanishing Hamiltonian..........................67 5.2.2 Linear Hamiltonian............................67 5.2.3 A Quadratic Hamiltonian........................68 V 5.3 Propagation with the Universal Propagator..................69 5.4 Classical Limit..................................71 5.4.1 Classical Solutions............................74 6. DISCUSSION AND CONCLUSIONS........................76 A. BICOHERENT STATES AND PATH INTEGRALS................79 B. THE (o,6,c) MODEL..................................82 BIBLIOGRAPHY .....................................83 BIOGRAPHICAL SKETCH ...............................85 vi 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 BICOHERENT STATES, PATH INTEGRALS, AND SYSTEMS WITH CONSTRAINTS By Gajendra Tulsian December, 1995 Chairman: Professor John R. Klauder Major Department: Physics In this work we discuss how dynamically constrained systems are handled in a path integral framework. We consider models with a finite number of degrees of freedom and with constraints that are either first class or second class. We first briefly review the techniques developed in the past to handle such systems. Then we describe in detail the method developed here which is summarized as follows: To account for the constraints we construct an appropriate projection operator. We use this projection operator, rather than the resolution of unity, at every time slice in building a path integral representation of the propagator. The derivation of the projection operator leads to the introduction of Bicoherent States and is an integral over properly weighted independent coherentstate bras and kets. The path integral representation of the propagator, built using bicoherent states, leads to a complex phasespace action. This complex action has twice as many 'labels' as the standard action, the imaginary part of which reduces to a surface term on the classical trajectories. Also, on the classical trajectories the real part of the action reduces to just the standard action. The projection operator leads to the correct measure in the path integral representation of the propagator. The measure which is path dependent is 'modulated' by the imaginary part of the action. Vll CHAPTER 1 INTRODUCTION Modern physical theories of fundamental significance tend to be gauge theories and such theories have been the focus of extensive studies for the better part of this century. For instance, Quantum electrodynamics, a theory with an Abelian gauge group has been investigated for over fifty years now [14]. Also, YangMills theories, models with nonAbelian gauge groups, have been at the center stage in the recent past [57]. It is a well known fact that dynamical systems with gauge symmetries have unphysical degrees of freedom. In the Hamiltonian formalism, their dynamics are described by constrained Hamiltonians. Hamiltonian systems with constraints are the main concern of the present thesis. In this brief study we shall limit ourselves to systems with a finite number of degrees of freedom so as to not lose sight of the main issues involved. In describing physical theories, one usually starts with an action principle since it is easy to build Lagrangians with desired symmetries, such as relativistic invariance. Then, as a first step toward quantization one passes to the Hamiltonian formalism at which stage one promotes classical variables to quantum operators. It must be noted though that there exist methods of quantization where one passes directly from the Lagrangian to the quantum theory, hence bypassing the Hamiltonian formalism. But these schemes work only for Lagrangians quadratic in the velocities. Moreover, for their verification one has to go back to the Hamiltonian formalism. In the present study we shall take the Hamiltonian route to quantization. We would now like to introduce some of the terminology used in the field of constraints. Constraints are classified depending on the manner in which they are expressed. If the constraints are expressed as equations connecting the Cartesian coordinates of the 1 2 system, they are Holonomic constraints. Nonholonomic constraints are either of two types: 1. They are expressed as inequalities for certain functions of the coordinates. 2. They are linear nonintegrable relations among the differentials of these coordinates. An example of a system having Holonomic constraints is a rigid body where the constraints are given by equations of the form (r, Tjf at = 0. (1.1) In chapter 5 we study the (a, 6, c) model which is a system with a Holonomic constraint. An example of a system with a nonholonomic constraint of type 1 is particles confined inside a box. Such constraints, which tend to appear in the macroscopic world, can be handled by introducing forces of constraints that prevent the inequality from being violated [8,9]. We are not interested in such systems here and shall not mention them anymore. Nonholonomic constraints of type 2 are called dynamical constraints and systems having such constraints will be the main focus of our study. Dynamically constrained systems were systematically studied by Dirac and we now briefly review the analysis of such systems [10]. The starting point of our discussion will be the action integral S J L(q,q)dt. (1.2) The classical trajectories of the system are those that make the action stationary and are determined by the EulerLagrange equations d ,dL s dL n n KT * fc") ~ a= n = l,2,..JV. (1.3) dt dqn dqn Here qn and qn are the generalized coordinates and velocities respectively and N is the number of degrees of freedom. The equations of motion (3) are second order differential equations and the dynamical information contained in each of them can 3 be equivalently stated with two independent first order equations. Dynamics in the Hamiltonian formulation is described by first order equations. To go over to the Hamiltonian formalism, we first introduce the canonically conjugate momentum variable defined by dL Pn = dqn (1.4) The p's as defined, above and the formalism. From the momenta defined in (4) one can construct a matrix called the Hessian, which is defined below. One finds that if the determinant of the Hessian vanishes, i.e. if oqnvqm (1.5) not all the pn's are independent. The model then is said to have constraints, i.e. there exist certain relations of the type These relations are called Primary Constraints. The rank of the Hessian is equal to N M if the primary constraints (6) are all independent. The primary constraints, which are identities when the definition pn (dL/dqn) is substituted into (6), must satisfy certain regularity conditions. These conditions can be stated as follows: In any variation for which Sq and 8p are O(e) the constraints must be such that S(j>m{q,p) are also of O(e). For example, of the following three ways of stating the same constraint, P = 0, P2 = 0, ^=0, (1.7) only the first one is acceptable [11,12]. We now introduce the canonical Hamiltonian, H = pnqn L. Here and in what follows, repeated indices are summed over unless otherwise stated. Notice that the 4 Hamiltonian defined in this way is not unique since we may add to it any linear combination of the constraints. Next, using pn = (dL/dqn) one can easily see that SH = qn6pn (dL/dqn)8qn. But the 6q's and the <5p's are not independent in this equation because of the constraint m{q,p) 0 relating them. Now, from the general methods of the calculus of variations for systems with constraints one finds dH + um OPn qn = dpn dL dH dqn = Pn dqn m Q undetermined coefficients. These equation are the Hamilton's equations of motion for a constrained system. These equations of motion can be written in a compact and elegant form using the Poisson bracket, which for two arbitrary functions f(q,p) and g(q,p) is defined by I ft \ i \\ df dg df dg n Q^ {f(q,ph9{q,p)} = kj,k ^ (1.9) Oqn OPn OPn Oqn Thus, using the Poisson bracket notation we can write the time development of an arbitrary function g(q,p) as ./ v dg dg . g{q,p) = ^Qn + JTPn, dqn dpn (1.10) = {g,H} + um{g,(j)m}. At this point we would like to introduce the idea of weak equality (). We accept as a rule that all Poisson brackets must be worked out before the constraint equations are imposed [10]. To remind us of this rule we write the constraint equations using a slightly different equality sign as Such equations are called weak equations to distinguish them from the usual or 'strong equation.' We can now write the evolutions equations for an arbitrary dynamical variable 5 concisely, using the idea of weak equality, as g(q,p)*{g,HT}. (1.12) In the expression above we have introduced, H? = H + um Let us now examine the consequences of the Hamilton equations of motion. We want the constraints to be zero at all times, i.e. we want The above equation can lead to one of several possibilities: It can reduce to an equation, independent of the u's, involving only the dynamical variables of the form X(,P) = 0. (1.14) If this equation is independent of the primary constraints then we have a new constraint in the theory and such constraints are called secondary constraints. They are referred to as secondary because the equations of motion have been used to arrive at them. Continuing in the same vein, we require that the new constraint hold at all times and so X {x,H} + um{X,m} = 0. (1.15) The above equation can lead to more constraints. In this manner we could obtain all constraints in the theory. We shall use the following notation to represent the secondary constraints Next, in the case where equations (13) and (15) are not independent of the um's these are conditions on the coefficients um. Thus, the coefficient um's are such that they satisfy the equation {^if} + wro{^^m}0. (1.17) 6 In the expression above the /s include both the primary and the secondary constraints, i.e. j = 1, ...,M + K, whereas the m's denote only the primary constraints. Equation (17) are nonhomogeneous linear equations in the unknowns um's and the general solution to these equation are um = Um{q,p) + VaVam(q,P)i (1.18) where Um(q,p) are particular solutions of (17) and Vam(q,p) are solutions of the corresponding homogeneous equation Vm{ HT = H + Um where We would now like to introduce two new terms. A function, f(q,p) is said to be firstclass if its Poisson bracket is weakly zero with all the constraints {/(,P),^(9,P)}0. (1.20) If the above is not true then f(q,p) is secondclass. It can be easily checked that our total Hamiltonian, fly = H + Um(})m +va Let us now examine the transformations generated by the firstclass constraints. Going back to our total Hamiltonian, recall that the coefficients va are totally arbitrary. So, for a general dynamical variable f{q,p) with initial value fo, its value at an infinitesimal time 6t later is f(6t) =f0+ 6t{f, H + Um i.e. there is an arbitrary function of time in the evolution of f(q,p) since it contains the coefficient va. Had we chosen a different coefficient such as v' we would have arrived 7 at a different value for f(St). The difference in the value of f(St) corresponding to the different coefficients, is Af(6t) = 8t(vav'a){f, Since va and v'a are arbitrary coefficients, in our analysis, the two different values f(St) and f'(St) for the dynamical variable f(q,p) must correspond to the same physical state. Thus, we conclude that firstclass primary constraints are generating functions of contact transformations that do not affect the physical state of the system. Such transformations are also called gauge transformations. From the discussion so far we have learned that the important classification of constraints is the one where they are classified as either firstclass or secondclass constraints. We also saw that the primary firstclass constraints are generators of gauge transformations that do not alter the physical state of the system. Thus, from the standpoint of dynamics the primary and secondary constraints must be treated on equal footing and therefore one postulates that all firstclass constraints, both primary and secondary, are generators of gauge transformations. Hence, we introduce an Extended Hamiltonian which includes all firstclass constraints each accompanied by an arbitrary coefficient HE = H + Um(f)m + ua We now address the question of quantizing a constrained Hamiltonian theory. First consider the case where there are only firstclass constraints present. We promote the dynamical variables, the p's and the g's, to operators. Then, we replace the 8 Poisson brackets of dynamical variables with commutation relations for the corresponding operators, i.e. { } ? (l/ih)[ ]. The Schrodinger equation is given by dxj) zh = H+, (1.24) with the following supplementary conditions on the wave functions (1.25) The wave functions that satisfy the above supplementary conditions are the physical states of the system. One finds that two additional conditions are needed for the quantization to be successful. Thus, for the supplementary conditions to be consistent we must have [ The commutator of the constraints acting on the wave functions will vanish if it can be written as [(f)j, 4>k] Cjkl*t>b We know that in the classical theory the <^>'s are all firstclass and so their Poisson brackets amongst themselves are linear combinations of some or all of the 's, i.e. { commute with the <^>'s and so the condition (26) need not be satisfied. If such ordering problems occur we might be able to achieve only a first approximation to the quantum theory, with quantities of order 0(h ) neglected in trying to get the c's to be on the left of 's. It turns out a second condition is required of the constraints as operators. The second condition arises as follows: As the wave functions which satisfy the supplementary condition (25) evolve in time according to the Schrodingers equation we want them to respect this condition at all times and this leads to the following requirements of the constraints to,#]0 = O. (127) 9 The equation above implies that [ Let us now consider the case where secondclass constraints are present. Recall that a constraint is secondclass when its Poisson bracket with at least one of the other constraints is weakly nonvanishing. We shall denote the secondclass constraints by Xs(<2SP)One finds that when the secondclass constraints are such that no linear combination of them can be converted to a firstclass constraint then the number of secondclass constraints is even [10]. The simplest example of two secondclass constraints is 9i 0, pi w 0. (1.28) In the above example it is quite obvious that the degree of freedom labeled by the variables ( matrix {xs>Xr}, i.e. evaluate the Poisson brackets and then set all the constraints equal to zero. It can be shown that the determinant of this matrix is nonvanishing and hence is invertible. Let the inverse of the matrix {x$>Xr} be C, i.e. Cr5{xs,X/} ~ $rl The Dirac bracket of two quantities (q,p) and rj(q,p) is defined as {t,V}DB = {t,fl) {t,Xs}Csr{Xr,ri}. (1.29) 10 The Dirac brackets satisfy all the identities satisfied by the Poisson brackets including the Jacobi identity [10]. Two things are noteworthy about the Dirac brackets: First, the equations of motion are as valid for the Dirac Brackets as for the original Poisson brackets, since for any f{q,p) {fi Ht}dB = {f,HT} {f,Xs}Car{Xr,HT} (1.30) ~{/,#t}. The above equation is true because {xt,Ht} weakly vanishes. Thus, we can write for the evolution of an arbitrary dynamical variable / & {f^H^DB Second, the Dirac bracket of any function (q,p) of the coordinates and the momenta vanishes with all of the x's since, {,Xl}DB = U,X/} {ZiXr}Crs{Xs,Xl} (1.31) = {&Xl}{*,Xr}*rl = 0. Thus, if we use Dirac brackets in dynamical equations we can set Xs(q,p) = 0 (1.32) as strong equations. Hence, if we modify the classical theory by replacing the Poisson brackets with Dirac brackets, the passage to quantum theory is achieved by making the commutation relation correspond to the Dirac brackets. Then, equation (32) is taken to be an equation between the operators in the quantum theory. In practice, it must be noted, operator solutions to (32) might not be possible. Thus far we have briefly introduced the language of constrained dynamics and described Dirac's prescription for operator quantization of such systems. Operator quantization is inelegant and not readily amenable to approximation schemes when one studies field theories. Hence, we would like to study the problem of handling theories with constraints in a path integral framework. The rest of this study is devoted to this problem. We have organized the thesis as follows: In chapter 2, we briefly review the existing 11 path integral techniques for handling systems with first and secondclass constraints. In chapter 3, we describe a new technique for constructing path integral representations of canonical coherentstate propagators using Bicoherent States for systems with firstclass constraints. Chapter 4 deals with the question of path integrals for systems with secondclass constraints. Here we extend to such systems the techniques of chapter 3. In chapter 5, we construct the "Universal propagator" for a particle confined to a circular path. This is the simplest system with an holonomic constraint. We show in appendix B that this system can be considered at as a model with secondclass constraints. CHAPTER 2 CONSTRAINTS AND PATH INTEGRALS In this chapter we briefly review the existing techniques for handling systems with constraints in a path integral framework. We will confine our attention to systems with a finite number of degrees of freedom as already noted. In section 2.1 we discuss Faddeev's scheme of quantization of systems with firstclass constraints [13,14]. In section 2.2 we summarize the generalization of Faddeev's results to the case when secondclass constraints are present. This generalization is due to Senjanovich [15]. In section 2.3 we review the methods of extended coherent states developed by Klauder and Whiting [16]. Extended coherent states introduce auxiliary variables which lead to a resolution of unity with a nonunique measure. These states, when used in the construction of path integral representations of propagators, lead to an action containing 'extra' pathspace variables which engender constraints in the classical limit. 2.1 Systems with Firstclass Constraint We now discuss Faddeev's quantization of systems with firstclass constraints. Consider a mechanical system of TV degrees of freedom described by the Lagrangian L = L(q,q). (2.1) If the above Lagrangian is to describe a constrained system, it must be singular, i.e., the determinant of its Hessian must vanish. We denote the constraints by {H, 13 Thus the constraint hypersurface M in the phase space T is a surface of dimension 2N j. It was noted in the introduction that all firstclass constraints generate gauge transformations. Now, only those functions whose equations of motion do not contain any arbitrary functions of time on the constraint hypersurface M are defined as observables Recall, the equation of motion of a arbitrary function f{q,p) is ^ = {/,#}+ (2.4) where the coefficients v; are functions of time. For f(q,p) to be an observable, the second term on the right hand side of the above equation must vanish on the constraint surface, i.e., we require {/> j} = djk The above equations can be viewed as a set of j first order differential equations on the manifold M with (3) serving as integrability conditions. Since f{q,p) satisfies a set of j first order differential equations, it is completely determined by its values on a submanifold of dimension 2N j j = 2(N j). It is convenient to take as such a manifold a surface T*, defined by the constraint equations (2) and j additional conditions xMiP) = i J = 1,2,..., J. (2.6) The above conditions are the gauge fixing conditions and they must be such that det{x;,0*} 7^0. (2.7) This allows the manifold T* to be an initial surface for the first order differential equations (5). It is also convenient to suppose that the gauge conditions mutually commute, i.e., {XjiXk} = o. (2.8) 14 If the gauge conditions meet the above requirements we can introduce a canonical transformation in T, such that in the new set of coordinates the gauge conditions take on the simple form Xj(q,p) = Pj, j = l,2,...,J. (2.9) The Pj's are a subset of the new canonical momentas. Equation (7) in the new coordinates becomes a f \\d(^k II u n det 11^0 (2.10) This equation can be solved for Qj. Thus, the surface T* is given by the equations R=0, Qj = Qj(g*,p*), (2.11) where the g*'s and p*'s are independent canonical variables on T 2.1.1 Propagator We now study path integral representations of the propagator. The basic assertion is that the matrix element of the evolution operator is given by '/ \ptqiH(q,p)]dt 0 T[dfi(q(t)p(t)) (2.12) t where the measure at each time slice in the integral above is given by 1 = ^2^yv^r To prove this assertion we transform to the coordinates {Qj,q*,Pj,p*) in the integral (12). Thus, with these coordinates the measure in (12) becomes 1 86 N m*)=(2^0^=^ 11^7 1 n )^c<^>>) ^c^)^^c^) (2.14) 15 which can be rewritten as J NJ j *j * dfi{t) = J] 6{Pj)6{Qj QjtfrfWQjdPj II T^TW^J (215) Now one can easily perform the (Qj,Pj) integrals, thanks to the delta functions. Thus, the propagator now becomes if[Y,PtttB*( o dq" Mdp'(t) n n ' 2tt proving the assertion. We mention the two main problems with the above arguments. First, adequate gauge conditions in some cases cannot be found. Although it might be possible to choose satisfactory gauge conditions locally, global gauge conditions are, more often than not, hard to obtain. This phenomenon is known as Gribov obstruction [7]. Second, the canonical transformations carried out in the proof of the assertion cannot be justified within the path integral formulation [17]. These are the two main reasons to study alternate schemes to handle constrained systems in a path integral framework. We now discuss the quantization of systems that have both firstclass and secondclass constraints. 2.2 Systems with Secondclass Constraints In this section we present Senjanovic's formalism of quantization of systems where secondclass constraints are present [15]. This is a generalization of Faddeev's results described in the previous section. Consider a problem with the following constraints ^a(<7,p) = 0, 6b(q,p) = 0, a=l,2,...,m, 6 = 1,2, ...,2n. (2.17) The constraints a's which are firstclass, and the secondclass 0&'s, determine the constraint manifold M. We assume the constraints are independent and irreducible in 16 the sense that an arbitrary function f(q,p) which vanishes on the constraint surface can be expressed as f(q,P) = Ca{q,p)a{q,p) + db{q,p)6b{q,p). (2.18) Since the a's are firstclass and the 6b's secondclass we have the following relations amongst the constraints {a, { (2.19) Also, the determinant of the Poisson brackets of the secondclass constraints amongst themselves on the surface M is nonvanishing, i.e., det\\{ea,eh}\\M^0. (2.20) For the system under consideration, since there are m firstclass constraints, we will have m gauge fixing conditions, one for each gauge generator, for reasons noted in the previous section. Thus, an observable f{q,p) is uniquely determined by its value on a submanifold r* of dimension (2/V m 2n) m 2(N m n). Hence, the submanifold T* is specified by the constraints (17) together with the gauge conditions Xa{q,p) = 0, a = 1,2, ...,m. (2.21) The gauge conditions must be such that they satisfy the following two equations {Xa,X6} = 0, det{xa,to}^0. (2.22) With the above gauge conditions the expression for the matrix element of the evolution operator is given by 0 Y[dn(q(t)p(t)) (2.23) t 17 where the integration measure at each time slice in the expression above is given by m 2n N = det {Xa, ^}[det {^c,^}]^ II ^(X)^(0a) ^(^c) a=l c=l i=l (2.24) We will not discuss the proof of this assertion here, which the interested reader can find in Senjanovich [15]. It is noteworthy though that the proof, as in the case of firstclass constraints, relies on the generally unjustifiable assumption that one can perform canonical transformations within a path integral. 2.3 Extended Coherent States In this section we briefly discuss the use of extended coherent states (ECS) in the construction of path integrals developed by Klauder and Whiting [16]. Traditionally, coherent states have been constructed using a minimum number of quantum operators and classical variables appropriate for the problem under consideration. In constructing ECS one uses auxiliary quantum operators and associated classical variables. Such states, when used in the construction of path integrals, lead to extra pathspace variables in the action. We will consider a simple model and using ECS show how the auxiliary variables could lead to the emergence of classical constraints. This section closely follows the contents of the paper by Klauder and Whiting mentioned above. 2.3.1 Basic Example of ECS The conventional canonical coherent states based on an irreducible, selfadjoint representation of an Heisenberg pair of operators Q and P that satisfy [Q,P] = i, are given by \p,q) = etiPelrt\r1) = U(p,q)\T1). (2.25) 18 Here \tj) is an arbitrary, normalized fiducial vector and (p,q) G R2. For any \rj) these states satisfy 1 / = J U(p,q)\rj)(v\UHP,q)^ (2.26) a resolution of unity in terms of equally weighted projection operators these states make. If H = H(P,Q) is the Hamiltonian for some quantum mechanical system, then the propagator admits a formal path integral expression in the form {p",q"\einT\p',q') = /e^'*1^^ (2.27) where (p(0), This prescription suggests the interpretation of / = J[i{p,q\^\p,q) (p,q\H(P,Q)\p,q)]dt (2.28) as a classical action for the system [18]. We now introduce an example of an extended coherent state. Let D = (QP + PQ)/2 denote the dilation generator that satisfies the commutation relations [Q,^] = iQ and [P, D] = iP. Together with the commutator [Q, P] = z, it follows that (P,Q,D) make a three parameter Lie algebra. We now define a unitary operator Wr) = e1^^, r>0, (2.29) and let ) = V(r)r/). One notices the following 1 = / U(p,q)m\U\p,q)d^ = I \p,q,r)(p,q,r\^ (2.30) z7t / z7t is true for all r. The states p, q,r) = U(p, q)V(r)\q) are an example of extended coherent states. Now if a(r) is an arbitrary measure on r such that J da(r) = 1, then it is obvious that != / p>gr)(pgH P *a T = / \p,q,r)(p,q,r\d/i(p,q,r). (2.31) 19 This nonuniqueness of the measure in the resolution of unity is a salient feature of extended coherent states and is exploited advantageously in path integrals. We now study the propagator in the ECS representation. Proceeding in a manner similar to the case of canonical coherent states, one can write a formal path integral expression for the propagator in the ECS representation as (p'\ q\ r"\emT\p', q\ r') = f el JKm.HIi^^ ^ r) (2.32) where the choice of the nonunique measure /i(p,#,r) can depend on time. From the path integral point of view the expression /d [i{p, of which variables are fundamental and which auxiliary; this division is embodied in the choice of the measure fi(p,q,r). 2.3.2 Classical Limit We will now consider a few simple examples to illustrate how constraints can arise in going to the classical limit from the ECS path integral. Using the notation (?) = (t/*t/), we define the following quantities P(r) = V*(r)PV(r), Q(r) = V*(r)QV(r), p* = p+ (P(r)), (2.34) q* = q+ (Q(r)), AQ = Q (Q), AP = P (P). Expressed in terms of these variables the action in (33) can be written as / = /[?V (Q(r))p ~ (D)(r/r) H(p\q\r)}dt (2.35) where in the equation above H(p*,q*,r) = (p,q,r\H(P,Q)\p,q,r). We will now study the equations of motions obtained from this action. 20 We notice that the term containing the time derivative of r in the action above is a total derivative and can be discarded as far as the equations of motion are concerned. Thus, for the Harmonic oscillator where 2H =P2 + Q2, we find 2H = p*2 + r2((AP)2) + (l/r2)((AQ)2). It is clear that the stationary variation of the action with respect to r leads to the constraint ft H = r((AP)2) (l/r3)((AQ)2) = 0. (2.36) So, even though the quantum corrections ((AP)2) and ((AQ)2) are of order ft, the solution for r obtained from 4 ((AP)2) ((AQ)2) (2.37) is effectively independent of ft and could be required to hold even in the limit ft > 0. Next, consider the case of the quartic potential, 2H P2 + Q4. The 'classical Hamiltonian' is given by 2H = p*2 + 9*4 + r2((AP)2) + ^H(AQ)2) + %((AQ)3) + \({AQ)4). (2.38) Again, extremisation of the action leads to the constraint (dH/dr) = 0, with the solution r = r(q*). The auxiliary variable r and the shifted canonical variable q* are inextricably intercoupled. Elimination of this constraint will not change the classical dynamics, but will clearly break contact with the quantum theory because there is no reason to suppose that any measure exists which can preserve the resolution of unity and also be compatible with the constraint at the same time. In the discussion above we went 'against the grain' by starting with a quantum theory and then obtaining a classical limit from it. We saw from the example considered that the choice of the coherent state extension can strongly effect what becomes the classical theory. It is evident though that the relation between the resultant classical theory and 21 the initial quantum theory is far from transparent. Also, the elimination of the classical constraint variable can effect the transition back to the quantum theory. In the next two chapters we discuss how Bicoherent States can be used to handle systems with constraints. We will consider models with firstclass constraints and those with secondclass constraints. For clarity of the formalism we will consider systems with a finite number of degrees of freedom as already noted. CHAPTER 3 BICOHERENT STATES AND FIRSTCLASS CONSTRAINTS In this chapter we describe a new technique for handling theories with firstclass constraints in a path integral framework. Recall that firstclass constraints are those whose Poisson brackets with all other constraints vanish on the constraint surface. We consider a toy model which has been studied before [19,20]. The model has a primary and a secondary constraint, both firstclass. We quantize the model and following Dirac identify the physical subspace [10]. Then, starting with an orthonormal basis in this subspace we build a projection operator which we use in the construction of a coherentstate pathintegral representation of the propagator for our model. The projection operator, which leads to the introduction of the bicoherent states, gives us the correct measure for the path integral and also leads to the desired classical limit. This chapter is organized as follows: In section 3.1, we introduce the classical and quantum description of the model and identify its physical subspace. In section 3.2, which contains the bulk of the formalism developed here, we construct the projection operator and use it to evaluate the propagator for the case of a quadratic potential. We then obtain the classical limit from this propagator. In section 3.3, we study the propagator for a quartic potential. Section 3.4 discusses the measure obtained for the path integral representation of the propagator. In appendix A, we summarize the main features of path integrals constructed using bicoherent states. 3.1 Toy Model We consider the dynamical system described by the Lagrangian L(x,x,y,y) = (x yTxf V(x) (3.1) 22 23 where x= (x1,X2), a twodimensional vector, and y are dynamical variables. Also, T = iri is a 2 x 2 matrix where 72 is a Pauli matrix. As a first step toward quantization we go over to the Hamiltonian formalism. The canonically conjugate momenta to the coordinates are p = = x yTx ox tt = = 0 dy (3.2) and the canonical Hamiltonian is h = \p2 + ^(x) + VP?* 0.3) Therefore, we have a mechanical system with one primary constraint n 0. We want our primary constraint to hold at all times, so, we require jr = {tt, #} = pTx = a = 0 (3.4) i.e., we have a secondary constraint, a = 0. We note that class since {cr, 7r} 0. Also, there are no further constraints in the problem because a {ct,H} = 0. Thus, our model which has two firstclass constraints has only one physical degree of freedom which can be identified as follows: Performing the canonical transformation (x,p) > {r,0,pr,po) where (r, 6) are polar coordinates and (pr,Pe) are momenta conjugate to them respectively, we find that (r,pr) are gauge invariant and can be taken as the physical variables. In the case where one is interested in the most general physically permissible motion one should allow for an arbitrary gauge transformation to be performed while the system is dynamically evolving in time. Hence, we add to our Hamiltonian the two firstclass constraints multiplied by their corresponding Lagrange multipliers and obtain the extended 24 Hamiltonian H = ^p2 + V(x) + ypTx + upTx + i;tt, (3.5) where u and v are Lagrange multipliers [12]. We use the extended Hamiltonian in the construction of the propagator and henceforth will refer to it as just the Hamiltonian unless otherwise specified. The transition to the quantum description of the system is made by promoting the dynamical variables x, p,j/,7r and the Lagrange multipliers u,v to operators. We will use the same symbol to represent the classical variables and their corresponding quantum operators since the quantity being referred to will be clear from the context. At present, we consider the harmonic oscillator potential V(x) = x2, and so, our Hamiltonian is H = ip2 + ^x2 + ypTx + upTx + vtt. (3.6) It is useful to express our quantum mechanical problem in a secondquantized represen tation. Thus, using creation and annihilation operators we define Xj~ 72 P3~ y/2i ' J = 1,2 (as + at) (a3a!) (04 + 1!) (a5 + at) y/2 y/2i y/2 x/2 (3.7) and adopt the normalordered Hamiltonian H:H: given by l, t ,t .(4 + 3)/ t t \ (a4 + a4)/ f t \ v2 v2 (3.8) >/2 >/2i where [ai,aj] = Sij, [ai,a}] 0 and i,j = 1,2,3,4,5. An orthonormal basis for the Hilbert space under consideration is given by the oscillator occupation number states (a\)1 (ahm (a\)n (al)r (a\)>, 4 v/! vm! vn! yr! vs! 25 Given the quantization prescription and the Hilbert space indicated above, we would now like to identify the physical subspace which respects the constraints as quantum operators. First, consider the subspace in which the operators (x, p) live, which in the occupation number representation corresponds to the space spanned by the vectors 4^4&0> = /,m); (310) in this subspace, the physical states \) are singled out by the condition a\ Thus, the physical basis in this subspace is obtained by applying to the vacuum state 0) polynomials in and that commute with c i(ajct2 #2&i) The only such independent invariant polynomial is (a\2 + a]>2) i.e., Ma}2 + <42)] = 0. (312) Hence, an orthonormal basis in the physical subspace under consideration is given by (aj2 + an)k \4>k) = K 1 ZJ 0)i k = 0,l,2,... (3.13) Notice that these states are also the energy eigenstates for the canonical Hamiltonian with a quadratic potential in (3) with eigenvalues 2,4,6,... in appropriate units [20]. Second, the physical states in the subspace where (y,7r) operate is similarly determined by the condition 7c\ invariant under the two gauge transformations are , v (af + 42)*. I0jb)= 2^! M'0) (3'14) These states form an orthonormal basis for the physical space which is a subspace of the Hilbert space spanned by the vectors in (9). 26 3.2 The Path Integral The principal object represented by the path integral is the propagator. Before proceeding to the construction of the propagator for our toy model with two firstclass constraints, we will briefly recall the construction of the propagator in the canonical coherentstate representation for systems with a single degree of freedom and without constraints. In the canonical coherentstate representation the propagator is given by (z"\eirH\z') = /(z"e^^)(.iVe^^_1)...(z1ewz') {[ fH i][\{pqqp)H{p,q))dt T>pVq, 71 (3.15) where H(p,q) (p,q\H\p,q). The states \z) = \p,q) are canonical coherentstates and are given by 9 oo ^> = c"T">.rrn>, (3.16) and z (q + ip)/y/2. Also, the state \n) is the nth excited harmonic oscillator eigenstate. In (15) the resolution of unity l = J2\n)(n\= I \z)(z\ (3.17) n=0 has been used at every time slice during the construction of the path integral [18]. The principal premise of this thesis is that in the construction of the path integral representation of the propagator for a constrained system, rather than the resolution of unity, one should use a projection operator which ensures that at every infinitesimal time step forward the evolving state is projected onto the physical subspace. 27 3.2.1 Projection Operator We shall now construct the appropriate projection operator. In the space spanned by the basis vectors \l,m,n) = MMlO.O.O) (3,8) V l\m\n\ the physical subspace is spanned by the orthonormal vectors (J2 + J2)k M= 2*ib? 0'M>' (3*19) as already noted. Hence, a projection operator which will project vectors onto the physical subspace is oo ^ = V>*}(0*. (3.20) In order to use this projection operator in a path integral, we will write it as an integral in a fashion similar to how the unit operator is written as 1 = J \z)(z\d2z/V, and to this end we note that (aj\a\lalm = (aj\a*l0*m, (3.21) where and h[,6) are canonical coherentstates. We now show how to write our projection operator in an integral representation: oo k=0 oo k=0 (3 22) d2ad2/3d2rj \ l \l l \ t il**L\ A ,\i A.I 7t3 in the above expression we have multiplied the projection operator by unity on either side and we obtain ^ / fafiftfnftfsft, ...... fc=0 oo d2ad2pd2r]d21d28d2Z. c .. (a*2 + /?*2)* (72 + 62)k (3.23) / "6liP^)(7^^l fc=0 2*Jfe! 2*ifc! x e 2 i(kl2+l^l2+M2+l7l2+62+kl2) 28 where in (23) we have used the fact (z0) = (0z) = exp(j22). Next, using the Kronecker delta function in the form 8m = Jexp{i(k l)9}d6/27r, we can write our projection operator as p = 2^ /6wh/?,7/)(7,ul kj=0 J 2* 2//! (3.24) I(a2 + /?2 + H2 + 72 + 2 + K2) xev e 2 and we notice that the summation above can be converted to an exponential. Finally, we get for our projection operator P = I ~6 laj.rj)^^^] x e 2 I(a2+/?2+H2+72 + 62 + k2)+1M^e,e+1Hlie' *2 0*2x /..2 ,,2, (3.25) The form of the projection operator in the above equation suggests the name Bicoherent states, where the term bicoherent alludes to the fact that the projection operator is represented by a weighted integral over independent coherentstate bras and kets. Going back to the extended Hamiltonian in (6), recall that all primary and secondary constraints in the language of Dirac, appear in the Hamiltonian accompanied by their respective Lagrange multipliers and in the quantization process the Lagrange multipliers are also promoted to operators. Thus, to account for the Lagrange multipliers our projection operator becomes P d2ad2pd2r1d21d28d2Zd2Pd2(j d6 7t8 27t x e 2 HMa+IJI2+M8^ (3.26) where in the above expression a\a,/?,7/,p,a) = p\a, fl,r},p,a) and as\a,(3,r],p,a) a a, /?, t/, />, a). One can easily verify that the operator P above satisfies the two defining properties of a projection operator, namely, P^ = P and P2 = P. We will use the projection operator in (26), in the construction of the path integral representation of the propagator in the next subsection. At this point, however, the interested reader may want to digress to the appendix A where we discuss the salient features of path integral representations constructed using bicoherent states. 29 3.2.2 Propagator We are now equipped with the necessary tools to derive the path integral representation of the propagator. We will calculate the matrix element of the evolution op erator between canonical coherentstates i.e., la", ft", rf1, p", a" iTH According to our premise, the propagator is / // nil II II II \<* ,P ,V ,p ,o iTH I Ql I I l\ I II nil II II II ( >P ,n >p icH PNe icH ...Pie icH I nl I I l\ a ,p ,rf ,p , Pn d2 an d2 pn d2 rjn d2 7t 8 2tt x I&n, Pn,Vn, Pn, &n) (in, ^n, Cn, Pn, & n (3.27) (3.28) (an2+/?n2+hn2+7n2+62+^2)+(<:^2)e'9" + ^^e'e" 2 c2 x e 2 the projection operator in (26). Also, (TV + l)c = T and n = 1,2, ...TV. Hence, the propagator now becomes / // nil II II II \<* >P >P j<* iTH I nl I I l\ a ,fi ,r) ,p ,a) f n N icH <*n, Pn, Vn, Pn, n) JJ dp, n=0 where the measure at each time slice is 71=1 dPn d2and2Pnd2rjnd2fnd28nd2nd2pnd2an dOn 7t 8 2tt x e ?2 a*2 i(kn2 + ^2 + r,2 + 7n2 + n2 + kn2) + ^^e^n + (^+i) 10 n (3.29) (3.30) and the boundary conditions are (7N+l,f)N+l,tN+liPN+l,<7N+l) = ,f3",v" ,p","), AhWbPO^o) = trf,P ,) For small e, we have, to order e, (7n+l Qn, Pn, Vn, Pn, &n) (3.31) (7n+l) n+l?n+l, Pn+1 j 0"n+l [1 ~ icW] an, /?n, 7?^, /)n, (Tn) (3.32) (7n+l ^ ,n] 30 where in the expression above n (7n+l, ^n+1? n+l, Pn+1, I^I^ti, fin, gnj In, gn) \7n+l j Thus, using the fact that [1 icH n+1 ,n] e ,cb+m we are lead to the following expression, provided the integrals exist, for the propagator / // nil II II II \fit ,p ,t? ,p ,a iTH I nl I I l\ a >P ,V ,p , ^ n=0 7i=l (3.34) The canonical coherentstate overlap at each time slice in the above expression is \7n+l > (3.35) x e id^+i pjl^^.,! i^hI^h! 2ha>hi i2!!^.,.! i^^i^^p^i^is^^^^^i^p^^^p) and we notice that the factor exp{(l/2)(7ri+i2 + ^n+ir2 + ^+i2 + pn+i2 + kn+i2 + 2 2 2 2 77. 2 2 + l^n2 + pn'J + knl2)}, exceP* at the end points, can be absorbed in the 2 measure d^. Hence, our propagator becomes a" J1 Ji \<* >P ^ >P ,<* iTH I nl I I l\ &,p ,y ,p,cr) CJ(T+I^,a' ",2,," <7 +ia'i2+i^i2+i,T+ip'r+i /2 I /2 I _/2 x TV en=0 II Vn i 77=1 (3.36) where the overall factor arose from the end points of the term absorbed in the measure, which at each time slice has changed slightly and is now given by dp' dL an dz f3n d2 rjn dL ~jn dL 8n dl n dz pn dz 2 '2 77, tt 8 2tt 2 x2 (3.37) x e i0 n Our goal is to express the right hand side of (36) as a path integral and in preparation toward this objective we rewrite part of the exponent in the integrand of this equation 31 as follows: n N i n=0 2 n=0 7n+l(n+l a) + {8* + l 6*)Pn 8*+1(0n+i /?) + ~ C)V n (3 3S) C+l(Vn+l ~ Vn) + (Pn+1 ~ Pn)Pn ~ P*+l(/>**+! Pn) + +l ~ 77 = 0 + fn+l^n+l + PnPn + Pn+lPn+1 + &^(Tn + t7*+1(7n+l}. In the above equation, the terms 7o,o>o,ow+i,/?w+i and vn+i have not yet been defined. The factors containing these terms cancel and so these terms can take on arbitrary values and are at our disposal. We shall assign them the following values: (70,^0,6) = and (a,v+i,/?/v+i,7//v+i) = (a",/?",77"). The choice of these special values will become clear presently. Going back to (38), we notice the second term on the right hand side of this equation can be absorbed in the measure, so, our propagator can be written as r 1 / // nil 11 11 11 iTH 1 ni 1 1 i\ I r\~^7117 *\ {a yp ,p ,0 e a ,/3 ,r) ,p ,a) = expj^l^ l(7n+i ~7n)an 77=0  7n+l(an+l Ctn) + K)Pn ~ *+i(/?n+l ~ Pn) + (&+1 Ct)Vn (3.39) n+lO/n+l ~ Vn) + {pn+l ~ Pn)Pn ~ Pn+l(Pn+l Pn) + +l ~ tT*)(7n TV 77=1 where the measure at each time slice gets modified again and now finally is d2and2f3nd27ind2'ynd26nd2(tnd2pnd2an d6n dpn ~x e TT8 27T *2+/g2)rtt ill**2 2 C ~T 2 (M2+/?n2 + 77n2 + 7nf^ (3.40) Thus, interchanging the order of the limit and the integration in (39) we write for the propagator, in a formal way, the form it takes over continuous and differentiate paths as I~" qn Jl iTH 1 qI 1 1 i\ (a ,p ,ri ,p ,cr e a ,p ,7/ ,p , 0 32 where V\i Y[dpn, and (H) in the expression above is given by n (H) H where the Hamiltonian H is given in (8). Hence, according to (41) our phasespace action for continuous and differentiable paths is T S= j{(^)[7*a y*a + 6*0 6*f3 + *V ~ t*V + 9*9 ~ 9*9 + y/2 y/2 i with the following boundary conditions (a(0),j9(0),i(0),p(0),r(0)) = (7(0U(0U(0),p(0W0)) (3.44) wr),^),^!),^),^)) = (7(n*(nn/*nKT)) ia ,p ,ri ,p ,a ) 3.2.3 Classical Limit We will now study the classical equations of motion obtained from our phasespace action. However, before we do so, we add to our action a total time derivative which of course will not effect the equations of motion and write it as T S = J {i[7*d + 6*0 + Cv + 9*9 + ~ [l*<> + 6*P o (3.45) + i^to /**) + PSa) + =j=]}dt. Notice our action is generally complex i.e., S = S\ + iS2. Varying S with respect to a, /?, 7/, 7*, 6*, and a*, while keeping their end points fixed, we get the following 33 equations of motion a 0 (g + rj) (/ + g) 6 (j+ r,*) (/ + p) p = 7=a=a, o T=77=7, % y/2 y/2 I y/2 y/2 (7*gj*g0 (* + *) ; (<**63*l) K + (3.46) P y/2 2 2 (7*^ (C 1?) (7 =  V2 2 Now, consider the paths a(t) and 7(f); using the boundary conditions in (44) and the evolution equations above we find that a(0) = 7(0) = a These are sufficient conditions for a(t) = 7(<) i.e., they evolve along identical paths. One can easily check that the pairs of paths (f3(t),6(t)) and (/(<),(t)) also start off with the same initial conditions and so we have (3(t) = 8(t) and rj(t) = (t). We will take up the equations for p and a later; these equations determine the Lagrange multipliers. To further our study of the classical equations let us define the complex quantities a,/?,7/,7,^,and a as follows: 7fe + *^ 'k^, { = ^^, (3.48) (8) Now (a(0,/?(<), >?(0) = (7 W,*^^ (94(0?P4(0 95(*)>P5(0'96(0>P6(0) Using this fact and the definitions in (48) we write the evolution equations (46) in terms of the variables (91,pi,92,P2,93,P3,97,P7,98,Ps) 34 as follows 91 = Pi + (93 + 97)92, Pi = 91 + (93 + 97)P2, 92 = P2 ~ (93 + 97)91, P2 = 92 (93 + qi)p\, (3.49) 93 = 98, P*3 = (P291 Pi92), 97 = 0, Pi = (P291 P192), 98 = 0, ps = pa in order to compare the above equations of motion with those for the Hamiltonian H ^(k2 + x2) + ykTx + AkTx + <;7r we introduce a slightly extended classical phasespace action S", which will give us classical equations of motion in onetoone correspondence with the equations in (49). The extended action is T S* ~ J+ 1*2*2 + ny + Px^ + Ptf] o (3.50) 12 2 + y + ykTx + XkTx + w)}dt where (A, q) are Lagrange multipliers and (pa,P<;) are their respective conjugate momenta. The evolution equations obtained from the action S' are x\ = ki + (y + \)x2, h = x\ + (y + A)&2, x2 = h {y + A)xi, k2 = x2 (y + A)fci, (3.51) j/ = C, 7r = (a?iA?2 hx2), A = 0, Pa = (xik2 hx2), <; = 0, Ps = 7t. Comparing equations in (49) and (51) we see that if we make the identification (9l,Pl,92,P2,93,P3,97,P7,98,P8) *> (^1, h, x2, fc2, ^3, &3, A, pA, P<) (3.52) the two sets of equations would be the same. Thus, we conclude that the action obtained from our quantum propagator gives us the desired classical evolution equations and hence the right classical limit. We will now note two interesting features about the classical limit of the formalism developed here. First, substituting the definitions (48) in our complex action S = S\+iS2 35 in (45), we obtain for the real part of our action T Sl = \ / {[(P4?l ?4Pl) + (P592 qm) + (P693 96P3) + (P797 ~ 97P7) 0 + (P8?8 qsps)] [{qm + pm) + (9592 + P5P2)] 9s(p3 + pe) (353) + ^(P6 P3)[(?59l + PBPl) ~ (9492 + P4P2)] + 7j{qe + ?3)[(P59l 95Pl)  (P492 94P2)] + qi[{P5qi qm) ~ (P492 qm)]}dt while the imaginary part of our complex action is given by T S2 = ^ J {[qm + P4P1 + 9592 + p5p2 + 969*3 + P6P3 + 9797 + P7P7 0 + 9898 + P8Ps] + [(pm qm) + (P592 95P2)] + 98(93 qe) ^.54)  ^{qe + 93)[(?492 + P4P2) (9591 + PbPl)] ^(P6 P3)[(P591 ?5Pl)  (pq2 94P2)] ~ 97[(9492 + P4P2) (9591 + P5Pl)]}dt. Extremising Si and 52 in equations (53) and (54) respectively, while keeping the end points of the paths in them fixed we can obtain equations of motion for the dynamical variables (91, pi,92, P2,93, P3,94, P4,95, P5,96, P6) We find that for each of these variables the evolution equations obtained from S\ is identical to the one obtained from S2. The second interesting fact is the following: we saw that on the classical trajectories (a(t), (3(t),r](t)) = (7(f), which is equivalently stated as (qi(t),Pl(t),q2(t)>p2{t),q${t),pz(t)) = (q(t),p4(t),q5(t),p5(t),qe(t),pe(t)). Substitut ing this fact in (53) for Si we find that, up to a total derivative, S\ on the classical trajectories reduces to T Si > J {[Pl9l + P292 + P393 + P797 + P89s] + Pi) 0  2(92 + P2) 93(P192 91P2) 97(P192 91P2) qm]}dt (3.55) 36 exactly the standard classical phasespace action in (50). The imaginary part 52 of the action with the above substitution becomes T 52 /[9191 + pipi + 9292 + P2P2 + 9393 0 + P3P3 + 9797 + P7P7 + 9898 + PSPs]^ (3.56) a complete surface term! So, 52 on the classical trajectories gives rise to only an overall 'phase factor' in the propagator. 3.2.4 Constraint Hypersurface Here we study the restrictions on the states over which the matrix element of the evolution operator is evaluated in the propagator. Consider the following equations from the set (49) 93 = 98, P3 = (P29i Pi92), 97 = 0, (3.57) P7 = (P29i Pi92), 98 = 0, ps = P3 In the classical description of the model we are studying we had the two constraints 7r = 0 and pTx (p\x2 P2#i) = 0. So, in (57) we want p% n = 0 and p'3 = p7 = 0. Thus, the solutions to these equations are 98 = Ci, ps = C2, 97 = C3, (3.58) P7 = C4, 93 = Ci* + C5, p3 = 0, where ci,<22,G3,C4, and C5 are real constants which are determined by the particular classical solution one is interested in. Also, note that pz{t) = (P291 Pi92) = 0 implies in particular p3(0) = 2(a'Rp'j a'j0'R) = 0 and p3(T) = 2(a"Rp} a'}^) = 0, where (<*r,0r) and (aj,(3j) are the real and imaginary parts of (a,/?) respectively. These restrictions on (a', ft) and (a",(3") can be stated alternatively as (a7 a'?*) = 0, (a"*0" a"/?"*) = 0. (3.59) Thus, in our propagator the states \a', f#\rf, p1 ,a') and \a",f3", t//;, p'\ a") are not arbitrary but must be restricted, as discussed above, to ensure that the system remains on the 37 constraint hypersurface in the classical limit. Hence, our propagator is / // nil II II III (a ,p ,7/ ,p , a ,p ,7/ ,p ,J y/2 V2 V2 iTH where Id if?) and (a",/?") are restricted as noted in (59). 3.3 The Quartic Potential (3.60) For completeness we shall consider the quartic potential V(x) = t(x2)2 and show that we again obtain the correct classical limit by following the quantization procedure outlined in this chapter. The Hamiltonian now is H = ^p2 + ^(x2)2+ypTx+upTx+v7r. In the secondquantized notation the normalordered form of our Hamiltonian is H =: (j)[(ai a\)2 + (a2 a\)2) + l[(a, + a\)2 + (a2 + J,)2]2 .(a3 + <4)/ t t \ K + al), t t x (a5 + as)(a33) x/2 x/2 72 \/2z (3.61) The propagator for canonical coherentstates is again / // nil II II II (a ,P ,rj ,p ,a iTH <*,P ,V ,p, la ,P ,r) ,/> ,cr icH PNe icH PNie ieH ...Pie ieH I nl I I l\ a ,P ,rj ,p ,a), (3.62) where Pn is the projection operator in (26). Interchanging the order of the limit and the integration as usual we formally write the propagator as / // nil II II II (a ,p ,9 ,/ , I nl I I l\ iS (3.63) where the action, which is complex, is given for continuous and differentiate paths up to a surface term by T s 1 {t[7*a + 6*0 + CV + P*P + 0 1 [(a + 7*)2 + (/? + ni *\2i2 16 *+7^j(7 P v/2 y/2i }dt (3.64) 38 with boundary conditions specified in (44). Extremising S we obtain the following equations of motion; variation with respect to 7* and a lead to variation with respect to 5* and /? leads to (3.65) 0 6 (3.66) + />], and finally variation with respect to and rj lead to (j*0S*a) K + cr) ? =+ y/2 2 V2 2 (3.67) Next, consider the trajectories a(t) and 7(f); using (65) and the boundary conditions in (44) we find that for these paths 1 a(0) = 7(0) = (0) = 7(0) = ^ + (Q' I^Ka' + a'*)2 + 09* + tf) 2i 4t (3.68) They have identical initial conditions and hence evolve along the same paths, i.e. a(t) 7(r). Similarly it can be confirmed easily that 3(t) S(t) and (t). So, using the fact that (a(t), 3(t),q(t)) (l(t),6(tU(t)) implies (qi(t),Pi(t)Mi)Mt)MthMt)) (q4(t),P4{t),qs(t),ps(t),q6(t),p6(t)), and the definitions in (48) we can write the evolution equations (6567) in terms of the variables (9i,Pi,<72,P2,93,P3,97,P7,98,P8) and we get the following equations 9i Pi + 92(93 + 97), 92 =P2 9i(93 + 97), Pi 9i(9i +92) +P2(93 + 97), 2 P3 (91P2 92Pl), P2. 92(9l +92) Pi(93 + 97), 93 = 0, 97 = 0, (3.69) P7 (91P2 92Pl), 98 0, j>8 P3 39 These are exactly the equations one would get from the classical phasespace action T S' = j {[kixi + k2x2 + ny + p\\ + p? (3.70) + j(x2)2 + ykTx + XkTx + ^]}dt a if one makes the identification (9lPl??2,P2,93,P3,97,P7,98,P8) <" *1,*2, &2, 33, &3, A,pA, (3.71) Hence, we see again that the action obtained from the quantum propagator gives us the desired classical equations of motion. Now, let us substitute (a(t),0(t), r)(t)) = {"f(t),6(t),(t)) and the definitions (48) in (64). One can obtain the real and imaginary parts of the action evaluated on the classical trajectories. We get for the real part, up to a total derivative, T Sl * /{[Pl9l + P2?2 + P393 + P797 + P89s] ~ ^(Pl + P2) 0 (3.72) + ?)2 (93 + 97)(Pl92 91P2) 98P3}<*< exactly the classical phasespace action of equation (70). The imaginary part reduces to T S2 /[919i + P1P1 + 9292 + P2P2 + 939*3 (3.73) 0 + P3P3 + 9797 + P7P7 + 9898 + PsPs]dt^ a surface term. Once again, we find that the complex action obtained from the quantum propagator gives us the correct classical evolution equations and when evaluated on the classical trajectories its real part is exactly equal to the classical phasespace action evaluated on the same paths and the imaginary part is just a surface term. 3.4 The Measure We would now like to make the point that the procedure developed here for con structing the path integral for the propagator is merely a recipe for obtaining the correct 40 measure. We begin by noting that for a system with three dynamical degrees of freedom the unit operator is given by 1 d2 z\d2 Z2d2 zz 7t * (3.74) but the unit operator can also be written as 1 d2z\d2Z2d2zz 7t 3 *lj z2, Z$){z\,Z2, Zs d2zd2z$d2z 7t 3 Z4,Z5,Z$)(Z4,Z5,Z6 d2z\d2Z2d2zz d2zd2z$d2z6 7t 3 7t 3 Zl,Z2,Z3)(z4,ZcnZ6 (3.75) x e 2 !(kil2+M2+M2+24 + 252r262)4^2442j25r2326 1 an integral over bicoherent states. So, for the Hamiltonian 7i = p2 + V(x) + ypTx + upTx + vir we are studying in this paper we could have written the unit operator as 1 d2ad2(3d271d21d28d2Zd2Pd2(j TT 8 (3.76) x e 2 T(ki2+i^i2+M2+i7r+i^i2+i^i2)+^7+^+^ Comparing this to the projection operator in (26) for our constrained system, which we reproduce below for convenience, P d2ad2pd2r}d21d28d2d2pd2(j dO 7t 8 2tt x e 2 i(iai2+i/?r+hi2+M2+i^i2+ki2H^ *2f/3*2)ce j )ce (3.77) we see that the only difference between the projection operator and the unit operator is the measure over which the bicoherent states are integrated. Recall now the expression we obtained for the propagator using this projection operator at every time slice / // nil II II II \ot ,p ,v ,P , iTH I nl I I l\ & ,P ,V ,p , (3.78) where the discrete form of the measure is given by Vu d2 an d2 (3n d2 rjn d2 ^n d2 8n d2 n d2 pn d2 an dOn 7t 8 2tt X n ( n 2> (a*2 + /3*2 (3.79) } 41 On the other hand had we used the resolution of unity as written in (76) instead, the expression for our propagator would have been / // nil II II II iTH I Ql I I l\ {a ,p ,1/ ,/> , if {t[7*rii(5*/?+r^+pV+^](^)} (3.80) unit * and the measure would be _ rr r d2and2f3nd2r)nd2 n (3.81) (anr' + nf+7/n2 + 7nf + M } So, we see that both procedures would have lead to the same action for continuous and differentiable path, but with quite different measures! Since the actions are the same they would yield identical classical equations of motion, but the different measures would give different spectrums to the quantization, only one of them being correct, of course. It must be noted that a general operator admits a bicoherent state representation according to 0= I \a)(a\O\0){0\^^. (3.82) 7T" Such operators have been dealt with, for example, by R. J. Glauber [21]. What is novel in the present thesis is the use of such representations in path integral constructions for which in all previous coherentstate applications only weighted coherentstate projection operators have been used. The term 'Bicoherent States' was coined by Prof. J. R. Klauder. CHAPTER 4 BICOHERENT STATES AND SECONDCLASS CONSTRAINTS In this chapter we extend the formalism of the previous chapter developed for systems with firstclass constraints, to systems with secondclass constraints. The principal premise of this formalism is that in the construction of the path integral representation of the propagator for a constrained system one should use, rather than the resolution unity, a projection operator at every time slice. The projection operator is such that it ensures the evolving state, at every infinitesimal time step forward of its evolution, is projected onto the physical subspace. The present chapter discusses two distinct models. In section 4.1 we state the Hamiltonian for the first model that is being investigated and impose a constraint by hand. We construct the path integral representation of the propagator using the proposition stated above and using the phasespace action obtained from this propagator extract the classical equations of motion, which we show are the desired classical evolution equations. For the second model, which is discussed in section 4.2, we show that it has one primary and three secondary constraints, all secondclass. We show our formalism again gives us the correct classical limit. 4.1 Toy Model 1 In this section we study the two dimensional Harmonic oscillator, described by the Hamiltonian H = \{p\ + p\) + + ?)' subject to constraints. Let us impose the constraint lt\ = (q\ #2) = 0 on our model. We want the constraint to hold at all times, so, we require <7l = { 43 i.e. our model has a secondary constraint 02 0. We find that &2 {(?2,H} = i.e. there are no further constraints in our model. Also, {o"i,cr2} = 2, hence, the two constraints are secondclass. In the quantization of systems with secondclass constraints one finds that the constraints can be imposed as operator equations once the Poisson brackets are replaced by Dirac brackets. Thus, for the model being studied here, since the canonical Hamiltonian can be written as H = ^[{pi P2)2 + ?>PiP2} + ^[(91 92)2 + 2qiq2], the physical Hamiltonian Hph = (11)^=0 may be written, preserving 1 <> 2 symmetry, as Hph = P1P2 + <7i<72 (4.2) We will use Hph to identify the physical subspace of our model. For ease of writing we will use the same symbols to represent the classical variables and their corresponding quantum operators, since the quantity being referred to shall be clear from the context. Thus, using creation and annihilation operators we write 9j = 7k* Pj y/2 rJ y/2i (4.3) [a,,aj] = 6ij, [auaj] = 0, i,j = 1,2, we get : Hph := a[d2 + a}2a\. The eigenstates of this Hamiltonian are easily found by noting the fact that [{a\a2 + a\ai),(a\ + a^)] = (a\ + a\) and thus, the normalized eigenstates are given by These eigenstates span the physical subspace. Therefore, a projection operator that can be used in the construction of the path integral representation of the propagator is 00 k=0 44 This projection operator will project the evolving states onto the physical subspace. But to be able to use this projection operator in the construction of the path integral representation of the propagator we will first need to write it as an integral and we proceed as follows: oo p k=0 oo *2*>0(,/0&){fc 7T 7t k=0 d2ad2pd21d28 (4.6) TT 4 a^)(7^(a^^)(^7,(5), where we have multiplied the projection operator by unity on either side in the first line of the above equation. The states a,/?) and 7,) are canonical coherent states where ai\a,P) a a,/3), a2a,/3) = /?a,/?), and ax\i,8) = 7\y,6), a2\~f,8) = %,<5). Next, using the fact that (0,0a,/3) = (a,/?0,0) = e 2(H2+I^I2) we can write the projection operator as P oo r /fc=cr d2ad20d21dHl (a* + /?*)* (7 + <5) 4 (V2)*Vfc! (>/2)*\/*! (4.7) xe 2 I(H2+/?2+72 + *2) In the above expression we see that the summation can be performed and so our projection operator can finally be written as P J 7T4 (4.8) a weighted integral over independent bras and kets, which are called bicoherent states [18]. 4.1.1 Path Integral We are now ready to construct the path integral representation for our canonical coherentstate propagator. As already noted, we should insert the projection operator constructed above at every time slice in the propagator rather than the resolution of unity, and so our propagator is given by ii \ iTH c!J) = (a",P"\e ieH PNe ieH ...Pie ieH (4.9) 45 where H : H : t t axa\ + a2a,2. Also, (N + l)e = T, and Pn, the projection operator at the nth time slice, is Pn '2 2 '2 7t 4 &n> Pn) {Hn j X e + /?n2 + 7n2 + gn2H(a^^)^t<") n 72 (4.10) with n = 1,2,... TV. Thus, our propagator becomes iTH n S JJ(7n+l9^n+l a.) n v, ra=0 n=l (4.11) where the measure at each time slice is d2an(P0nd2fnd28n _i( 2 + l^n2 + 7n2 + n2) + ^^^^ 7t 4 72 V2" (4.12) and the boundary conditions are (7jv+i,jv"+i) = (a'',f3") and (ao,/?o) = (a',/?'). In equation (11), for small e, we have, correct to order 6, (7 ,4 if?* &n,fin) (7n+i ~ ieH}\an,(3n) (7n+i, where in the expression above H n+l,n (l[n+l,{>n+l\'H\ Therefore, using the fact that (7n+l A+lK, Ai) = exp{(l/2)(7n+iI2 + Wn+1 2 + a 7? 2 + Ai2) + 7n\ian + our propagator, provided the integrals exist, becomes iTH n I(7n+12 + 6n+12 + an2 + /?^ n n v. (4.15) 71 = 0 71=1 Notice that the term exp{( 1/2)(7n+i 2 +  a 7? 2 + IAi2)}, except at the end points, can be absorbed in the measure and so our propagator now is given by / w nil iTM N 2 'f\2 i o//2 /2 E^n+ln+^+1/3n^+1,] * gn=0 (4.16) n=l 46 where the overall factor arose from the end points of the term absorbed in the measure, which at each time slice has changed slightly and is now given by , d2and2(3nd2lnd28n ^(aB'+/?j'+7ro'+fiB')+K^) fragaj /V Going back to (16), we note that the term Yl [7n+ian + ^n+i^] can ^e rewritten as follows, TV AT ^[7n+l<*n + <$n+lAi] = ^ ^{(7n+l 7nW 7rc+l(a"+l ~ <*n) 77 = 0 77 = 0 * I (4.18) 77 2 n=0 + ^ + ^n+1}. In the above expression the terms (70,^0) and (a/v+i,Pn+i) have not been defined yet. The factors containing these terms cancel and so these terms can take on arbitrary values We assign them the following values: (70,^0) = (#',/?') and (a/v+i,/3#+i) = (a",(3") The reason for the choice of these special values will become clear shortly. Notice that the second sum above can be absorbed in the measure, and so our propagator becomes, (a",/?"e ,rwa',/?')= / exp{^([(7*+1 7*)a 7*+1(an+i an) J n=0 Z + (*n+l ~ K)Pn ~ <$*+i(/?7i+l /?)] ~ itHn+i,n)} JJ dpn, 77 = 1 where the measure at each time slice has changed again and has now become d2and2pnd2lnd28n ^(c,^24^2^Tr,2H^2)H(^^) ^T^or^H^^ ^ 72 r"^. (4.20) Interchanging the order of the limit and integration in (19), we can formally write the propagator as an integral over continuous and differentiable paths as la\P"\eiTn\a',p') = I e 0 Va, (4.21) 47 where in the above expression X>// = Yld(in, and (H) for the model under investigation 77 is given by injK lg, 6\H\a, P) * ' = ~~7Fim~ = 7 a + P v422) Thus, the phasespace action according to (21) is T S = J{()[7*A 7*<* + 6*/3] [7*a + 8*(3]}dt, (4.23) o with the following boundary conditions (a(0),/?(0)) = (7(0)^(0)) = (a', ^) (a(ry(T)) = (7(T)5f(r)) = (aw,^) (4.24) 4.1.2 Classical Limit We will now study the classical equations of motion obtained from our phasespace action in (23). However, before we do so, we add to our action a total time derivative which of course will not effect the equations of motion and write it as T S = / {i[7*a + 6*j3] [7*a + 6*P]}dt. (4.25) o Our action, it is to be noted, is complex, i.e., S = S\ +iS2 Variation of 5 with respect to a,/?,7* and 8* while keeping the end points of these paths fixed gives us the following equations of motion a = ia, (3 = i(3, 7 = 17, 8 = %8. (4.26) Next, consider the paths a(t) and 7(f); using the boundary conditions (24) and the evolution equations above, we have a(0) = 7(0) = a a(0) = 7(0) = 10! (4.27) 48 The above equations imply a(t) = 7(2). Similarly, one finds that /3(t) and S(t) evolve along identical paths, i.e. (3(t) = 6(t). To further our study of the classical equations of motion we define the complex quantities a,/?,7, and 8 as follows: (91 + ipi) n (92 + ip2) {qs + m) c {q* + ipA) (A Notice, the equality (a(t),P(t)) = (i{t),6(t)) implies (qi(t),pi(t),q2{t),p2(t)) = (q$(t),Pz(t), qi{t),Pi(t)) Using this fact and the definitions (28) we can write the evolution equations (26) in terms of the variables (9i,pi,92,^2) as follows 91 = Pi, Pi = qu 92 = P2, P2 = 92. (4.29) For the model being studied in this section the standard classical phasespace action is given by T S' = J {[Ml + M2] + *?) + + 4)]}dt. (4.30) 0 The equations of motion obtained from 5' would be identical to the ones in (29) if we made the identification (91,Pi,92,P2) > (xi,h,X2,fo) (4.31) Thus, we see that the action obtained from our quantum propagator has given us the desired classical evolution equations and hence the correct classical limit. It is interesting to note that just as in the case of the first class constraints, as discussed in chapter 3, if we substitute the definitions (28) in (23) and obtain the evolution equations for the variables (91,pi,92,P2,93,P3,94,^4) from S\ and S2 we find that for each of these variables, the equation obtained from S\ is exactly the same as the one obtained from 52. Furthermore, on the classical trajectories where 49 (ttWw(*)>ft(*)>P2(*)) = (93(0^P3(0^4(0^P4(0)^ we find that si reduces to just the standard phasespace action 5' up to a surface term. Also, the imaginary part of the action S2, reduces to a pure surface term using the classical equations of motion. 4.1.3 Constraint Hypersurface We now find the states over which the matrix elements of the evolution operator may be evaluated in the propagator such that in the classical limit we remain on the constraint surface. Note that the constraints in our model are satisfied, at the classical level, if 9i(0 ~~ 92(0 = 0 and Pl(0 ~~ P2(0 0 Fr these equations to hold we must have 9i(0) = ?2(0), Pl(T)=p2(T), (4.32) ?i(o) = 92(o), Pi(r)=p2(r). Using the definitions (28), the equations of motion (29), and the boundary conditions (24) we find that the conditions (32) for the constraints to hold imply that ol (3' and a" = /?". Thus, in our propagator the states = W or \a",(3") are not arbitrary but must be restricted as discussed above and so the allowed states are = a',a') or a", a"). Also, for these allowed states we find that : u{ : = 0, i.e. the expectation values of the constraints as operators vanish. 4.2 Toy Model 2 Here we study a second model with four secondclass constraints described by the Lagrangian L(x,x,y,y)=1x2y(x2R2) (4.33) where x = [xx%x%)% a two dimensional vector, and y are dynamical variables. As a first step toward quantization we go over to the Hamiltonian formalism. The canonically conjugate momenta to the coordinates are 50 Therefore, the canonical Hamiltonian is given by H=lv2 + y{?eW). (4.35) Thus, we have a system with one primary constraint it = 0. We of course want our primary constraint to hold at all times, so, we require * = {tt, H} = (x2 R2) = ai = 0, (4.36) i.e., we have a secondary constraint a\ 0. We find that our system has only two further constraints; b\ { 7r = 0, we find that (P^x)2 Hph = (4.38) where pTx = (p\x2 P2%\) and in obtaining Hph we have used the fact that p2 can be written as (px)2 (pTx) P = + x2 x (4.39) We see that Hpk is proportional to the square of the zcomponent of the orbital angular momentum and its eigenstates are known. Now, let us promote x, p,j/,7r to operators and express them using creation and annihilation operators as _ K+a]) (aj a)) .in ] r Pj 7= > J 1 j V2 V2\ (4.40) (a3 + a3) (a3 aj) 51 where [a,,aj] = Sij and [aj,a] = 0 for z, j = 1,2,3. Using the definitions in (40) we J find that x\p2 x2p\ = i(a\ai a\a2). Transforming to new operators defined by A: = Lljji, M = (4.4.) we can write x\p2 x2p\ = i(a\a\ a\a2) {A\A\ a\a2), where the operators Ai satisfy the following commutation relations; [A,, a\] Sij and [A,,^] = 0 for i,j = 1,2. The states /,m) given by "m>= m\ wi J0'0'" <4'42) y (/ + my(l my. are eigenstates of (^Ai A^^), where / = 1,2,... and m = /,/ + 1,...,/ (see Schwinger's oscillator model for angular momentum [22]). But recall that our problem is being described using three dynamical degrees of freedom, namely (x,y), so, we write the eigenstates /,m) as follows I/, in) s U, m, 0) = 3l^:(t"tr ,10.CO), (4.43, (>/2)/+m(>/2)/m+ m)\(l m)\ where a3/,m,0) = 0. 4.2.1 Projection Operator We saw above that the eigenstates of Hvh are the states /,m) as written in (43). Hence, a projection operator that will bring an arbitrary state from the Hilbert space under consideration onto the physical subspace is given by P= \l,m){l,ml (444) m=l We will now show how to write the projection operator as an integral. We multiply P by unity on either side, n ^ [d2ad2pd2r] n v/ WI [d21d28d2x c w r P = E / ^^a,^,7/)(a^,7//,m)(/,m / 7 3 7,*,flM,fl m J d2ad20d2r}d2)d28d2i 7T6 52 In the above expression the states a, /?,//) and 7,<5, ) are eigenstates of annihilation operators a,, for z = 1,2,3. Using the fact that for canonical coherent states (z\0) (0\z) exp( ^z2), and the definition in (43) for the states /, to) we can write the projection operator as P m 7t 6 X (a* + z/?* (a 7H ? \/m 2V(^ + ")K/m)! (7 i)'+m(7 + i<5)/+m (4.46) The summation in the above expression can easily be done, so the projection operator is finally given by P 6 2/! (4.47) xe 2 i(m2+/?2+m2+m2+62+ki2) a weighted integral over bicoherent states. 4.2.2 Path Integral We are now ready to construct a path integral representation of the canonical coherent state propagator using our assertion that for a constrained system one should use an appropriate projection operator rather than the resolution of unity at every time slice. Thus, the propagator is given by / oil "I {a ,p ,// e iTH I qI i\ (a\(l'\f\e"nPNe"H...P1e"n\a\(3\T1'), where in the above expression 7i, the normal ordered Hamiltonian is H = Ip2 + y(*2 V) = k^T^ silo? + A2 + 4 4 1 2 . lf(3 + a3) '2l ^2 ][a\ 4 aj"4 + + at2] 2 + [(^3 + ai) \/2 1 2 ](aai + 0^2) R t t ,2(3 + a3) y/2 In (48), Pn, the projection operator at the nth time slice, is d2 a d2 5 d2 r? d2 t d2 d2 (4.48) (4.49) 53 As in section 4.1 we can write the propagator formally as an integral over continuous and differentiable paths which is given by T / a an a (a ,P ,V iTH lJ {5h*ay*a+8*/36*p+Z*VZ*v]{'H)}dt o Vu (4.51) In the expression above 2 +[ 1 + ^](7* + ^)JR2* + ??) 2 v/2 ' the discrete form of the measure is Vp d2and2/3nd2r]nd2jnd26nd2^n (a*njn + /3tSn) 21 6 2/! n x e (ln2 + ^2 + ton2 + 72 + *n2^^ and the boundary conditions are (a(0), 0(0), 1/(0)) = (7(0),*(0),(0)) = (a',/7,i/) (a(r),/?(r),,(r)) = (7(7V(ru(T)) = (",/? V) (4.52) (4.53) (4.54) 4.2.3 Classical Limit The phasespace action according to the propagator in (51), up to a surface term, is given by T S 0 (4.55) 1 y/2 2 ^2 Extremising this action, while keeping the end points of the paths in them fixed, we obtain the following equations of motion a 0 (l*+ <*)(?+V) (7*o) V2i 2i (* + fi)(? + V) (6* ~ P) 1 V2i 2i i 6 s/2i 2i y/2i 2i ' J 1 2V2 ;[a2+7*2 + 02 + ^2] + 1 ;K2 + 7Z + rZ + ^] + 2 *2 2 2\/2: (7*Q + i?2) y/2i da* + 6/3* R2) y/2i (4.56) 3 54 boundary (a(r), f3(t),r](t)) = ("f(t), 6(t),(t)) on the classical trajectories. Next, introducing the following definitions Q (gl + ipi) p (92 + 1p2) (Q3 + m) y/2 \/2 y/2 (45^ _ (4) r (g5 + *Ps) > (gg + ipe) 7~ v/2 x/2 *~ y/2 ' and using the fact that the equality (a(t), (3(t),r)(t)) = (7^), implies (9i(<)Pi(')^2(<)P2(*)^3(<),P3(<)) = (94(0,Pi(t),qs(t),p5(t),q6(t),p6(t)) we can write the equations of motion (56) in terms of the variables (91, pi,92, P2, 93, P3) as 91 = Pi. Pi = 2?i93, 92 = P2, (4.58) P2 = 29293, 93 = 0, p3 = #2 (9i + These are precisely the equations one would obtain from the standard classical phase space action 5" = / {[Ml + hx2 + Ma] [\{k\ + k\) + xz(x\ + x\ R2)]}dt (4.59) 0 if we make the identification (91,pi, 92,P2,93,P3) > (#i, &i, #2, &2, ^3, ^3) Thus, we see our quantization procedure has again given us the correct classical limit. 4.2.4 Constraint Hypersurface Here, as we did for model 1, we determine the restrictions on the states \a\ /?', 77') and a", /?", 77") over which the matrix element of the evolution operator can be evaluated so that we remain on the constraint surface in the classical limit. Recall the constraints (37) and the equations of motion (58); using the boundary conditions (54) we see that 7r = p3 = 0 implies r/r = rj'j = 0. Also, 93 = 0 implies 93 = c = y/2rj R where c is a real constant. Here (r)R,r]j) are the real and imaginary parts of 77 respectively. Next, the equation p'3 = a\ = R2 (q\ + q\) = 0 implies 55 R22(a% + 0g) = R22{a"R2 + p"R2) = 0. Also, the equation by = 2(qiq1+q2q2) = 0, requires that the following be true, a'Ra'j + P'RPj = aRa" + P"RP" = 0. The last constraint a3 = {p\ + p22) 2q3(q2 + q2) = 0, implies {a'f + ft2) 2c(a'j + 0%) = (af + p'}*) 2c{a"Rz + P'jf) = 0. Thus, in the propagator the states \ct,p,rf) and a",/3",r)") are not arbitrary but must be restricted as noted above. We summarize the restrictions on the state \a',f3',ri') below for convenience, R2 2(4 + P'l) = 0, o/ro/j + M = 0, (4.60) (a? + P?) 2c(a% + 0i) = 0, rf ^. With these restrictions on the states = la',/3',7/') and la",/?",?/"), we find that (0 : 7r : = 0 and : crt : = 0 for the constraints in (37). CHAPTER 5 E(2) COHERENT STATES In this chapter we consider a system with a Holonomic constraint. We will show how to construct path integrals for such systems. The formalism described here relies on the ability to be able identify the appropriate group for the specific system under consideration. The problem we consider is that of quantizing a system whose configuration space is a circle, hence, the Holonomic constraint can be expressed as (x2 + x\ R2) = 0. We will construct the Universal propagator for such a system. The universal propagator is such that it correctly propagates coherentstate representatives of state vectors independent of the fiducial vector used in the construction of the coherent states. A universal propagator was first introduced for the HeisenbergWeyl group by Klauder [23]. Subsequently the universal propagator has been written for the affine or ax + b group and for the SU(2) group [24,25]. For the quantization of systems whose configuration space is the (n1) sphere Sn_1, the canonical group that arises naturally is the ndimensional Euclidean group E(n)the semidirect product Rn()SO(n) of the abelian group Rn of translations in ndimensions with the group SO(n) of rotations in ndimensions [26]. In addition to the fact that a system whose configuration space is the (n1) sphere has a single Holonomic constraint, a theory of coherent states for E(n) could throw new light on the much discussed question of the precise definition of a path integral for a system whose configuration space is a curved manifold. We first review the construction of the universal propagator for the canonical coherent states [23]. This will set the stage for the construction of the universal propagator for the E(2) coherent states. Let P and Q denote an irreducible pair of selfadjoint Heisenberg 56 57 operators satisfying [Q,P] = i, and let \p,q,r,) = ei*peirt\ri) (5.1) denote a family of normalized states for a fixed fiducial vector \rj) with (77 r/) = 1, and (p,q) G R2. These states are canonical coherent states and they admit a resolution of unity in the form, dpdq W>9^7; = A (52) z7t for any //) when integrated over all phase space [18]. These states lead to a representation of Hilbert space H by bounded, continuous functions, ?Mp,9) = (P^iVM (5.3) defined for all \if>) e H. This representation is evidently dependent on the fiducial vector 77). An inner product in this representation is given by ( an integral which is independent of the fiducial vector \r]). The Schrodinger equation = **\m) (55) and its formal solution in terms of the evolution operator U(t) itH mf)) = u(tM^t')m% (5.6) are given in this representation, respectively, by igfr(p,q,t) = (p,q,v\H(P,Q)\m) (5.7) and ^0>",= / Kv(p\q'',t'',p\q\t')Up\q\t')^. (5.8) z7t 58 The integral kernel in the expression above is I fiducial vector, which nevertheless, propagates each of the ^v correctly, i.e., ^(p'W','") = J K{p", q", t", p\ q\ t'Wvip'i q\ 0 (5.10) for any \r]). This function is constructed in two steps. First, one observes that (iQWtiiP, 9) = (p,9,?IW), (q + iQ)*l>v(p,), (5.11) independently of \rj). Thus, if H = H(P,Q) denotes a Hamiltonian, it follows that Schrodinger's equation takes the form ii>1i(p,q,t)= (p,q,t\H(P,Q)\4>) d d (5'12> valid for all \t]). We note that the universal propagator is also a solution to Schrodinger's equation iK (p, 9, *; p\ 9', t') = H(i, q + i^)K(p, 9, *; p', 9', *') (513) Second, one interprets the resulting Schrodinger's equation as an equation for two canonical degrees of freedom. In this interpretation yi = q and y2 p are viewed as two "coordinates", and one is looking at the irreducible Schrodinger representation of a special class of twovariable Hamiltonians, ones where the classical Hamiltonian is restricted to have the form Hc(p\,y\ p2). Based on this interpretation a standard phasespace path integral solution may be given for the universal propagator between 59 sharp Schrodinger states. In particular, it follows, after some change of variables, that the universal propagator is given by the formal path integral K{j/9,(/9,f\Jd,tl)= I e'/l^^^^^^^^p^, (5.14) where x and k are the "momenta" conjugate to the "coordinates" p and q [23]. We will now show that it is possible to introduce an appropriate universal propagator for E(2) coherent states by following the construction outlined above. 5.1 E(2) Coherent States and their Propagators 5.1.1 E(2) Coherent State Representative of Functions on L2(S1rd0) If a group G acts linearly on a vector space V, the group law on the semidirect product V(s)G is defined to be (t>2,&)(t;i,0i) = iv2 + R(92)vu929i) (5.15) where R(g) is the operator representing g G G by its linear action on V. In the case of (2) = R2(s}0(2), a unitary representation of the group is associated with the commutation relations of the selfadjoint operators [X, Y] = 0, [X, J] = iY, [y, J] = iX, (5.16) where X and Y are two translation generators and J is the generator of rotations [27]. For the group E(2) there is a oneparameter family of unitarily inequivalent irreducible representations, each of which can be realized on the Hilbert space L2(S1rd0) of square integrable, periodic functions of an angle variable 9 with tt < 9 < it, (Xil>)(6) = rcosOrp(0), (Yj>)(6) = rsm 9^(0), (J^)(6) =i^(0). (5.17) The representation of the group itself can be taken as (U(a,brcU){9) = eiarcos$ihr8inet/>(0cY (5.18) 60 where (a, 6) and c are elements of R2 and SO(2), respectively. The parameter r that specifies the representation lies in the range 0 < r < oo and its square is the value of the Casimir operator X2 + Y2. We are going to consider those reducible representations that can be written as a direct integral of irreducible representations with the parameter varying such _2 a direct integral is to use the Hilbert space L of functions defined on the annulus 0 < R < r < R < oo, in the plane. So, with each fiducial vector \rj) G 2, we associate the family of coherent states, a,6,c,7?) = eiaReiaXelbYelcJ\V), (5.19) where the fiducial vector satisfies R (ri\ri) = J J \ri(r,e)\2rdrd6 = 1. (5.20) R S1 The factor etaR has been appended to the group action to ensure proper limiting behavior as R > oo. The map from vectors tf? G L (S ,d9) to functions ^(a,^, c) on E(2) is defined by R 0^(a,6,c) = (a, 6, c, rj\^) = J J (a,b,c,r)\r,6)(r,6\^)rdrd6 R S1 R (5.21) eiaR [ZjA12" / /r/>^c)elflrcosWrsinV()r, r/r m\ J J (R R2) v R S1 where we have used the resolution of unity R J j \r,0)(r,6\rdrd6 = / (5.22) R S1 in terms of the delta normalized eigenkets r, 6). Also, since G L2(Sl,d6), we must have 61 to ensure that r (00)= / J(^\r,6)(r,6\^)rdrd6 r S1 r 2 (5.24) / / \*/>(O)\2rdrd0 = J \^(0)\2d0 R2 R2 r s1 s1 Thus, the 'length' of a vector is OO OO 7T OO OO 7T i*[_2]I / ^(r^_c)ercostf+.6r8intf^rdr^2d^a?6)C) (5.25) (R R2) J \m\ 2 dO. s1 _2 In the expression above we have set d//(a,6,c) = [(R R2)/2][dadbdc/(27r)2], and used the fact that r 7T r 7T //,M^ = //*<)^ = l. (5.26, # 7T TT The functions ipTJ(a,b,c) are bounded and continuous, and the set of functions on E(2) obtained by this transformation on 0 E L2(S1^d9) is a proper subspace, say C, of all elements of L2(R2x51,dfi(a,6,c)). An inverse transform from 0,y(a, 6,c) E C, to 0 6 L2(S\d9) is given by 72 9]* / rdre'fl* / e"iarC08'"6rsin%(r,0 c)^(a, 6,c)^(a, 6,c) r = [sJ2" / rdr / (r,0a,6, c, rj)(a,b, c, 7/0)d//(a, 6, c) (5.27) (n i?2) y y 2 2 _*J* / rdr(r,0\il>) = 0(0). (R R?) 62 An inner product is given by OO OO 7T (4>\4>)= J J j OO OO 7T (5.28) S1 These properties demonstrate the unitarity of the coherentstate transformation between L2(S\d6) and C. Since C cL2{R2xS\dfi(a,b,c)) is isomorphic to L2(S\d6), the functions 0(0) = (00) and i/jv(a,b,c) = (a, 6,c,t/0) can be regarded as two different representations of the single abstract ket 0). Thus, in the braket notation, the basic result (25) can be written in the form OO OO 7T / / a, 6, c, 7/)(a, 6, c, r/d//(a, 6, c) = /, (5.29) OO OO 7T which is the usual form of the coherentstate resolution of unity, albeit for a reducible representation of E(2). 5.1.2 Surface Constant Fiducial Vectors To make full use of coherent states, we wish to have available the reproducing kernel property (a"rb",c",r,\rj>) = / (a", 6",c",7a', 6',c',7)(a#, 6;, c',i/0)^(a;, 6', c') (5.30) for arbitrary 0 G L2(S1,d0), and the idempotent property OO OO 7T (a",6",c",r/a,6,c,77)= / / J (a" ,b" ,c" ,r,\a\b\c',V) (5.31) OO OO 7T x (a', 6', c', 7/a, 6, c, rj)dfi(a, 6', c,). The inner products in equations (30) and (31) do not involve vectors that lie in L2(Sl,d6) alone, simply because (r, 0a, 6, c, 7/) depends on r. Consequently equations (30) and (31) 63 do not hold for a general fiducial vector \rj). The restricted class of vectors for which the above two relations hold can be seen as follows. First, we have / W iff ff I / \ ia"Rr ^ i (a ,6 ,c ,7/10) = e ] (R R2) R J J 9*(ri^ ~ c")e (5.32) ^ f Jl\ia"r cos 9+ib"r sin 9 ^{6)rdrd6 R Sl Second, we also desire to have (a",b",c",rj\i/>) = / (a",b",c",ri\a',b',c!,ri){a',b',c!,Tj\rJ;)dfi(a',b',c/,) ia"R { ei(a''a')rcosS+,(6''6')rsin%*(r^c'')7?(r^C')r^}x R {R R2) J J v R S1 R a"*r_2if / / eia"r cos Wr rin *(r> 0 { / 7/(r,flc')2(7? ~^ ^dc'}rdrdO. 7T (5.33) Hence, for (33) to equal (32) we see that our fiducial vectors need to satisfy the condition, 7T 7T / \rj(r,0c')\2dc' = I Iv^c'tfdc' = ^. (5.34) J J R R2 We will call such vectors "surfaceconstant" vectors. The conclusion is that if the reproducing kernel property (30) and the idempotency condition (31) are to hold, surface constant fiducial vectors must be employed, even though the resolution of unity holds for more general fiducial vectors. All surface constant vectors can be written in the form *M) = [_22 ]* ^ (5.35) (R R>) ( jr \(r,ct)\2da)* 7T for an arbitrary nonvanishing function (r, 0), for almost all r and 6 64 5.1.3 Propagators The abstract Schrodinger equation ity) = H\il>), (5.36) involving the selfadjoint Hamiltonian H, is formally solved with the aid of the evolution operator U(t) c lHt, namely W)) = e*t"t'>BW')). (5.37) In the E(2) coherentstate representation the time evolution is affected by an integral kernel TS ( II ill II ,11 I il I ,l\ I II iff It iH(t" t') I ll 1 \ /r on\ K(a ,o,c,r;a,o,c,r) = (a,6,c,7/e v ; a 6 c r/) (5.38) in the form iprj(a\ ft", c", t") = J Krj(an ,bn ,cn\th'\a\b\ Clearly Kv depends strongly on the fiducial vector as does i/>v. The universal propagator K(a",b",c",tn; a',b*,c',t') in contrast, is a single function independent of any fiducial vector, which, nevertheless, has the property that *l>r)(a" ,b"\c" = / /i^(a,,,6w,c,\t,,;a,,6,,c/,t')0Jy(a,, 6;,c/,<;)d/i(a/,6/,c,), (5.40) holds just as before for any choice of fiducial vector \rj). The functions Kv and K are qualitatively different as is clear from their behavior as t" > t'. In particular lim Kji(a',b" ,c"\t";a\b'\c ,t') = (a", 6", c7/a', c, 77), (5.41) which clearly retains a strong dependence on the fiducial vector \r)). On the other hand, if (40) is to hold for any \rj) we must require that lim K{a", 6", c", a', b', c', f') = J^^(a" a'W&" b')6{c" c'). (5.42) R R2 65 Now let us turn our attention to a suitable differential equation satisfied by the propagators Kv and K. It is straightforward to see that (z + R){a,b,c,ri\il>) = (a,b,c,r)\X\il>), (ilL)(a,b,c,Ti\tl>) = (a,b,c,V\Y\iJ>), (5.43) d d d (z ia + ib bR)(a,b,c,7]\ip) = (a, 6, c,rj\J0), hold quite independently of r/). Thus if H = H(X,Y,J) denotes a Hamiltonian it follows that Schrodinger's equation takes the form zV^a, 6, c, 0 = (a, 6, c, 17F, J)V>(t)> = a (5 44) # z + R, z, i^ ia + ibbR)%{a, 6, c, <), valid for any \rj). The propagators are also solutions of Schrodingers equation so it follows that iK*(a, b, c, t\ d, b\ c', t') = 01 (5.45) tt / a ^ r% ^ V d i ^ 7T~>\ iv / ? i 111 I 11 \ H(i + R, i, z za + z& bR)I\*(a, 6,c,<;a 6 ,c ,i ), where A'* denotes either Kn or A'. The initial conditions at t = t1', determines which function is under consideration. Equation (45) admits two qualitatively different interpretations: when Kv is under consideration, the operators (ijj + it!), (ijy) and (ij^ iajr + iftjj bR) refer to a single degree of freedom made irreducible by confining attention to the subspace C of L2{R2xS\dfi(a,b,c)) for a fixed \rj) and for all 0 G L2(S\d9). For A' a different interpretation is appropriate. 5.2 The Universal Propagator In contrast to the former case, when the universal propagator K is under consideration the resultant Schrodinger's equation (45) is interpreted as one appropriate to three 66 canonical degrees of freedom. In this view x\ = a, x2 6, and x$ = c denote three "coordinates", and one is looking at the irreducible Schrodinger representation of a special class of Hamiltonians, ones where the classical Hamiltonian is restricted to the form Hc(p\ + R,p2,p% + x\p2 x2p\ x2R), rather than the most general form #c(pi,P2,P3,^2,#3). In the case of K, based on the interpretation described above, a standard phasespace path integral solution may be given for the universal propagator between sharp Schrodinger states. In particular it follows that A [a 0 c ,c ; a,0,c t j = ei J[paa+pbb+pccH(pa+R,p\pc+apbbpabR)]fa (5.46) Here pa,pb, and pc are the "conjugate momenta" to the coordinates a, 6, c, respectively. Note that the special form of the Hamiltonian has been used. Also, (pa,pb) are continuous, while pc is a discrete variable. In the standard phasespace path integral there is always one more set of (pa,p ,pc) integrations (summations) compared to the (a, 6, c) family. This situation is made explicit in the regularized prescription for the path integral given, in standard notation, by 00 00 K(a",b",c",t"a',b',c',t,) = lim / JT ... V _ x _ o2 e0 v 11 11 pc =00 vc, = oo 2 2 N (5.47) t'Ebr+i('+i<)+p' ,(6+i6)+P'+i(ci+ic)^] JL JL rfpf. idpj, , (=o +* + TT^.^a.^.TT _il_lI /=! f=0 V ' where in the expression above (N + l)e = (t" t') and the boundary conditions are (oami,bff+i,cjv+i) = (a",6",c") and (a0,&o,co) = (a',tf,d). Also, fe4+Mwa + tl) ff) (5'48) 2 P'+* 2 ^' and the momenta pc = n, where n is an integer. We will now construct several examples of the universal propagator. 5.2.1 Vanishing Hamiltonian 67 The propagator, when the Hamiltonian is zero, is given by N 2(2tt)2 f ^2 'S^^0'41"0'^^^1"6'^'4^0'410'^ ^2 /?2 p~ 1 p\ x{^Mq1 2 2 /=1 /=o v ; 8tt at tv _2 a')*(6" V) I f[{Y, + 2l)} IIdc* R R2 1=0 mi l=\ S*2 6(a" a')6(b" b')6(c" c'). R2R2 (5.49) In the above we have used the fact that tt < c\ < tt and m/ Z. Actually, with this restriction on q, only the ra/ = 0 term contributes. It is evident we have obtained the correct result. 5.2.2 Linear Hamiltonian Next we consider the linear Hamiltonian H(X, Y, J) = aX + /3Y + ~/J. This case will help us in the proof of the fact that the universal propagator for a general Hamiltonian propagates all state vectors correctly. Thus, for the linear Hamiltonian js / // ill II ,11. I il I ,l\ if{paa+pbb+pcca(pa+R)tlpb^^ (5.50) We now make the following change of variable pc > pc apb + bpa + bR, and the 68 resultant regularized path integral becomes ts( 11 l/l ft ft! t 4*\ A (a 0 c ,1 ; a 0 c z J = 1 J \paa+Vbb+Vccapbc+bpac+bRca^ 2 n JLdp^dp* 2(27r) jgRT R2R2 X (2*) 2 e <= 7 7 1 7 x e /=0 87r2e~ic*RT a (3 o(a a COS7T + 6 sin7Tsin7T(cos 7T 1)) 7 7 0 a x (5(6" b' cos 7T a/sin7Tsin7T H(cos 7T 1)) 7 7 oo x V {6(c" c ^T + 2irm)}. m= 00 (5.51) Here we have set (t" t') T'. It can easily be verified that the above propagator satisfies the 'folding' property jK{a", 6", c", T + S] a!, &', c', S)K{a\ c', 5; a, 6, c, 0)d//(a', 6', c') (5.52) K{a", 6", c;,? T + 5; a, 6, c, 0). 5.2.3 A Quadratic Hamiltonian The last example we consider here is the quadratic Hamiltonian H(X, Y, J) = aJz. 69 For this Hamiltonian the propagator in the regularized form is given by ist a til a 411 1 ui I *i\ A (a 0 c t ;a ,6,c,i ) = i J\j>ad+pbb+pcca(pc+apbbpabR)2]dtpp^ 1 2>i+1 c')*(p,+ 1 + 5pi+ 13Pt+12R) 1 x e <=o 2 OO = 2e^a,)6[(a"2 + b"2) (a'2 + 6'2)]L_ V e^S* . (4ttiq>T)2 m=_oo The last factor in this result agrees with one obtained by Schulman using different (5.53) techniques [28]. The 6 function tells us that the particle is confined to move on a circle of radius r = (a'2 + b'2). 5.3 Propagation with the Universal Propagator Here we present an argument showing that the universal propagator evolves all wave functions of the system correctly. Let in the following, /(a,/?,7,T) = e%T(QX+Py+iJ). One can show that U( Cc L2(R2xSV//(a,6,c)) under C/(a,/?, 7,71) the following 0,M,c,T) = (a,b,c,V\U(aJ^T)\xl>) = e~iaRx eia#^ei(c7r)7ez'[& cos yTa sin 7T sin 7T^(cos 7T1)]Y x i[a cos 7T+6sin 7T sin 7Tf(cos 7T 10) (5.55) = 0([acos7T + 6sin7T sin7T + (cos 7T 1)1, 7 7 [bcos^T as'm^Tsin7r(COS7T 1)], [c + 27rn]). 7 7 70 Where n is an integer such that 7r < (c 7T + 2?Tn) < ir. From our discussion earlier, we know that there exists a universal propagator such that 0v(a, 6, c,T) = J K(a, 6, c, T; a', 6', c, 0)ipv(d, 6', c')dn(a', 6', c). (5.56) Thus, using ^(a, 6,c, T) from equation (55) we have 0r?([acos7T + 6sin7Tsin7T H(COS7T 1)], 7 7 [6COS7T a sin 77sin7T(C0S7T 1)], [c 7T + 27rn]) (5.57) 7 7 7i (a, 6, c, T; a', 6', c, 0)ipv(a', 6', c')dfi(a', 6', c'). If the above equation is to be valid for arbitrary 7/ and 0, we must require that K(a,b,c,T;a',b\c',0) = 2^ e~iaRTx R R2 6[d a cos 77 6sin7T Hsin7T(cos 7T 1)1 x 7 7 0 oc 6[b' bcos 77 + asin7T Hsin7T H(cos 7T l)]x 7 7 (5.58) 00 {(5(c'c + 7r27rm)}. 777 = OO This expression agrees exactly with our result obtained earlier for the Hamiltonian H(X, y, J) = aX + (3Y + 7J, and therefore establishes its validity. Now we show that the universal propagator for an arbitrary Hamiltonian evolves the state functions correctly. Let f/(a,/9,7,1) = [/(a, ,#,7). One can represent every bounded function of X, F, and J as a weak limit of finite linear combinations of the set ?7(a,^,7) [24, 25, 29]. In particular, we can represent every time evolution operator as such a weak limit of finite linear combinations of the /(af,/3,7). Let us denote by 77 j=0 and also let 77 Kn{a\ b1c', T\ a, 6, c, 0) = y SjKQj/3jJj (a, 6', c', 1; a, 6, c, 0). (5.60) ;=0 71 It follows that every time evolution operator is a weak limit of S, i.e., ( nKX) Then in particular we have (a,b,c,7]\eiTH\il>) = lim (a, 6,c,t/SJV>), V ^ G H. (5.62) 71>OG Now let us denote by A"(a, 6, c,T; a', c', 0) the universal propagator associated with the time evolution operator e *TH; Equation (62) can now be written as an equation for linear functionals, and takes the following form, K(d\ b', c, T\ a, 6, c, 0)iprj(a, 6, c)d/z(a, 6, c) 77?oo lim / Kn(a\ b', c, T; a, 6, c, 0)^(a, 6, c)d[i(a, 6, c), (5.63) for all ^(a,6, c) C. Hence, every propagator can be written as the weak* limit of an appropriate Kn, i.e. AT(a\ 6', c', T\ a, 6, c, 0) = w* lim 6', c', T; a, 6, c, 0). (5.64) 71?OO Although the point is clear from the foregoing, it is worth emphasizing that the universal propagator evolves any state in a way that leaves the choice of the fiducial vector \rj) invariant. 5.4 Classical Limit Although the universal propagator has been derived by identifying the relevant Schrodinger equation as one for three degrees of freedom, it should nevertheless be true that the classical limit refers to a single degree of freedom constrained to lie on a circle. This is possible because of the restricted form of the classical and quantum Hamiltonians. The classical action appropriate to the E(2) coherent state path integral is Ic\ = lim / [z(a, 6, c, rj\ a, 6, c, 77) (a, 6, c, rj\H(X, F, J)a, 6, c, rj)]dt. (5.65) h^o / dt 72 Usually one restricts the fiducial vector \tj) such that (771A"(77) > R, as R > 7?, and also that (//Kr/) = 0, (771^177) = 0. A surface constant fiducial vector that has the above properties is given by 9 x Rrcos6 9 1 Rr cos 6 V0(r,6) = [5]lr = [12 (5.66) \r2R2) (Je2/2rcosarfa) \r2 R2)> (2*I0(2J&))* where Iq is the Bessel function of zeroth order. Hence, under the standard assumptions given above the classical action reads, ^c/ = / [clRcos c+ bRs'mc H(Rcosc, i?sinc, 7?(a sin c 6cos c))]dt. (5.67) Extremal variation of this expression with respect to a, 6, and c holding the end points fixed leads to the equations of motion. The variation with respect to a and b give the same equation, hence the equations of motion are, = #3, aR sin c + bRcosc = H\R s'm c + #2 A! cos c + i/3 (aR cos c + bRs'm c), (5.68) where in the expression above ft = dH^x*\ (5.69) We note, if we set p = i?(asinc 6cose) that the equations of motion (68) can be written as ._ dH OH C ~ dp P (9c ' (5.70) which are just Hamilton's equations of motion appropriate to a single degree of freedom (constrained to move on a circle). We denote a generic solution of equation (70) by pc(t) and cc(t), i.e., P(t) = Pc(t) = R[ac(t) sincc(t) 6c(t)coscc(<)], c(<) = cc(t). (5.71) 73 However, when we deal with a more general fiducial vector ?/), such that (r,\X\V) = Xv, (r,\Y\V) = K, (t/I Jt/) = j, (5.72) are generally all nonvanishing, the classical action, in the limit of zero dispersion, becomes Icl = J[a(Xrj cos c Yfj sin c) + b(Yv cos c + sin c) + cJ^ H(XV cos c K sin c, cos c + Xv sin c, (5.73) Jrj + ^(a sin c b cos c) + Yq(a cos c + 6 sin c))]dt. In this expression JC^ly, and Jv are time independent constants. The term J(aR + cJ)dt = (a" a')R + (c" c') Jv is a pure surface term and will not affect the equations of motion; it could be eliminated simply by a phase change of the coherent states. Extremal variation holding the end points fixed leads to the equations of motion = #3, a^Xj) sin c + Y cos c) + b(Yv sin c Xv cos c) + H^aXfj cos c + bXv sin c aYv sin c + bYv cos c) (5.74) = Hi(Xr) sin c + Yjj cos c) + H2(YV sin c cos c). Now, if we set p = X^(a sin c fecos c) + Yr](a cos c + 6sin c) the equations of motion can be written as dH(Xv cos c Yff sin c, cos c + Xv sin c, p + Jv) <9p * dH oc Which have as their solutions c(t) = cc(t), (5.75) pc(t) = XJac(t) sin cc(t) bc(t) cos cc(t)) (5.76) + Yv(ac(t) cos cc(t) + bc(t) sin cc(t)) Jv. Observe that the generally nonvanishing values of X^Y^ and Jv are vestiges of the coherentstate representation induced by the fiducial vector t/) that remain even in the limit h ? 0. 74 5.4.1 Classical Solutions In the case of the universal propagator the expression that serves as the classical action is identified as Icl = J[pah + phb + pc6 H{pa + R,p\pc + apb bpa bR)]dt (5.77) Extremal variation of this expression holding the end points fixed leads to the following set of equations h = H1bHz, b = H2 + aH3, c = #3, pa = pbH3, ph = (pa + R)H3, pc = 0. The solution to the last three equations can be written as pa(t) = oc\ cos c(t) a2 sin c(t) i?, (5.78) pb(t) = a2 cos c(t) + a\ sin c(t), (5.79) Pc(t) = a3, where ai,a2, and a% are arbitrary constants of integration. The first two equations can be combined to give [apb b(pa + R)] = Hipb H2{f +R) = (5>80) at 0~ Hence we see that if we define p = apb b(pa + i?), the equations of motion can be written as p = H(ol\ cos c a2 sin c, a2 cos c + cc\ sin c,p + a^), a (581) c = H(ot\ cos c a2 sin c, a2 cos c + a\ sin c, p + 0:3), up which are seen to be Hamilton's equations for a single degree of freedom. The solution to the above equations can be written as P(t) Pc(t) = ac(t)[a2 cos cc(t) + ot\ sincc(?!)] bc(t)[ai cos cc(t) ot2 sin cc(t)] a3, (5.82) c(t) = cc(t). 75 Where ai,at2, and a% are arbitrary constants as mentioned before. Among all possible values of ai,a2, and c*3 are those that coincide with XV,YV,JV corresponding to the fiducial vector \rj). Thus, we find that the set of classical solutions obtained from the action appropriate to the universal propagator includes every possible solution of the classical equations of motion appropriate to the most general coherentstate propagator. The techniques discussed in this chapter may be extended to E(n) coherent states, suitable for quantizing a system whose configuration space is the sphere [27]. CHAPTER 6 DISCUSSION AND CONCLUSIONS We will now summarize the results of this thesis. Our goal had been to study methods of handling constrained systems in a functional integral framework. Dynamical constraints are one of two types; firstclass or secondclass constraints. We have studied three models, each with a finite number of degrees of freedom, and each with either firstclass constraints or secondclass constraints. Our 'tools' have been coherentstates, the formalism surrounding which is exceptionally rich. The principal assertion of this thesis is that, for constrained systems, in the construction of the path integral representation of a propagator one should use a projection operator rather than the resolution of unity at every time slice. The projection operator, for a given system, is constructed using a complete set of states in the physical subspace of the system. The main difference in the treatment of systems with either firstclass or secondclass constraints is the way in which the physical subspace is identified. Once the physical subspace of the system is identified and a complete set of states in this subspace found it is straightforward to construct the desired projection operator. The projection operator is used at each time slice of the propagator and as a state of the system evolves in time, the projection operator ensure that the state remains in the physical subspace at every infinitesimal time step of the evolution. In chapter 3 we studied a model with two firstclass constraints. We constructed the projection operator appropriate to the model. It is found that the projection operator can be written as a properly weighted integral over independent bras and kets which were called Bicoherent States. The use of the projection operator leads to the correct measure for the path integral. Also, the projection operator which is an integral over bicoherent 76 77 states leads to an action that is complex and has twice as many labels as the standard action. We found that in the classical limit among these labels, which number twice the normal labels, pairs of them follow identical trajectories which correspond to the classical trajectories. It is also found that on the classical trajectories the real part of the action reduces to the standard classical action while the imaginary part becomes just a surface term. The projection operator leads to a measure that is path dependent. This path dependent measure is 'modulated' by the imaginary part of the action The formalism studied here additionally has the desirable feature whereby one can turn on and off the constraints as needed. For instance, for the model studied in chapter three if we wish the system to evolve under the constrained Hamiltonian Hi = ^p2 + V(x) + ypTx + upTx + vn for an interval of time T\ with the two firstclass constraints, and subsequently to evolve under an unconstrained Hamiltonian say 7^2 = jP2 + V(x) for a period T2 the propagator is given by (6 1 ^ , CiT + C5 C3 + ZC4 Ci + ic2 iTtHi a _L c3 + ic* cl + ic2 > Ma^' y/2 y/2 V2 6 a'^v^' y/2 y/2 h where the factor (a",f3n\e~lT2n2\a', fi1) in the integrand above is evaluated in the usual manner by introducing resolutions of unity at each time slice, whereas the term (O'^.fl^.a^i,^^^!"^,^,^,^) would be evaluated as discussed in section 3.3. The Dirac delta function in the integrand above ensures that at time t = T\ the system is on the constraint surface. In chapter 4 we apply the formalism of bicoherent states to systems with second class constraints. In this chapter we have studied two models. The crucial step of the quantization process is the identification of the physical subspace. For systems with secondclass constraints one starts of by identifying Hph, on the way to identifying a complete set of vectors in the physical subspace. For the two models studied in this 78 chapter the physical subspaces were spanned by the eigenstates of Hph which were known. We used these states to form the projection operator which was used in the construction of the path integral representation of the propagator. So, in essence we wrote the path integrals for problems whose solution we already knew. But in the case where one does not know the eigenstates of Hph, all one has to do is find a suitable operator, say O, that commutes with Hpu and whose eigenstates are known. The eigenstates of O then span the physical subspace and can be used to construct the required projection operator and hence a path integral for the propagator. In conclusion, we note bicoherent states are quite versatile. In appendix A we saw that they can be used to construct path integrals for unconstrained systems. Thus, the quantization 'recipe' discussed here using bicoherent states amounts to obtaining the correct measure for the path integrals. So, one can have, in principle, an identical action in the path integral for a constrained and an unconstrained system but the measure would be entirely different. Also, all integrals are regular when bicoherent states are used just as in the case when coherent states are used. The formalism developed here and applied to systems with a finite number of degrees of freedom should be extendable to field theories. 79 APPENDIX A BICOHERENT STATES AND PATH INTEGRALS In this section we highlight the main features of path integrals constructed using bicoherent states. For simplicity we consider a system without constraints and with a single degree of freedom. To begin with, notice that the unit operator can be written as 1 1 2 d2a 7t a) (a d2(3 l/WI d2ad2$ (A.l) 7t 2 a)(fi\e i(a2 + /?2)+aV a weighted integral over bicoherent states. We will use this form of unity in our con struction of the path integral representation of the propagator. Consider the Hamiltonian H p2/2 + V(x). In the construction of the propagator we use the normal ordered Hamiltonian H = : H : Thus, the propagator is given by / w (a iTH a') n la ieH icH ajv_i)...(/?ie a ') n *A n1 n n (A.2) n=0 n+1 icH OLn) Y[ dt*n> 71=1 where (N + l)e = T and n = 1,2...N. In the equation above, the boundary conditions are (ao,/?/v+i) = (a',a") and the measure at each time slice is given by dPn d2and2fin L(\an\*+\0n\*)+a*0 7t 2 n (A.3) For small e, we evaluate, to order e, each term in the integrand in (A.2) as follows: (fin + 1 teTi Otn) (fin+x\[l ieH)\an), (/?n+ia)[l ieHn+in] ~ (j8n+ian)e itHn+in (A.4) where in the expression above is ^77 + 1,71 (0n+i\n\an) (/?+! a) (A.5) 80 Also, the overlap of coherentstates at each time slice is n + l Ocn) ei(/?n+l2 + On2)+/^+1an (A.6) Thus, provided the integrals exist, the propagator is given by la" iTH a') N N J(/?n+i2+a2)++1awfeiTl>+iin Yl dli"n. n=0 n=l (A.7) In the expression above we notice that the factor e HI&hiI +lQrl2)5 except at the end points, can be absorbed in the measure. So, our propagator becomes N la iTH a') e 2 + ,2, f E^"+la" "I ) I e= l'e//n + l,n] N n (A.8) 71=1 where the measure at each time slice has changed slightly and is now given by d2and2(3n (\an\*+\0n\*)+aMn 7t 2 (A.9) In our quest to express the right hand side of (A.8) as a path integral, we rewrite part of the exponent as follows, N N 7? 1 n=0 n=0 2 Wn + 1 77 77 + 1 {0ln + 1 <*n)] N (A. 10) 1 + S 9 ^a + ^n+lan+l] 77 = 0 Notice that in the above equation the terms (fio^^N+i) have not been defined yet. The factors containing these terms cancel and so they can take on arbitrary values. We assign them the following values: = (a', a"). Also, the second sum in (A. 10) can be absorbed in the measure and so our propagator is finally written as N I tl la iTH a') Elj[(^r^^lkln)]flfn+llB) N 6n=0 (A. 11) 77=1 where the measure at each slice is now dun 7t 2 7t 2 (A. 12) 81 Thus, in the limit N + oo, e 0 with (JV + l)c = 7\ the right side of (A.ll) can formally be written as an integral over continuous and differentiable paths as = ( 5^^)m)HV (A.13) 