The hamiltonian path integrals and the uniform semiclassical approximations for the propagator

.AUKALS

OF PHYSICS108. 165-197 (1977)

The Hamiltonian Uniform

Path Integrals and the

Semiclassical Approximations S. LEVIT

for the Propagator

AND U. SMILANSKY

Department of Nitclear Physics, Weizmann lnstitrite of Science, Rchhowt, Israel Received December 30, 1976

The generalized path expansion scheme is defined for path integration in phase-space. Within this framework we study the semiclassical limits to the propagator, both in the momentum and the coordinate representations. It is shown that the role played by the Morse operator in the Lagrangian formulation of the path integral method is taken by another differential operator of the Dirac type. The relevant properties of this operatorarediscussed. The semiclassical approximations are obtained by extending the results of catastrophe theory for the asymptotic evaluation of finite-dimensional integrals to the domain of path integration. Various forms of the uniform semiclassical approximations are obtained. Their validity and applicability are discussed. The method is illustrated by a solution of a simple example in which nongeneric catastrophe occurs.

1. INTRODUCTION

The present study was stimulated by the recent applications of the semiclassical techniques to the description of scattering phenomena. These techniques developed by Miller [l, 21 and Marcus [3] have been successful in the description of molecular collisions. Attempts to introduce similar techniques to treat nuclear scattering problems [4-61 have met with considerable difficulties. The strong and rapidly varying nuclear interactions produce a complicated structure in the space of classical trajectories. This structure affects the interference between the contributing trajectories and leads to the divergences and discontinuities in the results of the calculations. The prescriptions which were introduced and applied in describing molecular collisions, were found to be insufficient in the nuclear problems [6, 71. In order to overcome thesedifficulties, we undertook to derive the semiclassicalapproximation from the exact quanta1 expression. An appropriate basis is provided by the path integral formulation of quantum mechanics. The path integral formulation can be carried out using two different approaches. The original approach of Feynman [9] dealt with paths which were defined in coordinate-space and the Lagrangian description was used to define the dynamics of the system. An alternative approach was introduced by Garrod [IO] and Davies [II], who defined the paths in phase-space and used the Hamiitonian formulation to obtain their results. Copyright O 1977by Academic Press,Inc. X1 rights of reproduction in any form reserved.

165 ISSK

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The Hamiltonian approach is more general than the Lagrange formulation since it can describe a larger class of physical systems [12, 131. It has another important advantage in introducing a formal symmetry between the momenta and the coordinate variables. This fact has important consequences for semiclassical analysis [l]. Gutzwiller [14] gives a clear discussion of this point. Unfortunately, the Hamiltonian formulation was not discussed in the literature as extensively as the Lagrangian formulation. It is the purpose of this work to fill this gap, especially in as far as the semiclassical approximations are concerned. Throughout this paper most of the emphasis will be put on the derivation of the semiclassical approximations for the propagator in the momentum representation. The corresponding results for the coordinate representation will be quoted and discussed only when there are essential differences between the two representations. We shall start our discussion by adopting the path expansion method to the Hamiltonian formulation. From previous studies [15, 161 of the Lagrangian approach, it is known that the path expansion technique is most appropriate for deriving semiclassical approximations. It has the advantage of establishing a hierarchy of importance to the integration variables and it introduces the powerful tools of the Morse theory [17] in a natural way. The flexibility in the choice of the path expansion basis is also important for the applications. The semiclassical approximations are obtained performing the path integration by appropriate asymptotic methods. The common feature of these methods is the fact that the most important contributions to the path integral come from the vicinity of the classical paths which are consistent with the boundary conditions [l, 91. Since usually there exist several classical trajectories, the semiclassical propagator reflects the quanta1 interference between the various contributions. As long as the classical trajectories are well separated in phase-space, one can use the simple stationary phase approximation in order to obtain the semiclassical expressions. This approximation diverges when the classical paths tend to coalesce, that is, when the paths approach a focal surface [18]. Such situations are referred to as catastrophes [19, 201 and more sophisticated asymptotic integration methods should be applied. The treatment of catastrophes was given in the literature for finite-dimensional integrals [19, 201. Using the path expansion technique we are able to extend this treatment to the domain of path integration. The resulting expressions approximate the path integrals in a uniform way over the complete range of classically allowed transitions. The relevant information which is necessary for the semiclassical analysis is contained in the properties of the second variation of the action functional related to the classical paths. In the Lagrangian approach the results of the Morse theory [17,21] provide this information. There, the second variation depends on the eigenvalues of the Morse differential operator [15]. It is Sturm-Liouville-type operator which has a finite number of negative eigenvalues. This number defines the phase of the semiclassical propagator [14, 211. The Morse focal point theorem [17] relates it to the number of focal points along the classical path. We show that the situation is different in the Hamiltonian case. The second variation

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of the action depends on the eigenvalues of a Dirac-type operator [22] which has an infinite number of negative eigenvalues.The definition of the phase of the propagator is therefore not as simple as in the Lagrangian case.In Section 3.4 we show how this phase should be calculated. In the coordinate representation a simple practical prescription can be given. Unfortunately we were not able to find a similar prescription for the momentum representation. In order to handle the infinite products of eigenvalues appearing in the expression for the semiclassicalpropagator we prove in Appendix B a useful theorem which extends the results of a similar theorem proved in the Lagrangian case [23]. This theorem provides a closed expression for the propagator. In Section 3.5 we present the discussion of how the catastrophes should be treated by meansof the uniform semiclassicalapproximations to path integral. The discussion is based on the concept of an “unfolding of a singularity” [24. 351which we extend to the case of path integration. In certain situations the physical system itself provides us with the function which unfolds the singularities in the semi-classicalpropagator. In thesecasesthe propagator may be approximated by a finite-dimensional integral (Section 3.6). This representation is analogous to the Initial-Value Representation (IVR) formulated by Miller [l, 21 and Marcus [3]. Various uniform expressions used in the study of molecular collisions were all derived [26] on its basis and the validity of IVR was always taken for granted. Our derivation of the IVR enablesus to discussits limitations. We show that it might lead to seriouserrors if applied outside its range of validity [a]. This seems to explain the difficulties arising when the IVR was indiscriminately used for the nuclear scattering problems. Interesting examples of the catastrophes in momentum space arise when some symmetry is breaking in the Hamiltonian of the system. We discussa simple example of such a catastrophe. We conclude the paper by pointing out several directions in hhich the semiclassical approximation could be improved and extended.

2. PATH

INTEGRATION

IN THE HAMILTONIAN

FORMALISM

A general physical system with M degreesof freedom is described by its generalized coordinates q = (ql ,..., qM) and cooresponding momenta p = {pl ,..., pM]. The motion of the system is governed by the classical Hamiltonian H(p, q). In the Hamiltonian formalism the path integral expression for the propagator can be written in both coordinate and momentum representations. Consider first the propagation of the system in momentum space from the point p’ at the time t’ to the point p” at the time t”. The path integral expression for the propagator in this case [IO, 111is K(p”, t”; y’, t’) = J eiT[*cf)*“ct)lD[q(t), p(t)]

(2.1)

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where D[q(t), p(t)] is the “path differential”

SMILANSKY

[l 1, 131 and

Tk(t),p(t)l = (I/@ s,;” [-qj - H(p,

q)] dt

(2.2)

is the classical action functional in momentum space in units of +i. In the present context a path is represented by a 2M-dimensional vector function. Its first M components display the variation of the coordinates in time, whereas its last M components refers to the momenta. We shall consistently use the following notation. 2M-dimensional vectors and matrices will be denoted by Greek symbols, whereas latin latters will be reserved for M-dimensional objects. In the spirit of the Hamiltonian mechanics the coordinates and momenta are varied independently during the path integration in (2.1), the only restriction imposed is p(f) = pw P(f) = P’; (2.3) for all paths. The propagator K(q”, t”; q’, t’) in coordinate representation [lo, 111 with the action functional replaced by m(t),P(ol

is also given by (2.1)

= (l/fil s,;- [PP - H(PP 411 dt*

(2.4)

The boundary conditions on the paths are now q(f) = 4’;

q(f)

= q”.

(2.5)

Because of this similarity between the coordinate and momentum representations, both can be discussed in parallel. We shall explicitly treat the more interesting and less investigated momentum representation mentioning only the final results for the coordinate case. Recently, it has been noticed [13] that the function H(p, q) appearing in (2.2) and (2.4) should be taken as the Weyl transform of the quantum-mechanical Hamiltonian operator, which differs from the classical Hamiltonian function by terms of order h2. Since we address ourselves to the semiclassical limit of the propagator this difference will be discarded. It is convenient to choose the units such that q and p together with H and t be dimensionless so that the Plank constant fi disappears from (2.2) and (2.4). The path integration in (2.1) was defined by Garrod [lo] in a manner similar to Feynman’s poligon procedure [9, 21, 271 in the Lagrangian formulation, e.g.,

Another method for path integration was successfully used in the Lagrangian approach. This is the method of generalized path expansion [16,28] and in this section we wish to extend it to the integration in (2.1).

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Let

be an arbitrary this path

path satisfying

the boundary

form a linear space of 2M-component

condition

(2.3). The path variations

from

vector functions which satisf)

y(f)

= y(P)

The scalar product in this space is naturally

(2.6)

= 0.

defined by

(2.7) Let {,$‘)(t)j be any orthonormal basis in the path variation space. We show below that such a basis may be conveniently chosen as the normalized eigenfunctions of an arbitrary boundary value proboem of Dirac’s type 222). One can order then the set (x(“)(t)> according to the increasing absolute value of the corresponding eigenvalues. Every path which satisfies (2.3) can be expanded using the basis (x(“)(f)) and the reference path Y(t), (2.8)

= 2 it- {(q&) i&l 1’

-

Q&))

&j(t)

+ (pi(r)

- Pi(t)) $)(t)j~ dt,

(2.9)

where

p(p)

The Nth approximation

=

(p(p)

zzz

0,

(i

=

I,...,

to #(1) is obtained by truncating

M).

(2.10)

the series (2.8) (2. I I )

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The path integral in (2.1) is then defined by K(p”, t”; p’, t’) = Gym K,(p”,

t”; p’t’)

(2.12)

with (2.13) and (2.14) The normalization factor JN was introduced in Eq. (2.13) to provide a proper convergence of the limit (2.12). As in the Lagrangian case [16] it is independent on the choice of the reference path Y(t) and the set {X’“)(t)) in the limit N -+ co. Moreover, in analogy with the Garrod-Feynman limiting procedure JN is then identical for all physical systems with the same number of degrees of freedom. In the Lagrangian case we have extracted the analogous normalization factor by evaluating the propagator for a simple case and comparing it with the known result [16]. The simplest possible case there is the free motion, reflecting the fact that systems with the same mass tensor have identical normalization factors [29]. In the Hamiltonian case the simplest and most natural choice for extraction of the normalization factor is the system with identically vanishing Hamiltonian. The propagator in momentum space for this system is obviously constant in time and is given by K’O’(p”, t”; p’, t’) = 8(pM - p’). (2.15) We shall calculate K(O)(p”, t”; p’, t’) in the path expansion approach, Eq. (2.12). The comparison of the result with Eq. (2.15) will provide the expression for the normalization factor JN . The action integral for K(O) is T’O)

=

_

St” 41, dt = $(p’q’ - p”q”) + 4 1;” l)l(t) F(d/dt) $(t) dt t’

(2.16)

with

and the symplectic 2M

x

2M matrix

r = (; -J. Here lis the M x M unit matrix.

(2.17)

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We choose an arbitrary reference path Y((t) = (Q(t), P(t)) satisfying (2.3) and the basis (x(“)(l)} to consist of the normalized eigenfunctions of the following boundary value problem 2M 1

,=;I

@/y(f)

xj”‘(t)

=

(2.184

rpxl”‘(r),

k = l,..., M.

(2. I8b)

The differential operator @(O)(t) is defined as

@/9’(r) = rjj(d/d).

(2.19)

The matrix I’ is given by Eq. (2.17) and the lower M components of x(*)(t) are denoted by +)(t) as in Eq. (2.10). The boundary value problem (2.18) is of Dirac’s type and its normalized solutions form a complete set [22]. The eigenvalues $A” are easily found and we discuss them in detail in the next section. At this point it is important to notice that the problem (2.18) has an M-degenerate zero eigenvalue. We count this eigenvalue by the first M values of the index (Y.The M corresponding normalized eigenfunctions are

try(f) = (I” - f’)-

6,,f,

(i, Y = I)..., ill).

(2.20)

Inserting the expansion (2.11) in the Eq. (2.16) with (x(“)(t)) defined by (2.18) one gets @[Y(t),

a , ,..., a,,r] = ?-“‘[Q(t), f’(t)] + f a,ql, -1 3 2 a=1 Y-M-I1

&“‘u,2,

(2.21)

where

Tco)[Q(t),P(t)] = - 1’” Qp LI’

Notice that since Eq. (2.18) has an M-degenerate Eq. (2.21) starts from N = M + 1.

dt

(2.22)

zero eigenvalues the last sum of

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The Nth approximation

AND

SMILANSKY

to K(O) is given by

T(O’[Q(t), P(t)] - i

(2.24)

The existence of the zero eigenvalue of (2.18) leads to the appearance of the &function in the expression for the propagator K’O)(p”, t”; p’, t’). We show now that it is indeed a S-function in momentum space. Inserting (2.20) in Eq. (2.23) for qm it is obvious that for cxranging from 1 to M ?P# = (t” - t’)-l/2

cy.= l,..., M,

(p,’ - pz>,

(2.25)

and fi S(!PJ = (t” - t’)M/2 S(p’ - p”).

(2.26)

‘X=1

The remaining terms on the right-hand side of Eq. (2.24) have to be calculated only for p’ = p*. We show in Appendix A that in this case the sum in the exponent of Eq. (2.24) is equal to T’O)[Q(t), P(t)] when N + co. Using this and the result (2.26) in (2.24) we get from the comparison with (2.15) that the normalization factor JN (N -+ co) for the path integration of the Hamiltonian form in the momentum representation is J&l = [(27r)2 (t” - t’)]-M’2

ib+l

(gy2.

(2.27)

Here the product is taken over the nonzero eigenvalues of the operator @O)(t) given by Eq. (2.18). For a finite value of N one should add a correction term in the expression for JN , which vanishes in the limit of N + ao. Since we are always interested in this limit the correction term will be neglected throughout the following discussion. The expression (2.27) is the main result of this section. Together with the definitions (2.12) and (2.13) it provides a generalized path expansion scheme for performing the path integration in (2.1). One can define a similar scheme for the path integration in the coordinate representation based on Eq. (2.4) for the action functional. The normalization factor in this case is also given by Eq. (2.27) but the boundary conditions (2.18b) should be changed to up)(f) = ui”‘(t”) = 0, for the upper M components of x(“)(t).

i = I,..., A4

(2.28)

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173

It should be finally noticed that in the usual system of units one has to replace the factors 27r in Eq. (2.27) by 27ri.

3. SEMICLASSICAL

EXPANSIONS

3.1. Separated ClassicalPaths

In this section we study the semiclassical limit of the propagator. In this limit the main contribution to the path integral (2.1) comes from the neighborhood of the classical paths q(t) = 40(t) > p(r)

= p’?(t) c

3

i =

I,..., s,

(3.1)

which satisfy (3.2)

and the boundary conditions (2.3) (or (2.5) in the coordinate representation). Each classical path contributes a phase given by the action along the path and in addition a correction factor reflecting the small quanta1 motion around the path. Both the phase and the correction factor may be calculated classically [l, 14, 301. In general the propagator depends on various parameters such as the initial or final momenta, the propagation time or any parameters in the Hamiltonian. The positions of the S classical paths depend on these parameters and it may happen that for a certain value of the parameters some of the paths are nearly coincident. Under these conditions the quantum correction factors in the semiclassical expression diverge. This situation is referred to as a catastrophe [IS-201. Catastrophic phenomena were extensively studied in relation to the asymptotic evaluation of finite-dimensional integrals 124, 251. The path expansion procedure presented in the previous section enables us to extend the methods developed for finite-dimensional integrals to the domain of path integration. We shall start from the simplest case of well-separated classical paths. As will be seen later this case provides a basis for the more complicated situations. Consider the Nth approximant K,,, to the propagator given by Eq. (2.13). The stationary points of T, a, = at),

i =T 1,...) s;

\ := l,.... N:

defined by

~TXW), aI +...,aNI _ o> 2a,

(Y =:=I...., Iv)

(3.3)

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tend to the stationary points of the exact action Tin the limit of N -+ co. The idea is then to evaluate asymptotically the finite-dimensional integral which represents KN and afterwards to take the limit of N -+ co. When the stationary points of TN are well separated the well-known stationary phase method [26] gives the following result for K,,, KN = i I?$’ exp[iTk’]

(3.5)

i=l

where the reduced propagators

are introduced

(27ri)N/2 JN - {det(PT,/&z, &B)‘i’}1/2 ’

j$)

i = l,..., S.

(3.6)

Here the superscript (i) stands for a value calculated at the ith stationary point. The normalization factor JN is given by Eq. (2.27). The quantities T$’ in Eq. (3.5) tend to the action integrals T[q$(t),p$(t)] in the limit of N --f 03. When the stationary points of TN coalesce, some of the R$) diverge, indicating the occurrence of a catastrophe. The information on the catastrophe type is therefore contained in the reduced propagators. The phases of l?$’ are important for evaluating the interference between the different classical paths. We shall examine the properties of the reduced propagators in some detail in the next subsection. 3.2. The Reduced Propagator In this subsection we will be dealing with the reduced propagator X$1 corresponding to a particular classical path. We shall drop therefore the superscript (i) in the definition (3.6) of Kc’. The determinant which appears in Eq. (3.6) is obtained from the second variation of the action functional Tat the given classical path. The second variation is written explicitly as (PT)Cl = q&t)] =

-

f 2xi3ii i=l

f

[&(t)

yiyj + 2&(t)

y,xj + &(t)

xi+]

i,j=l

(3.7) Here the path variations from the classical path Ye’(t) are taken as (3.8) so that yj(t’) = y&“) = 0,

i = I,..., M.

(3.9)

HAMILTONIAN

In Eq. (3.7) the M x M-dimensional

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matrices {&}, {Cij}, and {Oi,} are (3.10a) (3.10b)

The superscript ( )ci indicates evaluation along the classical path. The second variation (3.7) is a quadratic functional with respect to the path variations f(t) and can be diagonalized. In the Lagrangian formulation this is achieved by means of the eigenvectors of the Morse differential operator which is of SturmLiouville type 115, 171. In the present case we introduce the following Dirac-type operator @ij(t)

=

rij(d/dt)

- Eij(t),

(i,j = I,..., 2M)

(3.11)

where the matrix r is given by Eq. (2.17) and the 2M x 2M matrix d(t) is expressed via the M ,,,: M matrices B, C, and D defined in (3.10). (3.12)

Performing the integration by parts one can easily express the functional in Eq. (3.7) as the expectation value of the operator @

I[&)]

(3.13)

The operator @ is self-conjugate in the space of path variations so that its eigenvalues are all real. Together with the corresponding eigenfunctions they are found by solving the boundary value problem

I=1

j=l

(i = l,..., 2M)

(3.14)

with the boundary condition (3.9). This linear system of 2M first-order differential equations is known in the literature as the Dirac system [22]. In our case it can be written in a less formal way.

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with the so-called secondary Hamiltonian

[31] defined by

It is seen that the boundary problem (3.14) is an extension of the Morse boundary problem [17] to the Hamiltonian formalism. Let (I) be the complete set of the normalized eigenfunctions of (3.14)

with the corresponding eigenvalues {+J. Expanding an arbitrary path variation

5(t) = i a0lx’W

(3.18)

ol=l

one obtains for the functional

(3.13) a(01

= fl

+aa,2.

(3.19)

We now turn to the expression (3.6) for the reduced propagator. The result (3.19) shows that as N -+ co the determinant in this expression is given by the product of the &. Thus one gets for the reduced propagator

(3.20)

= [27Ti(f - t’)]-M’2

The reduced propagator is given by the ratio of the infinite products of eigenvalues of two Dirac-type operators di and @(O)defined in Eq. (3.11) and Eq. (2.19), respectively. The limiting process in (3.20) converges because of the asymptotic dependence of & on a: [22]. 3.3. The Jacobi Fields

The ratio of the infinite products in Eq. (3.20) is not easy to handle. However, a useful relation for this ratio can be found which allows one to write it in a closed, form. We present this relation discussing first the absolute value of the reduced propagator. Consider the following system of differential equations ?f Q&t)

i-l

&(t) = 0,

i = l,..,, 2M,

(3.21a)

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where the operator Cpis defined in (3.11). M independent solutions p’(t)

x’“‘(t)

= i

k =-- I..... M.

y’“‘(t) i ’

(3.2lh) I’

of this system may be selected by imposing the following initial conditions at t xf,f’(t’)

= S& )

J&t’)

It, I?I z I,...: hf.

= 0;

(3.‘Ic)

According to the result (B4) of the theorem proved in Appendix B, the solutions of the initial-value problem (3.21a)-(3.21c) are simply related to the infinite ratio in Eq. (3.20), (3.22)

Using this relation we find for the absolute value of the reduced propagator 1fqt”, t’)l zzz(27pP

j d(f’* t’jj-1,’

(3.2) (3.23)

where A(t, t’) = det{y:)(t)).

(3.23)

The result (3.23) expresses the reduced propagator through the solutions of the differential system (3.21a). The equations of this system are analogous to the small disturbances equations in the Lagrangian case [21]. Any of its solution defines a so-called Jacobi field along the classical path [17]. In analogy with the Proposition I of Ref. 1211 it may be shown that the set of derivatives of the general solution of the Hamiltonian equations of motion with respect to its 2M constants of integration forms a complete set of Jacobi fields. The particular set of solutions defined by the initial conditions (3.21~) corresponds to the Hadamard functions [21] used in the Lagrangian case. Using our formalism it may be shown that in the coordinate representation another particular set of solutions should be used. This set corresponds to the commutator function in the Lagrangian case 1211 and is selected by the initial conditions xy(t’)

= 0;

j&y)

= s,, ,

h-, 111= I,.... 241.

(3.251

These conditions may be obtained formally from (3.21~) by interchanging x and ~3. The reduced propagator in the coordinate representation will be given by (3.23) with x$‘(t”) replacing y$(t”). This of course reflects the symmetric nature of the Hamiltonian form of the path integral method and coincides with the result obtained in the framework of Lagrangian formalism-[15, 211. It is clear that both sets of solutions form the complete set of the 2M Jacobi fields. The physical meaning of these fields is simple. Consider the set selected by the initial conditions (3.21~). Once the classical path satisfying the boundary conditions (2.3) is found one may consider the variations of this path due to the infinitesimal changes

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of the initial coordinates qcl(t’). It can be easily shown that the solutions of (3.21) are given by -d?tt> = t~~mtoPq&‘Ncl, (3.26)

Therefore (3.27)

and j &t”,

t’)j = (2~)-~/~ 1det(ap,(t”)/aq,(t’))cl

j-1/2.

This result expresses the absolute value of the reduced propagator Vleck determinant [32] in momentum space det{~2TCr(p”, p’)/ap: apj’}.

(3.28)

through the Van (3.29)

In the coordinate representation the classical path satisfies the conditions (2.5). The initial momenta pcl(t’) should now be varied and the result for K((t”, t’) will be given by Eq. (3.28) with p and q interchanged. Finally, it should be mentioned that in the usual system of units one should replace 2~ by 274 in Eq. (3.28). 3.4. Focal Points

Let us now return to the expression (3.20) and discuss the phase of &(t”, t’). The eigenvalues +L”’ of the unperturbed operator @O)(t) appearing in (3.20) are easily found from Eqs. (2.18) and (2.19). They are M-times degenerate and given by

where it = 0, I,..., co; k = -1, +I for n # 0, and k = 1 for n = 0. The index i = I,..., M counts the degenerate @. Thus, these eigenvalues are arranged in groups with the index n counting the groups and the additional indices k and i counting the 2M eigenvalues inside the given group. The group with II = 0 consists of only M eigenvalues. The above degeneracy is in general removed when one considers the eigenvalues +o! of the full operator G(t) (Eq. (3.1 I)). It is still convenient to use for them the same indexing as for the 4;” in view of the fact that (bol+ c#:’ for large j & (. For the limiting process in Eq. (3.20) we get

In the last expression we have separated the product of eigenvalues of the full operator Q(t) which go to zero at the limit of vanishing Hamiltonian.

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It is seen from Eq. (3.20) that apart from the trivial factor (2+)-hJ/2 the phase of &(t”. t’) is defined by the sign of the expression (3.31). The numerator of (3.31) oscillates from positive to negative values as N grows (unless M is even). However, for large N the denominator of (3.31) will oscillate coherently with the numerator yielding a definite sign for the limit. Thus, in general the phase of the reduced propagator is given by the excess of the negative eigenvalues & of the full operator @ over the negative $L”‘s of the operator PO). The simple example presented in Fig. 1 illustrates the general situation. When one passes from operator Q(O) to cli one finds the familiar gap in the spectrum of the Dirac-type operator with two branches of eigenvalues projecting in the positive and negative directions and reaching the unperturbed eigenvalues at high values of the index n. As long as no member of the upper branch enters the gap to become negative, the sign of the ratio will remain positive. Otherwise the balance between the upper and the lower branches will be destroyed and (3.31) will change its phase. I _ -10.

I

SPECTRA OF ~~~~~ENTIJM

I

SOME DIRAC OPERATORS REPRESENTATION)

-

0 0 0

- 5.,,

0

o

l

o

. x

l l ’

0 . x

0 l x

0

0

0 .

.

x

* x x

0 . x

0 . x

0 . x

0 . I

:

-

x

x x

Hz 0’. -&

_ o-

L I

I

Oo: -“:~~““o x

. a*

Oo x

l x

5.-_ _ .

T=2a x H=O P2 l H’T

.

0 ,,;$+“’

0 . x

. X.

0

x .

x

IO.5

n-

00 . I

0 . x

. X.

4 -

x

I

1

IO

15

y

FIG. 1. The spectra of three simple Dirac operators. Notice the appearance of the gap in the

spectrafor the cases where

H f

0.

This situation is in contrast to what happens in the Lagrangian case [15]. There, the phase is defined by the number of negative eigenvaluesof the Morse operator [ 171. Since this operator is of the Sturm-Liouville type, it can have only a finite number of negative eigenvalues. By the Morse theory [17] this number is equal to the number of focal points on the classical path counted with their multiplicites.

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We wish to examine the validity of a similar relation in the Hamiltonian case. At this point we discuss the coordinate and momentum representations separately. In the coordinate representation the notion of a focal point is the same as in the Lagrangian formulation [31]. In practice they can be found from the solutions of the system (3.21a) with the initial conditions (3.25). The focal points are given by the zeros of the determinant det{x$(t)} along the classical paths whereas the reduction of its rank defines the multiplicity of a focal point. Obviously in the coordinate representation both the Lagrangian and the Hamiltonian formulations have to give the same results. Indeed when the Hamiltonian is of the form H = 4 E BiiPipj

+ V(ql y-**>qw)

(3.32)

i,j=l

with a constant mass tensor Bz$, we show in Appendix C that the product in the denominator of (3.31) is simply related to the product of eigenvalues of the corresponding Sturm-Liouville-type operator, arising in the Lagrangian formulation. This result together with the Morse theory shows that the phase of the reduced propagator in the coordinate representation is given by (---VT/~). The integer v is equal to the number of focal points along the classical path counted with their multiplicity. In the momentum representation the situation is less clear. The focal points in this case are defined by the zeros of the determinant (3.24). In contrast to the coordinate representation, there may be focal points for any small time intervals. Moreover, this always happens for the important class of scattering problems, when the potential is zero at large distances. In such cases always some of the caustics (surfaces over which the focal points are spread [31]) pass through the initial point. The Morse theory is of no help in the momentum representation. We have been able to show that when H is given by (3.32) and the matrix Dij(t) = (PV/aq$q$l does not change sign in the interval t’ < t < t” the same result holds for the phase of the reduced propagator as in the case of the coordinate representation. We also have some indications that when Dij(t) is not definite (in the scattering problem, for instance) this result may be wrong. At this stage we are not able to make a more definite statement. Apart from influencing the phase of the reduced propagator the existence of the focal points in the paths space means that catastrophic phenomena may occur when the parameters of the propagator are varied. Indeed, when the operator @(t) has a zero eigenvalue the determinant (3.24) vanishes at t = t” and the reduced propagator diverges. In this case the end point of the classical paths lies on the caustic and the simple quadratic approximation (3.19) to the action functional is insufficient. One should then apply uniform approximations. They are discussed in the next subsection. 3.4. The Uniform Treatment In this subsection the uniform technique [19, 201 for asymptotic evaluation of finite-dimensional integrals in the presence of caustics will be extended to the domain

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of path integration. This extension is possible because the concept of an “unfolding of a singularity” [24, 251 which underlies the uniform technique depends on the caustic type and not on the number of integration variables. We shall again start with the Nth approximant K,,, to the propagator given by Eq. (2.13). We are now interested in the situation where the S stationary points of T,v (Eq. (3.3)) may coalesce when the parameters of which the propagator depends are varied. The number and the topology of the S stationary points of the function T,V in Eq. (2.13) do not change when N increases, once N is sufficiently large. It is then possible to find a function F of 1 variables b = (6, ,..., b,) which is simpler in form than T, . but identical as far as the number of stationary points and their topology is concerned [19,20]. The number 1 is independent of N. Since there are S stationary points which are involved in the coalescence, the function F depends on S -- 1 parameters A = (A, ,..., AsPI) [19]. Let us define a diagonal quadratic form which depends on the variables b,,, ,..., b,\, LN(bl.l

,..., bN) = -b;+,

- bf-, -

..a - b,’ + b;,, -;- .a. + b,?

(3.33)

The choice of p which determines the index of L, will be discussed below. Using the functions F and L, we introduce a change of variables in (2.13) b, -= b,(a, ,..., a,),

a! = I...., N;

(3.34)

such that the following mapping is constructed

Tv[yl(f), a, T...,a,1 = FC-4b, ,..., b,)

+ &v(bl,,

,..., b,) + F, .

The parameters A = (A, ,..., A,-,) and F,, are found from the requirement the stationary points both sides of Eq. (3.35) coincide, T,[Y(t),

al”) )...)

a!$] = F(A, b(“) 1 ,*-., b?‘) + Fi, ,

(k =z I,...) S).

(3.35)

that at (3.36)

The superscript k stands for a value calculated at the kth stationary point. This requirement guarantees that the mapping (3.35) is one-one and uniformly analytic in a neighborhood of the S stationary points [19, 201. We note that together with the function F the index vL (number of negative eigenvalues) of the quadratic form L, (3.33) should be chosen so that at the stationary points (kl = )$I + k = l,..., s (3.37) VT VL > vT and v, are the indices of the matrices {~2T/~a,&z,} and (a2F/abJbs}, respectively. In the new integration variables (3.34) the relation (2.13) is written KN = JNeiFo

s

G(b, ,..., bN)

x expGF(A,bl ,..., bd -I- GP) LN(blfl ,..-, bNN fi db,

(3.38)

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where G@,

s...,

bN>

=

(a,

I WW~bs>l

p

=

I)...,

N),

(3.39)

This expression is still exact. It has the advantage that all the complexity of the integral is included now in the functional behavior of the Jacobian G. We approximate this Jacobian by the first term of an expansion about the stationary points b(k) = (b’“’1 ,..., 6%)). Following Berry [19] we write

W, >...,bp,) cs go + ‘il t,MWW,

(3.40)

k=l

where the gk (k = O,..., S - 1) are independent comparing

of the {bs}. The gk are found from

(3.41)

a, /3 = l,..., I y, p = l,..., hJ I 1 i = I,..., s I to S-l

Gci’

=

go

+

c k=l

i =

&@F/&dk)(i),

l,..., S;

(3.42)

where again the superscript (i) means the evaluation at the ith stationary point. Using the approximation (3.40) in Eq. (3.38) and performing the integration over the variables of the quadratic form LN(bl+l ,..., bN) we get

where the integrals Iti)

are defined

~(O)(A)

=

(2rri)-Z/2

J" eiF(Ashs...sbE)

P(A)

= -@Z’O’/aAi),

(344a)

dzb,

i = l,..., S -

1.

(3.44b)

In order to pass to the limit of N + co we wish to transform Eq. (3.43) to a more convenient form. The solution of (3.42) can be expressed as gk

=

f i=l

Cf’&‘,

k = O,..., S - 1,

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with coefficients which depend on (i3F/aAi)(lc’ and are independent (3.45) into (3.43) we get KN as a sum over the stationary points KN = i

k=l

@$‘“‘(A,

of N. Substituting

(3.46)

FJ,

where the canonicalfunctions S-l

f”“‘(A, F,) _ ( det(a2F/Zb, 8bg)(k))1’2 eiFO c C~“‘l’i’(A),

(k = I,..., S)

(3.47)

i=O

are introduced and I?:) are the reduced propagators which have been already encountered in the simple stationary phase expression (3.5), We have studied their limit of N --f cc earlier in this paper. The canonical functions l(k)(A, F,) in the expression (3.47) depend on N only through the parameters A and F, . When N -j co T,[Y(f),

af”),..., $1

-

q&)(0,

(3.483

&~(tjl

and (3.35) which define A and F, become T[&)(t) , p$)(t)] = F(A , by’,..., bj”‘) + F,, ,

In summary we get that the uniform asymptotic in the case of S coalescing classical paths is K(p”, t”; p‘, t’) = f Ryp”,

k 7 l,.... s.

(3.49)

expression for the propagator

I”; p’, t’) 4’“‘(A, F,).

(3.50)

It may be easily checked that when the S classical paths are well separated I”‘“‘(A, F,) + exp(ir[&)(t),

#(t)]).

(3.51)

so that in this limit the simple stationary phase result is obtained. On the other hand, when the classical paths are nearly coincident the divergence of the reduced propagator is balanced by the similar divergence of det(PF/%,LVQ(~) in Eq. (3.47) for the canonical functions yielding a finite result. The expression (3.50) is uniformly valid on the caustics as well as far away from them. In the coordinate representation the derivation of the uniform approximation follows the same lines and is given by an expression analogous to (3.50). One should only replace the action functionals on the left-hand side of Eq. (3.49) by their values in coordinate space and use the proper expressions for the reduced propagators. This was discussed earlier in this section. The crucial point in the derivation of the uniform approximation (3.50) is to choose the mapping function F(A, b, ,..., b,) properly. In general this is not an easy task. As in the asymptotic evaluation of finite-dimensional integrals [20] one should examine the Taylor expansion of the action functional. This expansion should be performed at

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the values of the propagator parameters close to those at which the highest singularity occurs and around the path at which the S-classical paths coincide. The action functional in the path expansion scheme depends on an infinite number of variables {a,>. However, only the dependence on a finite and usually small number of variables is essential to define the required mapping. At this point Thorn’s classification of the elementary catastrophes [35, 361 is of great help. For instance, when only two coinciding classical paths are present the mapping function Fin many practical cases should be chosen as F(A, b) = &b3 + Ab (3.52) which leads to the well-known Airy approximation [37]. For the treatment of the situations with more complicated elementary catastrophes we refer the reader to Refs. [19, 201. 3.6. The Initial- Value Representation

We wish to show now that the problem of finding the uniform approximation for a path integral may be considerably simplified under certain conditions and even completely solved. This is possible because the physical system provides us with a natural mapping function which may be used in a uniform treatment. Consider the classical paths 4$ = 80,

4’9 PI

pi”’ = pF1(t, q’, p’),

(3.53) i = i ,.,,, M,

satisfying the initial conditions q%, P%

q’, P’) = qi’, q’, P’> = Pi!,

(3.54) i = I,..., M.

The family is obtained by varying the initial coordinates q’ while keeping the initial momenta fixed. The final coordinates and momenta of the family are now functions of q’ d(s’)

= qw,

P&‘)

4’7 PI

= PieYf, 4’3 P’L

(3.55) i = l,..., M.

The classical action calculated along a path is also a function of q’ %I’)

= %PfG7’>s P’)

(3.56) =

- fjl q?$icl - H(p”,

qcl)) dt.

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Consider the function FCq’) = SW) + f (P,%‘)

- &I cl,f(s’j

(3.57)

i=l

and assume that for all values of 9’ det( aqif(q’)/iqj’)

(3.58)

# 0.

One can easily show that the stationary points of the function F(q’) are given by the roots of the equations. p,f(q’) - p; = 0,

(3.59)

i =- I,.... M.

These are also the conditions which select the classical paths satisfying the boundary conditions (2.3). These paths are the stationary points of the action functional T[q(t), p(t)]. At each path the value of T equals the value of F(q’) at the corresponding stationary point. The function F(q’) is therefore a good candidate for the mapping function in the uniform treatment. The matrix of the second derivatives of F(q’) is (3.60)

At the stationary points the first term vanishes and the matrix i”F/Bq,‘i3q,’ is expressed through the Van Vleck matrix appearing in (3.27). Following the discussion of Section 3.4 this shows that the condition (3.37) may also be fulfilled. Using the function F(q’) for a mapping in Eq. (3.35) we can proceed as in Section 3.5 arriving at the following expression for the Nth approximant to the propagator Kni

=:

Gei{f(q’)+(l,‘2)~.,(hM+1,....bN)]

JN

fidq,’ ,--I

s

j-.-M-

r”r db,. 1

(3.61)

The quadratic form L,,, and the auxiliary integration variables bhf*l ,..., b, are introduced in the same spirit as in (3.38). G is the Jacobian for the change of the variables q*’ = qi’(a, ‘..., QN), b, z b&a, ,.-.’ UN),

i = I,.... M;

j=- M i l.....iV.

(3.62)

At the stationary points G is given by G’“’ ;=

I,‘2 I(

’

k = l,.... S.

(3.63)

From Eq. (3.60) we get det(PF(;(q’)/aq, 2qj’)‘k) = /&t ?!$$@ I

det ?!!!&$)‘“‘, I

(3.64)

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It was already shown that in the limit of N -+ co JN

i det PT,/aa,

aa,

(Id = (2T9’N-M’/2

jdet

( 'p$')

)rn'/-l'ze

(3.65)

Since in the semiclassical limit only the vicinity of the stationary points is important, we approximate the quantity J,G in the integral (3.61) using the Eqs. (3.63)-(3.65) as if they were valid for all values of q’. Performing the integration over the variables of the quadratic form L, in (3.61) and passing to the limit Iv --f co we get the following integral representation for the propagator K(P”, t”; p’, t’) cz s [det (v)I””

@‘)

$6 dq,‘.

This representation is similar to the Initial-Value Representation (IVR) used in recent studies [I, 31 of the semiclassical S-matrix. It also appears in optics [I91 under the name diffraction integral. It is satisfying to see how this result follows directly from the path integral approach. In the coordinate representation one may derive the initial-value representation in the same way by simply interchanging the roles of p and q in the previous discussion. The condition (3.58) is replaced in this case by

It should be also mentioned that one may similarly construct the integral representations using the final values of the coordinates or momenta. These representation are valid only when the condition of the type (3.58) are fuIfilled. When the conditions (3.58) are violated the mapping function F has additional stationary points. Even though the integrand of Eq. (3.66) vanishes at these spurious points, their existence means that the mapping induced by the function F is not single valued. Therefore in such situations the result (3.66) is strictly speaking incorrect. However, under certain conditions it may still provide a good answer. Consider the region where the physical stationary points defining the classical paths are concentrated. If the spurious stationary points remain far outside this region there exists a one-to-one mapping which covers the physical region. In other words the topology of this region is properly reproduced by the function F. The integral (3.66) may be used in this case if it is evaluated asymptotically, ignoring completely the spurious points. This point was not amply stressed in the literature. Attempts to use the initial-value representation in cases where the spurious stationary points interfere with the physical points, yielded completely wrong results [33]. The initial-value representation seems to have a wider range of applicability in the momentum representation. In order to claify this point, let us consider a problem with only one degree of freedom, and assume a Hamiltonian H = p2/2m + V(q).

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187

In the coordinate representation the role of the Van Vleck determinant is played by Zqf/+ and the determinant of Eq. (3.67) reduces to a@/+‘. Let us study the development of the quantities in time along a classical trajectory. At the initial point

at any point

d %lf(t)

--=--. Lit

f3pJ

1 @f(t) 112 ap,

Therefore, the zeros of apf(t)/+’ which define a spurious stationary point appear before the zeros of aqf(t)/ap’. The latter defines the points where the classical trajectory hits a caustic. Therefore for nontrivial problems where caustics are encountered the spurious stationary points are also present. In the momentum representation, the caustics are given by the zeros of apf(t)/bq’, and the spurious points are the zeros of aqf(t)/aq’. One can again write down the relations similar to (3.68) and (3.69) but they do not necessarily lead to the same conclusions. It can be shown that the decisive factor is whether the second derivative of the potential a2V/8q2 changes its sign along the classical trajectory. If it does, one can show that a caustic can be reached before a zero of aqf(t)/aq’ is encountered. 3.7. Nearly Free Motion in Momentum Space

Let us consider a situation when the Hamiltonian does not depend explicitly on some of the coordinates. This corresponds to a symmetry in the system and the propagator in the momentum representation contains a A-function. In this situation the operator @, Eq. (3.11) has a corresponding number of zero eigenvalues and the S-function can be extracted, say, by the method described in Section 2. When the symmetry of the Hamiltonian is broken by some perturbation the a-function in the propagator disappears. In the semiclassical limit one would seek for an approximation which approaches uniformly the A-function when the perturbation vanishes. We shall consider a simple one-dimensional example of such a system with the Hamiltonian H = (PV~) + EW, (3.70) where E is a small parameter. In Fig. 2 we show the family of classical paths when the potential is taken as a Gaussian function. Each path of the family starts with the initial momentum p’ at t = 0, while the initial coordinate is varied. The value of p’ is chosen so that (p’)2/2m >
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expression for the semiclassical propagator. Potentials with a more complicated structure than the Gaussian will yield a larger number of caustics. However, they will still collapse when E -+ 0. I

I

I

I

I Classical Trajectories ““rim,* initin,

2.4 -

I

05

I

I

1.0

I

I.5

2.0

I

For

I

2.5

t-

The timedependent classicaltrajectoriesp(t) corresponding to various initial coordinates xt . The full (dashed)lines correspond to the positive (negative)values of X, . Note the two branches of the envelope which emergefrom the initial point and define the classically allowed region. FIG.

2.

The situation is similar to the forward diffraction peak, studied by Berry [34] and it is known [19] that the standard results of Thorn’s theorem do not apply to such cases. The quanta1 expression for the propagator in momentum space for the Hamiltonian (3.70) can be calculated by perturbation theory [9], K(P", T; PI, 0) 1 _

ei((p’2/Zm)-(~“2j2nz))T

(3.71)

Here V(p) is the Fouirer transform of V(x). We show now that the initial-value representation (3.66) gives the correct answer in this case. To first order in Ethe classical trajectory determined by (3.70) and which starts with the initial values q’ and p’ at t = 0 is given by p(t) = P' - (dp')

d v(t),

q(t) = 4' + (P'h>

t -

E/P' lt ov(7)

(3.72a) d7,

(3.72b)

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where L!lV(t) = V(q’ + W/m) t) - Vq’).

The determinant

(3.73)

appearing in (3.66) is reduced to

Ed =1-L -zy P’ I’0T-?%’ n V(7)dT,

(3.74)

so that the condition (3.58) is fulfilled. For the function Fin (3.57) we get

T --E I-(p’/m> t)dt- (G ~P/P’) joT dJ’(T) dT sqq’

F(q’) = ---(pf2/2m) T + dp(q’ + (p//m) T)

(3.75)

0

with dp = p’ - p”. Using this in (3.66) and realizing that for allowed transitions dp is also of order E, one easily obtains the result (3.71). This simple example demonstrates the important role of the IVR in extracting the nonclassical singularities from the expression for the propagator, even though it was obtained by calculating classical quantities only. 4. CONCLUSION

In the preceding chapters we presented a study of the semiclassical limit of the propagator, based on phase-space integration. After presenting the path expansion method for evaluating path integrals, we showed how the uniform asymptotic approxitions for finite-dimensional integrals can be extended to the case of path integration. Based on these developments, we were able to derive various semiclassical approximations and discuss the ranges of their validity. There are a few problems of significant interest which can be handled using the present techniques, but which were not dealt with in this study. A notable example is the derivation of semiclassical amplitudes for classically forbidden ttansitions. For this purpose one should introduce complex-valued classical trajectories, and extend such concepts as the Morse operator, the Jacobi field, etc. into the complex domain. In a forthcoming publication, we discuss this problem and show how to enlarge the domain of applicability of the uniform approximations to include forbidden transitions. The unfolding technique used in Section 3.5, together with the path expansion procedure allows one to find, at least in principle, the higher terms in the semiclassical expansion. This would result in the complete asymptotic series for the propagator. It seems that in this way the recent results of De Witt-Morette [21] can be extended to arbitrary Hamiltonians as well as to cases where there exist several contributing classical trajectories. In the calculation of the S matrix, one usually uses the energy-dependent propagator rather than the time-dependent propagators discussed so far. Even though the two

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exact quanta1 representations are related to each other by a simple Fourier transformation, the corresponding connection between the semi-classical expressions is not so simple. This is due to the considerable difference in the structure of the caustics in the different representations. The derivation of the uniform approximations in the energy representation should therefore use the exact path integral expression as the starting point, rather than performing Fourier transforms on the expressions derived in the present work.

APPENDIX

A

In this appendix we evaluate the last sum in the exponent of Eq. (2.24). We should consider only the case when p” = p’. Using the explicit form (2.23) for pa this sum is written as

where x!“‘(;$+))(I$

?&‘(+)j[ d7,

(i = I,..., 2M).

(A2)

(i = I,..., 2M).

(A3)

As N -+ co, p’)(t) -+ q(t) which satisfies

Here the completeness relation g1 xii”‘(t) $‘(T)

= &S(t - T)

(A4)

was used. Inserting in Eq. (A3) the explicit form of x(@)(t)for 01 = l,..., M given by Eq. (2.20) we get that for p’ = p” the functions vi(t) satisfy

VW4

rl(t))i = (WdO W>)i 3

(i = l,..., 2M)

(A9

with the boundary conditions ri(t’) = I = 0, i = A4 + l,..., 2M, following from (A2) and (2.18). We solve (A5) under these conditions and after some algebra we obtain finally that for p’ = p”

; i+l

$$- e OL

s,r” - QPdt = T’“‘[Q,WI.

@WI

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B

APPENDIX

In this appendix a theorem is proved which allows to write in a closed form the ratio of the infinite products in Eq. (3.20). Consider the differential operator of the Dirac type @ij(CX,t) = .Tij(d/dt) -

(Bl)

YEjj(t)

defined for t’ .< t < r” and 0 < 01 < 1. {rij> is the 2M x 2M simplectic matrix defined in Eq. (2.17), and (~&t)) some real symmetric 2M x 2M matrix with elements depending smoothly on t. Let &(a) and x’“)(cx; t) be the eigenvalues and normalized eigenfunctions of the boundary value problem W’a) Z’,(cq

t’)

=

v,(cy;

t”)

=

0

fi

=-

1 ,...,

2n4),

(B2b)

where U,(OI; t), m = I,..., M are the lower M components of ~(a; t). Let ttk)(ol; t), k = l,..., M be the M-independent solutions of the initial-value problem “c” @ij(cq

t)

p(a;

f) = 0,

iB3a)

j=l

&x;

t')

=

s,,

;

yy(a;

t')

=

0,

i=

I ,...'

2M;

117,k = I,..., M.

(B3b)

Here ~$(a: t) and J&E; r) are the upper and the lower M components of ~(“)(cu; t), respectively. Then for every cythe following result holds

As was shown in Section 2 the operator Gpij(O; t) has an M-degenerate zero eigenvalue. The product in the denominator of the left-hand side of (B4) is formed over all nonzero $,(O). The convergence of the ratio of the infinite products in (B4) was discussed in Section 3.2. We prove now the relation (B4). It is clear that the two sides of this relation vanish simultaneously. Consider the functions

and f(a) = det{y,(“‘(a; r)j.

036)

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In a manner similar to Ref. [23] we wish to demonstrate first that (B7) for all LY. satisfyingf(ol) One easily gets

# 0, p(a) #

0.

1 dfC4 1 d4r(4 - da = ngl -$&j --XTf(a)

where the expression d+,(a)/dol = - “c” s,;” #‘(a; i.i has been used. The differentiation dej’“‘/da

t) &‘(a;

t) I,

dt

of Eq. (B3) with respect to 01gives the following

(B9)

equations for

y$&x; t)d@J;; O=z Eij(f) .&a;t),

j=l

1,..., 2M,

i=

k = I,..., M,

(BlOa)

with the initial conditions 4%;

0

da Using the Green’s function finds the solution of (BlO)

=

o

3

j

=

I,...,

2M;

k = l,..., M.

G$‘(t, T; a) of the boundary value problem

(Blob) (B2) one

where

The lower M components of dfi”‘(a; t)/dol at t = t” are dyz”‘(a; t “) = da

f P=l

$‘)(a;

t”) [“‘(a;

0),

i = I,..., M;

k = I,..., M.

(B13)

HAMILTONIAN

Applying

a rule for differentiating

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a determinant

one obtains (Rl3)

Using the spectral representation for the Green’s function G’“)(f, T; LX) and the fact that for r ( 7 it is a linear combination of L?)(Ix: t) one arrives at Eq. (BY?. The integration of this equation gives

where C is a constant independent of 01.In Ref. 1231 we fixed an analogous constant by comparing the values off(a) and f( 01) a t 01 = 0. In the present casef(ol) vanishes at 01 = 0 together with I and we compare the first nonvanisning derivatives of these functions. They are obviously of the order M. It can be easily shown that (Bl6) and

Solving Eq. (82) for ?I ---f 0 in a way similar to the perturbation theory for a degenerate state in quantum mechanics one gets a familiar secular equation

Using the explicit form (2.20) for ~(“‘(0; t), k = I ,..., M one gets from (B18) and (B19)

( B20) : (-1,p

(t” - r’)-M det [.[I” I,,

where the matrix (D,,(t)} is defined by Eq. (3.10~).

dtl.

k, 11

I...., M,

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The solution of Eq. (B10) for 01 = 0 can be easily found using the explicit form of J(“)(O; r). The lower M components of this solution are

Therefore

Using (B22) and (B20) together with (B16) and (B17) one finds the constant C in (Bl5) c = (t” - q-M (B23) which ends the proof of (B4). By the same method one proves the relation (B4) in connection to the coordinate representation, i.e., when the conditions (B2b) are imposed on the upper A4 components of ~(a; t) and the replacement y F? x is made in Eq. (B3b) and (B4).

APPENDIX

C

In this appendix we discuss the relation between the eigenvalues of the Dirac~ operator (3.11) and the corresponding Morse operator. We restrict ourselves to the system with a Hamiltonian of the form (3.32) in the coordinate representation. The boundary value problem (3.14) is written explicitly in this case as

-(du,/dt> -

;

Big(t) uj = +ui ,

j=l

u;(t’)

ua(t”)

=

=

0,

i = l,...,

M.

(Clb)

Equations (Cla) are easily transformed into

We wish to relate the product of the eigenvalues of (C2) to the corresponding boundary problem f A+(t) zzj + Xi& = 0, j=l

ii,

= iqt”) = 0,

(C3) i =

l,..., M

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INTEGRAL

with

l&(t) = -B,l(d2/dt2) -s Dij(t),

i. j -= 1,.... M.

(C4)

The normalized eigenfunctions of (C3) provide a basis in the space of the solutions of (C2). Substituting the truncated expansion i 7 l..... n!f

u,(t) = Nf C,iil”‘(t), I\=1

(0

in Eq. (C2), we get the secular equation det{@ S,, {- &$ / Dc't) / a) + (4 - h,)@ 1B j CL?;J= 0,

cx,p = 1,..., NM.

(C6)

The notations in (C6) were introduced in Appendix B. The left-hand side of Eq. (C6) is a polynomial of degree 2NM with respect to $. The eigenvalues & of (C2) are the roots of this polynomial in the limit of N --* X. Their product equals the value of the polynomial at #I = 0, thus,

(C7) = (- l)NM F X,1 det(P / B I a).

i a==1

,

A subspace of M eigenvectors of (Cl) are projected on the null solution of (C2) and therefore are not included in the product (C7). These are the vectors whose upper M components are identically zero and whose lower M components are the eigenvectors of the matrix (-B). The corresponding eigenvalues are the eigenvalues of (-I?), and they form the separate product of the “first” group of eigenvalues in Eq. (3.31). Since B is a constant matrix det@ j B I a) = (det BP.

Cc%)

Using the results obtained so far together with (3.30) in Eq. (3.20) we get the reduced propagator in the coordinate representation expressed via the eigenvalues of the Morse operator (C4) l?(t", t') = [2+(t"

- t’)]-“!z

(det B)-li2

,,,.' 2% (det B)--N/2 fi (m/t" - t')2x/EAE/1'z. n=1 1=1

(C9)

196

LEVIT AND SMILANSKY

This expression reproduces the result obtained for the propagator in the Lagrangian representation [ 15 1.

ACKNOWLEDGMENTS We thank Professor M. V. Berry for drawing our attention to the recent developments in catastrophe theory. We are grateful to Dr. K. M&ring for many valuable suggestions and comments. We would also like to thank Professor I. Kanai and Dr. T. Dreyfus for discussing some mathematical problems.

REFERENCES 1. W. H. MILLER, J. Chem. Whys. 53 (1970), 1949, 3578. 2. W. H. MILLER, Classical limit quantum mechanics and the theory of molecular collisions, Advun. Chem. Phys. 25 (1974), 69. 3. R. A. MARCUS, J. Chem. Phys. 54 (1971), 3965. 4. S. LEVIT, U. SMILANSKY, AND D. PELTE, Phys. Left. B 53 (1974), 39. 5. H. MA~SMANN AND J. 0. RASMUSSEN,Nucl. Phys. A 243 (1975), 155. 6. P. FR~~BRICH,Q. K. K. LIU, AND K. MBHRING, “Proceedings of the European Conference on Nuclear Physics with Heavy Ions, Caen, 1976,” Vol. 1, p. 96. 7. S. LEVIT AND U. SMILANSKY, unpublished. 8. H. KREEK AND R. A. MARCUS, J. Chem. Phys. 61 (1974), 3308. 9. R. FEYNMAN AND A. R. HIBBS, “Quantum Mechanics and Path Integrals,” McGraw-Hill, New York, 1965. 10. C. GARROD, Rev. Modern Phys. 38 (1966), 483. 11. H. DAVIES, Proc. Cambridge Philos. Sot. 59 (1963), 147. 12. W. TOBOCMAN, Nuovo Cimento 3 (1956), 1213. 13. M. M. MIZRAHI, J. Math. Phys. 16 (1975), 2201. 14. M. C. GUTZWILLER, .I. Math. Phys. (a) 8 (1967), 1979; (b) 10 (1969), 1004; (c) 11 (1970), 1791; (d) 12 (1971), 343. 15. S. LEVIT AND U. SMILANSKY, Ann. Physics 103 (1977), 198. 16. S. LEVIT AND U. SMILANSKY, preprint, Weizmann Institute of Science, WIS-76/22-Ph, Rehovot, Israel, 1976. 17. M. MORSE, “Variational Analysis,” Willey, New York, 1973. 18. L. S. SCHULMAN, Caustics and multivaluedness, in “Functional Integration and Its Applications” (A. M. Arthurs, Ed.), Proceedings of the International Conference, London, April 1976. 19. M. V. BERRY, Advan. Physics 25 (1976), 1. 20. J. N. L. CONNOR, Molecular Phys. 31 (1976), 33. 21. C. DEWIER-MORBTTE, Ann. Physics 97 (1976), 367. 22. N. M. LEVITAN AND I. S. SARGSIAN, “Introduction to the Spectral Theory,” Moskow, 1970. 23. S. LEVIT AND U. SMILANSKY, Proc. Amer. Math. Sot. (1977), in press. 24. V. I. ARNOL’D, Uspehi Mat. Nauk 28 (1973), 17 (English translation: Russian Math. Surveys 28(5) 25.

26. 27. 28. 29. 30.

(1973),

19).

J. J. DUISTERMAAT, Comm. Pure Appl. Math. 27 (1974), 207. J. N. L. CONNOR, Molecular Phys. 25 (1973), 181; 26 (1973), 1217. I. M. GEL’FAND AND A. M. YAGLOM, J. Math. Phys. 1 (1960), 48. B. DAVISON, Proc. Roy. Sot. A 225 (1954), 252. B. S. DEWITT, Rev. Modern Phys. 29 (1956), 377. P. PECHUKAS, Phys. Rev. 181 (1969), 166; 174.

HAMILTONIAN

31. 32.

33. 34. 35. 36. 37.

PATH

INTEGRAL

197

C. CARATHEODORY, “Calculus of Variations and Partial Differential Equations of the First Order, Part II,” Holden-Day, San Francisco, 1967. J. M. VAN VLECK, Proe. Nat. Acad. Sci. U. S. A. 14 (192Q 178. K. M~HRINC, private communication. M. V. BERRY, J. Phys. B2 (1969), 381. R. THOM, Topology 8 (1969), 313. G. WASSERMANN, “Stability of Unfoldings,” Vol. 393, Springer Mathematical Notes, 1974. M. V. BERRY AND K. E. MOUNT, Rep. Progr. Phys. 35 (1972), 315.