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Path Integral Solution of the Dirichlet Problem
Annals of Physics 5650 annals of physics 254, 397418 (1997) article no. PH965650
Path Integral Solution of the Dirichlet Problem J. LaChapelle Department of Physics and Center for Relativity, University of Texas, Austin, Texas 78712 Received April 29, 1996; revised July 8, 1996
A scheme for functional integration developed by CartierDeWitt-Morette is first reviewed and then employed to construct the path integral representation for the solution of the Dirichlet problem in terms of first exit time. The path integral solution is then applied to calculate the fixed-energy point-to-point transition amplitude both in configuration and phase space. The path integral solution can also be derived using physical principles based on Feynman's original reasoning. We check that the Fourier transform in energy of the fixedenergy point-to-point transition amplitude gives the well known time-dependent transition amplitude, and calculate the WKB approximation. 1997 Academic Press
1. INTRODUCTION Functional integrals are powerful tools for solving partial differential equations. Techniques of functional integration are particularly developed and refined in the context of stochastic processes. However, stochastic methods are not always applicable in many physical systems. For example, the solution of the Schrodinger equation is not obtained directly by stochastic means: first the diffusion equation is solved and then the solution is analytically continued. The same goes for the stationary Schrodinger equation. Also, stochastic paths are not differentiable, but many physical applications dictate the use of action functionals which require differentiability of the paths. There exists a scheme for functional integration due to CartierDeWitt-Morette ([1]) which in a sense generalizes and improves the stochastic approach. Relevant details are presented in Section 2. The underlying theme of their work is an emphasis on the domain of integration which is an infinite dimensional space of paths. Their scheme allows for rigorous rather than formal manipulations of path integrals. The functional integration formalism described here generalizes the stochastic approach in the sense that it can be used to solve a wider class of partial differential equations. For example, it is possible to solve both the Schrodinger and diffusion equations directly. In Section 3 we present a path integral solution of a generalized Dirichlet problemgeneralized because the characteristic form of the partial differential operator need not be positive definite in all cases. The theorem proved in Section 3 and the general theorem of Ref. [1] provide the functional integral solutions of elliptic and parabolic second order 397 0003-491697 25.00 Copyright 1997 by Academic Press All rights of reproduction in any form reserved.
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partial differential equations within the CartierDeWitt-Morette framework. Work on the hyperbolic case is in progress. Section 4 examines the configuration and phase space path integral solution for the point-to-point transition amplitude of the time-independent Schrodinger equation. Use of this familiar example is meant to check the general result of Section 3 and to illustrate some useful methods of manipulating functional integrals. The point-to-point transition amplitude of the time-independent Schrodinger equation (which depends on energy) can be used to study the eigenfunction spectrum of the associated partial differential equation. The task is to determine the poles and cuts of the transition amplitude in the complex energy plane. For the configuration space integral, it seems that one must first do the functional integral because the paths, which are fixed energy paths, have an energy dependence. However, the phase space integral might offer a way to study the pole and cut structure within the path integral since the paths are only restricted by a Lagrange multiplier. In other words, in the phase space integral the paths have no energy dependence before the path integral is done. This property of the phase space integral offers a new and possibly useful way to study the eigenfunction spectrum of a time-independent Schrodinger equation.
2. FUNCTIONAL INTEGRALS To construct a functional integral, we require three well-defined components: (1) a domain of integration which is an infinite dimensional function space, (2) an integrator defined on the domain of integration, and (3) an integrand or integrable functional. Given these three ingredients, a value can be assigned to the path integral. 2.1. The Domain Here we consider the domain of integration X to be a real, separable Banach space of pointed paths. 1 The paths x # X are taken to be L 2, 1 functions 2 which map a closed interval T/R into a target manifold. We denote the dual space of X by X$ and the duality by (x$, x) X where x$ # X$. For quantum mechanical applications, one is usually interested in a functional integral over all paths x: T N where T/R and N is an n-dimensional manifold. But, in general, the space of paths which take their values in a manifold N will not be a Banach space X. For suppose that we are interested in all paths which have the same fixed end-points at t=t a # T. Then x 1 +x 2 # X if and only if x(t a )=0 \x # X. This follows because the equality (x 1 +x 2 )(t a )=x 1(t a )+x 2(t a ) holds for 1 The term ``pointed paths'' means the paths x # X have one fixed end-point. For example, if T/R is the interval [t a , t b ], then all paths with fixed initial points x(t a )=x a and arbitrary final points would be pointed paths. 2 A function is L 2, 1 if T |dxdt| 2 dt< . We choose L 2, 1 functions because of their relevance to physics. Specifically, the Lagrangian density requires derivatives, and the kinetic energy must be finite.
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PATH INTEGRAL FOR DIRICHLET PROBLEM
399
vanishing end-points only. Therefore, the space of paths which have a nonvanishing fixed end-point is not a Banach space. However, if we restrict attention to the space of paths with only one end-point fixed (that is, a space of pointed paths) then the space of such paths is contractible. Consequently, it can be parametrized by a space of pointed paths which take their values in a flat manifold and have their fixed end-point at the origin. But this parameter space of paths is a Banach space. Hence, integration on the space of pointed paths can be defined in terms of integration on the parameter space ([1]). The situation is analogous to integration on finite dimensional manifolds. There one does not know how to integrate on a general manifold. Instead, one defines the integral by parametrizing points in the general manifold by points in flat manifolds where one knows how to integrate. For functional integrals, we do not know how to integrate over a general function space so we parametrize it with a Banach space where the rules of functional integration are established. The details of the parametrization follow. Let there be d linearly independent vector fields which generate a vector subspace D x of T x N at each x # N, and denote 2, 1 paths with fixed them by X (:) where : # [1, ..., d]. Let P D xa N denote the space of L D initial point x(t a )=x a # N \x # P xa N and such that x* (t)&Y(x(t)) # D x(t) where Y is some given vector field on N. Denote by P 0 R d =: Z a the space of L 2, 1 paths with fixed initial point z(t a )=0 # R d \z # Z a . 2, 1 and since [X (:)(x(t))] spans D x(t) , it follows Since the paths in P D xa N are L : that there exist functions z* (t) such that
{
x* (t)&Y(x(t))=X (:)(x(t)) z* :(t) x(t a )=x a
(2.1)
This differential equation associates a path z # Z a with each path x # P D xa N. If N is compact, the map x [ z can be inverted. That is, given some z # Z a , Eq. (2.1) has a unique solution x # P D xa N. We will denote the solution by x(t, z)=x a } 7(t, z) where 7(t, z): N N is a global transformation on N such that 3 x(t a , z)= x a } 7(t a , z)=x a . We now have a parametrization P: Z a P D xa N by z [ x. 2.2. The Integrator and Integrand It is not possible to define a measure for the type of path integrals used in quantum mechanics. However, it is possible to replace the notion of a measure with the notion of an integrator. In the CartierDeWitt-Morette scheme, an integrator, denoted by D3, Z x, is defined by the relation
|
3(x, x$) D3, Z x=Z(x$)
(2.2)
X
The integrator D3, Z x is defined implicitly by the functionals 3(x, x$) and Z(x$). 3 If we had chosen to parametrize with paths z # Z b , the solution of Eq. (2.1) would be x(t, z)= xa } 7(t$, z) where t$ :=t a +t b &t. Hence, x(t a , z)=x a } 7(t b , z)=x a as required.
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J. LACHAPELLE
It is possible to define a class 8 3 of integrands which is integrable with respect to D3, Z x by the relation F+(x)=
|
3(x, x$) d+(x$)
(2.3)
X$
where F + # 8 3 . 4 This construction of the functional integral implies
|
F +(x) D3, Z x=
X
|
Z(x$) d+(x$)
(2.4)
X$
The right-hand side of Eq. (2.4) should not be construed as a means to evaluate the left-hand side. Indeed, the measure + associated with a given integrand F + # 8 Q is not usually known. Equation (2.4) is a consequence of relations (2.2) and (2.3) and is useful for determining when the integral exists or for proving general theoremsneither of which requires the specific form of +. 2.2.1. Gaussian Integrator. By far the most prevalent choice for D3, Z x in quantum physics is related to the gaussian integrator. Assume there exists a continuous, symmetric, non-degenerate linear map D: X X$ and denote its inverse by the map G: X$ X. Then a quadratic form on X is defined by Q(x)=( Dx, x) X , and a quadratic form on X$ is defined by W(x$)=( x$, Gx$) X . For 3(x, x$), choose the functional exp[(&?s) Q(x)&2?i( x$, x) X ], and let Z(x$) be exp[&(?s) W(x$)] where s # C is an adjustable parameter. Then DQ, W x is characterized by
|
exp[(&?s) Q(x)&2?i( x$, x) X ] DQ, W x=exp[&?sW(x$)]
(2.5)
X
The integrator defined by Dw(x) :=exp[(&?s) Q(x)] DQ, W x
(2.6)
is called a gaussian integrator. Equation (2.5) can be viewed as a Fourier transform. That is, (Fw)(x$)=
|
exp[&2?i( x$, x) X ] Dw(x)=exp[&?sW(x$)]
(2.7)
X
Notice that at the origin of X$, the integrator has the normalization
|
X
4
Dw(x)=
|
exp[(&?s) Q(x)] DQ, W x=1 X
Since X$ is separable and complete, it is possible to define complex Borel measures + on X$.
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(2.8)
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PATH INTEGRAL FOR DIRICHLET PROBLEM
Gaussian integrators possess an important property under affine transformations. Consider an affine map A: X X by x [ Ax=Lx+b where L is an invertible linear map and b # X is a fixed element. Then the image of the integrator DQ, W x under A is given by DQ, W (Ax)= |Det L| DQ, W x
(2.9)
For gaussian integrators, one can define a class of functionals 8 Q which is integrable with respect to DQ, W x by the relation F+(x)=
|
exp[(&?s) Q(x)&2?i( x$, x) X ] d+(x$)
(2.10)
X$
so that to each measure + on X$ there is associated an integrand F + # 8 Q . 2.2.2. Delta Functional Integrator. Closely related to a gaussian integrator is the delta functional integrator. In order to define a delta functional integrator, we use the well-known fact that a delta function can be represented as a gaussian in the limit as the width goes to zero. The infinite dimensional analog is to take the limit s 0 in Eq. (2.5). Hence, lim s0
|
exp[&2?i( x$, x) X ] Dw(x)=: X
=:
| |
exp[&2?i( x$, x) X ] $(x) DQ, W x X
exp[&2?i( x$, x) X ] D$(x)
X
= lim exp[&?sW(x$)] s0
=1
(2.11)
defines a delta functional integrator. This definition is justified by the fact that it reduces to the finite dimensional result for the Fourier transform of a delta function. That is, let L x : X R n be the linear surjective map defined by x [ u=[u 1, ..., u n ] where u i =( x i$ , x) for a given x i$ # X$. Then it can be shown ([1]) that the left-hand side of Eq. (2.11) reduces to lim s0
|
Rn
=
i j n |det sW ij | &12 exp[(&?s) W &1 ij u u &2?i( u$, u) Rn ] d u
|
Rn
$ n(u) exp[&2?i( u$, u) Rn ] d nu
=1 We point out that this result is independent of n.
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(2.12)
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J. LACHAPELLE
The delta functional integrator also behaves like its finite dimensional counterpart for non-trivial integrable functionals:
|
F +(x) D$(x)=
X
=
|
F +(x) $(x) DQ, W x X
|
|
exp[(&?s) Q(x)&2?i( x$, x) X ] d+(x$) DQ, W x
| |
exp[(&2?s) Q(x)&2?i( x$, x) X ] DQ, W x d+(x$)
$(x)
X
X$
= lim s0
= lim s0
=
|
X$
|
X$
X
exp[& 12 ?sW(x$)] d+(x$)
d+(x$)
X$
=F +(0)
(2.13)
where the interchange of integrations in the third line can be rigorously justified, and the second and last lines follow from Eq. (2.10). Moreover, for non-trivial arguments of the delta functional, we have
|
F +(x) D$(Ax)=
X
=
|
F +(x) $(Ax) DQ, W x X
|
F +(A &1x ) $(x ) DQ, W (A &1x )
|
F +(A &1x ) $(x ) |Det L| &1 DQ, W (x )
X
=
X
= |Det L| &1 F +(x o )
(2.14)
where x o is a solution of Ax=Lx+b=0. In the second line, we used the affine transformation x [ A &1x, and in the third line Eq. (2.9) was used. The last line follows from Eq. (2.13). Equation (2.14) can be written in a short-hand notation as $(Ax)= |Det L| &1 $(x&x o ). This is obviously similar to the analogous finite dimensional result. 2.3. The Integral We are now in a position to give meaning to a functional integral of an integrable functional of pointed paths in a general manifold Ndenoted schematically by
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PATH INTEGRAL FOR DIRICHLET PROBLEM
403
PDx N F(x) Dx. First, construct the parametrization P: Z a P D xa N. Since we know a how to do a functional integral on Z a , simply define
|
PD x N
F(x) Dx :=
|
F(x(z)) D3, Z z
(2.15)
Za
a
where D3, Z z is characterized by Eq. (2.2) with X=Z a and F(x(z)) is an integrable functional defined by Eq. (2.3). (Henceforth, we will not make the + dependence of integrable functionals explicit.) Of course, for this definition to be useful, the righthand side must be independent of the parametrization used. That is, we must get the same answer if we use the parametrization P: Z a P D xa N by z~ [ x. This just means that the functional integral must be invariant under a change of variable of integration. 3. SOLUTION OF THE GENERALIZED DIRICHLET PROBLEM The solution of the Dirichlet problem in terms of a functional integral whose variable of integration is a stochastic process is well known (see, e.g., Refs. [6] and [7]). In fact, the stochastic strategy is quite similar to what we are presenting. However, there are important differences. Most obvious is the analytic nature of the paths: the trajectory of a stochastic process is only continuous whereas we deal with L 2, 1 paths. Another important difference is that our approach supplies solutions for a broader class of partial differential equations. In particular, for certain choices of the parameter s, the characteristic form of the second order differential operator need not be positive definitehence the section title ``... Generalized Dirichlet Problem.'' Our main result is contained in the following theorem. 2, 1 paths which have a fixed end-point Theorem. Let P D xa N be the space of L x(t a )=x a # N and whose velocity vectors are elements of D. Parametrize P D xa N by Z a . Let U be a bounded convex region in N with boundary U. Let T be the space of L 2, 1 monotonic functions {: T=[t a , t b ] [{(t a ), {(t b )]/R such that {(t a )=0. Assume that the parametrized paths reach the boundary U at t={(t b ), i.e., x({(t b ), } )=x {(tb ) # U. Let : N C m be any integrable C (N) bounded function. Then, for x a # U, the functional integral
9(x a ) :=
1 (x a } 7({(t b ), z)) exp & S(z; {(t b )) DQz , Wz zDQ{ , W{ { s Za_T
{
|
=
(3.1)
is a solution of the generalized homogeneous Dirichlet problem in U
{
_
s 2 :; G L L +sLY +V(x)+2?iE 4? tb X(:) X(;) lim
xa x{(tb ) # U
&}
9(x a )=0 x=xa
9(x a )=(x {(tb ) )
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(3.2)
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J. LACHAPELLE
Here LX(:) is the Lie derivative in the X (:) direction, and E # R (recall that s # C). The matrix G :; tb is given by
|
Za
$ 2 exp[&?sW(z$)] $z$:(t b ) $z$;(t b )
z :(t b ) z ;(t b ) exp[(&?s) Q z(z)] DQz , Wz z= =
s ( $ : , G$ ;tb ) 2? tb
=
s :; G (t b , t b ) 2?
=:
}
z$=0
s :; G 2? tb
(3.3)
It is the Green's function of Q z with boundary conditions G :;(t, u) | t=ta =0 and G :;(t, u) t | t=tb =0 evaluated at t=u=t b . The functional &(1s) S(z; {(t b )) is 1 1 & S(z; {(t b ))=(&?s) Q z(z; {(t b ))+ s s
|
{(tb )
[V(x(t, z))+2?iE] dt
(3.4)
{(ta )
where Q z(z; {(t b ))=
{(tb )
{(tb )
{(ta )
{(ta )
| |
z :(t) D :;(t, u) z ;(u) dt du
(3.5)
and D :;(t, u) depends on the nature of N. For example, if N is associated with the configuration space of a dynamical system, then
\
D :;(t, u)=$(t&u) h :;
d2 du 2
+
(3.6)
where h :; is real, symmetric and non-degenerate. If N is associated with the phase space of a Hamiltonian dynamical system and we choose canonical coordinates so that z : =(z iq , z ip ) where i # [1, ..., d2], then
D :;(t, u)=$(t&u)
} 2
\
0
d du 1 &$ ij m
&$ ij
d $ ij du
+
(3.7)
where } is some constant. The quadratic form Q { which characterizes DQ{ , W{ { is chosen to be Q{ =
1 (t b &t a )
|
tb
{* 2 dt
ta
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(3.8)
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PATH INTEGRAL FOR DIRICHLET PROBLEM
This theorem should be compared to the main theorem of Ref. [1] which provides the functional integral solution of the Cauchy problem. The major difference resides in the extra E term in the action functional and the extra functional integral over the { variable. Proof.
Equation (3.1) may be rewritten as 9(x a )=
|
exp T
2?i
{ s E({(t )&{(t ))= 9({(t ), x ) D b
a
b
a
Q{ , W{
{
(3.9)
where 9({(t b ), x a ) :=
|
Za
1 (x a } 7({(t b ), z)) exp & S (z) DQz , Wz z s
{
=
(3.10)
and 1 & S (z) :=(&?s) s
{(tb )
| | {(ta )
{(tb )
z :(t) D :;(t, u) z ;(u) dt du+
{(ta )
1 s
|
{(tb )
V(x(t, z)) dt
{(ta )
(3.11) Notice that the integrand in Eq. (3.9) depends on { at a single point (recall that {(t a )=0). That is, the integrand is a function of {(t b ). This fact can be exploited to transform the functional integral into a finite dimensional integral by utilizing Eq. (2.9). Consider the linear map L { : T R by { [ ( $ tb , {) ={(t b ), then by Eq. (2.9) we have 9(x a )=
|
2?i
{ s E(L ({))= 9(L ({), x ) |Det L | 2?i exp { s E(u)= 9(u, x ) du exp
{
{
a
&1
{
L { (T)
=
|
DQ{ , W{(L {({)) (3.12)
a
R
The second equality follows because our choice of normalization in Eq. (3.8) renders |Det L { | =1. Equation (3.12) says that 9(x a ) is the Fourier transform of 9(t, x a ). But it has been proved ([1]) that 9(t, x a ) is the solution of the Cauchy problem
{
s 2 :; G L L +sLY +V(x) 4? tb X(:) X(;) lim 9(t a , x a )=(x a )
_
&}
9(t b , x a )=s x=xa
tb ta
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9(t b , x a ) t b
(3.13)
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J. LACHAPELLE
Thus, taking the inverse Fourier transform of Eq. (3.12) and substituting into Eq. (3.13) gives
|
{
exp &
R
2?i E(u) s
=_
s 2 :; G L L +sLY +V(x)+2?iE 4? tb X(:) X(;)
&}
9(x a ) du=0 x=xa
(3.14) and, therefore,
_
s 2 :; G L L +sLY +V(x)+2?iE 4? tb X(:) X(;)
&}
9(x a )=0
(3.15)
x=xa
It remains to verify the limit in Eq. (3.2). For x a x {(tb ) # U expand (x {(tb ) )= (x a } 7({(t b ), z)) around x a . To facilitate the expansion, scale the time variable by {(t b ) &1. From Eq. (3.5), it follows that the paths z # Z {(ta ) get scaled by {(t b ) &12, i.e., {(t b ) &12 z(t)=: z(t{(t b )). Thus, after scaling, we obtain the space of paths z # Z 0 such that z: [0, 1] R d and z(0)=0. Under this time scaling, Eq. (2.1) becomes x* (t{(t b ))=X (:)(x(t{(t b ))) z* :(t{(t b ))+{(t b ) Y(x(t{(t b ))) ={(t b ) 12 X (:)(x(t{(t b ))) z* :(t{(t b ))+{(t b ) Y(x(t{(t b )))
(3.16)
Use Eq. (3.16) to expand (x {(tb ) ): (x {(tb ) )=(x a )+{(t b ) 12 LX(:) (x a ) z :(1)+{(t b ) LY (x a ) + 12 {(t b ) LX(:) LX(;) (x a ) z :(1) z ;(1)+O({(t b ) 32 ) =: (x a )+2(x a ; {(t b ))
(3.17)
Similarly, scale the time in &(1s) S(z; {(t b )) to obtain 1 1 & S(z; 1)=(&?s) Q z(z; 1)+ {(t b ) s s
|
1
[V(x(t, z))+2?iE] dt
(3.18)
0
Using Eqs. (3.17) and (3.18) in Eq. (3.1), again employing the linear map L { : T R by { [ ( $ tb , {), and taking the appropriate limit yields lim xa x{(tb )
9(x a )
= lim xa x{(tb )
=
| | R
| | R
Z0
lim Z0 xa x{(tb )
1 [(x a )+2(x a ; u)] exp & S(z; 1) DQz , Wz z du s
{ = 1 [(x )+2(x ; u)] exp & S(z; 1) D { s = a
a
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Qz , Wz
z du
(3.19)
407
PATH INTEGRAL FOR DIRICHLET PROBLEM
According to Eq. (3.17), the limit as x a x {(tb ) implies {(t b ) 0. Thus, to enforce this condition, use Eq. (3.17) to write (x {(tb ) )= lim
xa x{(tb )
(x {(tb ) )= lim [(x a )+2(x a ; u)] xa x{(tb )
=(x {(tb ) )+2(x {(tb ) ; u) $(u)
(3.20)
Putting this into Eq. (3.19) gives lim xa x{(tb )
9(x a )=
| | R
=
|
Z0
1 [(x {(tb ) )+2(x {(tb ) ; u)] $(u) exp & S(z; 1) DQz , Wz z du s Z0
{
=
(x {(tb ) ) exp[(&?s) Q(z; 1)] DQz , Wz z
=(x {(tb ) )
(3.21)
This completes the proof of the theorem.
4. APPLICATION: FIXED-ENERGY AMPLITUDES Studying fixed-energy amplitudes in detail serves three purposes. It yields useful explicit expressions, it affords physical insight into the meaning of Eq. (3.1), and it translates some of the CartierDeWitt-Morette scheme into perhaps more familiar terms. 4.1. Time-Independent Schrodinger Equation The theorem from the previous section can be used to give a path integral representation for the solution of the point-to-point 5 transition amplitude of the time-independent Schrodinger equation, i.e., the fixed-energy point-to-point transition amplitude. We will exhibit both a configuration and a phase space path integral representation. To simplify matters, only flat configuration space will be considered: however, the techniques exist to handle the general case (see, e.g., Refs. [1] and [2]). The partial differential equation to be solved is
{
2 2 { +V&E G(x b , x a ; E)=0 2m xa lim G(x b , x a ; E)=$(x b &x a )
_
&
&}
(4.1)
xa xb
First consider configuration space N=Q=R n with local coordinates [q i ] where i # [1, ..., n]. To obtain the configuration space path integral from the theorem of 5 In this section, we will restrict ourselves to point-to-point transition amplitudes although the formalism allows for more general cases.
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J. LACHAPELLE
Section 3, set G cs(q b , q a ; E)#9(x a ), q b # U, 2?iE# &E, X (:) =$ i: q i, Y=0, h :; =(2?m) &1 $ :; , s=i, and (x a } 7({(t b ), z))#$(q({(t b ))&q b ). 6 The space of paths P D qa Q is parametrized by Z a according to
{
q* i(t)=$ i: z* :(t) q(t a )=q a
(4.2)
so that q(t, z)=q a +z(t). Equation (3.1) becomes G cs(q b , q a ; E)=
|
$(q({(t b ))&q b ) exp Za_T
{
i S(z; {(t b )) DQz , Wz zDQ{ , W{ {
=
(4.3)
where i ?i i S(z; {(t b ))= Q(z; {(t b ))& = =
i
|
i
|
{(tb ) {(ta )
{
|
{(tb )
[V(q(t, z))&E] dt
{(ta )
m 2 |z* | &V(q(t, z))+E dt 2
=
{(tb )
[L(q(t, z))+E] dt
(4.4)
{(ta )
Now consider the corresponding phase space N=T *Q=T *R n with local coordinates [q i, p i ] where i # [1, ..., n]. For the phase space path integral, choose G ps(q b , q a ; E)#9(x a ), q b # U, 2?iE# &E, (X (:) , X (:$) )=($ i: q i, $ i:$ p i ), Y=0, }=(?) &1, s=i, and (x a } 7({(t b ), z))#$(q({(t b ))&q b ). The space of paths PD qa , pb T *Q is parametrized by Z a_Z b =: Z ab according to
{
q* i(t)=$ i: z* :q(t) q(t a )=q a
p* i(t)=$ i:$ z* :$ p (t) p(t b )=p b
(4.5)
Note that we have chosen a different fixed end-point for p and q to be consistent with the uncertainty principle. The solution of Eq. (4.5) is (q, p)(t, z)=(q a , p b )+ (z q , z p )(t). In order to ensure conservation of the action under the parametrization, it is necessary to choose a Lagrangian submanifold 7 of T*Q such that p b =0. In other words, the base space of T*Q is not necessarily identified with the zero section. Conservation of the action is imperative since we do not want the parametrization to change the dynamics or boundary conditions of the system. 6 Recall that in the CartierDeWitt-Morette scheme, the domain of integration is over a Banach space of pointed paths. Thus, the delta function fixes the remaining loose ends of the paths. 7 Roughly speaking, a Lagrangian submanifold of T *Q is a submanifold which can be identified with the configuration space.
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409
PATH INTEGRAL FOR DIRICHLET PROBLEM
According to Eq. (3.1), the transition amplitude is given by G ps(q b , q a ; E)=
|
$(q({(t b ))&q b ) exp Zab_T
{
i S(z; {(t b )) DQz , Wz(z q , z p ) DQ{ , W{ { (4.6)
=
with i ?i i S(z; {(t b ))= Q(z; {(t b ))& = = =
i i i
| | |
{(tb )
{(tb )
1
1
q
2 p
q p
2 p
p q
{(ta )
[V(q(t, z))&E] dt
{(ta )
{2 [z z* &z z* ]&2m z &V(q(t, z ))+E= dt 1 {z z* &2m z &V(q(t, z ))+E= dt p
{(ta )
|
{(tb )
q
q
{(tb )
[ p(t, z p ) q* (t, z q )&[H(q, p)(t, z)&E]] dt {(ta )
Now consider the map R: T 4 such that {(t)=
|
tb
3(t&t$) *(t$) dt$
(4.7)
ta
Accordingly, Eq. (3.8) becomes Q{ =
1 (t b &t a )
|
tb
{* 2 dt
ta
1 (t b &t a )
|
tb
* 2 dt=Q *
(4.8)
ta
and 4 is the space of L 2 functions. Under this map ([1]), the transition amplitude becomes G ps(q b , q a ; E)=
|
$(q(t b )&q b ) exp Zab_4
{
i S(z; *) DQz , Wz(z q , z p ) DQ* , W* *
=
(4.9)
where i i S(z; *)=
|
tb
[ p(t, z p ) q* (t, z q )&*(t)[H(q, p)(t, z)&E]] dt
(4.10)
ta
4.2. A Physical Derivation Besides using the theorem of Section 3 to derive fixed-energy point-to-point transition amplitudes, it is possible to obtain them from time-dependent amplitudes by Fourier transforming in time. Garrod ([3]) (see also Ref. [4]) used this approach to write down a path integral representation for fixed-energy amplitudes.
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410
J. LACHAPELLE
But it would be satisfying to obtain the path integral representation by applying Feynman's original prescription to fixed-energy paths. The purpose of this subsection is to derive a path integral representation of fixedenergy amplitudes from first principles. That is, we will use Feynman's reasoning to obtain both phase space and configuration space representations of fixed-energy amplitudes in quantum mechanics. Agreement with the theorem of Section 3 will validate the physical reasoning and provide a fixed-energy interpretation of Eq. (3.1). The goal is to write down a path integral representation of fixed-energy point-topoint transition amplitudes. Following Feynman's prescription for the case of fixedenergy paths, we will: (1) exponentiate the classical action functional for paths with fixed energy; (2) multiply this weight factor by the initial wave function; and (3) sum over all paths with fixed energy and appropriate boundary conditions. Our starting point is the phase space associated with a Hamiltonian system. In order to compare with the results of the previous subsection, we will specialize to the case of a cotangent bundle T*Q with coordinates (q i, p i ) where i # [1, ..., n]. Since we are interested in fixed-energy amplitudes, we will want to consider the subbundle H &1(E)=: M* T*Q where H: T*Q R is the Hamiltonian and E is some fixed energy. There are two action functionals which yield the classical equations of motion when the energy is fixed: one based on Hamilton's principle and the other based on the principle of least action. (The appendix contains a short review of the two principles.) Hamilton's principle yields an action functional on phase space. It contains a Lagrange multiplier which restricts the system to the sub-bundle M *. Specifically, we have S(q, p; *)=
|
tb
[ p } q* &*(t)[H(q, p)&E]] dt
(4.11)
ta
On the other hand, the principle of least action leads to a configuration space action functional. The paths all have constant energy, and the integration boundary is variable: S (q; {)=
|
{(tb )
(L+E) dt
(4.12)
{(ta )
where L is the Lagrangian obtained from H by the (inverse Legendre) transformation FH: T *Q TQ. 8 These are the actions which must be exponentiated in the phase space and configuration space path integrals respectively. For point-to-point transition amplitudes, the initial wave function is taken to be a delta functional. The delta functional serves to ``pin down'' one of the end-points 8 The (inverse Legendre) transformation FH: T *Q TQ is the (diffeomorphic) fiber derivative of the Hamiltonian H.
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PATH INTEGRAL FOR DIRICHLET PROBLEM
of the paths. Recall that the CartierDeWitt-Morette scheme does not confine us to point-to-point transitions, but we choose them for convenience. So, the phase space path integral will be a sum over the appropriate phase space paths, and the configuration space path integral will include the appropriate configuration space paths. The integration domains are described in detail below. We must also remember to do a functional integral over * and {. They have to be varied to obtain the correct equations of motion; so they represent degrees of freedom which must be integrated over. However, since there is no kinetic term associated with these variables, they are not dynamical degrees of freedom. These considerations lead us to propose the following path integral representations for the fixed-energy point-to-point transition amplitude: G ps(q b , q a ; E)=
|
exp M_4
i
{ S(q, p; *)= $(q(t )&q ) D(q, p) D*
(4.13)
i S (q; {) $(q({(t b ))&q b ) Dq D{
(4.14)
b
b
and G cs(q b , q a ; E)=
|
exp E_T
{
=
The domain of integration for the phase space integral is the space of pointed paths given by M_4=[(q, p): [t a , t b ] T *Q]_[*: [t a , t b ] R] such that q(t a )=q a , p(t b )=p b and *(t a )=0. Likewise, the domain of integration for the configuration space functional integral is the space of pointed paths E_T=[(q b {): [t a , t b ] ?(M)]_[{: [t a , t b ] [{(t a ), {(t b )]] where ?: TQ Q is the tangent bundle projection map and M :=FH(M*). The configuration paths satisfy q({(t a ))=(q b {) a , and {(t a )=0. It is clear that these domains are not Banach spaces so they must be parametrized. Also, the integrators must be characterized by specifying appropriate quadratic forms. This being done, the integrals can be given a precise meaning by way of Eq. (2.15). For the phase space paths, the parametrization is identical to Eq. (4.5). The integrator associated with D(q, p) is DQM , WM(q, p) where i ?i Q M (z; *)=
|
tb ta
{
pq* &
* 2 i p dt= 2m 2
=
|
tb
(z q , z p )
ta
0
\ddt
&ddt &*m
zq
+\z + dt
(4.15)
p
The integrator associated with D* is DQ* , W* * and we take Q* =
1 (t b &t a )
|
tb
* 2 dt
(4.16)
ta
Using Eq. (2.15), it follows that Eq. (4.13), under the inverse map R &1: 4 T, is equivalent to Eq. (4.6).
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412
J. LACHAPELLE
For the configuration space integral, we must choose the parametrization carefully. Since the paths q b { are constant energy paths, ?(M) is not necessarily flat and the simple parametrization of Eq. (4.2) will not do. However, Q is flat by assumption so the thing to do is to parametrize P D qa Q according to
{
dq i({) dz :({) =$ i: d{ d{
(4.17)
q({(t a ))=q o
Note that this parametrization depends on the function {. The integrator associated with Dq is DQE , WE q where ?i i Q E (z; {(t b ))=
|
{(tb ) {(ta )
i m 2 |q* | (t, z) dt= 2
|
{(tb ) {(ta )
m 2 |z* | dt 2
(4.18)
and the integrator associated with D{ is DQ{ , W{ { with Q{ =
1 (t b &t a )
|
tb
{* 2 dt
(4.19)
ta
Clearly Eqs. (4.14) and (4.3) are equivalent because of Eq. (2.15). 4.3. Recovering Time-Dependent Amplitudes To check the validity of Eqs. (4.3) and (4.6), the Fourier transforms in energy must give the time-dependent point-to-point transition amplitude K(q b , t b ; q a , t a ). So, taking the Fourier transform in energy of the phase space integral G ps(q b , q a ; E), we have
|
i exp & E(t b &t a ) G ps(q b , q a ; E) dE &
= =
{
i exp & E(t b &t a ) &
|
|
{ i $ | * dt&(t &t ) exp _ & { | i exp { | ( p } q* &H) dt= $ D
M_4
=
=
|
M
=|
M_4
tb
b
DQM , WM (q, p) DQ* , W* * exp tb
a
ta
ta
{
i S(q, p; *) $ qb dE
=
=
( p } q* &*H) dt $ qb DQM , WM (q, p) DQ* , W* *
tb
ta
qb
QM , WM
(q, p)
=K(q b , t b ; q a , t a )
(4.20)
where we have introduced the notation $ qb :=$(q(t b )&q b ). In the third equality we used Eq. (2.14) with A*= ttba *(t) dt&(t b &t a ), |Det L * | =1 and * o =1.
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413
PATH INTEGRAL FOR DIRICHLET PROBLEM
Similarly, for the configuration space integral G cs(q b , q a ; E) we find i exp & E(t b &t a ) G cs(q b , q a ; E) dE &
|
{
= = =
| | |
=
E_T
i i exp & E(t b &t a )+ S (q; {) $ (q b {)b DQE , WE qDQ{ , W{ { dE &
|
{
=
$[({(t b )&{(t a ))&(t b &t a )] exp E_T
exp Q
{
i
|
tb ta
{
i
|
{(tb ) {(ta )
=
L dt $ (q b {)b DQE , WE qDQ{ , W{ {
=
L dt $ qb DQE , WE q
=K(q b , t b ; q a , t a )
(4.21)
In the third equality we used Eq. (2.13) and the fact that {=Id implies E Q where Q=[q: [t a , t b ] Q] such that q(t a )=q a . 4.4. Recovering Garrod's Expression It is reassuring to reproduce Garrod's representation of G ps(q b , q a ; E). We first make a time reparametrization t [ t$ such that dt$=*(t) dt. The phase space action functional becomes
|
qb
p dq&
qa
|
t$(tb )
(H&E) dt$
(4.22)
t$(ta )
A second time reparametrization t$ [ t" such that [t$(t b )&t$(t a )] t"+t$(t a )=t$ brings Eq. (4.22) into the form
|
qb
p dq&[t$(t b )&t$(t a )] qa
|
1
(H&E) dt"
(4.23)
0
Following Garrod, define H := 10 H dt" and write Eq. (4.23) as
|
qb
p dq&[t$(t b )&t$(t a )](H &E)
(4.24)
qa
The next step is to insert |Det L * | =1 into the phase space integral G ps(q b , q a ; E). (Recall that L * : 4 R by * [ ttba *(t) dt=t$(t b )&t$(t a ).) This gives G ps(q b , q a ; E)=
|
=
|
M_L*(4)
i
{ S(q, p; *)= $ D (q, p) |Det L | D i exp p dq&(L *)(H &E) = {|
exp
M_4
qb
QM , WM
*
Q* , W*
*
qb
*
qa
_$ qb DQM , WM (q, p) DQ* , W*(L * *)
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(4.25)
414
J. LACHAPELLE
But the integral over L *(4) is just an integral over R which can be evaluated immediately yielding Garrod's result G ps(q b , q a ; E)=
|
$(H &E) exp M
{
i
|
qb qa
=
p dq $ qb DQM , WM (q, p)
(4.26)
4.5. WKB Approximation As a final exercise, we will calculate the WKB approximation of G ps(q b , q a ; E). Recall the exact equation for G ps(q b , q a ; E): G ps(q b , q a ; E)=
|
exp Zab_4
{
i S(z; *) $ qb DQz , Wz(z q , z p ) DQ* , W* *
=
(4.27)
To obtain the semiclassical expansion, first introduce a one-parameter family of paths (z q , z p ; *): P_T T *Q_R such that (z q , z p ; *)(\, t) | \=0 =(z qcl (t), z pcl (t); * cl (t)) where \ # P=[0, 1]/R. That is, expand the path (z q , z p ; *) # Z ab_4 around a classical path: z q(\, t)=z qcl (t)+\!(t) z p(\, t)=z pcl (t)+\'(t)
(4.28)
*(\, t)=* cl (t)+\_(t) This amounts to a change of integration variables in the functional integral. Since this parametrization is an affine map on Z ab_4, it follows from Eq. (2.9) that the integrators satisfy DQz , Wz(z q , z p )=DQz , Wz(\!, \')=: D\2Qz , \&2Wz `
(4.29)
DQ* , W* *=DQ* , W*(\_) Now consider the function S b (z q , z p ; *): P R by \ [ S b (z p , z p ; *)(\). Expanding this function in a Taylor series about \=0 yields
\ n d nS b (z q , z p ; *) (0) d\ n n=1 n!
S b (z q , z p ; *)(\)=S b (z q , z p ; *)(0)+ :
(4.30)
The WKB approximation results from evaluating Eq. (4.30) for the family of paths given by Eq. (4.28), keeping only terms up to O(\ 2 ), and putting \=- 2?. By definition, a classical path satisfies d(S b (z qcl , z pcl ; * cl ))(0)d\=0 so the action functional appropriate for the WKB approximation is 2 \ 2 d S b (z qcl , z pcl ; * cl ) S b (z q , z p ; *)(\)=S b (z qcl , z pcl ; * cl )+ 2 d\ 2
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(4.31)
415
PATH INTEGRAL FOR DIRICHLET PROBLEM
But, S b (z qcl , z pcl ; * cl )=
|
tb ta
z pcl (z qcl , E) z* qcl dt=
|
qb
p cl (q cl , E) dq cl qa
=: W(q b , q a ; E)
(4.32)
which follows from Eq. (4.11). Furthermore, it can be shown ([2]) that 2 \ 2 d S b (z qcl , z pcl ; * cl ) =? 2 d\ 2
|
tb ta
{ ( J(z
qcl
, z pcl ; * cl ) `, `) &2
_(t) dH(z qcl , z pcl ) dt \ d\
=
(4.33) where
J=
\
2H z iq z qj
&$ ij
2H $ ij + i t z p z qj
2H + i t z q z pj
+
2H z ip z pj
(4.34)
The map J(z qcl (t), z pcl (t); * cl ): T (zqcl (t), zpcl (t))(M*) T *(zqcl (t), zpcl (t))(M*) is the phase space Jacobi operator. In general there is a boundary term associated with the first and second variations, but they will not contribute because !(t a )=!(t b )=0 in our case. Therefore, under the change of variable defined by Eq. (4.28), the WKB approximation of G ps(q b , q a ; E) becomes (q b , q a ; E)=exp G WKB ps
{
i W(q b , q a ; E)
=|
{
exp ?iQ z(`)& Zab_4
i
|
tp
\ 2_(t)
ta
_$(\!(t b )) D\2Qz , \&2Wz `DQ* , W*(\_)
dH cl dt d\
=
(4.35)
where we have defined a new quadratic form Q z(`) :=
|
tb
ta
( J(z qcl , z pcl ) `, `) dt
(4.36)
The _ integral can be done by applying the same technique used in the previous subsection with the result
|
{
exp & 4
i
|
tb ta
\ 2_
dH cl dt DQ* , W*(\_)= d\
=
|
R
i
dH cl
{ \u d\ = du dH $ \ d\ +
exp &
=\ &1 =\ &1
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cl
(4.37)
416
J. LACHAPELLE
where dH cl := d\
|
1
0
dH cl dt" d\
(4.38)
and t" has been defined in the previous subsection. The last equality follows because dHd\=dHd\ and Eq. (4.26) says H =E for paths in Z ab . Hence, dHd\=0 is automatically satisfied. It remains to evaluate
|
Zab
exp[?iQ z(`)] $(\!(t b )) D\2Qz , \&2Wz `
(4.39)
By considering the linear map L: Z ab Z ab such that Q z =Q z b L and using Eq. (2.9), it can be shown ([1]) that
|
Zab
exp[?iQ z(`)] $(\!(t b )) D\2Qz , \&2Wz ` =i &Ind(Qz Qz ) |Det Q z Q z | &12
|
Zab
exp[?iQ z(`)] $(\!(t b )) DQz , Wz `
(4.40)
where the index Ind(Q z Q z ) equals the number of negative eigenvalues of Q z with respect to Q z . Notice that the integral on the right-hand-side now contains a gaussian integrator. The next step is to write $(\!(t b ))=
|
Rn
exp[&2?i\!(t b ) } u] d nu
(4.41)
substitute into Eq. (4.40), interchange integrations (which is allowed), and use Eq. (2.5) to get
|
Zab
exp[?iQ z(`)] $(\!(t b )) D\2Qz , \&2Wz ` =i &Ind(Qz Qz ) |Det Q z Q z | &12
|
_
Rn
exp[&?iW(z$q +\u$ tb , z$p )] d nu
}
(4.42) (z$q , z$p )=0
where (z$q , z$p ) # Z$ab and Z$ab is the dual of Z ab . A straightforward but involved calculation ([2]) gives
|
Rn
exp[&?iW(z$q +\u$ tb , z$p )] d nu
}
=\ &n |det iG(t b , t b )| &12 (4.43) (z$q , z$p )=0
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417
PATH INTEGRAL FOR DIRICHLET PROBLEM
where G(t b , t b ) is the Green's function of the Jacobi operator for the boundary conditions q cl (t a )=q a and p cl (t b )=p b . Finally, it can be shown ([1]) that |det iG(t b , t b )| |Det Q z Q z | = |det D VM | &1
(4.44)
where D VM is the van VleckMorette matrix of the action functional associated with Q z , i.e., S b (z qcl , z pcl ; * cl )=W(q b , q a ; E). It follows ([8]) that
det D VM =det
\
2W q a q b 2W E q b
2W q a E 2W E E
+
(4.45)
Putting together all the pieces, we obtain the final expression (q b , q a ; E)=i &Ind(Qz Qz )(2?) &(n+1)2 |det D VM | 12 exp G WKB ps
i
{ W(q , q ; E)= b
a
(4.46)
APPENDIX A.1. Hamilton's Principle The variational principle of Hamilton is usually associated with an action functional obtained from a Lagrangian defined on the tangent bundle TQ of configuration space. However, it works just as well for a suitably defined action functional of phase space variables. The variational calculus then yields Hamilton's equations of motion. In particular, to obtain the classical equations of motion for a Hamiltonian system with fixed configuration end-points, we impose the variational principle $S(q, p)=$
|
tb
[ p } q* &H(q, p)] dt=0
(A.1)
ta
such that the variations of the q's vanish at the end-points. If we want to resrict the system to a submanifold of constant energy M*=H &1(E), we simply make the replacement H *(H&E) where * is a Lagrange multiplier to obtain $S(q, p; *)=$
|
tb
[ p } q* &*(t)[H(q, p)&E]] dt=0
(A.2)
ta
Variation of this action functional yields Hamilton's equations (which will have a * dependence) with the additional constraint H=E. Thus the variation gives
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418
J. LACHAPELLE
Hamilton's equations for phase space paths with constant energy. Note that there is no harm in letting * be a function of t in Eq. (A.2). A.2. The Least Action Principle If one solves for the constraint in Eq. (A.2), the integrand becomes - 2m(E&V(q)) q* 2 (for the special case of H(q, p)=p 22m+V(q), and the action functional becomes a configuration space action functional. However, solving the constraint does not commute with the variational principle, and so it does not follow that variation of the resulting integral will yield the equations of motion of the restricted system. In other words, the equations of motion obtained by varying constrained paths are not the same as the equations of motion obtained by varying unconstrained paths supplemented with a constraint equation. In fact, in order to regain the correct equations of motion, it is necessary to vary the parametrization of the paths (see, e.g., Ref. [5]). The result is the least action principle which can be stated as $ E S (q; {)=$ E
|
{(tb )
- 2m(E&V(q(t))) q* (t) 2 dt=$ E
{(ta )
|
{(tb )
(L+E) dt=0
(A.3)
{(ta )
where $ E denotes a variation of paths with fixed end-points and fixed energy. The resulting variation yields Lagrange's equations of motion for configuration space paths with fixed energy.
ACKNOWLEDGMENT I thank C. DeWitt-Morette for valuable suggestions.
REFERENCES 1. P. Cartier and C. DeWitt-Morette, A new perspective on functional integration, J. Math. Phys. 36, No. 5 (1995), 2237. 2. J. LaChapelle, ``Functional Integration on Symplectic Manifolds,'' Ph.D. dissertation, University of Texas at Austin, 1995. 3. C. Garrod, Hamiltonian path integral methods, Rev. Mod. Phys. 38 (1966), 483. 4. M. C. Gutzwiller, Phase-integral approximation in momentum space and the bound state of an atom, J. Math. Phys. 8 (1967), 1979. 5. R. Abraham and J. E. Marsden, ``Foundations of Mechanics,'' BenjaminCummings, MA, 1981. 6. M. Freidlin, ``Functional Integration and Partial Differential Equations,'' Annals of Mathematics Studies, Vol. 109, Princeton Univ. Press, Princeton, NJ, 1985. 7. A. Friedman, ``Stochastic Differential Equations and Applications,'' Vols. 1 6 2, Academic Press, New York, 1975. 8. C. DeWitt-Morette, A. Maheshwari, and B. Nelson, Path integration in non-relativistic quantum mechanics, Phys. Rep. 50, No. 5 (1979), 255.
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Path Integral Solution of the Dirichlet Problem J. LaChapelle Department of Physics and Center for Relativity, University of Texas, Austin, Texas 78712 Received April 29, 1996; revised July 8, 1996
A scheme for functional integration developed by CartierDeWitt-Morette is first reviewed and then employed to construct the path integral representation for the solution of the Dirichlet problem in terms of first exit time. The path integral solution is then applied to calculate the fixed-energy point-to-point transition amplitude both in configuration and phase space. The path integral solution can also be derived using physical principles based on Feynman's original reasoning. We check that the Fourier transform in energy of the fixedenergy point-to-point transition amplitude gives the well known time-dependent transition amplitude, and calculate the WKB approximation. 1997 Academic Press
1. INTRODUCTION Functional integrals are powerful tools for solving partial differential equations. Techniques of functional integration are particularly developed and refined in the context of stochastic processes. However, stochastic methods are not always applicable in many physical systems. For example, the solution of the Schrodinger equation is not obtained directly by stochastic means: first the diffusion equation is solved and then the solution is analytically continued. The same goes for the stationary Schrodinger equation. Also, stochastic paths are not differentiable, but many physical applications dictate the use of action functionals which require differentiability of the paths. There exists a scheme for functional integration due to CartierDeWitt-Morette ([1]) which in a sense generalizes and improves the stochastic approach. Relevant details are presented in Section 2. The underlying theme of their work is an emphasis on the domain of integration which is an infinite dimensional space of paths. Their scheme allows for rigorous rather than formal manipulations of path integrals. The functional integration formalism described here generalizes the stochastic approach in the sense that it can be used to solve a wider class of partial differential equations. For example, it is possible to solve both the Schrodinger and diffusion equations directly. In Section 3 we present a path integral solution of a generalized Dirichlet problemgeneralized because the characteristic form of the partial differential operator need not be positive definite in all cases. The theorem proved in Section 3 and the general theorem of Ref. [1] provide the functional integral solutions of elliptic and parabolic second order 397 0003-491697 25.00 Copyright 1997 by Academic Press All rights of reproduction in any form reserved.
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partial differential equations within the CartierDeWitt-Morette framework. Work on the hyperbolic case is in progress. Section 4 examines the configuration and phase space path integral solution for the point-to-point transition amplitude of the time-independent Schrodinger equation. Use of this familiar example is meant to check the general result of Section 3 and to illustrate some useful methods of manipulating functional integrals. The point-to-point transition amplitude of the time-independent Schrodinger equation (which depends on energy) can be used to study the eigenfunction spectrum of the associated partial differential equation. The task is to determine the poles and cuts of the transition amplitude in the complex energy plane. For the configuration space integral, it seems that one must first do the functional integral because the paths, which are fixed energy paths, have an energy dependence. However, the phase space integral might offer a way to study the pole and cut structure within the path integral since the paths are only restricted by a Lagrange multiplier. In other words, in the phase space integral the paths have no energy dependence before the path integral is done. This property of the phase space integral offers a new and possibly useful way to study the eigenfunction spectrum of a time-independent Schrodinger equation.
2. FUNCTIONAL INTEGRALS To construct a functional integral, we require three well-defined components: (1) a domain of integration which is an infinite dimensional function space, (2) an integrator defined on the domain of integration, and (3) an integrand or integrable functional. Given these three ingredients, a value can be assigned to the path integral. 2.1. The Domain Here we consider the domain of integration X to be a real, separable Banach space of pointed paths. 1 The paths x # X are taken to be L 2, 1 functions 2 which map a closed interval T/R into a target manifold. We denote the dual space of X by X$ and the duality by (x$, x) X where x$ # X$. For quantum mechanical applications, one is usually interested in a functional integral over all paths x: T N where T/R and N is an n-dimensional manifold. But, in general, the space of paths which take their values in a manifold N will not be a Banach space X. For suppose that we are interested in all paths which have the same fixed end-points at t=t a # T. Then x 1 +x 2 # X if and only if x(t a )=0 \x # X. This follows because the equality (x 1 +x 2 )(t a )=x 1(t a )+x 2(t a ) holds for 1 The term ``pointed paths'' means the paths x # X have one fixed end-point. For example, if T/R is the interval [t a , t b ], then all paths with fixed initial points x(t a )=x a and arbitrary final points would be pointed paths. 2 A function is L 2, 1 if T |dxdt| 2 dt< . We choose L 2, 1 functions because of their relevance to physics. Specifically, the Lagrangian density requires derivatives, and the kinetic energy must be finite.
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399
vanishing end-points only. Therefore, the space of paths which have a nonvanishing fixed end-point is not a Banach space. However, if we restrict attention to the space of paths with only one end-point fixed (that is, a space of pointed paths) then the space of such paths is contractible. Consequently, it can be parametrized by a space of pointed paths which take their values in a flat manifold and have their fixed end-point at the origin. But this parameter space of paths is a Banach space. Hence, integration on the space of pointed paths can be defined in terms of integration on the parameter space ([1]). The situation is analogous to integration on finite dimensional manifolds. There one does not know how to integrate on a general manifold. Instead, one defines the integral by parametrizing points in the general manifold by points in flat manifolds where one knows how to integrate. For functional integrals, we do not know how to integrate over a general function space so we parametrize it with a Banach space where the rules of functional integration are established. The details of the parametrization follow. Let there be d linearly independent vector fields which generate a vector subspace D x of T x N at each x # N, and denote 2, 1 paths with fixed them by X (:) where : # [1, ..., d]. Let P D xa N denote the space of L D initial point x(t a )=x a # N \x # P xa N and such that x* (t)&Y(x(t)) # D x(t) where Y is some given vector field on N. Denote by P 0 R d =: Z a the space of L 2, 1 paths with fixed initial point z(t a )=0 # R d \z # Z a . 2, 1 and since [X (:)(x(t))] spans D x(t) , it follows Since the paths in P D xa N are L : that there exist functions z* (t) such that
{
x* (t)&Y(x(t))=X (:)(x(t)) z* :(t) x(t a )=x a
(2.1)
This differential equation associates a path z # Z a with each path x # P D xa N. If N is compact, the map x [ z can be inverted. That is, given some z # Z a , Eq. (2.1) has a unique solution x # P D xa N. We will denote the solution by x(t, z)=x a } 7(t, z) where 7(t, z): N N is a global transformation on N such that 3 x(t a , z)= x a } 7(t a , z)=x a . We now have a parametrization P: Z a P D xa N by z [ x. 2.2. The Integrator and Integrand It is not possible to define a measure for the type of path integrals used in quantum mechanics. However, it is possible to replace the notion of a measure with the notion of an integrator. In the CartierDeWitt-Morette scheme, an integrator, denoted by D3, Z x, is defined by the relation
|
3(x, x$) D3, Z x=Z(x$)
(2.2)
X
The integrator D3, Z x is defined implicitly by the functionals 3(x, x$) and Z(x$). 3 If we had chosen to parametrize with paths z # Z b , the solution of Eq. (2.1) would be x(t, z)= xa } 7(t$, z) where t$ :=t a +t b &t. Hence, x(t a , z)=x a } 7(t b , z)=x a as required.
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It is possible to define a class 8 3 of integrands which is integrable with respect to D3, Z x by the relation F+(x)=
|
3(x, x$) d+(x$)
(2.3)
X$
where F + # 8 3 . 4 This construction of the functional integral implies
|
F +(x) D3, Z x=
X
|
Z(x$) d+(x$)
(2.4)
X$
The right-hand side of Eq. (2.4) should not be construed as a means to evaluate the left-hand side. Indeed, the measure + associated with a given integrand F + # 8 Q is not usually known. Equation (2.4) is a consequence of relations (2.2) and (2.3) and is useful for determining when the integral exists or for proving general theoremsneither of which requires the specific form of +. 2.2.1. Gaussian Integrator. By far the most prevalent choice for D3, Z x in quantum physics is related to the gaussian integrator. Assume there exists a continuous, symmetric, non-degenerate linear map D: X X$ and denote its inverse by the map G: X$ X. Then a quadratic form on X is defined by Q(x)=( Dx, x) X , and a quadratic form on X$ is defined by W(x$)=( x$, Gx$) X . For 3(x, x$), choose the functional exp[(&?s) Q(x)&2?i( x$, x) X ], and let Z(x$) be exp[&(?s) W(x$)] where s # C is an adjustable parameter. Then DQ, W x is characterized by
|
exp[(&?s) Q(x)&2?i( x$, x) X ] DQ, W x=exp[&?sW(x$)]
(2.5)
X
The integrator defined by Dw(x) :=exp[(&?s) Q(x)] DQ, W x
(2.6)
is called a gaussian integrator. Equation (2.5) can be viewed as a Fourier transform. That is, (Fw)(x$)=
|
exp[&2?i( x$, x) X ] Dw(x)=exp[&?sW(x$)]
(2.7)
X
Notice that at the origin of X$, the integrator has the normalization
|
X
4
Dw(x)=
|
exp[(&?s) Q(x)] DQ, W x=1 X
Since X$ is separable and complete, it is possible to define complex Borel measures + on X$.
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(2.8)
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PATH INTEGRAL FOR DIRICHLET PROBLEM
Gaussian integrators possess an important property under affine transformations. Consider an affine map A: X X by x [ Ax=Lx+b where L is an invertible linear map and b # X is a fixed element. Then the image of the integrator DQ, W x under A is given by DQ, W (Ax)= |Det L| DQ, W x
(2.9)
For gaussian integrators, one can define a class of functionals 8 Q which is integrable with respect to DQ, W x by the relation F+(x)=
|
exp[(&?s) Q(x)&2?i( x$, x) X ] d+(x$)
(2.10)
X$
so that to each measure + on X$ there is associated an integrand F + # 8 Q . 2.2.2. Delta Functional Integrator. Closely related to a gaussian integrator is the delta functional integrator. In order to define a delta functional integrator, we use the well-known fact that a delta function can be represented as a gaussian in the limit as the width goes to zero. The infinite dimensional analog is to take the limit s 0 in Eq. (2.5). Hence, lim s0
|
exp[&2?i( x$, x) X ] Dw(x)=: X
=:
| |
exp[&2?i( x$, x) X ] $(x) DQ, W x X
exp[&2?i( x$, x) X ] D$(x)
X
= lim exp[&?sW(x$)] s0
=1
(2.11)
defines a delta functional integrator. This definition is justified by the fact that it reduces to the finite dimensional result for the Fourier transform of a delta function. That is, let L x : X R n be the linear surjective map defined by x [ u=[u 1, ..., u n ] where u i =( x i$ , x) for a given x i$ # X$. Then it can be shown ([1]) that the left-hand side of Eq. (2.11) reduces to lim s0
|
Rn
=
i j n |det sW ij | &12 exp[(&?s) W &1 ij u u &2?i( u$, u) Rn ] d u
|
Rn
$ n(u) exp[&2?i( u$, u) Rn ] d nu
=1 We point out that this result is independent of n.
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(2.12)
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J. LACHAPELLE
The delta functional integrator also behaves like its finite dimensional counterpart for non-trivial integrable functionals:
|
F +(x) D$(x)=
X
=
|
F +(x) $(x) DQ, W x X
|
|
exp[(&?s) Q(x)&2?i( x$, x) X ] d+(x$) DQ, W x
| |
exp[(&2?s) Q(x)&2?i( x$, x) X ] DQ, W x d+(x$)
$(x)
X
X$
= lim s0
= lim s0
=
|
X$
|
X$
X
exp[& 12 ?sW(x$)] d+(x$)
d+(x$)
X$
=F +(0)
(2.13)
where the interchange of integrations in the third line can be rigorously justified, and the second and last lines follow from Eq. (2.10). Moreover, for non-trivial arguments of the delta functional, we have
|
F +(x) D$(Ax)=
X
=
|
F +(x) $(Ax) DQ, W x X
|
F +(A &1x ) $(x ) DQ, W (A &1x )
|
F +(A &1x ) $(x ) |Det L| &1 DQ, W (x )
X
=
X
= |Det L| &1 F +(x o )
(2.14)
where x o is a solution of Ax=Lx+b=0. In the second line, we used the affine transformation x [ A &1x, and in the third line Eq. (2.9) was used. The last line follows from Eq. (2.13). Equation (2.14) can be written in a short-hand notation as $(Ax)= |Det L| &1 $(x&x o ). This is obviously similar to the analogous finite dimensional result. 2.3. The Integral We are now in a position to give meaning to a functional integral of an integrable functional of pointed paths in a general manifold Ndenoted schematically by
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PATH INTEGRAL FOR DIRICHLET PROBLEM
403
PDx N F(x) Dx. First, construct the parametrization P: Z a P D xa N. Since we know a how to do a functional integral on Z a , simply define
|
PD x N
F(x) Dx :=
|
F(x(z)) D3, Z z
(2.15)
Za
a
where D3, Z z is characterized by Eq. (2.2) with X=Z a and F(x(z)) is an integrable functional defined by Eq. (2.3). (Henceforth, we will not make the + dependence of integrable functionals explicit.) Of course, for this definition to be useful, the righthand side must be independent of the parametrization used. That is, we must get the same answer if we use the parametrization P: Z a P D xa N by z~ [ x. This just means that the functional integral must be invariant under a change of variable of integration. 3. SOLUTION OF THE GENERALIZED DIRICHLET PROBLEM The solution of the Dirichlet problem in terms of a functional integral whose variable of integration is a stochastic process is well known (see, e.g., Refs. [6] and [7]). In fact, the stochastic strategy is quite similar to what we are presenting. However, there are important differences. Most obvious is the analytic nature of the paths: the trajectory of a stochastic process is only continuous whereas we deal with L 2, 1 paths. Another important difference is that our approach supplies solutions for a broader class of partial differential equations. In particular, for certain choices of the parameter s, the characteristic form of the second order differential operator need not be positive definitehence the section title ``... Generalized Dirichlet Problem.'' Our main result is contained in the following theorem. 2, 1 paths which have a fixed end-point Theorem. Let P D xa N be the space of L x(t a )=x a # N and whose velocity vectors are elements of D. Parametrize P D xa N by Z a . Let U be a bounded convex region in N with boundary U. Let T be the space of L 2, 1 monotonic functions {: T=[t a , t b ] [{(t a ), {(t b )]/R such that {(t a )=0. Assume that the parametrized paths reach the boundary U at t={(t b ), i.e., x({(t b ), } )=x {(tb ) # U. Let : N C m be any integrable C (N) bounded function. Then, for x a # U, the functional integral
9(x a ) :=
1 (x a } 7({(t b ), z)) exp & S(z; {(t b )) DQz , Wz zDQ{ , W{ { s Za_T
{
|
=
(3.1)
is a solution of the generalized homogeneous Dirichlet problem in U
{
_
s 2 :; G L L +sLY +V(x)+2?iE 4? tb X(:) X(;) lim
xa x{(tb ) # U
&}
9(x a )=0 x=xa
9(x a )=(x {(tb ) )
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(3.2)
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J. LACHAPELLE
Here LX(:) is the Lie derivative in the X (:) direction, and E # R (recall that s # C). The matrix G :; tb is given by
|
Za
$ 2 exp[&?sW(z$)] $z$:(t b ) $z$;(t b )
z :(t b ) z ;(t b ) exp[(&?s) Q z(z)] DQz , Wz z= =
s ( $ : , G$ ;tb ) 2? tb
=
s :; G (t b , t b ) 2?
=:
}
z$=0
s :; G 2? tb
(3.3)
It is the Green's function of Q z with boundary conditions G :;(t, u) | t=ta =0 and G :;(t, u) t | t=tb =0 evaluated at t=u=t b . The functional &(1s) S(z; {(t b )) is 1 1 & S(z; {(t b ))=(&?s) Q z(z; {(t b ))+ s s
|
{(tb )
[V(x(t, z))+2?iE] dt
(3.4)
{(ta )
where Q z(z; {(t b ))=
{(tb )
{(tb )
{(ta )
{(ta )
| |
z :(t) D :;(t, u) z ;(u) dt du
(3.5)
and D :;(t, u) depends on the nature of N. For example, if N is associated with the configuration space of a dynamical system, then
\
D :;(t, u)=$(t&u) h :;
d2 du 2
+
(3.6)
where h :; is real, symmetric and non-degenerate. If N is associated with the phase space of a Hamiltonian dynamical system and we choose canonical coordinates so that z : =(z iq , z ip ) where i # [1, ..., d2], then
D :;(t, u)=$(t&u)
} 2
\
0
d du 1 &$ ij m
&$ ij
d $ ij du
+
(3.7)
where } is some constant. The quadratic form Q { which characterizes DQ{ , W{ { is chosen to be Q{ =
1 (t b &t a )
|
tb
{* 2 dt
ta
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(3.8)
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PATH INTEGRAL FOR DIRICHLET PROBLEM
This theorem should be compared to the main theorem of Ref. [1] which provides the functional integral solution of the Cauchy problem. The major difference resides in the extra E term in the action functional and the extra functional integral over the { variable. Proof.
Equation (3.1) may be rewritten as 9(x a )=
|
exp T
2?i
{ s E({(t )&{(t ))= 9({(t ), x ) D b
a
b
a
Q{ , W{
{
(3.9)
where 9({(t b ), x a ) :=
|
Za
1 (x a } 7({(t b ), z)) exp & S (z) DQz , Wz z s
{
=
(3.10)
and 1 & S (z) :=(&?s) s
{(tb )
| | {(ta )
{(tb )
z :(t) D :;(t, u) z ;(u) dt du+
{(ta )
1 s
|
{(tb )
V(x(t, z)) dt
{(ta )
(3.11) Notice that the integrand in Eq. (3.9) depends on { at a single point (recall that {(t a )=0). That is, the integrand is a function of {(t b ). This fact can be exploited to transform the functional integral into a finite dimensional integral by utilizing Eq. (2.9). Consider the linear map L { : T R by { [ ( $ tb , {) ={(t b ), then by Eq. (2.9) we have 9(x a )=
|
2?i
{ s E(L ({))= 9(L ({), x ) |Det L | 2?i exp { s E(u)= 9(u, x ) du exp
{
{
a
&1
{
L { (T)
=
|
DQ{ , W{(L {({)) (3.12)
a
R
The second equality follows because our choice of normalization in Eq. (3.8) renders |Det L { | =1. Equation (3.12) says that 9(x a ) is the Fourier transform of 9(t, x a ). But it has been proved ([1]) that 9(t, x a ) is the solution of the Cauchy problem
{
s 2 :; G L L +sLY +V(x) 4? tb X(:) X(;) lim 9(t a , x a )=(x a )
_
&}
9(t b , x a )=s x=xa
tb ta
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9(t b , x a ) t b
(3.13)
406
J. LACHAPELLE
Thus, taking the inverse Fourier transform of Eq. (3.12) and substituting into Eq. (3.13) gives
|
{
exp &
R
2?i E(u) s
=_
s 2 :; G L L +sLY +V(x)+2?iE 4? tb X(:) X(;)
&}
9(x a ) du=0 x=xa
(3.14) and, therefore,
_
s 2 :; G L L +sLY +V(x)+2?iE 4? tb X(:) X(;)
&}
9(x a )=0
(3.15)
x=xa
It remains to verify the limit in Eq. (3.2). For x a x {(tb ) # U expand (x {(tb ) )= (x a } 7({(t b ), z)) around x a . To facilitate the expansion, scale the time variable by {(t b ) &1. From Eq. (3.5), it follows that the paths z # Z {(ta ) get scaled by {(t b ) &12, i.e., {(t b ) &12 z(t)=: z(t{(t b )). Thus, after scaling, we obtain the space of paths z # Z 0 such that z: [0, 1] R d and z(0)=0. Under this time scaling, Eq. (2.1) becomes x* (t{(t b ))=X (:)(x(t{(t b ))) z* :(t{(t b ))+{(t b ) Y(x(t{(t b ))) ={(t b ) 12 X (:)(x(t{(t b ))) z* :(t{(t b ))+{(t b ) Y(x(t{(t b )))
(3.16)
Use Eq. (3.16) to expand (x {(tb ) ): (x {(tb ) )=(x a )+{(t b ) 12 LX(:) (x a ) z :(1)+{(t b ) LY (x a ) + 12 {(t b ) LX(:) LX(;) (x a ) z :(1) z ;(1)+O({(t b ) 32 ) =: (x a )+2(x a ; {(t b ))
(3.17)
Similarly, scale the time in &(1s) S(z; {(t b )) to obtain 1 1 & S(z; 1)=(&?s) Q z(z; 1)+ {(t b ) s s
|
1
[V(x(t, z))+2?iE] dt
(3.18)
0
Using Eqs. (3.17) and (3.18) in Eq. (3.1), again employing the linear map L { : T R by { [ ( $ tb , {), and taking the appropriate limit yields lim xa x{(tb )
9(x a )
= lim xa x{(tb )
=
| | R
| | R
Z0
lim Z0 xa x{(tb )
1 [(x a )+2(x a ; u)] exp & S(z; 1) DQz , Wz z du s
{ = 1 [(x )+2(x ; u)] exp & S(z; 1) D { s = a
a
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Qz , Wz
z du
(3.19)
407
PATH INTEGRAL FOR DIRICHLET PROBLEM
According to Eq. (3.17), the limit as x a x {(tb ) implies {(t b ) 0. Thus, to enforce this condition, use Eq. (3.17) to write (x {(tb ) )= lim
xa x{(tb )
(x {(tb ) )= lim [(x a )+2(x a ; u)] xa x{(tb )
=(x {(tb ) )+2(x {(tb ) ; u) $(u)
(3.20)
Putting this into Eq. (3.19) gives lim xa x{(tb )
9(x a )=
| | R
=
|
Z0
1 [(x {(tb ) )+2(x {(tb ) ; u)] $(u) exp & S(z; 1) DQz , Wz z du s Z0
{
=
(x {(tb ) ) exp[(&?s) Q(z; 1)] DQz , Wz z
=(x {(tb ) )
(3.21)
This completes the proof of the theorem.
4. APPLICATION: FIXED-ENERGY AMPLITUDES Studying fixed-energy amplitudes in detail serves three purposes. It yields useful explicit expressions, it affords physical insight into the meaning of Eq. (3.1), and it translates some of the CartierDeWitt-Morette scheme into perhaps more familiar terms. 4.1. Time-Independent Schrodinger Equation The theorem from the previous section can be used to give a path integral representation for the solution of the point-to-point 5 transition amplitude of the time-independent Schrodinger equation, i.e., the fixed-energy point-to-point transition amplitude. We will exhibit both a configuration and a phase space path integral representation. To simplify matters, only flat configuration space will be considered: however, the techniques exist to handle the general case (see, e.g., Refs. [1] and [2]). The partial differential equation to be solved is
{
2 2 { +V&E G(x b , x a ; E)=0 2m xa lim G(x b , x a ; E)=$(x b &x a )
_
&
&}
(4.1)
xa xb
First consider configuration space N=Q=R n with local coordinates [q i ] where i # [1, ..., n]. To obtain the configuration space path integral from the theorem of 5 In this section, we will restrict ourselves to point-to-point transition amplitudes although the formalism allows for more general cases.
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408
J. LACHAPELLE
Section 3, set G cs(q b , q a ; E)#9(x a ), q b # U, 2?iE# &E, X (:) =$ i: q i, Y=0, h :; =(2?m) &1 $ :; , s=i, and (x a } 7({(t b ), z))#$(q({(t b ))&q b ). 6 The space of paths P D qa Q is parametrized by Z a according to
{
q* i(t)=$ i: z* :(t) q(t a )=q a
(4.2)
so that q(t, z)=q a +z(t). Equation (3.1) becomes G cs(q b , q a ; E)=
|
$(q({(t b ))&q b ) exp Za_T
{
i S(z; {(t b )) DQz , Wz zDQ{ , W{ {
=
(4.3)
where i ?i i S(z; {(t b ))= Q(z; {(t b ))& = =
i
|
i
|
{(tb ) {(ta )
{
|
{(tb )
[V(q(t, z))&E] dt
{(ta )
m 2 |z* | &V(q(t, z))+E dt 2
=
{(tb )
[L(q(t, z))+E] dt
(4.4)
{(ta )
Now consider the corresponding phase space N=T *Q=T *R n with local coordinates [q i, p i ] where i # [1, ..., n]. For the phase space path integral, choose G ps(q b , q a ; E)#9(x a ), q b # U, 2?iE# &E, (X (:) , X (:$) )=($ i: q i, $ i:$ p i ), Y=0, }=(?) &1, s=i, and (x a } 7({(t b ), z))#$(q({(t b ))&q b ). The space of paths PD qa , pb T *Q is parametrized by Z a_Z b =: Z ab according to
{
q* i(t)=$ i: z* :q(t) q(t a )=q a
p* i(t)=$ i:$ z* :$ p (t) p(t b )=p b
(4.5)
Note that we have chosen a different fixed end-point for p and q to be consistent with the uncertainty principle. The solution of Eq. (4.5) is (q, p)(t, z)=(q a , p b )+ (z q , z p )(t). In order to ensure conservation of the action under the parametrization, it is necessary to choose a Lagrangian submanifold 7 of T*Q such that p b =0. In other words, the base space of T*Q is not necessarily identified with the zero section. Conservation of the action is imperative since we do not want the parametrization to change the dynamics or boundary conditions of the system. 6 Recall that in the CartierDeWitt-Morette scheme, the domain of integration is over a Banach space of pointed paths. Thus, the delta function fixes the remaining loose ends of the paths. 7 Roughly speaking, a Lagrangian submanifold of T *Q is a submanifold which can be identified with the configuration space.
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PATH INTEGRAL FOR DIRICHLET PROBLEM
According to Eq. (3.1), the transition amplitude is given by G ps(q b , q a ; E)=
|
$(q({(t b ))&q b ) exp Zab_T
{
i S(z; {(t b )) DQz , Wz(z q , z p ) DQ{ , W{ { (4.6)
=
with i ?i i S(z; {(t b ))= Q(z; {(t b ))& = = =
i i i
| | |
{(tb )
{(tb )
1
1
q
2 p
q p
2 p
p q
{(ta )
[V(q(t, z))&E] dt
{(ta )
{2 [z z* &z z* ]&2m z &V(q(t, z ))+E= dt 1 {z z* &2m z &V(q(t, z ))+E= dt p
{(ta )
|
{(tb )
q
q
{(tb )
[ p(t, z p ) q* (t, z q )&[H(q, p)(t, z)&E]] dt {(ta )
Now consider the map R: T 4 such that {(t)=
|
tb
3(t&t$) *(t$) dt$
(4.7)
ta
Accordingly, Eq. (3.8) becomes Q{ =
1 (t b &t a )
|
tb
{* 2 dt
ta
1 (t b &t a )
|
tb
* 2 dt=Q *
(4.8)
ta
and 4 is the space of L 2 functions. Under this map ([1]), the transition amplitude becomes G ps(q b , q a ; E)=
|
$(q(t b )&q b ) exp Zab_4
{
i S(z; *) DQz , Wz(z q , z p ) DQ* , W* *
=
(4.9)
where i i S(z; *)=
|
tb
[ p(t, z p ) q* (t, z q )&*(t)[H(q, p)(t, z)&E]] dt
(4.10)
ta
4.2. A Physical Derivation Besides using the theorem of Section 3 to derive fixed-energy point-to-point transition amplitudes, it is possible to obtain them from time-dependent amplitudes by Fourier transforming in time. Garrod ([3]) (see also Ref. [4]) used this approach to write down a path integral representation for fixed-energy amplitudes.
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J. LACHAPELLE
But it would be satisfying to obtain the path integral representation by applying Feynman's original prescription to fixed-energy paths. The purpose of this subsection is to derive a path integral representation of fixedenergy amplitudes from first principles. That is, we will use Feynman's reasoning to obtain both phase space and configuration space representations of fixed-energy amplitudes in quantum mechanics. Agreement with the theorem of Section 3 will validate the physical reasoning and provide a fixed-energy interpretation of Eq. (3.1). The goal is to write down a path integral representation of fixed-energy point-topoint transition amplitudes. Following Feynman's prescription for the case of fixedenergy paths, we will: (1) exponentiate the classical action functional for paths with fixed energy; (2) multiply this weight factor by the initial wave function; and (3) sum over all paths with fixed energy and appropriate boundary conditions. Our starting point is the phase space associated with a Hamiltonian system. In order to compare with the results of the previous subsection, we will specialize to the case of a cotangent bundle T*Q with coordinates (q i, p i ) where i # [1, ..., n]. Since we are interested in fixed-energy amplitudes, we will want to consider the subbundle H &1(E)=: M* T*Q where H: T*Q R is the Hamiltonian and E is some fixed energy. There are two action functionals which yield the classical equations of motion when the energy is fixed: one based on Hamilton's principle and the other based on the principle of least action. (The appendix contains a short review of the two principles.) Hamilton's principle yields an action functional on phase space. It contains a Lagrange multiplier which restricts the system to the sub-bundle M *. Specifically, we have S(q, p; *)=
|
tb
[ p } q* &*(t)[H(q, p)&E]] dt
(4.11)
ta
On the other hand, the principle of least action leads to a configuration space action functional. The paths all have constant energy, and the integration boundary is variable: S (q; {)=
|
{(tb )
(L+E) dt
(4.12)
{(ta )
where L is the Lagrangian obtained from H by the (inverse Legendre) transformation FH: T *Q TQ. 8 These are the actions which must be exponentiated in the phase space and configuration space path integrals respectively. For point-to-point transition amplitudes, the initial wave function is taken to be a delta functional. The delta functional serves to ``pin down'' one of the end-points 8 The (inverse Legendre) transformation FH: T *Q TQ is the (diffeomorphic) fiber derivative of the Hamiltonian H.
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PATH INTEGRAL FOR DIRICHLET PROBLEM
of the paths. Recall that the CartierDeWitt-Morette scheme does not confine us to point-to-point transitions, but we choose them for convenience. So, the phase space path integral will be a sum over the appropriate phase space paths, and the configuration space path integral will include the appropriate configuration space paths. The integration domains are described in detail below. We must also remember to do a functional integral over * and {. They have to be varied to obtain the correct equations of motion; so they represent degrees of freedom which must be integrated over. However, since there is no kinetic term associated with these variables, they are not dynamical degrees of freedom. These considerations lead us to propose the following path integral representations for the fixed-energy point-to-point transition amplitude: G ps(q b , q a ; E)=
|
exp M_4
i
{ S(q, p; *)= $(q(t )&q ) D(q, p) D*
(4.13)
i S (q; {) $(q({(t b ))&q b ) Dq D{
(4.14)
b
b
and G cs(q b , q a ; E)=
|
exp E_T
{
=
The domain of integration for the phase space integral is the space of pointed paths given by M_4=[(q, p): [t a , t b ] T *Q]_[*: [t a , t b ] R] such that q(t a )=q a , p(t b )=p b and *(t a )=0. Likewise, the domain of integration for the configuration space functional integral is the space of pointed paths E_T=[(q b {): [t a , t b ] ?(M)]_[{: [t a , t b ] [{(t a ), {(t b )]] where ?: TQ Q is the tangent bundle projection map and M :=FH(M*). The configuration paths satisfy q({(t a ))=(q b {) a , and {(t a )=0. It is clear that these domains are not Banach spaces so they must be parametrized. Also, the integrators must be characterized by specifying appropriate quadratic forms. This being done, the integrals can be given a precise meaning by way of Eq. (2.15). For the phase space paths, the parametrization is identical to Eq. (4.5). The integrator associated with D(q, p) is DQM , WM(q, p) where i ?i Q M (z; *)=
|
tb ta
{
pq* &
* 2 i p dt= 2m 2
=
|
tb
(z q , z p )
ta
0
\ddt
&ddt &*m
zq
+\z + dt
(4.15)
p
The integrator associated with D* is DQ* , W* * and we take Q* =
1 (t b &t a )
|
tb
* 2 dt
(4.16)
ta
Using Eq. (2.15), it follows that Eq. (4.13), under the inverse map R &1: 4 T, is equivalent to Eq. (4.6).
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412
J. LACHAPELLE
For the configuration space integral, we must choose the parametrization carefully. Since the paths q b { are constant energy paths, ?(M) is not necessarily flat and the simple parametrization of Eq. (4.2) will not do. However, Q is flat by assumption so the thing to do is to parametrize P D qa Q according to
{
dq i({) dz :({) =$ i: d{ d{
(4.17)
q({(t a ))=q o
Note that this parametrization depends on the function {. The integrator associated with Dq is DQE , WE q where ?i i Q E (z; {(t b ))=
|
{(tb ) {(ta )
i m 2 |q* | (t, z) dt= 2
|
{(tb ) {(ta )
m 2 |z* | dt 2
(4.18)
and the integrator associated with D{ is DQ{ , W{ { with Q{ =
1 (t b &t a )
|
tb
{* 2 dt
(4.19)
ta
Clearly Eqs. (4.14) and (4.3) are equivalent because of Eq. (2.15). 4.3. Recovering Time-Dependent Amplitudes To check the validity of Eqs. (4.3) and (4.6), the Fourier transforms in energy must give the time-dependent point-to-point transition amplitude K(q b , t b ; q a , t a ). So, taking the Fourier transform in energy of the phase space integral G ps(q b , q a ; E), we have
|
i exp & E(t b &t a ) G ps(q b , q a ; E) dE &
= =
{
i exp & E(t b &t a ) &
|
|
{ i $ | * dt&(t &t ) exp _ & { | i exp { | ( p } q* &H) dt= $ D
M_4
=
=
|
M
=|
M_4
tb
b
DQM , WM (q, p) DQ* , W* * exp tb
a
ta
ta
{
i S(q, p; *) $ qb dE
=
=
( p } q* &*H) dt $ qb DQM , WM (q, p) DQ* , W* *
tb
ta
qb
QM , WM
(q, p)
=K(q b , t b ; q a , t a )
(4.20)
where we have introduced the notation $ qb :=$(q(t b )&q b ). In the third equality we used Eq. (2.14) with A*= ttba *(t) dt&(t b &t a ), |Det L * | =1 and * o =1.
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PATH INTEGRAL FOR DIRICHLET PROBLEM
Similarly, for the configuration space integral G cs(q b , q a ; E) we find i exp & E(t b &t a ) G cs(q b , q a ; E) dE &
|
{
= = =
| | |
=
E_T
i i exp & E(t b &t a )+ S (q; {) $ (q b {)b DQE , WE qDQ{ , W{ { dE &
|
{
=
$[({(t b )&{(t a ))&(t b &t a )] exp E_T
exp Q
{
i
|
tb ta
{
i
|
{(tb ) {(ta )
=
L dt $ (q b {)b DQE , WE qDQ{ , W{ {
=
L dt $ qb DQE , WE q
=K(q b , t b ; q a , t a )
(4.21)
In the third equality we used Eq. (2.13) and the fact that {=Id implies E Q where Q=[q: [t a , t b ] Q] such that q(t a )=q a . 4.4. Recovering Garrod's Expression It is reassuring to reproduce Garrod's representation of G ps(q b , q a ; E). We first make a time reparametrization t [ t$ such that dt$=*(t) dt. The phase space action functional becomes
|
qb
p dq&
qa
|
t$(tb )
(H&E) dt$
(4.22)
t$(ta )
A second time reparametrization t$ [ t" such that [t$(t b )&t$(t a )] t"+t$(t a )=t$ brings Eq. (4.22) into the form
|
qb
p dq&[t$(t b )&t$(t a )] qa
|
1
(H&E) dt"
(4.23)
0
Following Garrod, define H := 10 H dt" and write Eq. (4.23) as
|
qb
p dq&[t$(t b )&t$(t a )](H &E)
(4.24)
qa
The next step is to insert |Det L * | =1 into the phase space integral G ps(q b , q a ; E). (Recall that L * : 4 R by * [ ttba *(t) dt=t$(t b )&t$(t a ).) This gives G ps(q b , q a ; E)=
|
=
|
M_L*(4)
i
{ S(q, p; *)= $ D (q, p) |Det L | D i exp p dq&(L *)(H &E) = {|
exp
M_4
qb
QM , WM
*
Q* , W*
*
qb
*
qa
_$ qb DQM , WM (q, p) DQ* , W*(L * *)
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(4.25)
414
J. LACHAPELLE
But the integral over L *(4) is just an integral over R which can be evaluated immediately yielding Garrod's result G ps(q b , q a ; E)=
|
$(H &E) exp M
{
i
|
qb qa
=
p dq $ qb DQM , WM (q, p)
(4.26)
4.5. WKB Approximation As a final exercise, we will calculate the WKB approximation of G ps(q b , q a ; E). Recall the exact equation for G ps(q b , q a ; E): G ps(q b , q a ; E)=
|
exp Zab_4
{
i S(z; *) $ qb DQz , Wz(z q , z p ) DQ* , W* *
=
(4.27)
To obtain the semiclassical expansion, first introduce a one-parameter family of paths (z q , z p ; *): P_T T *Q_R such that (z q , z p ; *)(\, t) | \=0 =(z qcl (t), z pcl (t); * cl (t)) where \ # P=[0, 1]/R. That is, expand the path (z q , z p ; *) # Z ab_4 around a classical path: z q(\, t)=z qcl (t)+\!(t) z p(\, t)=z pcl (t)+\'(t)
(4.28)
*(\, t)=* cl (t)+\_(t) This amounts to a change of integration variables in the functional integral. Since this parametrization is an affine map on Z ab_4, it follows from Eq. (2.9) that the integrators satisfy DQz , Wz(z q , z p )=DQz , Wz(\!, \')=: D\2Qz , \&2Wz `
(4.29)
DQ* , W* *=DQ* , W*(\_) Now consider the function S b (z q , z p ; *): P R by \ [ S b (z p , z p ; *)(\). Expanding this function in a Taylor series about \=0 yields
\ n d nS b (z q , z p ; *) (0) d\ n n=1 n!
S b (z q , z p ; *)(\)=S b (z q , z p ; *)(0)+ :
(4.30)
The WKB approximation results from evaluating Eq. (4.30) for the family of paths given by Eq. (4.28), keeping only terms up to O(\ 2 ), and putting \=- 2?. By definition, a classical path satisfies d(S b (z qcl , z pcl ; * cl ))(0)d\=0 so the action functional appropriate for the WKB approximation is 2 \ 2 d S b (z qcl , z pcl ; * cl ) S b (z q , z p ; *)(\)=S b (z qcl , z pcl ; * cl )+ 2 d\ 2
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(4.31)
415
PATH INTEGRAL FOR DIRICHLET PROBLEM
But, S b (z qcl , z pcl ; * cl )=
|
tb ta
z pcl (z qcl , E) z* qcl dt=
|
qb
p cl (q cl , E) dq cl qa
=: W(q b , q a ; E)
(4.32)
which follows from Eq. (4.11). Furthermore, it can be shown ([2]) that 2 \ 2 d S b (z qcl , z pcl ; * cl ) =? 2 d\ 2
|
tb ta
{ ( J(z
qcl
, z pcl ; * cl ) `, `) &2
_(t) dH(z qcl , z pcl ) dt \ d\
=
(4.33) where
J=
\
2H z iq z qj
&$ ij
2H $ ij + i t z p z qj
2H + i t z q z pj
+
2H z ip z pj
(4.34)
The map J(z qcl (t), z pcl (t); * cl ): T (zqcl (t), zpcl (t))(M*) T *(zqcl (t), zpcl (t))(M*) is the phase space Jacobi operator. In general there is a boundary term associated with the first and second variations, but they will not contribute because !(t a )=!(t b )=0 in our case. Therefore, under the change of variable defined by Eq. (4.28), the WKB approximation of G ps(q b , q a ; E) becomes (q b , q a ; E)=exp G WKB ps
{
i W(q b , q a ; E)
=|
{
exp ?iQ z(`)& Zab_4
i
|
tp
\ 2_(t)
ta
_$(\!(t b )) D\2Qz , \&2Wz `DQ* , W*(\_)
dH cl dt d\
=
(4.35)
where we have defined a new quadratic form Q z(`) :=
|
tb
ta
( J(z qcl , z pcl ) `, `) dt
(4.36)
The _ integral can be done by applying the same technique used in the previous subsection with the result
|
{
exp & 4
i
|
tb ta
\ 2_
dH cl dt DQ* , W*(\_)= d\
=
|
R
i
dH cl
{ \u d\ = du dH $ \ d\ +
exp &
=\ &1 =\ &1
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cl
(4.37)
416
J. LACHAPELLE
where dH cl := d\
|
1
0
dH cl dt" d\
(4.38)
and t" has been defined in the previous subsection. The last equality follows because dHd\=dHd\ and Eq. (4.26) says H =E for paths in Z ab . Hence, dHd\=0 is automatically satisfied. It remains to evaluate
|
Zab
exp[?iQ z(`)] $(\!(t b )) D\2Qz , \&2Wz `
(4.39)
By considering the linear map L: Z ab Z ab such that Q z =Q z b L and using Eq. (2.9), it can be shown ([1]) that
|
Zab
exp[?iQ z(`)] $(\!(t b )) D\2Qz , \&2Wz ` =i &Ind(Qz Qz ) |Det Q z Q z | &12
|
Zab
exp[?iQ z(`)] $(\!(t b )) DQz , Wz `
(4.40)
where the index Ind(Q z Q z ) equals the number of negative eigenvalues of Q z with respect to Q z . Notice that the integral on the right-hand-side now contains a gaussian integrator. The next step is to write $(\!(t b ))=
|
Rn
exp[&2?i\!(t b ) } u] d nu
(4.41)
substitute into Eq. (4.40), interchange integrations (which is allowed), and use Eq. (2.5) to get
|
Zab
exp[?iQ z(`)] $(\!(t b )) D\2Qz , \&2Wz ` =i &Ind(Qz Qz ) |Det Q z Q z | &12
|
_
Rn
exp[&?iW(z$q +\u$ tb , z$p )] d nu
}
(4.42) (z$q , z$p )=0
where (z$q , z$p ) # Z$ab and Z$ab is the dual of Z ab . A straightforward but involved calculation ([2]) gives
|
Rn
exp[&?iW(z$q +\u$ tb , z$p )] d nu
}
=\ &n |det iG(t b , t b )| &12 (4.43) (z$q , z$p )=0
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417
PATH INTEGRAL FOR DIRICHLET PROBLEM
where G(t b , t b ) is the Green's function of the Jacobi operator for the boundary conditions q cl (t a )=q a and p cl (t b )=p b . Finally, it can be shown ([1]) that |det iG(t b , t b )| |Det Q z Q z | = |det D VM | &1
(4.44)
where D VM is the van VleckMorette matrix of the action functional associated with Q z , i.e., S b (z qcl , z pcl ; * cl )=W(q b , q a ; E). It follows ([8]) that
det D VM =det
\
2W q a q b 2W E q b
2W q a E 2W E E
+
(4.45)
Putting together all the pieces, we obtain the final expression (q b , q a ; E)=i &Ind(Qz Qz )(2?) &(n+1)2 |det D VM | 12 exp G WKB ps
i
{ W(q , q ; E)= b
a
(4.46)
APPENDIX A.1. Hamilton's Principle The variational principle of Hamilton is usually associated with an action functional obtained from a Lagrangian defined on the tangent bundle TQ of configuration space. However, it works just as well for a suitably defined action functional of phase space variables. The variational calculus then yields Hamilton's equations of motion. In particular, to obtain the classical equations of motion for a Hamiltonian system with fixed configuration end-points, we impose the variational principle $S(q, p)=$
|
tb
[ p } q* &H(q, p)] dt=0
(A.1)
ta
such that the variations of the q's vanish at the end-points. If we want to resrict the system to a submanifold of constant energy M*=H &1(E), we simply make the replacement H *(H&E) where * is a Lagrange multiplier to obtain $S(q, p; *)=$
|
tb
[ p } q* &*(t)[H(q, p)&E]] dt=0
(A.2)
ta
Variation of this action functional yields Hamilton's equations (which will have a * dependence) with the additional constraint H=E. Thus the variation gives
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418
J. LACHAPELLE
Hamilton's equations for phase space paths with constant energy. Note that there is no harm in letting * be a function of t in Eq. (A.2). A.2. The Least Action Principle If one solves for the constraint in Eq. (A.2), the integrand becomes - 2m(E&V(q)) q* 2 (for the special case of H(q, p)=p 22m+V(q), and the action functional becomes a configuration space action functional. However, solving the constraint does not commute with the variational principle, and so it does not follow that variation of the resulting integral will yield the equations of motion of the restricted system. In other words, the equations of motion obtained by varying constrained paths are not the same as the equations of motion obtained by varying unconstrained paths supplemented with a constraint equation. In fact, in order to regain the correct equations of motion, it is necessary to vary the parametrization of the paths (see, e.g., Ref. [5]). The result is the least action principle which can be stated as $ E S (q; {)=$ E
|
{(tb )
- 2m(E&V(q(t))) q* (t) 2 dt=$ E
{(ta )
|
{(tb )
(L+E) dt=0
(A.3)
{(ta )
where $ E denotes a variation of paths with fixed end-points and fixed energy. The resulting variation yields Lagrange's equations of motion for configuration space paths with fixed energy.
ACKNOWLEDGMENT I thank C. DeWitt-Morette for valuable suggestions.
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