ANNALS
OF PHYSICS
199, 155-186 (1990)
Phase Space Path Integration of Integrable Quantum Systems ARLEN ANDERSON Department Salt
of Physics, Luke City,
University of Utah, Utah, 84113
AND SCOTT B. ANDERSON Department
of Physics, Montana Stare Bozemun, Montana 59717* Received
August
University,
9, 1989
A new method for exact evaluation of phase space path integrals for integrable quantum systems is presented. By making use of point canonical and other transformations to bring the Hamiltonian to a form linear in the coordinates, the path integral is changed so that the functional 4 integration may be done. This produces a momentum delta functional which can be evaluated to give an ordinary integral. This procedure is applied to find an expression for the exact propagator for a particle in the harmonic oscillator potential, the inverse quadratic potential, the Coulomb potential, the Morse potential, the l/cash’(x) potential, and for a free particle propagating on the 2-sphere. The latter two problems are solved for the first time wholly within the path integral formalism. 11“ 1990 Academic Press, Inc.
There are a number of quantum mechanical problems for which an exact expression for the time-dependent Schriidinger propagator can be given without the use of eigenfunctions [l-S]. Because these propagators have the form of a “sum indexed by classical paths,” they are assumed to have a close connection to the path integral, but the path integral has never been explicitly evaluated in any of these examples. In the case of a free particle propagating on a Lie group [ 1,2], it is known that the Gaussian approximation of the path integral gives the exact result, but it is a mystery why the contributions of the terms neglected in the approximation cancel. It would be very desirable to have a method for evaluating the path integral which could solve these problems. One common feature of these problems is that they are all integrable in terms of eigenfunctions, which are classical special functions. This is not surprising and * Present
address:
Logicon,
Long
Beach,
CA 90733-0471. 155 0003-4916/90
$7.50
CopyrIght c: 1990 by Academic Press, Inc. All rights oi reproductmn in any iorm reserved
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Ref. [S] suggests the source of the close connection between the eigenfunction and sum-over-paths representations of the propagator. The importance of this feature is that there are other potentials whose propagators are known in terms of classical special functions, e.g., gx -’ + w2x2/2, the Coulomb potential, and the Morse potential. The path integral for these problems has been evaluated directly [9-131. The methods used require direct treatment of the path integral in its time-sliced representation. Some rely on special concatenation identities which allow the shorttime propagator to be built up into the finite-time propagator. It would be useful to have a simpler method of evaluation which is more general in its application. In this paper, the “method of delta functionals” is presented as a means of evaluating phase space path integrals for the integrable quantum systems described above. This method was inspired by the discussion of path integration by Katz [ 141 and was developed by one of the authors [ 151. An important motivation for the development of the method of delta functionals was a desire to avoid explicit mention of time-slicing and the proliferation of unnecessary detail. The central observation of the method of delta functionals is that phase space path integrals involving Hamiltonians linear in the coordinate can be immediately evaluated. While a rigorous justification of this requires time-slicing, a heuristic explanation can be given as follows. Suppose one has the phase space path integral
After integrating by parts in the action to isolate q, one can do the q functional integral to give a delta functional,
s
!$fNb+F(~)l
exp(ip(T)q(T)-ip(O)q(O)).
(2)
The delta functional has the effect of selecting out the momentum space paths which satisfy the classical equations of motion. If the classical equations of motion can be solved for arbitrary initial conditions, one can do the functional p integral and it reduces to an ordinary integral over the initial conditions. The solution for p in terms of t and the initial conditions is substituted in the rest of the integrand and the functional integral has been evaluated. The method is made powerful because there exist point canonical (coordinate) and other transformations to convert a Hamiltonian to linear coordinate form. Unfortunately, these transformations are not general canonical transformations; in particular, the Hamiltonian remains quadratic in the momenta. As a result, only a sub-class of integrable systems can be solved by the method at this time. In particular, the equation of motion for p which appears in the argument of the delta functional is at most quadratic in p and this corresponds to the classical equation of motion for a harmonic oscillator. Nevertheless, the method of delta functionals is not trivial since the configuration space path integral that is formed
PHASESPACEPATHINTEGRATION
157
by integrating out the momenta is not quadratic in the coordinates and velocities. It is essential in the phase space path integral to integrate first the coordinates and then the momenta. The class of integrable systems which can be solved by the method of delta functionals appears to coincide with the list of potential problems exactly soluble in terms of special functions. In this paper, the problems of a particle in the Coulomb potential, gx-’ + w2x2/2, the Morse potential, g cash p2(x) and the free particle on the 2-sphere are all solved. These potentials all have forms listed by Olshanetsky and Perelomov [ 161 as characteristic of integrable systems. It is conjectured that the methods here will generalize to apply to the full class of quantum integrable systems based on Lie algebras described by Olshanetsky and Perelomov. It should be emphasized that while time-slicing is involved in deriving the method of delta functionals, it is not needed to apply the method in practice. This leads to a considerable simplification in the amount of technical detail involved in computations. 1. SYMMETRIC HAMILTONIAN
The path integral in phase space is defined from the Schrodinger equation with a particular operator ordering. The relation between the Schrodinger equation and the phase space path integral for different orderings has been discussed in detail by Mayes and Dowker [17]. We choose to work with a symmetric ordering of the Hamiltonian operator in which each term is written as an average of the term ordered with all of the d’s on the left and the term ordered with all of the a’s on the right
WA4)= 4c F,(d)G,(G) +G,(@)F,@).
(3)
It is convenient to define an operator 9’ as the operation of putting a function of J? and 4 into symmetrically ordered form. It is also convenient to define the operator “symm” to represent terms which have already been reduced to symmetric order. For example,
while
If H,.(p, q) is the symmetrically ordered operator Hamiltonian Y(H(b, 4)) with b and 4 treated as c-numbers, the phase space path integral takes the form
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where N is a possible normalization constant (including, for example, endpoint measure factors in curved space applications). Generally, effective potential terms will arise when an operator Hamiltonian is put into symmetrically ordered form and these must be included in the action of the phase space path integral [ 171. The philosophy followed in this paper will be to take the Hamiltonian operator in the Schrodinger equation as fundamental. Transformations will first be made on this operator and the correct phase space path integral will be reconstructed through its correspondence with the symmetric ordering of the Hamiltonian operator. In this way, the quantization of the original problem will be preserved. It is important to emphasize that a path integral is not well defined without specifying its time-slicing approximation [17]. While our intent is to avoid explicit mention of time-slicing when possible, this information is contained in the shorttime propagator K(b,
t+
At;
c, t) = (271)-lg(b)-1/4g(C)-1/4
[ dp
x exp(ip(h - c) - i dt(H(p,
b) + H(p, c))/2).
(5)
This has the limit 6(6 -c)/g”*(b) as At -0, where g”*(b)= (det g,(b))‘/’ is the measure density at the endpoint b in a curved space with metric g,. To be reassured of the correspondence between the functional integral and the Schrodinger equation, it is useful to derive the Schriidinger equation that K(b, T; a, 0) satisfies from the composition law for propagators K(b, T; a, 0) = j. K(b, r; c, t)K(c,
t; a, 0) gl’*(c)
dc
(6)
and the infinitesimal propagator with symmetric ordering (5). Since the measure g ‘I2 in which our operators are self-adjoint will not always be trivial, we include it for completeness. For analytic F(p), the following identity holds g(b)-1’4g(c)P1’4
s F(p)&-(‘)dp
=F -ig(b)-‘/‘$7(b)‘.” g(b)~1:4g(c)~‘i4ieiP(b~~)dp. > Taking T= t + At, this will be used to integrate the right hand side of (6) for small At. It is convenient to define
Expand H(p, b) as a power series in p, H(P,
b)=~fXW’. m
159
PHASESPACE PATH INTEGRATION
Note that here p and b are c-numbers and commute. By writing H(p, 6) with all of the h’s to the left and expanding the infinitesimal propagator to first order in At, one can use the identity (7) to write K(b, t + At; a, 0)
=
1-qffn(p,
b))(g(b)g(c))~“4~g1’2(c)dc~~e”~‘*~C1K(c,
- (g(b)g(c))~1’4~Sg”‘(C)
dc~$ei~C”~C)H(p,
t;a,O)
c)K(c, t; a, 0),
(8)
where HR(f, b) is the Hamiltonian with all of the P on the right. The integral over p in the first term is a representation of the delta function. The second term can be evaluated by expanding H(p, c) in a power series in p and applying the identity (7) to obtain 1 Pm(g(b)g(c))FL’4TS m
Again the p integral integrals gives
g”‘(c) de ~~e’P’t-“f,(c)K(c,
t; a, 0).
can be done to give a delta function. Evaluating
the final
l-fH,(p,b)At--iH,(P,b)At .
(9)
where HL(p, b) is the Hamiltonian with all of the P on the left. Rearranging taking the limit as At--f 0, one obtains the Schrodinger equation
and
K(b,t+At;a,O)=
(
i
8K(b, t; a, 0) = H,(p, b)K(b, t; a, 0), at
where H,(p, 6) = +(H,,(p, 6) + HR(p, b)) is the desired symmetrically Hamiltonian operator.
2. DELTA
(10) ordered
FUNCTIONAL
There are two key steps in the method presented here. First there is the functional q integration of a path integral with a Hamiltonian linear in q to obtain a delta functional of the momentum equation of motion. Then there is the functional p integration of this delta functional to leave an ordinary integral over the initial
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condition for the solution of the equation of motion. This procedure must be justified in the time-slicing approximation because this is where the symmetric ordering of the Hamiltonian operator manifests itself in the path integral. Once the time-slicing approximation of the path integral has been written down, the coordinate integration to give a delta functional is straightforward. It is in the integration of the delta functional that one finds an extra factor arising. In the final analysis this is a “composition” factor, depending on the remaining integration variable and the endpoint times, which is necessary for the propagator in its final form to satisfy a composition law. The detailed form of this composition factor depends on the choice of lattice approximation and hence of the operator ordering in the Hamiltonian. Consider a phase space path integral involving a Hamiltonian linear in the coordinate
The time-sliced the Hamiltonian
version of this path integral with symmetric is
K(b, T; a, 0) = b:m
j “ri’ fl=O
operator ordering of
dp,, Nfi’ dq, (27~~~ II=1
N-l xexp
i
1 n=O
(12)
Pn(qn+l-qn)-~f(Pn)(qn+l+qn)&-h(Pn)&
where E= T/N, go = a, qN = b, and the momentum between q,, and q,, + I. Doing the dq,, integrations delta functions, “N-1
Wb, T;a,O)=?i_muj
pn is understood
to lie in time gives a product of momentum
N-l
n
dp,(2n)-’
PI=0
n
6(~,-,-P,-~~(P,~,)&-~~(P,)&)
n=1 N-l
i(pN-lb-poa)-i
c
h(p,)E
(13)
There is one more dp, integration than delta function. This is why the functional will reduce to an ordinary integration. The momentum at any point along the path can be chosen as the integration variable which remains. This momentum will serve as the “initial” condition for solving the first order equation of motion. For definiteness, choose to integrate all but the pk momentum. Consider first the dp, + I integration. Here we are to integrate the delta function
p integration
PHASE
The argument identity
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INTEGRATION
of this delta function is a function of pk+, and one must use the
(14)
where pr are the roots of g(p) = 0 (and prime indicates differentiation), to reduce the delta function to one linear in pk+ , . We assume that there is a single root to the equation in the argument of the delta function corresponding to the solution of the momentum equation of motion, so the delta function reduces to 6 (Pk+ I -F(P,, 1 +f’(Pk+
Integrating
up one momentum
@+l)E)) I)&/2
.
at a time, the pn delta function is &Prl-
F(P/c, n&l)
1 +f’(PJ@
(15)
’
where F(p,, ns) is the solution of the momentum equation of motion at time nc given initial momentum pk at time kc A similar procedure can be used to integrate down from pk- I. Here a sign changes in the argument of the delta function changing the derivative of the argument at the root, and one finds a(~, - F(p,, 1 -f’(P,W
The result of integrating
out the momentum
n&))
(16)
.
delta functions is
K(b, 7,0,0)=!--~exp(i(p,~,bp,a)-i~~~h(p.)i
where pn = F(p,, ns) is the solution of the momentum equation of motion n& with initial condition pk at time ks. Taking the limit N+ co, the sums over f’(p,) become integrals
Sincep = -f(p)
is the momentum iln
at time
equation of motion, these can be integrated to give (
f(P(9)f(P(O)) f(Pk12
>
(18)
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ANDERSON AND ANDERSON
Using the chain rule,
dp(t) dp(t) -- 4, -=dpk dt
dt’
one has
fMT))= Exponentiating
yf
k
(18) gives the “composition
(Pk).
factor” as
(~~)“*.
(19)
The final result for the linear coordinate
path integral is then
K(b, T;a,0)=~~(~~)1’2exp(io(T)hp(0)n)ijg).
(20)
k
This is a central result in the method of delta functionals. One can verify it by direct substitution in the Schriidinger equation. We emphasize that the details of the composition factor (19) depend on the chosen symmetric operator ordering of the Hamiltonian.
3. SIMPLE HARMONIC
OSCILLATOR
To illustrate the use of the delta functional, the simple harmonic oscillator may be solved. The harmonic oscillator is of course well understood and easy to solve in many different ways. For this reason, it is valuable for the sake of comparison and for completeness to solve it again with this new method. The phase space path integral for the simple harmonic oscillator is K&b,
T; a, 0) = J”[ 91
exp (i J” pcj - i p2 -i
co*q*dt).
(21)
Shift the momentum by p =p - ioq to cancel the quadratic term in q. A total derivative term -iwqQ arises and can be integrated. Dropping the bar over the new momentum, the path integral takes the linear coordinate form K&b,
T; a, 0) = exp (~co2(b2-a2))~[~]exp(i~pQ-~p2+icoqpdt).
(22)
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PHASESPACE PATHINTEGRATION
Integrate the first term in the action by parts to isolate q and then do the functional integral over q to obtain the delta functional h[L’-iwp].
(23)
The path integral now takes the form K&b,
T;a,O)=exp
~~‘ib’.‘))j~a[ii-j~~]
x exp i(p(T)b-p(0)a)--il~~2dr
>
.
(241
The functional p integration is done using the results of Section 2. The equation of motion in the argument of the delta functional can be integrated, with the initial condition p0 at time t = T/2 chosen for convenience, p(po, t) =poeiw+
T’2).
(25)
Using (19), the composition factor is found to be unity. The integral of p* in the action is evaluated to give pi sin(oT)/w. The propagator after the functional p integration is given by the ordinary integral K&b,
T; a, 0) = exp i 02(b2 - a2)) i_“, 2 x exp ipo(&‘wTP - ae ~ iwm) - & pi sm(oT) ’
This is integrated to obtain the familiar
>
.
(26)
result
K,,(h,T;~,0)=(2nis~(wT))“2exp(2Si~~T)((b2+a2)co (27)
4. POINT CANONCAL
TRANSFORMATION
Few Hamiltonians are linear in the coordinate to begin with. To apply the method of delta functionals, one must be able to transform a more general Hamiltonian to linear coordinate form. Classically, one uses general canonical transformations to transform an integrable Hamiltonian to action-angle variables. That this can be done is a statement of classical integrability. It is not difIicult to prove that any system in linear coordinate form can be reduced to action-angle variables, provided the momentum equations of motion are integrable. There .must then exist 595/199/l-12
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general canonical transformations to transform an integrable Hamiltonian directly to linear coordinate form, Quantum mechanically, however, the use of canonical transformations which mix the coordinates and the momenta are not understood because it is not clear how to preserve the quantization structure reflected in the canonical commutation relations. As a result, they cannot be used. Point canonical transformations in which the coordinate is changed q =f(Q) are understood. This limits the class of transformations available to reduce an integrable Hamiltonian to linear coordinate form and presumably limits the class of systems soluble by the method of delta functionals at this time. One finds that in preserving the quantization of the original system after a point canonical transformation, effective potential terms must be added to the transformed system. Several authors have considered the problem of point canonical transformations in the path integral from as many perspectives, primarily relying on the time-sliced representation of the path integral. The approach taken here will be another still. The guiding philosophy is that the Schrijdinger equation is fundamental. The path integral is defined from a particular operator ordering of the Schrodinger equation as described in Section 1. After making a point canonical transformation of the Schrodinger operator, we will find that the transformed Schrodinger operator is not symmetrically ordered in the new variables and must be reordered. The reordering produces effective potential terms which appear in the action of the path integral in the new variables. Consider making a point canonical transformation, q =f(Q). To be canonical, one must have t-28)
Then p = P/‘(Q) + h(Q) is the most general transformation allowed for p. Note that shifting the momentum by a function of q, while the coordinate is not changed, is a canonical transformation. It will be convenient to take h(Q) = 0 when a point canonical transformation q =f( Q) is made and to shift the momentum separately. The shifting procedure is straightforward, without operator ordering complications (for Hamiltonians at most quadratic in the momenta). Under the change q =f(Q), the operator $ = -id/dq transforms to ~3= -if’(Q)-Id/de. We need to express this in terms of the new operators p and Q. Before the transformation, the inner product was
4. sll/*(q)x(q) After the transformation, the measure changes from dq to f’(Q) dQ. The momentum operator must be self-adjoint in this measure and it must satisfy the canonical
PHASE SPACE PATH INTEGRATION
commutation relation, [p, &] = -i. Let Q = Q, that is, multiplication Defining g”‘(Q) =f’( Q), this implies that j= is the self-adjoint
momentum
165 by Q.
-ig(Q)-“4(d/dQ)g(Q)“4 operator. As a result, 4 zg(Q)-‘~“~g(Q)--
l/4.
(29)
When the new variables are substituted into the Hamiltonian, the operator ordering of the Hamiltonian is no longer in symmetric form (3). Reordering the Hamiltonian produces an effective potential term. If H(& q) = $$’ + V(4), then the reordered Hamiltonian is
==(1/4)(g-‘P2+P2g-‘)+ The effective potential
V(f(i),,-$+$.
(30)
is identified as v,=
-xl+8g2
5. INVERSE QUADRATIC
9g’2 32g3’
(31)
OSCILLATOR
As an example of the use of point canonical transformations in the method of delta functionals, consider the particle in an inverse quadratic potential with Hamiltonian H(p,q)=~p’+$u2q2+gq-‘.
The path integral for the propagator
takes the form
K,p(b, t; a, 0) = j [ $$I
exp (i JOT~4 - HP, 4) dt).
(The coordinate space version of this path integral has been evaluated before [9, lo].) Make the point canonical transformation q = 2Q112,p = Q”‘P. The new measure is g’12(q) = Q - ‘j2. From (31), this gives an effective potential
The metric prefactors in the transformed
path integral are given by
(g(B)g(A))-“4=
(B.4)““,
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where B = b2/4 and A = a214 The path integral in the new variables is &(B,
T; A, 0) = (BA)“4
j [ 71 P&;P2Q-2w2Q-g$dr).
The l/Q term in the action can be eliminated becomes
(32)
by a shift of the momentum,
P = P - h/Q. The path integral
-iF2Q+iaP-2w2Q+
(2a2-g-lWdt 4Q
.
(33)
>
Choosing 2cr2= g + $ cancels the l/Q term. The term Q/Q is a total time derivative and can be integrated to give (B/A)“. Dropping the bar over the P, the path integral is then &(B,
T; A, 0) = (BA)1’4
(y,[%q Pa-iP2Q+izP-2w2Qdt
Integrating
over the coordinate
(34)
Q produces the delta functional 6[P+$P2+202].
(35)
The P functional integral will reduce to an ordinary integral over the initial condition PO = P(0) for solutions to the momentum equation of motion in the argument of the delta functional. It is convenient to change variables, P(t) = 2j‘(t)/‘(t), before solving the equation for P(t). The new equation is that of a harmonic oscillator, p+ coy-= 0, which has the general solution
f(t) =fo cos(wt)+jb +g%
(36)
One finds that P(t) equals P(t) =
PO cos(wt) - 20 sin(wt) Po(2w)-’ sin(ot) + cos(ot)’
(37)
PHASE
SPACE
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167
INTEGRATION
In converting the argument of the delta functional from the differential equation to its solution P(t), one acquires the composition factor
dP(T)dP(0)“2 f. dp,dp, > =-f(T)' As well, in the action, there remains an integral of P over T. This can be evaluated to be
From (36),
f(Tlmp .fi
sin(wT) + cos(oT).
,, 2.
(38)
Combining these with the above, the path integral after integrating tum delta functional is &(B,
(~)‘i~(~)‘+‘“,,Priin-,a,.
T; A, 0) = (B/l)“”
Changing variables from P, to u = P, + 2w cot(oT), &(B,
out the momen-
(39) the propagator
becomes
T; 4 0) = (BA)“”
-$exp(i2w
cot(oT)(B+A))
~(JJ”(&)‘+2^,lrduu~“+2”exp(-i(Au+Us~n~~T))). The following integral representation to evaluate this integral, - 27liI”( - i2az) = z”
(40)
of the modified Bessel function I, can be used
5 J‘-cc drt”+‘)exp(
-ia(*+c)).
This can be verified to hold for arg(a) = 0 and k(v) > - 1 by applying the modified Bessel equation under the integral and integrating the resulting total derivative. The propagator we are calculating must be valid for all T. A concern arises when one observes that the argument of the exponent in the integrand of (40) does not change sign when T -+ T+ 7c/o but that the argument of the prefactor (2o/sin(oT))’ + 2a does. Naive use of the Bessel function integral representation
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appears not to preserve the phase coming from this term. The Bessel integral may be evaluated after explicitly extracting the sign from w/sin(oT) and keeping the (real) absolute value in the integral. The result agrees with the naive evaluation of the integral because of the identity e i”mnz”(z) = I,( eiy) and there is no longer cause for concern. Changing variables back from A, B, to a, b, the propagator quadratic potential is K&b,
T; a, 0) = (ba)“’
o i sin(oT)
e(iw/2)cot(wTl(62+02~z
for the inverse
(41)
where 2~ = (2g+ i)1’2 (and Re(2cr) > - 1). This agrees with the result of Peak and Inomata [9]. Note that as g + 0, this becomes the propagator for a harmonic oscillator with a hard reflecting wall at the origin. 6. TIME RESCALINC
Just as it is useful to make a canonical transformation defining the coordinate q in terms of a new coordinate Q by q =f(Q), it is also sometimes useful to redefine time r in terms of a new time s. This transformation was first introduced by Duru and Kleinert [ 111 and later developed by Pak and Sokmen [13]. These authors used detailed time-slicing arguments. The argument here will be somewhat different, relying on the transformation of the underlying Schriidinger equation. It is most helpful to make the time-resealing transformation in the form f( q( t)) dt = ds.
(42)
Since the time redefinition depends implicitly on the path q(t), there is not a direct relation between s and t. To motivate the form of the transformed propagator, one begins with the energy Green’s function which is independent of time K,(b, a) = i sy; dT eiETK(b, r a, 0).
(43)
0
Heuristically, the intent is to interchange the time integration with the [dq] functional integration so that a path dependent time transformation can be made and then to interchange back. Since T is the time-endpoint and q(T) = 6, one has
PHASE
The exp(iET)
SPACE
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169
INTEGRATION
becomes path dependent after transformation
and so may not be interchanged past the [dq] integration. transformed propagator &b, S; a, 0) by K,(b, a) = i jox $$
since
One is led to define the
&b, S; a, 0).
Inverting the relation between the energy Green’s function and the original dependent propagator, one has
time-
= dE -iET ‘~2 dS w K(b, T; a, 0) = j_- ic 27~e j. f~ K(b, S; a, 0).
(45)
To construct the path integral representation of l?(b, S; a, 0), it is necessary to return to the original Schrijdinger equation and to find the transformed Schrodinger equation. This may then be ordered as described in Section 1 to give the transformed path integral. It is improper to try to change the time directly in the path integral for K(b, T; a, 0) in the heuristic argument above. To do so would leave out important effective potential terms produced by ordering the transformed Schrodinger operator. Applying H- E to the energy Green’s function in (44) gives 6(b - a) = i j’= dS(H-
E)f-‘(b)@b,
S; a, 0),
0
where the operators in H act on b. But, assuming lim,,, one has 6(b-
a) = i j-
dS(i&)lt(b,
@b, S; a, 0) = 6(b-
a),
S; a, 0).
0
Combining
these, one concludes that (H-
E)f-‘g(b,
S; a, 0)= it3,R(b, S; a, 0).
If the ordered H is given by 9’(H(p,
G))=isymm
(46)
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then the ordered transformed
AND
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Hamiltonian
is
‘*((H,,~1)=~symm(~~2)+fsymm(~~) -+a-’
+ 4g2f 2
vf-‘-Ef-‘.
The path integral for g is then
(1 S
xexp
i o P9-2gf .
1
df’
if’
P2-2gf2P-22
%f
-!+E& f f
>
(47)
To eliminate the term linear in p, a shift in the momentum p = p - if’/2f can be made. Integrating the total derivative which arises and dropping the bar over the p, the transformed propagator becomes &b, S; a, 0) = (g(b) g(u)) - 1’4 s
xexp Inserting
i (s
0
’ f” ’ P4-&f-s-4g2f-2
g’f’ ---
V
E (48)
this in (45), one has
xl[~]exp(i~osp4-Ipif-~-~-)I+Eds).
(49) 4gf
f
f
Note that this is now manifestly symmetric in the endpoints a and b. If f is chosen to cancel the g in the p2 term, the effective potential contribution reduces to V, = -g12/8g2
and the prefactor becomes (g(b) g(u))““.
(50)
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PHASESPACE PATH INTEGRATION
7. COULOMB POTENTIAL
As an application of the time resealing transformation, consider an S-wave (I = 0) particle in a Coulomb potential. This calculation has been done before by Duru and Kleinert in a different way [ 111. The propagator has the form
(51)
K,(b, ..,0)=,[y]exp(ijoTp4--$+$dt).
Make
the time resealing transformation
q ds = dt. In the notation
of Section 6,
f= 4-l and g= 1. The transformed path integral from (47) is &(b,S;a,0)=~[~]exp(i~osp~-p~+~+e2+Eqds).
(52)
The action is linear in the coordinate and the functional q integration giving a delta functional
This is the same as that in the inverse quadratic potential be evaluated similarly. The result is
K,(b, ~a,O)=j~,~j~
may be done
with -E = 2w2 and may
dSbe-iET+ie2.‘jm ~(&-)2.gi(p(s’bppw’, ~Km
where fo/f (S) is given by (38). Changing variables to u =f&“(S) integral, the final result is
(53)
and doing the p.
K&b, T; a, 0) = (ba)“’ joz dS jX dE ePiET+ir2Si71si$r*lS) -71 x ei2rucot(wS)(b
+ Y)Z,
- i4w(ba)‘/2
sin(oS) where 01=(-E/2) ‘I2 . One can evaluate the integrals function expansion.
>’
to get the familiar
8. MORSE POTENTIAL
The problem of a particle propagating
in the Morse potential
V(q) = Vo(e-284 - 21emP4)
(54)
eigen-
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can be solved by transforming to the problem of a particle in the inverse quadratic potential. This transformation is made with a point canonical transformation and a time resealing. The path integral for the propagator in the Morse potential is K,(b,
T; a, 0) = J [q]
exp (i JO*& -$
(55)
- VO(e-2PY - 21eCPq) &).
The point canonical transformation 2
--ln P
q=f(Q)=
Q
leads to the path integral
V,(Q”-2lQ’)-
V,(Q)dt
>
,
(56)
where g(Q) =f’(Q) = 4fl -*Q -* and V, is the effective potential (31). The new endpoints are related to the original ones by B = exp( - @/2), A = exp( -pa/a). A time resealing g(Q) ds = dt will cancel the l/g in the kinetic term. From Section 6, a prefactor of (g(B) g(A))“’ and a contribution of -g’*/8g* to the effective potential will arise. The net effective potential is
The time resealed path integral is K’,(B,
~~,0)=~(B~)-1/2$dSI_~e-iL’i[~] PC+-
P*
4Vo(Q2 -21) P’
4E 1 +p2Q2+8e2ds
>
. (57)
The functional integral in this expression is recognized as the propagator for the inverse quadratic potential, U(Q) = 02Q2/2 + gQ -*, where CO*= 8 VoflP2, g= -4Ep-* - i, and 21x= (-8Efi-*)“*. Using (41), one has R,(B,
r;a,O)=$(B~)~‘/*JbYdSJ-_~e
prET+i81”ofl-2SK,Q(B,
S; A, 0). (58)
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PHASESPACEPATHINTEGRATION
The inverse quadratic propagator carries the restriction Re(2cl) > - 1. This is satisfied for --n < arg(E) d 0 if 0 6 arg(/l) d 42 or rc6 arg(/?) 6 3rc/2 (mod 27~). This gives the result &,,(B, T; A, O)=;
1‘01dS j”,,
$+ET+nY’voP-2S
x exp y cot(wS)(B’ (’
isin;ms)
+ A”) ) f,, ($g)
(59)
which agrees with Duru [12]. Resealing S to S/w and changing back to the original variables b, a gives the final result for the propagator in the Morse potential K,(b,
T; a, 0) = f jr &
!“y
x
~~~~~~~~~~~~ -iwe-lb+u)/2
x exp where ~‘=81/,j-~
sin(S)
and 2~~=(-8EB-‘)“~(Re(2cl)>
9. UNCOMPLETING
3
(60)
-1).
THE SQUARE
Occasionally in the transformation of a Hamiltonian to linear coordinate form, one will reach a Hamiltonian which is linear except for a term p2q2. A trick called “uncompleting the square” can be applied in which an auxiliary functional integral is introduced. The basic idea follows from the identity exp( -ij~T~2,~dr)=j~,*drl
[dx];exp(ijoT&ctfdt)
(61)
The notation [dx],” indicates a functional integration over x with endpoints at x(0) = 0 and x(T) = x. This identity is easily proven by shifting i to complete the square, evaluating the resulting free particle propagator in x and integrating over the endpoint. Whenfis composed of factors which do not commute as operators, as in the case of interest wheref2 =p2q2, there is the possibility of effective potential contributions arising from ordering the Schrodinger operator from which the path integral is derived. To handle these properly, it is most direct to study the above identity in the Hamiltonian form. At the end, the auxiliary momenta will be integrated out to give the configuration space version.
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Suppose that one wants to uncomplete K(b, T; a, 0) = J [ y]
ANDERSON
the square g-‘(q)p*
in the path integral
exp (i IO’ +‘(qb2-h(dp-
WW).
(62)
First observe the fact that the integral over all positions of the final endpoint of the free particle propagator is unity
(63)
Unity in this form is inserted into the p, q path integral Shifting u in the functional integration so that
(62) to be transformed.
makes the new endpoint x path dependent, but since the endpoint is integrated over all values, this dependence can be removed by redefining it as well. The resulting x and [dx] integrations can be interchanged with the [dp dq/2n] integrations. The full Hamiltonian in the path integral becomes H=~~2+g-1’2(q)pji+~g~1(q)p2+h(q)p+
V(q).
(64)
The object is to complete the square of the 7i: term to absorb the g-‘p’ term. From Section 1, we know that the Hamiltonian in the Schrbdinger equation is the symmetrically ordered operator version of the Hamiltonian in the action in the path integral. Because of this non-trivial operator ordering in the Schrodinger equation, we must be careful with manipulations in the action of the path integral. To be safe, one can convert to the Schrodinger Hamiltonian and do the manipulations there. Denote by Symm the operation of taking a function of p, q to its symmetric ordering in terms of operators p, 4. Since a and @ commute with $ and i as operators, the Schrodinger Hamiltonian operator corresponding to the path integral with the Hamiltonian (64) is Symm(H)
= $Z’+ Symm(g-“*(q)p)i + $WMg-‘(q)p*)
Jn completing
+ Symm(h(q)p)
+ V4).
(65)
the square of 71, the key step is to recognize that
~Pymm(g-1’2(q)p))2= 4Symm(g-‘p2) + Y,(q),
(66)
PHASE
SPACE
where V,(q) is an effective potential
PATH
175
INTEGRATION
arising from the operator ordering (67 1
Using this, the full operator Hamiltonian Symm(H)
can be rewritten
= +(i + Symm(g-1’2(q)p))2
The path integral corresponding s=jT
+ Symm(h(q)p)
to this Hamiltonian
+ V(q) - V,(q).
has the action
-* nx+pg-~(it+g-“2(q)P)2-h(q)p-
V(q)+
V,(q)dt.
0
Shifting it in the path integral and integrating K(b, T;a,O)=j‘=
-_dxj
(68)
(69)
it out gives the final result
[dr];j[$$]exp(iJoTpQ+fi’
-g p”2pi-h(q)p-
V(q)+
?‘,(q)dt
>
.
(70)
Note that completing the ii square in the action in the path integral requires the subtraction of an effective potential term. Effectively this is because the square of the symmetric ordering of a term is not equal to the symmetric ordering of the square of the term. 10. V(q) = -/I* coshP2(q)/2 The propagator for a particle in the potential V(q) = -A2 cash -*(q)/2 can be found by reducing it to the problem of a particle in the Morse potential. This is done by making a point canonical transformation and uncompleting the square of the resulting P2Q2 term. After the Q functional integration is done and the delta functional evaluated, one finds that the auxiliary functional integration introduced in uncompleting the square is that of a particle in a Morse potential. Evaluating this gives the result. The propagator in the cash-*(q) potential is given by Kdb,
T;a,oi=~[~]exp(i~o*~~-~+2co~~2(q)dr).
The point canonical transformation tive potential
Q = sinh(q) with g(Q) = ( 1 + Q2) ~ 1 and effec3
‘k&Q)=,-,,,
(71)
1 +Q2)
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leads to the transformed L(&
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path integral
7;A,O)=(R(B)g(A))-‘ld~[~] xexp
(I
i
r 0
PO-
Let 2: = A2 + a. Shifting the momentum
P’( 1 + Q2)
2
A2+ a -2dr. +2(1+Q2) 8 >
(72)
by
- A P=P-l results in the cancellation of the (1 + Q’))’ potential term. A total derivative involving Q can also be integrated out. Dropping the bar over the P, the resulting propagator is &JB,
T; A, 0) = (g(B) g(A))-‘14 exp( -i&Jarctan(B)
- arctan(d
x[[~]exp(i{o’PQ-P2(1~Q2)+&P-~dr).
(73)
Using the results of Section 9, uncomplete the square of the P2Q2 term. An effective potential contribution (67) of -z is added to the potential cancelling the i already present. Denoting the prefactor in the propagator by C= (g(B)g(A))-1’4exp(-il,(arctan(B)-arctan(d))), the propagator
(74)
becomes
(75) After an integration by parts on the P& term, the Q functional be done giving a delta functional
integration
S[P+.tP-J.
(76)
Integrating the differential equation in the argument of the delta functional respect to the boundary conditions P(0) = PO, x(0) = 0, and x(T) = x, gives
P(t) = POexp( -x(t)). The functional
P integration
may be performed with the delta functional
may
with
(77) reducing
177
PHASE SPACE PATH INTEGRATION
this to an ordinary integration (19)
over the initial
condition
P,. A composition
factor
arises in the reduction and the solution of the differential equation is substituted for P(t) in the action. One finds i?<;.,(B,r;A,O)=Cf”
-%
dxep‘:‘/:X
2exp(iP,(Be-.‘-A))j
[dx]:
(78) Note that the x functional integration is the configuration space path integral for a particle in a potential of the Morse form, V(x) = Vo(ep28-Y- 2~e~fi-Y). Here, /3 = 1, V. = Pi/2 and I= 2, PO ‘. One has p &,(B,
T;A,O)=CSI
d.xe~-‘i’/mXz~ ~ x2
exp(iP,( Be -.’ - A)) K,( B, T; A, 0). (79)
Using the result (60) for the propagator for a particle in Morse potential, propagator for a particle in the cash-‘(q) potential is
the
where B = sinh(b), A = sinh(a), A, = (A2 + $)‘I’, and C is given in (74). It is a vigorous exercise in integration and manipulation of special functions to prove that this gives the correct eigenfunction expansion, but it can be done. If one is interested in the discrete part of the eigenfunction expansion of the propagator, this can be obtained by beginning with the discrete part of the eigenfunction expansion for the propagator for the Morse potential. Since the effective Morse potential in (78) has no bound states if PO< 0, the integration over PO becomes restricted to an integral from 0 to co.
11. FREE PARTICLE ON THE ~-SPHERE The solution of the problem of a free particle propagating on the two-dimensional sphere is an important step in the application of the method of delta functionals outside the context of one-dimensional potential problems. It has been
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known for some time that the Gaussian approximation of the path integral for a particle propagating on a Lie group is in fact exact [2]. This result can be obtained by solving the Schrddinger equation, but as yet no derivation wholly within the path integral has been given. It has also been found that one can solve the Schrodinger equation to obtain an exact “classical path” representation of the propagator on the n-dimensional sphere [7] and on other rank one and rank two symmetric spaces [S]. Again these results are not obtained by directly computing the path integral. The problem of a free particle propagating on a 2-sphere is also known as the rigid rotor problem. Peak and Inomata [9] give a configuration space path integral solution of this problem by applying a constraint to a free particle in three dimensions. They begin with an expression for the free particle action in polar coordinates for short times. This involves an angular term exp(u cos(0)) which comes from a dot product of velocity vectors where 8 is the angle between the vectors. They then use the well known identity exp(u cos(8)) = c I” 2 (21+ l)Pl(cOs(fI))Z,+,,,(u). ( > /=O Using the expansion of P,(cos(B)) in terms of spherical harmonics, and the orthogonality of the spherical harmonics, it is not difficult to do the functional angular integrations. Since the radial integrations are constrained by a delta function, they are not difficult either. In short order, the problem has been solved. This solution while obviously correct is not entirely satisfying because it depends importantly on the expansion of exp(u cos(0)) in terms of Legendre functions and their subsequent expansion in spherical harmonics. While these expansions are well known for this and similar problems, their analogs are not well known in less commonplace problems, for example, on arbitrary symmetric spaces. Certainly analogs exist for these other problems. The difficulty is to find explicit representations for them. As solving the relevant eigenvalue problem is difficult, one might hope that the problem could be finessed by going to the path integral. Clearly the path integral is no help if the methods for evaluating it require knowledge equivalent to that contained in the solution of the eigenvalue problem. The method of delta functionals provides a method of evaluating the path integral without a priori knowledge of special functions and their properties. Here its application is demonstrated for the particle propagating on the 2-sphere. As in the examples of one-dimensional potential problems, an integral representation of the propagator will be obtained. By evaluating the integrals in this representation, one can explicitly obtain the familiar eigenfunction expansion of the propagator in terms of Legendre functions. The Legendre functions appear in one of their integral representations. If they had not already been known, this would give the representation of the solutions to the eigenvalue problem. One point that should be noted is that the phase space path integral will be constructed from a symmetric ordering of the covariant Hamiltonian for a free particle
PHASE
SPACE
PATH
179
INTEGRATION
on the symmetric space. The effective potential term which arises is not covariant. It is also not the curvature term R/6 that is often included in a path integral in curved space. This point is discussed in depth by Mayes and Dowker [ 171 who prove that the non-covariant term differs from R/6 by higher order terms which do not contribute in a time-slicing evaluation of the path integral. Nevertheless we must use the non-covariant potential in our evaluation and not simply R/6. This emphasizes the importance of performing the symmetric ordering of the Hamiltonian before writing down the phase space path integral-the result generally is not the naive one. The covariant Hamiltonian for a free particle in a curved space is H = + g ~ l/4jji gi’s’/zjjj g ~ 114,
(81)
where jii= is the self-adjoint momentum cally ordering this Hamiltonian
-ig-‘J4aig1f4
(82)
in the measure density g”? = (det g1,)‘j2. Symmetrigives [ 171
Hs=a(~,~jgg”+g”ai~j+g~~-2g”4Ag~L’4)
which contains the non-covariant
(83)
effective potential
+~g$-$g1’4Ag-1/4,
(84)
where A = g-‘l’d , gVg’j2a, is the covariant Laplacian the phase space path integral is
on the space. From Section 1,
For the free particle on the 2-sphere, the line element is ds2 = (d/3)’ + sin”(O)(dq$)’
from which one finds the measure density g112= sin(O). The effective potential v,=
-‘+
8
is
.l
8 sm2(6)
The full phase space path integral is then K.def, #f, C O,, #i, 0) = (sin(O/) sin(Bj)))‘/?
elTi8
j[y][y] 2
p($+p+P”-
595:199/l-13
2
p:-i d* . 2 sin2(B) >
(87)
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Consider the 4, p, dependent part of the path integral first (88)
Integrate the pId term by parts to make the action linear in the coordinate then integrate to obtain a delta functional
4 and
.
The functional integral of the delta functional is easily done and denoting the initial condition for the solution of the equation in the argument of the delta functional by p, one obtains the ordinary integral
Substituting this in the expression for the full propagator, remaining 8, pe dependent functional integral to be
one identifies
the
(91)
First make the point transformation Q = cos(0), which gives new endpoints A = cos(oi) and B = cos(0J. This transformation produces an effective potential (31) (92)
A prefactor of (sin(8,.) sin(8,))1/2 is also produced. This will cancel the measure prefactor in the original expression for the path integral. The action in the new variables is (93)
Shifting the momentum to P = P - ip( 1 - Q2) ~ ’ cancels the potential produces a total Q derivative. Dropping the bar, one has -
4-d
1-Q’
+ 2 dt.
8
term and
(94)
The total derivative integrates to give -ip arctanh(Q) evaluated at the endpoints. Now, one uncompletes the square P2Q2. This requires some subtlety to avoid technical difficulties which would arise later on. The ob.ject is to introduce unity in
PHASE
SPACE
PATH
181
INTEGRATION
the form of an integral over the endpoint of the path integral for a free particle and then to shift the functional integration in that path integral to cancel the P2Q2 term. This leaves an action which is linear in the coordinate Q which can be integrated. The potential difficulty is that the resulting momentum equation of motion may be complex leading to oscillatory solutions which in turn give rise to an oscillatory potential for the auxiliary functional integral. While there is nothing obviously wrong with this, it is difficult to prove the correspondence between the final result and the known eigenfunction expansion of the propagator for the particle on the 2-sphere. We proceed to uncomplete the square P*Q* by using the complex conjugate of the identity in Section 9, Jm dbI[dx]iexp - 5>
!
-iji2/2dt
>
=l,
and shifting i to i + PQ. Nothing else in the procedure of Section 9 changes. An effective potential (67) of - i is produced cancelling the i already present. The theta part of the propagator becomes K,
= (sin(/jf)
eparctanh(Q)I~j’,,dbj
sin(fj,))“’
7
xexp
i
(J
0
2
.2
-?
2
Wl;[[~]
-4PQ+P&$+ipPdl
(95)
Doing the Q functional integral produces a delta functional S[P+ .+P]. Performing the P functional integral replaces P by the solution to the equation in the argument of the delta functional P(r) = Poet”‘).
and gives the composition
(96)
factor (97)
The result is K,
= (sin(ef)
X
sin(ei))‘/2ePa’c’anh’Q)~~ cyrn dbepbi2
rm 2exp(iP,(Be-‘-
A)) j [dx]; ipP,e-”
dt .
(98)
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The x functional integral is recognized as the complex conjugate of that of a particle in an inverted Morse potential, V(x) = Vo(ep2” - 2Je-“), where V,, = - P32, and 2il V. = ipPo. The inverted Morse potential is unbounded below and has no bound states. This difficulty can be remedied by rotating the contour of the P, integral from the positive real axis to the positive imaginary axis. No poles in P, are enclosed and the arc at infinity vanishes. The contour rotation changes the potential into the standard Morse potential. The integral over negative PO is discarded as not contributing to the result. (A rigorous justification for this is not known at this time, but our experience shows that only bound states contribute to propagators involving normalizable eigenfunctions and there are no bound states for P,
s
om%exp(-P,(Be-“-A)) j
[dx]:
---+pP,e-“dt
(99)
Using the non-tilde version of the complex conjugate of the Morse propagator (58) with o = 2P, and 2a = ( -8E)‘j2, and resealing S to S/o, one has K,*(b, T;O, 0)=-&eb’4j:
Using the non-tilde
dSjym
~eiLI.~i2psK&(ecb~2,
S/o; l,O).
(100)
form of (40), one has S/o; 1, 0)
Krp(e-b’2,
1 =iie
~b”~b’4exp(-iPocot(S)(e-b+
1))
Collecting
the results (90), (99), (lOO), and (lOl), and making the substitutions and ii = u/4, one has the following expression for the full propagator for a free particle on the 2-sphere (dropping the bars over the new variables), PO = sin(S)P,
=e
iT/8
O” s -2YLe
dE
cc
IET s
dp
-,27L
eip(h-dl)eparctah(Q)l::
- i2p.S
m s
-00
db
e-b(z+
I/2)
PHASESPACE PATH INTEGRATION
m idP,
X D 2~ s X
* J‘
~% duu-
P$ exp(iP,(e-‘(
183
-cos(S) + iB sin(S)) - cos(S) - iA sin(S)))
(1+2’)exp ( i ( u+ %p)).
(102)
In this expression, B = cos(O,), A = cos(Oi), and tl = ( - 2E)“‘. While this is an expression for the propagator in terms of finite-dimensional integrals, it is not immediately transparent as it is in terms of six such integrals. Nevertheless, it is the exact result and one can do the integrals to obtain the familiar eigenfunction expansion (see Appendix A). The structure of the expression is fairly straightforward. The P, and u integrals are done to give the kernel of an integral representation for a Legendre function which is completed by the integral over 6. The p integral gives a delta function relating the azimuthal variables df- di to S. The S integration then substitutes them into the relevant part of the argument of the Legendre function. Finally, the E integral is the usual conversion from the energy Green’s function to the time-dependent Green’s function. It will be a countour integral which will pick up contributions from the poles of the coefficients of the Legendre function giving the desired sum over an integral index of the Legendre function. 12. CONCLUSION The method of delta functionals has been presented. It has been shown that a phase space path integral whose Hamiltonian is linear in the coordinate can be immediately integrated. By using point canonical transformations, time resealing and uncompleting the square, Hamiltonians which are not linear in the coordinate to begin with may be made so. In the course of applying these transformations, effective potential terms arise from operator ordering in the Schrodinger equation. These must be carefully accounted for. This method has been applied to solve several of the classic integrable potential problems: the inverse quadratic potential, the Coulomb potential, the Morse potential, and the sech2(x) potential. In addition, the problem of a particle propagating on a 2-sphere has been solved. The method of solution would appear to extend naturally to solve the problems of a free particle propagating on a Lie group or a symmetric space. These problems can be solved in terms of the Schrddinger equation, but have not been tractable in the path integral outside of the Gaussian approximation. The method described here is important in that it approaches the problem of path integration from a different perspective than the method of Gaussian integration. It does so without emphasizing the details of time-slicing, except in the proof of some underlying results. This method is powerful because it draws a clear connection between the classical integrability of a system and its quantum integrability. If it were possible to extend the set of allowed transformations to include general
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canonical transformations (which mix coordinates and momenta), then this method might allow the quantum solution of all integrable classical systems. At this time, the restriction to point canonical transformations and a few other special transformations are the most serious limit upon the method. In future work, it will be important to consider not only how to broaden the admissible class of transformations but also how to apply this method to obtain approximate results.
APPENDIX
The multiple integral representation a 2-sphere can be evaluated to give terms of Legendre functions. Begin q=e-b, and q = qti-*. Dropping the becomes
X
A
(102) of the propagator for a free-particle on the well known eigenfunction expansion in with (102) and change variables, U = PO/u, bars over the new variables, the propagator
m idPo
s
o xexp(iP,(u*q(-cos(S)+iBsin(S))+u(q+
l)-cos(S)-iA
sin(S))), (103)
where B = cos(Of), A = cos(Oj), and CI= ( - 2E)‘/*. Doing the P, integral produces a term + iB sin(S)) + u(q + 1) - cos(S) - L4 sin(S)’
u2q( -cos(S)
Denote &?= - cos( S) + iB sin(S) and J&’ = - cos( S) - iA sin(S). The u integral may be done by closing the contour in the upper half u-plane. This encloses one pole of the expression produced from the PO integral and the result is 2n((q+
The full expression for the propagator ae,,
4f/,
=e
Z
iT/8
ei,
di3
1)2-4qdL8-1’2.
is now
0)
00 dE iET ’ dp s -,27c
eiP(~,-),)+parctanh(Q)lf:
s +2Tte
X
idqq”pl/z
((q+
1)2-4q%!&%-1’2.
m dS
-e
-12p.Y
s0 7c
(104)
PHASE
SPACE
PATH
INTEGRATION
185
The square root in the integrand can be rewritten as (42 - 2qz + 1) ~ 1’2,
where z=24%4?--1.
(105)
To simplify this, note the identity cos(S) f Xsin(S) = Jl
+X’
cos(S+ arctan(X
(106)
valid for any X. Using this in (105) and shifting S = S- (i/2) arctanh( Q) / 2 gives Z=2sin(0,)sin(8f)cos
(
S+!
2)
cos S--! -1, ( 2)
where A = i(arctanh(B)
+ arctanh(A)).
Note that shifting S also cancels the arctanh term in the p exponential. algebra reduces Z to Z = cos(ef) cos(Bj) + cos(2S) sin(8/) sin(B,).
Further (107)
Doing the p integration gives a delta function in terms of S and doing the S integration sets S= ( -#f+ di)/2. The q integral is recognized as an integral representation of the Legendre function
I
x dqqxP”*(l
-2qcos($)+qy”‘=7ccsc((cr+$r)P,-,,,(-cos(~)),
0
where COS(l//)=Z= The propagator
COS(+)
COS(Oi)
+ cOS($r-
fji) sin(0,.) sin(ej).
(108)
becomes
Recalling that cc= ( - 2E) ‘I2 , the E integral may be done. Change variables from runs down below the positive real c( axis and up the positive imaginary c( axis. For positive T, the contour may be closed in the upper right quadrant of the a-plane and simple poles of csc((a + 4)~) are picked up at c(= n - 4 where n is a positive integer. The result is E to ~1.The contour
186
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where E, = (n - 4)2/2. Shifting P,( -x) = (- l)“P,(x), one has K,(tlf,
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the index
dr, T; Oi, di, 0) = -& ‘$
of the
epi"'"+')"2(n
sum by one,
+
i)P,(Z).
and using
(111)
?I=0
with Z given by (108). This is the correct propagator for a free particle on the 2-sphere. By using the addition theorem for Legendre functions, it can be brought into the familiar eigenfunction expansion form as a product of eigenfunctions at each endpoint.
ACKNOWLEDGMENT This work
was supported
in part by NSF
Grant
No. PHY85-06583.
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