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Path integration of a quadratic action with a generalized memory
Volume
93A, number
PHYSICS
4
PATH INTEGRATION
D.C. KHANDEKAR
OF A QUADRATIC
ACTION WITH A GENERALIZED
1983
MEMORY
l, S.V. LAWANDE l and K.V. BHAGWAT 2
Bhabha Atomic Research Centre, Trombay. Bombay-400085. Received
10 January
LETTERS
28 September
India
1982
Path integration of an action representing a harmonic oscillator with a generalized memory is carried out within the framework of Feynman’s polygonal approach. The exact propagator obtained is in the form of an exponential integral over a single variable. Closed analytical results are available for special cases of the memory function.
In his path integral theory of an electron gas in a random potential tion with memory given by s[x(r)]
dr(+n?’
= j
-$$
0
j-
Bezak [l] has considered the quadratic
ac-
,x(t) -x(r’)l’dr’)
0
Bezak, however, used imaginary time - ip (0 being the inverse of temperature) for obtaining the partition function and only arrived at an approximate form of the path integral. Subsequently, Papadopoulos [2] obtained the path integral (in imaginary time) in an exact closed form by coupling the system to auxiliary external forces. An entirely new approach of path integration based on the theory of promeasures [3] was invoked by Maheshwari [4] to rederive Papadopoulos’ results [2]. Recently, Khandekar et al. [5] have shown that path integration of Bezak’s action can be carried out within the framework of Feynman’s polygonal approach [6]. Such a derivation is self-contained and does not require the knowledge of an auxiliary measure [4] or an artificial coupling to the external forces [2]. In the present paper, we exploit further the polygonal method to treat the following quadratic action with a generalized memory: T
(2) where F(u) is an arbitrary function of u and p and u have been introduced from dimensional considerations. Bezak’s action (1) results from (2) for the special case when F(u) = u2 and ,u/a2 = mf12/2 T. We show that the polygonal method is able to path integrate the action (2) resulting in an exact form for the propagator. The path integral is defined as I (T )=x K(x, T:xo, 0) = j9x(t) X(0)=X0
exp{(i/fi)S
where (7, x(t) is the usual Feynman 1 Theoretical Reactor Physics 2 Nuclear Physics Division.
L+)lI,
path differential
(3) measure. In the polygonal scheme eq. (3) takes the form [6]
Section.
0 031-9163/83/0000-0000/$03.00
0 1983 North-Holland
167
Volume
93A, number
PHYSICSLETTERS
4
10
January 1983 (4)
where KN(x, T;xg,
0) = AN 7
... 7
_-m
exp(iSN/fi)
Fll
dXj.
(5)
_m
Here S, is the discretized form of the action S [x(t)] of eq. (2):
SN =
~(~ (Xj
Xj_l)*
-
-
F2n2X~)
+
J
~~(~ ~ J
(6)
)
Xj)
withXj =X(tj),Xo'x(O),XN =X(T) =X, fj - f.J_1 = T,JN = E and A, = (rn/27riBe)N1* is the normalization factor in the Nth approximation. NOW, in order to evaluate KN we have to substitute expression (6) for S,,, in (5) and carry out the integration overXj (i = 1,2, ....N- 1). This appears to be a formidable task at first. However, with the following change of variables N-l Yu =X0,
YJ=
C
(j= 2, ....N-
Yj=Xj
xj9
l),
(7)
j=l
we can rewrite eq. (5) as N-l KN(x, T;x~,
O)=AN
J
...J
II
J=l
Here C is an (N+ l)-dimensional
‘=(;T
dYj exp [(im/2fiie) FC
Yl expKb.@>F((EY
1 + EYN)/o)~.
(8)
square matrix: (9)
1,’
where the matrices A and B are symmetric
matrices of dimensions
3 and (N-
2), respectively,
defined explicitly
by A={eij,lG(i,iG3),
eJl=y-l,
G!J*=e*r=O,
fZ13’U31=0,
l722=1,
fZ23=LZ3*=-1,
033=0, (10)
B={b+
1 G:,j_(N-2},
bjj= 2y
b,, =2(1+-r),
(2
bjk=b,j=r
blj=bjl=(l’Y)
bjj+l =bj+r,j=(y-1)
(3~~~N-2),
(2Gj
(2~j~N-2,k=j’2,...,N-2),
(11)
where y = 2 - Q2e2. The rectangular (3 X (N - 2)) matrix D is given by D={ddii,i=
djl = -(I
1,2,3;1
+y),
Gi
d,j=O (2
The matrix DT is the transpose ofD. 168
(l
d,,,_,=-1,
d,j=
1
(1
Volume
93A. number
Further,
PHYSICS
4
10 January
LETTERS
1983
y, the transpose of the vector Y, is defined as
y= (E, P),
F’= cY2,Y3,
c= (YN?YOJl),
Next, we define a transformation
(13)
...7YN-l).
Y = Z + W such that
i%JtBW=O.
(14)
Then we may write
kY=~AUt~BZt+DW. Consequently,
(15)
eq. (8) assumes the form
K/,&t T; x0, 0) N-2
=AN_fdYI exp[(im/2Re)(~AUt~DW)+(i~/ultz)F((Ey1
+ e~jN)/u)]l
The gaussian integral in eq. (16) may be easily performed KN(x, T;xO,0) = (m/2rrih)
IpI dzjexp[(im/2fie)zBZ].
(16)
to obtain
[e4(det B)lpV2$ d(eyr)exp[(im/2Re)P+
(ip/A)F((eyr
+ev~)/a)],
(174
where for short we write
P=i?AiJtcDW.
(17b)
It is now clear from eqs. (4) and (17) that the problem of evaluating the propagator lim,,, P/Eand lim,,, e4det B. For this purpose, we first write eq. (14) as a set of algebraic equations cl++1 -yw,+w,_r
=wr
-ywu+yu
reduces to obtaining
(O
where 0,. (1 < r G N - 2)are the components
(18)
of the vector W and
N-2 w-1
WO =Yl
‘YO,
Introducing
-
c s=2
US’
a more convenient
#N-l
variable pr = o,_~
P,+l-YPrtPr_l=P2-YPl+PO
(19)
=yN.
(1
we may rewrite eqs. (18) and (19) as
l),
(20)
with N-l
PO'YO,
Pl=q=Yl-
Recalling the definition equation P(f) + n2p(t)
fx Pp
s=2
(20
of y, it is easy to see that in the limit E + 0 eqs. (20) and (21) lead to the differential
= (p’ + dp),,o
along with the end-point
= c$c,
(22)
conditions
P(O)=Yg =x0, Pm=Ym=Y=x, and a self-consistency
PN=YN.
(23)
condition
169
PHYSICS LETTERS
Volume 93A, number 4
10 January 1983
T
ey, =
lim
EjO
jp(t)dt =j-
x(t)dt
(24)
=t(,
0
0
which is used to determine the constant C in eq. (22). Eq. (22) is readily solved with the conditions (23) and (24) to yield (25)
p(t)=asinflt+bcosClt+c,
where [(sin RT - QT)(x
a =
-x0)
(26)
- xgT)J/d,
+ (1 - cos RT)R(u
b = [(cos RT - 1)(x - x0) + R sin ClT(u - xoT)]/d,
(27)
c=xo
(28)
-b,
with -cosRT)-RTsinQT.
d=2(1
Next, using eqs. (lo)-(12), as
(29)
eq. (13) and eq. (17a), we write the P/E term in the exponential
of eq. (17) explicitly
N-2
=
[(l
-fi2e2)?& +P; -Yl(&
-2P,
+P(J
+fi*&+PlPO -PN_lPN]/~-
(30)
The second step in eq. (30) is obtained by inserting the value of y and using the relation between wy and pr. We now use the Taylor expansions p, = po + Ebo + f’2fio
+ O(G))
p2=po+2~fi0
+2c2po+O(e3),
ply-l
=P,-E&,,+~(E~),
(31,32,33)
to simplify the expression for P/E as P/e = [-&;-
(ijo + f12po)cyl
We noiv use eqs. (34) and eqs. (25)-(28) ;Fo P/e = [-C12cu - p(O)p(O) + 2(x +xo)u~(cosRT-
to arrive at the result:
+P(T)p(T)]
= R [(x2 t xi)(sin
RT - RTcos52T)
+ 2xox(RT
- sin RT)
1) + L?‘2,‘sinRT]/c/.
Next, we proceed to evaluate the determinant ofB by exploiting its secular equation. The eigenvalue equation for B reads as (B-
(34)
- PO& + p,y&,, + o(E2)].
h/)Q=O,
(35) the fact that it is related to the constant
term in
(36)
where Q = (q,, q2, .. . . qN_2) is the eigenvector of B corresponding to the eigenvalue h. Using the explicit form of the matrix B given in eq. (11). eq. (36) may be rewritten as a set of algebraic equations: ++I +4r_I
- 2 cos@q, =ql
with the boundary
- Y4rJ
(OGrGN-2),
(37)
conditions
N-2 90 = - c Ys, S=l 170
qN_1 = 0.
(38)
Volume
93A, number
PHYSICS
4
LETTERS
10 January
1983
The angle 4 in eq. (37) has been defined as 2c0s@=y-h.
(39)
It is easily verified that the general solution of the difference equation (37) may be expressed in terms of q. and ql
as
qr = {q. [X(sin rf$ - sin(r - 1) @- sin $j + 2(cos rf#~- cos $j cos $11 +qr(sinrg-sin(r+ The two boundary
l)~+sinf$)}/[2sin@(l conditions
-cos@)].
(40)
(38) then yield the following pair of homogeneous
equations
in q. and 4 1
[V2(4J)+ g2(@)140 + 4(@)4 1 = 0,
tv,(~)+gl(~)l40 + hl(4)ql = 0,
(4 1~42)
where fi(G)=
[sin@+sin@-
l)@-NsinGl/1C/(@),
g,(4)={[cos+(N- 1)Wn$Wlb-$$
f2(@~)= [sin@-
-Ncos@Hx($),
h l(G)= (N sin4 - sinNQ)lG(Q), h2(@)= [sin(N$(@)=2sin$(l-cos@),
lj~-sin(N-2)~-sin~l/rl,(~),
82(G)=[co@-I)@-
cos@l/x(~),
l)@ - sinN@~ + sin $]/IJ(I#J),
x(@)=(l-cos@).
Eqs. (41) and (42) lead to the characteristic
(45 946) (47948) (49350)
equation
determining
h which reads as
+&(@)I= 0.
G(4) = %(@)Pf1(43) +g1(@)1 - hl(@)N-i(G)
When h = 0 the lhs of eq. (5 1) reduces to det B. Further, cos~o=y/2=1-!Ll=e’/2,
(43,44)
(5 1)
from eq. (39) we have at X = 0
@o=@(h=O),
(52)
which yields sin2i$o/2 = Q2e2/4,
$Jo=~e+O(E2).
(53)
Thus, we have det B = GMo) = h2(G0)g1(@0) - hl(rJO)gZ(GO), where Go is given by (53). Substituting $($I()) = fi3e3 + 0(&t),
=
2[(cosRT
(5556)
- cos aT)/2
t O(~“)]/s2~e’,
- l)+ fi2~ sin RT+ O(E~)]/~~~E~,
Iz,(@~) = [(QT - sin nT) + O(E’)]/~~~E~, Expressions (57)-(60)
and noting that NE = T, we arrive at
x(@(r) = a2e2/2,
gl(Goj = 2[(sin QT - QT) t fie(l g2($oI
(54)
the value of Go in eqs. (45)-(50)
!z~(c$~)= [(l-
(57) (58)
cos~T)-(stesinQT)/2+O(e2)]/~3e3.
(59760)
are used in eq. (54) to yield
lim e4det f3 = d/Q4. e+O where d is given by (29). Combining in exact form
(61) eqs. (4), (17a), (24), (35) and (6 l), we arrive at the propagator K(x, T;xo, 0)
171
Volume
93A, number
K(x, T;xo, 0) = (mSI*/2niA)d-r’*~ + 2xxo(RT-
PHYSICS
4
d u exp{(imCl/2fid)[(x*
sinaT) + 2(x+xo)uS2(cosS2T-
10 January
LETTERS
+xi)(sinRT-
1983
QTcosClT)
1) t u*Q* sinfiT] + (i/.@)F(u/o)}.
(62)
It is easy to verify that for the two special cases, viz: (i) F(u) = 0 and (ii) F(u) = U, which represent respectively a free oscillator and an oscillator acted on by a constant force y/o, eq. (62) yields the well-known closed form expressions for the propagator [6]. Finally F(u) = u2 withp/c* = mR2/2T which corresponds to an oscillator with a memory [l] also leads to the closed form obtained previously [5,7]. The authors express their gratitude to Professor A. Maheshwari for useful discussions. References [I] [2] [3] [4] [5] [6] [7]
172
V. Bezak, Proc. R. Sot. (London) 315A (1970) 339. G. Papadopoulos, J. Phys. A7 (1974) 183. C. Dewitt-Morette, A. Maheshwari and B. Nelson, Phys. Rep. 50 (1979) 255. A. Maheshwari, J. Phys. A8 (197.5) 1019. DC. Khandekar, S.V. Lawande and K.V. Bhagwat, Lett. Math. Phys. 5 (1981) 501. R.P. Feynman and A.R. Hibbs, Quantum mechanics and path integrals (McGraw-Hill, A.K. Dhara, D.C. Khandekar and S.V. Lawande, 5. Math. Phys., to be published.
New York,
1965).
93A, number
PHYSICS
4
PATH INTEGRATION
D.C. KHANDEKAR
OF A QUADRATIC
ACTION WITH A GENERALIZED
1983
MEMORY
l, S.V. LAWANDE l and K.V. BHAGWAT 2
Bhabha Atomic Research Centre, Trombay. Bombay-400085. Received
10 January
LETTERS
28 September
India
1982
Path integration of an action representing a harmonic oscillator with a generalized memory is carried out within the framework of Feynman’s polygonal approach. The exact propagator obtained is in the form of an exponential integral over a single variable. Closed analytical results are available for special cases of the memory function.
In his path integral theory of an electron gas in a random potential tion with memory given by s[x(r)]
dr(+n?’
= j
-$$
0
j-
Bezak [l] has considered the quadratic
ac-
,x(t) -x(r’)l’dr’)
0
Bezak, however, used imaginary time - ip (0 being the inverse of temperature) for obtaining the partition function and only arrived at an approximate form of the path integral. Subsequently, Papadopoulos [2] obtained the path integral (in imaginary time) in an exact closed form by coupling the system to auxiliary external forces. An entirely new approach of path integration based on the theory of promeasures [3] was invoked by Maheshwari [4] to rederive Papadopoulos’ results [2]. Recently, Khandekar et al. [5] have shown that path integration of Bezak’s action can be carried out within the framework of Feynman’s polygonal approach [6]. Such a derivation is self-contained and does not require the knowledge of an auxiliary measure [4] or an artificial coupling to the external forces [2]. In the present paper, we exploit further the polygonal method to treat the following quadratic action with a generalized memory: T
(2) where F(u) is an arbitrary function of u and p and u have been introduced from dimensional considerations. Bezak’s action (1) results from (2) for the special case when F(u) = u2 and ,u/a2 = mf12/2 T. We show that the polygonal method is able to path integrate the action (2) resulting in an exact form for the propagator. The path integral is defined as I (T )=x K(x, T:xo, 0) = j9x(t) X(0)=X0
exp{(i/fi)S
where (7, x(t) is the usual Feynman 1 Theoretical Reactor Physics 2 Nuclear Physics Division.
L+)lI,
path differential
(3) measure. In the polygonal scheme eq. (3) takes the form [6]
Section.
0 031-9163/83/0000-0000/$03.00
0 1983 North-Holland
167
Volume
93A, number
PHYSICSLETTERS
4
10
January 1983 (4)
where KN(x, T;xg,
0) = AN 7
... 7
_-m
exp(iSN/fi)
Fll
dXj.
(5)
_m
Here S, is the discretized form of the action S [x(t)] of eq. (2):
SN =
~(~ (Xj
Xj_l)*
-
-
F2n2X~)
+
J
~~(~ ~ J
(6)
)
Xj)
withXj =X(tj),Xo'x(O),XN =X(T) =X, fj - f.J_1 = T,JN = E and A, = (rn/27riBe)N1* is the normalization factor in the Nth approximation. NOW, in order to evaluate KN we have to substitute expression (6) for S,,, in (5) and carry out the integration overXj (i = 1,2, ....N- 1). This appears to be a formidable task at first. However, with the following change of variables N-l Yu =X0,
YJ=
C
(j= 2, ....N-
Yj=Xj
xj9
l),
(7)
j=l
we can rewrite eq. (5) as N-l KN(x, T;x~,
O)=AN
J
...J
II
J=l
Here C is an (N+ l)-dimensional
‘=(;T
dYj exp [(im/2fiie) FC
Yl expKb.@>F((EY
1 + EYN)/o)~.
(8)
square matrix: (9)
1,’
where the matrices A and B are symmetric
matrices of dimensions
3 and (N-
2), respectively,
defined explicitly
by A={eij,lG(i,iG3),
eJl=y-l,
G!J*=e*r=O,
fZ13’U31=0,
l722=1,
fZ23=LZ3*=-1,
033=0, (10)
B={b+
1 G:,j_(N-2},
bjj= 2y
b,, =2(1+-r),
(2
bjk=b,j=r
blj=bjl=(l’Y)
bjj+l =bj+r,j=(y-1)
(3~~~N-2),
(2Gj
(2~j~N-2,k=j’2,...,N-2),
(11)
where y = 2 - Q2e2. The rectangular (3 X (N - 2)) matrix D is given by D={ddii,i=
djl = -(I
1,2,3;1
+y),
Gi
d,j=O (2
The matrix DT is the transpose ofD. 168
(l
d,,,_,=-1,
d,j=
1
(1
Volume
93A. number
Further,
PHYSICS
4
10 January
LETTERS
1983
y, the transpose of the vector Y, is defined as
y= (E, P),
F’= cY2,Y3,
c= (YN?YOJl),
Next, we define a transformation
(13)
...7YN-l).
Y = Z + W such that
i%JtBW=O.
(14)
Then we may write
kY=~AUt~BZt+DW. Consequently,
(15)
eq. (8) assumes the form
K/,&t T; x0, 0) N-2
=AN_fdYI exp[(im/2Re)(~AUt~DW)+(i~/ultz)F((Ey1
+ e~jN)/u)]l
The gaussian integral in eq. (16) may be easily performed KN(x, T;xO,0) = (m/2rrih)
IpI dzjexp[(im/2fie)zBZ].
(16)
to obtain
[e4(det B)lpV2$ d(eyr)exp[(im/2Re)P+
(ip/A)F((eyr
+ev~)/a)],
(174
where for short we write
P=i?AiJtcDW.
(17b)
It is now clear from eqs. (4) and (17) that the problem of evaluating the propagator lim,,, P/Eand lim,,, e4det B. For this purpose, we first write eq. (14) as a set of algebraic equations cl++1 -yw,+w,_r
=wr
-ywu+yu
reduces to obtaining
(O
where 0,. (1 < r G N - 2)are the components
(18)
of the vector W and
N-2 w-1
WO =Yl
‘YO,
Introducing
-
c s=2
US’
a more convenient
#N-l
variable pr = o,_~
P,+l-YPrtPr_l=P2-YPl+PO
(19)
=yN.
(1
we may rewrite eqs. (18) and (19) as
l),
(20)
with N-l
PO'YO,
Pl=q=Yl-
Recalling the definition equation P(f) + n2p(t)
fx Pp
s=2
(20
of y, it is easy to see that in the limit E + 0 eqs. (20) and (21) lead to the differential
= (p’ + dp),,o
along with the end-point
= c$c,
(22)
conditions
P(O)=Yg =x0, Pm=Ym=Y=x, and a self-consistency
PN=YN.
(23)
condition
169
PHYSICS LETTERS
Volume 93A, number 4
10 January 1983
T
ey, =
lim
EjO
jp(t)dt =j-
x(t)dt
(24)
=t(,
0
0
which is used to determine the constant C in eq. (22). Eq. (22) is readily solved with the conditions (23) and (24) to yield (25)
p(t)=asinflt+bcosClt+c,
where [(sin RT - QT)(x
a =
-x0)
(26)
- xgT)J/d,
+ (1 - cos RT)R(u
b = [(cos RT - 1)(x - x0) + R sin ClT(u - xoT)]/d,
(27)
c=xo
(28)
-b,
with -cosRT)-RTsinQT.
d=2(1
Next, using eqs. (lo)-(12), as
(29)
eq. (13) and eq. (17a), we write the P/E term in the exponential
of eq. (17) explicitly
N-2
=
[(l
-fi2e2)?& +P; -Yl(&
-2P,
+P(J
+fi*&+PlPO -PN_lPN]/~-
(30)
The second step in eq. (30) is obtained by inserting the value of y and using the relation between wy and pr. We now use the Taylor expansions p, = po + Ebo + f’2fio
+ O(G))
p2=po+2~fi0
+2c2po+O(e3),
ply-l
=P,-E&,,+~(E~),
(31,32,33)
to simplify the expression for P/E as P/e = [-&;-
(ijo + f12po)cyl
We noiv use eqs. (34) and eqs. (25)-(28) ;Fo P/e = [-C12cu - p(O)p(O) + 2(x +xo)u~(cosRT-
to arrive at the result:
+P(T)p(T)]
= R [(x2 t xi)(sin
RT - RTcos52T)
+ 2xox(RT
- sin RT)
1) + L?‘2,‘sinRT]/c/.
Next, we proceed to evaluate the determinant ofB by exploiting its secular equation. The eigenvalue equation for B reads as (B-
(34)
- PO& + p,y&,, + o(E2)].
h/)Q=O,
(35) the fact that it is related to the constant
term in
(36)
where Q = (q,, q2, .. . . qN_2) is the eigenvector of B corresponding to the eigenvalue h. Using the explicit form of the matrix B given in eq. (11). eq. (36) may be rewritten as a set of algebraic equations: ++I +4r_I
- 2 cos@q, =ql
with the boundary
- Y4rJ
(OGrGN-2),
(37)
conditions
N-2 90 = - c Ys, S=l 170
qN_1 = 0.
(38)
Volume
93A, number
PHYSICS
4
LETTERS
10 January
1983
The angle 4 in eq. (37) has been defined as 2c0s@=y-h.
(39)
It is easily verified that the general solution of the difference equation (37) may be expressed in terms of q. and ql
as
qr = {q. [X(sin rf$ - sin(r - 1) @- sin $j + 2(cos rf#~- cos $j cos $11 +qr(sinrg-sin(r+ The two boundary
l)~+sinf$)}/[2sin@(l conditions
-cos@)].
(40)
(38) then yield the following pair of homogeneous
equations
in q. and 4 1
[V2(4J)+ g2(@)140 + 4(@)4 1 = 0,
tv,(~)+gl(~)l40 + hl(4)ql = 0,
(4 1~42)
where fi(G)=
[sin@+sin@-
l)@-NsinGl/1C/(@),
g,(4)={[cos+(N- 1)Wn$Wlb-$$
f2(@~)= [sin@-
-Ncos@Hx($),
h l(G)= (N sin4 - sinNQ)lG(Q), h2(@)= [sin(N$(@)=2sin$(l-cos@),
lj~-sin(N-2)~-sin~l/rl,(~),
82(G)=[co@-I)@-
cos@l/x(~),
l)@ - sinN@~ + sin $]/IJ(I#J),
x(@)=(l-cos@).
Eqs. (41) and (42) lead to the characteristic
(45 946) (47948) (49350)
equation
determining
h which reads as
+&(@)I= 0.
G(4) = %(@)Pf1(43) +g1(@)1 - hl(@)N-i(G)
When h = 0 the lhs of eq. (5 1) reduces to det B. Further, cos~o=y/2=1-!Ll=e’/2,
(43,44)
(5 1)
from eq. (39) we have at X = 0
@o=@(h=O),
(52)
which yields sin2i$o/2 = Q2e2/4,
$Jo=~e+O(E2).
(53)
Thus, we have det B = GMo) = h2(G0)g1(@0) - hl(rJO)gZ(GO), where Go is given by (53). Substituting $($I()) = fi3e3 + 0(&t),
=
2[(cosRT
(5556)
- cos aT)/2
t O(~“)]/s2~e’,
- l)+ fi2~ sin RT+ O(E~)]/~~~E~,
Iz,(@~) = [(QT - sin nT) + O(E’)]/~~~E~, Expressions (57)-(60)
and noting that NE = T, we arrive at
x(@(r) = a2e2/2,
gl(Goj = 2[(sin QT - QT) t fie(l g2($oI
(54)
the value of Go in eqs. (45)-(50)
!z~(c$~)= [(l-
(57) (58)
cos~T)-(stesinQT)/2+O(e2)]/~3e3.
(59760)
are used in eq. (54) to yield
lim e4det f3 = d/Q4. e+O where d is given by (29). Combining in exact form
(61) eqs. (4), (17a), (24), (35) and (6 l), we arrive at the propagator K(x, T;xo, 0)
171
Volume
93A, number
K(x, T;xo, 0) = (mSI*/2niA)d-r’*~ + 2xxo(RT-
PHYSICS
4
d u exp{(imCl/2fid)[(x*
sinaT) + 2(x+xo)uS2(cosS2T-
10 January
LETTERS
+xi)(sinRT-
1983
QTcosClT)
1) t u*Q* sinfiT] + (i/.@)F(u/o)}.
(62)
It is easy to verify that for the two special cases, viz: (i) F(u) = 0 and (ii) F(u) = U, which represent respectively a free oscillator and an oscillator acted on by a constant force y/o, eq. (62) yields the well-known closed form expressions for the propagator [6]. Finally F(u) = u2 withp/c* = mR2/2T which corresponds to an oscillator with a memory [l] also leads to the closed form obtained previously [5,7]. The authors express their gratitude to Professor A. Maheshwari for useful discussions. References [I] [2] [3] [4] [5] [6] [7]
172
V. Bezak, Proc. R. Sot. (London) 315A (1970) 339. G. Papadopoulos, J. Phys. A7 (1974) 183. C. Dewitt-Morette, A. Maheshwari and B. Nelson, Phys. Rep. 50 (1979) 255. A. Maheshwari, J. Phys. A8 (197.5) 1019. DC. Khandekar, S.V. Lawande and K.V. Bhagwat, Lett. Math. Phys. 5 (1981) 501. R.P. Feynman and A.R. Hibbs, Quantum mechanics and path integrals (McGraw-Hill, A.K. Dhara, D.C. Khandekar and S.V. Lawande, 5. Math. Phys., to be published.
New York,
1965).