Volume 95A, number 6
PHYSICS LETrERS
9 May 1983
COMMENT ON
"PATH INTEGRATION OF A QUADRATIC ACTION WITH A GENERALIZED MEMORY"
E. BOSCO
Department of lnorganic and Physical Chemistry, Indian Institute of Science,
Bangalore - 560 012, India
Received 2 March 1983
Using the promeasure technique, we give an alternative evaluation of a path integral corresponding to a quadratic action
with a generalized memory.
Recently, Khandekar et al. [ 1] considered the path integration of a quadratic action with a generalized memory.
Using Feynman's polygonal method and the action functional * 1
S[x(t)l=
f (½k 2 _ ~a eo2x2) dt + laF(o -1 fT x(t) dt)
T
(1)
where F(u) is an arbitrary function of u, the mass m and h are normalised to unity, the path integral was evaluated
exactly. Here we rederive the same using the powerful technique of global integration on function spaces, originally developed by DeWitt-Morette [2]. This method is elegant, shorter and without any limiting procedure.
The path integral in question is:
X(tb)=X
K(x, t b ; x 0 , ta) = f
Dtx(t)] exp{iS[x(t)]},
X(ta)=X0
(2)
with S[x(t)] given by eq. (1).
After absorbing the harmonic part of the action into the promeasure ~o', the propagator in eq. (2) can be
written as [3]
K = Ko(X, tb;X0, ta) f
exp{i/aF(o-1 fT x(t) dt)} d w ' ( x ) .
~xxo
(3)
The function-space integral in eq. (3) is over ~bxxo, the space of continuous functions x E Cbxxo such that X(ta)
= x 0 and X(tb) = x defined over the interval T = [ta, tb]. Ko(X, tb;X0, ta) is the propagator for the harmonic oscillator.
Under the function-space translation ~bxxo ~ ~0 by x(t) ~ y ( t ) = x(t) +~(t), where cI'0 is the space of continuous functions y vanishing at t a and t b and ~(t) is the fixed path in ~xxo given by
~(t) =
x 0 sin co(t b - t) + x sin co(t - ta)
sin co T
(4)
,1 Even if the memory function F(a -I fT x(t) dt) is replaced by a more general one of the form F(a -1 fT ×(t) x(t) dt), the analysis will essentially be the same.
0 0 3 1 - 9 1 6 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 03.00 © 1983 North-Holland
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Volume 95A, number 6
PHYSICS LETTERS
9 May 1983
With the image w on ~0 of the promeasure co', the integral in (3) can be written
f
exp {i/aF(o -1 fT Ix(t) + ~(t)] dt)) dco(x) ~
(5)
The promeasure co is defined in terms of its Fourier transform ~rco~) on the space c/g dual of (I,0, which is the
space of bounded measures/a on T, and the covariance G(r, s):
~rco(/~) = exp[-~iW(/s)],
W(/a)= f d v ( r ) f d / a ( s ) G(r, s),
T
T
G(r,s) = [Y+(r - s) sin co(s - ta) sin w(t b - r) + Y - ( r - s) sin w(r - ta) sin w(t b - s)]/co sin coT.
The integrand in the function-space integral on ¢ 0 is a cylindrical function and hence the integral in eq. (5)
can be readily evaluated by a linear mapping r: ¢ 0 ~ R by x(t) ~ u = (X, x) where X is the Lebesgue measure on
the interval T. The image of dco under r is
[27ri W(~)]-l/2 exp[iu2/2W(~)] du ,
where
W(~.) = f dr f dsG(r,s)=d/co 3 sin coT,
T
T
and
d = 2(1 - c o s coT) - coT sin coT.
Thus
f exp{i/aF( or-1 f r [x(t) +~(t)] dt)} dco(x) = [2zriW(?,)]-l/2 f d u exp[ilaF(u/o + <X, ~)/o)] exp[½iu2/W(X)].
*o
(6)
On combining eqs. ( 3 ) - ( 6 ) and simplifying, we get
K(x, t b ;x0, ta) = (co2/27rix/~)
fdu exp('(ico/2d)[(x 2 + x02) (sin c o T -
coT cos toT)
+ 2xx0(co T - sin co T) + 2(x + Xo)U co(cos co T - 1) + u2co 2 sin coT] + ilaF(u/o)} ,
the expression obtained by Khandekar et al.
References
[1] D.C. Khandekar, S.V. Lawande and K.V. Bhagwat, Phys. Lett. 93A (1983) 167.
[2] C. DeWitt-Morette, Commun. Math. Phys. 28 (1972) 47; 37 (1974) 63;
C. DeWitt-Morette, A. Maheshwari and B. Nelson, Phys. Rep. 50 (1979) 255.
[3] A. Maheshwari, J. Phys. A8 (1975) 1019.
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