Approximations 559xx’x’’ x x’(a) (b)Fig. 14.6(a) Diagram of the real- and imaginary-time paths for the correlation function in
eqn. (14.6.3). (b) Same for the time correlation function ineqn. (14.6.4).
to compute the two real-time paths because the matrix elements ofexp(±iHˆt/ ̄h) are
not positive definite, and the sampling schemes of Section 7.3 break down. A pos-
sible alternative could be to devise a molecular dynamics approach withcomplex
variables (Gausterer and Klauder, 1986; Lee, 1994; Bergeset al., 2007), although no
known approach is stable enough to guarantee convergence of a path integral with a
purely imaginary discretized action functional.
Before proceeding, we note that there are two alternative quantum time correlation
functions that have important advantages overCAB(t). The first is a symmetrized
correlation functionGAB(t) defined by
GAB(t) =1
Q(N,V,T)
Tr[
AˆeiHˆτc∗/ ̄hBˆe−iHˆτc/ ̄h]
. (14.6.4)
Hereτcis a complex time variable given byτc=t−iβ ̄h/2. Although not equal to
CAB(t), the Fourier transform ofGAB(t)
G ̃AB(ω) =√^1
2 π∫∞
−∞dte−iωtGAB(t) (14.6.5)is related to the Fourier transform ofCAB(t) by
C ̃AB(ω) = eβ ̄hω/^2 G ̃AB(ω), (14.6.6)which provides a straightforward route to the determination of a spectrum, assuming
GAB(t) can be calculated. Eqn. (14.6.6) can be easily proved by performingthe traces
in the basis of energy eigenstates (see Problem 14.2). The advantage ofGAB(t) over
CAB(t) can be illustrated for a single particle in one dimension. We assume, again,
thatAˆandBˆare functions only of ˆxand compute the trace in the coordinate basis,
which gives