Path integral derivation 451can be obtained by evaluating the density matrix at an imaginary inverse temperature
β=it/ ̄h. In fact, if we allow time and temperature to be complex componentsof
a general complex time parameterθ=t+iβ ̄h, then the transformationst=−iβ ̄h
andβ=it/ ̄hcan be performed by rotations in the complexθ-plane from the real
axis to the imaginary axis, as shown in Fig. 12.5. These rotations are known asWick
Re θIm θt-i βℏFig. 12.5 Wick rotation in the complex time plane.rotationsafter the Italian physicist Giancarlo Wick (1909–1992), and they permit, in
principle, the determination of the propagator given the density matrix, and vice versa.
Since it is generally easier to work with a damped exponential rather than a complex
one, we shall derive the Feynman path integral for the canonical density matrix and
then exploit eqn. (12.2.4) to obtain a corresponding path integral expression for the
quantum propagator.
Let us denote the coordinate-space matrix elements of ˆρ(β) as
ρ(x,x′;β)≡〈x′|e−β
Hˆ
|x〉. (12.2.5)Note thatHˆis the sum of two operatorsK(ˆp) andU(ˆx) that do not commute with each
other ([K(ˆp),U(ˆx)] 6 = 0). Consequently, the operator exp(−βHˆ) cannot be evaluated
straightforwardly. However, as we did in Section 3.10, we can exploitthe Trotter
theorem (see eqn. (3.10.18) and Appendix C) to express the operator as
e−β(Kˆ+Uˆ)= lim
P→∞[
e−βU/ˆ^2 Pe−βK/Pˆ e−βU/ˆ^2 P]P
. (12.2.6)
Substituting eqn. (12.2.6) into eqn. (12.2.5) yields
ρ(x,x′;β) = lim
P→∞
〈x′|[
e−β
U/ˆ 2 P
e−β
K/Pˆ
e−βU/ˆ 2 P]P
|x〉. (12.2.7)Let us now define an operatorΩ byˆ
Ω = eˆ −βU/ˆ^2 Pe−βK/Pˆ e−βU/ˆ^2 P. (12.2.8)