Functional integrals 465small, then the complex exponential oscillates very slowly in moving from path to path.
Without frequent sign changes in exp(iA[x]/ ̄h), the paths in this region of path space
contribute significantly to the integral. On the other hand, there are other regions
of path space for whichAvaries significantly in moving from one path to another.
In this case, the exponential oscillates wildly, and the paths contribute negligibly
to the path integral because contributions from closely spaced paths tend to cancel
each other. More concretely, if we consider a pathx(s) and a slightly different path
x ̃(s) =x(s)+δx(s), whereδx(s) is a small variation inx(s), then these two paths will
contribute significantly to the path integral ifδA≡A[x+δx]−A[x] is small. This
condition is satisfied in a region whereA[x] is flat, that is whereδA/δx(s)≈0. Note
that sincex(0) =xandx(t) =x′, it follows thatδx(0) =δx(t) = 0. Indeed, the most
significant contribution occurs whenδA= 0. However, recall from Section 1.8 that the
conditionδA= 0 is precisely the condition that leads to the Euler–Lagrange equation
for the classical path:
δA= 0 ⇒d
ds(
∂L
∂x ̇(s))
−
∂L
∂x(s)= 0. (12.4.9)
ForL= (m/2) ̇x^2 (s)−U(x(s)), the equation of motion is the usual Newtonian form
m
d^2 x
ds^2=−
∂U
∂x. (12.4.10)
We see, therefore, that the action integral and the principle of action extremization
emerge naturally from the path integral formulation of quantum mechanics. This re-
markable fact tells us that the most important contribution to the path integral is the
region of path space around a classical path. The importance of paths that deviate
from classical paths depends on the extent to which quantum effects dominate in a
given system. For example, when a process occurs via quantum tunneling, paths that
deviate considerably from classical paths have a significant contribution to the path
integral since tunneling is a classically forbidden phenomenon. In other cases, where
quantum effects are less important but not negligible, it may be possible to compute
a path integral to a reasonable level of accuracy by performing anexpansion about a
classical path and working to a low order in the “quantum corrections.” This popular
approach is the basis ofsemiclassical methodsfor quantum dynamics. Finally, we note
that the solution to the Euler–Lagrange equation with endpoint conditionsx(0) =x
andx(t) =x′may not be unique. We noted in Section 1.8 that the solution of the
Euler–Lagrange equation subject to initial values forxand ̇xis unique; but as the
path integral requires that the paths satisfy endpoint conditions, there are contribu-
tion from regions in path space aroundeachclassical path satisfying the endpoint
conditions.
Eqn. (12.4.7) represents an integral over continuous real-time paths for the quan-
tum mechanical propagator. However, due to eqn. (12.2.4), we may perform a Wick
rotation and obtain functional integral expressions for the quantum mechanical den-
sity matrix and partition function in the canonical ensemble. This Wickrotation is
performed by substitutingt=−iβ ̄hinto eqn. (12.4.7). Let the paths now be parame-
terized by a variableτrelated tosbyτ=is. Whens=t=−iβ ̄h,τ=β ̄h, and the
action integral becomes