464 The Feynman path integral
=
∫x(t)=x′x(0)=xDx(s) exp{
i
̄h∫t0dsL(x(s),x ̇(s))}
=
∫x′xDxeiA[x]/ ̄h. (12.4.7)At this point, several comments are in order. Eqn. (12.4.7) revealsthat the functional
integral is truly an integral over all pathsx(s) that begin atxats= 0 and end at
x′ats=twith a weight exp(iA[x]/ ̄h) assigned to each path. The integral over paths
is illustrated in Fig. 12.8. The last line in eqn. (12.4.7) implies that the time label
x(s)(^0) t
x
x’
s
Fig. 12.8Representative continuous paths in the path integral for the quantum propagator
in eqn. (12.4.7).
sis irrelevant, since the propagatorU(x,x′;t) only depends on the endpoints of the
paths and the timetassociated with paths. Similarly, the actionA[x] is a function
only ofx,x′, andt,A(x,x′;t). Consequently, the path integral sometimes appears in
the literature as
U(x,x′;t) =
∫x(t)=x′
x(0)=x
Dx(·) eiA[x(·)]/ ̄h (12.4.8)
to indicate that the symbol used as the integration variable in the action integral
is irrelevant. Finally, we point out that eqn. (12.4.7) is exactly equivalent to eqn.
(12.2.23); the former is only a symbolic representation of the latter. The functional
integral notation provides a convenient and compact way of representing the more
complicated discrete path-integral expressions such as eqn. (12.2.23). Nevertheless,
as we will see shortly, functional integrals can be directly manipulated and used for
analytical calculations involving path integrals. Hence, the functional integral notation
serves both a notational and a practical purpose.
Eqn. (12.4.7) contains some fascinating physical content. First, the weight factor
exp(iA[x]/ ̄h) implies that in the space of all possible pathsx(s),x(0) =x,x(t) =x′,
the most important regions of the “path space” are those for which the action changes
very little upon moving from one path to another. Indeed, when thevariation inAis