One can try and properly normalize things so that this limit becomes an integral
∫
Dγ e
~iS[γ]
(35.3)
where now the pathsγ(t) are paths in the position space.
An especially attractive aspect of this expression is that it provides a simple
understanding of how classical behavior emerges in the classical limit as~→0.
The stationary phase approximation method for oscillatory integrals says that,
for a functionfwith a single critical point atx=xc(i.e.,f′(xc) = 0) and for
a small parameter, one has
1
√
i 2 π∫+∞
−∞dx eif/=1
√
f′′(c)eif(xc)/(1 +O())Using the same principle for the infinite dimensional path integral, withf=S
the action functional on paths, and=~, one finds that for~→0 the path
integral will simplify to something that just depends on the classical trajectory,
since by the principle of least action, this is the critical point ofS.
Such position-space path integrals do not have the problems of principle of
phase space path integrals coming from the Groenewold-van Hove theorem, but
they still have serious analytical problems since they involve an attempt to in-
tegrate a wildly oscillating phase over an infinite dimensional space. Away from
the limit~→0, it is not clear that whatever results one gets will be indepen-
dent of the details of how one takes the limit to define the infinite dimensional
integral, or that one will naturally get a unitary result for the time evolution
operator.
One method for making path integrals better defined is an analytic contin-
uation in the time variable, as discussed in section 12.5 for the case of a free
particle. In such a free particle case, replacing the use of equation 12.9 by equa-
tion 12.8 in the definition of the position space path integral, one finds that
this leads to a well-defined measure on paths, Wiener measure. More generally,
Wiener measure techniques can be used to define the path integral when the po-
tential energy is non-zero, getting results that ultimately need to be analytically
continued back to the physical time variable.
35.4 Advantages and disadvantages of the path integral
In summary, the path integral method has the following advantages:
- An intuitive picture of the classical limit and a calculational method for
“semi-classical” effects (quantum effects at small~). - Calculations for free particles or potentialsV at most quadratic inqcan
be done just using Gaussian integrals, and these are relatively easy to eval-
uate and make sense of, despite the infinite dimensionality of the space