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” efects (quantum efects at small ℏ).
• Calculations for free particles or potentials V at most quadratic in can be done just using Gaussian integrals, and these are relatively easy to evaluate and make sense of, despite the infinite dimensionality of the space of paths. For higher order terms in , one can get a series expansion by expanding out the exponential, giving terms that are moments of Gaussians so can be evaluated exactly.
• After analytical continuation, path integrals can be rigorously defined using Wiener measure techniques, and often evaluated numerically even in cases where no exact solution is known.
On the other hand, there are disadvantages:
• Some path integrals such as phase space path integrals do not at all have the properties one might expect for an integral, so great care is required in any use of them.
• How to get unitary results can be quite unclear. The analytic continuation necessary to make path integrals well-defined can make their physical interpretation obscure.
• Symmetries with their origin in symmetries of phase space that aren’t just symmetries of configuration space are dificult to see using the configuration space path integral, with the harmonic oscillator providing a good example. Such symmetries can be seen using the phase space path integral, but this is not reliable.
Path integrals for anticommuting variables can also be defined by analogy with the bosonic case, using the notion of fermionic integration discussed earlier. Such fermionic path integrals will usually be analogs of the phase space path integral, but in the fermionic case there are no points and no problem of continuity of paths. In this case the “integral” is not really an integral, but rather an algebraic operation with some of the same properties, and it will not obviously sufer from the same problems as the phase space path integral.
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正文：英文 · OCR 机器稿 · 待校对
核对状态：OCR 机器稿 · 待校对
原书 PDF · 印刷页 368、369、370、371、372、373、374、375、376、377、378
来源版本：2025-10-20
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