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Path integrals

Another application of imaginary time is in Feynman’s path integral approach to QM [Feynman, 1948]. This was originally formulated in real time and is used for non-relativistic Schrodinger wave mechanics and relativistic quantum field theory. For example, in Schrodinger mechanics, our interest is often the amplitude (y, tf | x, ti) for the particle to go from (t,?, x) to (tf, y). The path integral formulation expresses this as a sum

of sub-amplitudes, where the sum is taken over all possible continuous paths Г12,. . ., each of which runs from the initial spacetime event (ti, x) to the final spacetime event (tf, y). For each path Г, the corresponding Sr is calculated by the time integral of the Lagrangian over that path, that is,

Formally, the infinite sum (21.14) is written [Feynman & Hibbs, 1965] and interpreted as a functional integral.

The path integral does not specify which path the particle actually takes: according to Feynman’s prescription, all possible paths have to contribute to the amplitude. This is a direct realization of the principle of contextuality in QM that we have encountered before in this book: if we are not monitoring an SUO, we cannot say what it is doing, or exclude any accessible possibility.

In general, Feynman’s path integrals are not well-defined mathematically. This is related to the fact that, for real time, the arguments of the exponentials being integrated over in (21.16) are pure imaginary. One remedy has been the imaginary-time approach. In the 1920s, Norbert Wiener developed the theory of integration in function spaces, the so-called Wiener integral, in his study of

Brownian motion,[1] giving the technology to solve diffusion equations. In 1947, Kac realized that going to imaginary time in the Feynman path integral led to a Wiener integral, equivalent to turning the free particle real-time Schrodinger equation into the parabolic diffusion equation (21.7). This approach gives Euclidean (imaginary time) Green’s functions, discussed above for the harmonic oscillator.

The general strategy is to formulate a path integral in real time, rewrite it in imaginary time by performing a Wick rotation, calculate the Euclidean Green’s functions, and then perform a reverse Wick rotation back to real time. The real problem with this strategy is that it is not clear why this rigmarole should be necessary. Clearly, we do not really understand time.

  • [1] The diffusion of particles suspended in a liquid.
 
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