Question

Although this question is going to seem completely trivial to anyone with any exposure to path integrals, I'm looking to answer this precisely and haven't been able to find any materials after looking for about 40 minutes, which leads me to believe that it makes sense to ask the question here. In particular I'm looking for an answer wherin any mathematical claims are phrased as precisely as possible, with detailed proofs either provided or referenced. Also my search for a solution has led me to think I'm actually looking for a good explenation of Wick rotation, which I can't really claim to understand in detail. Any good references about this would be very welcome as well.

I'm looking to make sense of the following integral identity:
∫[infinity]−[infinity]dx exp(ia2x2+iJx)=(2πia)1/2exp(−iJ22a),a,J∈R Wikipedia (and various other sources) say that "This result is valid as an integration in the complex plane as long as a has a positive imaginary part." Clearly the left hand side does not exist in Lebesgue sense for real a,J. An answer to the question "Wick rotation in field theory - rigorous justification?" claims:

"it is convergent as a Riemann integral, thanks to some rather delicate cancellations. To make the integral well defined -- equivalently to see how these cancellations occur -- we need to supply some additional information. Wick rotation provides a way of doing this. You observe that the left hand side is analytic in t , and that the right hand side is well-defined if Im(t)<0. Then you can define the integral for real t by saying that it's analytic continued from complex t with negative imaginary part."

I want to see the gory details and all known motivation for the validity of this procedure for the kinds of applications where such integrals occur. Suggestions such as "include an iϵ
in order to make it finite" seem arbitrary. In that case one would have to motivate that prescription from the very start, that is within the modeling procedure that ends up giving that integral expression (which is likely the correct way to approach this). I'm also not sure how to interpret the right hand side, since it involves the square root of an imaginary number, which should involve some choice of branch cut, which I have never seen specified in connection to this formula.

Answer 1

The integral identity with exponential functions involves concepts like complex analysis and Wick rotation in quantum field theory, with a need for careful mathematical justification and selection of a branch cut for square roots of complex numbers.

Step-by-step explanation:

The integral identity in question is an example of a Gaussian integral, which is encountered frequently in quantum field theory and statistical mechanics. The use of imaginary numbers is fundamental in quantum mechanics, as indicated by wave functions like Y(x, t) = Aei(kx-wt), where the imaginary unit i = √-1 ensures compliance with the Schrödinger equation. Real functions such as sine and cosine waves cannot satisfy the Schrödinger equation; thus, the complex exponential form of the wave function is required.

To understand the integral mentioned and its issues with convergence, consider Wick rotation, which is a technique used to redefine the time coordinate in a complex plane to improve the behavior of the integral. By rotating the time axis into the complex plane, we analytically continue our functions to make certain integrals converge where they otherwise would not. However, this is not a trivial process and requires careful justification. The appearance of iε (where ε is a small positive value) in the prescriptions is to ensure the convergence of the integral and stems from the need to modify the physical problem in a way that makes mathematical sense.

The right-hand side of the integral identity involves the square root of a complex number, which indeed requires one to choose a branch cut to define the function properly. Physicists conventionally take the principal branch where the square root has a non-negative real part to maintain consistency with physical predictions.

For detailed proofs and deeper understanding, I recommend studying complex analysis, particularly residue calculus, and textbooks on quantum field theory, where such techniques are elucidated with mathematical rigor.