Zeta-function regularization in QFT for heat kernels

Question

When one is doing zeta-function regularization of the heat-kernel for QFT then one is doing these following steps,

• the integral over the imaginary time
• taking the trace of the heat-kernel or the short-distance limit
• the space-time volume integral

I would like to know as to what justifies the sequence in which these steps need to be done. I have generally seen it being done in the order as listed above but its more or less clear from working with examples that the answer clearly depends on which order on does it.

Or is the art of regularization about being able to choose the "right" sequence depending on the situation? But is there an argument as to why there should be only one particular sequence which will generate a finite answer?

Also in some examples it seems that the order needs to be changed whether one is regularizing the zero-modes of the theory or not. But that doesn't seem consistent! That would effectively mean doing a different regularization for different terms of the same expression!

Answer 1

Zeta-function regularization can be thought of as analytic regularization with a special choice of the subtraction scheme. Like any other regularization, there are going to be possible ambiguities that unless treated consistently across a calculation will make the results of a naive/minimal subtraction result incorrect. However, these ambiguities should always be able to be accounted for by a finite counter-terms.¹
So the art of regularization is in setting up a consistent subtraction scheme -- either by using a consistent minimal subtraction, using renormalization conditions, an R-operation, or by consistently using some implicit subtraction like zeta-function regularization. However, as you noted in your question, the latter option is not always so easy.

The zeta-function regularization approach to one-loop QFT calculations comes from the observation that $\partial_t H^{-t}|_{t\to0} = -\log(H)$. Then² $$\begin{align} \log\det(H) &= \mathrm{tr}\log H := -\zeta'_H(0)\,, \\ \zeta_H(s) &= \mathrm{tr} H^{-s} = \frac1{\Gamma(s)}\int_0^\infty\mathrm{d}t\, t^{s-1}\mathrm{tr}(\exp(-H t)) \end{align}$$ The heat kernel is $K(x,x'|t) = \exp(-H t)\delta(x,x')$ and taking its trace involves setting $x' \to x$ and integrating over all spacetime $x$ (and taking the trace over any group or flavour indices). Often the spacetime integral is not performed as you want the answer as an effective action.
$\zeta_H$ is called the zeta function of $H$ since $$ \zeta_H(s) = \mathrm{tr} H^{-s} = \sum_{n} \lambda_n^{-s} \,, $$ where the $\lambda_n$ are the eigenvalues of $H$. $\zeta_H(s)$ is basically just the analytically regularized one-loop integral and you could just as easily expand it in powers of $s$ to extract the divergent part ($s^{-1}$), the finite part ($s^0$) and the terms that vanish as $s\to0$. Zeta-function regularization just returns the $s^0$ part and throws away the rest - as you noted in your question, this is not always guaranteed to work in all cases and all order of operations. There is no reason to think that the implicit subtraction scheme of zeta-function regularization is better than any other subtraction scheme for analytic regularization.

Normally the coincidence limit $x'\to x$ is done before the propertime integral, since it makes things simpler. Also, the heat kernel is often calculated via momentum space and then it is possible to leave the momentum integral until after the propertime integral - this means you never have a position space expression for the heat kernel, but it can also make calculations simpler. If different results are obtained by different operation orders, then there is some sort of conditional convergence that is not fixed by your regularization scheme.

The result of a zeta-function regularized calculation or any other renormalized QFT calculation should not be taken as correct unless it satisfies a sensible choice of physically motivated renormalization conditions (and Ward identities etc...). Any other subtraction scheme is merely a convenient inbetween result.

Finally, as noted in https://physics.stackexchange.com/a/13045/429, the trick of writing $\log H$ as the derivative of some $H^{-n}$ is not unique. This non-uniqueness can be used to parameterize the ambiguity of zeta-regularization so that different methods can be compared and renormalization conditions more easily enforced. The inadequateness of the naive "zeta-function" regularization of heat kernels becomes clear in higher-loop calculations.

_{1. That said, I've done calculations where the ambiguity arises in a finite (higher-mass dimension) term that is not present in the classical action nor amenable to correction by any renormalizable counter-term. This ambiguity, being in a finite term, comes from conditional convergence in the one-loop integrals in either their unregularized or regularized forms. In such cases, I am not sure how to deal with the ambiguities...}

_{2. All equality signs are to be taken with a pinch of salt. They depend on the regularization and renormalization scheme etc...}

Answer 2

Whenever we need to evaluate the Trace of the Heat Kernel .. we can use the WKB approximation for the Theta function = Trace of the Heat kernel, namely for the one dimensional model

$ Trace(exp(-tH))= \sum _{n=0}^{\infty}exp(-tE_{n})= \iint_{R}dpdxexp(-tp^{2}-tV(x)) $