"Trace" of a distribution?

Question

In quantum field theory we have to sometimes take the "trace" of a distribution $M(x,y)$, $\text{tr}M\sim\int dx M(x,x)$. This happens for instance when we try to expand the determinant of a Dirac operator $\det{\not}D=\exp\text{tr}\log{\not}D.$

These traces are not well defined mathematically, because they are divergent. A simple example would be looking at the distribution $f(x)\delta(x-y)$, whose trace is taken to be $\delta(0)\int dx f(x)$. Somehow, physicists are able to get away with these infinite $\delta(0)$ factors and still get something sensible. In the cases I have seen, these factors are usually assumed to be related to the volume of space, taken to be infinite.

Since these objects are not of trace class, these calculations are not guaranteed to make sense. A trace is supposed to have the same value regardless of what basis you calculate it in, so is the trace of $M(x,y)$ in position space the same as it is in momentum space?

For that matter, how does one take the trace of a distribution like $f(x)\partial_{\mu}\delta(x-y)$, or an arbitrary number of derivatives acting on a delta function?

Mathematicians seem to think there is no way of doing it, but physicists have found ways of getting sensible results! There must be some way of defining it properly, right?

Answer 1

Yes, in some cases the trace is related to the volume of the space. However, it would be more appropriate to say that it is related to the cardinality of the space, in other words, the number of modes in the space. People usually get around the problem of infinite cardinality by reducing the number of modes to a finite number. This is done with the aid of some form of regularization. The quantities that would have been infinite then become finite and one can proceed to perform the calculation. In the final result, these quantities would cancel out if the result represents a physical quantity. If they don't then it usually means that there is something wrong. After they cancelled, one can undo the regularization that would have made these quantities infinite. Since they are gone, the result remains finite.

What all this really means is that even if you are working in a space where some of these traces would diverge, the results of calculations for physical quantities that one would for instance get from measurements would always be finite. The question is just how to perform the calculation with these infinities so that they would cancel. The reason why it is difficult to work with infinities is because they obey cardinal arithmetic. Regularization converts them into ordinal numbers so that one can handle them in the usual way with ordinal arithmetic. Then, after they cancelled out one can take the necessary limit to remove the regularization.

Hope it helps.