We introduce a concept of asymptotic principal values which enables us to handle rigorously singular integrals of higher-order poles encountered in the computation of various quantities based on correlation functions of a vacuum. Several theorems on asymptotic principal values are proved, and they are expected to become bases for investigating and developing some classes of regularization methods for singular integrals. We make use of these theorems for analyzing mutual relations between some regularization methods, including a method naturally derived from asymptotic principal values. It turns out that the concept of asymptotic principal values and the theorems for them are quite useful in this type of analysis, providing a suitable language to describe what is discarded and what is retained in each regularization method.

Physics of quantum vacuum fluctuations is one of the intriguing research topics expected to be developed through the interplay between theories, experiments, and practical applications.

Investigations of quantum vacuum fluctuations even stimulate the border area between physics and mathematics. As a typical example of this sort, we often encounter singular integrals in computing several quantities based on correlation functions of a vacuum in question. The occurrence of singularity or divergence is often a signal of surpassing the border of validity of a model by too much extrapolation. Furthermore, it could originate from deeper physical processes for which satisfactory consistent mathematics is still unavailable. How to handle singular integrals can be then a challenging topic, requiring both mathematical analysis and physical considerations.

Faced with singular integrals, we need to resort to some regularization method to get a finite result. The aim of this paper is to give an organized mathematical basis underlying some typical regularization methods and make clear their mutual relations. We introduce below a concept of

There are still various uncertainties to clear up in regularization methods, reflecting our lack of mathematical basis for handling infinities. In this situation, we cannot expect any universal regularization method, but we need to customize the method by try and error depending on the problem in question. It is far from the aim of this paper to judge which method is better than the others. It just tries to present a concrete mathematical basis for further considerations and developments of better regularization methods.

The organization of the paper is as follows. In Section

Let us consider as an example the measurement of the electromagnetic vacuum fluctuations in a half-space bounded by a perfectly reflecting infinite mirror. Recently the switching effect [

We thus take a model introduced in [

When the velocity of the probe-particle is much smaller than the light velocity

Within the time-period when the particle does not move so much, (

For simplicity, let us consider only the “sudden-switching” case; the measurement is switched on abruptly, stably continued for

It is obvious that the integral in (

For the present purpose, it suffices to show only the result of

This result is derived by a formula for an

Leaving the rigorous treatment of singular integrals for the next section, we here focus on the physical reason why the singularity of correlation functions occurs at

On the other hand, typical regularization procedures [

It turns out that the model given here is too simplified and should be modified taking into account the switching effect [

In view of the example in the previous section, it is clear that we sometimes need to estimate a singular integral whose integrand possesses a higher-order pole. In order to investigate various regularization methods later, we first need some concrete quantity corresponding to a singular integral for which all the information is retained and nothing is discarded. Then, the following asymptotic definition of a singular integral may be relevant.

Let

The asymptotic principal value is a generalization of the standard Cauchy principal value, corresponding to

Let us now introduce another asymptotic quantity.

With the same premises as in

It is easily shown that

By a Taylor-expansion in

The following definition is just for making formulas below concise.

We have

With these preparations, let us start with the following lemma.

For any function

Noting that

We now prove a formula which relates a multipole integral with a simple-pole integral.

For any function

(

(

(

Based on Theorem

We define

The “mild part”

We now have a formula which enables us to separate singular contributions from a multipole integral.

For any function

It is straightforward to show this formula due to Theorem

For any function

We compute

For any function

Due to Theorem

(

(

(

Based on the results in the previous section, let us now come back to the problem of regularization methods for singular integrals.

Let us consider a typical singular integral

The first method of regularization we consider is a

Due to Theorem

There is still room, however, to regard the method of partial integrals as a shorthand prescription of what we here call the

The method of infinitesimal imaginary part is based on well-known Dirac’s formula [

By differentiating the both-sides of (

It is notable that just the introduction of some infinitesimal imaginary part results in a tamable quantity such as

As far as one is evaluating

Another way of looking at this method is to pay attention to the L.H.S. (rather than the R.H.S.) of (

There is some subtle points in this method. One of them is to discard the imaginary part of the R.H.S. of (

Indeed, a simple example can be presented for which this kind of procedure fails. Let us consider an integral

On the other hand, the above-mentioned scheme makes a replacement

Quite interestingly, no contradiction occurs for

With these caveats in mind, let us now move to a new regularization method based on the asymptotic principal values.

Let us finally introduce a new regularization method based on the asymptotic principal values.

For the simple example in Section

Going back to the example of the integral

We see that Theorem

In this paper, we have focussed on singular integrals with a higher-order pole which frequently emerge in computing quantities based on two-point correlation functions of a vacuum.

To deal with this type of singular integrals, we have introduced the concept of

We have then proved several theorems on asymptotic principal values which are expected to serve as bases for studying regularization methods for singular integrals.

To see how asymptotic principal values can be made use of, we have selected three typical regularization methods and have analyzed their mutual relations with the help of theorems we have prepared. It has turned out that the concept of asymptotic principal values and related theorems are quite useful in this kind of analysis. Indeed, in terms of asymptotic principal values, it has been possible to describe without ambiguity what is discarded and what is retained in each regularization method.

No universal regularization method is available so far and we need to carefully select or invent a suitable method depending on the problem in question. For instance, we recall the example in Section

In any case it is significant to compare the results derived by different regularization methods in more detail for approaching to a more satisfactory mathematical theory of regularization procedures.

The author would like to thank C. H. Wu for various discussions.