This review is devoted to measure theoretical methods in the canonical quantization of scalar field theories. We present in some detail the canonical quantization of the free scalar field. We study the measures associated with the free fields and present two characterizations of the support of these measures. The first characterization concerns local properties of the quantum fields, whereas for the second one we introduce a sequence of variables that test the field behaviour at large distances, thus allowing distinguishing between the typical quantum fields associated with different values of the mass.

The phase space of the classical scalar field is a linear space and therefore constitutes an infinite dimensional analogue of the usual phase space for classical dynamics of a finite number of particles, namely, the cotangent bundle

The theory of representations of the Weyl relations can be seen as a problem in measure theory in infinite dimensional linear spaces. Just as in finite dimensions, it would be natural to look for representations in spaces of square integrable functions with respect to some measure on the classical configuration space, which is a space of functions in

In the present review we discuss in detail the canonical quantization of the free massive scalar field, following [

The canonical quantization of field theories involves the introduction of a convenient measure in an infinite dimensional space. In the case of the real scalar field the appropriate measure space is the dual of the Schwartz space. Measures in this space allow the construction of representations of the Weyl relations which are of the Schrödinger type. In particular, the measure associated with the free field is a Gaussian measure.

Let

The Fourier transform of a measure

A complex function

The Fourier transform of a measure on

Let

Note that, given a Gaussian measure

We are particularly interested in the case where the covariance is defined by certain types of linear operators on

We will say that a linear operator

It is clear that the bilinear form

In the canonical quantization of real scalar field theories in

It is a well established fact [

Given a quasi-invariant measure

The following proposition gives necessary and sufficient conditions for the equivalence of cyclic representations. We introduce the Weyl operators

Two cyclic representations

Representations (

Let then

Let us consider the free scalar field of mass

The operator

(i) The differential operator

(ii) The inverse operator

(iii) The operators

The evolution equations (

Let

In this section we discuss the question of unitary implementation of linear canonical transformations in the context of Gaussian representations of the Weyl relations for the real scalar field. We follow [

Let

Let

We now define the group of linear symplectomorphisms that does admit a natural quantization for a given Gaussian representation. Let then

Consider the group of (linear) orthogonal symplectomorphisms, that is, the group

Let

To prove the theorem, let us start by showing uniqueness. Suppose that

In this section we present the quantization of the free real scalar field of mass

As we saw in Section

The quantization of the coordinate functions

As we saw, the representation can be continuously extended to all

The fundamental aspect of the representation

On

This result gives a quantization of the dynamics or a quantization

In this section we show that the measure

The Euclidean group

The measure

One thus has a unitary action

Given a probability space

It follows from (

The action (

In fact, by linearity and continuity, it is sufficient to verify (

The subgroup (

We will now consider the relativistic invariance properties of the free field quantization, showing explicitly how a covariant formulation can be obtained from the above canonical quantization. Although presented here in heuristic form, one can give a precise meaning to the results in this section (see [

Let us consider the time-dependent Weyl operators:

We will now present a characterization of the support of the free field measure

Let

Let us then apply Theorem

So, one can say that the Fourier transform of

Note that although the value of mass

Nevertheless, as mentioned in Section

It is well known that the free field measures

In the inverse covariance

In order to obtain our formal result, let us consider the measurable functions

The push-forward of the free field measure

The measures

To prove the lemma we will rely on Theorem I.23 of [

Let

This proposition can be proven as follows. Consider the measurable sets

Finally, by combining Lemma

The sets

The second statement of the theorem, which in particular allows the construction of disjoint supports for

Measures in infinite dimensional spaces, both linear and nonlinear, are of major importance in quantum theory. They appear already in standard Quantum Mechanics, in the path integral formulation, but are even more relevant in the quantum theory of fields, both from the Euclidean path integral and from the canonical quantization perspectives (see, e.g., [

In all known cases the distributional extensions are crucial, given that, for measures of interest, the classical nondistributional configurations turn out to be (subsets of) sets of measure zero.

The construction of measures corresponding to field theories with interactions turned out to be a problem of major complexity, in particular for spacetime dimensions above

It is well known that the representations associated with free fields with different values of the mass are not unitarily equivalent. This nonequivalence reflects the fact that the measures corresponding to distinct masses are supported in different—in fact disjoint—sets of

Finally, let us mention that the study of the support of the free field measures is important not only for a good understanding of these models but also from the broader perspective of the construction of quantum theories with interactions. In the usual procedure, the introduction of interactions in a free model is constrained by the properties of the “typical free fields”: the local (distributional) behaviour leads to the well known ultraviolet divergences, whereas the long range behaviour is related to infrared divergences [

The author declares that there is no conflict of interests regarding the publication of this paper.