Let me enter the arena with the statement that there is a basic problem with the Wightman-Gårding system. To put it baldly the central concept, that of an observable- valued distribution, is defined(implicitly) in a way that is simultaneously too strong (leads to difficulties in constructing models) and too week (difficulties in proving theorems). At the heart of quantum theory lies the fact that the observables are represented by unbounded s.a. or normal operators. Since these play the role of the real and complex numbers in this context, it would seem to be imperative to develop a parallel theory there, including the basic ideas of real and complex analysis—regularity conditions (continuity, measurability, smoothness) of observable-valed functions and functions between observables, differentiation of such functions, series expansions (Taylor, Hermite, Fourier) and, most importantly for QFT, observable-valued distributions. It seems that I need to emphasise that these are a priori purely mathematical questions—in my opinion, highly interesting ones, completely independent of any relations to physics of any sort. Of course, the fact that there are such relations (not just in quantum theory, by the way) increases this interest.

The basic mathematical structures required for a rigorous development of QFT are the concepts of holomorphic functions and distributions with values in in the space of observables, i.e., the self-adjoint, resp., normal unbounded operators on a (say, separable) Hilbert space. I am arguing that a major stumbling block for a construction of the former is the lack of a suitable definition of these concepts.

Before proceeding to the constructive part of my posting, let me note that the Wightman-Gårding axioms implicitly give a definition of observable-valued distributions which assumes that all of the operators which occur have the same domain of definition. I would claim that this is much too strong as simple examples show—consider (in the holomorphic case) the family $T(z)=(z^n)$. This can be regarded as a family of complex sequences and hence of normal operators (by multiplication on $\ell^2$) dependent on $z$. It should plainly be, by definition, an observable-valued entire function but fails the stringent conditions imposed by the axioms on the domains of definition. A further example is the function $z\mapsto I+zA$ for an unbounded normal $A$. This should plainly be holomorphic but fails the condition on the domains of definition.

The axioms are often stated in a weaker form, demanding only that there be a dense subspace contained in the domains of the operators
but this is hopelessly inadequate—if you know a s.a. operator on a dense subspace, you know nothing. Think of the Laplace operator on some domain in euclidean space regarded as an operator on the smooth functions with compact support. Without boundary conditions, and there are many candidates, this is not s.a. In fact, this operator defined, say, on the test functions has uncountably many self adjoint extensions and if we label them with a complex variable $z$ we get a wildly pathological observable which is holomorphic since it is constant on a dense subspace.

We remark that we have displayed these functions in the context of holomorphicity for simplicity but the ideas apply equally in other contexts.

Given the inadequacy of this definition, it can be no surprise that there are no indications in the literature of a general theory of observable-valued distributions or holomorphic functions (the usual basic things—functoriality, differentiation, series developments—Taylor or Hermite, for example, and so on). I am prepared to place my head on the block and claim that there never can be such a mathematically rigorous theory based on the above definition.

Now for the positive remarks. I will concentrate on the case of holomorphic functions since this is the most transparent one but many other spaces of functions and distributions can be handled analogously. We begin with the remark that the extension of the concepts of smoothness of functions to the vector case (Banach spaces, locally convex spaces, even topological vector spaces, although the latter is rather delicate) was carried out over 50 years ago, for example by Schwartz and Grothendieck). in carrying this over the the case of observables, there are a number of stmbling blocks—the natural ingredients—topological and algebraic structures (addition and multiplication of observables) are no longer available, or rather they are but in much more delicate forms). The starting point is that the space of observables has a natural topology for which it is a polish space.

If $U$ is a complex domain (to be specific), the we denote by $H(U)$ the space of holomorphic functions on $U$ and its dual by $H(U)‘$. It helps, but is not essential, that the former is a nuclear Fréchet space and the latter has a concrete representation as a space of holomorphic functions on the complement of $U$ in the Riemann sphere (Sebastião e Silva, Köthe).

Grothendieck extended the work of the latter by showing that the space of holomorphic functions on $U$ with values say in a Banach space $E$ can naturally be identified with $L(H(U)‘,E)$, the continuous linear operators on the dual into $E$.

This makes it natural to define the observable-valued holomorphic functions on $U$ to be the space of continuous linear operators from $H(U)‘$ into the space of observables. Since the latter is not a tvs (or even a vector space), the latter requires some explication. The continuity creates no problems (the space of observables has a topology). Linearity uses the rather subtle concept of addition for unbounded s.a. or normal operators—for which see the standard literature, e.g., Reed and Simon. We say that a mapping with values in the space of observables is linear if $T(\alpha f)=\alpha T(f)$ for all $\alpha$ and $T$ and for each $f_1$ and $f_2$ the operators $T(f_1)$ and $T(f_2)$ have a sum, with the usual equality holding.

In the case of distributions, say tempered—the ones most used by physicists, the same definition applies, using operators on the Fréchet space of rapidly decreasing smooth functions.

We remark briefly that these spaces have the basic properties that one uses , e.g., for constructing and computing with models, in perturbation arguments, etc. and which fail under the Wightman-Gårding axioms.

Tho holomorphic functions contain the classical ones (i.e., with values in the bounded operators), they can be differentiated (I emphasise—not true in W-G—who ever heard of not being able to differentiate holomorphic functions?), have Taylor expansions and are functorial property in dependence of the domain of definition (which means that symmetry groups there lift to the fields).

With regard to distributions, similar remarks hold—they can de differentiated, expanded in terms of the Hermite basis, Fourier transformed, symmetries of the underlying euclidean space operate on them and they contain, as a special case, the classical ones (i.e., with bounded operators as values).

Since this posting has become uncomfortably long, I will close with the fact that these concepts have the properties which one would reasonably expect from a theory of observable-valued holomorphic functions or distributions and so could serve as the basis for a more rigorous approach to QFT.

Some quotes:

Kazhdan: Physics is very interesting. There are many, many interesting theorems. Unfortunately, there are no definitions.

Zeidler. In mathematics one never does calculations with quantities that do not exist.

Jost: The fact that intuition barely can guide us forces us to use standards of rigour usually frowned on in theoretical physics.

3more comments