nLab operator-valued distribution

Idea

An operator valued distribution is a distribution with values in a space of linear operators.

In mathematically rigorous constructions of perturbative quantum field theory – such as via the Wightman axioms or causal perturbation theory – the quantum observables which naively would be represented by would-be linear operators on some Hilbert space assigned to single points of spacetime, such as the field operator “$\Phi(x)$” in scalar field theory, are understood to yield such operators only after “smearing” or “adiabatic switching”, namely after “integrating” them against a bump function on spacetime. This realizes these objects as continuous linear functionals on the space of bump functions with values in linear operators, and hence as distributions with values not in the real numbers (as for ordinary distributions) but with values in operators. Where an ordinary distribution may be thought of as a “generalized function with values in the real numbers”, such an “operator valued distribution” may be thought of as a “generalized function with values in linear operators”. For example the object denoted $\mathbf{\Phi}(x)$ is understood as really being a distribution which to a bump function $b$ assigns the (well-defined) operator traditionally written $\int_X b(x) \mathbf{\Phi}(x) dvol$.

In causal perturbation theory/locally covariant perturbative AQFT these operator-valued distributions are just a means to an end: the actual algebra of observables is the Wick algebra of the free fields and its deformation to the interacting field algebra, both of which are (Fedosov-)formal deformation quantizations of algebras of ordinary smooth functions on phase space. But since the phase space in field theory is itself a mapping space (for the scalar field) or more generally a space of sections, such functions are conveniently expressed or constructed in terms of distributions.

Indeed the realization of these algebras via distributions makes transparent that

1. the restriction of the free field‘s Wick algebra to microcausal functionals is governed by the wave front set condition for the multiplication of distributions;

2. the renormalization freedom in defining the S-matrix/interacting field algebra is governed by the freedom of performing point-extensions of distributions .

This way the theory of distributions serves to give a mathematically rigorous construction of perturbative quantum field theory. The notorious “infinities” that allegedly “plague” perturbative quantum field theory in other approaches may be argued (vividly so in Scharf 95, p. 181-182) to be just mathematical errors in handling distributions properly.

Incidentally, in the context of causal perturbation theory/locally covariant perturbative AQFT the “operator-valued distributions” are in general just algebra-valued distributions, since a representation of these algebra elements (quantum observables) as actual linear operators on some Hilbert space is not required, in fact it may not generally exists in a suitably invariant sense on on curved spacetimes and if it exists it will not be unique, and even if a choice is made, most quantities of physical interest may be computed more directly from the algebra and a algebraic choice of state.

Definition

Let $V$ be a topological vector space and $X$ a smooth manifold. A $V$-valued distribution on $X$ is a continuous linear functional from the space $C^\infty_c(X)$ of bump functions to $V$.

If instead of $C^\infty_c(X)$ one uses the space $C^\infty(X)$ of all smooth functions then these are compactly supported distributions, or if one uses the Schwartz space $\mathcal{S}(X)$, then these are tempered distributions.

References

An early use of the concept of operator-valued distributions in perturbative quantum field theory is in the foundations of causal perturbation theory in

• Henri Epstein, Vladimir Glaser, The Role of locality in perturbation theory, Annales Poincaré Phys. Theor. A 19 (1973) 211.

• Nikolay Bogoliubov, A. A. Logunov, A. I. Oksak, I. T. Todorov, General principles of quantum field theory, Kluwer 1990

For review see

• Pierre Ca Grange, Ernst Werner, Quantum fields as operator valued distributions and causality (arXiv:math-ph/0612011)

• Günter Scharf, Finite Quantum Electrodynamics – The Causal Approach, Berlin: Springer-Verlag, 1995, 2nd edition

• Günter Scharf, Quantum Gauge Theories – A True Ghost Story, Wiley 2001