The God has created a man in order that he creates that the God fails to do

Sunday, 2 June 2013

Quantum field theory: generating functionals as a measure (from my Scientific Biography)

Third period (1978 - 1990) ...

The fact that a gravitational field by its physical nature is a Higgs field, drew my attention to a general problem of description of a Higgs vacuum. In united models of fundamental interactions, its occurrence is regarded as a kind of phase transition at a certain energy (temperature). I made various attempts to approach this problem [37,62], in particular, developed an idea that a Higgs vacuum is a source of a Higgs gravitational field [51]. However, any substantive theory of a Higgs vacuum still not constructed.

Working on the problem of Higgs vacuum, I met the fact that, in general, there is no mathematically correct formulation of quantum field theory, either axiomatic or in the form of the perturbation theory. The latter is formulated in terms of the so-called functional integrals, but they are not any true integrals, and their properties are defined by analogy with those on finite-dimensional spaces in order to reproduce Feynman diagrams. Pursuing the well-known GNS construction in quantum theory, I knew its expansion to unnormed involutive algebras and, in particular, to an algebra of free fields represented by probe (rapidly decreasing at infinity) functions. These functions form a nuclear space S. Continuous forms of this algebra are generalized functions that make up the dual space S'. A problem was with a description of a system of interacting fields, appeared and disappeared in some instants of time. Such a system is characterized by chronological forms on a space S, but they are not continuous. However, I have found that, after a Wick rotation to Euclidean fields, their chronological form (Euclidean Green function) are continuous. Moreover, they are derived from a generating functional which, by virtue of the well-known theorem, is a measure on a space of generalized functions S'. This construction provides a good mathematical basis of quantum field theory [54,59]. It was extended to fermion fields [57]. However, a problem lies in the fact that it is impossible to write these measures in an explicit form, with the exception of Gaussian measures for fields without interaction. In addition, properties of these measures differ from those adopted for functional integrals in perturbation quantum field theory. For example, there is no translationally invariant Lebesgue measure on S', and I even tried to use this fact for describing a Higgs vacuum [62].

Later I repeatedly returned to attempts to build a generating functional of quantum field theory as a measure, but so far unsuccessfully. However, in the language of measures I described nonequivalent representations of algebras of canonical commutation relations, modeled on nuclear spaces [105].


G. Sardanashvily, True functional integrals in Algebraic Quantum Field Theory, arXiv: hep-th/9410107

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