Model P(φ)4 quantum field theory: A nonstandard approach based on nonstandard pointwise-defined quantum fields

What is it about?

In mathematical physics, constructive quantum field theory is the field devoted to showing that quantum field theory can be defined in terms of precise mathematical structures. This demonstration requires new mathematics beyond classical functional analisys end classical internal set thery namely IST. The traditional basis of constructive quantum field theory is the set of Wightman axioms. Osterwalder and Schrader showed that there is an equivalent problem in mathematical probability theory. The examples with d < 4 satisfy the Wightman axioms as well as the Osterwalder–Schrader axioms. They also fall in the related framework introduced by Haag and Kastler, called algebraic quantum field theory.Verification Haag and Kastler axioms in physical dimension d=4 for self interacting scalar quantum fields that is more then 50 years unsolved problem.

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Why is it important?

A new non-Archimedean approach to interacted quantum fields is presented. In proposed approach, a field operator φ(x,t) no longer a standard tempered operator-valued distribution, but a non-classical operator-valued function. We prove using this novel approach that the quantum field theory with Hamiltonian P(φ)_4 exists and that the corresponding C^*­ algebra of bounded observables satisfies all the Haag-Kastler axioms except Lorentz covariance. We prove that the λ(φ^4 )_4 quantum field theory model is Lorentz covariant.