Algebra of distributions of quantum-field densities and space-time properties

TL;DR

This paper explores how the algebraic structure of quantum-field densities influences space-time properties, revealing restrictions on densities and implications for phenomena like chirality violation and matter-antimatter asymmetry.

Contribution

It establishes the conditions under which the algebra of quantum-field densities exists and links these to space-time structure and fundamental symmetry violations.

Findings

Quantum-field densities form an algebra only in Lorentzian spacetime.

Restrictions on densities imply a one-dimensional arrow of time.

Symmetry violations in densities explain matter-antimatter imbalance.

Abstract

In this paper we consider properties of the space-time manifold M caused by the proposition that, according to the scheme theory, the manifold M is locally isomorphic to the spectrum of the algebra A, M = Spec(A), where A is the commutative algebra of distributions of quantum-field densities. In order to determine the algebra A, it is necessary to define multiplication on densities and to eliminate those densities, which cannot be multiplied. This leads to essential restrictions imposed on densities and on space-time properties. It is found that the only possible case, when the commutative algebra A exists, is the case, when the quantum fields are in the spacetime manifold M with the structure group SO(3,1) (Lorentz group). The algebra A consists of distributions of densities with singularities in the closed future light cone subset. On account of the local isomorphism M = Spec(A), the quantum fields exist only in the space-time manifold with the one-dimensional arrow of time. In the fermion sector the restrictions caused by the possibility to define the multiplication on the densities of spinor fields can explain the chirality violation. It is found that for bosons in the Higgs sector the charge conjugation symmetry violation on the densities of states can be observed. This symmetry violation can explain the matter-antimatter imbalance.