Realizations of Causal Manifolds by Quantum Fields

TL;DR

This paper explores how quantum operators and fields can be viewed as realizations of causal spacetime manifolds, emphasizing the mathematical structure and implications for divergences in field theories.

Contribution

It introduces a framework linking quantum fields to causal manifolds parametrized by complex unitary operations, highlighting new realizations that avoid divergencies.

Findings

Realizations are parametrized by homogenous spaces $ ext{D}(n)$ with $n=1,2$.

Canonical field theories are inappropriate for the 4D causal manifold $ ext{D}(2)$ due to divergencies.

Faithful realizations are reducible but nondecomposable, including principal vectors without particle eigenvectors.

Abstract

Quantum mechanical operators and quantum fields are interpreted as realizations of timespace manifolds. Such causal manifolds are parametrized by the classes of the positive unitary operations in all complex operations, i.e. by the homogenous spaces $\D(n)=\GL(\C^n_\R)/\U(n)$ with $n=1$ for mechanics and $n=2$ for relativistic fields. The rank $n$ gives the number of both the discrete and continuous invariants used in the harmonic analysis, i.e. two characteristic masses in the relativistic case. 'Canonical' field theories with the familiar divergencies are inappropriate realizations of the real 4-dimensional causal manifold $\D(2)$. Faithful timespace realizations do not lead to divergencies. In general they are reducible, but nondecomposable - in addition to representations with eigenvectors (states, particle) they incorporate principal vectors without a particle (eigenvector) basis as exemplified by the Coulomb field.