Complex Structures on Jet Spaces and Bosonic Fock Space Dynamics for Causal Variational Principles

TL;DR

This paper develops a mathematical framework connecting causal variational principles with complex structures on jet spaces and describes how these lead to a unitary second-quantized dynamics on bosonic Fock spaces, including an approximation method.

Contribution

It introduces a method to endow jet spaces with an almost-complex structure, integrates it into a complex Hilbert space, and derives a second-quantized unitary dynamics for critical points of causal variational principles.

Findings

Jet spaces can be given an almost-complex structure based on conservation laws.

A complex Hilbert space framework is constructed from surface layer integrals.

A linear, norm-preserving evolution on bosonic Fock spaces is established.

Abstract

Based on conservation laws for surface layer integrals for critical points of causal variational principles, it is shown how jet spaces can be endowed with an almost-complex structure. We analyze under which conditions the almost-complex structure can be integrated to a canonical complex structure. Combined with the scalar product expressed by a surface layer integral, we obtain a complex Hilbert space $\mathfrak{h}$. The Euler-Lagrange equations of the causal variational principle describe a nonlinear time evolution on $\mathfrak{h}$. Rewriting multilinear operators on $\mathfrak{h}$ as linear operators on corresponding tensor products and using a conservation law for a nonlinear surface layer integral, we obtain a linear norm-preserving time evolution on bosonic Fock spaces. The so-called holomorphic approximation is introduced, in which the dynamics is described by a unitary time evolution on the bosonic Fock space. The error of this approximation is quantified. Our constructions explain why and under which assumptions critical points of causal variational principles give rise to a second-quantized, unitary dynamics on Fock spaces.