Complex structures on jet spaces and bosonic Fock space dynamics for causal variational principles

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 $(\mathfrak{h}, \langle . \vert . \rangle)$. 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.

Keywords

causal variational principles, surface layer integrals, complex structures, bosonic Fock spaces, Hilbert space structures on jets

2010 Mathematics Subject Classification

46C05, 49Q20, 49S05, 53C15, 81T20

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N.K.’s research was supported by the NSERC grant RGPIN 105490-2018.

Received 10 June 2020

Accepted 10 June 2020

Published 11 April 2021