Complex Structures on Jet Spaces and Bosonic Fock Space Dynamics for Causal Variational Principles. (arXiv:1808.03177v1 [math-ph])

Based on conservation laws for surface layer integrals for minimizers 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 norm-preserving time evolution on $\mathfrak{h}$. Rewriting multilinear operators on $\mathfrak{h}$ as linear operators on corresponding tensor products, 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 查看全文>>