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

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.

63 pages, LaTeX, 2 figures, details added and minor improvements (published version)

References in corpus (6)

• A Hamiltonian Formulation of Causal Variational Principles
• Causal Fermion Systems: An Elementary Introduction to Physical Ideas and Mathematical Concepts
• The Causal Action in Minkowski Space and Surface Layer Integrals
• Fermionic Fock Spaces and Quantum States for Causal Fermion Systems
• Two-Dimensional Area and Matter Flux in the Theory of Causal Fermion Systems
• Positive Functionals Induced by Minimizers of Causal Variational Principles

Cited by in corpus (19)

• Causal Fermion Systems: An Elementary Introduction to Physical Ideas and Mathematical Concepts
• Banach Manifold Structure and Infinite-Dimensional Analysis for Causal Fermion Systems
• The Causal Action in Minkowski Space and Surface Layer Integrals
• Fermionic Fock Spaces and Quantum States for Causal Fermion Systems
• The Linear Dynamics of Wave Functions in Causal Fermion Systems
• Two-Dimensional Area and Matter Flux in the Theory of Causal Fermion Systems
• A Mechanism of Baryogenesis for Causal Fermion Systems
• Causal Fermion Systems: Discrete Space-Times, Causation and Finite Propagation Speed
• Causal Fermion Systems: Classical Gravity and Beyond
• Modified Measures as an Effective Theory for Causal Fermion Systems
• Causal Fermion Systems and Octonions
• A Positive Mass Theorem for Static Causal Fermion Systems
• Causal Fermion Systems as an Effective Collapse Theory
• A Notion of Entropy for Causal Fermion Systems
• Local Algebras for Causal Fermion Systems in Minkowski Space
• Elliptic Methods for Solving the Linearized Field Equations of Causal Variational Principles
• Linear Bosonic Quantum Field Theories Arising from Causal Variational Principles
• A Positive Quasilocal Mass for Causal Variational Principles
• Holographic Mixing and Fock Space Dynamics of Causal Fermion Systems