Linearized fields for causal variational principles: existence theory and causal structure

@article{Dappiaggi2018LinearizedFF,
  title={Linearized fields for causal variational principles: existence theory and causal structure},
  author={Claudio Dappiaggi and Felix Finster},
  journal={arXiv: Mathematical Physics},
  year={2018}
}
The existence theory for solutions of the linearized field equations for causal variational principles is developed. We begin by studying the Cauchy problem locally in lens-shaped regions, defined as subsets of space-time which admit foliations by surface layers satisfying hyperbolicity conditions. We prove existence of weak solutions and show uniqueness up to vectors in the orthogonal complement of the jets used for testing. The connection between weak and strong solutions is analyzed. Global… Expand

Figures from this paper

Elliptic Methods for Solving the Linearized Field Equations of Causal Variational Principles
The existence theory is developed for solutions of the inhomogeneous linearized field equations for causal variational principles. These equations are formulated weakly with an integral operatorExpand
A class of conserved surface layer integrals for causal variational principles
In the theory of causal fermion systems, the physical equations are obtained as the Euler–Lagrange equations of a causal variational principle. Studying families of critical measures of causalExpand
The linear dynamics of wave functions in causal fermion systems
The dynamics of spinorial wave functions in a causal fermion system is studied. A so-called dynamical wave equation is derived. Its solutions form a Hilbert space, whose scalar product is representedExpand
A Positive Mass Theorem for Causal Fermion Systems
Asymptotically flat static causal fermion systems are introduced. Their total mass is defined as a limit of surface layer integrals which compare the measures describing the asymptotically flatExpand
Causal Fermion Systems: Discrete Space-Times, Causation and Finite Propagation Speed
  • F. Finster
  • Physics, Mathematics
  • Journal of Physics: Conference Series
  • 2019
The theory of causal fermion systems is a recent approach to fundamental physics. Giving quantum mechanics, general relativity and quantum field theory as limiting cases, it is a candidate for aExpand
Fermionic Fock Spaces and Quantum States for Causal Fermion Systems
It is shown for causal fermion systems describing Minkowski-type spacetimes that an interacting causal fermion system at time t gives rise to a distinguished state on the algebra generated byExpand
A notion of entropy for causal fermion systems
  • F. Finster
  • Physics, Mathematics
  • Letters in Mathematical Physics
  • 2021
A notion of entropy is introduced for causal fermion systems. This entropy is a measure of the state of disorder of a causal fermion system at a given time compared to the vacuum. The definition isExpand
Proposal 42: A New Storyline for the Universe Based on the Causal Fermion Systems Framework
Based on preliminary results from the Causal Fermion Systems framework regarding the matter-antimatter asymmetry in the universe, I propose a novel story line for the universe that would, if correct,Expand
Proposal 42.
Based on preliminary results from the Causal Fermion Systems framework regarding the matter-antimatter asymmetry in the universe, I propose a novel story line for the universe that would, if correct,Expand
Causal Fermion Systems: An Elementary Introduction to Physical Ideas and Mathematical Concepts
We give an elementary introduction to the theory of causal fermion systems, with a focus on the underlying physical ideas and the conceptual and mathematical foundations.
...
1
2
...

References

SHOWING 1-10 OF 36 REFERENCES
Perturbation theory for critical points of causal variational principles
The perturbation theory for critical points of causal variational principles is developed. We first analyze the class of perturbations obtained by multiplying the universal measure by a weightExpand
A Hamiltonian formulation of causal variational principles
Causal variational principles, which are the analytic core of the physical theory of causal fermion systems, are found to have an underlying Hamiltonian structure, giving a formulation of theExpand
Positive Functionals Induced by Minimizers of Causal Variational Principles
Considering second variations about a given minimizer of a causal variational principle, we derive positive functionals in space-time. It is shown that the strict positivity of these functionalsExpand
On the Support of Minimizers of Causal Variational Principles
A class of causal variational principles on a compact manifold is introduced and analyzed both numerically and analytically. It is proved under general assumptions that the support of a minimizingExpand
Noether-like theorems for causal variational principles
The connection between symmetries and conservation laws as made by Noether’s theorem is extended to the context of causal variational principles and causal fermion systems. Different notions ofExpand
Complex structures on jet spaces and bosonic Fock space dynamics for causal variational principles
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 underExpand
Symmetric Hyperbolic Linear Differential Equations
The present paper is concerned with symmetric systems of linear hyperbolic differential equations of the second order. The existence of a solution of Cauchy’s initial problem will be proved underExpand
Green-Hyperbolic Operators on Globally Hyperbolic Spacetimes
Green-hyperbolic operators are linear differential operators acting on sections of a vector bundle over a Lorentzian manifold which possess advanced and retarded Green’s operators. The most prominentExpand
A class of conserved surface layer integrals for causal variational principles
In the theory of causal fermion systems, the physical equations are obtained as the Euler–Lagrange equations of a causal variational principle. Studying families of critical measures of causalExpand
On Lorentzian causality with continuous metrics
We present a systematic study of causality theory on Lorentzian manifolds with continuous metrics. Examples are given which show that some standard facts in smooth Lorentzian geometry, such asExpand
...
1
2
3
4
...