# On the Mathematical Foundations of Causal Fermion Systems in Minkowski Space

@article{Oppio2019OnTM, title={On the Mathematical Foundations of Causal Fermion Systems in Minkowski Space}, author={Marco Oppio}, journal={arXiv: Mathematical Physics}, year={2019} }

The emergence of the concept of a causal fermion system is revisited and further investigated for the vacuum Dirac equation in Minkowski spacetime. After a brief recap of the Dirac equation and its solution space, in order to allow for the effects of a possibly nonstandard structure of spacetime at the Plank scale, a regularization by a smooth cutoff in momentum space is introduced, and its properties are discussed. Given an ensemble of solutions, we recall the construction of a local…

## 10 Citations

Local algebras for causal fermion systems in Minkowski space

- Mathematics
- 2020

A notion of local algebras is introduced in the theory of causal fermion systems. Their properties are studied in the example of the regularized Dirac sea vacuum in Minkowski space. The commutation…

A gauge fixing procedure for causal fermion systems

- Mathematics
- 2020

Causal fermion systems incorporate local gauge symmetry in the sense that the Lagrangian and all inherent structures are invariant under local phase transformations of the physical wave functions. In…

Banach manifold structure and infinite-dimensional analysis for causal fermion systems

- MathematicsAnnals of Global Analysis and Geometry
- 2021

A mathematical framework is developed for the analysis of causal fermion systems in the infinite-dimensional setting. It is shown that the regular spacetime point operators form a Banach manifold…

Fermionic Fock Spaces and Quantum States for Causal Fermion Systems

- PhysicsAnnales Henri Poincaré
- 2021

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 by…

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

- Mathematics
- 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…

A notion of entropy for causal fermion systems

- Physics, Computer ScienceLetters 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 is…

Existence of minimizers for causal variational principles on compact subsets of momentum space in the homogeneous setting

- MathematicsCalculus of Variations and Partial Differential Equations
- 2022

We prove the existence of minimizers for the causal action in the class of negative definite measures on compact subsets of momentum space in the homogeneous setting under several side conditions…

Introduction to Causal Fermion Systems

- Philosophy
- 2021

This paper is dedicated to give a concise introduction to the theory of causal fermion systems. After putting the theory of causal fermion systems into the historical context, we recall fundamental…

H\"older Continuity of the Integrated Causal Lagrangian in Minkowski Space

- Physics
- 2021

It is proven that the kernel of the fermionic projector of regularized Dirac sea vacua in Minkowski Space is L-integrable. The proof is carried out in the specific setting of a continuous…

## References

SHOWING 1-10 OF 32 REFERENCES

The Continuum Limit of Causal Fermion Systems

- Physics
- 2016

This monograph introduces the basic concepts of the theory of causal fermion systems, a recent approach to the description of fundamental physics. The theory yields quantum mechanics, general…

A Lorentzian Quantum Geometry

- Mathematics
- 2011

We propose a formulation of a Lorentzian quantum geometry based on the framework of causal fermion systems. After giving the general definition of causal fermion systems, we deduce space-time as a…

Dirac equation with external potential and initial data on Cauchy surfaces

- Mathematics
- 2014

With this paper, we provide a mathematical review on the initial-value problem of the one-particle Dirac equation on space-like Cauchy hypersurfaces for compactly supported external potentials. We,…

On the regularized fermionic projector of the vacuum

- Mathematics
- 2008

We construct families of fermionic projectors with spherically symmetric regularization, which satisfy the condition of a distributional MP-product. The method is to analyze regularization tails with…

Time Evolution of the External Field Problem in QED

- Mathematics
- 2009

We construct the time-evolution for the second quantized Dirac equation subject to a smooth, compactly supported, time dependent electromagnetic potential and identify the degrees of freedom…

Lectures on Nonlinear Hyperbolic Differential Equations

- Mathematics
- 1997

In this introductory textbook, a revised and extended version of well-known lectures by L. Hormander from 1986, four chapters are devoted to weak solutions of systems of conservation laws. Apart from…

Quantum Mechanics and Quantum Field Theory: A Mathematical Primer

- Physics
- 2011

Introduction Part I. Non-relativistic: 1. Mathematical prelude 2. Classical mechanics 3. Quantum mechanics 4. Single particle 5. Many particles 6. Statistical mechanics Part II. Relativistic: 7.…

Introduction to Fourier Analysis on Euclidean Spaces.

- Mathematics
- 1971

The authors present a unified treatment of basic topics that arise in Fourier analysis. Their intention is to illustrate the role played by the structure of Euclidean spaces, particularly the action…

Introduction to Functional Analysis

- Mathematics
- 1997

Preliminaries 1. Banach spaces and Metric Linear Spaces 2. Spectral of Theory Linear Operators 3. Frechet Spaces and their Dual Spaces