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Felix Finster

Publications and source records attributed to Felix Finster.

At least 19 recordsLinked to original sources
arXiv2026

The $\mathcal{L}$-Calculus for Causal Variational Principles: An Exterior Differential Calculus on Non-Smooth Spaces

A differential calculus for causal variational principles is developed, which generalizes the exterior calculus of differential forms and some of the associated differential topological structures to non-smooth spaces. Our calculus includes the exterior derivative, de Rham cohomology, glueing constructions (restrictions and extensions of differential forms, Mayer-Vietoris sequence), a K\"unneth formula and Poincar{\'e}'s lemma. Moreover, we prove versions of Stokes' theorem and the Gau{\ss} divergence theorem. The constructions and results are illustrated by several examples.

math-ph↗
arXiv2026

A Geometric Derivation of the Einstein Equations from the Causal Action Principle

The causal action principle for causal fermion systems is analyzed for a minimizing measure whose support is assumed to have the structure of a smooth manifold $\tilde{M}$. The concept of osculating vacua is introduced. It is shown that the Lagrangian induces on $\tilde{M}$ a Lorentzian metric. Moreover, the Euler-Lagrange equations of the causal action imply that the Ricci tensor must satisfy the Einstein equations of general relativity for an energy momentum tensor given in terms of a power expansion in the regularization length. The gravitational coupling constant is found to be the square of the regularization length. Our methods provide a systematic procedure for deriving corrections to the Einstein equations. The paper includes a self-contained introduction to causal variational principles and the causal action principle. Most geometric structures (connection, Riemannian metric and curvature) are introduced and analyzed in the general setting of causal variational principles for an arbitrary dimension of $\tilde{M}$. The Lorentzian setting works only for causal fermion systems and is worked out only in four spacetime dimensions.

math-ph↗
arXiv2026

The Fermionic Signature Operator in the Reissner-Nordstr\"om Geometry in Horizon-Penetrating Coordinates

We study the Dirac equation in the Reissner-Nordstr\"om geometry in horizon-penetrating coordinates up to the Cauchy horizon. A mass decomposition theorem is proved, which gives a covariant representation of the spacetime inner product that naturally involves the fermionic signature operator and the fermionic flux operator. We compute their spectra and show that both are bounded symmetric operators on the solution space $\mathcal{H}_m$ of the massive Dirac equation. The corresponding fermionic projector state is constructed and shown to satisfy the Hadamard condition. Lastly, we give some physical interpretations of the fermionic flux operator.

math-ph↗
arXiv2026

The Continuum Limit Analysis of Causal Fermion Systems for Curved Spacetimes

We construct the causal fermion system for globally hyperbolic spacetimes starting in the framework of algebraic quantum field theory. The fermionic projector is identified with the one-particle density operator of a quasi-free Hadamard state. The ultraviolet regularization is built into the fermionic projector via a chart-independent $i\varepsilon$-regularization scheme. The continuum limit analysis is developed in globally hyperbolic spacetimes. It is shown that the Euler-Lagrange equations of the causal action principle are satisfied in this setup if and only if the coupled Einstein-Dirac equations hold.

math-ph↗
arXiv2026

Causal Fermion Systems, Non-Commutative Geometry and Generalized Trace Dynamics

We compare the structures and methods in the theory of causal fermion systems with generalized trace dynamics and non-commutative geometry. Although the three theories differ on many aspects, they agree in that the geometric structure to be recovered in the continuum limit is not the bare spacetime but a suitable fiber bundle. Furthermore, the comparison leads us to the conclusion that the key innovation in causal fermion systems lies in the manner in which the relation between different spacetime points is encoded. The role of Synge's classical world function $\sigma(x,y)$ that encodes the geodesic distance between any two points in the manifold, is taken by a generalized two-point correlator. We show that this idea can be transferred to non-commutative geometry and generalized trace dynamics.

math-ph↗
arXiv2025

A Collapse Mechanism Without Heating

It is shown that the collapse model derived from the theory of causal fermion systems does not give rise to a heating of the probe, up to higher order corrections in the Planck length. Experimental consequences are worked out. The connections to the continuous spontaneous localization (CSL) model, models involving a second-quantized background and the events-trees-histories (ETH) formulation of quantum theory are discussed.

quant-ph↗
arXiv2025

Construction of Currents in Causal Fermion Systems

This paper presents a novel and systematic formalism for deriving classical field equations within the framework ofcausal fermion systems, explicitly accounting for higher-order corrections such as quantum effects and those arising from spacetime discreteness. Our method, which also generalizes to non-abelian gauge fields and gravitation, gives a systematic procedure for evaluating the linearized field equations of causal fermion systems. By probing these equations with specific wave functions and employing Taylor expansions, we reformulate them as a family of tensorial equations of increasing rank. We show that, for rank one, this approach recovers the established classical dynamics corresponding to Maxwell's equations. In addition, the approach gives rise to higher-rank tensorial equations, where the second-rank equations are expected to encode the Einstein equations, and higher-rank tensors potentially reveal new physics and systematic corrections.

math-ph↗
arXiv2025

The Cauchy Problem for Symmetric Hyperbolic Systems with Nonlocal Potentials

In this paper, we investigate the initial value problem for symmetric hyperbolic systems on globally hyperbolic Lorentzian manifolds with potentials that are both nonlocal in time and space. When the potential is retarded and uniformly bounded in time, we establish well-posedness of the Cauchy problem on a time strip, proving existence, uniqueness, and regularity of solutions. If the potential is not retarded but has only short time range, we show that strong solutions still exist, under the additional assumptions that the uniform bound in time is sufficiently small compared to the range in time and that its kernel decays sufficiently fast in time with respect to the zero-order terms of the system. Furthermore, we present a counterexample demonstrating that when the uniform bound is too large compared to the time range, solutions may fail to exist. As an application, we discuss Maxwell's equations in linear dispersive media on ultrastatic spacetimes, as well as the Dirac equation with nonlocal potential naturally arising in the theory of causal fermion systems. Our paper aims to represent the starting point for a rigorous study for the Cauchy problem for the semiclassical Einstein equations.

math.AP↗
arXiv2025

The Relative Fermionic Entropy in Two-Dimensional Rindler Spacetime

The fermionic relative entropy in two-dimensional Rindler spacetime is studied using both modular theory and the reduced one-particle density operators. The methods and results are compared. A formula for the relative entropy for general Gaussian states is derived. As an application, the relative entropy is computed for a class of non-unitary excitations.

math-ph↗
arXiv2025

A Canonical Construction of the Extended Hilbert Space for Causal Fermion Systems

It is shown that second variations of the causal action can be decomposed into a sum of three terms, two of which being positive and one being small. This gives rise to an approximate decoupling of the linearized field equations into the dynamical wave equation and bosonic field equations. A concrete construction of homogeneous and inhomogeneous solutions of the dynamical wave equation in time strips is presented. In addition, it is show that the solution space admits a positive definite inner product which is preserved under the time evolution. Based on these findings, a canonical construction of the extended Hilbert space containing these solutions is given.

math-ph↗
arXiv2025

Baryogenesis in Conformally Flat Spacetimes

Based on a baryogenesis mechanism originating from the theory of causal fermion systems, we analyze its main geometric and analytic features in conformally flat spacetimes. An explicit formula is derived for the rate of baryogenesis in these spacetimes, which depends on the mass $m$ of the particles, the conformal factor $\Omega$ and a future directed timelike vector field $u$ (dubbed the regularizing vector field). Our analysis covers Friedmann-Lema{\^i}tre-Robertson-Walker, Milne and Milne-like spacetimes. It sets the ground for concrete, quantitative predictions for specific cosmological spacetimes.

math-ph↗
arXiv2025

Action-Driven Flows for Causal Variational Principles

We introduce action-driven flows for causal variational principles, being a class of non-convex variational problems emanating from applications in fundamental physics. In the compact setting, H\"older continuous curves of measures are constructed by using the method of minimizing movements. As is illustrated in examples, these curves will in general not have a limit point, due to the non-convexity of the action. This leads us to introducing a novel penalization which ensures the existence of a limit point, giving rise to approximate solutions of the Euler-Lagrange equations. The methods and results are adapted and generalized to the causal action principle in the finite-dimensional case. As an application, we construct a flow of measures for causal fermion systems in the infinite-dimensional situation.

math-ph↗
arXiv2024

Causal Fermion Systems: An Introduction to Fundamental Structures, Methods and Applications

This textbook 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 relativity and quantum field theory as limiting cases and is therefore a candidate for a unified physical theory. From the mathematical perspective, causal fermion systems provide a general framework for describing and analyzing non-smooth geometries and "quantum geometries." The dynamics is described by a novel variational principle, the causal action principle. The book includes a detailed summary of the mathematical and physical preliminaries. It explains the physical concepts behind the causal fermion system approach from the basics. Moreover, all the mathematical objects and structures are introduced step by step. The mathematical methods used for the analysis of causal fermion systems and the causal action principle are introduced in depth. Many examples and applications are worked out. The textbook is addressed to master and graduate students in mathematics or physics. Furthermore, it serves as a reference work for researchers working in the field.

math-ph↗
arXiv2024

Holographic Mixing and Fock Space Dynamics of Causal Fermion Systems

A limiting case is considered in which the causal action principle for causal fermion systems describing Minkowski space gives rise to the linear Fock space dynamics of quantum electrodynamics. The quantum nature of the bosonic field is a consequence of the stochastic description of a multitude of fluctuating fields coupled to non-commuting operators, taking into account dephasing effects. The scaling of all error terms is specified. Our analysis leads to the concept of holographic mixing, which is introduced and explained in detail.

math-ph↗
arXiv2024

Notions of Fermionic Entropies of a Causal Fermion System

The fermionic von Neumann entropy, the fermionic entanglement entropy and the fermionic relative entropy are defined for causal fermion systems. Our definition makes use of entropy formulas for quasi-free fermionic states in terms of the reduced one-particle density operator. Our definitions are illustrated in various examples for Dirac spinors in two- and four-dimensional Minkowski space, in the Schwarzschild black hole geometry and for fermionic lattices. We review area laws for the two-dimensional diamond and a three-dimensional spatial region in Minkowski space. The connection is made to the computation of the relative entropy using modular theory.

math-ph↗
arXiv2024

Baryogenesis in Minkowski Spacetime

Based on a mechanism originally suggested for causal fermion systems, the present paper paves the way for a rigorous treatment of baryogenesis in the language of differential geometry and global analysis. Moreover, a formula for the rate of baryogenesis in Minkowski spacetime is derived.

math-ph↗