Felix Finster

Felix Finster
Verified
Felix verified their affiliation via an institutional email.
Verified
Felix verified their affiliation via an institutional email.
  • Dr. rer. nat. habil.
  • Professor at University of Regensburg

About

242
Publications
10,602
Reads
How we measure 'reads'
A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more
3,039
Citations
Current institution
University of Regensburg
Current position
  • Professor

Publications

Publications (242)
Article
Full-text available
The Cauchy problem for the massive Dirac equation is studied in the Reissner–Nordström geometry in horizon-penetrating Eddington–Finkelstein-type coordinates. We derive an integral representation for the Dirac propagator involving the solutions of the ordinary differential equations which arise in the separation of variables. Our integral represent...
Preprint
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.
Preprint
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 so...
Preprint
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 direc...
Article
Full-text available
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...
Preprint
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 gener...
Article
Full-text available
A new inequality for a nonlinear surface layer integral is proved for minimizers of causal variational principles. This inequality is applied to obtain a new proof of the positive mass theorem with volume constraint. Next, a positive mass theorem without volume constraint is stated and proved by introducing and using the concept of asymptotic align...
Preprint
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 fer...
Preprint
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 operato...
Article
Full-text available
The authors compare the structures and methods in the theory of causal fermion systems with approaches to fundamental physics based on division algebras, in particular the octonions. It is found that octonions and, more generally, tensor products of division algebras come up naturally to describe the symmetries of the vacuum configuration of a caus...
Article
Full-text available
It is shown that, in the non-relativistic limit, causal fermion systems give rise to an effective collapse theory. The nonlinear and stochastic correction terms to the Schrödinger equation are derived from the causal action principle. The dynamics of the statistical operator is described by a deterministic equation of Kossakowski–Lindblad form. Mor...
Preprint
Full-text available
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...
Preprint
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.
Preprint
Full-text available
The fermionic R\'enyi entanglement entropy is studied for causal diamonds in two-dimensional Minkowski spacetime. Choosing the quasi-free state describing the Minkowski vacuum with an ultraviolet regularization, a logarithmically enhanced area law is derived.
Article
Full-text available
We define and analyze the fermionic entanglement entropy of a Schwarzschild black hole horizon for the regularized vacuum state of an observer at infinity. Using separation of variables and an integral representation of the Dirac propagator, the entanglement entropy is computed to be a prefactor times the number of occupied angular momentum modes o...
Preprint
It is shown that, in the non-relativistic limit, causal fermion systems give rise to an effective collapse theory. The nonlinear and stochastic correction terms to the Schr\"odinger equation are derived from the causal action principle. The dynamics of the statistical operator is described by a deterministic equation of Kossakowski-Lindblad form. M...
Article
Full-text available
The present paper is a modest attempt to initiate the research program outlined in this abstract. We propose that general relativity and relativistic MOND (RelMOND) are analogues of broken electroweak symmetry. That is, SU(2)R×U(1)YDEM→U(1)DEM (DEM stands for dark electromagnetism), and GR is assumed to arise from the broken SU(2)R symmetry and is...
Article
Full-text available
We compare the structures of the theory of causal fermion systems (CFS), an approach to unify Quantum Theory with General Relativity, with those of modified measure theories (MMT), which are a set of modified gravity theories. Spacetimes with MMT can be obtained as the continuum limit of a CFS. This suggests that MMT could serve as effective descripti...
Preprint
Full-text available
A new inequality for a nonlinear surface layer integral is proved for min-imizers of causal variational principles. This inequality is applied to obtain a new proof of the positive mass theorem with volume constraint. Next, a positive mass theorem without volume constraint is stated and proved by introducing and using the concept of asymptotic alig...
Preprint
Full-text available
We consider the fermionic entanglement entropy for the free Dirac field in a bounded spatial region of Minkowski spacetime. In order to make the system ultraviolet finite, a regularization is introduced. An area law is proven in the limiting cases where the volume tends to infinity and/or the regularization length tends to zero. The technical core...
Preprint
The linearized field equations for causal fermion systems in Minkowski space are analyzed systematically using methods of functional analysis and Fourier analysis. Taking into account a direction-dependent local phase freedom, we find a multitude of homogeneous solutions. The time evolution of the inhomogeneous equations is studied. It leads to the...
Preprint
Full-text available
We compare the structures of the theory of causal fermion systems (CFS), an approach to unify Quantum Theory with General Relativity, with those of modified measure theories (MMT), which are a set of modified gravity theories. Spacetimes with MMT can be obtained as the continuum limit of a CFS. This suggests that MMT could serve as effective descri...
Article
Full-text available
It is shown that the theory of causal fermion systems gives rise to a novel mechanism for dark matter and dark energy. This mechanism is first worked out for cubical subsets of Minkowski space with periodic boundary conditions. Then it is studied in Friedmann-Lemaˆıtre-Roberson-Walker spacetimes. The magnitude of the effect scales like one over the l...
Preprint
The Cauchy problem for the massive Dirac equation is studied in the Reissner-Nordstr\"om geometry in horizon-penetrating Eddington-Finkelstein-type coordinates. We derive an integral representation for the Dirac propagator involving the solutions of the ordinary differential equations which arise in the separation of variables. Our integral represe...
Preprint
We define and analyze the fermionic entanglement entropy of a Schwarzschild black hole horizon for the vacuum state of an observer at infinity. Using separation of variables and an integral representation of the Dirac propagator, the entanglement entropy is computed to be a numerical factor times the number of occupied states on the event horizon.
Article
The homogeneous causal action principle on a compact domain of momentum space is introduced. The connection to causal fermion systems is worked out. Existence and compactness results are reviewed. The Euler–Lagrange equations are derived and analyzed under suitable regularity assumptions.
Preprint
We give a novel construction of global solutions to the linearized field equations for causal variational principles. The method is to glue together local solutions supported in lens-shaped regions. As applications, causal Green's operators and cone structures are introduced.
Article
Full-text available
We prove quantitative versions of the following statement: If a solution of the $$1+1$$ 1 + 1 -dimensional wave equation has spatially compact support and consists mainly of positive frequencies, then it must have a significant high-frequency component. Similar results are proven for the $$3+1$$ 3 + 1 -dimensional wave equation.
Preprint
It is shown that the theory of causal fermion systems gives rise to a novel mechanism for dark matter and dark energy. This mechanism is first worked out for cubical subsets of Minkowski space with periodic boundary conditions. Then it is studied in Friedmann-Lema\^itre-Roberson-Walker spacetimes. The magnitude of the effect scales like one over th...
Article
Full-text available
The bosonic signature operator is defined for Klein–Gordon fields and massless scalar fields on globally hyperbolic Lorentzian manifolds of infinite lifetime. The construction is based on an analysis of families of solutions of the Klein–Gordon equation with a varying mass parameter. It makes use of the so-called bosonic mass oscillation property w...
Article
Full-text available
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 operator which is shown to be bounded and symmetric on a Hilbert space endowed with a suitably adapted weighted L2\documentclass[12pt]{minimal} \usepackage{amsmath}...
Preprint
This paper is dedicated to a detailed analysis and computation of quantum states of causal fermion systems. The mathematical core is to compute integrals over the unitary group asymptotically for a large dimension of the group, for various integrands with a specific scaling behavior in this dimension. It is shown that, in a well-defined limiting ca...
Article
Full-text available
It is shown that the theory of causal fermion systems gives rise to a novel mechanism of baryogenesis. This mechanism is worked out computationally in globally hyperbolic spacetimes in a way which enables the quantitative study in concrete cosmological situations.
Preprint
The homogeneous causal action principle in a compact domain of momentum space is introduced. The connection to causal fermion systems is worked out. Existence and compactness results are reviewed. The Euler-Lagrange equations are derived and analyzed under suitable regularity assumptions.
Article
Full-text available
It is shown that the linearized fields of causal variational principles give rise to linear bosonic quantum field theories. The properties of these field theories are studied and compared with the axioms of local quantum physics. Distinguished quasi-free states are constructed.
Preprint
The numerical analysis of causal fermion systems is advanced by employing differentiable programming methods. The causal action principle for weighted counting measures is introduced for general values of the integer parameters $f$ (the particle number), $n$ (the spin dimension) and $m$ (the number of spacetime points). In the case $n=1$, the causa...
Preprint
Full-text available
The bosonic signature operator is defined for Klein-Gordon fields and massless scalar fields on globally hyperbolic Lorentzian manifolds of infinite lifetime. The construction is based on an analysis of families of solutions of the Klein-Gordon equation with a varying mass parameter. It makes use of the so-called bosonic mass oscillation property w...
Preprint
It is shown that the linearized fields of causal variational principles give rise to linear bosonic quantum field theories. The properties of these field theories are studied and compared with the axioms of local quantum physics. Distinguished quasi-free states are constructed.
Preprint
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 operator which is shown to be bounded and symmetric on a Hilbert space endowed with a suitably adapted weighted $L^2$-scalar product. Guided by the procedure in the...
Preprint
It is shown that the theory of causal fermion systems gives rise to a novel mechanism of baryogenesis. This mechanism is worked out computationally in globally hyperbolic spacetimes in a way which enables the quantitative study in concrete cosmological situations.
Article
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 fermionic and bosonic field operators. The proof of positivity of the state is given, and representations are constructed.
Article
Full-text available
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 given both in the finite- and infinite-dimensional settings. General properties of the entropy are analyzed.
Preprint
In this short review, we explain how and in which sense the causal action principle for causal fermion systems gives rise to classical gravity and the Einstein equations. Moreover, methods are presented for going beyond classical gravity, with applications to a positive mass theorem for static causal fermion systems, a connection between area chang...
Article
Full-text available
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 endowed with a canonical Fréchet-smooth Riemannian metric. The so-called expedient differential calculus is introduced with the purpose of treating deriva...
Article
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 represented by a conserved surface layer integral. We prove under general assumptions that the initial value problem for the dynamical wave equation admits a uniq...
Preprint
Full-text available
This chapter is an up-to-date account of results on globally hyperbolic spacetimes, and serves several purposes. We begin with the exposition of results from a foundational level, where the main tools are order theory and general topology, we continue with results of a more geometric nature, and we conclude with results that are related to current...
Preprint
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 given both in the finite and infinite-dimensional settings. General properties of the entropy are analyzed.
Preprint
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 endowed with a canonical Fr\'echet-smooth Riemannian metric. The so-called expedient differential calculus is introduced with the purpose of treating deri...
Preprint
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 fermionic and bosonic field operators. Positivity of the state is proven and representations are constructed.
Preprint
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 represented by a conserved surface layer integral. We prove under general assumptions that the initial value problem for the dynamical wave equation admits a uniq...
Chapter
This chapter is an up-to-date account of results on globally hyperbolic spacetimes and serves as a multitool; we start the exposition of results from a foundational level, where the main tools are order-theory and general topology, we continue with results of a more geometric nature, and we finally reach results that are connected to the most recen...
Article
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 relations are worked out, and the differences to the canonical commutation relations are discussed. It is shown that the spacetime point operators assoc...
Article
We prove the existence of minimizers of causal variational principles on second countable, locally compact Hausdorff spaces. Moreover, the corresponding Euler–Lagrange equations are derived. The method is to first prove the existence of minimizers of the causal variational principle restricted to compact subsets for a lower semi-continuous Lagrangi...
Article
Full-text available
The notions of two-dimensional area, Killing fields and matter flux are introduced in the setting of causal fermion systems. It is shown that for critical points of the causal action, the area change of two-dimensional surfaces under a Killing flow in null directions is proportional to the matter flux through these surfaces. This relation generaliz...
Article
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 the present paper, it is explained and worked out in detail that, despite this local gauge freedom, the structures of a causal fermion system give ris...
Preprint
Full-text available
We prove quantitative versions of the following statement: If a solution of the $1+1$-dimensional wave equation has spatially compact support and consists mainly of positive frequencies, then it must have a significant high-frequency component. Similar results are proven for the $3+1$-dimensional wave equation.
Article
Full-text available
The fermionic projector state is a distinguished quasi-free state for the algebra of Dirac fields in a globally hyperbolic spacetime. We construct and analyze it in the four-dimensional de Sitter spacetime, both in the closed and in the flat slicing. In the latter case we show that the mass oscillation properties do not hold due to boundary effects...
Preprint
Full-text available
After reviewing the theory of "causal fermion systems" (CFS theory) and the "Events, Trees, and Histories Approach" to quantum theory (ETH approach), we compare some of the mathematical structures underlying these two general frameworks and discuss similarities and differences. For causal fermion systems, we introduce future algebras based on causa...
Preprint
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 relations are worked out, and the differences to the canonical commutation relations are discussed. It is shown that the spacetime point operators assoc...
Chapter
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.
Preprint
We prove the existence of minimizers of causal variational principles on second countable, locally compact Hausdorff spaces. Moreover, the corresponding Euler-Lagrange equations are derived. The method is to first prove the existence of minimizers of the causal variational principle restricted to compact subsets for a lower semi-continuous Lagrangi...
Article
Full-text available
Quantum-gravity corrections (in the form of a minimal length) to the Feynman propagator for a free scalar particle in RD are shown to be the result of summing over all dimensions D′≥D of RD′, each summand taken in the absence of quantum gravity.
Book
This book focuses on a critical discussion of the status and prospects of current approaches in quantum mechanics and quantum field theory, in particular concerning gravity. It contains a carefully selected cross-section of lectures and discussions at the seventh conference “Progress and Visions in Quantum Theory in View of Gravity” which took plac...
Preprint
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 flat space-time and a vacuum space-time near spatial infinity. Our definition does not involve any regularity assumptions; it even applies to singular or general...
Chapter
These lecture notes are concerned with linear stability of the non-extreme Kerr geometry under perturbations of general spin. After a brief review of the Kerr black hole and its symmetries, we describe these symmetries by Killing fields and work out the connection to conservation laws. The Penrose process and superradiance effects are discussed. De...
Article
Full-text available
The support of minimizing measures of the causal variational principle on the sphere is analyzed. It is proven that in the case τ>3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{...
Article
Full-text available
We consider the problem of essential self-adjointness of the spatial part of the Klein–Gordon operator in stationary spacetimes. This operator is shown to be a Laplace–Beltrami type operator plus a potential. In globally hyperbolic spacetimes, essential self-adjointness is proven assuming smoothness of the metric components and semi-boundedness of...
Preprint
Quantum-gravity corrections (in the form of a minimal length) to the Feynman propagator for a free scalar particle in $\mathbb{R}^D$ are shown to be the result of summing over all dimensions $D'\geq D$ of $\mathbb{R}^{D'}$, each summand taken in the absence of quantum gravity.
Preprint
Full-text available
The notions of two-dimensional area, Killing fields and matter flux are introduced in the setting of causal fermion systems. It is shown that for critical points of the causal action, the area change of two-dimensional surfaces under a Killing flow in null directions is proportional to the matter flux through these surfaces. This relation generaliz...
Preprint
Full-text available
We consider the problem of essential self-adjointness of the spatial part of the Klein-Gordon operator in stationary spacetimes. This operator is shown to be a Laplace-Beltrami type operator plus a potential. In globally hyperbolic spacetimes, essential selfadjointness is proven assuming smoothness of the metric components and semi-boundedness of t...
Article
Full-text available
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 a unified physical theory. The dynamics is described by a novel variational principle, the so-called causal action principle. The causal action principle doe...
Preprint
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.
Preprint
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 the present paper it is explained and worked out in detail that, despite this local gauge freedom, the structures of a causal fermion system give rise...
Article
The structure of the solution space of the Dirac equation in the exterior Schwarzschild geometry is analyzed. Representing the space-time inner product for families of solutions with variable mass parameter in terms of the respective scalar products, a so-called mass decomposition is derived. This mass decomposition consists of a single mass integr...
Preprint
The fermionic projector state is a distinguished quasi-free state for the algebra of Dirac fields in a globally hyperbolic spacetime. We construct and analyze it in the four-dimensional de Sitter spacetime, both in the closed and in the flat slicing. In the latter case we show that the mass oscillation properties do not hold due to boundary effects...
Article
Full-text available
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 causal variational principles, a class of conserved surface layer integrals is found and analyzed.
Preprint
Full-text available
The structure of the solution space of the Dirac equation in the exterior Schwarzschild geometry is analyzed. Representing the space-time inner product for families of solutions with variable mass parameter in terms of the respective scalar products, a so-called mass decomposition is derived. This mass decomposition consists of a single mass integr...
Preprint
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 a unified physical theory. The dynamics is described by a novel variational principle, the so-called causal action principle. The causal action principle doe...
Preprint
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 uniq...
Preprint
These lecture notes are concerned with linear stability of the non-extreme Kerr geometry under perturbations of general spin. After a brief review of the Kerr black hole and its symmetries, we describe these symmetries by Killing fields and work out the connection to conservation laws. The Penrose process and superradiance effects are discussed. De...
Article
Full-text available
We consider the massive Dirac equation in the non-extreme Kerr geometry in horizon-penetrating advanced Eddington--Finkelstein-type coordinates and derive a functional analytic integral representation of the associated propagator using the spectral theorem for unbounded self-adjoint operators, Stone's formula, and quantities arising in the analysis...
Preprint
The support of minimizing measures of the causal variational principle on the sphere is analyzed. It is proven that in the case $\tau>\sqrt{3}$, the support of every minimizing measure is contained in a finite number of real analytic curves which intersect at a finite number of points. In the case $\tau>\sqrt{6}$, the support is proven to have Haus...
Preprint
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...
Article
It is proven that for smooth initial data with compact support outside the event horizon, the solution of every azimuthal mode of the Teukolsky equation for general spin decays pointwise in time.
Preprint
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 causal variational principles, a class of conserved surface layer integrals is found and analyzed.
Article
In Section 5.1 in [1] it is incorrectly claimed that condition (A) is equivalent to the vanishing of the operator B in the expansion. © 2017 Springer International Publishing AG, part of Springer Nature
Article
The Lagrangian of the causal action principle is computed in Minkowski space for Dirac wave functions interacting with classical electromagnetism and linearized gravity in the limiting case when the ultraviolet cutoff is removed. Various surface layer integrals are computed in this limiting case.
Preprint
Full-text available
The Lagrangian of the causal action principle is computed in Minkowski space for Dirac wave functions interacting with classical electromagnetism and linearized gravity in the limiting case when the ultraviolet cutoff is removed. Various surface layer integrals are computed in this limiting case.
Article
Full-text available
The fermionic signature operator is constructed in Rindler space-time. It is shown to be an unbounded self-adjoint operator on the Hilbert space of solutions of the massive Dirac equation. In two-dimensional Rindler space-time, we prove that the resulting fermionic projector state coincides with the Fulling-Rindler vacuum. Moreover, the fermionic s...

Network

Cited By