# The Theory of Causal Fermion Systems

## Current Research Projects

Here we list our current research projects (see here for a thematic overview). If you are interested in working on any of these problems and want to get more information, do not hesitate to contact us. After each title, it is indicated if the problem is related to mathematics (M) or physics (P). At the end of each project you find a list with names of people who are interested in and familiar with the research question.

# Connection between the Causal Action Principle and a Linear Dynamics on Fock Spaces (M, P)

The dynamics of a causal fermion system is described by the Euler-Lagrange equations corresponding to the causal action principle. There are first hints that this dynamics can be formulated in terms of a linear time evolution on Fock spaces. So far, the connection has been made in specific limiting cases under additional assumptions. The goal is to work out the connection in full generality, without referring to any limiting cases. The hope is that this analysis will eventually give a full correspondence to standard quantum field theory, up to small quantifiable corrections.

*People interested in this question: Claudio Dappiaggi, Felix Finster, Niky Kamran, Marco Oppio, Jürgen Tolksdorf*

# Entanglement and Entanglement Entropy (M, P)

Entanglement is a fundamental manifestation of the quantum nature of our world. In the context of causal fermion systems, the following questions arise:

- A direct connection to the usual notions of entanglement can be made using the Fock space structures (see the mathematics section → Connection to Quantum Field Theory). However, it would be interesting to understand more directly what entanglement means for the basic structures of a causal fermion systems. For example, how can entanglement be expressed in terms of the universal measure $\rho$?
- Considering a spatial subsystem, there is the notion of the entanglement entropy of the subsystem. The question is again, how this entropy can be defined and understood directly from the causal fermion systems. It seems that the conservation laws for surface layer integrals will be important for this analysis.
- In curved spacetime, the notion of entropy should be closely related to the area of black holes (black hole entropy) and to the information paradox. The precise connections still need to be explored. This part is closely related to the project thermodynamics of causal fermion systems and connections to area laws.

*People interested in these questions: Felix Finster, Magdalena Lottner, Moritz Reintjes*

# Total Energy and Momentum of Asymptotically Flat Causal Fermion Systems (M)

In order to get a deeper understanding of the nature of the gravitational interaction as described by the causal action principle, it is an important task to define and analyze notions which generalize the ADM mass and momentum of an asymptotically flat Lorentzian manifold. For static causal fermion systems, notions of asymptotic flatness and total mass have been introduced and analyzed in [pmt19]. The next step is to consider time-dependent systems.

*People interested in this question: Felix Finster, Johannes Wurm*

# Thermodynamics of causal fermion systems and connections to area laws (M, P)

The basic question is what the causal action principle tells us about the nature of thermodynamics in connection with area laws and black holes. A first step in this direction is [cfi19] where notions of area, area change and matter flux were introduced, and it is shown that they satisfy an equation reminiscent of a relation in classical general relativity due to Ted Jacobson. But a lot remains to be understood and worked out. In particular, is there a notion of temperature? Which concepts of classical thermodynamics can be extended to causal fermion systems? It also seems interesting to explore the connection to Padmanabhan’s Thermodynamic Gravity.

*People interested in this question: Eric Curiel, Felix Finster, José Isidro, Magdalena Lottner, Claudio Paganini, Johannes Wurm *

# Intrinsic definition of the black hole horizon of a causal fermion system (M, P)

Starting from the Kruskal extension of the Schwarzschild spacetime, one can follow the procedure in Minkowski space to obtain a corresponding causal fermion system. The question is how the event horizon can be characterized intrinsically in the causal fermion system and how the resulting notions can be extended to more general causal fermion systems and generic black holes.

*People interested in this question:** Felix Finster, Olaf Müller *

# Dynamical Gravitational Coupling (M, P)

Causal fermion involve an ultraviolet regularization, and the scale of this regularization determines the gravitational constant (see the mathematics sections → Minkowski space as a causal fermion system and → Connection to General Relativity). As first observed in the unpublished preprint [dgc16] and worked out in more detail in [reghadamard17], the regularization length is not fixed but changes dynamically, at least if the Dirac equation is respected by the regularization. This suggests that the gravitational coupling constant should not be a “constant” but should also change dynamically in spacetime. In [dgc16] a first proposal is made for how the Einstein equations should be modified. However, these modifications are derived under the hypothesis that the Dirac equation holds even on microscopic length scales. This hypothesis is questionable and must either be justified or disproved directly from the causal action principle. If the Dirac equation turns out to be violated, the corrections must be quantified.

*People interested in this question: Felix Finster, Margarita Kraus, Maximilian Jokel*

# The Matter/Anti-Matter Asymmetry (P)

As outlined in the research project on dynamical gravitational coupling above, in the presence of gravitational fields the regularization length as well as the detailed form of the regularization of the causal fermion system changes dynamically. This also means that the number of states per spatial volume needed to form a regularized Dirac sea configuration may change in time. On the other hand, the number of states per spatial volume is determined by current conservation as expressed by a conserved surface layer integral. Combining these results, one finds that, as time evolves, there should be an “excess” or “shortage” of states to fill the Dirac sea. This should give rise to particles and anti-particles, respectively. We thus obtain a mechanism of generation of matter. This could explain the matter/anti-matter asymmetry in our universe. First ideas for possible applications to cosmology are given in [Pa18]. Moreover, in [FJP21] the mechanism is derived mathematically and worked out computationally. But the detailed physical consequences remain to be studied.

*People interested in this question: Felix Finster, Maximilian Jokel, Claudio Paganini *

# Existence Theory in the Infinite-Dimensional Setting (M)

If the Hilbert space ${\mathscr{H}}$ is finite-dimensional and the total volume $\rho({\mathscr{F}})$ is finite, the existence theory was developed in [continuum08]. Although this result is sufficient from the physical perspective, it seems an important task to prove existence also in cases when $\text{dim}\, {\mathscr{H}} = \infty$ and $\rho({\mathscr{F}})=\infty$. Under the additional assumption of translation symmetry, a partial existence result was obtained in [continuum08, Section4]. The only other work in this direction is [noncompact20], where existence was proved for a causal variational principle in a locally compact setting with infinite total volume. The general case that ${\mathscr{F}}$ is not locally compact is wide open.

*People interested in this question: Felix Finster, Christoph Langer*

# Connections Between Local Curvature and the Global Geometry of Causal Fermion Systems (M)

In [lqg11] notions of connection and curvature were introduced for causal fermion systems of spin dimension two. Moreover, the topology and global issues were studied in [topology14]. However, the connections between these structures have not yet been explored. In particular, is there an intrinsic formulation of the Gauss-Bonnet theorem and generalizations thereof?

*People interested in this question: Felix Finster, Niky Kamran, Saeed Zafari*

# Infinite-Dimensional Jets and Resulting Analytic Structures (M)

It has turned out to useful to describe perturbations of measures by *jets *consisting of a scalar function on $M$ and a vector field on ${\mathcal{F}}$ along $M$. This jet formalism was introduced in [jet16] and is convenient for the formulation and analysis of surface layer integrals [osi18] the linearized field equations [linhyp18]. It can also be used for a perturbative treatment of nonlinear variations [perturb17] and is a suitable starting point for Fock space constructions [FK18].

So far, the jet formalism has been developed under the assumption that ${\mathcal{F}}$ is a smooth manifold. If the underlying Hilbert space ${\mathcal{H}}$ is infinite-dimensional, however, the regular points of ${\mathcal{F}}$ for am infinite-dimensional Banach manifold. Consequently, the jet spaces at every spacetime point are infinite-dimensional. As a consequence, the whole formalism needs to be developed from scratch, taking into account subtle topological and differentiability issues.

The paper [FL21] is a first step in this direction.

*People interested in this question: Felix Finster, Magdalena Lottner, Olaf Müller, Marco Oppio*