# The Theory of Causal Fermion Systems

## Research

Below we discuss active research topics and list current projects and open questions related to them. The links direct you to a more detailed description, including a list of people who are interested in the problem. Let us know if you have any questions or find a problem interesting!

## Current Research Projects

Here we list the research topics which we are currently working on. Since some of these topics are long-term projects, at present we are often concerned only with certain aspects or specific problems related to these topics. Typically, for each topic there are interesting open questions to think about. Email us if you have any ideas or would like to get involved!

## Quantum Field Theory and Entanglement

It has been shown that in certain limiting cases, causal fermion systems gives rise to quantum field theory [qft13, FK18]. However, this limiting case involves approximations and simplifying assumptions which still need to be justified carefully. Also, the constructions in [qft13] are based on structures obtained in the continuum limit. It would be more convincing conceptually to work instead exclusively with the inherent structures of the causal fermion system, in particular with the conserved surface layer integrals. The more recent paper [FK18] is a first step in this direction. For more details on the present status see the mathematics section

→ Connection to Quantum Field Theory.

Current research projects related to this topic:

People involved in this project: Claudio Dappiaggi, Felix Finster, Niky Kamran, Magdalena Lottner, Moritz Reintjes, Jürgen Tolksdorf

## Gravity, Entropy and Black Holes

It has been shown that the causal action principle gives rise to Einstein’s gravity in the continuum limit (for more details see the mathematics section → Connection to General Relativity). It is an important task to understand the connection to gravity in more detail and more directly, without taking the continuum limit.

Research projects and open questions related to this topic:

People involved in this project: Eric Curiel, Felix Finster, José Isidro, Magdalena Lottner, Johannes Wurm

## Phenomenology I

The theory is now at a stage where it is clear that it can reproduce our current description of the physical world in suitable limiting cases. The theory does already make a number of interesting additional post-dictions such as the number of fermionic generations and explaining the weakness of gravity. However, for any scientific theory it is crucial to obtain unique and testable predictions that make the theory falsifiable. For the Theory of Causal Fermion Systems the question is, of course, what changes when we go to finite regularization length.

Current research projects related to this topic:

People involved in this project: Felix Finster, Maximilian Jokel, Margarita Kraus, Claudio Paganini

## Mathematical Foundations I

Despite the solid mathematical foundations of the theory, there is a wide range of mathematical questions regarding the foundations of causal fermion systems that can and should be investigated further. These require techniques from different areas of modern mathematics including functional analysis, measure theory and calculus of variations.

Current research projects in the different areas:

#### Calculus of Variations

#### Geometry

#### Infinite-Dimensional Analysis

People involved in this project: Felix Finster, Magdalena Lottner, Saeed Zafari

## Future Perspectives

Here we list ideas for research projects which we are not yet working on. Sometimes because we did not yet have the time or the courage to get started, but sometimes also because the projects still need to be concretized. Any ideas and feedback are welcome!

## The Standard Model and Beyond

It has been worked out in [cfs16] that the gauge fields of the Standard Model can be derived from the causal action principle. However, there is one piece missing, namely a derivation of the Higgs particle. Upon successful completion of this task, an interesting question is: How rigid is the Standard Model in the Theory of Causal Fermion Systems? That means, which extensions of the Standard Model are possible within the mathematical structures of causal fermion systems? In particular, it would be interesting whether the gauge groups of other theories, such as supersymmetry can be realized.

Research projects and open questions related to this topic:

## Connection to Quantum Gravity

In the continuum limit, causal fermion systems give rise to classical gravity (see the mathematics section → Connection to General Relativity). Once the above project Quantum Field Theory and entanglement will be completed, one should also analyze whether and how these methods and results apply to gravity. The good news is that in the framework of causal fermion systems, all bosonic fields are described on the same footing as variations of the universal measure. This gives the hope that all the methods and results for electromagnetic fields carry over to gravity. But of course, many important questions have not been addressed so far.

To our personal opinion, it is not necessary to make the connection to the mathematical formalism of quantum gravity. Instead, it seems sufficient to address the physical effects (once there are any). For this reason, it seems too early to formulate specific research topics.

## Conceptional Aspects

While the mathematics of the Theory of Causal Fermion Systems is quite mature, the understanding of the physical interpretation of the structures is still incomplete. In itself, this is a very vague question, of course, but in practice it boils down to understanding the minimal physical assumptions that enter the theory and how this differs from other approaches. Moreover, it is desirable to deepen the understanding of the relationship between the fundamantal structures and the emergent description in the continuum limit.

Research projects and open questions related to this topic:

## Phenomenology II

We here continue the research topic phenomenology with a list of ideas and open questions for the future:

## Spacetime Structures

Most of the results in the Theory of Causal Fermion Systems have been derived in Minkowski space or perturbations thereof. Accordingly, a study of more general spacetimes is necessary. This includes possible questions of topological nature as well as questions regarding the global structure of causality.

Research projects and open questions related to this topic:

## Mathematical Foundations II

We here continue the research topic mathematical foundations with a list of ideas and open questions for the future, again ordered by area.