# The Theory of Causal Fermion Systems

## Future Perspectives

Here we list open questions, both from the physical (P) and mathematical (P) perspective (see here for a thematic overview). If you are interested to work on any of these problems and want to get more information, do not hesitate to contact us. At the end of each question you find a list with names of people who are interested in the particular question and familiar with causal fermion systems.

# Description of the Higgs Particle (P)

This is the remaining puzzle piece to complete the Standard Model in the continuum limit. It is notable, though, that the left handed SU(2) gauge field turns out to be massive even, with the derivation of the Higgs still missing.

It is known that the Higgs field cannot be obtained with less than two fermionic sectors (see [cfs16, Section 3.8.5]). However, in models with more than two sectors, there are degrees of freedom which can be identified with those of a Higgs field (see [cfs16, Section 5.5]). These degrees of freedom are scalar fields which describe a dynamical change of the masses of the Dirac particles. So far, the dynamics of these scalar fields has not yet been worked out. The reason is that, for doing so, the continuum limit analysis must be carried out to one degree lower on the light cone. This is a laborious task, which has not yet been started.

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

# Rigidity of the particle model (P)

If we assume for the moment that the Higgs field can be obtained in the continuum limit, then we can describe the Standard Model of particle physics as a causal fermion system without the need to introduce a single new particle. This is compatible with all experiments, but it leaves cosmological questions open. In particular, the question of dark matter remains unanswered.

This raises the following question: Which particle systems can we get if we add more fermionic sectors? What are the consequences if we allow for right handed neutrinos? And so on.

In a sense, this is not a single question because the number of fermionic sectors that can be added is in principle unrestricted. However, once you fix the number of fermionic sectors, the gauge group of the resulting particle system should be highly constrained if not unique.

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

# The Measurement Problem and the Collapse of the Wave Function (M, P)

The causal action principle gives rise to corrections to the linear dynamics of Quantum Theory:

- Nonlinear corrections come about because the Euler-Lagrange equations are nonlinear equations.
- Stochastic corrections should arise if the unknown microscopic structure of spacetime and its fluctutations are described effectively with the help of random variables.

There is hope that, combining these effects, the quantum dynamics described by the causal action principle should be similar to that of continuous dynamical collapse theories. A first proposal in this direction has been made in Chapter 5 of Johannes Kleiner’s PhD thesis. It seems an important task to work out these effects quantitatively and to compare them with the existing collapse theories. An interesting feature is that, despite the nonlinear and stochastic evolution, the conservation laws for surface layer integrals hold exactly. In particular, probability is conserved in the collapse.

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

# What are the Physical Principles satisfied by the Theory of Causal Fermion Systems? (P)

- The principle of absolute causality and relative locality.
- The correspondence principle.
- The weak holographic principle.
- The quantum equivalence principle.

*People interested in this question: Claudio Paganini*

# Is there a Complete Set of Observables? (P)

In the paper comparing causal fermion systems to the ETH approach to Quantum Theory [FFOP20], the entire algebra of one-particle observables is constructed from the spacetime operators. This suggests a split into two categories of observables: Observables of the first kind are the spacetime operators, while observables of the second kind are non trivial combination of several spacetime point operators.

It would be interesting to construct the complete set of observables for a one particle Dirac state in the effective description in the continuum limit. In particular, the goal is to find the correct operator whose expectation value coincides in the continuum limit with the probability density of the Dirac field. This operator is expected to be an observable of the first kind. The more challenging part is to find an operator expression which has the momentum distribution for the same effective Dirac field as an effective description in the continuum limit. Note that this is expected to be an observable of the second kind. (This split of observables would in fact be comparable to the situation in Bohmian mechanics (see for example here), where the observables of “second kind” are referred to as “emergent” observable. However, the Theory of Causal Fermion Systems comes without all the other notions associated with the Bohmian picture (like point particles, particle trajectories or pilot waves).

*People interested in this problem: Claudio Paganini*

# Interpretation of the Causal Action Principle (P)

Most objects in the framework of causal fermion system have a satisfactory interpretation in terms of physical concepts. The events are obtained as the local correlation operators on a particular Hilbert space. The support of the universal measure gives spacetime.

However, at the heart of the framework lies the causal action principle with a specific Lagrangian. The Lagrangian has many interesting properties, but a clear interpretation of the physical content of the Lagrangian is still missing. The Lagrangian is intimately related to the causal structure, as points with spacelike separation do not contribute to the action, and the largest contributions come from a neighborhood of the null cone.

There are several possible interpretations of which so far none has struck us as entirely satisfying:

- Just taking it for what it is, the action tries to
*minimize timelike separation*(in the sense that the more pairs of points are spacelike separated, the better). - What enters the action are the eigenvalues of the product of two spacetime point operators. This operator product can be understood as encoding the correlation of the physical wave functions between the two spacetime points. The Lagrangian can be understood in encoding this “correlation” in a non-negative number. With this perspective, the action tries to
*minimize the correlations*between spacetime points. This seems remiscent and might be related to the concept in thermodynamics of “maximizing entropy”. - If the operator product is self-adjoint, then the Lagrangian can be written as the variance with respect to a particular state on the Hilbert space, and the boundedness constraint is related to the mean value. For all but a few products, however, this relationship fails, but one can still try to interpret the Lagrangian as a suitable generalization of some sort of
*“spectral variance”*. - Another interpretation is related to the fact that the boundedness constraint is related to a possible definition of a causal distance function (see [nrstg17, Section 5.1]). With this perspective, minimizing the action might be related to “minimizing the fluctuations in the causal structure”. This would also give the Euler Lagrange equation a somewhat natural feel (the variance of the causal distance of an event to the rest of spacetime has to be constant). Also, it makes intuitively sense that the largest contribution to the Lagrangian would be supported near the light cone as the relative change in the causal distance due to small fluctuations will be biggest there.

It seems an interesting question to work out these different interpretations in more detail and to compare them.

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

# Linking Causal Fermion Systms to the ETH Formulation of Quantum Theory (P)

Fröhlich’s Events Trees Histories (ETH) approach to quantum mechanics is formulated in the Heisenberg picture. In the recent paper [FFOP20] the mathematical structures were compared and, despite striking similarities, there are substantial differences. It is an interesting question whether there exists a way to merge these ideas into a new theory.

*People interested in this question: Felix Finster, Marco Oppio, **Claudio Paganini*

# The Discrepancy of the Muon Magnetic Moment (P)

The motivation for this question is clear: If the discrepancy between standard model prediction and observation persists in the g-2 experiment, we have to figure out whether this the experimental findings can be explained by the modifications to the Standard Model as predicted by the Theory of Causal Fermion Systems.

There are reasons why one might expect non-trivial modifications: The framework of causal fermion systems requires the existence of three generations of fermions to be able to remove the vacuum polarization singularity. This implies that the three generations do influence each other at high frequencies. It seems at least plausible that this might have an effect on the prediction for the muon magnetic moment.

Moreover, there are claims that taking General Relativity into account could resolve the tension. The fact that General Relativity enters as a next-to-next-to-leading-order effect in the Theory of Causal Fermion Systems might give a systematic justification why General Relativity should be taken into account.

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

# Effects due to the Sectorial Projection (P)

The description of the Standard Model in the framework of causal fermion systems shows a few major differences to the usual description. One difference is that the wave functions with different generation index are not necessarily orthogonal. The reason is that, in order to obtain the correct effective gauge groups, one must take the sum (but not direct sum) of the different isospin components. This is implemented technically in the so-called *sectorial projection (*for details see[cfs16, Section 3.4]). The sectorial projection leads to a new type of current which comes into play if a quantum-mechanical wave is in a superposition of two particles which have the same quantum numbers except for the generation (like for example a superposition of electron and muon). It is unclear to us if such superpositions exist, if they could be created in a lab, and if the above-mentioned current can be detected.

*People interested in this question: Felix Finster*

# Effects due to the Microlocal Chiral Transformation (P)

Another major difference between the description of the Standard Model in the framework of causal fermion systems and the usual description is that the logarithmic poles of the kernel of the fermionic projector on the light cone (which are related to the usual logarithmic divergences of the vacuum polarization loops) are not removed with a renormalization procedure, but instead by slightly modifying all fermionic states. As a consequence, the fermionic states no longer satisfy the Dirac equation. This procedure is referred to as the *microlocal chiral transformation* (for details see [cfs16, Section 3.7]). The microlocal chiral transformation should give rise to corrections which, at least in principle, could be measurable. However, their effect has not yet been worked out in sufficient detail for making concrete predictions.

*People interested in this question: Felix Finster*

# Effects due to the Non-Causal Low and High Energy Contributions (P)

Another major difference between the description of the Standard Model in the framework of causal fermion systems and the usual description has to do with the normalization of the states of the Dirac sea contained in the fermionic projector. If the normalization is treated carefully in the perturbation expansion, then the light-cone expansion gives rise to smooth contributions to the kernel of the fermionic projector which also enter the resulting field equations (for details see [cfs16, Section 2.2]). It is unclear whether and how these contributions are taken into account in the usual procedure of perturbative quantum field theory.

*People interested in this question: Felix Finster*

# Are Generic Einstein Manifolds Critical Points of the Causal Action (M, P)

In order to get a connection between the causal action principle and classical physics. one needs to represent Minkowski space as a causal fermion system. The involed local correlation map relies on the solutions to the vacuum Dirac equation and a splitting into the positive and negative frequency subspaces. In curved spacetimes, this split is not canonical. A local correlation map from a general globally hyperbolic Lorentzian manifolds into a causal fermion system has been obtained in [nrstg17]. However, it is unclear whether the resulting causal fermion system is a critical point of the causal action.

Current maps are based on the construction of Hadamard states and the fermionic signature operator (for details see [FMR15] in the case of an external potential in Minkowski space). For Lorentzian manifolds which are not globally hyperbolic, it is not clear how to construct a corresponding causal fermion system.

A tool that might be useful is the Weyl quantization procedure and the associated symbol calculus. The advantage is that the frequency splitting can be performed in the tangent space where it is well-defined and canonical.

*People interested in this question:*

*Hadamard approach: Claudio Dappiaggi, Felix Finster, Claudio Paganini*

*Weyl quantization approach: Claudio Dappiaggi, Claudio Paganini*

# Closed Causal Curves (M, P)

The causal structure of a causal fermion system is defined by spectral properties of the product of spacetime point operators. In the example of causal fermion systems describing the Minkowski space, this causal structure gives back the usual notions in Minkowski space asymptotically as $\varepsilon \searrow 0$. Without taking this limit, not much is known on the causal structure (see [reg06] and [CFI19, Appendix A] for some work in this direction). More specifically, the following questions are open

- The causal relations are defined between spacetime points $x,y \in M$, but more generally for any points $x, y \in {\mathcal{F}}$. It is unclear what the resulting causal structure of ${\mathcal{F}}$ is.
- Clearly, the causal relations are closely related to the causal action principle. Intuitively speaking, the causal action principle tries to make as many pairs of points as possible spacelike separated, while respecting the constraints. But it is unclear what this implies on the resulting causal structure. Are there minimizers with closed causal curves? Or are such closed causal curves suppressed by the causal action principle? Does this happen at least on microscopic scales?

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

# Singular Structure of Minimizing Measures (M)

In the example of the causal variational principle on the sphere (i.e. $\dim {\mathscr{H}}=2$, spin dimension $n=1$ and prescribed eigenvalues), the numerical analysis in [support10] gave a strong indication that in the parameter range $\tau > \sqrt{2}$, every minimizing measure is discrete (more precisely, is a weighted counting measure). This numerical finding was partly proven in this paper and, more recently, in [sphere18]. It would be interesting to generalize these results to higher dimensions. Ultimately, the quest is to understand in which situations minimizers of the causal action principle are discrete.

*People interested in this question: Felix Finster, Heiko von der Mosel, Daniela Schiefeneder*

# Describing Minimizers with Methods of Homogenization and $\Gamma$-Convergence (M)

A minimizing measure of the causal action describes a regularized spacetime. Thus on the large scale, spacetime should look like Minkowski space or a Lorentzian manifold. On the small scale, however, spacetime could have a quite different microstructure. In order to simplify the problem by disregarding the microstructure, one would like to understand the behavior if the length scale $\varepsilon$ of the regularization tends to zero. For systems in Minkowski space, this procedure is carried out in the continuum limit analysis.

From the perspective of pure analysis and the calculus of variations, these problems can also be restated as follows: The regularization length is determined by the constant $C$ in the boundednes constraint (→ basic definitions). The limit $\varepsilon \rightarrow 0$ corresponds to the limit $C \rightarrow \infty$. The question is whether and in which sense a sequence of minimizers converges in this limit. The hope is that, after suitable rescalings, methods of homogenization theory and $\Gamma$-convergence can be applied.

*People interested in this question: Felix Finster*

# Approximation Methods for the Euler-Lagrange Equations and the Linearized Field Equations (M, P)

In Quantum Theory, very good approximation methods have been developed (WKB approximation, mean field theory, Hartree-Fock approximation, Galerkin method, density functional theory, etc.). It is only due to these approximation methods that one has gained a good understanding of the general behavior of solutions of, for example, the many-particle Schrödinger equation. In the Theory of Causal Fermion Systems, similar approximation methods have not yet been developed. Any ideas or approaches in this direction would be highly welcome!

*People interested in this question: Felix Finster*

# Well-Posedness of the Cauchy Problem for the Euler-Lagrange Equations (M)

For the linearized field equations, the Cauchy problem has been studied in [linhyp18] adapting methods of hyperbolic PDEs. The Cauchy problem for the Euler-Lagrange equations is considerably harder because of the nonlinearities. The hope is that methods of nonlinear hyperbold PDEs can be used. It should be noted that the paper [cauchy13] is devoted to an attempt to solve the Cauchy problem with a constrained variational principle. However, this method has the disadvantages that it does not give unique solutions and that implementing the initial data with constraints affects the form of the Euler-Lagrange equations. Moreover, the methods in [linhyp18] seem more suitable because they are much closer to hyperbolic PDEs.

*People interested in this question: Felix Finster, Simone Murro*

# Semi-Classical Analysis of the Linearized Field Equations and the Notion of Geodesics (M)

At present, there is no general notion of a geodesic in causal fermion systems. There are two obstacles:

- Since spacetime could be discrete, it is not sufficient to work with continuous curves, one should allow for “discrete paths” formed by a sequence of spacetime points (similar as done in [lqg11, Section 5.5]).
- In Lorentzian geometry, the geodesics can be obtained as the path of suitable “wave packets” being solutions of linear wave equations (like the scalar wave equation or the Klein-Gordon equation). The connection can be made precise with methods of semi-classical analysis or Ehrenfest-type theorems. In the setting of causal fermion systems, the analog of the linear wave equations are the linearized field equations (for details see [linhyp18]). Therefore, a sensible notion of geodesic should harmonize with a suitable “semi-classical limit” of the linearized field equations. For null geodesics, the construction of causal cones in [linhyp18, Section4.1] might be a good starting point.

The hope is that, by suitably combining all these notion and methods, it should be possible to get a closer connection between the geometric and analytic structures of a causal fermion system.

*People interested in this question: Felix Finster*

# Cone Structures and Global Hyperbolicity (M)

A notion of global hyperbolicity for causal fermion systems was introduced in [linhyp18, Section4.5]. However, it is unclear whether and if yes how this notion is related to the existence of global foliations by Cauchy surfaces. Following ideas from Lorentzian geometry, it seems a good starting point to work with transitive cone structures as introduced in [linhyp18, Section4.1]. Understanding these connections better would build a stronger bridge between analytic structures and geometric concepts.

*People interested in this question: Felix Finster*

# Generalization of the Continuum Limit Analysis to Curved Spacetimes (M)

So far, the analysis of the continuum limit has been developed in Minkowski space. In order to analyze systems involving strong gravitational fields (like black holes), it is important to generalize this analysis to curved spacetimes. The light-cone expansion and its regularization has been extended to curved spacetime in [reghadamard17]. It remains to generalize the computation rules of the continuum limit analysis (contraction rules, integration-by-parts rules) and the weak evaluation on the light cone to curved spacetimes (for details in Minkowski space see [pfp06, Chapter 4] or [cfs16]). For the time being, it seems a good idea to restrict attention to globally hyperbolic spacetimes.

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

# Spectral Geometry for the Fermionic Signature Operator and Index Theory (M)

The fermionic signature operator ${\mathscr{S}}$ defined by

\[ {\mathscr{S}} = \int_M x\: d\rho(x) \]

is a symmetric operator on the Hilbert space ${\mathcal{H}}$ which encodes certain aspects of the global geometry of a causal fermion system. It can also be introduced in a Lorentzian spacetime (see [finite13], in particular Section 4 for the connection to causal fermion systems) and gives a setting for spectral geometry with Lorentzian signature (see [drum14] for the case of Lorentzian surfaces). From the perspective of causal fermion systems, the point of interest is that the geometric content in ${\mathscr{S}}$ immediately carries over to geometric information on the causal fermion system. One question is to analyze how the resulting geometric notions relate to the “Lorentzian quantum geometry” in [lqg11]. Another problem is analyze how the chiral index of the fermionic signature operator introduced in [index14] is related to the geometric properties of the causal fermion system.

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

# Minimization Problems for Causal Fermion Systems with Symmetries (M)

Even in the compact setting (i.e. if ${\mathcal{H}}$ is finite-dimensional and $\rho$ has finite total volume), causal variational principles involve an infinite number of degrees of freedom. But by building in suitable symmetries, one can restrict attention to a finite number of degrees of freedom. This makes it possible to construct minimizers numerically. A first step in this direction is [ssymm07]. A more advanced model where spacetime is diffeomorphic to $\mathbb{R} \times \text{SO}(3)$ has been worked out and is waiting for being implemented numerically.

*People interested in this question: Felix Finster*

# Analysis of State Stability Including Neutrinos (M, P)

In the state stability analysis one considers a Lorentz invariant minimization problem where one varies the masses of the elementary Dirac particles. This minimization problem is a consequence of the causal action principle. In [vacstab07] this analysis is carried out numerically for one sector (i.e. a system composed of three Dirac seas describing the charged leptons $e$, $\mu$ and $\tau$). This analysis should be extended to include the quarks and neutrinos. These changes the methods and results considerably, mainly because, due to the different masses of neutrinos and the charged leptons, the operator ${\mathcal{M}}$ in the distributional ${\mathcal{M}}P$-product also has a scalar component.

The goal is to compute or at least to obtain constraints on the possible values of the rations of the masses of elementary particles.

*People interested in this question: Felix Finster*