# The Theory of Causal Fermion Systems

## Relation to Other Theories

*Thermodynamic Gravity*

A mesoscopic approach to quantum gravity, based on thermodynamic considerations has been developed by T. Padmanabhan (see for example, here and references therein).

The central idea is that it is irrelevant what the underlying quantum gravitational degrees of freedom are, you merely assume they exist and then try to derive global corrections from it. In order to obtain the ﬁeld equations, a variation principle on the light cone is derived from a thermodynamic argument. The variation then gives the Einstein ﬁeld equations on the light cone. Demanding that the stress energy tensor is divergence-free leads to the full Einstein ﬁeld equations up to a integration constant (the cosmological constant) which has to be determined by observation. It is then claimed that the cosmological constant can be related to the ﬂuctuations in the CMB and the energy density of matter and radiation observed in the universe, in good agreement with observation.

Formulating the variation principle for the Einstein Field equations on the light cone is motivated by the fact that, in this way, the variation principle for gravity becomes invariant under adding a constant to the Lagrangian (this is true for the equations of motion of all other ﬁelds in the standard model). Note, however, that this leads to a situation where we have two variation principles that are very diﬀerent in spirit. There are several parallels with the framework of causal fermion systems. In the continuum limit, all ﬁeld equations (not only gravity) are derived from an analysis on the light cone. In this way, the conceptional ideas by Padmanabhan are respected while maintaining a single form of the variation principle (in fact, the framework of causal fermion systems does even more, as all ﬁelds are derived from the same Lagrangian).

Note that two particular technical features in Padmanabhan’s considerations show up in the CFS framework as well:

- Points in the framework of causal fermion systems have a non-zero self separation. Padmanabhan modiﬁes Singh’s world function by adding a non-zero term $\sigma(x,y)^2 = \sigma(x,y)^2 + L^2$, where $\sigma$ is the geodesic distance between $x$ and $y$.
- The metric that corresponds to the modiﬁed world function is everywhere singular. The space time point operators in the continuum limit of the CFS framework become singular operators.

Over all, there seem to be interesting conceptional and technical parallels between the thermodynamic Gravity approach and the Theory of Causal Fermion Systems.

*Non-Commutative Geometry*

The Theory of Causal Fermion Systems has a few similarities to Connes’ non-commutative geometry:

- In both approaches the Dirac operator plays a central role.
- It is a general idea behind both approaches that geometry is encoded in operators.

However, there are also major differences:

- One conceptual difference is that, instead of the Dirac operator as in non-commutative geometry, in the framework of causal fermion systems it is more the “ensemble of occupied wave functions” as encoded in the universal measure which plays the central role.

More precisely, the difference can be understood as follows:In non-commutative geometry, the key input for the underlying geometrical structure is a $C^*$ algebra represented by operators on a Hilbert space. This algebra holds information about the differential and topological structure of the manifold whereas, roughly speaking, the Dirac operator holds the metric information.

For causal fermion systems, the key input is the Hilbert space and the universal measure which lives on a subset of the bounded linear operators on the Hilbert space. Roughly speaking, the operators encode the geometric structure while the support of the measure determines the topology of the spacetime.

- Connes’ spectral action seems to be very different from the causal action principle. Indeed, in non-commutative geometry, the key input for the dynamics is the spectral action which takes the spectrum of the Dirac operator as an input. For a given model, the fermions give the Hilbert space on which the algebras are represented. For a causal fermion systems, however, the dynamics is determined by the spectra of products of operators in the support of the universal measure.
- So far, non-commutative geometry has been developed mainly on compact Riemannian manifolds. The adaptation or generalization of the concepts to Lorentzian signature is not straightforward (for attempts in this direction see [BB16/1, BB16/2, Fr11, M02, S01]). The Theory of Causal Fermion Systems, on the other hand, was developed for spacetimes with Lorentzian signature. This is also reflected by the causal action principle, which is intimately connected to the causal structure of spacetime.

On a technical level, there are a few more similarities. In order to compute the spectral action (Dixmier trace), one considers the asymptotics of the large eigenvalues as encoded in the behavior of the resolvent near the diagonal. This expansion is the elliptic analog of the light-cone expansion (or, more generally, Hadamard expansion) used for analyzing the causal action in the continuum limit. Moreover, there are a few similarities of how the standard model is described. In non-commutative geometry, in the model by Connes-Chamseddine-Marcolli [CCM06] (see also [CC07] and [BF14]) one prescribes the entire fermionic content of the model in mind. For the standard model, this would be Up, Down, electron, neutrino, a left and a right handed copy of each of them and then an anti particle for all of those. So 32 fermions in total per generation. Spacetime points are mapped to elements in a non-commutative algebra. The bosonic interactions between the fermions, including the Higgs, can then be derived.

In the framework of causal fermion systems, on the other hand, we assume eight fermionic sectors to exist (one for every quark colour and isospin and one for the leptons and one for the neutrinos) from that one can derive the standard model gauge ﬁelds as emerging phenomena. In the frauemwork of causal fermion systems, each sector automatically contains the anti-particles as well. The framework can model a Standard Model where the chiral symmetry is broken in the neutrino sector. In the Theory of Causal Fermion Systems, the spactime point operators are in general non-commutative. It is unclear whether this non commutativity relates in any way to the ideas in non commutative geometry.

It might be helpful to explore in more detail how these two frameworks relate and where they diﬀer.

*Constructive Gravity*

Constructive Gravity was developed by Schuller and collaborators (see here for an introduction). The basic idea behind the approach is that if we demand that the evolution problem (Cauchy problem) be well posed (existence of a unique local solution for given initial data), then the structure of the Lagrangian in the matter sector highly constrains the possible form of the Lagrangian in the geometric/gravity sector.

A direct connection with the setting of causal fermion systems does not exist. Nevertheless, ideas and methods from constructive gravity might help to constrain extensions of General Relativity and corresponding matter models within the Theory of Causal Fermion Systems.

*Renormalizable Acausal Theories with Gravity
*

In an approach by Anselmi and collaborators (see for example [AH06]) it is argued that gravity can be made renormalizable if one is willing to introduce microscopic violations of causality. This concept seems to fit to causal fermion systems, because the normalization of the fermionic states gives rise to nonlocal effects on the Compton scale (for details see see [cfs16, Section 2.2]). At present, however, it is not clear if a connection can be made on the technical level.*Causal Sets*

Causal Sets were initiated by Sorkin (for an introduction to the ideas see Sorkin or Dowker). It is similar to causal fermion systems on the conceptional level in that it asserts the causal structure to be fundamental. However, the way this assertion is implemented is substantially different from the causal structure of a causal fermion system.

A causal set is deﬁned to be a locally ﬁnite partially ordered set, or in other words a set $C$ together with a relation $\prec$, called “precedes”, which satisfy the following axioms:

(1) if $x \prec y$ and $y \prec z$ then $ x \prec z$, $\forall x,y,z \in C$ (transitivity);

(2) if $x \prec y$ and $y \prec x$ then $x = y$ $\forall x,y \in C$ (non-circularity);

(3) for any pair of ﬁxed elements $x$ and $z$ of $C$, the set $\{y|x \prec y \prec z\}$ of elements lying between $x$ and $z$ is ﬁnite.

In a causal fermion system, the causal relations are not necessarily transitive except in the continuum limit, and discreteness of spacetime is not required. Furthermore, the set $\mathscr{F}$ on which the causal relations are defined contains a lot more structure. While Sorkin expresses the hope that non gravitational degrees of freedom might be encoded in the causal sets, in the CFS theory this has been proven to be the case.

An interesting parallel, however, is that in both approaches the measure takes a special role. In the Theory of Causal Fermion Systems, it is explicitly introduced as the dynamic entity describing the physical system. In Causal Set Theory, it has to be recovered from “number counting” as the volume element can not be recovered from the causal structure alone. Thus in principle, both theories are compatible with non-Riemannian measures.

*Algebraic Quantum Field Theory*

*Events, Trees, Histories (ETH) Approach to Quantum Theory*

The ETH approach developed by Jürg Fröhlich and collaborators (see for example the review [Fr19]) is intended to represent a completion of Quantum Theory enabling one to solve fundamental conceptual problems, such as the infamous measurement problem. The central postulate of the ETH approach is the so-called *Principle of Diminishing Potentialities*, which enables one to propose a precise definition of *isolated open systems*, come up with a compelling notion of *events* featured by such systems, and to understand the nature of the stochastic time evolution of states in quantum theory. Among central ingredients of the ETH approach are algebras, $\mathcal{E}_{\geq t}$, generated by bounded operators representing potential events that may happen at or after some future time $t$.

There are some striking similarities between the ETH approach to Quantum Theory and causal fermion systems in their mathematical structure. However, there are also essential differences. For a detailed comparison see [FFOP20].

*Canonical Quantum Gravity / Loop Quantum Gravity*

*String Theory*

It is unclear whether the framework of causal fermion systems relates in any way to String Theory. The Theory of Causal Fermion Systems can reproduce the Standard Model without postulating additional particles. It is an interesting open question (link) whether extensions to the Standard Model (including Super Symmetry) can be incorporated in the framework of causal fermion systems.

*Further Approaches*

Besides the here-mentioned approaches there is number of further approaches that we are not familiar enough to comment on. This includes, but is not limited to: Asymptotic safety, causal dynamical triangulation, group field theory, E8 theory and the approach based on octonions.

### Claudio Paganini

Author