The Theory of Causal Fermion Systems

We begin our introduction to the physical aspects of the Theory of Causal Fermion Systems with this well-known quote for a very good reason: the core of what we present on this website is not only a new model for our physical reality, but a completely new mathematical framework within which this novel theory can be formulated.

We first give a broad sketch of the mathematical framework before we explain how one can construct a particular physical model based on this framework. Learning a new mathematical framework takes time, there is no way around that, but here we try to give you the gist of it, so you can get a rough intuition for the physical implications.
(An introduction focusing on the mathematical ideas can be found here).

The Mathematical Framework

When we set out to look for a good mathematical structure, we first have to remind ourselves what is supposed to be achieved. So what are the central elements of the theories we try to unify? Two of the core features of General Relativity are that it is background independent and diffeomorphism invariant. Hence, spacetime, which in mathematical terms is a Lorentzian manifold, does not exist inside any external entity, it is everything there is, and any two observers inside a spacetime agree on the physical laws of the universe independent of their state of motion and independent of what coordinate system they choose to describe the world around them.

Quantum Theories, on the other hand, usually start with a set of selfadjoint linear operators on a Hilbert Space, where the Hilbert space is usually taken as a suitable space of functions defined in a given background spacetime.

The mathematical framework of causal fermion systems merges these concepts in a novel manner in the concept of an operator manifold with an intrinsic causal structure. We will now give a rough idea of how this is achieved.

The setup of the mathematical framework starts with an abstract Hilbert space as a background structure. It turns out that a suitable subset of the linear operators acting on this Hilbert space has in fact a manifold structure. We will refer to this operator manifold as the space of all possible events for a physical system. We are then able to formulate causal relations between events that tells us whether they are timelike, spacelike or lightlike separated. These relations are defined via the structure of the eigenvalues of operator products to ensure basis independence, which takes the place of observer independence in General Relativity.

Finally, we can formulate an action principle over the space of possible events. The causal relations between possible events enter the action principle in a crucial manner, and spacelike separated events do not contribute to the action. Physical systems are then realized as minimizers of this so-called causal action principle. It selects which events are actually realized in a particular physical system, as the system will in general not be supported in the entire operator manifold. One intuitive way of looking at it the causal action principle is that a minimizer tries to maximize the number of pairs of events that are spacelike separated.

Thus the causal action principle selects a set of events based on their causal relations. This set is then referred to as spacetime. One can then define a rich set of emergent objects with respect to this selected set of events.

Last but not least, every vector in the abstract Hilbert Space can be represented by a wave function in the emergent spacetime. This representation eventually gives a usual “Hilbert space formed of wave functions in spacetime,” giving a connection to the standard formalism of quantum theory. Moreover, we can derive the geometric structure from the net of causal relations between all events which are realized in a particular physical system.

How the Theory gives rise to a Model

The crucial step for a physical theory is that it must be able to model our reality. The Standard Model of particle physics is one particular Quantum Field Theory, and the concordance model of cosmology is one particular model in Einstein’s General Relativity. We now explain how to construct a particular causal fermion system to model the world we live in. Only when we succeed in this step, the mathematical framework becomes a candidate for a physical theory.

This program has been carried out in detail for the simplest spacetime geometry, namely Minkowski space. In order to obtain a causal fermion system, we first have to choose a Hilbert space. The space of negative-energy solutions of the Dirac equation (i.e. the Dirac sea) turns out to be a good choice. In the next step, we must find a way to assign an operator to every point in Minkowski space. Here the local correlation operator does the job. For every pair of vectors in the Hilbert space it returns their correlation at a particular point in spacetime. However, for this construction to be well-defined, we need to introduce a regularization on a length scale $\varepsilon$. Then the local correlation operator associates to every point of Minkowski space a symmetric linear operator on a Hilbert space. By varying the spacetime point, we obtain the local correlation map, which maps Minkowski space to the linear operators and thereby defines a Causal Fermion System. In this way, one can describe Minkowski space by a causal fermion system. Finally, in order to recover the structures of Minkowski space from the structures of the causal fermion system, we have to consider the limiting case when the regularization parameter $\varepsilon$ tends to zero. This is the so called continuum limit for causal fermion systems. Taking this limit is in fact highly non-trivial and involves a lot of subtle analysis. However, currently the study of the continuum limit is our best tool to obtain statements about the real world if modelled by a causal fermion system.

As a side remark, it is worth noting that the Dirac sea vacuum is to be seen as an effective model describing a particular minimizing causal fermion system. It is one particular physical system that we can describe as a minimizing causal fermion system. But we should really only think of it as an effective description, in the sense that it describes only the macroscopic structure of spacetime, whereas its microscopic structure on the Planck scale is essentially unknown. This is an absolutely crucial point, as in the following we will always start from the “wrong” direction, with a local correlation map of an effective description, to crank out the physical properties of that system in the continuum limit. The idea of the Dirac Sea did, however, play an important role in the conception of the causal fermion systems framework, and most of the existing literature is written with that point of view in mind. A more detailed motivation for why it is a natural starting point can be found here.

The World as we know it

When setting up a new fundamental theory, the first question we have to answer is: how do the familiar theories and valuable concepts  from these theories emerge from the new fundamental description. At this point, it is worth noting that the Theory of Causal Fermion Systems fits into the “emergent gravity” corner among the various approaches to unification. Hence in the strict sense, the theory does not give us a quantization of gravity. In fact, it turns out that all the classical fields are emergent as an effective description. But let us better proceed step by step.

On a heuristic level, the physical world as we experience it emerges from perturbations of the local correlation map, which we can think of as describing perturbations of the Dirac sea.  To give you an idea for how the bosonic fields emerge, we start again from the “wrong” direction:  Suppose we look at the Dirac sea in Minkowski  space  and  turn  on  a  small  electromagnetic  field.   Since  the Dirac equation couples to the electromagnetic field, this leads to a “deformation” of all  the  states  in  the  Dirac  Sea.  This  again  leads  to  a  modification  of  the  causal correlations and hence of the causal action.

Now in the Theory of Causal Fermion Systems, the logic is essentially reversed (here you find a detailed discussion of the logic underlying the emergence of the classical field equations in the Continuum limit). Minimizing the causal action determines the causal correlations of the states in the Dirac Sea. The bosonic fields then emerge as an effective description of the collective “deformation” of the states in the Dirac Sea. The field equations for the bosonic fields then emerge from the Euler Lagrange equations of the causal action principle in the limiting regime when the regularization length tends to zero. Namely, we demand for the singularities in the continuum limit to be well behaved. This concept is the work horse behind all the results discussed below. One of the interesting observations from the analysis of the electromagnetic field is that, in order to remove the singularity related to the vacuum polarization,  we need to assume that the Hilbert space involves the negative-energy states of at least three generations of fermions.

A Sketch of the Standard Model and Gravity

In this final section we provide a brief and heavily condensed overview of physically relevant results obtained in the framework of causal fermion systems. Note that most of these results have so far only been obtained by analyzing causal fermion systems in a neighborhood of Minkowski space (this is explained in more detail here).

The fermionic one-particle state and gravity emerge in a similar fashion as the electromagnetic field when considering different perturbations of the Dirac sea. In order to obtain the Standard Model of particle physics, we put in the number of fermionic sectors (eight) and the fact that the chiral symmetry is broken in the neutrino sector.  The requirement that the Euler-Lagrange equations are well-behaved and satisfied in the continuum limit requires for the coupling between the eight fermionic sectors to obey the gauge group of the standard model.  This leads to a so-called spontaneous block formation where two sectors pair up, one for each isospin, and the remaining sectors encode the charged leptons and neutrinos. At this point we have to mention that the continuum limit for the Higgs field has not been worked out yet.  It is clear from the considerations in the book [cfs16] that for the Higgs field to emerge it takes at least two fermionic sectors. Under the assumption that the Higgs field can be derived, we thus get the full standard  model  including  a good  theoretical  motivation  for  the  fact  that  we  observe three generations of fermions in each sector.

In this last paragraphs we discuss a couple of assorted facts that seem relevant with respect to various open questions in theoretical physics.

It is interesting to observe that the coupling constants for the gauge fields of the standard  model  are  of  order  one  in  the  continuum  limit.   Gravitation,  on  the  other hand,  enters  the  scheme  as  a  next  to  next  to  leading  order  effect  and  hence  the coupling constant is of order epsilon squared. (For a simpler example of next-to-leading order effects in classical theories see eg. the gravitational Spin Hall effect)

This is interesting with respect to the hierarchy problem.   We  collected a number  of  quantum  gravity wish lists from  the  literature and comment on those points where the Theory of Causal Fermion Systems might provide an answer, or at least novel insights. It can also be shown that the spacetime emerging in the continuum limit has exactly one time dimension.  Furthermore, the maximal dimension of the Clifford Algebra is five-dimensional.  If we reserve one dimension for chirality, this tells us that the emerging spacetime has at most three spatial dimensions.