The theory of causal fermion systems is an approach to describe fundamental physics.
It provides a unification of the weak, the strong and the electromagnetic forces with gravity at the level of classical field theory.
This concept also makes it possible to generalize notions of differential geometry to the non-smooth setting.
Taking the concept seriously that the states of the Dirac sea form an integral part of the physical system, one finds that many structures (like the causal and metric structures as well as the bosonic fields) can be recovered from the wave functions of the sea states.
The observable information on the distribution of the wave functions in spacetime is encoded in the local correlation operators
In order to make the wave functions into the basic physical objects, one considers the set
The resulting structures, namely a Hilbert space together with a measure on the linear operators thereon, are the basic ingredients of a causal fermion system.
Moreover, taking the abstract definition as the starting point, causal fermion systems allow for the description of generalized "quantum spacetimes."
The connection to conventional physical equations is obtained in a certain limiting case (the continuum limit) in which the interaction can be described effectively by gauge fields coupled to particles and antiparticles, whereas the Dirac sea is no longer apparent.
As will be outlined below, this definition is rich enough to encode analogs of the mathematical structures needed to formulate physical theories.
In particular, a causal fermion system gives rise to a spacetime together with additional structures that generalize objects like spinors, the metric and curvature.
Moreover, it comprises quantum objects like wave functions and a fermionic Fock state.
[7] Inspired by the Langrangian formulation of classical field theory, the dynamics on a causal fermion system is described by a variational principle defined as follows.
In contemporary physical theories, the word spacetime refers to a Lorentzian manifold
This implies additional inherent structures that correspond to and generalize usual objects on a spacetime manifold.
This corresponds to the physical notion of causality that spatially separated spacetime points do not interact.
In contrast to the structure of a partially ordered set, the relation "lies in the future of" is in general not transitive.
Being an operator from one spin space to another, the kernel of the fermionic projector gives relations between different spacetime points.
This fact can be used to introduce a spin connection The basic idea is to take a polar decomposition of
It is possible to express the usual conservation laws for charge, energy, ... in terms of surface layer integrals.
Based on the conservation laws for the above surface layer integrals, the dynamics of a causal fermion system as described by the Euler–Lagrange equations corresponding to the causal action principle can be rewritten as a linear, norm-preserving dynamics on the bosonic Fock space built up of solutions of the linearized field equations.
and taking the wedge product of the corresponding wave functions gives a state of an
Causal fermion systems incorporate several physical principles in a specific way: Causal fermion systems have mathematically sound limiting cases that give a connection to conventional physical structures.
must consist of continuous sections, typically making it necessary to introduce a regularization on the microscopic scale
finite-dimensional in order to ensure the existence of the fermionic Fock state as well as of minimizers of the causal action), such that the corresponding wave functions go over to a configuration of interacting Dirac seas involving additional particle states or "holes" in the seas.
For example, for a simplified model involving three elementary fermionic particles in spin dimension two, one obtains an interaction via a classical axial gauge field
Likewise, for a system involving neutrinos in spin dimension 4, one gets effectively a massive
[1] For the just-mentioned system involving neutrinos,[2] the continuum limit also yields the Einstein field equations coupled to the Dirac spinors, up to corrections of higher order in the curvature tensor.
Fermionic loop diagrams arise due to the interaction with the sea states, whereas bosonic loop diagrams appear when taking averages over the microscopic (in generally non-smooth) spacetime structure of a causal fermion system (so-called microscopic mixing).
[3] The detailed analysis and comparison with standard quantum field theory is work in progress.