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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. Moreover, it gives quantum mechanics as a limiting case and has revealed close connections to quantum field theory. Therefore, it is a candidate for a unified physical theory. Instead of introducing physical objects on a preexisting spacetime manifold, the general concept is to derive spacetime as well as all the objects therein as secondary objects from the structures of an underlying causal fermion system. This concept also makes it possible to generalize notions of differential geometry to the non-smooth setting. In particular, one can describe situations when spacetime no longer has a manifold structure on the microscopic scale (like a spacetime lattice or other discrete or continuous structures on the Planck scale). As a result, the theory of causal fermion systems is a proposal for quantum geometry and an approach to quantum gravity. Causal fermion systems were introduced by Felix Finster and collaborators.
The physical starting point is the fact that the Dirac equation in Minkowski space has solutions of negative energy which are usually associated to the Dirac sea. 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. This leads to the idea that the wave functions of all occupied states (including the sea states) should be regarded as the basic physical objects, and that all structures in spacetime arise as a result of the collective interaction of the sea states with each other and with the additional particles and "holes" in the sea. Implementing this picture mathematically leads to the framework of causal fermion systems.
More precisely, the correspondence between the above physical situation and the mathematical framework is obtained as follows. All occupied states span a Hilbert space of wave functions in Minkowski space
(where
The above construction can also be carried out in more general spacetimes. Moreover, taking the abstract definition as the starting point, causal fermion systems allow for the description of generalized "quantum spacetimes." The physical picture is that one causal fermion system describes a spacetime together with all structures and objects therein (like the causal and the metric structures, wave functions and quantum fields). In order to single out the physically admissible causal fermion systems, one must formulate physical equations. In analogy to the Lagrangian formulation of classical field theory, the physical equations for causal fermion systems are formulated via a variational principle, the so-called causal action principle. Since one works with different basic objects, the causal action principle has a novel mathematical structure where one minimizes a positive action under variations of the universal measure. 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.
In this section the mathematical framework of causal fermion systems is introduced.
A causal fermion system of spin dimension
The measure
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.[1]
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.
Given a Hilbert space
Moreover, the spectral weight
The Lagrangian is introduced by
The causal action is defined by
The causal action principle is to minimize
Here on
The constraints prevent trivial minimizers and ensure existence, provided that
In contemporary physical theories, the word spacetime refers to a Lorentzian manifold
For a causal fermion system
With the topology induced by
For
This notion of causality fits together with the "causality" of the above causal action in the sense that if two spacetime points
Let
distinguishes the future from the past. In contrast to the structure of a partially ordered set, the relation "lies in the future of" is in general not transitive. But it is transitive on the macroscopic scale in typical examples.[3][4]
For every
is an indefinite inner product on
A wave function
On wave functions for which the norm
is finite (where
Together with the topology induced by the norm
To any vector
(where
The kernel of the fermionic projector
(where
which has the dense domain of definition given by all vectors
As a consequence of the causal action principle, the kernel of the fermionic projector has additional normalization properties[5] which justify the name projector.
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
where the tangent space
Similarly, the metric connection gives rise to metric curvature. These geometric structures give rise to a proposal for a quantum geometry.[3]
A minimizer
(with two Lagrange parameters
For the analysis, it is convenient to introduce jets
hold for any test jet
Families of solutions of the Euler-Lagrange equations are generated infinitesimally by a jet
to be satisfied for all test jets
The Euler-Lagrange equations describe the dynamics of the causal fermion system, whereas small perturbations of the system are described by the linearized field equations.
In the setting of causal fermion systems, spatial integrals are expressed by so-called surface layer integrals.[5][6][7] In general terms, a surface layer integral is a double integral of the form
where one variable is integrated over a subset
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.[8] In the so-called holomorphic approximation, the time evolution respects the complex structure, giving rise to a unitary time evolution on the bosonic Fock space.
If
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.
Starting on any globally hyperbolic Lorentzian spin manifold
(where
one obtains a causal fermion system. For the local correlation operators to be well-defined,
The Euler-Lagrange equations corresponding to the causal action principle have a well-defined limit if the spacetimes
Taking the non-relativistic limit of the Dirac equation, one obtains the Pauli equation or the Schrödinger equation, giving the correspondence to quantum mechanics. Here
Likewise, for a system involving neutrinos in spin dimension 4, one gets effectively a massive
For the just-mentioned system involving neutrinos,[10] the continuum limit also yields the Einstein field equations coupled to the Dirac spinors,
up to corrections of higher order in the curvature tensor. Here the cosmological constant
Starting from the coupled system of equations obtained in the continuum limit and expanding in powers of the coupling constant, one obtains integrals which correspond to Feynman diagrams on the tree level. 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).[12] The detailed analysis and comparison with standard quantum field theory is work in progress.[8]