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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.^[1][2] Moreover, it gives quantum mechanics as a limiting case and has revealed close connections to quantum field theory.^[3][4] 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.^[5][6] 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.

Motivation and physical concept

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 ${\displaystyle {\hat {M}}}$. The observable information on the distribution of the wave functions in spacetime is encoded in the local correlation operators ${\displaystyle F(x),x\in {\hat {M}},}$ which in an orthonormal basis ${\displaystyle (\psi _{i})}$ have the matrix representation

${\displaystyle {\big (}F(x){\big )}_{j}^{i}=-{\overline {\psi _{i}(x)}}\psi _{j}(x)}$

(where ${\displaystyle {\overline {\psi }}}$ is the adjoint spinor). In order to make the wave functions into the basic physical objects, one considers the set ${\displaystyle \{F(x)\,|\,x\in {\hat {M}}\}}$ as a set of linear operators on an abstract Hilbert space. The structures of Minkowski space are all disregarded, except for the volume measure ${\displaystyle d^{4}x}$, which is transformed to a corresponding measure on the linear operators (the "universal measure"). 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.

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.

General mathematical setting

In this section the mathematical framework of causal fermion systems is introduced.

Definition of a causal fermion system

A causal fermion system of spin dimension ${\displaystyle n\in \mathbb {N} }$ is a triple ${\displaystyle ({\mathcal {H}},{\mathcal {F}},\rho )}$ where

• ${\displaystyle ({\mathcal {H}},\langle .|.\rangle _{\mathcal {H}})}$ is a complex Hilbert space.
• ${\displaystyle {\mathcal {F}}}$ is the set of all self-adjoint linear operators of finite rank on ${\displaystyle {\mathcal {H}}}$ which (counting multiplicities) have at most ${\displaystyle n}$ positive and at most ${\displaystyle n}$ negative eigenvalues.
• ${\displaystyle \rho }$ is a measure on ${\displaystyle {\mathcal {F}}}$.

The measure ${\displaystyle \rho }$ is referred to as the universal 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.^[7]

The causal action principle

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 ${\displaystyle ({\mathcal {H}},\langle .|.\rangle _{\mathcal {H}})}$ and the spin dimension ${\displaystyle n}$, the set ${\displaystyle {\mathcal {F}}}$ is defined as above. Then for any ${\displaystyle x,y\in {\mathcal {F}}}$, the product ${\displaystyle xy}$ is an operator of rank at most ${\displaystyle 2n}$. It is not necessarily self-adjoint because in general ${\displaystyle (xy)^{*}=yx\neq xy}$. We denote the non-trivial eigenvalues of the operator ${\displaystyle xy}$ (counting algebraic multiplicities) by

${\displaystyle \lambda _{1}^{xy},\ldots ,\lambda _{2n}^{xy}\in {\mathbb {C} }.}$

Moreover, the spectral weight ${\displaystyle |.|}$ is defined by

${\displaystyle |xy|=\underset{i=1}{\overset{2n}{\sum <!--\; \sum \; -->}}|{\mathrm{\lambda <!--\; \lambda \; -->}}_i^{xy}|\text{and}|(xy{)}^2|}$Zdroj:https://en.wikipedia.org?pojem=Causal_fermion_system
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