# The Theory of Causal Fermion Systems

## Connection to Quantum Field Theory

### Prerequisites

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### Related Topics

So far, the connection between causal fermion systems and quantum field theory has been made on two levels:

## Description Starting from the Continuum Limit

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So far, the connection between causal fermion systems and quantum field theory has been made on two levels. In [qft13] the starting point are the effective field equations obtained in the continuum limit. In order to go beyond classical fields, the procedure is to complement the description by the so-called *mechanism of microscopic mixing*. This mechanism is based on the fact that causal fermion systems do allow for the description of systems which on the microscopic scale are composed of complicated mixtures of wave functions which may satisfy the Dirac equation for different bosonic potentials. Describing this mixing with random matrices, one gets an effective description of the interaction by a unitary time evolution on bosonic and fermionic Fock spaces.

While being a first step in the right direction, the procedure in [qft13] is not fully convincing for the following reasons:

- Working with random matrices reflects our lack of knowledge of the microscopic structure of spacetime. However, it is not so clear whether the microstructure really is “random” or whether it has more, yet unknown, structure. In other words, the use of random matrices still needs a convincing justification.
- The procedure requires assumptions (synchronization, instantaneous recombinations) which are ad-hoc and lack a fundamental explanation.
- Instead of starting from the effective field equations, one should better work on a more fundamental level with the Euler-Lagrange equations of the causal action principle.
- Similarly, the constructions are based on current conservation for the Dirac equation. In order to obtain a more fundamental description, one should work instead with the conservation laws for surface layer integrals.
- Finally, the procedure in [qft13] gives a connection to quantum field theory, but it does not seem to make it possible to go beyond quantum field theory in the sense of overcoming the shortcomings of present quantum field theory or of giving corrections to it.

## Description Starting from the Causal Action Principle

The above shortcoming were the motivation for getting a connection between causal fermion systems and quantum field theory starting directly from the causal action principle. The constructions make essential use of all the inherent structures of a causal fermion systems, in particular of the conservation laws for surface layer integrals and the related Fock space structures.

Restricting attention to *bosonic* interactions, these constructions are carried out in [fockbosonic18]. Indeed it is shown that the dynamics as described by the Euler-Lagrange equations can be reformulated in terms of a norm-preserving linear operator on ${\mathcal{F}}^* \otimes {\mathcal{F}}$. In the so-called *holomorphic approximation*, this operator is invariant on ${\mathcal{F}}$ and its dual, In this case, the time evolution can be described similar to quantum field theory by a unitary operator on the Fock space,

\[ U(t) \::\: {\mathcal{F}} \rightarrow {\mathcal{F}} \:. \]

In this procedure, the remaining task is to justify the holomorphic approximation. Moreover, it remains to include the *fermions. *These problems are currently under investigation. Justifying the holomorphic approximation again uses the mechanism of microscopic mixing. But instead of using random matrices, one analyzes the Euler-Lagrangian microscopically, taking into account the so-called *fragmentation* of the measure as well as the mechanism of *holographic mixing*.

The holomorphic approximation also gives rise to quantum entanglement. The detailed connections and consequences are topics of current research.

### Felix Finster

Author