# The Theory of Causal Fermion Systems

## Quantum Field Theory

## Quantum Field Theory

It is the general concept that the theory of causal fermion systems incorporates quantum field theory. In other words, no “quantization procedure” is needed. Instead, the interaction described by the causal action principle should give rise to interacting quantum field theory in a suitable limiting case, involving both quantized fermionic and quantized bosonic fields.

So far, this concept has been worked out in several stages. In [qft13] quantum electrodynamics (i.e. the fully quantized Maxwell-Dirac system) was derived from the causal action principle, however based on the effective field equations obtained in the continuum limit. More recently, the connection to quantum field theory was established starting from the fundamental structures of the causal fermion system (more precisely, the linearized field equations, the dynamical wave equation, the conservation laws for surface layer integrals and the related Fock space structures):

- Restricting attention to
*bosonic*interactions, in [FK18] it is shown that the dynamics as described by the causal action principle can be reformulated in terms of a norm-preserving linear operator on Fock spaces. In the so-called*holomorphic approximation*, the time evolution can be described similar to quantum field theory by a unitary operator on the Fock space. - In [FK21] the construction was generalized to include fermions. Moreover, an abstract construction of a
*quantum state*$\omega^t$ at time $t$ is given, being a positive linear mapping from the field algebra to the complex numbers. This state can be represented by a density operator on the usual Fock space generated by the bosonic and fermionic field operators. - In [FKR22], it is shown that this quantum state allows for the description of general entangled states. This analysis explains in detail how quantum entanglement is encoded in a causal fermion system.

The next step will be to analyze the time evolution of the state. This is a topic of ongoing research.

→ Quantum field theory and entanglement on the research pages

More details on the connection to quantum field theory can be found at

→ Connection to quantum field theory in the mathematics section.

There is also a video of a seminar talk on the quantum state of a causal fermion system given in the *Colloquium “Mathematical Physics Regensburg-Munich”* in February 2021.

### Felix Finster

Author