The Theory of Causal Fermion Systems

The Standard Model and General Relativity

Prerequisites

What is a causal fermion system
The causal action principle
The continuum limit
The logic underlying the derivation of the classical field equations

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Why the Dirac sea?
What is fundamental?

The Standard Model of Elementary Particles

In [cfs, Chapters 3-5] the continuum limit is worked out for systems of increasing complexity, leading to systems which reproduce all the interactions of the Standard Model plus gravity. In order to define these models, one must specify the causal fermion system of the vacuum. The output is the structure of the bosonic interaction including the effective field equations.

More precisely, specifying the causal fermion system of the vacuum entails:

  • Specify the number of direct summands of Dirac wave functions (the so-called sectors; the spin dimension is always four times the number of sectors)
  • Specify the number of Dirac seas in each sectors (the number of generations)
  • Specify the mass of each Dirac sea
  • Break the chiral symmetry of one sector (the neutrino sector)

In order for the resulting vacuum causal fermion system to satisfy the Euler-Lagrange equations, the masses must be the same in all sectors except for the neutrino sector, where one gets conditions for how the chiral symmetry is broken. Allowing for bosonic potentials, one gets the following results:

  • The gauge group of the effective bosonic potentials
  • Effective field equations
  • In order to get deterministic equations, the number of generations must be equal to three.
  • The field equations of left-handed bosonic potentials are  massive.
  • The linearized Einstein equations

In particular, the model in [cfs, Chapter 5] gives the Lagrangian of the Standard Model after spontaneous symmetry breaking. There are also scalar degrees of freedom which can be identified with the Higgs field (see [cfs, Sections 3.8.5 and 5.5]). However, the corresponding field equations have not yet been worked out.

→ Quantum field theory

General Relativity

In [cfs16, Chapter 4] it is shown that in the continuum limit, the Euler-Lagrange equations give rise to the Einstein equations, up to possible higher order corrections in curvature (which scale in powers of $(\delta^2\:\text{Riem})$, where $\delta$ is the Planck length and $\text{Riem}$ is the curvature tensor). The gravitational coupling constant $G \sim \delta^2$ is determined by the length scale $\delta$ of the microscopic spacetime structure.

For details and an outline of other methods to clarify the connections  see the mathematics section
→ Connection to General Relativity

Felix Finster

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