The Theory of Causal Fermion Systems
The Causal Action Principle
The core of every modern physical theory is an action principle that selects the physically relevant configurations as being critical points of a suitable action functional. In General Relativity, for example, among all possible Lorentzian metrics on a manifold, only those are considered to be physical that are critical points of the Einstein-Hilbert action, and hence satisfy Einstein’s field equations. Quantum Field Theory, on the other hand, is based on an action defined as a spacetime integral of a Lagrangian.
In the context of causal fermion systems, we want to formulate an action principle over the set of linear operators
Similar as already explained in the formulation of the causal relations on
Just a quick reminder: For a finite rank operator the algebraic multiplicity of an eigenvalue is the order of the factor in the characteristic polynomial. This does not necessarily agree with the geometric multiplicity, i.e. the dimension of the eigenspace to a particular eigenvalue.
To make a long story short, we will just present the causal action principle at the heart of the Causal Fermion Systems framework and explain the significance of its various constituent parts in detail afterward.
Lagrangian
Causal action
It is worth noting that the Lagrangian in the second formulation is intimately related to the definition of causality in the causal fermion system. In particular, one sees immediately that spacelike separated points do not contribute to the Lagrangian. A physical system is then given by a measure
volume constraint:
trace constraint:
boundedness constraint:
The need for the volume constraint is pretty clear: if we remove that constraint there will always be a trivial minimizer with measure zero (as the Lagrangian is strictly positive). The situation for the trace constraint is similar. If
A measure
→ The Euler-Lagrange equations
→ The linearized field equations
In the Theory of Causal Fermion systems, the Euler-Lagrange equations play the role of the usual physical equations (like Maxwell’s equations, Einstein’s equations or the Yang-Mills equations). The precise connection is obtained in the so-called continuum limit. Moreover, these Euler-Lagrange equations are also expected to describe the dynamics of quantum fields, as is outlined in the mathematics section
→ Connection to quantum field theory

Claudio Paganini
Author