Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Canonical Construction of the Extended Hilbert Space for Causal Fermion Systems

Felix Finster, Patrick Fischer

TL;DR

This work develops a canonical framework for extending the Hilbert space of physical wave functions in causal fermion systems by exploiting an approximate decoupling of the linearized field equations. It shows that second variations of the causal action decompose as $\delta^2\mathcal{S}=\delta^2\mathcal{S}^{\mathrm{lfe}}+\delta^2\mathcal{S}^{Q}+\mathcal{R}$ with the first two terms nonnegative and the remainder $\mathcal{R}$ small, enabling a decoupled dynamical wave equation and bosonic sector up to small coupling. The paper constructs inhomogeneous and homogeneous solutions within time strips, proves a positive definite, time-invariant commutator inner product on the solution space, and uses these ingredients to canonically complete the extended Hilbert space $\mathscr{H}^{\mathrm{f}}\subset\mathscr{H}_\rho$ with unitary time evolution. A perturbative treatment of the residual coupling terms provides a controlled path from the decoupled equations to the full linearized system, linking the CFS framework with conventional quantum-field-like dynamics. The results offer a principled route to a canonical extended Hilbert space that accommodates Green-type solutions and preserves a positive inner product under evolution, with potential implications for connecting CFS to standard QFT formalisms.

Abstract

It is shown that second variations of the causal action can be decomposed into a sum of three terms, two of which being positive and one being small. This gives rise to an approximate decoupling of the linearized field equations into the dynamical wave equation and bosonic field equations. A concrete construction of homogeneous and inhomogeneous solutions of the dynamical wave equation in time strips is presented. In addition, it is show that the solution space admits a positive definite inner product which is preserved under the time evolution. Based on these findings, a canonical construction of the extended Hilbert space containing these solutions is given.

A Canonical Construction of the Extended Hilbert Space for Causal Fermion Systems

TL;DR

This work develops a canonical framework for extending the Hilbert space of physical wave functions in causal fermion systems by exploiting an approximate decoupling of the linearized field equations. It shows that second variations of the causal action decompose as with the first two terms nonnegative and the remainder small, enabling a decoupled dynamical wave equation and bosonic sector up to small coupling. The paper constructs inhomogeneous and homogeneous solutions within time strips, proves a positive definite, time-invariant commutator inner product on the solution space, and uses these ingredients to canonically complete the extended Hilbert space with unitary time evolution. A perturbative treatment of the residual coupling terms provides a controlled path from the decoupled equations to the full linearized system, linking the CFS framework with conventional quantum-field-like dynamics. The results offer a principled route to a canonical extended Hilbert space that accommodates Green-type solutions and preserves a positive inner product under evolution, with potential implications for connecting CFS to standard QFT formalisms.

Abstract

It is shown that second variations of the causal action can be decomposed into a sum of three terms, two of which being positive and one being small. This gives rise to an approximate decoupling of the linearized field equations into the dynamical wave equation and bosonic field equations. A concrete construction of homogeneous and inhomogeneous solutions of the dynamical wave equation in time strips is presented. In addition, it is show that the solution space admits a positive definite inner product which is preserved under the time evolution. Based on these findings, a canonical construction of the extended Hilbert space containing these solutions is given.

Paper Structure

This paper contains 25 sections, 15 theorems, 212 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Causal Fermion Systems and the Reduced Causal Action Principle
  4. The Physical Wave Functions and the Wave Evaluation Operator
  5. The Restricted Euler-Lagrange Equations
  6. The Linearized Field Equations in Wave Charts
  7. The Conserved Commutator Inner Product
  8. Second Variations with Separated Supports
  9. General Second Variations
  10. Perturbation Expansion of the Eigenvalues
  11. Second Variations of the Effective Action
  12. Approximate Decoupling of the Linearized Field Equations
  13. An Estimate in Time Strips
  14. Evaluation in the Static Setting
  15. Construction of Solutions of the Dynamical Wave Equation
  16. ...and 10 more sections

Key Result

Proposition 2.3

Let $\rho$ be a minimizer of the reduced causal action principle. Then the local trace is constant in spacetime, meaning that Moreover, there are parameters $\mathfrak{r}, \mathfrak{s}>0$ such that the function $\ell$ defined by is minimal and vanishes in spacetime, i.e.

Figures (3)

  • Figure 1: A variation with separated supports.
  • Figure 2: Construction of homogeneous solutions in a time strip.
  • Figure 3: The convolution ${\hat{\mathcal{M}}} * \hat{P}$.

Theorems & Definitions (33)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • Proposition 4.1
  • Proposition 4.2
  • proof
  • ...and 23 more