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The empirical laws for Majorana fields in a curved spacetime

Hideyasu Yamashita

TL;DR

The paper tackles how to formulate observable empirical laws for Majorana fields in curved spacetime by embedding the problem in the CAR algebra framework, thereby avoiding reliance on a fixed vacuum. It builds a curved-spacetime extension of fermionic quantum field theory by defining Majorana/Dirac fields as CAR algebras over test-function spaces tied to spinor bundles, with observables coming from the Majorana (real) or Dirac (complex) structures. Through finite- and infinite-dimensional analyses, it derives explicit empirical-law forms, notably the 1-path/2-path densities (Sorkin densities) and their holonomy interpretations, and then provides concrete, dimension-specific laws such as a fundamental two-operator probability in the Majorana case. The work culminates in a general, dimension-agnostic empirical-law framework that bridges algebraic QFTCS, geometric quantization notions (holonomy), and practical verifiability of predictions for Majorana fields in curved backgrounds. This contributes a principled route to identify verifiable predictions in QFTCS and enhances understanding of fermionic observables beyond flat spacetime.

Abstract

This article is a sequel to our previous paper (arXiv:2511.12311), where we considered the conceptual problem on the empirical laws for the Klein\textendash Gordon quantum field theory in curved spacetime (QFTCS), and we will consider the similar problems for the Majorana field on curved spacetime here. A ``law'' in theoretical physics is said to be observable or empirical only if it can be verified/falsified by some experimental procedure. The notion of empiricality/observability becomes far more unclear in QFTCS, than in QFT in Minkowski (flat) spacetime (QFTM), mainly because QFTCS lacks the notion of vacuum. This could potentially undermine the status of QFTCS as a physical (not only mathematical) theory. We consider this problem for the Majorana field in curved spacetime, and examine some examples of the empirical laws.

The empirical laws for Majorana fields in a curved spacetime

TL;DR

The paper tackles how to formulate observable empirical laws for Majorana fields in curved spacetime by embedding the problem in the CAR algebra framework, thereby avoiding reliance on a fixed vacuum. It builds a curved-spacetime extension of fermionic quantum field theory by defining Majorana/Dirac fields as CAR algebras over test-function spaces tied to spinor bundles, with observables coming from the Majorana (real) or Dirac (complex) structures. Through finite- and infinite-dimensional analyses, it derives explicit empirical-law forms, notably the 1-path/2-path densities (Sorkin densities) and their holonomy interpretations, and then provides concrete, dimension-specific laws such as a fundamental two-operator probability in the Majorana case. The work culminates in a general, dimension-agnostic empirical-law framework that bridges algebraic QFTCS, geometric quantization notions (holonomy), and practical verifiability of predictions for Majorana fields in curved backgrounds. This contributes a principled route to identify verifiable predictions in QFTCS and enhances understanding of fermionic observables beyond flat spacetime.

Abstract

This article is a sequel to our previous paper (arXiv:2511.12311), where we considered the conceptual problem on the empirical laws for the Klein\textendash Gordon quantum field theory in curved spacetime (QFTCS), and we will consider the similar problems for the Majorana field on curved spacetime here. A ``law'' in theoretical physics is said to be observable or empirical only if it can be verified/falsified by some experimental procedure. The notion of empiricality/observability becomes far more unclear in QFTCS, than in QFT in Minkowski (flat) spacetime (QFTM), mainly because QFTCS lacks the notion of vacuum. This could potentially undermine the status of QFTCS as a physical (not only mathematical) theory. We consider this problem for the Majorana field in curved spacetime, and examine some examples of the empirical laws.
Paper Structure (14 sections, 15 theorems, 141 equations)

This paper contains 14 sections, 15 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. CAR algebra
  3. Definition of the Majorana and Dirac fields in curved spacetime
  4. Dirac matrices
  5. The Dirac/Majorana field in Minkowski space
  6. Quantum fields in curved spacetime
  7. The simplest examples of empirical laws in a CAR algebra
  8. Finite-dimensional case
  9. Infinite-dimensional case
  10. Empirical laws: a general consideration
  11. Finite-dimensional CAR algebra
  12. Empirical laws in the $n=2$ case
  13. Empirical laws in general dimension
  14. Empirical law for the Majorana field

Key Result

Proposition 3.1

Let $\gamma_{0},\dots,\gamma_{3}\in{\rm Mat}(4,\mathbb{C})$ and $\gamma_{0}',\dots,\gamma_{3}'\in{\rm Mat}(4,\mathbb{C})$ be two sets of the above gamma matrices. Then there exists $L\in GL(4,\mathbb{C})$ such that $\gamma_{\mu}'=L\gamma_{\mu}L^{-1}$, $\mu=1,...,4$. Such $L$ is unique up to scalar f

Theorems & Definitions (23)

  • Definition 2.1
  • Definition 2.2
  • Proposition 3.1: Pauli's fundamental theorem
  • Proposition 3.2
  • Proposition 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Lemma 4.1
  • proof
  • Lemma 5.1: Sorkin additivity
  • ...and 13 more