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The fermionic massless modular Hamiltonian

Francesca La Piana, Gerardo Morsella

TL;DR

The paper derives explicit one-particle modular Hamiltonians for massless fermionic fields (Weyl, Dirac, Majorana) in the unit double cone, by leveraging conformal symmetry and previous wedge/double-cone results. It shows that the modular Hamiltonian is the second quantization of the corresponding one-particle generator and provides an explicit spacetime-formulation expression $i\log \Delta_{O_1} \Psi_0(\boldsymbol{x}) = -\pi[(1 - r^2) \partial_k - x_k] \gamma^0 \gamma^k \Psi_0(\boldsymbol{x})$ with $r=|\boldsymbol{x}|$, along with the Weyl-form modular action on Cauchy data. The work extends to Dirac and Majorana fields, giving their one-particle modular Hamiltonians and a detailed relative-entropy formula for Majorana states localized in the unit ball, expressed both in momentum space and via the energy density $T_{00}$. This connects modular theory, conformal symmetry, and quantum information measures in free fermionic QFT and offers a path toward massive theories through the Cauchy-data framework.

Abstract

We provide an explicit expression for the modular hamiltonian of the von Neumann algebras associated to the unit double cone for the (fermionic) quantum field theories of the 2-component Weyl (helicity 1/2) field, and of the 4-component massless Dirac and Majorana fields. To this end, we represent the one particle spaces of these theories in terms of solutions of the corresponding wave equations, and obtain the action of the modular group on them. As an application, we compute the relative entropy between the vacuum of the massless Majorana field and one particle states associated to waves with Cauchy data localized in the spatial unit ball.

The fermionic massless modular Hamiltonian

TL;DR

The paper derives explicit one-particle modular Hamiltonians for massless fermionic fields (Weyl, Dirac, Majorana) in the unit double cone, by leveraging conformal symmetry and previous wedge/double-cone results. It shows that the modular Hamiltonian is the second quantization of the corresponding one-particle generator and provides an explicit spacetime-formulation expression with , along with the Weyl-form modular action on Cauchy data. The work extends to Dirac and Majorana fields, giving their one-particle modular Hamiltonians and a detailed relative-entropy formula for Majorana states localized in the unit ball, expressed both in momentum space and via the energy density . This connects modular theory, conformal symmetry, and quantum information measures in free fermionic QFT and offers a path toward massive theories through the Cauchy-data framework.

Abstract

We provide an explicit expression for the modular hamiltonian of the von Neumann algebras associated to the unit double cone for the (fermionic) quantum field theories of the 2-component Weyl (helicity 1/2) field, and of the 4-component massless Dirac and Majorana fields. To this end, we represent the one particle spaces of these theories in terms of solutions of the corresponding wave equations, and obtain the action of the modular group on them. As an application, we compute the relative entropy between the vacuum of the massless Majorana field and one particle states associated to waves with Cauchy data localized in the spatial unit ball.
Paper Structure (6 sections, 13 theorems, 132 equations, 1 figure)

This paper contains 6 sections, 13 theorems, 132 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Weyl field modular group in the spacetime formulation
  3. Weyl field modular group in the Cauchy data formulation
  4. Modular group for massless Dirac and Majorana fields
  5. Estimates for Thm. \ref{['thm:modham']}
  6. Computations for the relative entropy of the Majorana field

Key Result

Lemma 2.1

For any $\lambda \in \mathbb{R}$ and $x\in {\mathbb R}^4\setminus S_\lambda$, the Jacobian determinant of $\nu_{\lambda}$ is given by

Figures (1)

  • Figure 1: The $O_1$ modular flow on the $x_0, x_1$ plane. The trajectories are continuous on the conformal compactification of Minkowski space. Those outside $O_1$ are singular on Minkowski, and their different branches share the same color.

Theorems & Definitions (25)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 15 more