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The Fermionic Massless Modular Hamiltonian.

Francesca La Piana1, Gerardo Morsella2

  • 1Department of Mathematics, University of Oslo, P.O. Box 1053, 0316 Blindern, Oslo, Norway.

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This study derives the modular Hamiltonian for quantum field theories, including Weyl, Dirac, and Majorana fields. It calculates relative entropy, offering insights into quantum information in curved spacetime.

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Area of Science:

  • Quantum Field Theory
  • Mathematical Physics
  • Quantum Information Theory

Background:

  • Modular Hamiltonians are crucial for understanding quantum entanglement and information in quantum field theories.
  • Previous studies have explored modular structures in simpler quantum systems, but explicit calculations for relativistic fermionic fields remain challenging.
  • Von Neumann algebras associated with spacetime regions provide a framework for defining local observables and their associated modular properties.

Purpose of the Study:

  • To derive an explicit expression for the modular Hamiltonian of von Neumann algebras related to the unit double cone.
  • To analyze the modular properties of specific fermionic quantum field theories: 2-component Weyl, 4-component massless Dirac, and Majorana fields.
  • To compute the relative entropy between the vacuum state and one-particle states in the massless Majorana field theory.

Main Methods:

  • Representing one-particle spaces using solutions to the respective wave equations.
  • Determining the action of the modular group on these one-particle spaces.
  • Applying these methods to calculate the relative entropy for localized Cauchy data.

Main Results:

  • An explicit formula for the modular Hamiltonian is provided for the specified fermionic quantum field theories in the unit double cone.
  • The action of the modular group on the one-particle spaces of Weyl, Dirac, and Majorana fields is explicitly obtained.
  • The relative entropy between the vacuum and specific one-particle states of the massless Majorana field is computed.

Conclusions:

  • The study successfully provides explicit expressions for modular Hamiltonians in relativistic fermionic quantum field theories.
  • The findings offer a concrete method for investigating quantum information properties, such as relative entropy, in these theories.
  • This work contributes to a deeper understanding of the interplay between quantum field theory, von Neumann algebras, and quantum information in geometric settings.