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Modular Operator for Null Plane Algebras in Free Fields.

Vincenzo Morinelli1, Yoh Tanimoto1, Benedikt Wegener1

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Communications in Mathematical Physics
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We analyzed quantum field theory algebras in null plane regions. The study shows one-particle structure decomposition and validates quantum null energy conditions (QNEC) in free fields, calculating relative entropy for null cut algebras.

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Area of Science:

  • Quantum Field Theory
  • Mathematical Physics
  • Conformal Field Theory

Background:

  • Observables in quantum field theory are localized in specific spacetime regions.
  • The null plane is a significant setting for studying quantum field theory due to its unique geometric properties.
  • Understanding the structure of algebras generated by local observables is crucial for fundamental insights into quantum field theory.

Purpose of the Study:

  • To investigate the algebraic structure of observables in quantum field theory localized in the null plane.
  • To analyze the decomposition of the one-particle structure and modular operator in a scalar free field theory.
  • To establish the validity of the quantum null energy condition (QNEC) for null cut algebras and compute relative entropy.

Main Methods:

  • Consideration of algebras generated by observables in quantum field theory localized in null plane regions.
  • Decomposition of the one-particle structure into a continuous direct integral of lightlike fibres for a scalar free field theory.
  • Application of modular operator decomposition.
  • Computation of the relative entropy of null cut algebras with respect to the vacuum and coherent states.

Main Results:

  • The one-particle structure in a scalar free field theory on the null plane is shown to decompose into a continuous direct integral of lightlike fibres.
  • The modular operator is shown to decompose accordingly with the one-particle structure.
  • A specific form of the quantum null energy condition (QNEC) is demonstrated to be valid in free fields, involving causal completions of null plane half-spaces (null cuts).
  • The relative entropy of null cut algebras was computed for the vacuum and certain coherent states.

Conclusions:

  • The study provides a detailed analysis of quantum field theory algebras on the null plane.
  • The findings confirm the validity of the quantum null energy condition (QNEC) in free field theories within this geometric context.
  • The calculations of relative entropy offer insights into the properties of null cut algebras and their relation to quantum information theory.