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Curious QNEIs from QNEC: New Bounds on Null Energy in Quantum Field Theory

Jackson R. Fliss, Andrew Rolph

Abstract

We derive new families of quantum null energy inequalities (QNEIs), i.e. bounds on integrated null energy, in quantum field theories in two and higher dimensions. These are universal, state-independent lower bounds on semi-local integrals of $\langle T_{vv} \rangle$, the energy-momentum flux in a null direction, and the first of this kind for interacting theories in higher dimensions. Our ingredients include the quantum null energy condition (QNEC), strong subadditivity of von Neumann entropies, defect operator expansions, and the vacuum modular Hamiltonians of null intervals and strips. These results are fundamental constraints on null energy in quantum field theories.

Curious QNEIs from QNEC: New Bounds on Null Energy in Quantum Field Theory

Abstract

We derive new families of quantum null energy inequalities (QNEIs), i.e. bounds on integrated null energy, in quantum field theories in two and higher dimensions. These are universal, state-independent lower bounds on semi-local integrals of , the energy-momentum flux in a null direction, and the first of this kind for interacting theories in higher dimensions. Our ingredients include the quantum null energy condition (QNEC), strong subadditivity of von Neumann entropies, defect operator expansions, and the vacuum modular Hamiltonians of null intervals and strips. These results are fundamental constraints on null energy in quantum field theories.

Paper Structure

This paper contains 21 sections, 135 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Two-dimensional quantum null energy inequalities
  3. Fewster-Hollands bound from 2d QNEC
  4. An family of 2d QNEIs
  5. Integrating QNEC
  6. Bounds on first derivatives of entropy
  7. Upper bound.
  8. Lower bound.
  9. Upper and lower bound together.
  10. Modular energies of null intervals
  11. NB on higher dimensions:
  12. Back to entanglement entropy derivative bounds
  13. QNEI ss
  14. Higher-dimensional quantum null energy inequalities
  15. QNEC in higher dimensions
  16. ...and 6 more sections

Figures (4)

  • Figure 1: The regions $X$, $Y$, $Z$, $A$ and $B$ as used in \ref{['eq:SSAv1']} to prove \ref{['eq:upperbound']}.
  • Figure 2: The regions $X$, $Y$, $Z$, $A$ and $C$ as used in \ref{['eq:lowerbound']} to prove \ref{['eq:sacbo']}.
  • Figure 3: The regions $A$, $B$, and $C$ in the setup for the higher dimensional derivation. The turquoise region is the domain of support of $g(u,v)$ in the $(u,v)$ plane and is extended uniformly in the remaining $\mathbb R^{d-2}$ directions.
  • Figure 4: A plot of \ref{['eq:hvdel']}, which is $h(v)$ for the choice of $v_B(v) = v-\delta$ and $v_C(v) = v+\delta$. We take $g$ to be a normalised Gaussian.