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Lectures on quantum energy inequalities

Christopher J. Fewster

Abstract

Quantum field theory violates all the classical energy conditions of general relativity. Nonetheless, it turns out that quantum field theories satisfy remnants of the classical energy conditions, known as Quantum Energy Inequalities (QEIs), that have been developed by various authors since the original pioneering work of Ford in 1978. These notes provide an introduction to QEIs and also to some of the techniques of quantum field theory in curved spacetime (particularly, the use of microlocal analysis together with the algebraic formulation of QFT) that enable rigorous and general QEIs to be derived. Specific examples are computed for the free scalar field and their consequences are discussed. QEIs are also derived for the class of unitary, positive energy conformal field theories in two spacetime dimensions. In that setting it is also possible to determine the probability distribution for individual measurements of certain smearings of the stress-energy tensor in the vacuum state.

Lectures on quantum energy inequalities

Abstract

Quantum field theory violates all the classical energy conditions of general relativity. Nonetheless, it turns out that quantum field theories satisfy remnants of the classical energy conditions, known as Quantum Energy Inequalities (QEIs), that have been developed by various authors since the original pioneering work of Ford in 1978. These notes provide an introduction to QEIs and also to some of the techniques of quantum field theory in curved spacetime (particularly, the use of microlocal analysis together with the algebraic formulation of QFT) that enable rigorous and general QEIs to be derived. Specific examples are computed for the free scalar field and their consequences are discussed. QEIs are also derived for the class of unitary, positive energy conformal field theories in two spacetime dimensions. In that setting it is also possible to determine the probability distribution for individual measurements of certain smearings of the stress-energy tensor in the vacuum state.

Paper Structure

This paper contains 39 sections, 5 theorems, 165 equations, 5 figures.

Table of Contents

  1. Introduction and scope
  2. Summary of main conventions
  3. Quantum (energy) inequalities
  4. The classical energy conditions
  5. Examples
  6. Violation of Energy Conditions in QFT
  7. An example of a QEI and its consequences
  8. Scaling behaviour
  9. Bounds on the duration of negative energy density
  10. Quantum interest
  11. Application: A priori bounds on Casimir energy densities
  12. Some history and references
  13. Some methods of Quantum Field Theory in Curved spacetime
  14. The Klein--Gordon field
  15. Phase space
  16. ...and 24 more sections

Key Result

Theorem 2.1

If ${\boldsymbol{M}}$ is globally hyperbolic then, to each $f\in C_0^\infty({{\boldsymbol{M}}})$ there exists $\phi^\pm\in C^\infty({\boldsymbol{M}})$, with $\textrm{supp}\, \phi^\pm\subset J^\pm(\textrm{supp}\, f)$, solving the inhomogeneous problem Moreover, $\phi^{+/-}$ is the unique (distributional) solution to eq:KG_inhom whose support is past/future-compact (i.e., the support has compact in

Figures (5)

  • Figure 1: A spacetime plot of the energy density in a vacuum $+$$2$-particle superposition state FeRo03. Dark areas represent negative values.
  • Figure 2: $Q_3(x)$
  • Figure 3: Spacetime diagram of the Casimir plate set-up.
  • Figure 4: If $\ell\to\infty$ in the shaded region then $\widehat{\phi \omega_2} (l,l')\to 0$ rapidly, regardless of $\ell'$.
  • Figure 5: The probability density $P(x)$ plotted for $c=1$, where $x=\pi\tau^2\omega$

Theorems & Definitions (8)

  • Theorem 2.1
  • Theorem 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3
  • Lemma 3.4