Macroscopic effects of the quantum trace anomaly

Abstract

The low energy effective action of gravity in any even dimension generally acquires non-local terms associated with the trace anomaly, generated by the quantum fluctuations of massless fields. The local auxiliary field description of this effective action in four dimensions requires two additional scalar fields, not contained in classical general relativity, which remain relevant at macroscopic distance scales. The auxiliary scalar fields depend upon boundary conditions for their complete specification, and therefore carry global information about the geometry and macroscopic quantum state of the gravitational field.The scalar potentials also provide coordinate invariant order parameters describing the conformal behavior and divergences of the stress tensor on event horizons. We compute the stress tensor due to the anomaly in terms of its auxiliary scalar potentials in a number of concrete examples, including the Rindler wedge, the Schwarzschild geometry, and de Sitter spacetime. In all of these cases, a small number of classical order parameters completely determine the divergent behaviors allowed on the horizon, and yield qualitatively correct global approximations to the renormalized expectation value of the quantum stress tensor.

Scalar gravitational waves in the effective theory of gravity

“…Because it is asymmetrical in the invariants E and (C 2 , L i ), two additional scalar fields would be necessary to render the minimal non-local action (2.9) into a local form [39]. On the other hand, if one adds to (2.14) the Weyl invariant terms…”

“…Since around flat spacetime the only energy scale available is that of the perturbations, one must expect the other half of the solutions of the fourth order anomaly EFT to couple only when their energy scale approaches the Planck energy of 10 19 GeV, by which point the EFT approach to gravity has clearly broken down. In curved spacetime backgrounds possessing additional length scales or horizons, the breakdown of the EFT approach may occur at lower energy scales, such as in the vicinity of horizons [39,58]. Issues arising from fourth order anomaly effective action (2.14) including the quantization and Hamiltonian of the conformalon field will be taken up in a separate publication, inasmuch as they do not affect the conclusions of scalar gravitational waves with classical gauge invariant potentials obeying the second order eq.…”

“…As in the case of the QCD axial anomaly, the underlying QFT need not be exactly conformally invariant, or the axial and/or conformal invariance may be spontaneously broken. Notwithstanding any of the fields contributing to b and b ′ in (1.4) having a finite mass m ≪ M P l , their anomalous vacuum contributions to the effective action must still be taken into account for intermediate energies m ≪ k ≪ M P l , in the low energy EFT of gravity far below the Planck scale [32,39,40].…”

“…The form and value of (3.1) given by (3.2)-(3.4) depend in its tracefree parts quadratic in ϕ upon the completion of the square by addition of the Weyl invariant terms (2.11). If these Weyl invariant terms are absent from the full effective action, the slightly more complicated two-field effective action and stress tensor of the anomaly given in [39] should be used in place of the tracefree quadratic terms in (3.2) and (3.1).…”

“…In that case the scalar conformalon ϕ does not introduce any genuinely 'new' degrees of freedom, but rather re-expresses in a convenient form certain two-particle correlations present in the underlying QFT vacuum [28-30, 49, 50]. For example, the Casimir effect and vacuum polarization effects in fixed geometries such as the Schwarzschild and de Sitter backgrounds may be computed for a large variety of states for fields of arbitrary spin by use of S anom and its corresponding stress tensor (3.2) [39]. On the other hand, since the form of the trace anomaly (1.2) is prescribed by locality of QFT and general coordinate invariance, the effective action (2.9) associated with it may be regarded as a necessary part of the general effective action of low energy gravity, based on dimensional scaling and invariance principles alone.…”