Macroscopic Effects of the Quantum Trace Anomaly

TL;DR

This paper treats gravity as a low-energy effective field theory augmented by the quantum trace anomaly of massless fields, arguing that nonlocal anomaly terms must be retained because they are infrared-relevant. By introducing two auxiliary scalar fields, φ and ψ, the authors recast the nonlocal anomalous action into a local form, deriving two conserved tensors that contribute to the semiclassical stress-energy and depend on global boundary conditions. They compute the resulting stress-energy in two- and four-dimensional settings, with detailed analyses for conformally flat spacetimes and near-horizon geometries such as Schwarzschild and de Sitter, where horizon divergences are governed by horizon-order parameters. The framework provides a covariant, horizon-aware description of macroscopic quantum effects, connects with standard quantum states (Hartle-Hawking, Unruh, Boulware), and offers a practical route to compare with numerical quantum field results in curved backgrounds. Overall, the trace anomaly-induced auxiliary-field EFT presents a consistent infrared modification of general relativity that encodes macroscopic quantum coherence and potentially affects horizon structure and backreaction.

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.

Figures (5)

• Figure 1a: The expectation value $\langle T_t^{\ t} \rangle$ of a conformal scalar field in the Boulware state in Schwarzschild spacetime, as a function of $s=\frac{r-2M}{M}$ in units of $\pi^2 T_H^4/90$. The solid curve is Eq. (\ref{['Tanom']}) with (\ref{['Boulcond']}) and $c_{_H} = -\frac{7}{20}, d_{_H} = \frac{55}{84}$, the dashed curve is the analytic approximation of BroOtt, and the points are the numerical results of JenMcOtt.
• Figure 1c: The tangential pressure $\langle T_\theta^{\ \theta} \rangle$ of a conformal scalar field in the Boulware state in Schwarzschild spacetime. The axes and solid and dashed curves and points are as in Fig. \ref{['fig:TBOtt']}.
• Figure 2a: The expectation value $\langle T_t^{\ t} \rangle$ of a conformal scalar field in the Hartle-Hawking state in Schwarzschild spacetime as a function of $s=\frac{r-2M}{M}$ in units of $\pi^2 T_H^4/90$. The solid curve is Eq. (\ref{['Tanom']}) with $c_{\infty}=0.035, c_{_H}=-1.5144, p=1.5144, q=q'=2, d_\infty=1.0262, d_{_H}=1.0096, p'=-1.0096$, the dashed curve is the analytic approximation of FroZel and the data points are the numerical results of How.
• Figure 2b: The expectation value $\langle T_t^{\ t} \rangle$ of a massless Dirac field in the Hartle-Hawking state in Schwarzschild spacetime. The solid curve is Eq. (\ref{['Tanom']}) with $c_{\infty}=0.035, c_{_H}=-1.571 = -p, q=q'=2, d_\infty=0.6059, d_{_H}=0.6109 = -p'$, the dashed curve is the analytic approximation of FroZel and the data points are the numerical results of CHOAG.
• Figure 3a: The expectation value $\langle T_t^{\ t} \rangle$ of a massless, conformal field in the Unruh state in Schwarzschild spacetime. The solid curve is Eq. (\ref{['Tanom']}) with $c_{\infty}= 0, c_{_H} = -1.6500, p = 1.9192, q=q'=2, d_\infty = 1.4441, d_{_H} = 1.3244, p' = -1.0552$, and the dashed curve is the polynomial approximation of Vis which is an accurate fit to the numerical results of Elst.