A Postquantum Theory of Classical Gravity?

The effort to discover a quantum theory of gravity is motivated by the need to reconcile the incompatibility between quantum theory and general relativity. Here, we present an alternative approach by constructing a consistent theory of classical gravity coupled to quantum field theory. The dynamics is linear in the density matrix, completely positive, and trace preserving, and reduces to Einstein’s theory of general relativity in the classical limit. Consequently, the dynamics does not suffer from the pathologies of the semiclassical theory based on expectation values. The assumption that general relativity is classical necessarily modifies the dynamical laws of quantum mechanics; the theory must be fundamentally stochastic in both the metric degrees of freedom and in the quantum matter fields. This breakdown in predictability allows it to evade several no-go theorems purporting to forbid classical quantum interactions. The measurement postulate of quantum mechanics is not needed; the interaction of the quantum degrees of freedom with classical space-time necessarily causes decoherence in the quantum system. We first derive the general form of classical quantum dynamics and consider realizations which have as its limit deterministic classical Hamiltonian evolution. The formalism is then applied to quantum field theory interacting with the classical space-time metric. One can view the classical quantum theory as fundamental or as an effective theory useful for computing the backreaction of quantum fields on geometry. We discuss a number of open questions from the perspective of both viewpoints.

Popular Summary

The holy grail of physics—a quantum theory of gravity—has remained elusive for almost a century. Einstein’s theory of gravity, known as general relativity, posits that matter curves space-time. Since matter has a quantum nature, the prevailing assumption has been that space-time must also conform to quantum theory. However, gravity is different to the other forces, because it alone determines a universal background geometry and causal structure in which all other forces act. Perhaps the challenges in constructing a satisfactory quantum theory of gravity indicate that the quantization of space-time is a flawed approach. We take a different direction by developing a hybrid theory that keeps space-time classical while modifying quantum theory to allow for an intrinsic breakdown in predictability that is mediated by space-time itself.

This work, together with a companion paper, derives the most general form of consistent dynamics that couples quantum systems with classical ones. Although there have been many attempts at constructing classical-quantum dynamics, they tend to suffer from a breakdown in causality or lead to negative probabilities. The framework derived here achieves consistency by including the effect of correlations between the classical and quantum systems and allows for arbitrary backreaction of the quantum system on the classical one. We then apply this framework to derive a consistent theory of general relativity and quantum matter fields.

Although our primary motivation is in constructing a consistent theory of gravity and quantum mechanics, the approach considered here has several other consequences. The theory does not require the controversial measurement postulate of traditional quantum theory. Rather, the interaction of the quantum degrees of freedom with classical space-time necessarily causes the quantum systems to localize. The theory also has implications for the black-hole information paradox and has an experimental signature, owing to the inherent stochastic and unpredictable nature of space-time curvature.

Figure 1

The prediction of the semiclassical Einstein equation is depicted in the right panel, compared with what we expect to occur based on the statistical interpretation of the density matrix in the left panel. If a planet is in a statistical mixture of being in two possible locations (“$L$” and “$R$”), then we expect the gravitational field to become correlated with it. A test particle therefore falls toward one of the planets. If one treats the semiclassical Einstein equations as fundamental, the test particle falls down the middle, which is indeed the average trajectory of its path in the left-hand panel. The planets are also attracted toward the place they might have been. The systems in the linear theory considered here exhibit the behavior depicted in the left-hand panel. The case where we attempt to put the planet in a superposition of the two locations is discussed further in Refs. [29, 30]. If the gravitational field is suitably different in the two cases, in the sense of how distinguishable the field configurations are when noise is added to them, then there will be a limit on how coherent the superposition can be [30]. In the case where there is coherent superposition, the initial trajectory of the test mass will be insufficiently correlated with the planet’s position to enable a determination of where the planet is. However, after the decoherence time, it may be possible to determine the planet’s location from the trajectory of the test mass, since its trajectory will become correlated to the planet’s location [29]. This is also what one expects from a fully quantum theory, with the stochasticity of the test mass being due to vacuum fluctuations of the metric.

Figure 2

Consider a variation of the gedanken experiment proposed by Feynman [12, 35] and Aharonov [36] in which a massive particle travels through two slits as a plane wave. Quantum theory predicts an interference pattern on the screen behind the slits which enables a determination of the particle’s momentum. The local Newtonian potential it creates can be measured everywhere. At late times, the state of the electromagnetic field far from the slits will be slightly different depending on which slit the particle goes through, but these two states need not be orthogonal. The quantum nature of the electromagnetic field prevents an experimenter from distinguishing these two nonorthogonal states and while the electromagnetic interaction might lead to some decoherence of the particle, if the interaction is small enough the interference pattern remains. On the other hand, if the gravitational field is classical and the interaction deterministic, then one could imagine measuring the gravitational field close to the particle and during its flight, without disturbing the field to unambiguously determine which path the particle went through. One could therefore never get interference fringes (or the uncertainty principle would be violated). Two classical states cannot be different but nonorthogonal. Actually measuring the gravitational field is not required since classical systems are undisturbed; even if we cannot actually measure the field to the required accuracy [48], it is impossible to write down a pure quantum state of the particle which is correlated with a classical gravitational field determined by the particle’s position. Since the theory presented here modifies quantum theory so that the interaction is stochastic, the gravitational field need not become unambiguously correlated with the particle’s path and the interference pattern is restored. Probability densities over classical states can be different and nonorthogonal.