Is time fundamental?

The Schrödinger equation describes how a state changes with time, where time is a parameter attached to a clock that is not part of the quantum system.

General relativity has no preferred time coordinate. Choosing one is a choice of description, and the physics must not depend on it.

Bringing the two together means quantising a theory whose own equations say the time labelling is arbitrary. Done directly, the result is a constraint equation rather than an evolution equation.

The constraint was written down in 1967 by Bryce DeWitt, following work by John Wheeler. Applied to the whole universe it says that the quantum state is annihilated by the operator that would have generated time evolution.

Read literally, the state does not change. That is the problem of time, and it arrives immediately, from taking general relativity's indifference to the time coordinate and quantising it in the obvious way.

The equation has severe mathematical problems of its own and is not regarded as a finished theory. The problem it exposes is not confined to that formulation. Isham traced it to an inconsistency between the basic frameworks of the two theories, and Anderson’s 2012 review counts eight facets of it, the frozen state being one.

One response is that time is emergent: the fundamental description is timeless, and what we experience as evolution is a correlation between parts of a system, one of which can be used as a clock for the others. This is Rovelli’s position, and his thermal time hypothesis, which makes the flow we notice an artefact of the statistical state the world happens to be in, is a proposal within it rather than a result. Gambini and Pullin, building the same idea on real quantum clocks, describe their version as a solution to the problem of time.

Another is that the formalism is incomplete and a correct theory of quantum gravity will restore a notion of time, by identifying one before quantising rather than after.

A third is that the equation itself is the wrong starting point. Isham, reviewing the whole field, allowed that the Wheeler-DeWitt equation “may be completely the wrong way of formulating a quantum theory of gravity”.

No experiment discriminates between the positions, and the obstacle is not only the energy scale. Isham’s point is that most of the problem is not tied to the high-energy behaviour of quantum gravity at all, since the interpretative difficulties turn up already in finite-dimensional minisuperspace models. The case is argued on coherence rather than data.

The nearest thing is a testing programme for quantum spacetime in general, not for this question. It looks for Planck-scale effects where a tiny departure accumulates: photons from gamma-ray bursts crossing cosmological distances, and ultra-high-energy cosmic rays. Amelino-Camelia, reviewing that work, grants that the regime where the effects would be dramatic, the quantum black hole, is the one hardest to get at.

Even if the physics turns out to be timeless, the experience of a moving present has to be accounted for by something, and that something is a brain in a universe with a thermodynamic gradient.

No established account connects the two levels. Thermal time proposes where the flow comes from, not how it comes to be felt. A timeless fundamental description and a lived present are not obviously in conflict, and nobody has shown how the second follows from the first.