The Problem of Time in Quantum Gravity

General relativity made spacetime dynamical; quantum mechanics governs the microworld. Uniting them into a theory of quantum gravity — quantizing the gravitational field, i.e. spacetime geometry itself — is the great unsolved problem of fundamental physics, and it produces a philosophical shock unmatched elsewhere in the subject: in the most straightforward approaches, time disappears from the fundamental equations altogether. This is the "problem of time." It is not a technicality but a conceptual crisis: the two ingredient theories treat time so differently that combining them threatens to leave no room for time as a fundamental feature of the world at all. If they are right, the temporal question receives its most radical answer — that time is not fundamental but emergent, a feature of an approximate, low-energy description of a timeless underlying reality.

This page explains how time goes missing and surveys the responses; it presupposes general relativity and quantum mechanics.

Two theories, two treatments of time

The problem arises from a clash in how the ingredient theories treat time:

In quantum mechanics, time is a fixed, external background parameter $t$ . It is not an observable (there is no "time operator"); it is the independent variable against which the quantum state evolves, via the Schrödinger equation, $i ℏ \partial_{t} ∣ ψ ⟩ = \hat{H} ∣ ψ ⟩$ . Time stands outside the quantum system as a given, absolute backdrop — much as it did for Newton.
In general relativity, there is no fixed background time. Time is part of the dynamical spacetime metric, which bends, stretches, and has no preferred global slicing (as the relativity of simultaneity and general relativity pages stressed). The theory is diffeomorphism-invariant: coordinate time is pure gauge, mere labelling, with no physical content.

So one theory needs an external absolute time; the other denies there is any such thing. When we try to quantize gravity — to apply quantum rules to the spacetime metric itself — these incompatible treatments collide.

The Wheeler–DeWitt equation and the frozen formalism

The clash becomes concrete in the canonical approach to quantum gravity. Quantizing general relativity in the standard (Hamiltonian) way, one finds that its diffeomorphism invariance forces the total Hamiltonian — the generator of time evolution — to be a constraint that vanishes: $\hat{H} = 0$ . Applying the quantization recipe yields the Wheeler–DeWitt equation,

$\hat{H} ∣Ψ ⟩ = 0,$

for the quantum state $∣Ψ ⟩$ of the entire universe (a "wavefunction of the universe" defined on the space of possible 3-geometries). Compare the ordinary Schrödinger equation: where that has $i ℏ \partial_{t} ∣ ψ ⟩$ on the left, driving evolution in time, the Wheeler–DeWitt equation has nothing — no $\partial_{t}$ at all. The quantum state of the universe does not evolve. It is a single, static, timeless solution. The formalism is, in the standard phrase, "frozen."

This is the problem of time in its sharpest form: our best attempt at a fundamental theory contains no time variable, no evolution, no becoming. Time, which seemed the most basic feature of the world, has vanished from the deepest equation we can write. The passage that the A-theorist wanted to make fundamental is not even present as a parameter.

Responses: where did time go?

The interpretive responses fall into families, none yet decisive:

Time is emergent / relational (the leading view). Time is not fundamental but emerges at an approximate, semiclassical level. The universe's timeless quantum state contains correlations between subsystems; one subsystem (a physical clock — a molecule, a pulsar, the expansion of the universe) can serve as an internal clock against which the rest changes. Time is then read off from change in relational quantities, not presupposed. This revives, at the quantum level, the Machian and Barbourian idea that "time is change." The Page–Wootters mechanism makes it precise: the global state is static (satisfies $\hat{H} ∣Ψ ⟩ = 0$ ), yet conditional on a clock subsystem reading $t$ , the rest of the universe is in the state ordinary Schrödinger evolution would assign at $t$ . Apparent time evolution is entanglement between a clock and the world, within a globally timeless state — a striking vindication of the relationist instinct about time.
Julian Barbour's timelessness. Barbour embraces the conclusion fully: there simply is no time. Reality is a static configuration space of "Nows" (instantaneous 3-geometries with their matter content), each a complete momentary snapshot; the appearance of a flowing time and a remembered past is generated by the special structure of these Nows (his "time capsules," configurations that encode records suggestive of a past). Change is real (Nows differ); time as a container or dimension is not. This is the most radically B-theoretic — indeed no-theoretic — metaphysics of time on offer.
Time is fundamental; the formalism is misleading. A minority hold that time must be retained as basic and that the frozen formalism signals a defect in the canonical quantization or in general relativity's interpretation — e.g. that there is, after all, a preferred foliation (a neo-Lorentzian global time), perhaps the same one some interpretations of quantum non-locality seem to want. On this line the problem of time is a reason to modify the physics, not to eliminate time.
Different approaches, different verdicts. In loop quantum gravity spacetime geometry is discrete at the Planck scale and time is recovered relationally; in causal set theory the fundamental structure is a discrete causal order from which spacetime (and a notion of "becoming") is meant to emerge — a view congenial to a real, if discrete, temporal arrow; string theory typically retains a background time in its perturbative formulation but must confront the problem in its non-perturbative and cosmological regimes. (These programmes are surveyed on the physics QFT in curved spacetime and quantum gravity page.)

The philosophical stakes

The problem of time turns metaphysical positions into physical hypotheses:

If time is emergent/relational (the mainstream reading), then the deepest level of reality is timeless, and time — along with the passage, the arrow, and perhaps the distinction of space from time — is a feature of an approximate description valid only at low energies and large scales. The substantival container of time would be not merely denied but derived away.
The status of the thermodynamic arrow and the Past Hypothesis becomes obscure: what does "low-entropy initial condition" mean if there is no fundamental time to be "initial" in? The arrow may have to be reconstructed within the emergent-time description.
The A-theory's objective passage, already strained by special relativity, is put under still greater pressure: a fundamentally timeless physics has no place for a moving present, and even the B-series order may be emergent rather than basic.

That said, all of this is provisional. There is no accepted theory of quantum gravity, and the problem of time is partly an artifact of particular approaches (canonical quantization). It may look entirely different in the final theory — or dissolve. What is clear is that the reconciliation of general relativity and quantum mechanics cannot leave the naïve, Newtonian picture of time intact.

Where this sits

The problem of time is the frontier of the temporal question, where the tensions built up across the section come to a head: the relativity of simultaneity's denial of a global present, general relativity's dynamical and foliation-free spacetime, and quantum mechanics's external-parameter time all collide in the Wheeler–DeWitt "frozen" formalism. Its leading resolution — emergent, relational time — is the ultimate vindication of the Machian/Barbourian programme and the most radical answer the field offers to "what is time?" It also destabilises the arrow and its cosmological ground, the subject of the final physics page: time, cosmology, and the beginning.