Bryce DeWitt, and the Equation With No Time In It

By Art Knott II

In 1965, a physicist named John Wheeler had two hours to kill at the Raleigh-Durham airport. He called a colleague at UNC Chapel Hill named Bryce DeWitt and asked him to come sit with him. DeWitt brought a paper with him — someone else's work, on how to write general relativity in a form suited to quantization — and in the terminal, the two of them worked out an equation together. Wheeler thought they had found the equation of quantum gravity. He wanted to call it the Einstein-Schrödinger equation, after the men whose theories it married. DeWitt disagreed. He called it, at various points, the Wheeler equation, and later, more bluntly, that damned equation. Everyone else split the difference and called it the Wheeler-DeWitt equation. All three names describe the same object. Only one of them tells you anything true about what it says.

What it says is this. Two of physics' best theories do not fit together. General relativity says gravity is the shape of space and time. Quantum mechanics says everything else is a set of probabilities that changes over time. To unify them, you need a quantum description of the shape of space itself. DeWitt wrote one down. In ordinary quantum mechanics, the Schrödinger equation says how a system's state changes as time passes — the rate of change equals an energy term acting on the state. DeWitt applied the same recipe to the geometry of the universe, and got, in its compressed form: an energy operator acting on the state of everything, set equal to zero.

There is no t in that equation. Nothing plays the role of time. The equation meant to describe the quantum state of the entire universe does not contain a variable for when. Physicists have called this the problem of time since 1967, and it remains open. Carlo Rovelli's reading of it, from 2015, is the sharpest I have found: the missing t is not a quantum oddity bolted onto classical physics. It was already implicit in Einstein's theory. The coordinate we call time in general relativity has no physical meaning by itself. What the theory actually tracks is how quantities change relative to one another. DeWitt's equation just makes that unavoidable. It is not describing things moving against a clock. It is describing relations between parts of a system — one of which we happen to read as a clock.

That distinction is not decoration. It can be built. In 1983, Don Page and William Wootters showed how time could appear inside a universe whose total state never changes: split the universe into two pieces, let one serve as a clock, and the other, correlated with it, will look like it evolves — from the inside. From outside, watching the whole system at once, nothing moves. In 2014, a lab in Turin built exactly this with two entangled photons. Read one photon as a clock, and the other appears to change. Measure both together, and nothing changes at all. It is worth being precise about what this experiment does and does not show. Two photons are not a universe. It demonstrates that the mechanism is coherent — that "change" can be a fact about a relation rather than a property an object carries alone. It does not demonstrate that our universe runs this way.

Here is the part of the DeWitt story that keeps me honest about him, because thinkers are worth studying at the place where they contradict themselves, not only where they were right. Hugh Everett, in 1957, proposed what he called the "relative state" formulation of quantum mechanics — his emphasis, correctly, on relative. A system's state means something only in relation to another system's state. It is a fully structural idea, no less than DeWitt's own equation. And it was DeWitt, in 1970, who took Everett's obscure paper to a wide audience — and renamed it. He called it "many worlds." The name conjures a stack of separate universes branching off behind you, which is not what Everett described. DeWitt wrote down the most relational equation in twentieth-century physics, and then gave someone else's relational idea the most object-heavy name it could have received. The label outran the structure. It shaped, and in my view distorted, sixty years of how the public understands the theory. If I am going to credit DeWitt with dissolving the wall between observer and observed, I have to also credit him with building a new one, out of language, in the very paper meant to tear the old one down.

A plant will not settle this argument, but it clarifies what "relational" costs nothing to accept once you see it working somewhere ordinary. A farmer does not predict when corn will tassel by counting days on a calendar. She counts heat — the mean temperature each day, above a baseline, summed across the season. This is called growing degree days, and it is a better predictor of a plant's development than the date is. The plant is not keeping track of anything, and it does not know its own heat total; its rate of development is simply set by temperature, day after day. But the practical upshot is the same shape as DeWitt's physics, at a scale you can watch from your porch: the "when" of flowering is not a fact stamped onto the calendar. It is a relation between the organism and its conditions, and if you want to predict it, the calendar is the less useful clock.

DeWitt did not go as far as some of his admirers, including me at times, would like him to have gone. He was a realist about the universal wavefunction — he believed it was a real, complete description of a real universe, full stop. He never argued that objects are illusions, or that only relations exist. That further step belongs to a philosophical position called structural realism, which holds, in its strongest form, that structure is not merely what we know of the world but all there is to know, because there may be nothing else underneath. DeWitt supplies the physics that makes such a position worth taking seriously. He does not supply the position itself, and it would be dishonest of me to hand him credit for an argument he never made.

It would also be dishonest to leave out where the whole picture is shakiest. The Wheeler-DeWitt equation is mathematically troubled — the ordering of its operators and its handling of infinities are unresolved, and much of the theoretical physics community, string theorists especially, has moved past it rather than through it. The Page-Wootters mechanism has a known objection, raised by Karel Kuchař in 1992, that its method gives inconsistent answers when you ask about probabilities across two different times; a fix has been proposed, but the matter is not settled. And the oldest objection to structural realism itself has never gone away: there are no relations without things to relate. DeWitt's equation can walk you to the edge of that question. It cannot answer it for you.

I keep returning to DeWitt because I think he is the closest a working physicist came to writing down, formally, what I have spent years circling in my own reading: that a thing's state is not something it holds alone, that the whole looks static only until you ask what it looks like from inside one of its own parts, and that what we call time is one name among several a relation can go by. I hold that further step as mine, not his — a working framework, not settled physics, and I would rather say so plainly than borrow a dead man's authority for it. DeWitt removed a wall. He did not tell us what to build with the space he cleared. That part, it turns out, is a different job — for him, for Everett, and eventually, however small the contribution, for the rest of us standing at that particular equation, wondering where the clock went.