Stone's theorem

In plain terms

A system evolving in time is described by a family of transformations, one for each duration, which compose the way durations add, so evolving for two seconds and then for three is the same as evolving for five. Physics wants to differentiate that family, to speak of the rate at which the state changes, and to call the quantity that generates the change the energy, or Hamiltonian. But differentiating presupposes smoothness, and nobody said the evolution had to be smooth. Assuming it feels harmless, and in fact it's most of the conclusion.

Stone's theorem says you don't need to assume it. Require only that the evolution be continuous, meaning that states at nearby times are nearby and nothing jumps, and smoothness follows on its own. So does the generator, a single fixed quantity from which you can rebuild the whole family of transformations by compounding, much as a fixed interest rate generates the growth of a balance over any period. The analogy has one limit, as in quantum mechanics the generator turns the state rather than growing it, which is why total probability stays fixed.

So the Hamiltonian isn't an ingredient added to quantum mechanics by hand. It's extracted from the bare requirement that time evolution doesn't teleport.

In CSD

The programme needs the generator to be earned and not posited. If you posit a Hamiltonian you've smuggled in the dynamics you claimed to derive, and the reconstruction would be decorative. This theorem is how the smuggling is avoided, as continuity alone gives the generator, and the Schrodinger equation, the equation of motion of quantum mechanics, is then a conclusion rather than a starting point.

The full chain has two links. Wigner rigidity says a flow preserving transition probabilities must act by unitary transformations at each instant, and Stone's theorem says a continuous one-parameter family of unitary transformations is generated by a single fixed operator. Those are the corpus's two unconditional necessities, and the second pillar of the programme, the derivation of the dynamics, stands on exactly them.

Mathematically

A strongly continuous one-parameter group of unitary matrices is t maps to exp(tA) for a skew-Hermitian A, and the hypothesis is continuity only. The C-one version proved first carried a differentiability assumption, so the strengthening that removes it is the part with content, since assuming differentiability is uncomfortably close to assuming the answer.

One library note is worth knowing. Mathlib carries no Stone theorem at this pin, as the names beginning with Stone there are Stone-Weierstrass and Stone duality, so the finite-dimensional case is supplied here.

The name

Marshall Stone (1903-1989) was born in New York, the son of a future Chief Justice of the United States, and took his doctorate at Harvard under George Birkhoff. The theorem on one-parameter unitary groups dates from 1932, the same era as his duality and approximation theorems.

His second act was administrative and just as consequential. He moved to the University of Chicago in 1946 and rebuilt a fading department into arguably the world's best in about five years, hiring Weil, Chern, Zygmund and Mac Lane in a spree still called the Stone Age. He's one of the few first-rank mathematicians who was also a first-rank chairman.

Related entries

• Wigner rigidityPreserve the overlaps between quantum states and you have no freedom left, as the rearrangement must be a unitary or an anti-unitary transformation whether or not you intended it to be.
• Schrodinger equationIn CSD the Schrodinger equation isn't postulated, as it's what any continuous motion that preserves the measurable relations between quantum states is forced to look like.
• Schrodinger from recordsCSD doesn't postulate the Schrodinger equation, as preserving the statistics of records forces the dynamics to be unitary, unitarity together with continuity forces a Hamiltonian, and the exponential form of the equation is a theorem at the end of that chain.
• Hermitian operatorA single symmetry condition on an operator guarantees that every number it can produce in a measurement is real, and it delivers the whole machinery of outcomes and eigenstates at no extra cost.
• Bargmann invariantOne phase survives every change of convention, the phase of the product of three overlaps taken around a cycle, and it's also the fingerprint that tells a unitary symmetry from an anti-unitary one.