Hilbert space

In plain terms

Quantum mechanics needs three things from the space its states live in. Superposition, the ability to combine two states into a third, requires that states can be added like arrows, which makes them vectors. Probabilities require lengths and angles, which come from an inner product, a rule that assigns a number to every pair of vectors and thereby measures how long each is and how far apart they point. And calculus, with its limits and convergence, requires that a sequence of states closing in on something actually closes in on a state within the space rather than leaking out of it, and this property is called completeness. A Hilbert space is exactly a space with all three, vectors, an inner product and completeness.

In finitely many dimensions completeness comes for free, and a Hilbert space is nothing more exotic than ordinary Euclidean space with complex coordinates. In infinitely many dimensions completeness is a genuine constraint and the source of most of the subject's analytical difficulties.

The standard formulation of quantum mechanics identifies the theory with this stage, so states are vectors in Hilbert space and that's where the theory happens, or so the textbook says.

In CSD

CSD's sharpest departure from textbook framing is here. On CSD's account the physical world doesn't live in Hilbert space, as physical states are points of the constraint surface, the programme's underlying state space. The Hilbert space, or more precisely the space of its rays (a vector and all its rescalings count as one physical state, so only the direction matters), is bookkeeping built above that substrate, and it records what an observer can know and carries the regions of possible outcomes and the probabilities attached to them.

To say that quantum mechanics happens in Hilbert space is, on this account, like saying that the weather happens on the weather map, which is true about the representation and silent about the thing represented. The programme's analysis of the fibre, the structure that sits above each point of the space of rays, pushes further, as the space of rays is a lossy projection of the substrate and the content that forms records lives above it. Whether or not you accept the programme, this is the single point to grasp about how it reads the formalism.

Mathematically

A Hilbert space is a complete complex inner product space. The corpus is finite- dimensional throughout, so completeness is free, and the operative object is CP^(N-1), the space of rays, since vectors differing by a phase are one physical state.

Infinite dimension, with its unbounded operators, domains and continuous spectra, is deliberately out of scope, and the continuous-variable strand approximates it at an explicit finite cutoff instead.

The name

David Hilbert (1862-1943) was born in Konigsberg and moved in 1895 to Gottingen, which he made the centre of world mathematics for a generation. His 1900 Paris address, with its twenty-three problems, set the century's agenda, and his formalist dream of mathematics proved complete and consistent by finite means was bounded for ever by Godel in 1931.

Two footnotes have teeth. The space isn't his coinage, as von Neumann named it while building the rigorous foundations of quantum mechanics in Hilbert's own department. And when the Nazi education minister asked him, in 1934, whether Gottingen mathematics had suffered from the removal of the Jews, Hilbert answered that it hadn't suffered, as it no longer existed.

Related entries

• Euclidean spaceOrdinary geometry, extended to complex numbers, is the stage on which every finite quantum system in the corpus stands.
• 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.
• Projective sectorQuantum mechanics is written on a curved projective space, and in CSD that space isn't fundamental, as it's the sector the arena projects onto, with its geometry forced by symmetry.
• Bloch sphereThe states of the simplest quantum system form an ordinary ball, which is a lovely accident of low dimension and the single most misleading picture in quantum mechanics.
• Kronecker productTwo quantum systems become one by multiplying their descriptions rather than adding them, which is why quantum resources multiply where classical ones merely add.