nLab state




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Measure and probability theory



A state of a system in physics is a description of everything that is true (or known, or relevant, depending on the circumstances) about the physical system.

In the Bayesian interpretation of physics, the state of a system is not a property of reality but instead indicates an observer's knowledge about the system. A pure state gives maximal information about the system (which amounts to complete information in classical mechanics but not generally in quantum mechanics), while a mixed state is more general. A mixed state can be decomposed into a probability distribution on the space of pure states, although this decomposition is unique only for classical systems. In a frequentist interpretation of probability, a mixed state can describe only a statistical ensemble of systems; the real world is in one (generally unknown) pure state (possibly with additional hidden variables in the quantum case, depending on the interpretation of quantum physics).

States in the Schrödinger picture describe the state of the system at any given time and are subject to time evolution?, while in the Heisenberg picture a single state describes the entire history of the system (with time evolution applying to the observables instead).

In statistical physics, a microstate? is a complete description of all of the particles in a system, while a macrostate? considers only the macroscopic properties known to thermodynamics.


The precise mathematical notion of state depends on what mathematical formalization of mechanics is used.

Algebraic definition

Quite generally, both in classical physics as well as in quantum physics, one may define states as assignments of expectation values to observables in an algebra of observables. This is the definition used in quantum probability theory (which subsumes ordinary probability theory). See at

for details.

In classical mechanics

In classical Lagrangian mechanics, a pure state is a point in the state space? of the system, giving all of the (generalised) positions? and velocities. In classical Hamiltonian mechanics, a pure state is a point in the phase space of the system, giving the positions and momenta. In either case, a mixed state is a probability distribution on the space of pure states.

More generally, a classical state is a linear function ρ:A\rho\colon A \to \mathbb{R} on the Poisson algebra AA underlying the classical mechanical system which satisfies positivity and normalization.

In geometric quantization

In Hilbert-space quantum mechanics

In quantum mechanics given by a Hilbert space HH, a pure state is a ray in HH, which we often call the Hilbert space of states. Strictly speaking, the space of states is not HH but (H{0})/(H \setminus \{0\})/\mathbb{C}, or equivalently S(H)/U(1)S(H)/\mathrm{U}(1). A mixed state is then a density matrix on HH.


In AQFT, a quantum mechanical system is given by a C *C^*-algebra AA, and a quantum state is usually defined as a linear function ρ:A\rho\colon A \to \mathbb{C} which satisfies positivity and normalization; see states in AQFT and operator algebra.

Arguably, the correct notion of state to use is that of quasi-state; every state gives rise to a unique quasi-state, but not conversely. However, when either classical mechanics or Hilbert-space quantum mechanics is formulated in AQFT, every quasi-state is a state (at least if the Hilbert space is not of very low dimension, by Gleason's theorem). See also the Idea-section at Bohr topos for a discussion of this point.


In the FQFT formulation of quantum field theory, a physical system is given by a cobordism representation

Bord n S𝒞. Bord_n^S \to \mathcal{C} \,.

In this formulation the (n-1)-morphism in 𝒞\mathcal{C} assigned to an (n1)(n-1)-dimensional manifold Σ n1\Sigma_{n-1} is the space of states over that manifold. A state is accordingly a generalized element of this object.

Pure and mixed states

In statistical physics, a pure state is a state of maximal information, while a mixed state is a state with less than maximal information. In the classical case, we may say that a pure state is a state of complete information, but this does not work in the quantum case; from the perspective of the information-theoretic or Bayesian interpretation of quantum physics, this inability to have complete information, even when having maximal information, is the key feature of quantum physics that distinguishes it from classical physics.

See pure state.


Here are some toy examples of spaces of states.

For an impossible system, the space of states is empty; for a trivial system (with a unique way to be), the space of states is the point. This unique state is pure.

For a classical bit, a system with two distinct ways to be, the space of states is a line segment; a state is given by a real number tt with 0t10 \leq t \leq 1. This tt is the probability that the system is in the first state, with 1t1 - t the probability that it is in the second; the two pure states correspond to t=0t = 0 and t=1t = 1. This space naturally comes equipped with a metric structure in which its length is actually 22; it’s the line segment from (0,1)(0,1) to (1,0)(1,0) in the Lebesgue space l 1(2)l^1(2).

For a quantum bit, a qubit, a state is given by a self-adjoint 22-by-22 complex matrix

(a bic b+ic d) \begin{pmatrix} a & b - \mathrm{i} c \\ b + \mathrm{i} c & d \end{pmatrix}

with unit trace (so d=1ad = 1 - a) and nonnegative determinant (so a 2+b 2+c 2aa^2 + b^2 + c^2 \leq a). However, the metric structure from viewing this as a subset of 3\mathbb{R}^3 is misleading, so we use wad=2a1w \coloneqq a - d = 2 a - 1 instead, along with u2bu \coloneqq 2 b and v2cv \coloneqq 2 c to keep the same normalization. Then the density matrix may be written

12(1+w uiv u+iv 1w)=12(I+uσ x+vσ y+wσ z), \frac 1 2 \begin{pmatrix} 1 + w & u - \mathrm{i} v \\ u + \mathrm{i} v & 1 - w \end{pmatrix} = \frac 1 2 (\mathrm{I} + u \sigma_x + v \sigma_y + w \sigma_z) ,

where the σ\sigmas are the Pauli matrices. This already has unit trace; the inequality for the determinant is

u 2+v 2+w 21. u^2 + v^2 + w^2 \leq 1 .

The pure states are those satisfying u 2+v 2+w 2=1u^2 + v^2 + w^2 = 1, forming a unit sphere, known in this context as the Bloch sphere? (so the full space of all states is the Bloch ball).

quantum probability theoryobservables and states

duality between \;algebra and geometry

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A\phantom{A}topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand-KolmogorovAlg op\overset{\text{<a href="">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}A\phantom{A}A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg C *,comm op\overset{\text{<a class="existingWikiWord" href="">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}A\phantom{A}A\phantom{A}comm. C-star-algebraA\phantom{A}
A\phantom{A}noncomm. topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cptNCTopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg C * op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}A\phantom{A}A\phantom{A}general C-star-algebraA\phantom{A}
A\phantom{A}algebraic geometryA\phantom{A}A\phantom{A}NCSchemes Aff\phantom{NC}Schemes_{Aff}A\phantom{A}A\phantom{A}almost by def.TopAlg fin op\overset{\text{<a href="">almost by def.</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{fin} A\phantom{A}A\phantom{A}fin. gen.A\phantom{A}
A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}noncomm. algebraicA\phantom{A}
A\phantom{A}NCSchemes AffNCSchemes_{Aff}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg fin,red op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}A\phantom{A}A\phantom{A}fin. gen.
A\phantom{A}associative algebraA\phantom{A}A\phantom{A}
A\phantom{A}differential geometryA\phantom{A}A\phantom{A}SmoothManifoldsSmoothManifoldsA\phantom{A}A\phantom{A}Milnor's exerciseTopAlg comm op\overset{\text{<a href="">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}A\phantom{A}A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}supergeometryA\phantom{A}A\phantom{A}SuperSpaces Cart n|q\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}A\phantom{A}A\phantom{A}Milnor's exercise Alg 2AAAA op C ( n) q\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }A\phantom{A}A\phantom{A}supercommutativeA\phantom{A}
A\phantom{A}formal higherA\phantom{A}
A\phantom{A}(super Lie theory)A\phantom{A}
ASuperL Alg fin 𝔤A\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}AALada-MarklA sdgcAlg op CE(𝔤)A\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}A\phantom{A}differential graded-commutativeA\phantom{A}
A\phantom{A} (“FDAs”)

in physics:

A\phantom{A}Poisson algebraA\phantom{A}A\phantom{A}Poisson manifoldA\phantom{A}
A\phantom{A}deformation quantizationA\phantom{A}A\phantom{A}geometric quantizationA\phantom{A}
A\phantom{A}algebra of observablesA\phantom{A}space of statesA\phantom{A}
A\phantom{A}Heisenberg pictureA\phantom{A}Schrödinger pictureA\phantom{A}
A\phantom{A}higher algebraA\phantom{A}A\phantom{A}higher geometryA\phantom{A}
A\phantom{A}Poisson n-algebraA\phantom{A}A\phantom{A}n-plectic manifoldA\phantom{A}
A\phantom{A}En-algebrasA\phantom{A}A\phantom{A}higher symplectic geometryA\phantom{A}
A\phantom{A}BD-BV quantizationA\phantom{A}A\phantom{A}higher geometric quantizationA\phantom{A}
A\phantom{A}factorization algebra of observablesA\phantom{A}A\phantom{A}extended quantum field theoryA\phantom{A}
A\phantom{A}factorization homologyA\phantom{A}A\phantom{A}cobordism representationA\phantom{A}

Last revised on December 11, 2022 at 12:20:11. See the history of this page for a list of all contributions to it.