nLab quantum state




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A quantum state is a state of a system of quantum mechanics.


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

Hilbert spaces

In the simple formulation over a Hilbert space HH, a pure state is a ray in HH. Thus, the pure states form the space (H{0})/(H \setminus \{0\})/\mathbb{C}, where we mod out by the action of \mathbb{C} on H{0}H \setminus \{0\} by scalar multiplication; equivalently, we can use S(H)/U(1)S(H)/\mathrm{U}(1), the unit sphere in HH modulo the action of the unitary group U(1)\mathrm{U}(1). Often by abuse of language, one calls HH the ‘space of states’.

The mixed states are density matrices on HH. Every pure state may be interpreted as a mixed state; taking a representative normalised vector |ψ{|\psi\rangle} from a ray in Hilbert space, the operator |ψψ|{|\psi\rangle}{\langle\psi|} is a density matrix.

In principle, any quantum mechanical system can be treated using Hilbert spaces, by imposing superselection rules that identify only some self-adjoint operators on HH as observables. Alternatively, one may take a more abstract approach, as follows.


In AQFT, a quantum mechanical system is given by a C *C^*-algebra AA, giving the algebra of observables. Then a state on AA is

  • a \mathbb{C}-linear function ρ:A\rho\colon A \to \mathbb{C}

  • such that

    • it is positive: for every aAa \in A we have ρ(a *a)0\rho(a^\ast a) \geq 0 \in \mathbb{R};

    • it is normalized: ρ(1)=1\rho(1) = 1.

See also state in AQFT and operator algebra.

If HH is a Hilbert space, then the bounded operators on HH form a C *C^*-algebra H\mathcal{B}H, and states on the Hilbert space correspond directly to states on H\mathcal{B}H. Classical mechanics can also be formulated in AQFT; the classical space of states XX gives rise to a commutative von Neumann algebra L (X)L^\infty(X) as the algebra of observables.

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.

quantum probability theoryobservables and states

classical mechanicssemiclassical approximationformal deformation quantizationquantum mechanics
order of Planck's constant \hbar𝒪( 0)\mathcal{O}(\hbar^0)𝒪( 1)\mathcal{O}(\hbar^1)𝒪( n)\mathcal{O}(\hbar^n)𝒪( )\mathcal{O}(\hbar^\infty)
statesclassical statesemiclassical statequantum state
observablesclassical observablequantum observable


Discussion in algebraic quantum field theory:

See also:

Last revised on December 20, 2023 at 09:58:47. See the history of this page for a list of all contributions to it.