nLab quantum state

Contents

Context

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Measure and probability theory

Contents

Idea

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

Definitions

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

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 FQFT

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

References

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.