nLab quantum state

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

The mixed states are density matrices on $H$. Every pure state may be interpreted as a mixed state; taking a representative normalised vector ${|\psi\rangle}$ from a ray in Hilbert space, the operator ${|\psi\rangle}{\langle\psi|}$ 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 $H$ as observables. Alternatively, one may take a more abstract approach, as follows.

In AQFT

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

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

• such that

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

  • it is normalized: $\rho(1) = 1$.

See also state in AQFT and operator algebra.

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

$$Bord_n^S \to \mathcal{C} \,.$$

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

Related concepts

• potentiality

• quantum state tomography

quantum probability theory – observables and states

• states

  • classical state

  • quantum state

    • space of states (in geometric quantization)

    • state on a star-algebra, quasi-state

    • qbit, Bell state

      dimer, tensor network state

    • quantum state preparation

    • probability amplitude, quantum fluctuation

    • pure state

      • wave function

      • bra-ket

      • Bell state

    • quantum superposition, quantum interference

    • quantum entanglement

    • quantum measurement

      wave function collapse

      Born rule

      deferred measurement principle

      quantum reader monad

      measurement problem

    • superselection sector

    • mixed state, density matrix

      entanglement entropy

      holographic entanglement entropy

    • coherent quantum state

  • ground state, excited state

  • quasi-free state

  • Fock space, second quantization

  • vacuum, vacuum state

    • Hadamard state

    • vacuum diagram

    • vacuum expectation value, vacuum amplitude, vacuum fluctuation

    • vacuum energy

    • vacuum polarization

    • interacting vacuum

    • thermal vacuum, KMS state

    • vacuum stability

      • false vacuum, tachyon, Coleman-De Luccia instanton

    • theta vacuum

    • perturbative string theory vacuum

      • non-geometric string theory vacuum

      • landscape of string theory vacua

  • entangled state

  • tensor network state

    • matrix product state

    • tree tensor network state

• observables

  • quantum observable, beable

  • algebra of observables, star-algebra

  • Bohr topos

  • quantum operator (in geometric quantization)

  • quantum operation, quantum effect, effect algebra

  • in quantum field theory

    • local observable

    • polynomial observable

      • linear observable

        • field observable

    • regular observable

    • microcausal observable

    • normal-ordered product, time-ordered products, retarded product

    • Wick algebra

    • scattering amplitude

    • interacting field algebra of observables, Bogoliubov's formula

• GNS construction

• theorems

  • order-theoretic structure in quantum mechanics

    • Gleason's theorem

    • Alfsen-Shultz theorem

    • Harding-Döring-Hamhalter theorem

    • Kochen-Specker theorem

    • Nuiten's lemma

  • Wigner's theorem

  • no-cloning theorem

  • Bell's theorem

	classical mechanics	semiclassical approximation	…	formal deformation quantization	quantum mechanics
order of Planck's constant $\hbar$	$\mathcal{O}(\hbar^0)$	$\mathcal{O}(\hbar^1)$	$\mathcal{O}(\hbar^n)$	$\mathcal{O}(\hbar^\infty)$
states	classical state	semiclassical state			quantum state
observables	classical observable				quantum observable

References

Discussion in algebraic quantum field theory:

• Nikolay Bogolyubov, A. A. Logunov, A. I. Oksak, I. T. Todorov, G. G. Gould, Algebra of Observables and State Space [doi:10.1007/978-94-009-0491-0_6], Chapter in: General principles of quantum field theory, Mathematical Physics and Applied Mathematics 10, Kluwer (1990) [doi:10.1007/978-94-009-0491-0]

See also:

• Roland Omnès, §3 of: The Interpretation of Quantum Mechanics, Princeton University Press (1994) [ISBN:9780691036694]