nLab quantum state

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Measure and probability theory

measure theory

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 $H$, a pure state is a ray in $H$. Thus, the pure states form the space $\left(H\setminus \left\{0\right\}\right)/ℂ$, where we mod out by the action of $ℂ$ on $H\setminus \left\{0\right\}$ by scalar multiplication; equivalently, we can use $S\left(H\right)/\mathrm{U}\left(1\right)$, the unit sphere? in $H$ modulo the action of the unitary group $\mathrm{U}\left(1\right)$. 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 $\mid \psi ⟩$ from a ray in Hilbert space, the operator $\mid \psi ⟩⟨\psi \mid$ is a density matrix.

In principle, any quantum mechanical system can be treated using Hilbert spaces, by imposing superselection rule?s 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 $ℂ$-linear function $\rho :A\to ℂ$

• such that

• it is positive: for every $a\in A$ we have $\rho \left({a}^{*}a\right)\ge 0\in ℝ$;

• it is normalized: $\rho \left(1\right)=1$.

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

${\mathrm{Bord}}_{n}^{S}\to 𝒞\phantom{\rule{thinmathspace}{0ex}}.$Bord_n^S \to \mathcal{C} \,.

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

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

Revised on March 21, 2013 20:01:00 by Urs Schreiber (89.204.138.15)