# nLab quantum state

Contents

## Surveys, textbooks and lecture notes

#### 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 $(H \setminus \{0\})/\mathbb{C}$, 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: $\rho(1) = 1$.

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.

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

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