nLab
state

Idea

A state is a configuration of a system in physics, together with enough information about its time evolution? (typically, the first derivative is sufficient) to allow one to propagate it arbitrarily.

In the Bayesian interpretation of physics, the state of a system is not a property of reality but instead indicates an observer's knowledge about the system. A pure state? gives maximal information about the system (which amounts to complete information in classical mechanics but not generally in quantum mechanics), while a mixed state? is more general. A mixed state can be decomposed into a probability distribution on the space of pure states, although this decomposition is unique only for classical systems. In a frequentist interpretation of probability, a mixed state can describe only a statistical ensemble of systems; the real world is in one (generally unknown) pure state (possibly with additional hidden variables in the quantum case, depending on the interpretation of quantum physics).

States in the Schrödinger picture describe the state of the world at any given time and are subject to time evolution?, while in the Heisenberg picture a single state describes the entire history of the world.

Definitions

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

In classical mechanics

In classical Lagrangian mechanics, a pure state is a point in the state space? of the system, giving all of the (generalised) positions? and velocities?. In classical Hamiltonian mechanics, a pure state is a point in the phase space of the system, giving the positions and momenta. In either case, a mixed state is a probability distribution on the space of pure states.

More generally, a classical state is a linear function $\rho :A\to ℝ$ on the Poisson algebra $A$ underlying the classical mechanical system which satisfies positivity and normalization.

In Hilbert-space quantum mechanics

In quantum mechanics given by a Hilbert space $H$, a pure state? is a ray in $H$, which we often call the Hilbert space of states. Strictly speaking, the space of states is not $H$ but $(H\setminus \{0\})/ℂ$, or equivalently $S(H)/\mathrm{U}(1)$. A mixed state is then a density matrix on $H$.

In AQFT

In AQFT, a quantum mechanical system is given by a $C^{*}$-algebra $A$, and a quantum state is usually defined as a linear function $\rho :A\to ℂ$ which satisfies positivity and normalization; see states in AQFT and operator algebra.

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

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

In this formulation the (n-1)-morphism in $𝒞$ 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.

Pure and mixed states

In statistical physics, a state may encode less than maximal information. Fairly generally, we have a complex $C^{*}$-algebra $A$ of observables and a state is a positive-semidefinite linear operator $\rho :A\to ℂ$ such that $\rho (1)=1$. A state $\rho$ is pure if, whenever we have $\rho =a\sigma +b\tau$ for $\sigma ,\tau$ also states and $0\le a,b\le 1$, then either $a=0$ (hence $b=1$) or $b=0$ (hence $a=1$); conversely, a state $\rho$ is mixed if we ever have $\rho =a\sigma +b\tau$ for $\sigma ,\tau$ also states and $0<a,b<1$.

If $A$ is the algebra of continuous complex-valued functions on some compactum $X$, then the pure states on $A$ correspond precisely to the points in $X$; so pure states here are the states of classical mechanics (at least for a compact phase space). Mixed states, however, correspond more generally to Radon probability measures on $X$, with the pure states as the Dirac delta measure?s.

On the other hand, if $A$ is the algebra of all bounded operators on some Hilbert space $H$, then the pure states on $A$ correspond precisely to the rays in $H$. Mixed states, however, correspond more generally to density matrices on $H$, with the pure states those matrices of the form $\mid \psi ⟩⟨\psi \mid$ for some unit vector $\mid \psi ⟩$.

In each case, a pure state is a state of maximal information, while a mixed state is a state with less than maximal information. In the classical case, we may say that a pure state is a state of complete information, but this does not work in the quantum case; from the perspective of the information-theoretic or Bayesian interpretation of quantum physics, this inability to have complete information, even when having maximal information, is the key feature of quantum physics that distinguishes it from classical physics.

Related concepts

• state

  • classical state, quantum state

  • state in AQFT and operator algebra

• observable

  • algebra of observables

  • GNS construction