nLab classical state




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

In principle, a pure state in classical mechanics specifies completely all information about the state of the system, while a mixed state is a probability measure on the space of pure states. This space of pure states may be identified with the state space? in Lagrangian mechanics or with the phase space in Hamiltonian mechanics.


We give a definition in a very general context.

For AA a commutative unital associative algebra that encodes a system of classical mechanics (say, the associative algebra underlying a Poisson algebra), a classical state is an \mathbb{R}-linear function

ρ:A \rho\colon A \to \mathbb{R}

that satisfies

  • normalization ρ(1)=1\rho(1) = 1;

  • positivity for all aAa \in A we have ρ(a 2)0\rho(a^2) \geq 0.

This is essentially the definition of quantum state, but formulated for commutative algebras and over the real numbers.

If we take AA to be a **-algebra over the complex numbers, then we may take ρ\rho to be a \mathbb{C}-linear function from AA to \mathbb{C} instead.

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

quantum probability theoryobservables and states

Last revised on July 10, 2022 at 15:42:19. See the history of this page for a list of all contributions to it.