nLab
classical state

Context

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Contents

Idea

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 my be identified with the state space? in Lagrangian mechanics or with the phase space in Hamiltonian mechanics.

Definition

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.

order of \hbar𝒪( 0)\mathcal{O}(\hbar^0)𝒪( 1)\mathcal{O}(\hbar^1)𝒪( n)\mathcal{O}(\hbar^n)𝒪( )\mathcal{O}(\hbar^\infty)

Last revised on April 27, 2013 at 21:28:03. See the history of this page for a list of all contributions to it.