# nLab classical state

## Surveys, textbooks and lecture notes

#### Measure and probability theory

measure theory

probability theory

# 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 $A$ 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

$\rho\colon A \to \mathbb{R}$

that satisfies

• normalization $\rho(1) = 1$;

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

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

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

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

Revised on April 27, 2013 21:28:03 by Ingo Blechschmidt (46.244.223.210)