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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 my 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 $A$ 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
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 mechanics | semiclassical approximation | … | formal deformation quantization | quantum mechanics | |
---|---|---|---|---|---|
order of Planck's constant $\hbar$ | $\mathcal{O}(\hbar^0)$ | $\mathcal{O}(\hbar^1)$ | $\mathcal{O}(\hbar^n)$ | $\mathcal{O}(\hbar^\infty)$ | |
states | classical state | semiclassical state | quantum state | ||
observables | classical observable | quantum observable |