nLab phase and phase space in physics

Idea

In physics the term phase appears prominently in two superficially different uses:

Inside the term “phase space”, introduced by Boltzmann in 1872 in the context of classical mechanics, it refers to the instantaneous position coordinates and momenta of a particle; its Bewegungsphase, it phase of motion. (Nolte, p. 4).

But the term “phase” re-appeared, independently, with the advent of quantum physics: it refers to the complex phase of the complex numbers that wave functions take values in.

But it turns out that in the semiclassical approximation to quantum physics the phase of wave functions (or rather its derivative) precisely parametrizes points in phase space! Even though this is historically a pure coincidence of terminology, fortunately it matches to some extent.

This relation is discussed in some detail at semiclassical state. It is an old observation that drove the pre-history of quantum mechanics at a time when this was discovered by drawing analogies with wave optics:

the derivative of the phase (as in: complex numbers) of a wave function is (locally, or else in semiclassical approximation) the momentum of the corresponding particle (of which the wave function is the quantum state). This momentum is, in addition to the spatial dependence of the wave function, what parameterizes phase space.

More precisely: a stationary semiclassical state of a particle propagating on some Riemannian manifold $(X,g)$ (and subject to some force given by some scalar potential but not charged under a gauge field) is a wave function of the form

$$\psi \colon (x,t) \mapsto \exp(i S(x)/\hbar) a(x) \exp(-i \omega t) \,,$$

where $S$ and $A$ are real-valued functions. The de Rham differential of the complex phase-function

$$S\;\colon\; X \to \mathbb{R}$$

appearing here defines a map from $X$ into its phase space, which here is the cotangent bundle over $X$:

$$\mathbf{d}S \;\colon\; X \to T^\ast X \,.$$

The phase-space image of $X$ defined this way is the level set of the given Hamiltonian $H \colon T^* X \to \mathbb{R}$ at energy $E = \hbar \omega$.

So the de Rham differential $\mathbf{d}S$ of the complex phase function $S$ of a semiclassical wave function precisely parametrizes this semiclassical state as a submanifold in phase space.

References

The origin of the term “phase” in “phase space” is discussed in

• David Nolte, The tangled tale of phase space, Physics Today (April 2010) (pdf)

The close relation between complex phases of semiclassical wave functions and phase space points is discussed for instance in the introduction (around p. 9) of

• Sean Bates, Alan Weinstein, Lectures on the geometry of quantization (pdf)