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I’m continuing my series of articles for the Princeton Companion to Mathematics with my article on phase space. This brief article, which overlaps to some extent with my article on the Schrödinger equation, introduces the concept of phase space, which is used to describe both the positions and momenta of a system in both classical and quantum mechanics, although in the latter one has to accept a certain amount of ambiguity (or non-commutativity, if one prefers) in this description thanks to the uncertainty principle. (Note that positions alone are not sufficient to fully characterise the state of a system; this observation essentially goes all the way back to Zeno with his arrow paradox.)
Phase space is also used in pure mathematics, where it is used to simultaneously describe position (or time) and frequency; thus the term “time-frequency analysis” is sometimes used to describe phase space-based methods in analysis. The counterpart of classical mechanics is then symplectic geometry and Hamiltonian ODE, while the counterpart of quantum mechanics is the theory of linear differential and pseudodifferential operators. The former is essentially the “high-frequency limit” of the latter; this can be made more precise using the techniques of microlocal analysis, semi-classical analysis, and geometric quantisation.
As usual, I will highlight another author’s PCM article in this post, this one being Frank Kelly‘s article “The mathematics of traffic in networks“, a subject which, as a resident of Los Angeles, I can relate to on a personal level :-) . Frank’s article also discusses in detail Braess’s paradox, which is the rather unintuitive fact that adding extra capacity to a network can sometimes increase the overall delay in the network, by inadvertently redirecting more traffic through bottlenecks! If nothing else, this paradox demonstrates that the mathematics of traffic is non-trivial.