Why is it interesting?

Here we touch one of the central themes of this book, the metaplectic representation of the symplectic group. It is a deep and fascinating subject of mathematics, unfortunately unknown to most physicists. It is however essential to the understanding of the relationship between classical and quantum mechanics […] While it is true that Schrödinger’s argument was not rigorous (it was rather a “sleepwalker” argument 1 ), all the mathematically “forbidden” steps he took ultimately lead him to his famous equation (6.3). But it all worked so well, because what he was discovering, using rudimentary and awkward mathematical methods, was a property of pure mathematics. He in fact discovered the metaplectic representation of the symplectic group. […]

The metaplectic representation yields an algorithm allowing to calculate the solutions of Schrödinger’s equation from the classical trajectories. Conversely, the classical trajectories can be recovered from the knowledge of the wave function. Both classical and quantum motion are thus deduced from the same mathematical object, the Hamiltonian flow.

[…]

We will in fact see that both classical and quantum mechanics rely on the same mathematical object, the Hamiltonian flow, viewed as an abstract group. If one makes that group act on points in phase space, via its symplectic representation, one obtains Hamiltonian mechanics. If one makes it act on functions, via the metaplectic representation, one obtains quantum mechanics. It is remarkable that in both cases, we have an associated theory of motion: in the symplectic representation, that motion is governed by Hamilton’s equations.

chapter 6 and 7 in The Principles of Newtonian and Quantum Mechanics by M. Gosson

Layman

Student

A good introduction can be found in chapter 6 in The Principles of Newtonian and Quantum Mechanics by M. Gosson.

See also Quantum Theory, Groups and Representations by P. Woit

Researcher

Common Question 1

Common Question 2

Examples

Example1

Example2:

History