Schrödinger's equation is classical. Cheers!

Institut für Mathematik
der Universität Wien
Nordbergstrasse 15
1090 Vienna

A standard belief among physicists is that Schrödinger's equation is typically "quantum" and cannot be derived from classical mechanics. To witness Feynman's famous sentence "Where did that [the Schrödinger equation] come from? Nowhere. It came out of the mind of Schrödinger, invented in his struggle to find an understanding of the experimental obser- vations in the real world." Similar statements abound in the physical liter- ature; they are found in both introductory and advanced text on quantum mechanics, and we can read them on the web in various blogs and forums. However, they are strictly speaking not true. Firstly, it has been known for quite a long time by mathematicians working in the …eld of Harmonic Analysis that the classical and quantum dynamics are equivalent for Hamiltonians giving rise to linear flows (this property is a consequence of the metaplectic representation of the symplectic group).
Secondly, a misinterpretation of a famous "no-go"-theorem of Groenewold and Van Hove has then led physicists to believe that this equivalence cannot be pushed beyond the case of Hamiltonians with linear flows. We will show that the Schrödinger equation for a nonrelativistic spin- less particle is a classical equation, mathematically rigorously equivalent to Hamilton's equations, and hence essentially a classical equation. This will be done without invoking any additional physical assumption, but by insisting on the symplectic covariance of quantization and of Hamiltonian flows, and by using Stone's theorem on strongly continuous one-parameter groups of operators.
Admittedly, an equation is not a physical theory, so we certainly do not claim that QuantumMechanics emerges from our argument! But then, where does quantum physics enter the scene? We will explore some possibilities.

[1] M. de Gosson and B. Hiley. Imprints of the Quantum World in Clas- sical Mechanics, Found. Phys, 41(9), 1415-1436, (2010)

[2] M. de Gosson. Symplectic Methods in Harmonic Analysis and Appli- cations to Mathematical Physics, Birkhäuser (2011)