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--- advanced_tools:group_theory:representation_theory:metaplectic_representation [2017/07/04 09:17]
jakobadmin [Why is it interesting?]
+++ advanced_tools:group_theory:representation_theory:metaplectic_representation [2018/04/08 16:14] (current)
63.143.42.253 ↷ Links adapted because of a move operation
@@ Line 12: / Line 12: @@
 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
+discovered the metaplectic representation of the symplectic group. [...]
-<cite>chapter 6 in The Principles of Newtonian and Quantum Mechanics by M. Gosson</cite>
+**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 [[formalisms:hamiltonian_formalism|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.
+<cite>chapter 6 and 7 in The Principles of Newtonian and Quantum Mechanics by M. Gosson</cite>
 </blockquote>

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