advanced_tools:group_theory:representation_theory:metaplectic_representation
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advanced_tools:group_theory:representation_theory:metaplectic_representation [2017/07/04 09:14] jakobadmin created |
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 |
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| - | While it is true that Schrödinger’s argument was not rigorous (it was | + | 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 | 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 | 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 | so well, because what he was discovering, using rudimentary and awkward | ||
| mathematical methods, was a property of pure mathematics. He in fact | 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> | </blockquote> | ||
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| <tabbox Student> | <tabbox Student> | ||
| - | <note tip> | + | A good introduction can be found in chapter 6 in The Principles of Newtonian and Quantum Mechanics by M. Gosson. |
| - | In this section things should be explained by analogy and with pictures and, if necessary, some formulas. | + | |
| - | </note> | + | See also [[https://www.math.columbia.edu/~woit/QM/qmbook.pdf|Quantum Theory, Groups and Representations]] by P. Woit |
| - | + | ||
| <tabbox Researcher> | <tabbox Researcher> | ||
advanced_tools/group_theory/representation_theory/metaplectic_representation.1499152490.txt.gz · Last modified: 2017/12/04 08:01 (external edit)
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