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I have been trying to understand quantum mechanics as a unitary representation of spacetime symmetries.

  1. My first question is: Can Schroedinger equation be derived from the unitary representation of Galilean group?

  2. My second question is: Can we derive Dirac equation and Klein-Gordon equation from representation theory of Poincare algebra?

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  • $\begingroup$ The answer to the second one is yes: the Klein-Gordon equation can be derived from the representation theory of the Poincare group. I'll work on finding a reference, it's been a while since I've looked at this. $\endgroup$
    – Moya
    Aug 8, 2015 at 3:04
  • $\begingroup$ For the Schrodinger equation, if the Hamiltonian does not depend on time explicitly, the equation is simply a 'unitary evolution'. In such a case, QM is unitary representation of time translation. How do I generalize this idea? $\endgroup$
    – Xiaoyi Jing
    Aug 8, 2015 at 3:05
  • $\begingroup$ Thank you so much for your help, Moya. It is very hard for me to find such a reference. I hope that there is a theorem showing that for any connected Lie group, there is an associated 'wave function' corresponding to a specific representation of the Lie group. $\endgroup$
    – Xiaoyi Jing
    Aug 8, 2015 at 3:11
  • $\begingroup$ The Schrodinger equation has nothing to do with the Galilean group. Among other things, there's no guarantee that an arbitrary Hamiltonian has Galilean symmetry. $\endgroup$
    – Qiaochu Yuan
    Aug 8, 2015 at 6:59
  • $\begingroup$ @Qiaochu Yuan. I should have mentioned that I meant the Hamiltonian of Non-Relativistic QM. $\endgroup$
    – Xiaoyi Jing
    Aug 8, 2015 at 15:47

1 Answer 1

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Yes, you can derive the Klein-Gordon, (free) Dirac, (free) Maxwell, linearized Einstein vacuum, etc., equations from the representation theory of the Poincaré group.

Yes, you can derive the ordinary non-relativistic (free) Schrödinger equation from the representation theory of the Galilei group (i.e., the represenation theory gives you the well known explicit form of the free non-relativistic hamiltonian).

The definitive and ultimate reference (assuming you want rigorous mathematics, which for this type of topics is the only way I recommend) for this topic is: "Geometry of Quantum Theory", by V.S.Varadarajan: http://www.amazon.com/Geometry-Quantum-Veeravalli-Seshadri-Varadarajan/dp/0387493859

I warn you, though, it's very hardcore material, not for the faint of heart.

Complementary references are: Folland's "Quantum field theory, a tourist guide for mathematicians" and Folland's "A Course in Abstract Harmonic Analysis".

Also, Theory of group representations and applications - A.O.Barut, R. Raczka.

Valter Moretti's book (he posts in this site) "Spectral Theory and Quantum Mechanics" also has useful material on quantum symmetries (Wigner's and Kadison's symmetries and theorems, Bargmann's theorem, stuff on central group extensions and also some stuff on the Galilei group).

Some online articles and notes: http://arxiv.org/abs/0809.4942v1; http://www.staff.science.uu.nl/~ban00101/lecnotes/repq.pdf

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  • $\begingroup$ Hi QuantumLattice. Thank you so much for your answer. I was surprised that there would be someone answering my question. $\endgroup$
    – Xiaoyi Jing
    Aug 10, 2015 at 17:22

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