Differential geometric approach to quantum mechanics

Question

Plenty of books/papers have been written about differential geometry in relation with general relativity, string theory, classical/quantum/gauge field theory and classical mechanics (Mathematical Methods of Classical Mechanics by V. I. Arnold comes to mind). I was wondering if people have investigated non-relativistic quantum mechanics using differential geometry and if this approach has been fruitful? Any good resources/books that discuss this method would be greatly appreciated.

Comments

Yeah, the approach basically uses Kahler manifolds --- for a review of this "geometric description", see Ashtekar and Schilling "Geometrical Formulation of Quantum Mechanics" arXiv:gr-qc/9706069.

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That looks interesting, thanks!

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For a geometrical perspective on quantum information theory, and just quantum theory and quantum states in general, look up The Geometry of Quantum States.

Answer 1

I do not understand well the question. Are you asking whether the issue of (first) quantization on manifolds has been investigated? If this is the question the answer is positive. Several researchers focused on that problem. The point is that the standard procedure based on Stone-von Neumann theorem generally does not hold for manifolds different form ${\mathbb R}^n$. The algebra of elementary observables (that generated by position and momentum in $\mathbb R^n$ is very difficult to define). Naive approaches face the technical problem of  symmetric operators which should represent observables but are not (essentially) self adjoint.  The situation becomes simpler when the space is homogeneous, i.e. when there is a (at least topological) group acting transitively on the space. The action can be defined either  in terms of isometries, conformal transformations  or diffeomorphisms. In this case  there are procedures, in particular due to Isham and Landsman and collaborators (Letters in Mathemattical Physics 20:11-18, 1990, Nuclear Physics B365 (1991) 121-160) leading to  a *-algebra of essentially self adjoint operators defining observables in a suitable Hilbert space associated with the manifold. These procedures include (generalizations of) anyons theory and Aharonov Bohm quantization as well as the standard quantization procedure in $\mathbb R^n$.

Comments

Thanks! I have only recently starting to recognize how important differential geometry appears to be in physics, and I'm getting more and more interested in it. I am basically trying to understand in what level of details we can describe quantum mechanics using differential geometry. So "(first) quantization on manifolds" sounds really interested, although I will be lying if I say I understood most of your message ;) (this is due to my lack of knowledge). But if I have some spare time then I will try to look up some of the terms you have mentioned.

Answer 2

I'd say the answer to this is clearly: geometric quantization.

I'd argue that this is the theory of quantization, everything else is an approximation to it (in particular the popular algebraic deformation quantization is an approximation to full non-perturbative geometric quantization). And it is all based on differential geometry -- on higher differential geometry, actually, if one includes quantization of local field theory -- I have some technical details on this here.

For an exposition of how differential geometry and Lie theory yield the theory of quantization see here.

Comments

Thanks! I've saved your pdf file on my computer and hope to be able to tackle it after I have learnt more differential geometry. I'm currently reading "Geometry, Topology and Physics" by Mikio Nakahara, but I have the feeling I may need to buy a more advanced book on diff. geometry before I can better appreciate your paper ;).

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@Hunter, my remark on field theory was for completeness, to drive home the point that this really eventually gives the full story, not just some fragment. But I gather you want to and probably should start with just quantum mechanics for the time being, hence with just standard geometric quantization. A fairly comprehensive and commented list of literature on that is here. That lists introductions, and textbooks, and original articles and recent developments. Dig around a bit and see what suits your needs.

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@UrsSchreiber Thanks, that is indeed exactly what I was looking for!

Answer 3

An excellent book is Marsden and Ratiu, Introduction to mechanics of symmetry .

It exposes the symplectic geometry underlying much of classical mechanics and even some of quantum mechanics.

Comments

Nice, and it is free as well! Thanks