Quantum mechanics on a manifold

In quantum mechanics the state of a free particle in three dimensional space is $L^2(\mathbb R^3)$, more accurately the projective space of that Hilbert space. Here I am ignoring internal degrees of freedom otherwise it would be $L^2(\mathbb R^3)\otimes S$, but let's say it is not that time of the month. The observables are operators in that space and the dynamics is describe by the Schrodinger equation or in other equivalent ways. If we have more particles then it is $L^2(\mathbb R^{3N})$. My question is: are there any examples where one considers a system with configuration space a general manifold $M$, instead of $\mathbb R^3$, say a system of particles (a particle) with some restrictions, so that the state space is $L^2(M)$. There might be physical reasons why this is of no interest and I would be interested to here them. What I am interested in is seeing is specific (or general) examples worked out in detail. For example a system with a given Hamiltonian, where one can explicitly find the spectrum. Or if that is too much to ask for an example where the system has very different properties from the usual case. Say a particle living on the upper half plane with the Lobachevsky geometry, may be some connection to number theory! I am aware that there is quantum field theory on curved spacetime, I am interested in quantum mechanics.

Edit: Just a small clarification. The examples I would like to see do not have to come from real physics, they can be toy models or completely unrealistic mathematical models. Something along the lines: take your favorite manifold $M$ and pretend this is the space we live in, what can we say about QM in it. The choice of $M$ doesn't have to do anything with general relativity. As I said the upper half plane is interesting or quotients of it by interesting discrete groups or generalizations $\Gamma\backslash G(\mathbb R)/K$ or anything at all. The answers so far are interesting. Hoping to see more.

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Just a note: for identical particles, the manifold is actually $\mathbb R^{3N}/S_N$. This is responsible for the Fermi and Bose statistics in 3+1 dimensions. – dbrane Feb 24 '11 at 2:25
@dbrane Shouldn't it be $(\mathbb R^{3N}-\Delta)/S_N$, where $\Delta$ are the singular points where two particles occupy the same position? – Heidar Feb 24 '11 at 13:19
@4tnemele: indeed --- otherwise the paths would still be contractible and one doesn't get nontrivial topologies. – genneth Feb 24 '11 at 13:48
@4tnemele @genneth Yes, sorry, missed that out – dbrane Feb 24 '11 at 14:39

Here is an overview of quantization methods: http://arxiv.org/abs/math-ph/0405065

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As far as I understand it there are essentially two ways in which you can study quantum mechanics on a manifold with some curvature. Classically speaking these two ways lead to the same physics but in a quantum mechanical approach they are distinct.

The first approach is to think of a particle moving "freely" through three-dimensional space, but subject to external forces that confine the particle to some submanifold. The particle lives, in some sense, in a confining potential which defines the manifold. The phase space of the particle is, from the start, the usual phase space associated with the three-dimensional space. However, the external potential limits the particle to some subspace of this phase space.

The second approach is to work with generalized coordinates, as is done in Lagrangian mechanics. The coordinates of the particle are then a parametrization of the submanifold. What's important here is that there is no reference to the coordinates of the three-dimensional space. An example is the pendulum, which can be described solely in terms of the angle the pendulum makes with the z-axis.

Classically there is no distinction between the two approaches. This no longer holds when you move to quantum mechanics. If you follow the first approach, using some confining potential to keep the particle on the manifold, you will deal with the uncertainty principle that prohibits the exact localization of the particle onto the manifold. Because of this principle the particle will never be fully screened from the larger dimensional space. You can still systematically set up the quantization procedure, though. The advantage of this approach is that quantization works in the usual way (you work with cartesian coordinates, after all). The resolution is to essentially split up the wavefunction and the Schroedinger equation (S.E.) in contributions due to the confining potential and a sort-of effective S.E. for the remaining part of the wavefunction. The effective S.E. then contains two effective potentials due to the Mean curvature and Gauss curvature of the corresponding manifold.

This is a very important feature: a cylinder, for instance, has no Gauss curvature, only a mean curvature. In the second approach you will find that there is no distinction between two cylinders with different mean curvatures, because in this approach only the Gauss curvature pops up. Take for instance a particle living on a 1D line. You only require one coordinate to describe this line, so for the the second approach all systems are equivalent. But in the first approach you have to specify in what way the line is embedded in the higher dimensional space, and how the particle is confined to the lower-dimensional space.

The second approach might feel more natural, if you think like a mathematician. In this approach you require a way to quantize generalized coordinates -- which is a lot more subtle than ordinary quantization. The problem that plagues this approach is the so-called ordering problem. Essentially you want to replace the momentum label by a derivation operator $p \rightarrow -i\hbar\nabla$. Furthermore, there's also the choice of parametrization of the manifold, which should ofcourse have no effect on the underlying physics (similar to general relativity). The ordering problem states that you do not know a priori which way the classical (commuting) variables have to be ordered before you replace them by their quantum mechanical (non-commuting) counterparts. What's even worse, because of the curvature of the space the derivative operator also contains some ambiguity. There is an ambiguity in the choice of your momentum operator and your Hamiltonian (and any other functions). Many quantum mechnical Hamiltonians have the same classical limit, and the equivalence principle (i.e. linking quantum mechanics to classical physics) does not dictate which is best. For instance, the kinetic operator $\nabla^2$ can be defined using the canonical Laplacian or the Laplace-Beltrami operator. Still, there is some work out there which motivates a generalized equivalence principle (see e.g. Kleinert) and results in a consistent quantization procedure.

Both approaches have interesting feature, but the first one is actually a bit more physical. The reason is that in condensed matter you deal with confining potentials due to some ionic lattice. Take for instance graphene, which is a two-dimensional surface. As it turns out, this surface is not completely flat but will always form some ripples. These deformations of the surface can be interpreted as if the electrons (or Dirac fermions, if you want to use the effective theory) live on a curved manifold embedded in a three dimensional surface. This leads to hilarious applications, such as the existence of wormholes in Graphene. But in the end the curvature has a very physical manifestation in the electronic properties of the system.

= The first of these approaches, which uses a confining potential, is discussed in these papers by Costa:

The second approach is treated in this review paper by B.S. De Witt: http://link.aps.org/doi/10.1103/RevModPhys.29.377

See also the book by Kleinert, who has a whole chapter on it using a Path Integral approach: http://www.amazon.com/Integrals-Quantum-Mechanics-Statistics-Financial/dp/9814273562

Graphene wormholes: http://arxiv.org/abs/0909.3057

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Interesting links, I'll look into them. – MBN Feb 24 '11 at 16:19
What a great review on the subject! This is the theme of my masters dissertation, I'm already very familiar with Costa and De Witt's works. Would you have any more advices on this subject? – Diego Mar 1 '11 at 11:29
@Diego The only other reference I am aware of is Dirac's book. There are some other approaches as well, which make use of the Free energy functional on a curved background (see e.g. rmp.aps.org/abstract/RMP/v82/i2/p1301_1 ). But if you understand Costa's work than I think the next step is to get your hands dirty on some specific application. In principle any electronic / spin system lends itself for such an approach. – Olaf Mar 1 '11 at 14:35
excellent answer, +1. I would like to ask (although it probably deserves a separate question), in general don't we have known transformations between eigenstates obtained from different operator orderings with the same classical limit? – lurscher Mar 18 '11 at 17:38
@lurscher I'm not familiar with it. Would definitely be worthy of a seperate question. – Olaf Mar 20 '11 at 19:05

When you study angular momentum in QM, this is a case of particle on a sphere. The wavefunctions are the spherical harmonics, the Hamiltonian is $L^2$, etc. I can think about many other examples where the configuration system of some QM system would be curved (e.g. some group manifold or coset space), so I don't think there is any physical reasons not to look at such examples.

For more sophisticated set of examples, there are many studies on supersymmetric quantum mechanics on various manifolds, starting out with this paper Supersymmetry and Morse Theory by Ed Witten. The connection between SUSY QM and QFT on a manifold and the topology (or sometimes even the geometry) of the underlying manifold became a bit of an industry since then.

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Good, Moshe. Just to add a simple comment. Condensed matter physics has lots of projects looking at electrons in various bizarre situations - like quantum mechanics on graphs scholar.google.com/scholar?q=quantum+mechanics+on+graphs and the corresponding curved 2D versions. Electrons may be confined in any 1D or 2D submanifolds of our nearly flat 3D space, so people study it from this perspective. – Luboš Motl Feb 24 '11 at 7:20
@Luboš: Yes, also the Fermi surface may have interesting geometry, just trying to keep things simple. – user566 Feb 24 '11 at 7:28

One can consider another generalisation of the $L^2(\mathbb R^3)$ model by noting that $\mathbb R^3$ is simply the configuration space, $Q$, of a single particle. There is a field called Geometric Quantization in which the base manifold is extended from a Configuration space onto a full Symplectic Manifold $(M,\omega)$.

The idea is that all of the Hamiltonian geometry can be encoded in the symplectic 2-form $\omega$. Thus one can talk about Poisson brackets {f,g}, classical observables and so on. The symplectic manifold is the natural geometric space for the study of classical mechanics of any system. Symplectic manifolds might have the form $M=T^*Q$ - as a cotangent bundle (and thus for a single particle be diffeomorphic to $\mathbb R^6$). However they can arise in other cases, for example via constraint reduction, or independently as solutions of field equations. In finite dimensional cases the symplectic manifolds are 2N dimensional.

The Hilbert space is then constructed on top of this. This procedure involves introducing a complex line bundle (locally $U \times \mathbb C$) B over the space M. There are certain topological conditions needed to ensure existence of the sections of this bundle (which are related to the old Bohr quantization conditions). When the sections exist a pairing operator can be introduced and a Hilbert space constructed.

The condition that the wave function $\Psi$ be in a representation (say the position representation) is encoded by introducing what is known as a Polarization on M. This is a foliation of M subject to certain conditions. The sections need to be constant along these foliations. This geometric process results in constructing familiar position and momentum expressions for the wave function, and in a sense rebuilds the configuration space if desired.

However one can often introduce a Complex structure $J$ such that $J^2=-1$ on M which when compatible with the symplectic form $\omega$ results in some further properties. Firstly this introduces a metric on M, and secondly we have $(M,\omega)$ become a Kahler manifold.

So now the "phase space" of the classical system is a Kahler manifold. Furthermore the Polarization conditions mentioned above result in $\Psi(z)$ - a holomorphic function of z. As a concrete example

$z = x + ip$

would be the holomorphic coordinate in 2 (real) D. This complex holomorphic representation for elementary examples was introduced by Bergmann in the 1940s, but in the Geometric Quantization context it is the simplest of the Kahler examples. In these Kahler examples the Coherent states play a fundamental part.

In terms of non-trivial manifolds another interesting class of examples from Geometric Quantization are from symmetry (Lie group) examples. Here the classical manifold is constructed from the group manifold itself (by examining coadjoint orbits). As a specific example $SU(2)$ has as classical manifold $S^2$. That is the sphere of radius s is the classical phase space for the rotational degrees of freedom of an elementary particle with spin s.

All of this can be a uniform framework for studying the process of quantization and the implications of non-trivial topologies classically (Bohm-Aharanhov, Berry phase, etc).

One text is Geometric Quantization.

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You may have a look at deformation quantization.

See for example:

Bayen, F. ; Flato, M.; Frønsdal, C.; Lichnerowicz, A. ; Sternheimer, D.: Quantum Mechanics as a Deformation of Classical Mechanics. In: Lett. Math. Phys. 1 (1977), S. 521–530

Bayen, F. ; Flato, M. ; Frønsdal, C. ; Lichnerowicz, A. ; Sternheimer, D.: Deformation Theory and Quantization. In: Ann. Phys. 111 (1978), S. 61–151

for the original papers.

See for example http://omnibus.uni-freiburg.de/~sw12/Download/intro.pdf for a short elementary introduction. http://iopscience.iop.org/1742-6596/103/1/012002 may be also interesting as introduction.

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Suppose you wanted to talk about the "quantum mechanical version" of a rotating rigid body which does not "go anywhere" (classically, the center of mass is stationary). Then we would probably consider states in L^2(SO(3,R)).

A general rule is that if you have a classically described system and you want to know what the "quantum mechanical version" of it is, you let M be the configuration manifold (ie, that manifold which describes the "position" of the classical system and whose cotangent bundle is the phase space manifold) and you take your states in L^2(M).

For the purposes of real physics, this is not always a useful thing. After all, classical systems (probably) don't exist, so there is not necessarily fundamental value in knowing how to go from "classical to quantum". It may however say something very interesting about how the classical limit arises and about the nature of quantum decoherence.

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The quantum mechanical motion of a particle on a curved manifold $X$ is called an example of a "non-linear sigma-model". See on the nLab at sigma-model for a bunch of further discussion. This is a very fundamental notion of quantum physics. Since the spacetime which we inhabit is in general curved, any quantum particle propagating in that spacetime is given by a non-linear sigma-model.

This has an interesting relation to deep questions in geometry: namely one may ask to which extent one can recover the curved geometry of some manifold from the physics of a quantum particle that propagates on it, hence from its space of states, and from the energy levels -- hence the spectrum -- of its Hamiltonian. This "spectral geometry" question has famously been solved by Alain Connes, by the notion of a "spectral triple". Such as triple is, in physics terms, nothing but

1. a Hilbert space of states

2. a Hamiltonian for a quantum particle (or rather a Dirac operator for a spinning particle);

3. an algebra of of spatial observables densely embedded into the Hilbert space.

The fundamental theorem of spectral triples -- hence of quantum mechanics on curved manifolds -- is that one can recover the Riemannian geometry of the "target manifold" from this data. In turn the fundamental impact of this observation is: there are also "spectral triples" and hence quantum mechanical particles as above which do not come from smooth curved target manifolds. So while these are not given by ordinary geometry, one can still understand them as describing quantum particle motion on generalized spaces, namely on spaces in noncommutative geometry. From this point of view "non-commutative geometry" is whatever a quantum particle "sees" as its "probes" its target space. See on the nLab at Spectral geometry and gravity.

This perspective on non-linear sigma-models is crucial for understanding modern developments such as in string theory. Because next we can ask what it means for a string to propagate on a curved manifold. One finds that now the data is a kind of higher dimensional analog of the previous data, which one might call a 2-spectral triple, commonly modeled by structures such as vertex operator algebras. So now for the quantum string one can ask the same reverse-engeneering question as for the quantum particle: given some string quantum mechanics with such-such-such energy spectrum and such-and-such algebras of observables, can we reconstruct the curved spacetime that the string must be propagating on?

Indeed one can -- precisely up to the famous "string dualities". This is how string theory connects to phenomenology, by infering from the string's quantum mechanics the effective background structure that it must be propagating through.

For some reason the close similarity between Connes spectral ("noncommutative") geometry description of quantum particles on curved spacetime and perturvative string theory is not widely advertized. It became particularly striking when in 2006 Connes and Barret noticed that the only way to get the chiral fermion structure correct in a "spectral standard model" built this way is to consider a non-commutative KK-compactification where the fiber space has K-theoretic dimension equal to 6 (see the commented references here). This is of course the same answer as in string theory, albeit derived here from different assumptions.

In any case, describing curved geometry in terms of the quantum mechanics of particles (and strings and branes etc. ) propagating on it is a deep topic in mathematics and physics.

But since the question seems to be really about on which spaces the wavefunctions are sections of some bundle, one should look a bit further: in general a wavefunction is a polarized section of a prequantum line bundle over a reduced covariant phase space. Now, phase spaces in quantum mechanics and in quantum field theory mostly always come out as the cotangent bundles $T^\ast X$ of the configuration space. But it is important to remember that in general there are (gauge) symmetries in the system, and that the actual phase space is the quotient of this cotangent bundle by these symmetries. This in general leads to geometrically and topolically non-trivial phase spaces.

In particular, the phase space of "internal" degrees of freedom of quantum particles are generically curved and compact. The most basic example is the phase space for "rotors" and "spinors", hence for the spin degree of freedom of fermions. These are the 2-sphere (with its round curved metric). See at geometric quantization of the 2-sphere for more on this.

Or if the particle is "nonabelian charged" (for instance if it is a quark) then the internal degrees of freedom are given by a phase space which is a coadjoint orbit of the given gauge Lie group. Details on such compact curved phase spaces are on the nLab at orbit method.

In conlcusion, curved target spaces and phase spaces are more the rule than the exception, and their quantization connects to important and deep problems not just in physics but also in geometry and in mathematics in general.

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