My question is inspired by the analogy of the Berry phase in the spin coherent state representation of a rotator and the Aharonov-Bohm phase of a magnetic monopole (see e.g., Section 1.8.3 in http://www.physics.ubc.ca/~berciu/TEACHING/PHYS503/section1.pdf).

Consider a physical system invariant under a (continuous) group of transformations G. The aim is to define the corresponding quantum mechanical Hilbert space H ``in the richest possbile way'', that is, a representation which has as many state vectors as group elements. In case of G=SO(3) it is the 2-sphere of spin coherent states. The Aharonov-Bohm analogy says that H is a manifold with a connection corresponding to vector potential of some specific kind (in the example of spin it is the vector potential which gives the magnetic field of the monopole). Once we know that connection, can we classify the possible global topologies of representations of G as the angular momentum representations are classified by requiring the uniqueness of the Berry phase of a closed loop?

Can such a manifold be constructed for any G (from a sufficiently interesting class of groups)? My feeling is that this should correspond to some theorem in topolgy/representation theory. I'm very ignorant in these fields of math, and would appreciate a hint on where to look further.

  • 4
    $\begingroup$ It can be done for most (compact) Lie groups. See Perelomov's construction of "generalized coherent states". A good reference is Ali and Antoine-Gazeau's text on coherent states. $\endgroup$
    – user346
    May 2 '11 at 6:38

There are essentially three ways to generalize the notion of coherent state, based on extensions of the concept of coherent states for the case of the harmonic oscillator.

  1. The first is to note that in the case of the harmonic oscillator, the coherent states can be obtained by acting on the vacuum state with translations in position and momentum space. This is the approach that Perelomov generalizes by replacing the action with arbitrary Lie groups.
  2. The second approach is to note that coherent states of the harmonic oscillator are saturating the Heisenberg inequalities, i.e. they have minimal uncertainty in both momentum and position. This can be generalized as well to give other kinds of coherent states which are not necessarily equivalent to Perelomov generalized coherent states.
  3. Then there is a third approach, that generalizes the annihilation/creation operators for any Hamiltonian with discrete spectrum and builds an overcomplete set of states out of the eigenstates of the annihilation operator.

All three methods are mentioned in J.R. Klauder and B. Skagerstam, "Coherent States", World Scientific Publishing Co. Pte. Ltd., Singapore, 1985.

  • $\begingroup$ There's also coherent states that are eigenstates of the annihilation operator... $\endgroup$
    – Simon
    May 18 '11 at 13:08
  • $\begingroup$ What annihilation operator? I think when you define that, you basically end up with Perelomov's approach. $\endgroup$ May 18 '11 at 19:10
  • $\begingroup$ Assuming you have a discrete set of energy eigenstates (a fairly restrictive assumption), you can always define an annihilation operator. Then the group coherent states and annihilation operator eigenstates are not necessarily the same. I looked at this for my 4th year project, which seems like a long time ago. If you're interested you can find a version of my thesis at my mendeley profile. I investigated the infinite square well in some detail. $\endgroup$
    – Simon
    May 19 '11 at 1:02
  • $\begingroup$ @Simon, OK, I checked out your thesis, but it turns out this is the third approach I mention. I'll edit my text a bit later to make it clearer. $\endgroup$ May 19 '11 at 7:34
  • $\begingroup$ Thank you! Perelomov's construction is indeed what I've been looking for: we get a space on which the representation of G/H acts which the same topological properties as G/H. And the fiber bundle representing H is identified with the quantum-mechanical phase in the "phase space" created in this construction. $\endgroup$
    – Slaviks
    Aug 15 '11 at 6:46

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