Multidimensional Quantum-Classical Correspondence

    The problem of quantum-classical correspondence is a classic since the introduction of the Bohr model of the atom. A look at some of the people involved in this problem in this century is really impressive - contributions were made by Einstein, Sommerfeld, Brillouin, Keller, Kramers, Jeffreys, Wentzel, Jordan Heisenberg, Dirac, Pauli, VanVleck, Gutzwiller and many others which we will hopefully get to later during this very thin caricature of a subtle issue.

    So, what is this all about? One way of stating the problem is as follows:

    Given a classical system and its quantum counterpart, what is the correspondence (mapping) between the classical objects and the quantum objects?

    In other words, if we take a particular quantum object or observable, for example the discrete energy eigenvalues of a bound system, then what is the classical structure that gives rise to the discreteness of the energy spectrum? Remember, classically energy is a continuous quantity and quantum theory endows discreteness to the energy spectrum of bound systems. For example, the simple harmonic oscillator has energy levels with a constant spacing proportional to a fundamental constant known as the Planck's constant. Where does this discreteness come from? Clearly there is no Planck's constant in classical mechanics and hence what possible classical structure could give rise to this experimentally observed dicrete energy spectrum. One could ask similar questions pertaining to "quantum objects" like phase shifts, tunneling, wavefunctions, etc. The quotation marks are meant to emphasize that as we proceed it becomes clear that there are well defined classical analogs of these various quantum structures.

    Now, given a classical Hamiltonian one first needs the quantum Hamiltonian. This procedure of going from a classical Hamiltonian (structure) to a quantum Hamiltonian is known as "quantization". Dirac proposed a possible quantization which must be familiar if you have had quantum mechanics. I will not go into explaining Dirac quantization here but mention that it can be derived from rather general arguments of space-time symmetries. A natural question is whether Dirac quantization is the only possible quantization - this is a subtle question and mathematicians have been worrying over this for a while now. I am certainly not qualified to understand or explain this in clear and simple terms.
    Let us just assume for the present purposes that given a classical Hamiltonian we have a well defined quantum Hamiltonian. Armed with the quantum Hamiltonian we can now do quantum dynamics (pick your favorite method: Heisenberg's matrix mechanics, Schroedinger's wave mechanics, Dirac's approach, Feynman path integrals) and calculate a variety of observables of interest. Typical objects of interest for bound state systems are energy spectrum, density of states, quantum numbers, transition amplitudes from one state (surface) to another state (surface) etc. Most of the observables depend on the Planck constant. It turns out, interestingly enough, that the dimensionality of the Planck constant is the same as that of the classical action i.e, momentum times position. In fact the proper setting for classical dynamics is the "phase space" (p,q) where p denotes momentum and q denotes position. Thus, as the ratio of Planck's constant to the classical action of a trajectory approaches zero somehow quantum objects should go over to analogus classical objects. This limit is a key to understanding quantum-classical correspondence. Unfortunately, the limit is more often than not very highly singular and requires careful study. For example, if one were to take such a limit for the quantum wavefunction then there is an essential singularity implying that the classical object analogus to a wavefunction might not make any sense!
    As it turns out, the classical structure corresponding to a quantum wavefunction is a collection of classical trajectories in phase space as opposed to the intutive guess of a single classical trajectory. In general, quantum observables tend to have classical limits and objects that are not quantum observables have singular limits. I have also intentionally stayed away from discussing long time limits. The validity and usefulness of semiclassical mechanics is intricately tied into the two limits of small Planck's constant and long times. The fact that these two limits do not commute makes the correspondence problem interesting and difficult at the same time.

    The "somehow" represents a kind of bridge between classical and quantum mechanics encompassing the discipline of semiclassical mechanics. As pointed out earlier, the quantum objects cannot be written down as a convergent series in Planck's constant with the first term of the series being the required classical structure.
    Question: Is it possible to give a classical interpretation to the coefficients of the series? At first order in Planck's constant there is a beautiful geometrical interpretation for the coefficients via symplectic geometry. It is not clear what kind of geometical, if any, interpretation one could provide for higher orders.

    Hence, semiclassical mechanics is necessarily an asymptotic theory. Naturally, in semiclassical theories one is concerned with a family of classical trajectories. Each one of the trajectories has an amplitude and phase (classical action) assosciated with it thus incorporating the idea of superposition/interference. This is nothing new in the sense that Huygens wave theory for light already had most of these ideas! (The analogy with optics is not accidental). There has been considerable progress in understanding this correspondence for systems with one degree of freedom (d = 1). This is partially due to the fact that classically d = 1 case is always integrable. In other words, there is no possibility of chaotic dynamics. More generally, for any dimensionality, if the classical system is integrable then the correspondence is understood fairly well.
    In the case of an N-dim integrable system, the classical structures of importance are N-tori in the 2N-dim phase space. These tori correspond to quantum states and the N independent cycles on the N-torus correspond to the N quantum numbers. Wavefunction of a specific quantum state can be "assosciated" to a particular N-torus via a proper generalization of semiclassical mechanics.

    In the case of nonintegrable classical dynamics, this correspondence is not clear. Answers to questions like what is the quantum analog of chaos in classical dynamics? are still lacking. If there are no tori in phase space at all (strong chaos) then it turns out that the important classical structures are periodic orbits and the quantum density of states can be assosciated with these periodic orbits. However, the hard part of the correspondence problem has to do with mixed systems. These systems are rather generic in classical model Hamiltonians for chemical systems. The underlying phase space structures are extremely rich and complicated. Extracting useful dynamic information about the system requires identifying all the relevant classical phase space structures and their possible quantum analogs. These mixed systems are the focus of my current research.
    [You might ask, why should anyone bother with all this? Is it just fancy stuff or is there anything to be learnt from it? Why not just do quantum mechanics as, afterall, it is the correct theory? I have my own reasons as I am sure you can think of some of your own. If you are interested in discussing more about this, you are more than welcome to email me!]