I read some while ago that, currently, eleven different formulations of quantum mechanics exist. Is this correct / accurate? If yes, can someone provide a pointer(s) (i.e. link(s)) to the various formulations?
Answer
The mathematical formulation of QM, as defined by Dirac, is a closed thing--- there is always a Hilbert space of states, and operators which act on the Hilbert space to produce physical changes. This mathematical scheme, however, is very general, and when you write down the description of specific quantum systems, you have to make some assumptions about the Hilbert space structure and the form of the linear operators. There are general classes of systems which are best defined in different formal schemes, I suppose that these formalisms are the quantum mechanical formulations you are talking about.
The historical formulations of quantum mechanics were matrix mechanics and wave mechanics, which are two pictures of the same mathematical structure. The two are unified and contained in Dirac's transformation theory formalism, which is what people learn today as "Quantum mechanics." This formulation includes all the others in a certain sense, because it defines quantum mechanics.
The path-integral formulation came later, and it is also mostly equivalent to the ordinary formulation, but as it is generally presented, it makes the extra assumption that the Hamiltonian is quadratic in the momentum, so that the Lagrangian can be expressed simply in terms of the trajectory. This assumption is not absolutely necessary, but it is mathematically convenient, and it is correct for nearly all applications, and when it is not, like the string world-sheet action in Nambu-Goto form, it can often be made correct using auxiliary variables. The path integral makes unitarity nontrivial, it is trivial in Dirac's formulation.
The path integral links quantum systems to non-quantum systems by analytic continuation, and this is surprising when looking at the pure Dirac formalism, so the path integral should count as a second formulation, truly different from Dirac's. The path integral formulation produces a natural description of gauge theories, which is very inconvenient in Dirac form.
The third formulation is more recent, and this is the PT-symmetric quantum mechanics. In abstract terms, PT symmetric QM is again just ordinary Dirac QM, just as path-integral QM is also Dirac QM. But it should count as a new formulation, because the metric on Hilbert space is defined dynamically, from the Hamiltonian, and you would never find it starting with a naive metric. The naive metric on Hilbert space makes the Hamiltonian seem to be non-Hermitian, and to verify that there is a Hermitian Hamiltonian requires work.
This formulation is only about 10 years old, and is very actively studied, but I think it is probably the most fundamental formulation, considering.
- Dirac QM
- Path integral
- PT-symmetric QM
You also could formulate PT symmetric QM as a path integral, maybe you would count this as a fourth. But this gives three options. In addition, you can choose to formulate each of these as either acting on the Hilbert space of states, or on the space of all density matrices
- States
- Density matrices
The density matrix formulation is arguably more fundamental. It subsumes the formulation in terms of Wigner functions. So this is a choice of two options. Multiplying gives 6.
Different theories
There are different theories which are closely related to QM, which are generally not viable. These are most of the objective collapse theories, or nonlinear state evolution theories. But one deformation of quantum mechanics at least, is not ruled out by anything except theoretical principles, and this is the superoperator formulation
- Superoperator formulation
The superoperator formulation generalizes the notion of Hamiltonian to the most general operator on the density matrix, rather than the state-vector. This formulation allows for a description of instantaneous decoherence, and it is developed in the 1970s in an obscure book which I read a long time ago, and whose author escapes me (I am sorry, it doesn't google for me).
Hawking also advocated a direct density matrix formulation as part of his program of information loss in black holes, but I don't think he cited the earlier book I am talking about (probably because he wasn't aware of it--- I stumbled across it in a library years ago). The name "superoperator" is not standard--- it is the name given in quantum information theory to the operator which multiplies the density matrix to give its time evolution. This is the quantum analog of the Fokker-Planck equation.
Here is a refernce with "nine formulations", with somewhat different ideas of what constitutes a formulation: http://www-physique.u-strasbg.fr/cours/l3/divers/meca_q_hervieux/Articles/Nine_form.pdf
No comments:
Post a Comment