Geometric quantization (nonfiction)
In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory.
It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory and the quantum theory remain manifest.
For example, the similarity between the Heisenberg equation in the Heisenberg picture of quantum mechanics and the Hamilton equation in classical physics should be built in.
One of the earliest attempts at a natural quantization was Weyl quantization, proposed by Hermann Weyl in 1927. Here, an attempt is made to associate a quantum-mechanical observable (a self-adjoint operator on a Hilbert space) with a real-valued function on classical phase space. The position and momentum in this phase space are mapped to the generators of the Heisenberg group, and the Hilbert space appears as a group representation of the Heisenberg group. In 1946, H. J. Groenewold considered the product of a pair of such observables and asked what the corresponding function would be on the classical phase space. This led him to discover the phase-space star-product of a pair of functions.
The modern theory of geometric quantization was developed by Bertram Kostant and Jean-Marie Souriau in the 1970's. One of the motivations of the theory was to understand and generalize Kirillov's orbit method in representation theory.
More generally, this technique leads to deformation quantization, where the ★-product is taken to be a deformation of the algebra of functions on a symplectic manifold or Poisson manifold. However, as a natural quantization scheme (a functor), Weyl's map is not satisfactory. For example, the Weyl map of the classical angular-momentum-squared is not just the quantum angular momentum squared operator, but it further contains a constant term 3ħ2/2. (This extra term is actually physically significant, since it accounts for the nonvanishing angular momentum of the ground-state Bohr orbit in the hydrogen atom. As a mere representation change, however, Weyl's map underlies the alternate Phase space formulation of conventional quantum mechanics.
The geometric quantization procedure falls into the following three steps:
- Prequantization
- Polarization
- Metaplectic correction
Prequantization produces a natural Hilbert space together with a quantization procedure for observables that exactly transforms Poisson brackets on the classical side into commutators on the quantum side. Nevertheless, the prequantum Hilbert space is generally understood to be "too big". The idea is that one should then select a Poisson-commuting set of n variables on the 2n-dimensional phase space and consider functions (or, more properly, sections) that depend only on these n variables. The n variables can be either real-valued, resulting in a position-style Hilbert space, or complex-valued, producing something like the Segal–Bargmann space.
A polarization is a coordinate-independent description of such a choice of n Poisson-commuting functions.
The metaplectic correction (also known as the half-form correction) is a technical modification of the above procedure that is necessary in the case of real polarizations and often convenient for complex polarizations.
In the News
Fiction cross-reference
Nonfiction cross-reference
- Hilbrand J. Groenewold (nonfiction) - Hilbrand Johannes "Hip" Groenewold (1910–1996) was a Dutch theoretical physicist who pioneered the largely operator-free formulation of quantum mechanics in phase space known as phase-space quantization.
- Bertram Kostant (nonfiction) - one of the principal developers of the theory of geometric quantization
- Mathematics (nonfiction)
- Quantum mechanics (nonfiction)
- Jean-Marie Souriau (nonfiction)
External links:
- Geometric quantization @ Wikipedia