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.
Origins 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 1970s. One of the motivations of the theory was to understand and generalize Alexandre Kirillov's orbit method in representation theory.
Types The geometric quantization procedure falls into the following three steps: prequantization, polarization, and 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 analytic, 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.
Prequantization Suppose ( M , ω ) {\displaystyle (M,\omega )} is a symplectic manifold with symplectic form ω {\displaystyle \omega } . Suppose at first that ω {\displaystyle \omega } is exact, meaning that there is a globally defined symplectic potential θ {\displaystyle \theta } with d θ = ω {\displaystyle d\theta =\omega } . We can consider the "prequantum Hilbert space" of square-integrable functions on M {\displaystyle M} (with respect to the Liouville volume measure). For each smooth function f {\displaystyle f} on M {\displaystyle M} , we can define the Kostant–Souriau prequantum operator
Q ( f ) := − i ℏ ( X f + 1 i ℏ θ ( X f ) ) + f . {\displaystyle Q(f):=-i\hbar \left(X_{f}+{\frac {1}{i\hbar }}\theta (X_{f})\right)+f.}
where X f {\displaystyle X_{f}} is the Hamiltonian vector field associated to f {\displaystyle f} . More generally, suppose ( M , ω ) {\displaystyle (M,\omega )} has the property that the integral of ω / ( 2 π ℏ ) {\displaystyle \omega /(2\pi \hbar )} over any closed surface is an integer. Then we can construct a line bundle L {\displaystyle L} with connection whose curvature 2-form is ω / ℏ {\displaystyle \omega /\hbar } . In that case, the prequantum Hilbert space is the space of square-integrable sections of L {\displaystyle L} , and we replace the formula for Q ( f ) {\displaystyle Q(f)} above with
Q ( f ) = − i ℏ ∇ X f + f , {\displaystyle Q(f)=-i\hbar \nabla _{X_{f}}+f,}
with ∇ {\displaystyle \nabla } the connection. The prequantum operators satisfy
[ Q ( f ) , Q ( g ) ] = i ℏ Q ( { f , g } ) {\displaystyle [Q(f),Q(g)]=i\hbar Q(\{f,g\})}
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