Geometric quantization

The geometric quantization is an attempt to define a mapping between classical and quantum observables, on the one hand, like any quantization corresponds to the below three axioms Paul Dirac and the other hand is formulated in terms of differential geometry (in particular, independent of the choice of certain coordinates).

Definition

An important component of geometric quantization is to map

In this formula is the symplectic gradient or Hamiltonian vector field of a function on the space of classical solutions of a physical theory (eg mechanics, field theory ) and the triangular icon ( " nabla " ) is a covariant derivative in a complex - dimensional vector bundle over this space, and is a section of this bundle. Now, the beam is designed so that its curvature and symplectic 2 form the space of the conventional solutions are identical ( up to a constant ). It follows then that the mapping satisfies the three axioms Paul Dirac:

1 ) is linear over the real numbers,

2) When a constant function, then, the corresponding multiplication operator,

3) transferred (up to constant), the Poisson bracket of the space of classical solutions in the commutator of the corresponding operators.

After the introduction of this figure ( " Präquantisierung " ) must be chosen nor a measure on the space of classical solutions found and a polarization.

Advantages and Disadvantages

A major advantage of geometric quantization is its independence from the chosen coordinate and its geometric clarity. One disadvantage is associated with the calculus of mathematical difficulties, in particular the lack of a suitable measure for the infinite-dimensional spaces in the case of field theories.