Geometric quantization

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

definition

An important part of the geometric quantization is the mapping

${\ displaystyle Q: f \ mapsto Q (f)}$

${\ displaystyle Q (f) (\ psi): = - i \ hbar \ nabla _ {\ operatorname {sgrad} (f)} \ psi + f \ cdot \ psi}$

In this formula, the symplectic gradient or Hamiltonian vector field of a function on the space of classical solutions of a physical theory (z. B. mechanics , field theory ), and the triangular symbol ( " Nabla ") a covariant derivative in a complex-dimensional vector bundle over this space, and is a cut of that bundle. Now the bundle is constructed in such a way that its curvature and the symplectic 2-form are the same in the space of the classical solutions (except for one constant). It then follows that the map fulfills Paul Dirac's three axioms : ${\ displaystyle \ operatorname {sgrad} (f)}$ ${\ displaystyle f}$ ${\ displaystyle \ nabla}$ ${\ displaystyle \ psi}$

1) is linear over the real numbers, ${\ displaystyle Q}$

2) If is a constant function, then the corresponding multiplication operator is, ${\ displaystyle f}$ ${\ displaystyle Q (f)}$

3) converts (except for constant) the Poisson bracket of the space of the classical solutions into the commutator of the corresponding operators. ${\ displaystyle Q}$

After the introduction of this mapping (“pre-quantization”), a measure must be found in the space of the classical solutions and a polarization selected.

advantages and disadvantages

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

literature

Nicholas Michael John Woodhouse: Geometric Quantization , Oxford University Press 1993, ISBN 0-19-853673-9