In geometric quantization
In the context of geometric quantization Planck’s constant appears as the inverse scale of the symplectic form.
For instance in the simple case that phase space is with standard coordinates , then the normalization of the symplectic form actually needed in physics is
This is because after geometric quantization of this form the observables will obey
and this is supposed to be
Accordingly, it follows that if is a prequantum line bundle for , then its -fold tensor product with itself, for , is a line bundle with curvature . By the above this corresponds to rescaling
This implies in particular
a global rescaling of the periods of the symplectic form may be absorbed in a rescaling of Planck’s constant, see at geometric quantization of non-integral forms;
for a given prequantum line bundle the limit of the tensor powers as tends to infinity roughly corresponds to taking a classical limit. See also (Donaldson 00).
In Chern-Simons theory Planck’s constant corresponds to the inverse level of the theory, hence the inverse of the characteristic class that defines the theory, regarded as an element in .
Similarly for infinity-Chern-Simons theory. For instance ordinary spin group Chern-Simons theory may be taken to have as the fundamental value , because the first Pontryagin class that defines the theory is divisible by 2, the prequantum 3-bundle that defines the theory of the moduli stack of -principal connections is
Similarly for 7-dimensional String 2-group infinity-Chern-Simons theory the fundamental value is , with the extended Lagrangian being
See at higher geometric quantization for more on this.
- Simon Donaldson, Planck’s constant in complex and almost-complex geometry, XIIIth International Congress on Mathematical Physics (London, 2000), 63–72, Int. Press, Boston, MA, 2001
Revised on September 1, 2013 20:16:54
by Urs Schreiber