nLab
Planck's constant

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 $T^{*}ℝ\simeq {ℝ}^2$ with standard coordinates $\{p,q\}$, then the normalization of the symplectic form $\sim dp\wedge \mathrm{dq}$ actually needed in physics is

$$\omega =\frac{1}{\hslash }dp\wedge dq.$$\omega = \frac{1}{\hbar} d p \wedge d q \,.

This is because after geometric quantization of this form the observables will obey

$$[\hat{q},\hat{p}]=i({\omega }_{p,q}{)}^{-1}$$[\hat q, \hat p] = i (\omega_{p,q})^{-1}

and this is supposed to be

$$\cdots =i\hslash .$$\cdots = i \hbar \,.

Accordingly, it follows that if $(E,\nabla )$ is a prequantum line bundle for $\omega$, then its $k$-fold tensor product with itself, for $k\in \mathbb{N}$, is a line bundle $(E^{\otimes k},{\nabla }_k)$ with curvature $k\omega$. By the above this corresponds to rescaling

$$\hslash \to \hslash /k.$$\hbar \to \hbar / k \,.

So for $(E,\nabla )$ a given prequantum line bundle the limit of the tensor powers $(E^{\otimes k},{\nabla }_k)$ as $k$ tends to infinity roughly corresponds to taking a classical limit. See also (Donaldson)

Examples

Chern-Simons theory

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 $\mathbb{Z}$.

Similarly for infinity-Chern-Simons theory. For instance ordinary spin group Chern-Simons theory may be taken to have as the fundamental value $\hslash =2$, 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 $\mathrm{Spin}$-principal connections is

$$\frac{1}{2}{\hat{p}}_1:B{\mathrm{Spin}}_{\mathrm{conn}}\to B^{3}U(1{)}_{\mathrm{conn}}.$$\tfrac{1}{2}\hat \mathbf{p}_1 : \mathbf{B}Spin_{conn} \to \mathbf{B}^3 U(1)_{conn} \,.

Similarly for 7-dimensional String 2-group infinity-Chern-Simons theory the fundamental value is $\hslash =6$, with the extended Lagrangian being

$$\frac{1}{6}{\hat{p}}_2:B{\mathrm{String}}_{\mathrm{conn}}\to B^{7}U(1{)}_{\mathrm{conn}}.$$\tfrac{1}{6}\hat \mathbf{p}_2 : \mathbf{B}String_{conn} \to \mathbf{B}^7 U(1)_{conn} \,.

See at higher geometric quantization for more on this.

Related concepts

• theory (physics)

References

• 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