# Contents

## 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 $\left\{p,q\right\}$, then the normalization of the symplectic form $\sim dp\wedge \mathrm{dq}$ actually needed in physics is

$\omega =\frac{1}{\hslash }dp\wedge dq\phantom{\rule{thinmathspace}{0ex}}.$\omega = \frac{1}{\hbar} d p \wedge d q \,.

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

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

and this is supposed to be

$\cdots =i\hslash \phantom{\rule{thinmathspace}{0ex}}.$\cdots = i \hbar \,.

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

$\hslash \to \hslash /k\phantom{\rule{thinmathspace}{0ex}}.$\hbar \to \hbar / k \,.

So for $\left(E,\nabla \right)$ a given prequantum line bundle the limit of the tensor powers $\left({E}^{\otimes k},{\nabla }_{k}\right)$ 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 $ℤ$.

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

$\frac{1}{2}{\stackrel{^}{p}}_{1}:B{\mathrm{Spin}}_{\mathrm{conn}}\to {B}^{3}U\left(1{\right)}_{\mathrm{conn}}\phantom{\rule{thinmathspace}{0ex}}.$\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

$\frac{1}{6}{\stackrel{^}{p}}_{2}:B{\mathrm{String}}_{\mathrm{conn}}\to {B}^{7}U\left(1{\right)}_{\mathrm{conn}}\phantom{\rule{thinmathspace}{0ex}}.$\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.

## 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

Revised on January 6, 2013 19:43:05 by Urs Schreiber (89.204.138.182)