nLab canonical momentum

Idea

The archetypical example of a mechanical system is a particle propagating on a manifold $\Sigma$. The phase space of this particular system happens to be canonically identified with the cotangent bundle $X \coloneqq T^* \Sigma$ of $\Sigma$. Here the covector $(x,p)$ in $X$ over a point $x \in \Sigma$ is physically interpretd as describing a state of the system where the particle is at position $x \in \Sigma$ and has momentum (essentially: speed) as given by $p$.

Therefore locally for coordinate patch $\phi : \mathbb{R}^{2n} \simeq \mathbb{R}^n \times \mathbb{R}^n \to T^* \Sigma$ the $2n$-canonical coordinates of the Cartesian space $\mathbb{R}^n$ are naturally thought of as decomposed into $n$ “canonical coordinates” on the first $n$ factors and a set of “canonical momenta”, being the canonical coordinates on the second $\mathbb{R}^n$-factor.

Notice that “canonical” here refers (at best) to the canonical coordinates of the Cartesian space $\mathbb{R}^n$ once $\phi$ has been chosen. The choice of $\phi$ however is arbitrary. Hence, despite the (standard) term, there is nothing much canonical about these “canonical coordinates” and “canonical momenta”.

In general, the phase space of a physical system is a symplectic manifold which need not be a cotangent bundle as for the particle sigma-model.

But locally over a coordinate patch every symplectic manifold looks like $\mathbb{R}^{2n} \simeq \mathbb{R}^n \times \mathbb{R}^n$ such that under this identification the symplectic form reads $\sum_{i = 1}^n d q_i \wedge d p^i$, for $\{q_i\}$ the canonical coordinates on one $\mathbb{R}^n$ and $\{p^i\}$ for the other.

Therefore generally, in the context of mechanics, with such a local identification one calls $p^i$ the canonical momentum of the coordinate (or sometimes “canonical coordinate”) $q_i$.

Globally the notion of canonical momenta may not exist at all. The notion that does exist globally is that of a polarization of a symplectic manifold. See there for more details.

Properties

On a symplectic vector space

Discussion of how there is a flat connection on the bundle of spaces of quantum states over the space of choices of polarizations of a symplectic vector space and how this reproduces the traditional relation between canonical coordinates and canonical momenta by Fourier transform is in (Kirwin-Wu 04).

Related concepts

• momentum

• canonical coordinates, canonical transformation

• phase space

• Hamiltonian mechanics

• geometric quantization

References

• Marc Henneaux, Claudio Teitelboim, §1.1.1 in: Quantization of Gauge Systems, Princeton University Press (1992) [ISBN:9780691037691, jstor:j.ctv10crg0r]

• William Kirwin, Siye Wu, Geometric Quantization, Parallel Transport and the Fourier Transform, Comm. Math. Phys. 266 (2006), no. 3, pages 577 – 594 (arXiv:math/0409555)