nLab
canonical momentum

Context

Symplectic geometry

Geometric quantization

Contents

Idea

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

Therefore locally for coordinate patch ϕ: 2n n× nT *Σ\phi : \mathbb{R}^{2n} \simeq \mathbb{R}^n \times \mathbb{R}^n \to T^* \Sigma the 2n2n-canonical coordinates of the Cartesian space n\mathbb{R}^n are naturally thought of as decomposed into nn “canonical coordinates” on the first nn factors and a set of “canonical momenta”, being the canonical coordinates on the second n\mathbb{R}^n-factor.

Notice that “canonical” here refers (at best) to the canonical coordinates of the Cartesian space n\mathbb{R}^n once ϕ\phi has been chosen. The choice of ϕ\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 2n n× n\mathbb{R}^{2n} \simeq \mathbb{R}^n \times \mathbb{R}^n such that under this identification the symplectic form reads i=1 ndq idp i\sum_{i = 1}^n d q_i \wedge d p^i, for {q i}\{q_i\} the canonical coordinates on one n\mathbb{R}^n and {p i}\{p^i\} for the other.

Therefore generally, in the context of mechanics, with such a local identification one calls p ip^i the canonical momentum of the coordinate (or sometimes “canonical coordinate”) q iq_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 transformation? is in (Kirwin-Wu 04).

References

  • 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)

Revised on September 13, 2013 19:29:18 by Urs Schreiber (82.169.114.243)