15 Hamiltonian Vector Fields and the Moment Map

A basic feature of Hamiltonian mechanics is that, for any function $f$ on phase space $M$, there are parametrized curves in phase space that solve Hamilton’s equations

$${\dot{q}}_j=\frac{\partial f}{\partial p_j}{\dot{p}}_j=-\frac{\partial f}{\partial q_j}$$

and the tangent vectors of these parametrized curves provide a vector field on phase space. Such vector fields are called Hamiltonian vector fields. There is a distinguished choice of $f,$ the Hamiltonian function $h,$ which gives the velocity vector fields for time evolution trajectories in phase space.

More generally, when a Lie group $G$ acts on phase space $M$, the infinitesimal action of the group associates to each element $L\in \mathfrak{g}$ a vector field $X_L$ on phase space. When these are Hamiltonian vector fields, there is (up to a constant) a corresponding function ${\mu }_L$ . The map from $L\in \mathfrak{g}$ to the function ${\mu }_L$ on $M$ is called the moment map, and such functions play a central role in both classical and quantum mechanics. For the case of the action of $G={\mathbf{R}}^3$ on $M={\mathbf{R}}^6$ by spatial translations, the components of the momentum arise in this way, for the action of $G=SO(3)$ by rotations, the angular momentum.

Conventional physics discussions of symmetry in quantum mechanics focus on group actions on configuration space that preserve the Lagrangian, using Noether’s theorem to provide corresponding conserved quantities (see chapter 35). In the Hamiltonian formalism described here, these same conserved quantities appear as moment map functions. The operator quantizations of these functions provide quantum observables and (modulo the problem of indeterminacy up to a constant) a unitary representation of $G$ on the state space $H$. The use of moment map functions rather than Lagrangian-derived conserved quantities allows one to work with cases where $G$ acts not on configuration space, but on phase space, mixing position and momentum coordinates. It also applies to cases where the group action is not a “symmetry” (i.e., does not commute with time evolution), with the functions ${\mu }_L$ having non-zero Poisson brackets with the Hamiltonian function.

中文目录

Chapter contents

• 15.1 Vector fields and the exponential map
• 15.2 Hamiltonian vector fields and canonical transformations
• 15.3 Group actions on M and the moment map
• 15.4 Examples of Hamiltonian group actions
• 15.5 The dual of a Lie algebra and symplectic geometry
• 15.6 For further reading

Previous chapter: 14 The Poisson Bracket and Symplectic Geometry

来源与版本

正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

原书 PDF · 印刷页 171、172、173、174、175、176、177、178、179、180、181、182、183、184

来源版本：2025-10-20

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837

OCR 来源 SHA-256：c4bd5d0938766c14d748148c06d1d3db5fdb93a4163ce17a632b1b7c023ee003

OCR 产物 SHA-256：c4bd5d0938766c14d748148c06d1d3db5fdb93a4163ce17a632b1b7c023ee003