For a fourth provocative slogan about quantum mechanics I’ve chosen:
Use the moment map, not Noether’s Theorem.
Pretty much every physics textbook these days explains the way symmetry principles work as:
- Start with an action functional, invariant under a Lie group G.
- Use Noether’s theorem to get a conserved charge (for each element of the Lie algebra of G).
There’s a short (slightly mystifying) calculation always given to derive this. I’d like to argue that this is really not the best way to think about the implications of having a Lie group act on a physical system, that for this it’s better to take the Hamiltonian point of view. There the way symmetry principles work is:
- For a function on phase space (or on a general symplectic manifold) you get a vector field. This is just Hamilton’s equations, giving the vector field for time evolution corresponding to any Hamiltonian function.
- The infinitesimal action of G on phase space gives a vector field for each element of the Lie algebra of G. The moment map takes an element of the Lie algebra to a function on phase space (the one corresponding to the vector field).
I’m ignoring some subtleties here having to do with the relation between vector fields and functions not being quite one-to-one.
All of the basic examples of conservation laws in physics come about this way. The action of time translation gives the Hamiltonian function, space translation the momentum, rotations give the angular momentum, and phase transformations give charge. You can get these either as moment maps, or using Noether’s theorem.
The moment map however gives you much more, with phase space providing structure that is not visible just from the action. A simple example is the harmonic oscillator in 3 variables. SO(3) rotations act on the configuration variables, preserving the action, so Noether’s theorem gives you 3 conserved quantities, the angular momentum variables. The moment map point of view however gives you much more. The phase space is 6 dimensional (3 positions + 3 momenta) and the Lie group Sp(6,R) of linear symplectic transformations acts on it, with a subgroup U(3) preserving the Hamiltonian. The U(3) includes the SO(3) rotations as a subgroup, but it is much larger (9 dimensions vs. 3), so the moment map gives you many more conserved quantities. After quantization, you learn that energy eigenstates are U(3) representations, telling you much more about them than what angular momentum tells you.
The moment map point of view also gives you quantities corresponding to the directions in Sp(6,R) that are not in U(3). In the quantum theory these act on the full state space (not preserving energy eigenstates) and your state space is a representation of (a double cover of) this group.
For the simplest possible harmonic oscillator, in one-dimension, Noether’s theorem doesn’t really tell you anything. The moment map point of view says that there is an Sp(2,R) acting on phase space, with a U(1) subgroup preserving the Hamiltonian. The moment map is just the Hamiltonian itself. In the quantum theory you find that the harmonic oscillator state space is a representation of (a double cover of) Sp(2,R), with the U(1) action on states characterized by integers, which correspond to the energy. This integrality is the essence of the “quantum” in “quantum mechanics”, and it’s quite invisible to Noether’s theorem, but a basic fact of the moment map point of view.
In some sense this is an argument for the Hamiltonian vs. Lagrangian point of view in general. The relation between the two is that, given a Lagrangian, one constructs a symplectic structure on the space of solutions of the variational problem, and thus a Hamiltonian formalism. Noether’s conserved quantities are then examples of moment maps. The problem is that typically this requires the use of constraints and the quite tricky constrained Hamiltonian formalism.
The positive argument for the Lagrangian point of view is that it comes into its own in the relativistic setting, making Lorentz invariance easy to handle by the Noether’s theorem method. This is quite true, with the standard version of the Hamiltonian formalism distinguishing the time direction and breaking Lorentz invariance. There is however a less well-known “covariant phase space” point of view, where one tries to work with the space of solutions of the equations of motion as one’s phase space. Only if one identifies a solution with its initial data at a fixed time does one distinguish the time direction. I’ve recently enjoyed reading Igor Khavkine’s review article, which in particular does a great job of explaining the history of this line of thinking.
The Lagrangian also comes with the extremely seductive point of view on quantization of the path integral. This point of view works very well for dealing with Yang-Mills theory, and I spent much of my early career convinced that all there was to quantization was figuring out how to make sense of integrating over the exponential of the action. I’m now much more aware of the advantages of the Hamiltonian point of view, especially in terms of understanding quantum theory as representation theory. In some sense what one really wants is to understand quantization in a way that takes advantage of both points of view, but the relationship between them is quite non-trivial.
The discussion here has been far too wordy for most people to make sense of. If you want to understand any of this, you need equations. Luckily, I’ve provided lots of them and many details here, see chapters 12 and 13 for the moment map, chapter 19-22 for the harmonic oscillator.