# Noether Theorem in Terms of Symplectic Geometry

For the classical (=field theoretical) Noether theorem there exist a well known approach using symplectic geometry. Recall, that in classical terminology the conserved current (a $$1+3$$-vector time+spatial keeping physical framework) $$j^{\mu}$$ is related to the conserved charge through $$Q= \int_V j^0 dV.$$

In symplectic geometry the role of conserved charge $$Q$$ is replaced by so-called momentum map $$\Phi: P \to \mathfrak{g}^*$$ where in physical context $$P$$ is the phase manifold and $$\mathfrak{g}^*$$ the dual of the Lie algebra of the symmetry group $$G$$, which we consider to be a Lie group acting on $$P$$. Now the symplectic version of Noether lemma says, that $$\Phi$$ can be related to the conserved charge $$Q$$ in classical sense, by the fact that if $$\zeta \in \mathfrak{g}$$ is any generator of the Lie group of symmetries, then - reinterpreting $$\Phi$$ as a map $$\phi: P \times \mathfrak{g} \to \mathbb{R}$$ the restriction of $$\Phi(\zeta,-)$$ to the Hamilton integral curves - in english: the "physical" paths on the phase space; ie solutions of Hamilton equation - is constant, so can interpreted as "the" conserved charge.

Question: What is the counterpart for "conserved current" $$j^{\mu}$$ in this symplectic formulation of Noether theorem?

• In phase space we have a symplectic form $\Omega$ which is a closed and non-degenerate two-form. A vector field $X$ is Hamiltonian if and only if $\Omega(X,) = -X(Q)$ where $Q$ is the canonical charge. Now, in field theory, $\Omega$ is an integral of a $(d-1)$-form called symplectic density $\omega$, over a Cauhy slice. Likewise, the charge will be an integral of a $(d-1)$-form over a Cauchy slice. The Hodge dual of this of this form is the current.
– Gold
Jul 11 at 17:04
• @Gold: I'm a bit confused about used notations: by definition a vector field is a section $X: P \to TP$ of the tangent bundle $TP$ of $P$, so it gets as arguments elements from $P$. Could you remind me what do you mean by "canonical charge" $Q$ precisely as function, especially what are it's domain & codomain? The point is that I'm not understand the notion $X(Q)$. Jul 11 at 17:42
• $X(Q)$ is the action of the vector field on the function $Q$. The same as $dQ(X)$. In this discussion we are invoking the Covariant Phase Space. So the phase space is (subspace of) the space of solutions to the classical equations. In particular, a vector field $X$ is just a variation.
– Gold
Jul 11 at 18:17

Let us first note that a proper Noether current exists only in the context of field theories: In the world of particle mechanics and its finite-dimensional phase spaces there is no current.

So an answer must be specific to a field-theoretic formalism, and your symplectic space here is some space of sections of a field bundle $$\pi: F\to M$$ over a spacetime $$M$$. Unfortunately it turns out that a merely "symplectic setting" is not enough for doing Hamiltonian field theory in a covariant way, and people generally turn to multisymplectic geometry to formalize Hamiltonian field theory. So the starting point in the question is already somewhat unsuitable, as it is tethered to the "ordinary" symplectic viewpoint.

In this approach, we have some bundle $$\pi : B\to M$$ over a spacetime $$M$$ and $$F$$ carries a "multisymplectic form" $$\Omega\in \Omega^{n+2}(B)$$, where $$\mathrm{dim}(M) = n+1$$. States/trajectories of our theory are sections of $$\pi$$. Note that for $$\mathrm{dim}(M) = 1$$, this makes $$\omega$$ an ordinary symplectic 2-form, recovering the point particle setting of $$\mathbb{R} \times T^\ast Q \to \mathbb{R}$$, where trajectories are functions of 1-dimensional time $$\mathbb{R}$$ valued in phase space.

Very roughly, we get a multimomentum map $$\mu$$ that fulfills $$\iota(X_g)\Omega = \mathrm{d}\mu(g)$$ for any $$g\in\mathfrak{g}$$, which means $$\mu(g)\in \Omega^{n}(B)$$. When you pull this back along special sections $$s: M\to B$$ corresponding to solutions of the equations of motion, you find that $$\mathrm{d}s^\ast \mu(g) = 0$$, i.e. $$s^\ast\mu(g)\in\Omega^{n}(M)$$ is a conserved current (the Noether current) for the trajectory $$s$$ (if you want your current to be a 1-form/vector, take the Hodge dual).

For details, see "Multisymplectic geometry, covariant Hamiltonians, and water waves", in particular its chapter 5, by Marsden and Shkoller

There are several approaches. E.g.

1. If one starts from a Hamiltonian field theory with a symplectic target phase space $$P$$ (which seems to be OP's setup), perhaps conceptionally the simplest is to

• construct a Hamiltonian action principle, cf. e.g. this Phys.SE post;

• and then apply the standard Noether's theorem to this action principle to obtain the Noether current $$J^{\mu}$$.

For the point-mechanical case, see e.g. this related Phys.SE post.

2. If one starts from a Lagrangian field theory (which in practice is more natural although OP does not seem to take that road), then Noether's theorem is readily available, and then one can construct a covariant Hamiltonian field theory with a conserved symplectic 2-form current, see e.g. Ref. [CW] and this Phys.SE post.

References:

• [CW] C. Crnkovic & E. Witten, Covariant description of canonical formalism in geometrical theories. Published in Three hundred years of gravitation (Eds. S. W. Hawking and W. Israel), (1987) 676.