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It's been a while since I've thought about this stuff so take this with a grain of salt. Also, I'm only familiar with this in the context of a finite dimensional phase space, whereas the phase space is infinite-dimensional in the example you give and I'm not sure what extra subtleties that introduces. But the finite-dimensional case might still provide some useful intuition.

Those two caveats aside, I hope the following is of some use:

We can define a canonical symplectic form on phase space in terms of the Lagrangian. The symplectic form gives us a bijective correspondence between tangent vector fields and 1-forms (it works the same as with Riemannian manifolds, the key is just that we have a perfect pairing on tangent spaces). We also have a Poisson bracket operation [A, B] between scalar fields A and B. [A, B] is the Lie derivative of B along the tangent vector field corresponding to the exterior derivative of A (obtained using the correspondence between 1-forms and tangent vector fields provided by the symplectic form).

A continuous symmetry is a flow that preserves the Hamiltonian H and the symplectic form. To a continuous symmetry, we can associate a unique tangent vector field that generates it, which corresponds to a 1-form (which happens to be closed). We can then integrate that 1-form to get a scalar field, which I will call S. We have that [S, H] = 0, this basically says that the vector field corresponding to S generates a symmetry. But the Poisson bracket is anticommutative, so [H, S] = 0, implying that S is a conserved quantity (since the vector field corresponding to H generates the time-evolution flow). Therefore, continuous symmetries correspond to conserved quantities.

It's been a while since I've thought about this stuff so take this with a grain of salt. Also, I'm only familiar with this in the context of a finite dimensional phase space, whereas the phase space is infinite-dimensional in the example you give and I'm not sure what extra subtleties that introduces. But the finite-dimensional case might still provide some useful intuition.

Those two caveats aside, I hope the following is of some use:

We can define a canonical symplectic form on phase space in terms of the Lagrangian. The symplectic form gives us a bijective correspondence between tangent vector fields and 1-forms (it works the same as with Riemannian manifolds, the key is just that we have a perfect pairing on tangent spaces). We also have a Poisson bracket operation {A, B} between scalar fields A and B. {A, B} is the Lie derivative of B along the tangent vector field corresponding to the exterior derivative of A (obtained using the correspondence between 1-forms and tangent vector fields provided by the symplectic form).

A continuous symmetry is a flow that preserves the Hamiltonian H and the symplectic form. To a continuous symmetry, we can associate a unique tangent vector field that generates it, which corresponds to a 1-form (which happens to be closed). We can then integrate that 1-form to get a scalar field, which I will call S. We have that {S, H} = 0, this basically says that the vector field corresponding to S generates a symmetry. But the Poisson bracket is anticommutative, so {H, S} = 0, implying that S is a conserved quantity (since the vector field corresponding to H generates the time-evolution flow). Therefore, continuous symmetries correspond to conserved quantities.

It's been a while since I've thought about this stuff so take this with a grain of salt. Also, I'm only familiar with this in the context of a finite dimensional phase space, whereas the phase space is infinite-dimensional in the example you give and I'm not sure what extra subtleties that introduces. But the finite-dimensional case might still provide some useful intuition.

Those two caveats aside, I hope the following is of some use:

We can define a canonical symplectic form on phase space in terms of the Lagrangian. The symplectic form gives us a bijective correspondence between tangent vector fields and 1-forms (it works the same as with Riemannian manifolds, the key is just that we have a perfect pairing on tangent spaces). We also have a Poisson bracket operation [A, B] between scalar fields A and B. [A, B] is the Lie derivative of B along the tangent vector field corresponding to the exterior derivative of A (obtained using the correspondence between 1-forms and tangent vector fields provided by the symplectic form).

A continuous symmetry is a flow that preserves the Hamiltonian and the symplectic form. To a continuous symmetry, we can associate a unique tangent vector field that generates it, which corresponds to a 1-form (which happens to be closed). We can then integrate that 1-form to get a scalar field, which I will call S. We have that [S, H] = 0, this basically says that the vector field corresponding to S is a symmetry. But the Poisson bracket is anticommutative, so [H, S] = 0, implying that S is a conserved quantity (since the vector field corresponding to H generates the time-evolution flow).

It's been a while since I've thought about this stuff so take this with a grain of salt. Also, I'm only familiar with this in the context of a finite dimensional phase space, whereas the phase space is infinite-dimensional in the example you give and I'm not sure what extra subtleties that introduces. But the finite-dimensional case might still provide some useful intuition.

Those two caveats aside, I hope the following is of some use:

We can define a canonical symplectic form on phase space in terms of the Lagrangian. The symplectic form gives us a bijective correspondence between tangent vector fields and 1-forms (it works the same as with Riemannian manifolds, the key is just that we have a perfect pairing on tangent spaces). We also have a Poisson bracket operation [A, B] between scalar fields A and B. [A, B] is the Lie derivative of B along the tangent vector field corresponding to the exterior derivative of A (obtained using the correspondence between 1-forms and tangent vector fields provided by the symplectic form).

A continuous symmetry is a flow that preserves the Hamiltonian H and the symplectic form. To a continuous symmetry, we can associate a unique tangent vector field that generates it, which corresponds to a 1-form (which happens to be closed). We can then integrate that 1-form to get a scalar field, which I will call S. We have that [S, H] = 0, this basically says that the vector field corresponding to S generates a symmetry. But the Poisson bracket is anticommutative, so [H, S] = 0, implying that S is a conserved quantity (since the vector field corresponding to H generates the time-evolution flow). Therefore, continuous symmetries correspond to conserved quantities.

It's been a while since I've thought about this stuff so take this with a grain of salt. Also, I'm only familiar with this in the context of a finite dimensional phase space, whereas the phase space is infinite-dimensional in the example you give and I'm not sure what extra subtleties that introduces. But the finite-dimensional case might still provide some useful intuition.

Those two caveats aside, I hope the following is of some use:

We can define a canonical symplectic form on phase space in terms of the Lagrangian. The symplectic form gives us a bijective correspondence between tangent vector fields and 1-forms (it works the same as with Riemannian manifolds, the key is just that we have a perfect pairing on tangent spaces). Continuous symmetries are flows on phase space which preserve the symplectic structure. To a continuous symmetry, we can associate a tangent vector field (its derivative, sort of), and therefore a 1-form (via the symplectic form). This 1-form happens to be closed (I don't remember why, might edit this post later if I remember), and therefore we can get a scalar field by integrating it, at least locally. This scalar field is the conserved quantity.

This is all closely related to the Poisson bracket. With the Poisson bracket, any scalar field S can also be thought of as a mapping from scalar fields to scalar fields (the [S, _] operator). This mapping satisfies the Leibniz rule, and is just the derivative operator corresponding to some tangent vector field. Any continuous symmetry (which is a particular type of tangent vector field) can be obtained this way, and the conservation law is just a special case of the Poisson bracket identity [S, S] = 0.

You ask about time translation invariance; in this case the flow is the time evolution flow on phase space and the scalar field is the Hamiltonian.

It's been a while since I've thought about this stuff so take this with a grain of salt. Also, I'm only familiar with this in the context of a finite dimensional phase space, whereas the phase space is infinite-dimensional in the example you give and I'm not sure what extra subtleties that introduces. But the finite-dimensional case might still provide some useful intuition.

Those two caveats aside, I hope the following is of some use:

We can define a canonical symplectic form on phase space in terms of the Lagrangian. The symplectic form gives us a bijective correspondence between tangent vector fields and 1-forms (it works the same as with Riemannian manifolds, the key is just that we have a perfect pairing on tangent spaces). We also have a Poisson bracket operation [A, B] between scalar fields A and B. [A, B] is the Lie derivative of B along the tangent vector field corresponding to the exterior derivative of A (obtained using the correspondence between 1-forms and tangent vector fields provided by the symplectic form).

A continuous symmetry is a flow that preserves the Hamiltonian and the symplectic form. To a continuous symmetry, we can associate a unique tangent vector field that generates it, which corresponds to a 1-form (which happens to be closed). We can then integrate that 1-form to get a scalar field, which I will call S. We have that [S, H] = 0, this basically says that the vector field corresponding to S is a symmetry. But the Poisson bracket is anticommutative, so [H, S] = 0, implying that S is a conserved quantity (since the vector field corresponding to H generates the time-evolution flow).