Noether's theorem states that, for every continuous symmetry of an action, there exists a conserved quantity, e.g. energy conservation for time invariance, charge conservation for $U(1)$. Is there any similar statement for discrete symmetries?
Symmetry – Is There an Equivalent to Noether’s Theorem for Discrete Symmetries?
discretegroup-theorymathematical physicsnoethers-theoremsymmetry
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Nope, this operation is not a symmetry in the physics sense. A symmetry transformation is a transformation that changes or mixes the values of the basic degrees of freedom such as positions and momenta $x(t)$, $p(t)$ in mechanics or the values of fields such as $\vec E(\vec x,t)$ and $\vec B(\vec x,t)$ in electromagnetism.
The Hamiltonian is not an independent variable or a basic degree of freedom; it is a function of them. You're not changing the values of any quantities that evolve with time; instead, you're changing some formulae for derived and in principle unnecessary auxiliary objects in the theory (in this case the Hamiltonian), claiming that other formulae are preserved. This is not a symmetry so there is no conservation law associated with this operation. You're not "rotating" real objects which is what symmetry transformations should do: you're just redefining auxiliary, derived objects on the paper.
Incidentally, the operation you mention fails to be "harmless" in general relativity because the energy is a source of gravitational field - curvature of space - so if you move it by a constant, you do change physics.
Noether's (first) Theorem is really not about Lie groups but only about Lie algebras, i.e., one just needs $n$ infinitesimal symmetries to deduce $n$ conservation laws.
Lie's third theorem guarantees that a finite-dimensional Lie algebra can be exponentiated into a Lie group, cf. e.g. Wikipedia & n-Lab.
If one is only interested in getting the $n$ conservation laws one by one (and not so much interested in the fact that the $n$ conservation laws often together form a representation of the Lie algebra $L$), then one may focus on a 1-dimensional Abelian Lie subalgebra $u(1)\cong \mathbb{R}$.
In the context of field theory, there should be Lie algebra homomorphisms from the Lie algebra $L$ to the Lie algebra of vector fields on the field configuration space (so-called vertical transformations) and to the Lie algebra the vector fields of spacetime (so-called horizontal transformations).
That the action functional $S[\phi]$ possesses a symmetry (quasisymmetry) means that the appropriate Lie derivatives of $S$ wrt. the above vector fields should vanish (be a boundary term), respectively.
Note that the Noether currents & charges do not always form a representation of the Lie algebra $L$. There could e.g. appear central extensions, cf. this and this Phys.SE posts.
Best Answer
For continuous global symmetries, Noether theorem gives you a locally conserved charge density (and an associated current), whose integral over all of space is conserved (i.e. time independent).
For global discrete symmetries, you have to distinguish between the cases where the conserved charge is continuous or discrete. For infinite symmetries like lattice translations the conserved quantity is continuous, albeit a periodic one. So in such case momentum is conserved modulo vectors in the reciprocal lattice. The conservation is local just as in the case of continuous symmetries.
In the case of finite group of symmetries the conserved quantity is itself discrete. You then don't have local conservation laws because the conserved quantity cannot vary continuously in space. Nevertheless, for such symmetries you still have a conserved charge which gives constraints (selection rules) on allowed processes. For example, for parity invariant theories you can give each state of a particle a "parity charge" which is simply a sign, and the total charge has to be conserved for any process, otherwise the amplitude for it is zero.