Emmy Noether's theorem in simpler terms

Question

I'd like to understand Noether's theorem and its contents as to what it implies in a bit simpler terms. I am familiar with mathematics unto Calculus 1,2,3 and some linear algebra and group theory. I am familiar with that for each symmetry there is a conservation law.

1. Now is there a one-one correspondence between these two always?
2. What about laws like conservation of charge?
3. And conservation of energy?
4. And conservation of mass?
5. Considering our view of nature was changed as Special Relativity was introduced and thus, what earlier the theorem would have asserted as mass and energy to be originating from different symmetries then how did we reconcile these two as just one symmetry?

Answer 1

Two Noether's theorems, arXive:physics/9807044 Roughly speaking, given a continuous (or approximately continuous - Taylor series) symmetry, there must be an associated conserved current (property). Given a conserved property, there must be a symmetry.

Homogeneous time - conserved mass-energy;
Homogeneous space - conserved linear momentum;
Isotropic space - conserved angular momentum;
Note that parity is absolutely discontinuous and is therefore outside Noether's grasp.
$U(1)$ gauge transformation - electric charge;
$U(1)$ gauge transformation - lepton generation number;
$U(1)$ gauge transformation - hypercharge;
$U(1)_Y$ gauge transformation - weak hypercharge;
$U(2)$ [not $U(1)\times SU(2)$] - electroweak force;
$SU(2)$ gauge transformation - isospin;
$SU(2)_L$ gauge transformation - weak isospin;
$P\times SU(2)$ - G-parity;
$SU(3)$ "winding number" - baryon number;
$SU(3)$ gauge transformation - quark color;
$SU(3)$ (approximate) - quark flavor;
$SU(2)\times SU(3)$ $\left[U(1)\times SU(2)\times SU(3)\right]$ - Standard Model.

A conserved quantity derives from each symmetry commuting with time, and the reverse. A divergence-free current (conserved property) arises if the Lagrangian or the action is invariant under continuous transformation. To each continuous symmetry of an action there corresponds a conserved quantity because of the Euler-Lagrange equations of the Lagrangian, and the reverse. To each gauge symmetry of an action there corresponds an identity among Euler-Lagrange equations of the Lagrangian, and the reverse.

A physical system with a Lagrangian invariant with respect to the symmetry transformations of a Lie group has, in the case of a group with a finite (or countably infinite) number of independent infinitesimal generators, a conservation law for each such generator, and certain "dependencies" in the case of a larger infinite number of generators (General Relativity and the Bianchi identities). The reverse is true.

A symmetry can be broken explicitly - a term in the action or equations of motion may not be invariant. A symmetry can be broken anomalously - not all classical theory symmetries exist in the corresponding quantum theory. Quantum field theory anomaly spoils renormalizability. Anomaly absence in the Standard Model is crucial. A symmetry can be broken spontaneously if it is an exact symmetry of the equations of motion but not of a particular solution therein. Noether's theorem holds if the symmetry is not broken explicitly. Conservations can be relaxed in subsystems displaying reduced symmetry (Born scattering approximation, Fermi's golden rule, Snell's law).

General Relativity does not have Newtonian conservation laws.

Answer 2

First, the theorem states that for any system that can be described by minimizing the action, then if the action has a symmetry there must be a conserved quantity in the system. It does not say that for any conserved quantity there must be a symmetry.

There is no symmetry for the conservation of mass (which was an assumption). Later Einstein showed the mass and energy are related. And the invariance of mass falls out of the invariance of 4-vector dot products, which is unrelated to Noether's theorem.

Energy was shown to be conserved since there is a time-translation symmetry in the action. Conservation of charge and current comes from the Gauge symmetry in the electromagnetic potentials.