Hello, L.D. Edmonds.

Gauge theory is a type of theory in physics that describes subatomic particles and their interactions using mathematical functions and fields. It is based on the principle that the equations of motion of a physical system should not depend on the choice of coordinates or reference frame. This means that the system has some symmetries that can be expressed by certain transformations of the fields and parameters. These transformations are called gauge transformations, and the symmetries are called gauge symmetries or gauge invariances.

Gauge theory uses quantum mechanics and special relativity to account for the quantum and relativistic nature of the subatomic world. In gauge theory, the Lagrangian is a function that summarizes the dynamics of the system and is invariant under gauge transformations. The Lagrangian contains terms for the fields that describe the particles (such as electrons, quarks, etc.) and terms for the fields that describe the forces (such as photons, gluons, etc.). These force fields are called gauge fields, and their quanta are called gauge bosons.

Noether's theorem is a mathematical result that relates symmetries and conservation laws. It states that for every continuous symmetry of a physical system, there exists a corresponding conserved quantity (such as energy, momentum, angular momentum, etc.). Noether's theorem can be applied to gauge theories to derive conserved currents and charges associated with gauge symmetries.

However, Noether's theorem is not equivalent to gauge theory. Gauge theory is a more general framework that encompasses different types of symmetries and forces. Not all symmetries are gauge symmetries, and not all conserved quantities are related to gauge symmetries. For example, rotational symmetry is not a gauge symmetry, but it leads to conservation of angular momentum. On the other hand, electric charge is conserved because of a gauge symmetry (U(1)) in electromagnetism.

The other methods you mentioned for finding constants of motion are also useful in physics, but they are not specific to gauge theory. Finding operators that commute with a Hamiltonian is a technique from quantum mechanics that can be used to find observables that do not change in time. Deriving implications directly from the governing equations is a general approach that can be applied to any physical system.

None of the mentioned.

A gauge theory is a theory that is invariant under transformations, whose parameters depend on the point in spacetime. A (classical) theory is defined either by its equations of motion, or the Lagrangian, or the Hamiltonian, it doesn’t matter.

The typical example of a gauge theory is classical electrodynamics, that is invariant (i.e. its equations, Maxwell's equations are invariant) under global Lorentz transformations and translations-whose parameters, that define the boosts, the rotations and the translations of the coordinates t and x (that, in turn, induce transformations of the fields) don’t depend on the point in spacetime-and transformations of the fields, that don't affect the spacetime coordinates, but that do depend on the point in spacetime, namely δΑ_i(t,x)=grad_i f(t,x) and δΦ(t,x)=-df(t,x)/dt, where A_i and Φ are the vector and scalar potential and f(t,x) is an arbitrary, smooth, function of the point, (t,x), in spacetime. These transformations leave the magnetic field, B = curl A and electric field, E = -dA/dt-grad Φ and, thus, Maxwell's equations invariant (the sources in Maxwell's equations are invariant under gauge transformations). They were called gauge transformations by Hermann Weyl in 1918 (they were discovered, in fact, by Ludvig Lorenz, in 1867).

One reason why it's, not only, useful but necessary to introduce the potentials is that matter couples to the potentials, not directly to the fields, in electrodynamics.

Indeed a standard exercise is to show that the only equations, containing no more than two derivatives in time, and that are invariant under global Lorentz transformations and these local transformations, are Maxwell's equations, expressed in terms of the potentials: The symmetry under the transformations imposes the form of the equations.

Maybe the best strategy is by example, How it is used in connection to EM theory used

for QM, Using the gauge fields Phi and A, making up a 4 vector. Then the phase shifting effect of

these fields can be explored, as in Bohm Aharonov, or in determining the quantum levels induced by the magnetic

field in a gas of electrons. Then you can change A by a gradient of some scalar field, and in some cases the result is

invariant. Particle physics also has exampes, but they are much harder to study.

The H has form (p-qA).(p-qA)/2m + q Phi in nonrelativistic.

As indicated by Feyman all this stuff may still have its problems, but the formalism is widly used in the literature.

An alternative is just wires with current moving accross magnetic fields, with energy considerations.

(my favorite)

In classical physics Phi is merely and energy shift, and A has not real effect. E and B are then the Physical

fields.

Thank you all for the good advice. I was aware of the E&M example of gauge transformations but was not sure if all examples are similar. I think I get what Stam Nicolis said, that "A gauge theory is a theory that is invariant under transformations, whose parameters depend on the point in spacetime." Instead of the transformation being of the coordinates, it is a function of the coordinates. For the E&M example, the transformation that is called a gauge transformation is not a coordinate transformation but, instead, is a transformation of potentials by adding suitably defined functions of the coordinates. If my interpretation is correct then my question has been answered.

Indeed, you understood correctly: Gauge transformations act on the fields, not the points of spacetime.