The topic considered here is the Klein-Gordon equation governing some scalar field amplitude, with the field amplitude defined by the property of being a solution of this equation. The original Klein-Gordan equation does not contain any gauge potentials, but a modified version of the equation (also called the Klein-Gordon equation in some books for reasons that I do not understand) does contain a gauge potential. This gauge potential is often represented in the literature by the symbol Ai (a four-component vector). Textbooks show that if a suitable transformation is applied to the field amplitude to produce a transformed field amplitude, and another suitable transformation is applied to the gauge potential to produce a transformed gauge potential, the Lagrangian is the same function of the transformed quantities as it is of the original quantities. With these transformations collectively called a gauge transformation we say that the Lagrangian is invariant under a gauge transformation. This statement has the appearance of being justification for the use of Noether’s theorem to derive a conservation law. However, it seems to me that this appearance is an illusion. If the field amplitude and gauge potential are both transformed, then they are both treated the same way as each other in Noether’s theorem. In particular, the theorem requires both to be solutions of their respective Lagrange equations. The Lagrange equation for the field amplitude is the Klein-Gordon equation (the version that includes the gauge potential). The textbook that I am studying does not discuss this but I worked out the Lagrange equations for the gauge potential and determined that the solution is not in general zero (zero is needed to make the Klein-Gordon equation with gauge potential reduce to the original equation). The field amplitude is required in textbooks to be a solution to its Lagrange equation (the Klein-Gordon equation). However, the textbook that I am studying has not explained to me that the gauge potential is required to be a solution of its Lagrange equations. If this requirement is not imposed, I don’t see how any conclusions can be reached via Noether’s theorem. Is there a way to justify the use of Noether’s theorem without requiring the gauge potential to satisfy its Lagrange equation? Or, is the gauge potential required to satisfy that equation without my textbook telling me about that?
Things are much simpler than stated. The Klein-Gordon equation is that for a single, real-valued, field that transforms as a scalar under Lorentz transformations. It’s possible to consider the equations for two such fields, where the equations remain invariant, when the two scalars transform linearly under rotation of rhe fields.
This isn't true for any way of coupling the fields together, of course.
The parameter of this rotation is the same at every point in spacetime. This is a continuous transformation and Noether’s theorem implies the existence of a conserved current.
Now one may ask, what would happen, if the transformation were different at each point. Then the equations of motion aren’t invariant. They can become invariant, once more, however, if we introduce an additional field. It is then possible to show that the equations of motion for the scalars remain invariant under rotations of the field indices, that depend on the spacetime point, provided the additional field, also, changes. This invariance of the equations of motion implies the existence of a conserved current. The justif8cation is the same: The equations of motion are invariant under a continuous transformation, Noether’s theorem implies the existence of a conserved current.Full stop.
The current is conserved whatever the external field, provided it transforms in a particular way, that’s all that’s required. The dynamics of this field, in turn, is uniquely specified by demanding global Lorentz invariance, the local invariance just discussed, and that the equations of motion contain not more than two derivatives. These are then recognized as Maxwell’s equations for the gauge potential.
It seems useful and interesting in the symmetry-conservation correlations under local gauge transformations in the world of field theory
Stam Nicolis said:"These are then recognized as Maxwell’s equations for the gauge potential."
So the gauge potential is required to satisfy some equation. It is not enough for the transformed potential to be related to the original potential via a particular transformation law. In addition to a transformation law, they must also satisfy their Lagrange equations. This is consistent with my expectation. I worked through the derivation of Noether's theorem in detail and concluded that both the field amplitude and the gauge potential must satisfy their Lagrange equations in order for Noether's theorem to apply.
No; the gauge potential is, just, required to transform in a certain way, in order for the equations for the scalar fields to remain invariant, under transformations, that depend on the point in spacetime.
It need not have any independent dynamics, for Noether's theorem to apply-it can describe a fixed electromagnetic field, that doesn't react to the presence of the scalars, but affects them. Noether's thorem simply states that, if the equations of motion for the scalars are invariant under a continuous group of transformations, then there exists a conserved current. That's all. For any gauge potential the assumptions are true, therefore the result follows.
It's a separate issue, to specify the dynamics of the gauge field. The interesting statement is that the symmetries fix this dynamics uniquely-for the case at hand, if this gauge field is dynamical, it must satisfy Maxwell's equations, where the current is the current made up of the scalars and is conserved, indeed, as a consequence of Noether's theorem.
"Noether's thorem simply states that, if the equations of motion for the scalars are invariant under a continuous group of transformations, then there exists a conserved current. That's all."
If you review the derivation of Noether's theorem you will find another requirement. The varied functions must be extremals, i.e., satisfy Lagrange's equations, i.e., the equations of motion. Transformation properties alone, with no other requirements, will make the Lagrangian an invariant. But to obtain a conserved current we need not only invariance of the Lagrangian but also that the varied functions satisfy Lagrange's equations. I reviewed the proof of Noether's theorem for the specific case of the Klein-Gordon equation with gauge potentials and found that Noether's theorem will not apply unless the gauge potentials also satisfy their Lagrange equations.
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