What is the relevance of gauge symmetries in a gauge theory?

https://arxiv.org/abs/1107.4664

Simon Friederich (2012) writes that the ontology of gauge theories has received considerable attention from philosophers in recent years. There is a debate about spontaneous symmetry breaking: is it a genuine phenomenon of nature or is it just imagination of humans.

The author of this blog has had a feeling that Baron Munchausen has written the "derivation" of a gauge field from a global symmetry of a lagrangian.

For example, in the Dirac equation we can multiply a solution by any complex number whose absolute value is 1, and we get a new solution. We say that there in the lagrangian is a gauge symmetry, whose group is U(1), which are the rotations of the complex plane around the origin.

Turning a global gauge symmetry into a local symmetry

Let us then rotate our coordinate axes of the complex plane by a different amount at different points of spacetime, but in a continuous way. If we had a solution of the Dirac equation, that solution is spoiled by the rotations. To restore the lagrangian to its original value, we introduce a 4-vector potential A. If we choose A in a suitable way and replace the partial derivatives in the lagrangian with gauge covariant derivatives

       D_μ = ∂ / ∂x_μ - A_μ,

then we can restore the original value of the lagrangian.

We say that we have turned the global symmetry into a local symmetry by introducing A. We are free to rotate the local complex plane, as long as we compensate with a suitably chosen A.

We then claim that the new lagrangian describes an electron under a potential A.

A philosophical question is, if in nature there existed a "global symmetry" which was turned into a "local symmetry", or is this just a story which we construct in our human minds?

As we have noted in various blog entries in the past year, the new lagrangian with a 4-vector potential A does not describe the interaction of an electron correctly because it fails to take into account the change of the inertial mass of the electron under a static electric field potential. It looks like the story with symmetries is just human imagination, and furthermore, the story is incorrect.

The role of degrees of freedom in a field

Let us think about waves in a tense string. A degree of freedom in the string is the vertical position of a point in the string, at various spacetime locations. The position is given as a real number z(t, x).

We start the construction of a wave equation from this degree of freedom.

Let us imagine that the string has an infinite length. The vertical elevation of whole string is not important for waves. What counts is the differences of the vertical position at various spacetime points. There is a global symmetry

       z(t, x) -> z(t, x) + C

for the solutions, where C is a constant real number.

We could now let C vary according to the spacetime point (t, x), and introduce a gauge field A to compensate for the difference in the lagrangian of our string.

The existence of a global symmetry was the result of two things:

1. We chose a real number z as the degree of freedom.

2. We wrote a typical wave equation where only the differences in the value of the degree of freedom z matter.

Rather than saying that we constructed A from a global gauge symmetry, we could say that we constructed the gauge field A from the degree of freedom which we chose.

We were lucky to find a suitable degree of freedom which describes the string well. The existence of a global symmetry is an automatic consequence from this degree of freedom and the normal way how waves behave. The global symmetry is not interesting in itself.