What is an Intuitive example of a Gauge Symmetry?

Question

Can anyone give an intuitive example of what a gauge symmetry is? I am new to this concept and would like to understand it better!

Answer 1

Another example of of a Gauge Symmetry can be found in the basic $V=mgh$. Here you can have your "ground" anywhere you want. This freedom reflects the key idea in gauge theory.

Answer 2

A Gauge Symmetry (not talking about large gauge transformations) refers to mathematical symmetries that are not physical but rather redundancies in our formulation.

It is a quite rich subject, but I will limit myself to intuitive answers.

Consider the magnetic field. We know that it is given by the curl of a vector potential

$$\mathbf{B = \nabla \times A}$$

Now notice that if you add a curl-less to A, the magnetic field stays the same, that is the magnetic field is insensitive to curl-less terms. This means that we have a symmetry where any transformation of A in the form of $\mathbf{A} \rightarrow \mathcal{A} + \nabla f$, where f is just a scalar function, leaves our physics invariant.

In physics it is efficient and effective to describe these symmetries by gauge groups, so groups that describe these redundancies, that is groups under whose action our model stays physically the same. The standard Model is described by $SU(3)\times SU(2) \times U(1) $, which all describe different aspects of the theory.

Gauge theory is a very rich subject that has shaped the way we approach physics.

Reading Recommendations:

Large Gauge Transformation (Gauge Transformations that cannot be thought of as redundancies.

Yang Mills (Wikipedia)

Introduction to Gauge Theory for Beginners

Answer 3

Take a potential that only depends on the distance of an object’s center from a center point. Then you can consider a rotation of your object. If this transformation acts locally, so the rotation for example is dependent o. You spacetime location, then you have a gauge symmetry and the rotations are gauge transformations.

Answer 4

1. A toy model of a gauge theory is $$ Z ~\propto~\int \! dx ~dy~ e^{iS(x)}, \tag{71.8}$$ cf. Ref. 1. The action $S(x)$ and the path integral $Z$ are invariant under a gauge transformation $$y\quad\longrightarrow\quad y^{\prime}~=~y+f(x).$$ The $y$ variable is redundant/unphysical. See Ref. 1 for further details on how to gauge-fix the theory.

2. In more realistic gauge theories, it is more difficult (if not impossible) to cleanly separate physical and unphysical DOF, but the underlying idea is the same.

References:

1. M. Srednicki, QFT, 2007; chapter 71. A prepublication draft PDF file is available here.