Gauge Transformation: Definition & Uses

[captain]

what exactly constitutes a gauge transformation? is it a transformation using a differential operator?

 

[eendavid]

A gauge transformation is a transformation of the fields which doesn't change the physical observables. An example is adding a constant to the electrostatic potential. Demanding that such a (global) gauge symmetry exists as a local symmetry has given importants hints for theoretical physics. Such a symmetry requires changing differential operaters in gauge-covariant differential operators, which is probably what you heard of.

 

[captain]

are gauge fields the fields that you can use gauge transformations on?

 

[OOO]

are gauge fields the fields that you can use gauge transformations on?

Gauge transformations can be thought of as a kind of "internal" rotations. These rotations act on a kind of "vector", i.e. these are the things you use gauge transformations on.

If you also allow local gauge transformations, the rotations may vary from point to point in space and time. But such local gauge transformations are only self-consistent, if you define what it means to compare two "vectors" which are some distance apart in spacetime. This comparison amounts to "moving" one of the two vectors to the other.

For that you have to introduce a "new" field which describes how this transport should be achieved: the gauge field. The gauge field can be thought of as a prescription to do a gauge transform along a specified path from the first to the second vector's position. Generally the result depends on the path you are taking. Only if the physical field strengths are zero the result does not depend on the path.

Definining a gauge field is equivalent to defining a peculiar form of derivative (one which depends on the physical properties of the spacetime region under consideration): the covariant derivative. If you do a local gauge transformation, the gauge field will need to be transformed as well, so as to keep the whole thing consistent.