# Gauge Invariance for field of *Uncharged* particles?

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1. Jan 8, 2016

### referframe

A complex classical field Φ of particles is, by itself, invariant under global phase changes but not under local phase changes. It is made gauge invariant by coupling it with the EM potential, A, by substituting the covariant derivative for the normal partial derivative in the Lagrangian. But if the particles represented by Φ have zero electrical charge, the covariant derivative is the same as the normal derivative.

Does that mean that a classical field of uncharged particles cannot be made gauge invariant?

2. Jan 8, 2016

### Orodruin

Staff Emeritus
If it is uncharged it is not affected by the gauge transformation and therefore the covariant derivative is equivalent to the partial derivative. The kinetic term is still gauge invariant, because the field does not transform under gauge transformations.

3. Jan 9, 2016

### referframe

I think I understand. Are you saying that gauge invariance is irrelevant for classical fields representing a system of uncharged particles?

4. Jan 9, 2016

### Orodruin

Staff Emeritus
It is not irrelevant. It is just that an uncharged field does not transform under gauge transformations. If you have a local symmetry, the Lagrangian still needs to be invariant under gauge transformations, but this implies the covariant derivative being equal to the parial derivative. In general, the covariant derivative is given by $D_\mu = \partial_\mu - i g A^a_\mu \tau^a$ where $\tau^a$ is the representation of the gauge group generator in the relevant representation. An uncharged field corresponds to transforming under the trivial representation where $\tau^a = 0$.