# Unitary transformation behind gauge transformation

It is very well-known that for bosonic operators a Gauge transformation can always be associated with it $$a\rightarrow e^{i\phi}a.$$ Obviously this is a Unitary transformation. Something like $$a^{\prime}=\mathcal{U}^{\dagger}a\mathcal{U}$$

I want to know what is $\mathcal{U}$?

• I don't understand your question. First, are we talking about $\mathrm{U}(1)$ gauge or arbitrary gauge theories? Second, what do you mean by "giving rise" to the phase factor? The "phase factor" is, if we are in the fundamental representation, as you seem to imply, just the element of the gauge group corresponding to the gauge trafo. – ACuriousMind Aug 5 '14 at 16:15
• I am considering U(1) gauge. But I want to know what is the form of this U(1)? – Pratyay Ghosh Aug 5 '14 at 16:29
• $\mathrm{U}(1)$ is the circle group. I'm still uncertain what your question really is. You might be interested in What is the basis of gauge theory? – ACuriousMind Aug 5 '14 at 16:38

$$\mathcal{U}=e^{-i\phi a^{\dagger}a}$$