# Why do we require that the gauge condition $\alpha(x)$ falls off at infinity?

Let's say we are working in QED. The lagrangian is

$$$$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\bar{\psi}\big(i\not{D}-m\big)\psi$$$$

where $$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$$ and $$\not{D}=\gamma^\mu\big(\partial_\mu+iqA_\mu)$$. Here $$A_\mu$$ is the EM gauge field and $$q$$ is the charge of the electron. This lagrangian has a $$U(1)$$ gauge symmetry

$$$$\begin{cases} \psi\rightarrow\psi'=e^{iq\alpha(x)}\psi \\ \bar{\psi}\rightarrow\bar{\psi}'=e^{-iq\alpha(x)}\bar{\psi} \\ A_\mu\rightarrow A_\mu'=A_\mu-\partial_\mu\alpha(x) \end{cases}$$$$

where $$\alpha(x)$$ is a smooth arbitrary function of spacetime coordinate $$x^\mu$$. In many textbooks (for example Tong's lectures on QFT on p.125 eq. 6.11) it is required that $$\alpha(x)$$ falls off at infinity. Now, I understand this requirement for physical fields like $$F_{\mu\nu}$$ since we want finite energy, charge, etc. and if the fields are non-zero at infinity then by integration we would find an infinite total charge. However, I don't understand why we also require that $$\alpha(x)$$ dies as $$\vec{r}\rightarrow\infty$$. Regardless of the behaviour of $$\alpha(x)$$ and wether its behaviour makes $$A_\mu$$ take a constant value at infinity (or even blow up), the physical fields $$\vec{E}$$ and $$\vec{B}$$ will still be zero at the boundary as long as the gauge-independent part of $$A_\mu$$ falls off nicely.

Comment: Just for context, I'm rethinking the foundamentals of gauge theory because I'm studying Strominger's lectures on the infrared structure of gravity [1] where he relies heavily on gauge transformations at infinity.

• I think Tong imposes the condition to have a "real" gauge symmetry, i.e., symmetries that do not change the physics. However, if you want to have nontrivial symmetries (which change the physics) you allow for more general fall-off (as Strominger does.). – ungerade Dec 7 '19 at 20:44
• Btw. Tong clarifies this on page 138. – ungerade Dec 7 '19 at 20:47