# How to reconcile scaling dimension and gauge transformation in 2+1D U(1) Maxwell theory

+ 0 like - 0 dislike
186 views

In 2+1D (free) $U(1)$ Maxwell theory, power-counting in the action implies that the gauge potential $A_{\mu}$ should have scaling dimension 1/2. This is borne out by the propagator $\langle F_{\mu \nu}(x)F_{\lambda \sigma}(0) \rangle \sim 1/x^3$ or, more schematically, by the (gauge-dependent) propagator $\langle A_{\mu}(x) A_{\nu}(0) \rangle \sim 1/x$. However, the gauge transformation rule $A_{\mu} \rightarrow A_{\mu} + \partial_{\mu} \alpha$ means that $A_{\mu}$ should have dimension 1, as $\alpha$ must be dimensionless so that $e^{i\alpha}$ is a well-defined element of the gauge group $U(1)$.

Why do these power counting arguments give different results? Is that fact meaningful?

Note that in Chern-Simons theory or in 3+1D $U(1)$ Maxwell theory, there are no such issues; power-counting in the action also leads to $A_{\mu}$ having dimension 1, consistent with the gauge transformation law.

Thanks!

If your convention is such that  $A_{\mu} \rightarrow A_{\mu} + \partial_{\mu} \alpha$  is the gauge transformation for gauge fields, then on the matter field the gauge transformation should be a multiplication by $e^{ig \alpha}$ where $g$ is the gauge coupling. Remember the gauge coupling is dimensionful in 2+1 dimension, which means $\alpha$ is also dimensionful.
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$y$\varnothing$icsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.