# A Naive Question about Gauge Theory

I am suffering from a question I encountered from the lecture notes of gauge theory by David Tong. The problem comes from page 67 on the gauge fixing in back-ground gauge method. In David Tong's lecture notes, the gauge field $$A$$ is decomposed into a background gauge field $$\bar{A}$$ plus a perturbation part $$a$$ (In David Tong's lecture notes, the perturbative part is denoted as $$\delta A$$, I would rather denote it as $$a$$, instead.). i.e.

$$A=\bar{A}+a$$

Then David Tong claimed that under a gauge transformation $$U$$, the two parts transform as

$$\bar{A}^{U}=U^{-1}dU+U^{-1}\bar{A}U,\quad\quad a^{U}=U^{-1}aU.$$

Or, infinitesimally,

$$\delta\bar{A}=\bar{D}\omega,\quad\quad \delta a=[a,\omega],$$

where $$\bar{D}=d+[\bar{A},\,\,\,]$$ is the covariant derivative with respect to the background gauge field $$\bar{A}$$.

My question is the following:

The above definition of gauge transformation seems different from the one used in Peskin. In the book "An Introduction to Quantum Field Theory" by Peskin, on page 534, Chapter 16, Peskin claims that the background gauge field $$\bar{A}$$ should be regarded as fixed, the Lagrangian has a local gauge redundancy implemented by transformations on $$a$$:

$$a^{U}=U^{-1}aU+U^{-1}\bar{D}U,$$

or infinitesimally,

$$\delta a=\bar{D}\omega+[a,\omega]\equiv D_{A}\omega.$$

On page 535, Peskin (equation 16.99) said that the transformation (16.99), which is equivalent to the "gauge transformation" in Tong's lecture notes, is just a local symmetry, which shouldn't be a gauge transformation at all.

Is David Tong wrong here?

• When using background fields, there are many possible choices for how gauge fields should transform. You only know how the total field transforms. You are free to choose how the different parts transform. Tong is using a different choice from Pesking -- they are both fine, and there are many other possible choices too. – AccidentalFourierTransform Mar 2 at 15:43
• Maybe post this as an answer? – MannyC Mar 2 at 15:57
• @AccidentalFourierTransform Thank you for your answer. Would you answer another related question I encountered from Chern-Simons theory? The choice that Tong made here does not work for Chern-Simons theory. physics.stackexchange.com/q/419746/185558 – The Last Knight of Silk Road Mar 2 at 15:58