To preface this, I know very little about Standard model physics and nonabelian gauge theory, so please correct me if my understanding is incorrect. I was reading about the Standard Model, and I stumbled upon the following statement which seemed to hold true for all the examples given: Nonabelian gauge theories necessarily have nonlinear equations of motion. Correspondingly, abelian gauge theories are necessarily linear.

The example offered by the book I was reading was of a three-component SU(2) gauge field (this is not an actual Standard Model field, of course) $\mathbf{W}_\mu$ for which the Lagrangian included a term of the form $\mathbf{ G_{\mu\nu} G^{\mu\nu} }$ with $\mathbf{G}_{\mu\nu} = \mathbf{W}_\mu \times \mathbf{W}_\nu$. In my understanding, this leads to nonlinear equations of motion and results in the the boson coupling to itself. I understand how it works in this case, but I'm not sure how to think about this generally.

My question: Is this true in full generality? Moreover, what sorts of nonlinear interaction terms would appear in the Lagrangian for a given nonabelian gauge group? Does it relate in some way to "commutators" of the group action on a field?


If the gauge group is abelian, then its Lie algebra is trivial in the sense that all elements commute, so if $T_a$ are a set of generators, then $[T_a,T_b]=0$. If the gauge group is nonabelian, then $$ [T_a,T_b]=C^c_{ab}T_c $$ for some constants $C^c_{ab}=-C^c_{ba}$.

Now, the Lagrangian for gauge theories is usually (note that one may consider GR a gauge theory, and this is certainly not true for GR - it is true for Yang-Mills type theories basically) given by $$ \mathcal L\sim F^2 $$ where $$ F^a_{\mu\nu}=\partial_\mu A^a_\nu-\partial_\nu A_\mu^a+C^a_{bc}A^b_\mu A^c_\nu. $$ There might be a factor of $2$ or $1/2$ before the $C$ .

Now, if you square this, then from the first two terms, you get terms of the schematic form $(\partial A)^2$. These are quadratic in the field derivatives, and will produce linear equations of motion. On the other hand, the last term will produce quartic self-interaction $\sim A^4$, which means that the $A$ gauge field is self-interacting, and will produce nonlinear equations of motions.

However as you can see, this comes from the $C^a_{bc}A^b_\mu A^c_\nu$ term, which is absent if and only if the theory is abelian.


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