Motivation for adjoint representation in the Standard Model

Question

While learning about the group theory machinery of the Standard Model, I was surprised to find out that the way the group acts on the particle content is that of the two most "natural" representations, that being the fundamental/ trivial representation and that of the adjoint representation. The fundamental representation did not surprise me too much. It seems to me that if you have a matrix group (even if you don't, every Lie group can be embedded in one), the first representation that comes to mind is just to let the matrix act on its underlying vector space. The adjoint representation is more mystifying to me.

It still seems natural to me that the adjoint representation should be considered. After all, for a general abstract Lie group, the Lie algebra is a vector space that's associated to the group by default. What is still not clear to me is why it should be used to describe gauge bosons.

I'm not asking some sort of question like "why does the universe use these representations". What I want to know is the thought process of the physicists who came to use this kind of theory. What are the reasons, if they exist, why one would suspect that the adjoint representation is a good candidate to describe bosons?

It may be that this is just something that we noticed from the associated field theories and as such is something that we inferred from experience, and is just the way it is. The only answer I could find on here seems to point in this direction. I found this satisfactory from the aesthetic/ formal point of view. But was this, historically, the only clue that the adjoint representation should be used?

In either case, if the first question doesn't have a meaningful answer, I'd still appreciate if someone can refer me to the first articles and work where the adjoint representation was employed.

Answer 1

Nonabelian gauge groups that you are considering are all conceptual descendants of the SU(2) of isospin (1954), a classic paper you may have gone through. Generalization to higher N groups SU(N) is virtually effortless.

The simplest option for fermions, basic for any QFT of physical import, is to put these in the fundamental representations (hence the N-dimensional ones). For their kinetic term to be made an invariant (singlet) of the gauge group, since there is only one derivative in the kinetic term, you need a gauge boson completion in a representation saturating the $\bar{N}\otimes N= 1\oplus (N^2-1) $; where I am denoting irreps by their dimensionality.

So, with Yang and Mills, you are readily led to the $(N^2-1)$-dimensional adjoint representation, in a schematic completion term of the type $\bar F A F$. Most QFT texts review this in their geometric introduction to gauge groups.

• Note, however, that for any representation R, not just the fundamental, the fermion bilinear $\bar \psi^i~ _R\!T^a_{ij} \partial_\mu \psi^j$ arising out of infinitesimal transformations of the kinetic term is in the adjoint, and so provides a singlet upon saturation with a gauge field $A_\mu^a$ in the adjoint, a gauge invariant term $ \bar \psi( \partial_\mu -ig ~_R\!T^a A^a_\mu)\psi$. (For SU(2), you know that adding/combining two spin s objects necessarily includes an adjoint (spin-1) object in its Clebsch decomposition.)

So, regardless of the particular irrep of the fermion, this type of coupling ensures gauge invariance of the kinetic term, when the gauge boson universally transforms in the adjoint!

It is further straightforward to generalize this to bosons and more elaborate nonlinear couplings.

(You might even invent hypothetical fermions of isospin 1 coupling to scalars (bosons) of isospin 2 into a singlet Yukawa term. Not a gauge coupling!)

Answer 2

If you have a (Lie) symmetry, there will be generators. In particular, in the case of Lie symmetries, the generators - say $\sigma_x,\sigma_y,\sigma_z$ - connect states in the Hilbert space that carries a representation (irreducible or not) of the group. You can see the entries of matrix representation of the generators as the result of “bosons” transforming one state into another. The generators thus have a very natural interpretation as “objects in the theory that change states”.

It is the a simple matter to recognize that the generators also carry a representation of the algebra, and this by definition is the adjoint representation.

It doesn’t mean there are no irreps other than the adjoint; in the case of $su(2)$ above you can clearly have irreps of dim. $2j+1$ and thus the generators have a representation by $(2j+1)\times(2j+1)$ matrices acting on the $2j+1$-dimensional Hilbert space. The generators (as abstract objects) still transform amongst themselves (under commutation) as elements of a $3$-dim irreps (for $su(2)$) since there are (in this case) $3$ generators.

In general, (tensor) operators can maps states between irreps. For instance, the operator $\hat x$ acting on an hydrogen ket $\vert n\ell m\rangle$ will given another state $\vert n’\ell’m’\rangle$ where you can have $\ell\ne \ell’$.

Generators on the other hand cannot change the irrep label: $\hat L_x\vert n\ell m\rangle$ must produce another state with the same $\ell$ value. So in this way the adjoint representation couples in a very specific way with other irreducible representations: generators cannot change the irrep label.