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For a given gauge group $G$, we see three kinds of representations of $G$ used in quantum field theory: the fundamental representation, the adjoint representation, and the trivial representation. I am confused what justifies the use of each representation.

  • Adjoint Representation: It is my understanding that the gauge fields always transform using the adjoint representation. Since the adjoint representation acts on the Lie algebra $\mathfrak{g}$ of $G$, the number of gauge fields/gauge bosons is always equal to the dimension of $G$ and this dimension determines the number of colors in that theory. For example in the strong interaction $G = SU(3)$ which has dimension $8$, so there are 8 gauge bosons in this theory (i.e. gluons), one for each dimension.
  • Fundamental Representation: The covariant derivative gives the interaction between matter fields and gauge fields. We use the fundamental representation to describe how a matter field transforms with a gauge transformation. This is why, for example, in the strong interaction quarks have 3 colors because $SU(3)$ naturally acts on $\mathbb{C}^3$.
  • Trivial Representations: If a matter field does not interact with a gauge field, for example electrons and gluons, then the matter field is equipped with a trivial representation of that gauge group.

Assuming the above is correct, then the trivial representation makes sense for matter fields that do not interact with certain gauge fields but what justifies the use of the adjoint representation or the fundamental representation?

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    $\begingroup$ See physics.stackexchange.com/q/824867 $\endgroup$
    – Mitchell Porter
    Commented Oct 27 at 22:15
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    $\begingroup$ There are, I think, at least three different meanings for the phrase “fundamental representation”. Regardless of which one you intend, I don’t think there is any requirement in QFT that matter fields have to be in a fundamental rep. It just happens that this is the case in the Standard Model. $\endgroup$
    – Ghoster
    Commented Oct 27 at 22:47
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    $\begingroup$ If you think that “fundamental rep” means “defining rep” or “smallest faithful rep” then you should also learn about the definition in terms of “fundamental weights”. I think the latter is the standard definition for mathematicians, and in my opinion physicists should follow that usage since there are better names for the other two definitions. $\endgroup$
    – Ghoster
    Commented Oct 27 at 22:53

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  1. Assuming that you are talking about a local gauge theory, the invariance of the Lagrangian under local gauge transformations necessarily requires the presence of (spin 1) gauge fields transforming with respect to the adjoint representation of the local gauge group $\rm G$, which fixes the number of gauge fields. Assuming furthermore renormalizability of the theory, the self interactions of the gauge fields and their couplings to all other fields of the model are uniquely determined.

  2. Note that this is not the case for global symmetries (example: chiral flavour group $\rm SU(2)_L \times SU(2)_R$ of QCD in the limit of massless up and down quarks), where the presence of associated gauge fields is not required.

  3. It is not clear what you mean by "fundamental" representation. Are you referring to the smallest faithful representation? Are you aware of the fact that in an $\rm SO(10)$ GUT (grand unified theory) the spin $1/2$ matter fields are sitting in the 16-dimensional irreducible representation of $\rm SO(10)$? Do you know that in model building it is quite common to employ several different (irreducible) representations in the Higgs sector (spin $0$ fields) and not necessarily the smallest faithful representation only?

  4. Consulting suitable text-book chapters on model building and/or grand unification might be helpful for further details.

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  • $\begingroup$ Thank you. By the fundamental representation I meant, for example SU(N) being represented on $\mathbb{C}^N$. $\endgroup$
    – CBBAM
    Commented Oct 30 at 2:42

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