# Are there two different adjoint representations used in the SM?

We know from Lie representation theory that the Lie algebra is a vector space. Therefore a representation of the Lie group can be transformation of this vector space itself which we call the Adjoint Representation. An element of this vector space, is itself represented by a matrix. For example, in the case of $$SU(3)$$, to show the adjoint representation, we write $$V\psi(x)V^{-1}$$, where $$V$$ is a $$3*3$$ special unitary matrix, $$\psi(x)$$ is also a $$3*3$$ (traceless) matrix, i.e. an element in the Lie algebra vector space. So we are multiplying three $$3*3$$ matrices. A new vector, or $$3*3$$ matrix, is produced, which belongs to or is an element in the Lie algebra vector space.

But in a SM textbook I see another definition/use of the Adjoint Representation. This time we consider the (abstract) commutation relations among the generators of the $$SU(3)$$ and interpret them as a vector acts on the another to produce a new vector. In other words, use the structure constants as elements to build a $$8*8$$ matrix to represent the Lie algebra vectors: $$L_{1},...L_{8}$$. Now if we consider an octet, or an 8 component column object, $$\psi(x)=(\psi_{1}, ..., \psi_{8})^T$$ each of $$\psi_{i}$$ a fermion or Dirac spinor, with four complex components, i.e. spinor indices suppressed, we can write for example $$L_{1}\psi(x)$$ to make or show a transformation of the fermion fields.

Now my question is that in the example of $$SU(3)$$ are these two different Adjoint Representations related somehow, as one of them involves the operation of $$3*3$$ matrices and the another one involves the operation of $$8*8$$ matrices?

My first impression is that in the first case, the 3 dimensional one, we are using the adjoint representation of the group using the Lie algebra as a vector space whereas in the second case, the 8 dimensional one, we are using the adjoint representation of the Lie algebra using the Lie algebra itself as a vector space. And both of them are useful in writing the Lagrangian density of the SM. Is this correct?

• For 3 * 3 matrices see this lecture note page 169: thphys.uni-heidelberg.de/~weigand/QFT2-14/SkriptQFT2.pdf For an 8 dimensional matrix see this lecture note page 4 its.caltech.edu/~xcchen/img/Ph129b2020/lecture/lecture0305.pdf
– VVM
Nov 24, 2020 at 10:35
• Eqs 6.32 and 6.33 where $\psi(x)$ and $V \psi(x)V^{-1}$ are defined as $3*3$ matrices
– VVM
Nov 24, 2020 at 11:12
• You are writing the same expression in group and algebra space respectively. Write your group expressions as exponentials of algebra elements and utilize the BCH expansion, and Lie’s theorem. Read up on Lie groups or post in math.se. Nothing physics here! Nov 24, 2020 at 11:40

Thus, your conjugation map $$V\psi V^{-1}$$ amounts to $$e^L \psi e^{-L}= \operatorname{Ad}_{L} \psi= \exp( \operatorname{ad }_L ) ~~\psi\\ =\psi +[L,\psi]+ [L,[L,\psi]]~/2!+[L, [L,[L,\psi]]]~/3!+...$$ in the Lie algebra, for all L in the Lie algebra.