# Physical significance of the reality of an ${\bf N}$ representation: how the nature of interactions is affected?

Background The fundamental representation of $${\rm SU(N)}$$ is denoted by $${\bf N}$$ and the conjugate of the fundamental is denoted by $${\bar{\bf N}}$$. If the representation $${\bf N}$$ is related to $${\bar{\bf N}}$$ via a similarity transformation (i.e., equivalent), $${\bf N}$$ is called a real representation. For example, the $${\bf 2}$$ of $${\rm SU(2)}$$ is a real representation while the $${\bf 3}$$ of $${\rm SU(3)}$$ is not.

Sub-context $$1$$ In the Standard Model, the left-handed lepton and quark fields belong to the $${\bf 2}$$ of $${\rm SU(2)_L}$$ and their antiparticle fields belong to $${\bar{\bf 2}}$$.

Question $$1$$ What does the reality of $${\bf 2}$$ tell us about the weak interaction?

Sub-context $$2$$ In the theory of strong interactions, quantum chromodynamics, the quark of a given flavor but three different colors belong to the $${\bf 3}$$ of $${\rm SU(3)}$$ and their antiparticle field belong to $$\bar{{\bf 3}}$$ which is not equivalent to $${\bf 3}$$.

Question $$2$$ Also, how does the fact $${\bf 3}$$ of $${\rm SU(3)}$$, not being a real representation, affect the strong interaction of quarks?

• Is your "real" and "conjugate " meaning connected with real and complex numbers? – anna v Sep 12 at 3:23
• @annav en.wikipedia.org/wiki/… – probably_someone Sep 12 at 3:23
• @probably_someone thanks – anna v Sep 12 at 3:26

Specifically, $$(\phi_1, \phi_2)$$ is a doublet, and so is its strictly equivalent conjugate, $$(\phi_2^*, -\phi_1^*)$$. As a result, if you pick your vacuum to be $$\langle (\phi_1, \phi_2)\rangle =(0,v)$$, then $$\langle(\phi_2^*, -\phi_1^*)\rangle=(v,0)$$. So you may dot either the Higgs doublet or its conjugate to a fermion weak doublet in your (independent!) $$SU(2)_L$$-invariant Yukawa couplings; and thereby give (independent) masses to both the lower and upper components of your fermion doublet, so both the d and u quarks! (A very good thing.)
An ancillary advantage of the equivalence is that the symmetric d - coefficient in the anticommutator of two generators vanishes for SU(2), so the anomaly coefficient of the doublet and the equivalent antidoublet amount to the same thing. ∴ There are no "unmixed" SU(2)$$^3$$ chiral anomalies in the SM, which would otherwise invalidate this gauge symmetry.