# What is the connection between Lagrangian symmetry and particle multiplets?

I am struggling to see the connection between the symmetries of Lagrangians and particle multiplets. If I had three quark fields arranged in a vector $\Psi(x) = (u(x),d(x),s(x))$, these three fields have a combined Lagrangian

$$\mathcal{L}= \bar{\Psi}(x)(i\gamma^\mu\partial_\mu-m)\Psi(x)$$

I can see that this Lagrangian has an $SU(3)$ symmetry. When I quantise my theory, I promote my fields $u,d,s$ to quantum operators on an infinite dimensional Fock space. The interpretation of the operators $\hat{u},\hat{d},\hat{s}$ is that they are particle creation operators that create up, down and strange quarks respectively. However, I have also read that quarks live in the fundamental representation of the Lie algebra of $SU(3)$.

## My question:

From my studies of quantum field theory, I thought particles live in infinite dimensional Fock spaces, not finite dimensional representation spaces. How do I arrive at the particle multiplets from the symmetry of the Lagrangian?

• That makes sense, thank you. Should I really view the field operators as $\phi(x) \otimes u$ etc. Where $\phi(x)$ acts on the Fock space and tells us where to create the particle, while $u$ tells us what type of particle to create and that acts on the internal 3D Hilbert space? – Matt0410 Jun 4 '18 at 14:33