The Eightfold Way — Question 10

Question 10

Representation dimension counts flavor weights with multiplicity

For an SU(3) representation labeled by nonnegative integers (p,q), its dimension is $d=(p+1)(q+1)(p+q+2)/2$. Consider (1,1), (3,0) and (0,3).

1. Calculate each dimension and identify the octet and decuplet dimensions.

2. If every flavor state in the (3,0) representation has spatial spin 3/2, count flavor-spin basis states and explain why equal dimension does not equate (3,0) with (0,3).

Question 10 - Solution

Representation dimension counts flavor weights with multiplicity

(a) Method and calculation

$$\boxed{d(1,1)=2(2)(4)/2=8}.$$$$\boxed{d(3,0)=4(1)(5)/2=10=d(0,3)}.$$

(b) Interpretation and further calculation

$$\boxed{N_{\mathrm{flavor,spin}}=10(2J+1)=10(4)=40}.$$$$(3,0)\ne(0,3),\quad\boxed{\text{equal dimension does not imply identical weights}}.$$

Physical check

The dimension formula counts internal flavor states, including repeated weights. It does not determine spatial spin or the masses.