- #1

CAF123

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## Homework Statement

Consider a ##j=1, SU(2)## representation (or fundamental ##S0(3)## representation). Suppose that ##a_i, b_i## and ##c_i## (i=1,2,3) are vectors transforming under this representation i.e ##a_i' = [\rho_1 (x)]_{ij} a_j = \rho_{ij} a_j## and similarly for b and c. Consider $$I_1 (\vec{a}, \vec{b}) = \delta_{ij} a_i b_j\,\,\,\,\,\,\,\,\,\,I_2(\vec{a}, \vec{b}, \vec{c}) = \epsilon_{ijk} a_i b_j c_k$$

1)Deduce the conditions on ##\delta_{ij}## and ##\epsilon_{ijk}## for which ##I_1## and ##I_2## are invariants under the transformation.

## Homework Equations

Want to show that ##I_1 (\vec{a}, \vec{b}) = I_1 (\vec{a'}, \vec{b'}) ## and similarly for ##I_2##. and ##\rho_1## is an element of the ##SO(3)## group.

## The Attempt at a Solution

So need to show that ##I_1 (\vec{a}, \vec{b}) = I_1(\rho\vec{a}, \rho \vec{b})## under the transformation ##\rho##. I also know that $$\rho_j (x)_{mm'} = \langle j,m | e^{ix^a T_a}| j,m' \rangle$$ where ##T_a## are the Lie algebra elements.

To be invariant, under the transformation ##\delta_{ij}' a_i' b_j' = \delta_{ij}a_i b_j##. I am looking for a hint in the right direction to perhaps use what I have written down here. Thanks.

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