quadratic Casimir invariant

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Here we assume a compact Lie group, so that we can write \operatorname{tr}\left(t^a t^b\right) = d_r \delta^{ab}\, in some representation r\,. For any simple Lie algebra, the quantity

t^2 \equiv t^a t^a\,,

commutes with all the other elements of the algebra:

\left[t^a, t^b t^b\right]\, = (i f^{abc} t^c )t^b + t^b (i f^{abc} t^c)\,,
= i f^{abc} \left\{t^c, t^b\right\}\,
= 0\,, since f^{abc}\, is antisymmetric in its last two indices.

This object is an invariant of the algebra, known as the quadratic Casimir invariant or quadratic Casimir operator or simply the quadratic Casimir. The (irreducible) matrix representation of t^2\, is therefore proportional to the identity matrix (since t^2\, commutes with all t^a\,, and with \mathbf{1}\,, it commutes with all g \in G\,, and by Schur's lemma, t^2 \propto \mathbf{1}\,):

t^a t^a = c_2(r) \mathbf{1}\,,

where r\, labels the representation. Also, c_2(r)\, is sometimes labeled as c_r\,

In the adjoint representation,

(T^c T^c)_{ab} = -f^{cad} f^{cdb} = c_A \delta^{ab} \,.

If t^a\, are suitably normalized, so that f^{abc}\, is completely antisymmetric, then this can be written as

f^{acd}f^{bcd} = \operatorname{Tr}\,(T^aT^b) = c_A \delta^{ab}\,.

Now, in a particular irreducible representation r\,,

\operatorname{tr}(t^a t^a) = c_r \dim(r)\,,

but also

\operatorname{tr}(t^a t^a) = d_r \delta^{aa} = d_r \dim(G)\,.

Therefore we obtain the formula

\frac{d_r}{c_r} = \frac{\dim r}{\dim G}\,.

References

[1]

  1. ↑ M.E. Peskin and D.V. Schroeder (1995). An Introduction to Quantum Field Theory. Addison-Wesley, Reading, Massachusetts. ISBN 978-0201503975. 
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