Is there a way to prove that the Kronecker delta $\delta_{ij}$ is indeed the only isotropic second order tensor (i.e. invariant under rotation), i.e. so we can write $T_{ij} = \lambda \delta_{ij}$ for some constant $\lambda$?
By rotational invariance I mean:
$$
T_{ij} = T^\prime_{ij} = R_{ip} R_{jq} T_{pq}\text{,}
$$
where the matrices $R_{ij}$ are orthogonal.
It is very straightforward to show that $\delta_{ij}$ is invariant, but how can I show that it is unique?
Best Answer
There is a simpler proof at http://www.damtp.cam.ac.uk/user/reh10/lectures/nst-mmii-chapter3.pdf, pg 61, that proceeds as follows
Starting with an isotrophic 2nd order tensor written in matrix form Tij, its coordinates are rotated 90 deg. around the 3 axis to get T'ij, then by comparing T'ij (which equals Tij as T is isotrophic) and Tij and concluding by inspection that T11 = T22, T13 = T23 = -T13 = 0, T31 = T32 = -T31 = 0, and then rotating the coordinates 90% about the 2 axis to conclude that T11 = T33 and T21 = T32 = 0 and T21 = T23 = 0.
That is, the proof is simplicity itself and is given in detail in the reference.