Kronecker Delta

The simplest interpretation of the Kronecker delta is as the discrete version of the Delta Function defined by

$$\delta_{ij} \equiv \cases{ 0 & for $i \not = j$\cr 1 & for $i = j$.\cr}$$ (1)

It has the Complex Generating Function

$$\delta_{mn} = {1\over 2\pi i} \int z^{m-n-1}\,dz,$$ (2)

where $m$ and $n$ are Integers. In 3-space, the Kronecker delta satisfies the identities

$$\delta_{ii} = 3$$ (3)

$$\delta_{ij}\epsilon_{ijk} = 0$$ (4)

$$\epsilon_{ipq}\epsilon_{jpq} = 2\delta_{ij}$$ (5)

$$\epsilon_{ijk}\epsilon_{pqk} = \delta_{ip}\delta_{jq}-\delta_{iq}\delta_{jp},$$ (6)

where Einstein Summation is implicitly assumed, $i,j=1, 2, 3$, and $\epsilon$ is the Permutation Symbol.

Technically, the Kronecker delta is a Tensor defined by the relationship

$$\delta_l^k {\partial x_i'\over\partial x_k} {\partial x_l\ov... ...l x_k\over\partial x_j'} = {\partial x_i'\over\partial x_j'}.$$ (7)

Since, by definition, the coordinates $x_i$ and $x_j$ are independent for $i \not = j$,

$${\partial x_i'\over\partial x_j'} = {\delta'}_j^i,$$ (8)

so

$${\delta'}_j^i = {\partial x_i'\over\partial x_k} {\partial x_l\over\partial x_j'}\delta_l^k,$$ (9)

and $\delta_j^i$ is really a mixed second Rank Tensor. It satisfies

$${\delta_{ab}}^{jk} = \epsilon_{abi}\epsilon^{jki} = \delta_a^j\delta_b^k-\delta_a^k\delta_b^j$$ (10)

$$\delta_{abjk}=g_{aj}g_{bk}-g_{ak}g_{bj}$$ (11)

$$\epsilon_{aij}\epsilon^{bij} = {\delta_{ai}}^{bi}= 2\delta_a^b.$$ (12)

See also Delta Function, Permutation Symbol