Working with generalized Kronecker Deltas

Question

Given a Manifold with 6 dimensions with a Minkowski metric, how to define a generalized Kronecker Delta in this manifold, e.g. $\delta_{abc}^{cde}$?

My goal is to simplify, for example, the following expression: $\delta_{a_{1}c_{1}d_{1}e_{1}}^{b_{1}c_{2}d_{2}e_{2}}\delta_{a_{2}c_{2}d_{2}e_{2}}^{b_{2}c_{3}d_{3}e_{3}}\delta_{a_{3}c_{3}d_{3}e_{3}}^{b_{3}c_{4}d_{4}e_{4}}\delta_{a_{4}c_{4}d_{4}e_{4}}^{b_{4}c_{1}d_{1}e_{1}}$

In 6 dimensions with a Minkowski metric and using Einstein summation convention. How would I do that using Mathematica?

Answer 1

I would define it like so:

KDelta[a_, b_, c_, a1_, a2_, a3_,] := Det[({
     {KroneckerDelta[a, a1], KroneckerDelta[a, a2], KroneckerDelta[a, a3]},
     {KroneckerDelta[b, a1], KroneckerDelta[b, a2], KroneckerDelta[b, a3]},
     {KroneckerDelta[c, a1], KroneckerDelta[c, a2], KroneckerDelta[c, a3]}
})];

Then refer to it by means of

KDelta[a, b, c, a1, a2, a3]

Answer 2

To generalize Mark's answer, you can use Outer to construct the matrix.

Surprisingly, I don't think you actually want to use the built-in KroneckerDelta because in Einstein summation notation, KroneckerDelta[a,a] should evaluate to the dimension of the space a labels, but in Mma it's automatically 1. If not, you can always define δ=KroneckerDelta.

 kδ[upper_,lower_]/;Length[upper]==Length[lower]:=Det[Outer[δ,upper,lower]]

Then kδ[{a,b,c,d},{α,β,γ,ζ}] evaluates to

  δ[a, ζ] δ[b, γ] δ[c, β] δ[d, α]
- δ[a, γ] δ[b, ζ] δ[c, β] δ[d, α]
- δ[a, ζ] δ[b, β] δ[c, γ] δ[d, α]
+ δ[a, β] δ[b, ζ] δ[c, γ] δ[d, α]
+ δ[a, γ] δ[b, β] δ[c, ζ] δ[d, α]
- δ[a, β] δ[b, γ] δ[c, ζ] δ[d, α]
- δ[a, ζ] δ[b, γ] δ[c, α] δ[d, β]
+ δ[a, γ] δ[b, ζ] δ[c, α] δ[d, β]
+ δ[a, ζ] δ[b, α] δ[c, γ] δ[d, β]
- δ[a, α] δ[b, ζ] δ[c, γ] δ[d, β]
- δ[a, γ] δ[b, α] δ[c, ζ] δ[d, β]
+ δ[a, α] δ[b, γ] δ[c, ζ] δ[d, β]
+ δ[a, ζ] δ[b, β] δ[c, α] δ[d, γ]
- δ[a, β] δ[b, ζ] δ[c, α] δ[d, γ]
- δ[a, ζ] δ[b, α] δ[c, β] δ[d, γ]
+ δ[a, α] δ[b, ζ] δ[c, β] δ[d, γ]
+ δ[a, β] δ[b, α] δ[c, ζ] δ[d, γ]
- δ[a, α] δ[b, β] δ[c, ζ] δ[d, γ]
- δ[a, γ] δ[b, β] δ[c, α] δ[d, ζ]
+ δ[a, β] δ[b, γ] δ[c, α] δ[d, ζ]
+ δ[a, γ] δ[b, α] δ[c, β] δ[d, ζ]
- δ[a, α] δ[b, γ] δ[c, β] δ[d, ζ]
- δ[a, β] δ[b, α] δ[c, γ] δ[d, ζ]
+ δ[a, α] δ[b, β] δ[c, γ] δ[d, ζ]

You may also want something like einsteinSum,

Clear[einsteinSum];
einsteinSum[Plus[a_,addends__]]:=einsteinSum[a]+einsteinSum[Plus[addends]];
einsteinSum[Times[factors___,δ[a_,a_]]]:=δ[a,a]einsteinSum[Times[factors]];
einsteinSum[Times[factors___,δ[a_,b_]]]/;Not[FreeQ[Times[factors],a]]:=einsteinSum[Times[factors]//.a->b];
einsteinSum[Times[factors___,δ[a_,b_]]]/;Not[FreeQ[Times[factors],b]]:=einsteinSum[Times[factors]//.b->a];
einsteinSum[Times[factors__]]:=einsteinSum[Distribute[Times[factors]]]
einsteinSum[f_]:=f;

and which you can specialize further (order matters in defining the patterns!). It takes a very long time to do something as complex as your example of four four-index generalized Kroneckers (and it doesn't give it in some pretty compact form, which may exist, and which might be discoverable with patterns).