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I am doing some calculation with summation and the Kronecker symbol. Here are my steps :

$Assumptions = 
  k1 ∈ Reals &&  k2 ∈ Reals &&  k3 ∈ Reals &&  p1 ∈ Reals &&  p2 ∈ Reals &&  p3 ∈ Reals
k = {k1, k2, k3};
p = {p1, p2, p3};
d[i_, j_] := KroneckerDelta[i, j]
proj[i_, j_, k1_, k2_, k3_] := 
  d[i, j] - 
   (d[i, 1]*k1 + d[i, 2]*k2 + d[i, 3]*k3)*
   (d[j, 1]*k1 + d[j, 2]*k2 + d[j, 3]*k3)/
   (k1^2 + k2^2 + k3^2)
test1 = proj[i, j, k1, k2, k3]*proj[i, j, p1, p2, p3];
test2 = Sum[Sum[test1, {i, 1, 3}], {j, 1, 3}]
test2 // Expand

To explain the steps:

1) I define $\vec{k}$ and $\vec{p}$ with real components.
2) I define a projector $P_{ij} \left( \vec{k} \right) = \delta_{ij} - \frac{k_i k_j}{k^2}$.
3) I compute a summation on the repeated subscript.

After the last step, I have a relatively big expression, the product of k and p components. It looks like $$3-\frac{a}{a+b+c} - \frac{b}{a+b+c} - \frac{c}{a+b+c} +...-...$$ The a, b and c stands for k1, k2 and k3 (or p1, 2, 3).

Now the question: why doesn't Mathematica make the simplification because, as anyone can see, the preceding expression can be simplified to $2 +...-...$

Is the problem linked to the Expand operation? How can I make the simplification I want?. I thought of using /. to do it, but that doesn't work either.

I hope someone will understand my question!

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You can try using Simplify and related, but what you think it's a simpler expression will not necessarily match Mathematica's. –  b.gatessucks Feb 15 '13 at 10:23
+1 just for formatting your code:) Simplify@test2 works nicely and returns (2 k2 k3 p2 p3 + 2 k1 p1 (k2 p2 + k3 p3) + k1^2 (2 p1^2 + p2^2 + p3^2) + k2^2 (p1^2 + 2 p2^2 + p3^2) + k3^2 (p1^2 + p2^2 + 2 p3^2))/((k1^2 + k2^2 + k3^2) (p1^2 + p2^2 + p3^2)) –  Ajasja Feb 15 '13 at 10:34

3 Answers 3

up vote 1 down vote accepted

You have to explicitly tell Mathematica to simplify expressions. You can do this using Simplify or FullSimplify


(2 k2 k3 p2 p3 + 2 k1 p1 (k2 p2 + k3 p3) + 
   k1^2 (2 p1^2 + p2^2 + p3^2) + k2^2 (p1^2 + 2 p2^2 + p3^2) + 
   k3^2 (p1^2 + p2^2 + 2 p3^2))/((k1^2 + k2^2 + k3^2) (p1^2 + p2^2 + 
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Thank you for your answers. The problem when I use Simplify is that the form of the result is not "usable". I mean in the sense I want to use it. Because the calculation must lead to $1 - (\vec{k}.\vec{p})^2/(k^2*p^2)$. And with simplify, Mathematica doesn't do the factorization I want. I think I have to find an other way to do it, I may already have an idea. I thought I forget an assumption to enable Mathematica to do automatically this simplification. But it won't do what I didn't ask ! –  Lalylulelo Feb 15 '13 at 10:52

Not sure precisely what you want, but maybe it's this:

x = test2 // Expand;
x1 = PolynomialReduce[
   Numerator[Together[x]], {Denominator[Together[x]]}, Variables[x]];
x2 = PolynomialReduce[x1[[2]], {k1^2 + k2^2 + k3^2}, Variables[x]];
x3 = PolynomialReduce[x2[[2]], {p1^2 + p2^2 + p3^2}, Variables[x]];
y = x1[[1, 1]] + x2[[1, 1]]/(p1^2 + p2^2 + p3^2) + 
  x3[[1, 1]]/(k1^2 + k2^2 + k3^2) + 
  x3[[2]]/((p1^2 + p2^2 + p3^2) (k1^2 + k2^2 + k3^2))

Here's the output in reverse order; there's an issue with copy/paste:

$\frac{2 \text{k1} \text{k2} \text{p1} \text{p2}+2 \text{k1} \text{k3} \text{p1} \text{p3}+2 \text{k2}^2 \text{p2}^2+\text{k2}^2 \text{p3}^2+2 \text{k2} \text{k3} \text{p2} \text{p3}+\text{k3}^2 \text{p2}^2+2 \text{k3}^2 \text{p3}^2}{\left(\text{k1}^2+\text{k2}^2+\text{k3}^2\right) \left(\text{p1}^2+\text{p2}^2+\text{p3}^2\right)}+\frac{-\text{k2}^2-\text{k3}^2}{\text{k1}^2+\text{k2}^2+\text{k3}^2}+\frac{-\text{p2}^2-\text{p3}^2}{\text{p1}^2+\text{p2}^2+\text{p3}^2}+2$

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If you know approximately what denominators are obtained, you coud use command Collect[expression, factor].

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