# Calculation of tensor structure

I would like to calculate following type of contraction:

$$P_{ij}^{p-k} P_{mn}^{k} ((p_l - k_l) \delta_{ij} + k_j \delta_{il}) (k_n \delta_{lm} - p_m\delta_{ln}) = ?$$ where p,k are d-dimensional vectors and $$P_{ij}^{k} = \delta_{ij} - \frac{k_ik_j}{k^2}, \quad P_{ij}^{p-k} = \delta_{ij} - \frac{(p_i-k_i)(p_j-k_j)}{(p-k)^2}$$ are projection operators.

First I tried to define projector operators as a function of vector variable k

P[k_] = 1 - TensorProduct[k, k]/k.k


but I cannot figure out how to define a vector function.

Is this even a good idea or is there a better way how to deal with this?

• I think you are better off implementing your own tensor quantities. I.e. sc /: sc[{a_}, {b_}] sc[t_, {a_}] := sc[t, {b}]; so that for example sc[{i},{j}]sc[p,j] will evaluate to sc[p,i]. Btw these projectors look familiar, are you doing some conformal stuff? Commented Nov 22, 2015 at 1:11
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• Thank you! But no, I am working with stochastic field theory. I am trying to solve tensor structures which appears when you write down field theoretical formulation for navier-stokes equations and try to study its critical dynamics (loop corrections, renormalization, RG) Commented Nov 22, 2015 at 18:26
• Your expression in the tensor above has a misplaced index somewhere — the index $i$ appears three times. If you correct this, I can probably write up some code that will do some of what you want. Commented Nov 23, 2015 at 16:28
• In the corrected version, the index $l$ appears three times, so it's still not a valid tensor contraction. Commented Aug 4, 2017 at 20:25

I have a few suggestions, but I'm not sure about their efficiency. I'm sure that if this is the only thing that you want to compute my suggestion is good enough.

What I do when I have to compute something like this is to use a lot of replacement rules and a dot product. First, define

P[k_,i_,j_]:=delta[i,j] - k[i] k[j]/k^2

For the P[p-k,i,j] term just use a replacement rule (one of many):

P[p - k, i, j]//. {(a_ - b_)[c_] -> a[c] - b[c]}

gives you the desired output.

Now you only need to use replacement rules, for example

k[i] delta[i,j]-> k[j]

is achieved with rule={p_[a_] delta[a_, b_] -> p[b]}

for k^2 rule2={p_[i_] p_[i_]->p^2}

rule3={delta[i_,i_]->Dim}

You have to make sure to add all the replacement, I just added a few, you need also the case

k[i] delta[j,i]-> k[j]

but now you know how. Finally, you can either keep the use of k[i] p[i] or define a dot product

k[i] p[i]->dot[p,k]

I go for the second, always. If you follow my advise, define P as

P[k_,i_,j_]:=delta[i,j] dot[k,k] - k[i] k[j]

now, when you face your expression, keep in mind that you have to divide for (k^2 (k-p)^2). Why do I use the dot? Because (p-k)^2 is not p^2+k^2-2pk. Mathematica will take p[i]p[i] k[j]k[j] to be the same as p[i]k[i] p[j] k[j] which is wrong. Now it is easier to simplify.

Saludos!