# Simple tensor manipulations in mathematica

I want to perform simple calculations involving kronecker deltas without specifying the dimension of the space I am in.

For example, I would like to automatically calculate (note the implied Einstein summation convention): $$\delta_{i,j}\delta_{j,k}=\delta_{i,k}$$ and $$\delta_{i,i}=d$$ This is straightforward to do by hand, but the expressions involved are lengthy, so this is tediuous: e.g. I want to evaluate expressions sort of like the following, but a bit more complicated: $$(\delta_{a,b}\delta_{c,d} \delta_{e,f}+3\delta_{a,d}\delta_{c,f} \delta_{e,b})(\delta_{i,j}\delta_{c,d} \delta_{e,f}+3\delta_{i,d}\delta_{c,f} \delta_{e,j})=?$$

I would think this is easy to do, but I cannot figure out how.

You can use the new in M10 Tensor* function to do this, although it is a bit long-winded. First, lets define your identity matrix:

id = IdentityMatrix[d];


Next, we can represent your individual terms using TensorProduct and TensorTranspose. For example,

$\delta _{a,b} \delta _{c,d} \delta _{e,f}$:

TensorProduct[id, id, id];


$\delta _{a,d} \delta _{e,b} \delta _{c,f}$:

TensorTranspose[TensorProduct[id, id, id], {1, 4, 3, 6, 5, 2}];


and similarly for the others. Finally, we want to contract the expression:

res = TensorContract[
TensorProduct[
(
TensorProduct[id, id, id] +
3 TensorTranspose[TensorProduct[id, id, id], {1, 4, 3, 6, 5, 2}]
),
(
TensorProduct[id, id, id] +
3 TensorTranspose[TensorProduct[id, id, id], {1, 4, 3, 6, 5, 2}]
)
],
{{3, 9}, {4, 10}, {5, 11}, {6, 12}}
];


Unfortunately, TensorReduce is not able to simplify this further. However, we can make use of my TensorSimplify package to help out. Install the paclet with::

PacletInstall[
"TensorSimplify",
"Site"->"http://raw.githubusercontent.com/carlwoll/TensorSimplify/master"
]


and then load the package with:

<<TensorSimplify


Then:

TensorSimplify[res]


6 IdentityMatrix[d]\[TensorProduct]IdentityMatrix[d] + d^2 IdentityMatrix[d]\[TensorProduct]IdentityMatrix[d] + 9 d TensorTranspose[ IdentityMatrix[d]\[TensorProduct]IdentityMatrix[d], {1, 3, 2, 4}]

In terms of kronecker deltas, the above is equivalent to:

$$d^2 \delta _{a,b} \delta _{i,j}+9 d \delta _{a,i} \delta _{b,j}+6 \delta_{a,b} \delta _{i,j}$$

• excellent answer and a very useful package. Can I ask how you got from the Mathematica expression of the TensorSimplify command to restoring the subscripts in Kronecker deltas? I see the {1,3,24} in the end which indicates the ordering of the term proportional to 9d, but the other expressions are blank. Thanks – DiSp0sablE_H3r0 Mar 27 '18 at 9:34
• Thanks. The non-trivial part appears to be your package, which seems the most important. I was previously trying to do something involving replacement rules acting on symbolic expressions. But I think this will work. – simple_Guest Mar 27 '18 at 14:25
• @Konstantinos For a rank 4 tensor T, TensorTranspose[T, {1, 2, 3, 4}] is equivalent to T. So, imagine that there is a TensorTranspose` wrapper on the other terms. – Carl Woll Mar 27 '18 at 16:55