How do I use tensors in Mathematica 7?

[Dox]

Hi there!

I'm tring to check some calculations of the propagator of a gauge field using the $R_\xi$ gauge fixing.

Since the propagator has a matricial structure,

$\Delta_{\mu\nu}=\frac{1}{(p^0)^2+E_p^2}\left[\delta_{\mu\nu}-\frac{1-\xi}{(p^0)^2+E_p^2}p_\mu p_\nu\right],$​

I'd like to check it using that fact. Using the command line

Array[(KroneckerDelta[#1,#2]-(1- xi) p[#1] p[#2]/(p[0]^2+ep^2))/(p[0]^2+ep^2) & , {4,4}, {0,0}]

I was able to write the needed expression, but then I'd like to compute the residues of ANY component... without losing the matricial structure. Is this possible?

All my guesses are fraud.For example... if I try to give a name to the array

M=Array[(...)]

it does not recognize the matrix structure.Whatever help is useful. Thanks.

 

[NateD]

The best way to compute the residues of any component is to use the function Residue[expr, var]. For example, to compute the residue of the (1, 2) element of the matrix M, you can useResidue[M[[1,2]], p[0]]

 

[oswaler]

Hi there,

Thank you for your question. To use tensors in Mathematica 7, you can use the built-in tensor functions such as TensorProduct, TensorTranspose, and TensorContract. These functions allow you to manipulate tensors and perform operations such as multiplication, transposition, and contraction.

In your specific case, you can use the TensorProduct function to create a tensor with the desired structure. For example, you can create a tensor with components \Delta_{\mu\nu} as follows:

TensorProduct[KroneckerDelta[\[Mu], \[Nu]], (1/(p[0]^2 + Ep^2)) - (1 - xi)/(p[0]^2 + Ep^2)^2 p[\[Mu]] p[\[Nu]]]

To compute the residues of any component, you can use the TensorContract function with the appropriate indices. For example, to compute the residue of the first component, you can use:

TensorContract[%, {\[Mu], \[Nu]}]

As for giving a name to the array, you can use the Set function to assign a name to the tensor. For example:

M = TensorProduct[KroneckerDelta[\[Mu], \[Nu]], (1/(p[0]^2 + Ep^2)) - (1 - xi)/(p[0]^2 + Ep^2)^2 p[\[Mu]] p[\[Nu]]]

I hope this helps. Let me know if you have any further questions. Good luck with your calculations!