Index brackets

Indices can be associated to tensors, as in

$T_{\mu\nu}$ , but it often also happens that we want to associate indices to a sum or product of tensors, without writing all indices out explicitly. Examples are

$\begin{equation*} (A + B + C)_{\alpha\beta}\,,\quad\text{or}\quad (\psi \Gamma_{m n} \Gamma_{p})_{\beta}\,. \end{equation*}$ Here the objects

$A$ ,

$B$ ,

$C$ and

$\Gamma$ are matrices, while

$\psi$ is a vector. Their explicit components are labelled with

$\alpha$ and

$\beta$ indices, but the notation above keeps most of these vector indices implicit.

Cadabra can deal with such expressions through a construction which is called the "indexbracket". It is possible to convert from one form to the other by using the combine and expand algorithms. Combining terms goes like this:

ex:= (\Gamma_r)_{\alpha\beta} (\Gamma_{s t u})_{\beta\gamma}; combine(_);

$\displaystyle{}\left(\Gamma_{r}\right)\,_{\alpha \beta} \left(\Gamma_{s t u}\right)\,_{\beta \gamma}$

\indexbracket(\Gamma_{r})_{\alpha \beta} \indexbracket(\Gamma_{s t u})_{\beta \gamma}

$\displaystyle{}\left(\Gamma_{r} \Gamma_{s t u}\right)\,_{\alpha \gamma}$

\indexbracket(\Gamma_{r} \Gamma_{s t u})_{\alpha \gamma}

ex:= (\Gamma_r)_{\alpha\beta} Q_\beta; combine(_);

$\displaystyle{}\left(\Gamma_{r}\right)\,_{\alpha \beta} Q_{\beta}$

\indexbracket(\Gamma_{r})_{\alpha \beta} Q_{\beta}

$\displaystyle{}\left(\Gamma_{r} Q\right)\,_{\alpha}$

\indexbracket(\Gamma_{r} Q)_{\alpha}

If the index bracket has only one index, either the first or the last argument should be a matrix, but not both:

A::Matrix. {m,n,p}::Indices(vector). ex:= (A B)_{m}; expand(_);

$\displaystyle{}\left(A B\right)\,_{m}$

\indexbracket(A B)_{m}

$\displaystyle{}\left(A B\right)\,_{m}$

\indexbracket(A B)_{m}

If the index bracket has two indices, all arguments should be matrices,

{A,B}::Matrix. {m,n,p}::Indices(vector). ex:= (A B)_{m n}; expand(_);

$\displaystyle{}\left(A B\right)\,_{m n}$

\indexbracket(A B)_{m n}

$\displaystyle{}A_{m p} B_{p n}$

A_{m p} B_{p n}

If there are more arguments inside the bracket, these of course all need to be matrices (and are assumed to be so by default).