I want to know if there is a way of typing into Mathematica an expression like the following,
$$\epsilon^{\mu \nu \lambda} f^{abc} A^a_\mu A^b_\nu A^c_\lambda + g\epsilon^{\mu \nu \lambda} A^a_\mu \partial_\lambda A^a_\nu + \bar{\psi}(\gamma^\mu(\partial_\mu + gA^a_\mu T^a))\psi$$
(..repeated indices are understood to be summed over..)
where $g$ is a number, $A^a_\mu$ can be thought of as matrices with $a,b,c = {1,2,3,..,N}$ for some $N$ and $\mu, \nu, \lambda = \{ 0,1,2\}$. So the $\partial_\mu$ are partial derivatives as $\partial_\mu = \frac{\partial}{\partial x^\mu}$. $f^{abc}$ is a set of numbers depending on the values of a,b and c and it is completely cyclic and anti-symmetric in it. $\epsilon^{\mu \nu \lambda}$ evaluates to $0$ if any two or more of its indices are equal and evaluates to 1 or -1 depending on whether the three distinct entries are in cyclic or anti-cyclic order.
$\psi$ should also be thought of as a matrix $\psi^a_i$ where $i,j = \{0,1,2\}$. $\gamma^\mu$ are a chosen set of $3\times 3$ matrices. Each of $T^a$ is a $N \times N$ matrix. Then the terms involving $\psi$ when expanded out look like, $$\bar{\psi}\gamma^\mu \partial _\mu \psi = (\psi^\dagger)^a_i (\gamma^0\gamma^\mu \partial_\mu )_{ij}\psi_j^a \quad{ \rm and }\quad\bar{\psi}\gamma^\mu A^a_\mu T^a \psi = (\psi^\dagger)^a_i(\gamma^\mu)_{ij}(A^c_\mu T^c)^{ab} \psi^b_j$$
- I would like to be able to input the above expression into Mathematica without having to explicitly specify the numbers $f^{abc}$ and the matrices $T^a$. I would like to be able to manipulate the expression with the matrices $T,A,\psi$ and the numbers $f^{abc}$,$g$ as being variables.
If the above is possible then I would eventually like to do something like write $A^a_\mu = B^a_\mu + C^a_\mu$ and $\psi^a_i = \eta ^a _i + \xi ^a _i$ and expand the expression in terms of B,C,$\eta$ and $\xi$.
