a field-theory motivated approach to computer algebra


Make indices explicit on an expression with implicit indices.
In Cadabra you can write expressions which are understood to have indices suppressed, in order to get a cleaner notation. This is often used for vector/matrix notation, or when dealing with spinors. In order to inform Cadabra about these implicit indices, you use the ImplicitIndex property (which is also necessary to prevent Cadabra from moving these objects through each other when sorting products into canonical form). The explicit_indices algorithm can then make these indices explicit, which can sometimes make them easier to work with, for example when doing substitutions.
In the following example we define two sets of indices, and several objects which are assumed to have implicit indices.
{m,n,p}::Indices(spacetime, position=fixed); {a,b,c,d,e,f,g,h}::Indices(spinor, position=fixed); \sigma^{p}::ImplicitIndex(\sigma^{p a}_{b}); \psi::ImplicitIndex(\psi_{a}); \chi::ImplicitIndex(\chi^{a});
\(\displaystyle{}\text{Attached property Indices(position=fixed) to }\left[m, n, p\right].\)
\(\displaystyle{}\text{Attached property Indices(position=fixed) to }\left[a, b, c, d, e, f, g, h\right].\)
\(\displaystyle{}\text{Attached property ImplicitIndex to }\sigma^{p}.\)
\(\displaystyle{}\text{Attached property ImplicitIndex to }\psi.\)
\(\displaystyle{}\text{Attached property ImplicitIndex to }\chi.\)
The following is a valid expression for a spinor bilinear,
ex:= \psi \sigma^{m} \sigma^{n} \chi;
\(\displaystyle{}\psi \sigma^{m} \sigma^{n} \chi\)
\psi \sigma^{m} \sigma^{n} \chi
We can now make the indices explicit using
\(\displaystyle{}\psi_{a} \sigma^{m a}\,_{b} \sigma^{n b}\,_{c} \chi^{c}\)
\psi_{a} \sigma^{m a}_{b} \sigma^{n b}_{c} \chi^{c}
This also works when there are trace operators, as is illustrated in the following example.
Tr{#}::LaTeXForm("{\rm Tr}"). Tr{#}::Trace(indices=spinor); ex:= Tr(\sigma^{m} \sigma^{n} + \sigma^{n} \sigma^{m});
\(\displaystyle{}\text{Attached property Trace to }{\rm Tr}\left(\#\right).\)
\(\displaystyle{}{\rm Tr}\left(\sigma^{m} \sigma^{n}+\sigma^{n} \sigma^{m}\right)\)
Tr(\sigma^{m} \sigma^{n} + \sigma^{n} \sigma^{m})
\(\displaystyle{}\sigma^{m a}\,_{b} \sigma^{n b}\,_{a}+\sigma^{n a}\,_{b} \sigma^{m b}\,_{a}\)
\sigma^{m a}_{b} \sigma^{n b}_{a} + \sigma^{n a}_{b} \sigma^{m b}_{a}
Copyright © 2001-2019 Kasper Peeters