ParamD

[Ahmed YOUSSEF]

Dear all,
I have 3 questions and I will be very thankful if you can help me.

1) Is it possible to tell Xact to commute ParamD and the covariant derivative?

2)How can I implement an automatic rule of the form (cd is the covariant derivative):

MakeRule[{ cd[a] [ ParamD[μ][α[]]], n[a] ParamD[μ,μ][α[ ] ] ] } MetricOn -> All,
ContractMetrics -> True]]?

The following command does not seem to work

AutomaticRules[α, MakeRule[{cd[a][ParamD[μ][α[]]], n[a] ParamD[μ, μ][α[]]}, MetricOn -> All,
ContractMetrics -> True]].

3) I need to declare two tensors as non commuting operators (say two generators of a Lie algebra M[a,b] and M[c,d]). Is this possible?

Thank you for your time,
Best

Ahmed YOUSSEF

[JMM]

Hi,

> 1) Is it possible to tell Xact to commute ParamD and the covariant derivative?

Yes. When you have two types of derivatives you need to sort them into
some preferred order. For example you might want to have all ParamD
derivatives outside the covariant derivatives (this is wrong of course
if the Christoffel of covd depends on any of the parameters ps):

ParamD /: covd_?CovDQ[i_][ ParamD[ps__][expr_] ] := ParamD[ps]
[ covd[i][expr] ]

or you might want the opposite:

ParamD[ps__][ covd_?CovDQ[i_][ expr_ ] ] := covdQ[i][ ParamD[ps]
[ expr ] ]

If you think you will use a different convention in different places
then define delayed rules and use them as desired. The fact that there
are two possibilities is what makes this not being automatic in
xTensor.

> 2)How can I implement an automatic rule of the form (cd is the covariant derivative):
>
> MakeRule[{ cd[a] [ ParamD[μ][α[]]], n[a] ParamD[μ,μ][α[ ] ] ] } MetricOn -> All,
>   ContractMetrics -> True]]?
>
> The following command does not seem to work
>
> AutomaticRules[α, MakeRule[{cd[a][ParamD[μ][α[]]], n[a] ParamD[μ, μ][α[]]}, MetricOn -> All,
>   ContractMetrics -> True]].

Well, it does work, but perhaps not as you expect. There are two
commands in that construction. First you have MakeRule, which
constructs the rule you want. But then you try to associate it to the
symbol α, and that symbol is too deep (this is the famous level-1
restriction of upvalues in Mathematica). Hence AutomaticRules puts the
rule into a global variable of xTensor, called $Rules, which you must
use by hand whenever you need this rule. Instead, what you can do is
associating your rule to cd, which is not that deep. You can also
declare the rule by hand:

cd[a_][ ParamD[μ][α[]] ] := n[a] ParamD[μ, μ][α[]]

which will associate it to cd as well. Remember that MakeRule is just
a convenience function to help in constructing simple cases. With
Mathematica's language you can construct any rule, far beyond
MakeRule's capabilities.

> 3) I need to declare two tensors as non commuting operators (say two generators of a Lie algebra M[a,b] and M[c,d]). Is this possible?  

This is a very general question. In xAct there is no predefined
concept of operator action. And tensor products are handled via
abstract indices and the Times product. Hence the only type of
operator action is stringing the operators through index contractions
in a formal inner space (say with indices A,B,...). Think of the
elements of the Lie algebra as represented with matrices whose indices
are A,B,... etc. You would work with objects having both indices
a,b,... and A,B,...

Cheers,
Jose.