Exterior derivative and complex-valued forms

[g.lomo...@gmail.com]

Hello,

I'm working with complex valued forms and sometimes I need to take exterior derivatives of complex conjugate of some expression; however, in this case the result the mathematica gives is:

Diff[ Conjugate[a] ]= Diff[ a ]*Conjugate'[ a ]

where Conjugate' should be the derivative of conjugate with respect to its argument. However, I would like to obtain something like:

Diff[Conjugate[ a] ]    or   Conjugate[Diff[ a ]] 

In the same way, Exterior derivative and Real (or Imaginary) part don't commute; In fact in some expression like:

Re(Diff[ a ])

It doesn't enter the real part in the exterior derivative, getting

Diff[Re(a)]

Someone knows how to solve this kind of problems? thank to everybody,

Gabriele

[Leo Stein]

Hi Gabriele,

I hope you have seen the option 'Dagger' which can be used when defining a tensor or differential form. By default it's Dagger->Real, but if you specify Dagger->Complex, then you will get a pair of complex-valued tensors or forms. Then, within the xAct suite, we use Dagger[] instead of Conjugate[] to perform complex conjugation. Thus you can write something like

  Dagger[ Diff[ w[] ] ]

and it will automatically get turned into

  Diff[ w†[] ].

Best

Leo

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