How to use \dagger ?

Question

Following is my code :

{\mu,\nu}::Indices(vector).
\partial{#}::PartialDerivative.
{u,g_R,g_L,h}::Depends(\partial{#}).
\dagger::Symbol.
{u,g_R,g_L,h,(u)^\dagger}::NonCommuting.
ex:=i{(u)^{\dagger} (\partial_{\mu}(u)-i r_{\mu} u)-u (\partial_{\mu}((u)^\dagger)-i \ell_{\mu} 
(u)^\dagger)};
substitute(ex,$u->g_R u h^\dagger$);
product_rule(_);

I want to make

 (A B C...)^\dagger=... C^\dagger B^\dagger A^\dagger 
 U^\dagger U=U U^\dagger=1
 (U^\dagger)^\dagger=U

can be true, how to achieve this?

Answer 1

We are in the process of making this properly supported, but in the meantime, give this function a shot (you need to build Cadabra from source for this, the functionality is not yet in any of the binary packages):

def expand_conjugate(ex):
   tst:= (A??)^{\dagger};
   for node in ex:
      if tst.matches( node ):
         rep=$P$
         lst=[]
         for prod in node["\\prod"]:
            for factor in prod.factors():
               lst.append($ @(factor) $)
         for factor in list(reversed(lst)):
            rep.top().append_child($ @(factor)^{\dagger} $)
         rep.top().name=r"\prod"
         node.replace(rep)
   return ex

If you stick the above in a cell and evaluate it, you can then do

\dagger::Symbol;
ex:= (A B C)^{\dagger} + Q + (D E)^{\dagger};
expand_conjugate(ex);

to produce

$$C^{\dagger} B^{\dagger} A^{\dagger} + Q + E^{\dagger} D^{\dagger}$$

There will be better support for generic conjugation operations in Cadabra soon.

For the other two lines, try simple substitution with a rule of the type

rl:= { (A?^{\dagger})^{\dagger} = A?, 
         ( (A??)^{\dagger} )^{\dagger} = A??,
         A?^{\dagger} A? = 1,
         A? A^{\dagger} = 1,
         (A??)^{\dagger} A?? = 1,
         A?? (A??)^{\dagger} = 1 };

You can then do e.g.

\dagger::Symbol;
{A,B,U}::NonCommuting;
ex:= ( U^{\dagger}  )^{\dagger} + U^{\dagger} U 
        + (A B)^{\dagger} + ( (A B)^{\dagger} )^{\dagger};
substitute(ex, rl);

Hope this helps.

Comments

How to make "repeat" operation become effective? The other two functions also don't work, i.e.:

 U^\dagger U=U U^\dagger=1
(U^\dagger)^\dagger=U

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I have updated my original answer to your question, let me know if anything else is missing.

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Thanks, Kasper. There seem to be some bugs in your code , Cadabra reports "kernel crash". PS1: I also wish that Cadabra could know following rule in next version:

i^\dagger=-i

and

H^\dagger=H

where H is Hermitian operator( including real number).

PS2: By the way, do you konw how to achieve following operation?

a b c... d e...->d e... a b c...

Following is my attempt:

ex:=a b c... d e...;
substitute(_,$A?? d B??-> d B?? A??$);

it does't work. I want to implement cyclicity of trace

tr(a b c)=tr(c a b)=tr(b c a)

PS3: I don't find any easy ways to compute the traces of gamma matrices in cadabra, are there any plans to do this work?

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You need the latest from github for this to work.