[GiNaC-list] Simplify clifford expresion: Works?

[Sheplyakov Alexei]

Hello!

On Mon, May 28, 2007 at 09:44:44AM +0200, Javier Ros Ganuza wrote:

>  cout << "subs Rotation R * e00 * R^\\dag = " <<
> canonicalize_clifford((R * e00 * clifford_star(R)).expand()) << endl;
> 
>  cout << "subs Rotation R * e00 * R^\\dag = " <<
> canonicalize_clifford((R * e00 *
> clifford_star(R)).expand()).subs(sin(wild())*cos(wild())==sin(2*wild())/2) << endl;
>  
>  cout << "subs Rotation R * e00 * R^\\dag = " <<
> canonicalize_clifford((R * e00 *
> clifford_star(R)).expand()).subs(sin(wild())*cos(wild())==sin(2*wild())/2,subs_options::algebraic) << endl;
> 
> Gives no subs at all
> 
> subs Rotation R * e00 * R^\dag = e~0*cos(1/2*theta)^2
> +2*e~1*sin(1/2*theta)*cos(1/2*theta)-e~0*sin(1/2*theta)^2
> subs Rotation R * e00 * R^\dag = e~0*cos(1/2*theta)^2
> +2*e~1*sin(1/2*theta)*cos(1/2*theta)-e~0*sin(1/2*theta)^2
> subs Rotation R * e00 * R^\dag = e~0*cos(1/2*theta)^2
> +2*e~1*sin(1/2*theta)*cos(1/2*theta)-e~0*sin(1/2*theta)^2

Interesting. This program works for me:

$ cat substest.cpp
#include <iostream>
#include <stdexcept>
#include <ginac/ginac.h>
using namespace std;
using namespace GiNaC;

int main(int argc, char** argv) {
	symbol x("x"), p("p"), eps("eps");
	varidx mu(symbol("mu"), 4-2*eps),
	       nu(symbol("nu"), 4-2*eps);

	ex test = dirac_gamma(mu)*dirac_gamma(nu)*sin(x/2)*cos(x/2)
		+ lorentz_g(mu, nu)*pow(sin(x), 2);
	cout << test << " ==> " << endl;
	cout << test.subs(sin(wild())*cos(wild())==sin(2*wild())/2,
			subs_options::algebraic) << endl;
	return 0;
}
$ g++ substest.cpp -lginac
$ ./a.out
(gamma~mu*gamma~nu)*cos(1/2*x)*sin(1/2*x)+eta~mu~nu*sin(x)^2 ==> 
eta~mu~nu*sin(x)^2+1/2*(gamma~mu*gamma~nu)*sin(x)
$ pkg-config --modversion ginac
1.3.7
 
> I understand that
> 
> 
>  cout << "subs Rotation R * e00 * R^\\dag = " <<
> canonicalize_clifford((R * e00 * clifford_star(R)).expand()) << endl;
> 
>  cout << "subs Rotation R * e00 * R^\\dag = " <<
> canonicalize_clifford((R * e00 *
> clifford_star(R)).expand()).subs(2*sin(1/2*theta)*cos(1/2*theta)==sin(theta)) << endl;

In C[++] 1/2 means integer division, so 1/2*theta is zero. So your
rule is subs(0==sin(theta))...
  
>  cout << "subs Rotation R * e00 * R^\\dag = " <<
> canonicalize_clifford((R * e00 *
> clifford_star(R)).expand()).subs(2*sin(1/2*theta)*cos(1/2*theta)==sin(theta),subs_options::algebraic) << endl;
>  
> 
> Produces nonsense substitutions:
> 
> subs Rotation R * e00 * R^\dag = -e~0*sin(1/2*theta)^2
> +cos(1/2*theta)^2*e~0+2*cos(1/2*theta)*e~1*sin(1/2*theta)
> subs Rotation R * e00 * R^\dag = -e~(sin(theta))*sin(1/2*theta)^2
> +2*cos(1/2*theta)*e~1*sin(1/2*theta)+cos(1/2*theta)^2*e~(sin(theta))
> subs Rotation R * e00 * R^\dag = -e~(sin(theta))*sin(1/2*theta)^2
> +2*cos(1/2*theta)*e~1*sin(1/2*theta)+cos(1/2*theta)^2*e~(sin(theta))
 
... and you got exactly what you asked for: e~0 turned into
e~(sin(theta)

> For example: what does mean: -e~(sin(theta)) ?

Something like \gamma~(4-2*eps) in dimensional regularisation? :)

> Also in this simple expresion, even expand() is giving a bad result
> 
> cout << "out: " << ((2*sin(1/2*theta)*cos(1/2*theta) + 1).expand()) <<
> endl; 

Again, 1/2 is 0, so the expression is 1, ...
 
>  cout << "out: " << ((2*sin(1/2*theta)*cos(1/2*theta) +
> 1).expand()).subs(2*sin(1/2*theta)*cos(1/2*theta)==sin(theta)) << endl; 

... and the rule is to replace zero with sin(theta) ...
  
>  cout << "out: " << ((2*sin(1/2*theta)*cos(1/2*theta) +
> 1).expand()).subs(2*sin(1/2*theta)*cos(1/2*theta)==sin(theta),subs_options::algebraic) << endl; 
> 
>  cout << "out: " << ((2*sin(1/2*theta)*cos(1/2*theta) +
> 1).expand()).subs(sin(wild())*cos(wild())==sin(2*wild())/2) << endl; 
>  
>  cout << "out: " << ((2*sin(1/2*theta)*cos(1/2*theta) +
> 1).expand()).subs(sin(wild())*cos(wild())==sin(2*wild())/2,subs_options::algebraic) << endl; 
>  
> 
> out: 1
> out: 1
> out: 1
> out: 1
> out: 1

... and (obviously) there are no matches.


Best regards,
 Alexei

-- 
All science is either physics or stamp collecting.

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