1: load cantens$
*** enlarging fasl space by 25000 items
*** .. redefined
2: in "selfgra.tst";
%%%%%%%%%%%%%%%%%%% A. Burnel and H. Caprasse %%%%%%%%%%%%%%%%%%%%%%
%
% Application of CANTENS.RED
% Date: 15/09/98
%
% Computes the gluon contribution to the gluon self-energy in the
% "finite" theory
% contains initially 18 terms which are reduced to 10 by cantens
% in a dm-dimensional Minkowski space and 8 terms in a 4-dimensional
% Minkowski space.
%
% *** Will look much nicer if run in the GRAPHIC mode
%
% LOADING CANTENS
load cantens$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Structure definitions, Minkowski space X internal symmetry space
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
off onespace;
% to be allowed to work within several subspaces
define_spaces wholespace={dm+di,signature=1};
t
define_spaces mink={dm,signature=1};
t
%,indexrange=0 .. 3};
define_spaces internal={di,signature=0};
t
%,indexrange=4 .. 11};
%
% Memberships of indices:
mk_ids_belong_space({mu1,mu2,nu1,nu2,tau},mink);
t
mk_ids_belong_space({a1,a2,b1,b2,c1,c2},internal);
t
%%%%%%%%%%%%%%%%
% Used Tensors %
%%%%%%%%%%%%%%%%
%% variables x1,x2 and xi=x1-x2,
%% aa, gluon field
%% dd, contracted gluon field
%% which appears inside the expression
%% a is the antisymmetric structure constant of SU3.
%% It is called "a" to assure that it appears first
%% inside REDUCE expressions and to assure that they
%% factorize in front of the output expression.
%
tensor aa,dd,a,x1,x2,xi;
t
% tensor declaration
make_variables x1,x2,xi;
t
% variable declaration
% declare to which subspace the declared tensors belong to.
make_tensor_belong_space(x1,mink);
mink
make_tensor_belong_space(x2,mink);
mink
make_tensor_belong_space(xi,mink);
mink
make_tensor_belong_space(a,internal);
internal
antisymmetric a;
% antisymmetry of structure constant.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% building of starting expression to be manipulated and simplified.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
es1:=g^2*a(a1,b1,c1)*a(a2,b2,c2);
a1 b1 c1 a2 b2 c2 2
es1 := a *a *g
as1:=-aa(x1,nu1,-b1)*aa(x2,nu2,-b2)*df(df(dd(xi,mu1,-c1,mu2,-c2),xi(nu
1)),xi(nu2))
*dd(xi,-mu1,-a1,-mu2,-a2);
nu1 nu2
as1 := - aa (x1)*aa (x2)*dd (xi)
b1 b2 mu1 a1 mu2 a2
mu1 mu2 nu1 nu2
*df(dd (xi),xi ,xi )
c1 c2
as2:=-aa(x1,nu1,-b1)*aa(x2,nu2,-b2)*df(dd(xi,mu1,-c1,mu2,-a2),xi(nu1))
*df(dd(xi,-mu1,-a1,-mu2,-c2),xi(nu2));
nu1 nu2
as2 := - aa (x1)*aa (x2)
b1 b2
nu2
*df(dd (xi),xi )
mu1 a1 mu2 c2
mu1 mu2 nu1
*df(dd (xi),xi )
c1 a2
as3:=aa(x1,nu1,-b1)*df(aa(x2,mu2,-c2),x2(nu2))*df(dd(xi,mu1,-c1,nu2,-b
2),xi(nu1))
*dd(xi,-mu1,-a1,-mu2,-a2);
nu1 mu2 nu2
as3 := aa (x1)*dd (xi)*df(aa (x2),x2 )
b1 mu1 a1 mu2 a2 c2
mu1 nu2 nu1
*df(dd (xi),xi )
c1 b2
as4:=aa(x1,nu1,-b1)*df(aa(x2,mu2,-c2),x2(nu2))*df(dd(xi,mu1,-c1,-mu2,-
a2),xi(nu1))
*dd(xi,-mu1,-a1,nu2,-b2);
nu1 nu2 mu2 nu2
as4 := aa (x1)*dd (xi)*df(aa (x2),x2 )
b1 mu1 a1 b2 c2
mu1 nu1
*df(dd (xi),xi )
c1 mu2 a2
as5:=-aa(x1,nu1,-b1)*aa(x2,mu2,-a2)*df(dd(xi,mu1,-c1,nu2,-b2),xi(nu1))
*df(dd(xi,-mu1,-a1,-mu2,-c2),xi(nu2));
nu1 mu2
as5 := - aa (x1)*aa (x2)
b1 a2
nu2
*df(dd (xi),xi )
mu1 a1 mu2 c2
mu1 nu2 nu1
*df(dd (xi),xi )
c1 b2
as6:=-aa(x1,nu1,-b1)*aa(x2,mu2,-a2)*df(df(dd(xi,mu1,-c1,-mu2,-c2),xi(n
u1)),xi(nu2))
*dd(xi,-mu1,-a1,nu2,-b2);
nu1 mu2 nu2
as6 := - aa (x1)*aa (x2)*dd (xi)
b1 a2 mu1 a1 b2
mu1 nu1 nu2
*df(dd (xi),xi ,xi )
c1 mu2 c2
as7:=-df(aa(x1,mu1,-c1),x1(nu1))*aa(x2,nu2,-b2)*df(dd(xi,nu1,-b1,mu2,-
c2),xi(nu2))
*dd(xi,-mu1,-a1,-mu2,-a2);
nu2
as7 := - aa (x2)*dd (xi)
b2 mu1 a1 mu2 a2
mu1 nu1 nu1 mu2 nu2
*df(aa (x1),x1 )*df(dd (xi),xi )
c1 b1 c2
as8:=-df(aa(x1,mu1,-c1),x1(nu1))*aa(x2,nu2,-b2)*df(dd(xi,-mu1,-a1,mu2,
-c2),xi(nu2))
*dd(xi,nu1,-b1,-mu2,-a2);
nu2 nu1
as8 := - aa (x2)*dd (xi)
b2 b1 mu2 a2
mu1 nu1 mu2 nu2
*df(aa (x1),x1 )*df(dd (xi),xi )
c1 mu1 a1 c2
as9:=df(aa(x1,mu1,-c1),x1(nu1))*df(aa(x2,mu2,-c2),x2(nu2))*dd(xi,nu1,-
b1,nu2,-b2)
*dd(xi,-mu1,-a1,-mu2,-a2);
nu1 nu2
as9 := dd (xi)*dd (xi)
mu1 a1 mu2 a2 b1 b2
mu1 nu1 mu2 nu2
*df(aa (x1),x1 )*df(aa (x2),x2 )
c1 c2
as10:=df(aa(x1,mu1,-c1),x1(nu1))*df(aa(x2,mu2,-c2),x2(nu2))*dd(xi,nu1,
-b1,-mu2,-a2)
*dd(xi,-mu1,-a1,nu2,-b2);
nu2 nu1
as10 := dd (xi)*dd (xi)
mu1 a1 b2 b1 mu2 a2
mu1 nu1 mu2 nu2
*df(aa (x1),x1 )*df(aa (x2),x2 )
c1 c2
as11:=-df(aa(x1,mu1,-c1),x1(nu1))*aa(x2,mu2,-a2)*df(dd(xi,-mu1,-a1,-mu
2,-c2),xi(nu2))
*dd(xi,nu1,-b1,nu2,-b2);
mu2 nu1 nu2
as11 := - aa (x2)*dd (xi)
a2 b1 b2
mu1 nu1 nu2
*df(aa (x1),x1 )*df(dd (xi),xi )
c1 mu1 a1 mu2 c2
as12:=-df(aa(x1,mu1,-c1),x1(nu1))*aa(x2,mu2,-a2)*df(dd(xi,nu1,-b1,-mu2
,-c2),xi(nu2))
*dd(xi,-mu1,-a1,nu2,-b2);
mu2 nu2
as12 := - aa (x2)*dd (xi)
a2 mu1 a1 b2
mu1 nu1 nu1 nu2
*df(aa (x1),x1 )*df(dd (xi),xi )
c1 b1 mu2 c2
as13:=-aa(x1,mu1,-a1)*aa(x2,nu2,-b2)*df(dd(xi,nu1,-b1,mu2,-c2),xi(nu2)
)
*df(dd(xi,-mu1,-c1,-mu2,-a2),xi(nu1));
mu1 nu2
as13 := - aa (x1)*aa (x2)
a1 b2
nu1
*df(dd (xi),xi )
mu1 c1 mu2 a2
nu1 mu2 nu2
*df(dd (xi),xi )
b1 c2
as14:=-aa(x1,mu1,-a1)*aa(x2,nu2,-b2)*dd(xi,nu1,-b1,mu2,-a2)
*df(dd(xi,-mu1,-c1,-mu2,-c2),xi(nu1),xi(nu2));
mu1 nu2 nu1 mu2
as14 := - aa (x1)*aa (x2)*dd (xi)
a1 b2 b1 a2
nu1 nu2
*df(dd (xi),xi ,xi )
mu1 c1 mu2 c2
as15:=aa(x1,mu1,-a1)*df(aa(x2,mu2,-c2),x2(nu2))*dd(xi,nu1,-b1,nu2,-b2)
*df(dd(xi,-mu1,-c1,-mu2,-a2),xi(nu1));
mu1 nu1 nu2 mu2 nu2
as15 := aa (x1)*dd (xi)*df(aa (x2),x2 )
a1 b1 b2 c2
nu1
*df(dd (xi),xi )
mu1 c1 mu2 a2
as16:=aa(x1,mu1,-a1)*df(aa(x2,mu2,-c2),x2(nu2))*dd(xi,nu1,-b1,-mu2,-a2
)
*df(dd(xi,-mu1,-c1,nu2,-b2),xi(nu1));
mu1 nu1 mu2 nu2
as16 := aa (x1)*dd (xi)*df(aa (x2),x2 )
a1 b1 mu2 a2 c2
nu2 nu1
*df(dd (xi),xi )
mu1 c1 b2
as17:=-aa(x1,mu1,-a1)*aa(x2,mu2,-a2)*df(dd(xi,-mu1,-c1,-mu2,-c2),xi(nu
1),xi(nu2))
*dd(xi,nu1,-b1,nu2,-b2);
mu1 mu2 nu1 nu2
as17 := - aa (x1)*aa (x2)*dd (xi)
a1 a2 b1 b2
nu1 nu2
*df(dd (xi),xi ,xi )
mu1 c1 mu2 c2
as18:=-aa(x1,mu1,-a1)*aa(x2,mu2,-a2)*df(dd(xi,-mu1,-c1,nu2,-b2),xi(nu1
))
*df(dd(xi,nu1,-b1,-mu2,-c2),xi(nu2));
mu1 mu2
as18 := - aa (x1)*aa (x2)
a1 a2
nu2 nu1
*df(dd (xi),xi )
mu1 c1 b2
nu1 nu2
*df(dd (xi),xi )
b1 mu2 c2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% building of the gluon contribution to gluon self-energy %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
es:=es1*for i:=1:18 sum mkid(as,i);
a1 b1 c1 a2 b2 c2 2 mu1 mu2
es := a *a *g *( - aa (x1)*aa (x2)
a1 a2
nu1 nu2
*dd (xi)
b1 b2
nu1 nu2 mu1
*df(dd (xi),xi ,xi ) - aa (x1)
mu1 c1 mu2 c2 a1
mu2 nu2 nu1
*aa (x2)*df(dd (xi),xi )
a2 mu1 c1 b2
nu1 nu2 mu1
*df(dd (xi),xi ) - aa (x1)
b1 mu2 c2 a1
nu2 nu1 mu2
*aa (x2)*dd (xi)
b2 b1 a2
nu1 nu2 mu1
*df(dd (xi),xi ,xi ) - aa (x1)
mu1 c1 mu2 c2 a1
nu2 nu1
*aa (x2)*df(dd (xi),xi )
b2 mu1 c1 mu2 a2
nu1 mu2 nu2 mu1
*df(dd (xi),xi ) + aa (x1)
b1 c2 a1
nu1 mu2 nu2
*dd (xi)*df(aa (x2),x2 )
b1 mu2 a2 c2
nu2 nu1 mu1
*df(dd (xi),xi ) + aa (x1)
mu1 c1 b2 a1
nu1 nu2 mu2 nu2
*dd (xi)*df(aa (x2),x2 )
b1 b2 c2
nu1 nu1
*df(dd (xi),xi ) - aa (x1)
mu1 c1 mu2 a2 b1
mu2 nu2
*aa (x2)*dd (xi)
a2 mu1 a1 b2
mu1 nu1 nu2 nu1
*df(dd (xi),xi ,xi ) - aa (x1)
c1 mu2 c2 b1
mu2 nu2
*aa (x2)*df(dd (xi),xi )
a2 mu1 a1 mu2 c2
mu1 nu2 nu1 nu1
*df(dd (xi),xi ) - aa (x1)
c1 b2 b1
nu2
*aa (x2)*dd (xi)
b2 mu1 a1 mu2 a2
mu1 mu2 nu1 nu2 nu1
*df(dd (xi),xi ,xi ) - aa (x1)
c1 c2 b1
nu2 nu2
*aa (x2)*df(dd (xi),xi )
b2 mu1 a1 mu2 c2
mu1 mu2 nu1 nu1
*df(dd (xi),xi ) + aa (x1)
c1 a2 b1
mu2 nu2
*dd (xi)*df(aa (x2),x2 )
mu1 a1 mu2 a2 c2
mu1 nu2 nu1 nu1
*df(dd (xi),xi ) + aa (x1)
c1 b2 b1
nu2 mu2 nu2
*dd (xi)*df(aa (x2),x2 )
mu1 a1 b2 c2
mu1 nu1 mu2
*df(dd (xi),xi ) - aa (x2)
c1 mu2 a2 a2
nu2 mu1 nu1
*dd (xi)*df(aa (x1),x1 )
mu1 a1 b2 c1
nu1 nu2 mu2
*df(dd (xi),xi ) - aa (x2)
b1 mu2 c2 a2
nu1 nu2 mu1 nu1
*dd (xi)*df(aa (x1),x1 )
b1 b2 c1
nu2 nu2
*df(dd (xi),xi ) - aa (x2)
mu1 a1 mu2 c2 b2
mu1 nu1
*dd (xi)*df(aa (x1),x1 )
mu1 a1 mu2 a2 c1
nu1 mu2 nu2 nu2
*df(dd (xi),xi ) - aa (x2)
b1 c2 b2
nu1 mu1 nu1
*dd (xi)*df(aa (x1),x1 )
b1 mu2 a2 c1
mu2 nu2
*df(dd (xi),xi ) + dd (xi)
mu1 a1 c2 mu1 a1 mu2 a2
nu1 nu2 mu1 nu1
*dd (xi)*df(aa (x1),x1 )
b1 b2 c1
mu2 nu2 nu2
*df(aa (x2),x2 ) + dd (xi)
c2 mu1 a1 b2
nu1 mu1 nu1
*dd (xi)*df(aa (x1),x1 )
b1 mu2 a2 c1
mu2 nu2
*df(aa (x2),x2 ))
c2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Are some terms identical ? %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
es:=canonical es;
a1 a2 b1 b2 c1 c2 2 mu1 mu2
es := a *a *g *( - aa (x1)*aa (x2)
b2 a1
nu1 nu2
*dd (xi)
c1 a2
nu1 nu2 mu1
*df(dd (xi),xi ,xi ) + aa (x1)
mu1 c2 mu2 b1 b2
mu2 nu1 nu2
*aa (x2)*dd (xi)
a1 c1 a2
mu2 nu1 mu1
*df(dd (xi),xi ,xi ) + aa (x1)
mu1 c2 nu2 b1 b2
mu2 nu1 nu2
*aa (x2)*dd (xi)
a1 c1 a2
mu1 nu2 mu1
*df(dd (xi),xi ,xi ) - aa (x1)
nu1 c2 mu2 b1 b2
mu2 nu1 nu2
*aa (x2)*dd (xi)
a1 c1 a2
mu1 mu2 mu1
*df(dd (xi),xi ,xi ) + aa (x1)
nu1 c2 nu2 b1 b2
mu2 nu2
*aa (x2)*df(dd (xi),xi )
a1 mu1 c2 b1 nu1
nu2 mu1
*df(dd (xi),xi ) - aa (x1)
nu1 c1 mu2 a2 b2
mu2 nu2
*aa (x2)*df(dd (xi),xi )
a1 mu1 c2 b1 nu1
mu2 mu1
*df(dd (xi),xi ) - aa (x1)
nu1 c1 nu2 a2 b2
mu2 mu1
*aa (x2)*df(dd (xi),xi )
a1 nu1 c1 nu2 a2
nu1 mu1
*df(dd (xi),xi ) + aa (x1)
c2 mu2 b1 nu2 b2
mu2 mu1
*aa (x2)*df(dd (xi),xi )
a1 nu1 c1 nu2 a2
nu1 nu2 mu2 mu1
*df(dd (xi),xi ) - aa (x1)
c2 b1 b2
mu2 nu1
*dd (xi)*df(aa (x2),x2 )
c1 a1 nu1 a2 nu2
mu2 mu1
*df(dd (xi),xi ) + aa (x1)
mu1 c2 nu2 b1 b2
mu2 nu1
*dd (xi)*df(aa (x2),x2 )
c1 a1 nu1 a2 nu2
mu1 mu1
*df(dd (xi),xi ) + aa (x1)
mu2 c2 nu2 b1 b2
mu2 nu1 nu2 nu1
*dd (xi)*df(aa (x2),x2 )
c1 a1 a2
mu2 mu1
*df(dd (xi),xi ) - aa (x1)
mu1 c2 nu2 b1 b2
mu2 nu1 nu2 nu1
*dd (xi)*df(aa (x2),x2 )
c1 a1 a2
mu1 mu1
*df(dd (xi),xi ) + aa (x2)
mu2 c2 nu2 b1 b2
mu2 nu1
*dd (xi)*df(aa (x1),x1 )
a1 c1 mu2 a2 nu2
nu1 mu1
*df(dd (xi),xi ) - aa (x2)
nu2 b1 mu1 c2 b2
mu2 nu1
*dd (xi)*df(aa (x1),x1 )
a1 c1 mu2 a2 nu2
mu1 mu1
*df(dd (xi),xi ) - aa (x2)
nu2 b1 nu1 c2 b2
mu2 nu1 nu2 mu2
*dd (xi)*df(aa (x1),x1 )
a1 c1 a2
nu1 mu1
*df(dd (xi),xi ) + aa (x2)
nu2 b1 mu1 c2 b2
mu2 nu1 nu2 mu2
*dd (xi)*df(aa (x1),x1 )
a1 c1 a2
mu1 mu1 mu2
*df(dd (xi),xi ) - dd (xi)
nu2 b1 nu1 c2 b2 a1
nu1 nu2 mu1
*dd (xi)*df(aa (x1),x1 )
c1 a2 nu1 c2
nu2 mu1 mu2
*df(aa (x2),x2 ) + dd (xi)
mu2 b1 b2 a1
nu1 nu2 mu1
*dd (xi)*df(aa (x1),x1 )
c1 a2 nu1 c2
mu2
*df(aa (x2),x2 ))
nu2 b1
length es;
18
% no simplification
tensor dc;
t
% new tensor
make_tensor_belong_space(dc,mink);
mink
% belongs to Minkowski space
make_partic_tens(rho,metric);
t
% "rho" is a metric tensor
make_tensor_belong_space(rho,internal);
internal
% in the internal space
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% rewriting rule and subsequent simplification %
% dd(mu1,mu2,a,b)=>rho(a,b)*dc(mu1,mu2) %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
ddrule:={dd({~xi},~a,~b,~c,~d)=>rho(b,d)*dc({xi},a,c)};
~a ~b ~c ~d b d a c
ddrule := {dd (~xi) => rho *dc (xi)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% simplification after application of the rule %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
es:=(es where ddrule);
a1 a2 b1 b2 c1 c2 2 mu1 mu2
es := a *a *g *( - aa (x1)*aa (x2)
b2 a1
nu1 nu2 nu1 nu2
*dc (xi)*df(dc (xi),xi ,xi )*rho
mu1 mu2 a2 c1
mu1 mu2 nu1 nu2
*rho + aa (x1)*aa (x2)*dc (xi)
b1 c2 b2 a1
mu2 nu1
*df(dc (xi),xi ,xi )*rho *rho +
mu1 nu2 a2 c1 b1 c2
mu1 mu2 nu1 nu2
aa (x1)*aa (x2)*dc (xi)
b2 a1
mu1 nu2
*df(dc (xi),xi ,xi )*rho *rho -
nu1 mu2 a2 c1 b1 c2
mu1 mu2 nu1 nu2
aa (x1)*aa (x2)*dc (xi)
b2 a1
mu1 mu2
*df(dc (xi),xi ,xi )*rho *rho +
nu1 nu2 a2 c1 b1 c2
mu1 mu2 nu2
aa (x1)*aa (x2)*df(dc (xi),xi )
b2 a1 mu1 nu1
nu2
*df(dc (xi),xi )*rho *rho -
nu1 mu2 a2 c1 b1 c2
mu1 mu2 nu2
aa (x1)*aa (x2)*df(dc (xi),xi )
b2 a1 mu1 nu1
mu2
*df(dc (xi),xi )*rho *rho -
nu1 nu2 a2 c1 b1 c2
mu1 mu2 mu1
aa (x1)*aa (x2)*df(dc (xi),xi )
b2 a1 nu1 nu2
nu1
*df(dc (xi),xi )*rho *rho +
mu2 nu2 a2 c1 b1 c2
mu1 mu2 mu1
aa (x1)*aa (x2)*df(dc (xi),xi )
b2 a1 nu1 nu2
nu1 nu2 mu2
*df(dc (xi),xi )*rho *rho -
a2 c1 b1 c2
mu1 mu2 nu1
aa (x1)*dc (xi)*df(aa (x2),x2 )
b2 nu1 a2 nu2
mu2
*df(dc (xi),xi )*rho *rho +
mu1 nu2 a1 c1 b1 c2
mu1 mu2 nu1
aa (x1)*dc (xi)*df(aa (x2),x2 )
b2 nu1 a2 nu2
mu1
*df(dc (xi),xi )*rho *rho +
mu2 nu2 a1 c1 b1 c2
mu1 mu2 nu1 nu2 nu1
aa (x1)*dc (xi)*df(aa (x2),x2 )
b2 a2
mu2
*df(dc (xi),xi )*rho *rho -
mu1 nu2 a1 c1 b1 c2
mu1 mu2 nu1 nu2 nu1
aa (x1)*dc (xi)*df(aa (x2),x2 )
b2 a2
mu1
*df(dc (xi),xi )*rho *rho +
mu2 nu2 a1 c1 b1 c2
mu1 mu2 nu1
aa (x2)*dc (xi)*df(aa (x1),x1 )
b2 mu2 a2 nu2
nu1
*df(dc (xi),xi )*rho *rho -
nu2 mu1 a1 c1 b1 c2
mu1 mu2 nu1
aa (x2)*dc (xi)*df(aa (x1),x1 )
b2 mu2 a2 nu2
mu1
*df(dc (xi),xi )*rho *rho -
nu2 nu1 a1 c1 b1 c2
mu1 mu2 nu1 nu2 mu2
aa (x2)*dc (xi)*df(aa (x1),x1 )
b2 a2
nu1
*df(dc (xi),xi )*rho *rho +
nu2 mu1 a1 c1 b1 c2
mu1 mu2 nu1 nu2 mu2
aa (x2)*dc (xi)*df(aa (x1),x1 )
b2 a2
mu1
*df(dc (xi),xi )*rho *rho -
nu2 nu1 a1 c1 b1 c2
mu1 mu2 nu1 nu2 mu1
dc (xi)*dc (xi)*df(aa (x1),x1 )
nu1 c2
nu2
*df(aa (x2),x2 )*rho *rho +
mu2 b1 a1 b2 a2 c1
mu1 mu2 nu1 nu2 mu1
dc (xi)*dc (xi)*df(aa (x1),x1 )
nu1 c2
mu2
*df(aa (x2),x2 )*rho *rho )
nu2 b1 a1 b2 a2 c1
%
es:=canonical es;
a1 a2 b1 b2 2 mu1 mu2
es := a *a *g *( - aa (x1)*aa (x2)
a1 a2 b1 b2
nu1 nu2 nu1 nu2
*dc (xi)*df(dc (xi),xi ,xi ) +
mu1 mu2
mu1 mu2 nu1 nu2
aa (x1)*aa (x2)*dc (xi)
b1 b2
mu2 nu1 mu1
*df(dc (xi),xi ,xi ) + aa (x1)
mu1 nu2 b1
mu2 nu1 nu2
*aa (x2)*dc (xi)
b2
mu1 nu2 mu1
*df(dc (xi),xi ,xi ) - aa (x1)
nu1 mu2 b1
mu2 nu1 nu2
*aa (x2)*dc (xi)
b2
mu1 mu2 mu1
*df(dc (xi),xi ,xi ) + aa (x1)
nu1 nu2 b1
mu2 nu1
*aa (x2)*df(dc (xi),xi )
b2 mu1 nu2
nu1 mu1 mu2
*df(dc (xi),xi ) - aa (x1)*aa (x2)
nu2 mu2 b1 b2
nu1 mu2
*df(dc (xi),xi )*df(dc (xi),xi ) -
mu1 nu2 nu2 nu1
mu1 mu2 nu2
aa (x1)*aa (x2)*df(dc (xi),xi )
b1 b2 nu1 mu2
nu1 nu2 mu1 mu1 mu2
*df(dc (xi),xi ) + aa (x1)*aa (x2)
b1 b2
mu2 nu1 nu2 mu1
*df(dc (xi),xi )*df(dc (xi),xi ) +
nu1 nu2
mu1 mu2 nu1
aa (x1)*dc (xi)*df(aa (x2),x2 )
b1 nu1 b2 nu2
mu2 mu1 mu2 nu1
*df(dc (xi),xi ) - aa (x1)*dc (xi)
mu1 nu2 b1
mu1
*df(aa (x2),x2 )*df(dc (xi),xi ) -
nu1 b2 nu2 mu2 nu2
mu1 mu2 nu1 nu2 nu1
aa (x1)*dc (xi)*df(aa (x2),x2 )
b1 b2
mu2 mu1 mu2 nu1
*df(dc (xi),xi ) + aa (x1)*dc (xi)
mu1 nu2 b1
nu2 nu1 mu1
*df(aa (x2),x2 )*df(dc (xi),xi ) -
b2 mu2 nu2
mu1 mu2 nu1
aa (x2)*dc (xi)*df(aa (x1),x1 )
b1 mu2 b2 nu2
nu1 mu1 mu2 nu1
*df(dc (xi),xi ) + aa (x2)*dc (xi)
nu2 mu1 b1
mu1
*df(aa (x1),x1 )*df(dc (xi),xi ) +
mu2 b2 nu2 nu2 nu1
mu1 mu2 nu1 nu2 mu2
aa (x2)*dc (xi)*df(aa (x1),x1 )
b1 b2
nu1 mu1 mu2 nu1
*df(dc (xi),xi ) - aa (x2)*dc (xi)
nu2 mu1 b1
nu2 mu2 mu1
*df(aa (x1),x1 )*df(dc (xi),xi ) -
b2 nu2 nu1
mu1 mu2 nu1 nu2 mu1
dc (xi)*dc (xi)*df(aa (x1),x1 )
nu1 b1
nu2 mu1 mu2 nu1 nu2
*df(aa (x2),x2 ) + dc (xi)*dc (xi)
mu2 b2
mu1 mu2
*df(aa (x1),x1 )*df(aa (x2),x2 ))
nu1 b1 nu2 b2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Particular gauge: %
% case of Fermi gauge : dc(mu1,mu2)=g(mu1,mu2)*dc %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
make_partic_tens(delta,delta);
t
% delta tenseur defined with name "delta"
% eta tenseur introduced with name "eta":
make_partic_tens(eta,eta);
t
make_tensor_belong_space(eta,mink);
mink
% rule for the choice of gauge:
dcrule:={dc({~xi},~a,~c)=>eta(a,c)*dc(xi)};
~a ~c a c xi
dcrule := {dc (~xi) => eta *dc }
% rewriting of the expression
res:=(es where dcrule);
a1 a2 b1 b2 2 mu1 mu2
res := a *a *g *( - aa (x1)*aa (x2)
a1 a2 b1 b2
mu1 mu2 nu1 nu2
*dc(xi)*df(dc(xi),xi ,xi )*eta *eta +
nu1 nu2
mu1 mu2
aa (x1)*aa (x2)*dc(xi)
b1 b2
mu1 nu2 nu1 nu2
*df(dc(xi),xi ,xi )*eta *eta +
mu2 nu1
mu1 mu2
aa (x1)*aa (x2)*dc(xi)
b1 b2
mu2 nu1 nu1 nu2
*df(dc(xi),xi ,xi )*eta *eta -
mu1 nu2
mu1 mu2
aa (x1)*aa (x2)*dc(xi)
b1 b2
nu1 nu2 nu1 nu2
*df(dc(xi),xi ,xi )*eta *eta -
mu1 mu2
mu1 mu2 nu1
aa (x1)*aa (x2)*delta *df(dc(xi),xi )
b1 b2 mu1 nu2
mu2 mu1
*df(dc(xi),xi )*eta + aa (x1)
nu1 nu2 b1
mu2 nu1
*aa (x2)*delta *df(dc(xi),xi )
b2 mu1 nu2
nu1 mu1
*df(dc(xi),xi )*eta + aa (x1)
mu2 nu2 b1
mu2 mu1 mu2
*aa (x2)*df(dc(xi),xi )*df(dc(xi),xi )
b2
nu1 nu2 mu1 mu2
*eta *eta - aa (x1)*aa (x2)
nu1 nu2 b1 b2
mu1 nu2 nu1 nu2
*df(dc(xi),xi )*df(dc(xi),xi )*eta *eta
mu2 nu1
mu1
- aa (x1)*dc(xi)*df(aa (x2),x2 )
b1 nu1 b2 nu2
mu1 mu2 nu1 mu1
*df(dc(xi),xi )*eta *eta + aa (x1)
mu2 nu2 b1
mu2
*dc(xi)*df(aa (x2),x2 )*df(dc(xi),xi )
nu1 b2 nu2
mu2 nu1 mu1
*eta *eta + aa (x1)*dc(xi)
mu1 nu2 b1
nu2 nu1 mu1
*df(aa (x2),x2 )*df(dc(xi),xi )*eta
b2 mu2 nu2
mu2 nu1 mu1
*eta - aa (x1)*dc(xi)
b1
nu2 nu1 mu2
*df(aa (x2),x2 )*df(dc(xi),xi )*eta
b2 mu1 nu2
mu2 nu1 mu1
*eta + aa (x2)*dc(xi)
b1
mu1
*df(aa (x1),x1 )*df(dc(xi),xi )*eta
mu2 b2 nu2 nu1 nu2
mu2 nu1 mu1
*eta - aa (x2)*dc(xi)
b1
nu1
*df(aa (x1),x1 )*df(dc(xi),xi )*eta
mu2 b2 nu2 mu1 nu2
mu2 nu1 mu1
*eta - aa (x2)*dc(xi)
b1
nu2 mu2 mu1
*df(aa (x1),x1 )*df(dc(xi),xi )*eta
b2 nu1 nu2
mu2 nu1 mu1
*eta + aa (x2)*dc(xi)
b1
nu2 mu2 nu1
*df(aa (x1),x1 )*df(dc(xi),xi )*eta
b2 mu1 nu2
mu2 nu1 2 mu1
*eta - (dc(xi)) *df(aa (x1),x1 )
nu1 b1
nu2 mu1 mu2 nu1 nu2
*df(aa (x2),x2 )*eta *eta +
mu2 b2
2 mu1
(dc(xi)) *df(aa (x1),x1 )
nu1 b1
mu2 mu1 mu2 nu1 nu2
*df(aa (x2),x2 )*eta *eta )
nu2 b2
% simplification
res:=canonical res;
a1 a2 b1 b2 2 mu1
res := a *a *g *( - aa (x1)*aa (x2)
a1 a2 b1 mu1 b2
mu2 mu1
*dc(xi)*df(dc(xi),xi ,xi ) - aa (x1)
mu2 b1
mu2 mu1 mu2
*aa (x2)*dc(xi)*df(dc(xi),xi ,xi )*dm + 2
b2
mu1 mu2
*aa (x1)*aa (x2)*dc(xi)
b1 b2
mu1 mu2 mu1 mu2
*df(dc(xi),xi ,xi ) + aa (x1)*aa (x2)
b1 b2
mu1 mu2 mu1
*df(dc(xi),xi )*df(dc(xi),xi )*dm - aa (x1)
b1
mu2 mu1 mu2
*aa (x2)*df(dc(xi),xi )*df(dc(xi),xi ) -
b2
mu1
aa (x1)*dc(xi)*df(aa (x2),x2 )
b1 mu1 b2 mu2
mu2 mu1
*df(dc(xi),xi ) + aa (x1)*dc(xi)
b1
mu2 mu1 mu2
*df(aa (x2),x2 )*df(dc(xi),xi ) +
b2
mu1
aa (x2)*dc(xi)*df(aa (x1),x1 )
b1 mu1 b2 mu2
mu2 mu1
*df(dc(xi),xi ) - aa (x2)*dc(xi)
b1
mu2 mu1 mu2 2
*df(aa (x1),x1 )*df(dc(xi),xi ) + (dc(xi))
b2
mu1 mu2
*df(aa (x1),x1 )*df(aa (x2),x2 ) -
b1 mu2 mu1 b2
2 mu1
(dc(xi)) *df(aa (x1),x1 )
b1 mu2
mu1
*df(aa (x2),x2 ))
mu2 b2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% last rewriting rule: %
% second derivative of dc(xi) with %
% respect to xi tensor is zero %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
dalrule:={df(dc(xi),xi(~a),xi(-~a))=>0};
xi ~a
dalrule := {df(dc ,xi ,xi ) => 0}
~a
res:=(res where dalrule);
a1 a2 b1 b2 2 mu1 mu2
res := a *a *g *( - aa (x1)*aa (x2)
a1 a2 b1 b2
mu1 mu2 mu1
*dc(xi)*df(dc(xi),xi ,xi )*dm + 2*aa (x1)
b1
mu2 mu1 mu2
*aa (x2)*dc(xi)*df(dc(xi),xi ,xi ) +
b2
mu1 mu2 mu1
aa (x1)*aa (x2)*df(dc(xi),xi )
b1 b2
mu2 mu1 mu2
*df(dc(xi),xi )*dm - aa (x1)*aa (x2)
b1 b2
mu1 mu2 mu1
*df(dc(xi),xi )*df(dc(xi),xi ) - aa (x1)
b1
mu2
*dc(xi)*df(aa (x2),x2 )*df(dc(xi),xi ) +
mu1 b2 mu2
mu1 mu2 mu1
aa (x1)*dc(xi)*df(aa (x2),x2 )
b1 b2
mu2 mu1
*df(dc(xi),xi ) + aa (x2)*dc(xi)
b1
mu2
*df(aa (x1),x1 )*df(dc(xi),xi ) -
mu1 b2 mu2
mu1 mu2 mu1
aa (x2)*dc(xi)*df(aa (x1),x1 )
b1 b2
mu2 2 mu1
*df(dc(xi),xi ) + (dc(xi)) *df(aa (x1),x1 )
b1 mu2
mu2 2
*df(aa (x2),x2 ) - (dc(xi))
mu1 b2
mu1 mu1
*df(aa (x1),x1 )*df(aa (x2),x2 ))
b1 mu2 mu2 b2
canonical res - res;
0
% gives 0
length res;
10
dm:=4;
dm := 4
% particularization to 4-dimensional Minkowski space
res4:=res;
a1 a2 b1 b2 2 mu1 mu2
res4 := a *a *g *( - 2*aa (x1)*aa (x2)
a1 a2 b1 b2
mu1 mu2 mu1
*dc(xi)*df(dc(xi),xi ,xi ) + 3*aa (x1)
b1
mu2 mu1 mu2
*aa (x2)*df(dc(xi),xi )*df(dc(xi),xi ) -
b2
mu1
aa (x1)*dc(xi)*df(aa (x2),x2 )
b1 mu1 b2 mu2
mu2 mu1
*df(dc(xi),xi ) + aa (x1)*dc(xi)
b1
mu2 mu1 mu2
*df(aa (x2),x2 )*df(dc(xi),xi ) +
b2
mu1
aa (x2)*dc(xi)*df(aa (x1),x1 )
b1 mu1 b2 mu2
mu2 mu1
*df(dc(xi),xi ) - aa (x2)*dc(xi)
b1
mu2 mu1 mu2 2
*df(aa (x1),x1 )*df(dc(xi),xi ) + (dc(xi))
b2
mu1 mu2
*df(aa (x1),x1 )*df(aa (x2),x2 ) -
b1 mu2 mu1 b2
2 mu1
(dc(xi)) *df(aa (x1),x1 )
b1 mu2
mu1
*df(aa (x2),x2 ))
mu2 b2
length res4;
8
% 8 is the correct number of terms.
end;
3: quit;
Quitting