Artifact 74dfed976a37644b3f5926baaa682df29afb1cc2173b19d8349d2fcb70bf300e:
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r37/packages/assist/selfgra.rlg
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[f2fda60abd]
at
2011-09-02 18:13:33
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— Some historical releases purely for archival purposes
git-svn-id: https://svn.code.sf.net/p/reduce-algebra/code/trunk/historical@1375 2bfe0521-f11c-4a00-b80e-6202646ff360 (user: arthurcnorman@users.sourceforge.net, size: 47097) [annotate] [blame] [check-ins using] [more...]
- Executable file
r38/packages/assist/selfgra.rlg
— part of check-in
[f2fda60abd]
at
2011-09-02 18:13:33
on branch master
— Some historical releases purely for archival purposes
git-svn-id: https://svn.code.sf.net/p/reduce-algebra/code/trunk/historical@1375 2bfe0521-f11c-4a00-b80e-6202646ff360 (user: arthurcnorman@users.sourceforge.net, size: 47097) [annotate] [blame] [check-ins using]
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