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r36/xlog/DUMMY.LOG
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2011-09-02 18:13:33
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REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ... % test of DUMMY package version 1.0 running in REDUCE 3.5 % DATE: 15 February 1994 % Authors: A. Dresse <adresse@ulb.ac.be> % H. Caprasse <caprasse@vm1.ulg.ac.be> % % Case of commuting operator: % operator co1,co2; % declare dummy indices % first syntax : base <name> % dummy_base dv; dv % dummy indices are dv1, dv2, dv3, ... exp := co2(dv2)*co2(dv2)$ c_exp := canonical(exp); 2 c_exp := co2(dv1) exp := dv2*co2(dv2)*co2(dv2)$ c_exp := canonical(exp); 2 c_exp := co2(dv1) *dv1 exp := c_exp * co1(dv3); 2 exp := co1(dv3)*co2(dv1) *dv1 c_exp := canonical(exp); 2 c_exp := co1(dv2)*co2(dv1) *dv1 % aliases for dummy indices % dummy_names i,j,k; t canonical(c_exp); 2 co1(j)*co2(i) *i % remove dummy_names clear_dummy_names; t % Case of anticommuting operators % operator ao1, ao2; anticom ao1, ao2; % product of anticommuting operators with FREE indices a_exp := ao1(s1)*ao1(s2) - ao1(s2)*ao1(s1); a_exp := ao1(s1)*ao1(s2) - ao1(s2)*ao1(s1) a_exp := canonical(a_exp); a_exp := 2*ao1(s1)*ao1(s2) % the indices are summed upon, i.e. are DUMMY indices a_exp := ao1(dv1)*ao1(dv2)$ canonical(a_exp); 0 a_exp := ao1(dv1)*ao1(dv2) - ao1(dv2)*ao1(dv1); a_exp := ao1(dv1)*ao1(dv2) - ao1(dv2)*ao1(dv1) a_exp := canonical(a_exp); a_exp := 0 a_exp := ao1(dv2,dv3)*ao2(dv1,dv2)$ a_exp := canonical(a_exp); a_exp := ao1(dv2,dv1)*ao2(dv1,dv3) a_exp := ao1(dv1)*ao1(dv3)*ao2(dv3)*ao2(dv1)$ a_exp := canonical(a_exp); a_exp := - ao1(dv1)*ao1(dv2)*ao2(dv1)*ao2(dv2) % Case of non commuting operators % operator no1, no2, no3; noncom no1, no2, no3; n_exp := no3(dv2)*no2(dv3)*no1(dv1) + no3(dv3)*no2(dv1)*no1(dv2) + no3(dv1)*no2(dv2)*no1(dv3); n_exp := no3(dv1)*no2(dv2)*no1(dv3) + no3(dv2)*no2(dv3)*no1(dv1) + no3(dv3)*no2(dv1)*no1(dv2) n_exp:=canonical n_exp; n_exp := 3*no3(dv3)*no2(dv2)*no1(dv1) % Case of mixed commutation properties % *** % The example below displays a restriction of the package i.e % The non commuting operators are ASSUMED to COMMUTE with the % anticommuting operators. % *** exp := co1(dv1)*ao1(dv2,dv1,dv4)*no1(dv1,dv5)*co2(dv3)*ao1(dv1,dv3); exp := co1(dv1)*co2(dv3)*(ao1(dv2,dv1,dv4)*no1(dv1,dv5)*ao1(dv1,dv3)) canonical(exp); - co1(dv1)*co2(dv2)*ao1(dv2,dv1)*ao1(dv4,dv1,dv3)*no1(dv5,dv1) exp := c_exp * a_exp * no3(dv2)*no2(dv3)*no1(dv1); 2 exp := - co1(dv2)*co2(dv1) *dv1*ao1(dv1)*ao1(dv2)*ao2(dv1)*ao2(dv2)*no3(dv2) *no2(dv3)*no1(dv1) can_exp := canonical(exp); 2 can_exp := - co1(dv2)*co2(dv1) *dv1*ao1(dv1)*ao1(dv2)*ao2(dv1)*ao2(dv2) *no3(dv2)*no2(dv3)*no1(dv1) % Case of internal symmetries of operators % operator as1, as2; antisymmetric as1, as2; clear_dummy_base ; t dummy_base s; s % First a non trivial expression: asc_exp:=as1(s1,s2)*as1(s1,s3)*as1(s3,s4)*co1(s3)*co1(s4)+ 2*as1(s1,s2)*as1(s1,s3)*as1(s3,s4)*co1(s2)*co1(s4)$ canonical asc_exp; - as1(s1,s2)*as1(s1,s3)*as1(s3,s4)*co1(s3)*co1(s4) % Second simple illustrative examples: clear_dummy_base; t dummy_base dv; dv exp1 := ao1(dv1)*ao1(dv2)$ canonical(exp1); 0 exp2 := as1(dv1,dv2)$ canonical(exp2); 0 canonical(exp1*exp2); - as1(dv1,dv2)*ao1(dv1)*ao1(dv2) canonical(as1(dv1,dv2)*as2(dv2,dv1)); - as1(dv1,dv2)*as2(dv1,dv2) operator ss1, ss2; symmetric ss1, ss2; exp := ss1(dv1,dv2)*ss2(dv1,dv2) - ss1(dv2,dv3)*ss2(dv2,dv3); exp := ss1(dv1,dv2)*ss2(dv1,dv2) - ss1(dv2,dv3)*ss2(dv2,dv3) canonical(exp); 0 exp := as1(dv1,dv2)*as1(dv3,dv4)*as1(dv1,dv4); exp := as1(dv1,dv2)*as1(dv1,dv4)*as1(dv3,dv4) canonical(exp); 0 % Indeed the result is equal to half the sum given below: % exp + sub(dv2 = dv3, dv3 = dv2, dv1 = dv4, dv4 = dv1, exp); 0 exp1 := as2(dv3,dv2)*as1(dv3,dv4)*as1(dv1,dv2)*as1(dv1,dv4); exp1 := - as1(dv1,dv2)*as1(dv1,dv4)*as1(dv3,dv4)*as2(dv2,dv3) canonical(exp1); as1(dv1,dv2)*as1(dv1,dv3)*as1(dv3,dv4)*as2(dv2,dv4) exp2 := as2(dv1,dv4)*as1(dv1,dv3)*as1(dv2,dv4)*as1(dv2,dv3); exp2 := as1(dv1,dv3)*as1(dv2,dv3)*as1(dv2,dv4)*as2(dv1,dv4) canonical(exp2); as1(dv1,dv2)*as1(dv1,dv3)*as1(dv3,dv4)*as2(dv2,dv4) canonical(exp1-exp2); 0 % Indeed: % exp2 - sub(dv1 = dv3, dv2 = dv1, dv3 = dv4, dv4 = dv2, exp1); 0 % Declare the internal symmetry properties of the Riemann tensor % operator r; symtree (r, {!+, {!-, 1, 2}, {!-, 3, 4}}); canonical( r(x,y,z,t) ); - r(t,z,x,y) canonical( r(x,y,t,z) ); r(t,z,x,y) canonical( r(t,z,y,x) ); - r(t,z,x,y) exp := r(dv1, dv2, dv3, dv4) * r(dv2, dv1, dv4, dv3)$ canonical(exp); 2 r(dv4,dv3,dv2,dv1) exp := r(dv1, dv2, dv3, dv4) * r(dv1, dv3, dv2, dv4)$ canonical(exp); r(dv4,dv2,dv3,dv1)*r(dv4,dv3,dv2,dv1) dummy_names i,j,k,l; t canonical(exp); r(l,j,k,i)*r(l,k,j,i) exp := r(i,j,k,l)*ao1(i,j)*ao1(k,l); exp := r(i,j,k,l)*ao1(i,j)*ao1(k,l) canonical(exp); 0 exp := r(k,i,l,j)*as1(k,i)*as1(k,j); exp := as1(i,k)*as1(j,k)*r(k,i,l,j) canonical(exp); - as1(i,j)*as1(i,k)*r(l,j,k,i) clear_dummy_names; t clear_dummy_base; t exp := co1(dv3)$ c_exp := canonical(exp); c_exp := co1(dv3) end; (TIME: dummy 420 420)