/** @file exam_clifford.cpp * * Here we test GiNaC's Clifford algebra objects. */ /* * GiNaC Copyright (C) 1999-2008 Johannes Gutenberg University Mainz, Germany * * This program is free software; you can redistribute it and/or modify * it under the terms of the GNU General Public License as published by * the Free Software Foundation; either version 2 of the License, or * (at your option) any later version. * * This program is distributed in the hope that it will be useful, * but WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the * GNU General Public License for more details. * * You should have received a copy of the GNU General Public License * along with this program; if not, write to the Free Software * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA */ #include #include "ginac.h" using namespace std; using namespace GiNaC; const numeric half(1, 2); static unsigned check_equal(const ex &e1, const ex &e2) { ex e = normal(e1 - e2); if (!e.is_zero()) { clog << "(" << e1 << ") - (" << e2 << ") erroneously returned " << e << " instead of 0" << endl; return 1; } return 0; } static unsigned check_equal_simplify(const ex &e1, const ex &e2) { ex e = normal(simplify_indexed(e1) - e2); if (!e.is_zero()) { clog << "simplify_indexed(" << e1 << ") - (" << e2 << ") erroneously returned " << e << " instead of 0" << endl; return 1; } return 0; } static unsigned check_equal_lst(const ex & e1, const ex & e2) { for (unsigned int i = 0; i < e1.nops(); i++) { ex e = e1.op(i) - e2.op(i); if (!e.normal().is_zero()) { clog << "(" << e1 << ") - (" << e2 << ") erroneously returned " << e << " instead of 0 (in the entry " << i << ")" << endl; return 1; } } return 0; } static unsigned check_equal_simplify_term(const ex & e1, const ex & e2, idx & mu) { ex e = expand_dummy_sum(normal(simplify_indexed(e1) - e2), true); for (int j=0; j<4; j++) { ex esub = e.subs( is_a(mu) ? lst ( mu == idx(j, mu.get_dim()), ex_to(mu).toggle_variance() == idx(j, mu.get_dim()) ) : lst(mu == idx(j, mu.get_dim())) ); if (!(canonicalize_clifford(esub).is_zero())) { clog << "simplify_indexed(" << e1 << ") - (" << e2 << ") erroneously returned " << canonicalize_clifford(esub) << " instead of 0 for mu=" << j << endl; return 1; } } return 0; } static unsigned check_equal_simplify_term2(const ex & e1, const ex & e2) { ex e = expand_dummy_sum(normal(simplify_indexed(e1) - e2), true); if (!(canonicalize_clifford(e).is_zero())) { clog << "simplify_indexed(" << e1 << ") - (" << e2 << ") erroneously returned " << canonicalize_clifford(e) << " instead of 0" << endl; return 1; } return 0; } static unsigned clifford_check1() { // checks general identities and contractions unsigned result = 0; symbol dim("D"); varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), rho(symbol("rho"), dim); ex e; e = dirac_ONE() * dirac_ONE(); result += check_equal(e, dirac_ONE()); e = dirac_ONE() * dirac_gamma(mu) * dirac_ONE(); result += check_equal(e, dirac_gamma(mu)); e = dirac_gamma(varidx(2, dim)) * dirac_gamma(varidx(1, dim)) * dirac_gamma(varidx(1, dim)) * dirac_gamma(varidx(2, dim)); result += check_equal(e, dirac_ONE()); e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(nu.toggle_variance()) * dirac_gamma(mu.toggle_variance()); result += check_equal_simplify(e, pow(dim, 2) * dirac_ONE()); e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(mu.toggle_variance()) * dirac_gamma(nu.toggle_variance()); result += check_equal_simplify(e, 2*dim*dirac_ONE()-pow(dim, 2)*dirac_ONE()); e = dirac_gamma(nu.toggle_variance()) * dirac_gamma(rho.toggle_variance()) * dirac_gamma(mu) * dirac_gamma(rho) * dirac_gamma(nu); e = e.simplify_indexed().collect(dirac_gamma(mu)); result += check_equal(e, pow(2 - dim, 2).expand() * dirac_gamma(mu)); return result; } static unsigned clifford_check2() { // checks identities relating to gamma5 unsigned result = 0; symbol dim("D"); varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim); ex e; e = dirac_gamma(mu) * dirac_gamma5() + dirac_gamma5() * dirac_gamma(mu); result += check_equal(e, 0); e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma5() + dirac_gamma(mu); result += check_equal(e, 0); return result; } static unsigned clifford_check3() { // checks traces unsigned result = 0; symbol dim("D"), m("m"), q("q"), l("l"), ldotq("ldotq"); varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), rho(symbol("rho"), dim), sig(symbol("sig"), dim), kap(symbol("kap"), dim), lam(symbol("lam"), dim); ex e; e = dirac_gamma(mu); result += check_equal(dirac_trace(e), 0); e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho); result += check_equal(dirac_trace(e), 0); e = dirac_gamma5() * dirac_gamma(mu); result += check_equal(dirac_trace(e), 0); e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu); result += check_equal(dirac_trace(e), 0); e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho); result += check_equal(dirac_trace(e), 0); scalar_products sp; sp.add(q, q, pow(q, 2)); sp.add(l, l, pow(l, 2)); sp.add(l, q, ldotq); e = pow(m, 2) * dirac_slash(q, dim) * dirac_slash(q, dim); e = dirac_trace(e).simplify_indexed(sp); result += check_equal(e, 4*pow(m, 2)*pow(q, 2)); // cyclicity without gamma5 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) - dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(mu); e = dirac_trace(e); result += check_equal(e, 0); e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap) * dirac_gamma(lam) - dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap) * dirac_gamma(lam) * dirac_gamma(mu); e = dirac_trace(e).expand(); result += check_equal(e, 0); // cyclicity of gamma5 * S_4 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) - dirac_gamma(sig) * dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho); e = dirac_trace(e); result += check_equal(e, 0); // non-cyclicity of order D-4 of gamma5 * S_6 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap) * dirac_gamma(mu.toggle_variance()) + dim * dirac_gamma5() * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap); e = dirac_trace(e).simplify_indexed(); e = (e / (dim - 4)).normal(); result += check_equal(e, 8 * I * lorentz_eps(nu.replace_dim(4), rho.replace_dim(4), sig.replace_dim(4), kap.replace_dim(4))); // one-loop vacuum polarization in QED e = dirac_gamma(mu) * (dirac_slash(l, dim) + dirac_slash(q, 4) + m * dirac_ONE()) * dirac_gamma(mu.toggle_variance()) * (dirac_slash(l, dim) + m * dirac_ONE()); e = dirac_trace(e).simplify_indexed(sp); result += check_equal(e, 4*((2-dim)*l*l + (2-dim)*ldotq + dim*m*m).expand()); e = dirac_slash(q, 4) * (dirac_slash(l, dim) + dirac_slash(q, 4) + m * dirac_ONE()) * dirac_slash(q, 4) * (dirac_slash(l, dim) + m * dirac_ONE()); e = dirac_trace(e).simplify_indexed(sp); result += check_equal(e, 4*(2*ldotq*ldotq + q*q*ldotq - q*q*l*l + q*q*m*m).expand()); // stuff that had problems in the past ex prop = dirac_slash(q, dim) - m * dirac_ONE(); e = dirac_slash(l, dim) * dirac_gamma5() * dirac_slash(l, dim) * prop; e = dirac_trace(dirac_slash(q, dim) * e) - dirac_trace(m * e) - dirac_trace(prop * e); result += check_equal(e, 0); e = (dirac_gamma5() + dirac_ONE()) * dirac_gamma5(); e = dirac_trace(e); result += check_equal(e, 4); // traces with multiple representation labels e = dirac_ONE(0) * dirac_ONE(1) / 16; result += check_equal(dirac_trace(e, 0), dirac_ONE(1) / 4); result += check_equal(dirac_trace(e, 1), dirac_ONE(0) / 4); result += check_equal(dirac_trace(e, 2), e); result += check_equal(dirac_trace(e, lst(0, 1)), 1); e = dirac_gamma(mu, 0) * dirac_gamma(mu.toggle_variance(), 1) * dirac_gamma(nu, 0) * dirac_gamma(nu.toggle_variance(), 1); result += check_equal_simplify(dirac_trace(e, 0), 4 * dim * dirac_ONE(1)); result += check_equal_simplify(dirac_trace(e, 1), 4 * dim * dirac_ONE(0)); // Fails with new tinfo mechanism because the order of gamme matrices with different rl depends on luck. // TODO: better check. //result += check_equal_simplify(dirac_trace(e, 2), canonicalize_clifford(e)); // e will be canonicalized by the calculation of the trace result += check_equal_simplify(dirac_trace(e, lst(0, 1)), 16 * dim); return result; } static unsigned clifford_check4() { // simplify_indexed()/dirac_trace() cross-checks unsigned result = 0; symbol dim("D"); varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), rho(symbol("rho"), dim), sig(symbol("sig"), dim), lam(symbol("lam"), dim); ex e, t1, t2; e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(mu.toggle_variance()); t1 = dirac_trace(e).simplify_indexed(); t2 = dirac_trace(e.simplify_indexed()); result += check_equal((t1 - t2).expand(), 0); e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(mu.toggle_variance()) * dirac_gamma(lam); t1 = dirac_trace(e).simplify_indexed(); t2 = dirac_trace(e.simplify_indexed()); result += check_equal((t1 - t2).expand(), 0); e = dirac_gamma(sig) * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(nu.toggle_variance()) * dirac_gamma(mu.toggle_variance()); t1 = dirac_trace(e).simplify_indexed(); t2 = dirac_trace(e.simplify_indexed()); result += check_equal((t1 - t2).expand(), 0); e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(mu.toggle_variance()) * dirac_gamma(sig) * dirac_gamma(nu.toggle_variance()); t1 = dirac_trace(e).simplify_indexed(); t2 = dirac_trace(e.simplify_indexed()); result += check_equal((t1 - t2).expand(), 0); return result; } static unsigned clifford_check5() { // canonicalize_clifford() checks unsigned result = 0; symbol dim("D"); varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), lam(symbol("lam"), dim); ex e; e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu); result += check_equal(canonicalize_clifford(e), 2*dirac_ONE()*lorentz_g(mu, nu)); e = (dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(lam) + dirac_gamma(nu) * dirac_gamma(lam) * dirac_gamma(mu) + dirac_gamma(lam) * dirac_gamma(mu) * dirac_gamma(nu) - dirac_gamma(nu) * dirac_gamma(mu) * dirac_gamma(lam) - dirac_gamma(lam) * dirac_gamma(nu) * dirac_gamma(mu) - dirac_gamma(mu) * dirac_gamma(lam) * dirac_gamma(nu)) / 6 + lorentz_g(mu, nu) * dirac_gamma(lam) - lorentz_g(mu, lam) * dirac_gamma(nu) + lorentz_g(nu, lam) * dirac_gamma(mu) - dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(lam); result += check_equal(canonicalize_clifford(e), 0); return result; } /* We make two identical checks with metrics defined through a matrix in * the cases when used indexes have or have not variance. * To this end we recycle the code through the following macros */ template unsigned clifford_check6(const matrix &A) { unsigned result = 0; matrix A_symm(4,4), A2(4, 4); A_symm = A.add(A.transpose()).mul(half); A2 = A_symm.mul(A_symm); IDX v(symbol("v"), 4), nu(symbol("nu"), 4), mu(symbol("mu"), 4), psi(symbol("psi"),4), lam(symbol("lambda"), 4), xi(symbol("xi"), 4), rho(symbol("rho"),4); ex mu_TOGGLE = is_a(mu) ? ex_to(mu).toggle_variance() : mu; ex nu_TOGGLE = is_a(nu) ? ex_to(nu).toggle_variance() : nu; ex rho_TOGGLE = is_a(rho) ? ex_to(rho).toggle_variance() : rho; ex e, e1; /* checks general identities and contractions for clifford_unit*/ e = dirac_ONE(2) * clifford_unit(mu, A, 2) * dirac_ONE(2); result += check_equal(e, clifford_unit(mu, A, 2)); e = clifford_unit(IDX(2, 4), A) * clifford_unit(IDX(1, 4), A) * clifford_unit(IDX(1, 4), A) * clifford_unit(IDX(2, 4), A); result += check_equal(e, A(1, 1) * A(2, 2) * dirac_ONE()); e = clifford_unit(IDX(2, 4), A) * clifford_unit(IDX(1, 4), A) * clifford_unit(IDX(1, 4), A) * clifford_unit(IDX(2, 4), A); result += check_equal(e, A(1, 1) * A(2, 2) * dirac_ONE()); e = clifford_unit(nu, A) * clifford_unit(nu_TOGGLE, A); result += check_equal_simplify(e, A.trace() * dirac_ONE()); e = clifford_unit(nu, A) * clifford_unit(nu, A); result += check_equal_simplify(e, indexed(A_symm, sy_symm(), nu, nu) * dirac_ONE()); e = clifford_unit(nu, A) * clifford_unit(nu_TOGGLE, A) * clifford_unit(mu, A); result += check_equal_simplify(e, A.trace() * clifford_unit(mu, A)); e = clifford_unit(nu, A) * clifford_unit(mu, A) * clifford_unit(nu_TOGGLE, A); result += check_equal_simplify_term(e, 2 * indexed(A_symm, sy_symm(), nu_TOGGLE, mu) *clifford_unit(nu, A)-A.trace()*clifford_unit(mu, A), mu); e = clifford_unit(nu, A) * clifford_unit(nu_TOGGLE, A) * clifford_unit(mu, A) * clifford_unit(mu_TOGGLE, A); result += check_equal_simplify(e, pow(A.trace(), 2) * dirac_ONE()); e = clifford_unit(mu, A) * clifford_unit(nu, A) * clifford_unit(nu_TOGGLE, A) * clifford_unit(mu_TOGGLE, A); result += check_equal_simplify(e, pow(A.trace(), 2) * dirac_ONE()); e = clifford_unit(mu, A) * clifford_unit(nu, A) * clifford_unit(mu_TOGGLE, A) * clifford_unit(nu_TOGGLE, A); result += check_equal_simplify_term2(e, 2*indexed(A_symm, sy_symm(), nu_TOGGLE, mu_TOGGLE) * clifford_unit(mu, A) * clifford_unit(nu, A) - pow(A.trace(), 2)*dirac_ONE()); e = clifford_unit(mu_TOGGLE, A) * clifford_unit(nu, A) * clifford_unit(mu, A) * clifford_unit(nu_TOGGLE, A); result += check_equal_simplify_term2(e, 2*indexed(A_symm, nu, mu) * clifford_unit(mu_TOGGLE, A) * clifford_unit(nu_TOGGLE, A) - pow(A.trace(), 2)*dirac_ONE()); e = clifford_unit(nu_TOGGLE, A) * clifford_unit(rho_TOGGLE, A) * clifford_unit(mu, A) * clifford_unit(rho, A) * clifford_unit(nu, A); e = e.simplify_indexed().collect(clifford_unit(mu, A)); result += check_equal_simplify_term(e, 4* indexed(A_symm, sy_symm(), nu_TOGGLE, rho)*indexed(A_symm, sy_symm(), rho_TOGGLE, mu) *clifford_unit(nu, A) - 2*A.trace() * (clifford_unit(rho, A) * indexed(A_symm, sy_symm(), rho_TOGGLE, mu) + clifford_unit(nu, A) * indexed(A_symm, sy_symm(), nu_TOGGLE, mu)) + pow(A.trace(),2)* clifford_unit(mu, A), mu); e = clifford_unit(nu_TOGGLE, A) * clifford_unit(rho, A) * clifford_unit(mu, A) * clifford_unit(rho_TOGGLE, A) * clifford_unit(nu, A); e = e.simplify_indexed().collect(clifford_unit(mu, A)); result += check_equal_simplify_term(e, 4* indexed(A_symm, sy_symm(), nu_TOGGLE, rho)*indexed(A_symm, sy_symm(), rho_TOGGLE, mu) *clifford_unit(nu, A) - 2*A.trace() * (clifford_unit(rho, A) * indexed(A_symm, sy_symm(), rho_TOGGLE, mu) + clifford_unit(nu, A) * indexed(A_symm, sy_symm(), nu_TOGGLE, mu)) + pow(A.trace(),2)* clifford_unit(mu, A), mu); e = clifford_unit(mu, A) * clifford_unit(nu, A) + clifford_unit(nu, A) * clifford_unit(mu, A); result += check_equal(canonicalize_clifford(e), 2*dirac_ONE()*indexed(A_symm, sy_symm(), mu, nu)); e = (clifford_unit(mu, A) * clifford_unit(nu, A) * clifford_unit(lam, A) + clifford_unit(nu, A) * clifford_unit(lam, A) * clifford_unit(mu, A) + clifford_unit(lam, A) * clifford_unit(mu, A) * clifford_unit(nu, A) - clifford_unit(nu, A) * clifford_unit(mu, A) * clifford_unit(lam, A) - clifford_unit(lam, A) * clifford_unit(nu, A) * clifford_unit(mu, A) - clifford_unit(mu, A) * clifford_unit(lam, A) * clifford_unit(nu, A)) / 6 + indexed(A_symm, sy_symm(), mu, nu) * clifford_unit(lam, A) - indexed(A_symm, sy_symm(), mu, lam) * clifford_unit(nu, A) + indexed(A_symm, sy_symm(), nu, lam) * clifford_unit(mu, A) - clifford_unit(mu, A) * clifford_unit(nu, A) * clifford_unit(lam, A); result += check_equal(canonicalize_clifford(e), 0); /* lst_to_clifford() and clifford_inverse() check*/ realsymbol s("s"), t("t"), x("x"), y("y"), z("z"); ex c = clifford_unit(nu, A, 1); e = lst_to_clifford(lst(t, x, y, z), mu, A, 1) * lst_to_clifford(lst(1, 2, 3, 4), c); e1 = clifford_inverse(e); result += check_equal_simplify_term2((e*e1).simplify_indexed(), dirac_ONE(1)); /* lst_to_clifford() and clifford_to_lst() check for vectors*/ e = lst(t, x, y, z); result += check_equal_lst(clifford_to_lst(lst_to_clifford(e, c), c, false), e); result += check_equal_lst(clifford_to_lst(lst_to_clifford(e, c), c, true), e); /* lst_to_clifford() and clifford_to_lst() check for pseudovectors*/ e = lst(s, t, x, y, z); result += check_equal_lst(clifford_to_lst(lst_to_clifford(e, c), c, false), e); result += check_equal_lst(clifford_to_lst(lst_to_clifford(e, c), c, true), e); /* Moebius map (both forms) checks for symmetric metrics only */ matrix M1(2, 2), M2(2, 2); c = clifford_unit(nu, A); e = clifford_moebius_map(0, dirac_ONE(), dirac_ONE(), 0, lst(t, x, y, z), A); /* this is just the inversion*/ M1 = 0, dirac_ONE(), dirac_ONE(), 0; e1 = clifford_moebius_map(M1, lst(t, x, y, z), A); /* the inversion again*/ result += check_equal_lst(e, e1); e1 = clifford_to_lst(clifford_inverse(lst_to_clifford(lst(t, x, y, z), mu, A)), c); result += check_equal_lst(e, e1); e = clifford_moebius_map(dirac_ONE(), lst_to_clifford(lst(1, 2, 3, 4), nu, A), 0, dirac_ONE(), lst(t, x, y, z), A); /*this is just a shift*/ M2 = dirac_ONE(), lst_to_clifford(lst(1, 2, 3, 4), c), 0, dirac_ONE(); e1 = clifford_moebius_map(M2, lst(t, x, y, z), c); /* the same shift*/ result += check_equal_lst(e, e1); result += check_equal(e, lst(t+1, x+2, y+3, z+4)); /* Check the group law for Moebius maps */ e = clifford_moebius_map(M1, ex_to(e1), c); /*composition of M1 and M2*/ e1 = clifford_moebius_map(M1.mul(M2), lst(t, x, y, z), c); /* the product M1*M2*/ result += check_equal_lst(e, e1); return result; } static unsigned clifford_check7(const ex & G, const symbol & dim) { // checks general identities and contractions unsigned result = 0; varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), rho(symbol("rho"), dim), psi(symbol("psi"),dim), lam(symbol("lambda"), dim), xi(symbol("xi"), dim); ex e; clifford unit = ex_to(clifford_unit(mu, G)); ex scalar = unit.get_metric(varidx(0, dim), varidx(0, dim)); e = dirac_ONE() * dirac_ONE(); result += check_equal(e, dirac_ONE()); e = dirac_ONE() * clifford_unit(mu, G) * dirac_ONE(); result += check_equal(e, clifford_unit(mu, G)); e = clifford_unit(varidx(2, dim), G) * clifford_unit(varidx(1, dim), G) * clifford_unit(varidx(1, dim), G) * clifford_unit(varidx(2, dim), G); result += check_equal(e, dirac_ONE()*pow(scalar, 2)); e = clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(nu.toggle_variance(), G) * clifford_unit(mu.toggle_variance(), G); result += check_equal_simplify(e, pow(dim*scalar, 2) * dirac_ONE()); e = clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(mu.toggle_variance(), G) * clifford_unit(nu.toggle_variance(), G); result += check_equal_simplify(e, (2*dim - pow(dim, 2))*pow(scalar,2)*dirac_ONE()); e = clifford_unit(nu.toggle_variance(), G) * clifford_unit(rho.toggle_variance(), G) * clifford_unit(mu, G) * clifford_unit(rho, G) * clifford_unit(nu, G); e = e.simplify_indexed().collect(clifford_unit(mu, G)); result += check_equal(e, pow(scalar*(dim-2), 2).expand() * clifford_unit(mu, G)); // canonicalize_clifford() checks, only for symmetric metrics if (ex_to(ex_to(ex_to(clifford_unit(mu, G)).get_metric()).get_symmetry()).has_symmetry()) { e = clifford_unit(mu, G) * clifford_unit(nu, G) + clifford_unit(nu, G) * clifford_unit(mu, G); result += check_equal(canonicalize_clifford(e), 2*dirac_ONE()*unit.get_metric(nu, mu)); e = (clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(lam, G) + clifford_unit(nu, G) * clifford_unit(lam, G) * clifford_unit(mu, G) + clifford_unit(lam, G) * clifford_unit(mu, G) * clifford_unit(nu, G) - clifford_unit(nu, G) * clifford_unit(mu, G) * clifford_unit(lam, G) - clifford_unit(lam, G) * clifford_unit(nu, G) * clifford_unit(mu, G) - clifford_unit(mu, G) * clifford_unit(lam, G) * clifford_unit(nu, G)) / 6 + unit.get_metric(mu, nu) * clifford_unit(lam, G) - unit.get_metric(mu, lam) * clifford_unit(nu, G) + unit.get_metric(nu, lam) * clifford_unit(mu, G) - clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(lam, G); result += check_equal(canonicalize_clifford(e), 0); } else { e = clifford_unit(mu, G) * clifford_unit(nu, G) + clifford_unit(nu, G) * clifford_unit(mu, G); result += check_equal(canonicalize_clifford(e), dirac_ONE()*(unit.get_metric(mu, nu) + unit.get_metric(nu, mu))); e = (clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(lam, G) + clifford_unit(nu, G) * clifford_unit(lam, G) * clifford_unit(mu, G) + clifford_unit(lam, G) * clifford_unit(mu, G) * clifford_unit(nu, G) - clifford_unit(nu, G) * clifford_unit(mu, G) * clifford_unit(lam, G) - clifford_unit(lam, G) * clifford_unit(nu, G) * clifford_unit(mu, G) - clifford_unit(mu, G) * clifford_unit(lam, G) * clifford_unit(nu, G)) / 6 + half * (unit.get_metric(mu, nu) + unit.get_metric(nu, mu)) * clifford_unit(lam, G) - half * (unit.get_metric(mu, lam) + unit.get_metric(lam, mu)) * clifford_unit(nu, G) + half * (unit.get_metric(nu, lam) + unit.get_metric(lam, nu)) * clifford_unit(mu, G) - clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(lam, G); result += check_equal(canonicalize_clifford(e), 0); } return result; } unsigned exam_clifford() { unsigned result = 0; cout << "examining clifford objects" << flush; result += clifford_check1(); cout << '.' << flush; result += clifford_check2(); cout << '.' << flush; result += clifford_check3(); cout << '.' << flush; result += clifford_check4(); cout << '.' << flush; result += clifford_check5(); cout << '.' << flush; // anticommuting, symmetric examples result += clifford_check6(ex_to(diag_matrix(lst(-1, 1, 1, 1)))); result += clifford_check6(ex_to(diag_matrix(lst(-1, 1, 1, 1))));; cout << '.' << flush; result += clifford_check6(ex_to(diag_matrix(lst(-1, -1, -1, -1))))+clifford_check6(ex_to(diag_matrix(lst(-1, -1, -1, -1))));; cout << '.' << flush; result += clifford_check6(ex_to(diag_matrix(lst(-1, 1, 1, -1))))+clifford_check6(ex_to(diag_matrix(lst(-1, 1, 1, -1))));; cout << '.' << flush; result += clifford_check6(ex_to(diag_matrix(lst(-1, 0, 1, -1))))+clifford_check6(ex_to(diag_matrix(lst(-1, 0, 1, -1))));; cout << '.' << flush; result += clifford_check6(ex_to(diag_matrix(lst(-3, 0, 2, -1))))+clifford_check6(ex_to(diag_matrix(lst(-3, 0, 2, -1))));; cout << '.' << flush; realsymbol s("s"), t("t"); // symbolic entries in matric result += clifford_check6(ex_to(diag_matrix(lst(-1, 1, s, t))))+clifford_check6(ex_to(diag_matrix(lst(-1, 1, s, t))));; cout << '.' << flush; matrix A(4, 4); A = 1, 0, 0, 0, // anticommuting, not symmetric, Tr=0 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0; result += clifford_check6(A)+clifford_check6(A);; cout << '.' << flush; A = 1, 0, 0, 0, // anticommuting, not symmetric, Tr=2 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0; result += clifford_check6(A)+clifford_check6(A);; cout << '.' << flush; A = 1, 0, 0, 0, // not anticommuting, symmetric, Tr=0 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0; result += clifford_check6(A)+clifford_check6(A);; cout << '.' << flush; A = 1, 0, 0, 0, // not anticommuting, symmetric, Tr=2 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0; result += clifford_check6(A)+clifford_check6(A);; cout << '.' << flush; A = 1, 1, 0, 0, // not anticommuting, not symmetric, Tr=4 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1; result += clifford_check6(A)+clifford_check6(A);; cout << '.' << flush; symbol dim("D"); result += clifford_check7(minkmetric(), dim); cout << '.' << flush; varidx chi(symbol("chi"), dim), xi(symbol("xi"), dim); result += clifford_check7(delta_tensor(xi, chi), dim); cout << '.' << flush; result += clifford_check7(lorentz_g(xi, chi), dim); cout << '.' << flush; result += clifford_check7(indexed(-2*minkmetric(), sy_symm(), xi, chi), dim); cout << '.' << flush; result += clifford_check7(-2*delta_tensor(xi, chi), dim); cout << '.' << flush; return result; } int main(int argc, char** argv) { return exam_clifford(); }