X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=check%2Fexam_clifford.cpp;h=53b57906557727df751f9bf6b3146f69b4514f9e;hp=06396e1a0fb5dd8b09011a85ad48c7a587b357b4;hb=20bd3aad103c4b97f899b29c059caf9e78f49dae;hpb=b60d73e12181182b5afd3d37363d99151628b92b diff --git a/check/exam_clifford.cpp b/check/exam_clifford.cpp index 06396e1a..53b57906 100644 --- a/check/exam_clifford.cpp +++ b/check/exam_clifford.cpp @@ -3,7 +3,7 @@ * Here we test GiNaC's Clifford algebra objects. */ /* - * GiNaC Copyright (C) 1999-2004 Johannes Gutenberg University Mainz, Germany + * GiNaC Copyright (C) 1999-2005 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 @@ -17,16 +17,18 @@ * * 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., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA + * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA */ #include "exams.h" +const numeric half(1, 2); + static unsigned check_equal(const ex &e1, const ex &e2) { - ex e = e1 - e2; + ex e = normal(e1 - e2); if (!e.is_zero()) { - clog << e1 << "-" << e2 << " erroneously returned " + clog << "(" << e1 << ") - (" << e2 << ") erroneously returned " << e << " instead of 0" << endl; return 1; } @@ -35,15 +37,55 @@ static unsigned check_equal(const ex &e1, const ex &e2) static unsigned check_equal_simplify(const ex &e1, const ex &e2) { - ex e = simplify_indexed(e1) - e2; + ex e = normal(simplify_indexed(e1) - e2); if (!e.is_zero()) { - clog << "simplify_indexed(" << e1 << ")-" << e2 << " erroneously returned " - << e << " instead of 0" << endl; + 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, varidx & mu) +{ + ex e = expand_dummy_sum(normal(simplify_indexed(e1) - e2), true); + + for (int j=0; j<4; j++) { + ex esub = e.subs(lst(mu == idx(j, mu.get_dim()), mu.toggle_variance() == 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 @@ -194,7 +236,9 @@ static unsigned clifford_check3() 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)); - result += check_equal_simplify(dirac_trace(e, 2), e); + // 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; @@ -262,105 +306,163 @@ static unsigned clifford_check5() return result; } + static unsigned clifford_check6(const matrix & A) { varidx 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 G = A; - - matrix A2(4, 4); - A2 = A.mul(A); + matrix A_symm(4,4), A2(4, 4); + A_symm = A.add(A.transpose()).mul(half); + A2 = A_symm.mul(A_symm); + ex e, e1; - + bool anticommuting = ex_to(clifford_unit(nu, A)).is_anticommuting(); int result = 0; // checks general identities and contractions for clifford_unit - e = dirac_ONE() * clifford_unit(mu, G) * dirac_ONE(); - result += check_equal(e, clifford_unit(mu, G)); + 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(varidx(2, 4), G) * clifford_unit(varidx(1, 4), G) - * clifford_unit(varidx(1, 4), G) * clifford_unit(varidx(2, 4), G); + e = clifford_unit(varidx(2, 4), A) * clifford_unit(varidx(1, 4), A) + * clifford_unit(varidx(1, 4), A) * clifford_unit(varidx(2, 4), A); result += check_equal(e, A(1, 1) * A(2, 2) * dirac_ONE()); - e = clifford_unit(nu, G) * clifford_unit(nu.toggle_variance(), G); + e = clifford_unit(nu, A) * clifford_unit(nu.toggle_variance(), A); result += check_equal_simplify(e, A.trace() * dirac_ONE()); - e = clifford_unit(nu, G) * clifford_unit(nu, G); - result += check_equal_simplify(e, indexed(G, sy_symm(), nu, nu) * 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, G) * clifford_unit(nu.toggle_variance(), G) * clifford_unit(mu, G); - result += check_equal_simplify(e, A.trace() * clifford_unit(mu, G)); + e = clifford_unit(nu, A) * clifford_unit(nu.toggle_variance(), A) * clifford_unit(mu, A); + result += check_equal_simplify(e, A.trace() * clifford_unit(mu, A)); - e = clifford_unit(nu, G) * clifford_unit(mu, G) * clifford_unit(nu.toggle_variance(), G); - result += check_equal_simplify(e, 2*indexed(G, sy_symm(), mu, mu)*clifford_unit(mu, G) - A.trace()*clifford_unit(mu, G)); + e = clifford_unit(nu, A) * clifford_unit(mu, A) * clifford_unit(nu.toggle_variance(), A); + if (anticommuting) + result += check_equal_simplify(e, 2*indexed(A_symm, sy_symm(), mu, mu)*clifford_unit(mu, A) - A.trace()*clifford_unit(mu, A)); + + result += check_equal_simplify_term(e, 2 * indexed(A_symm, sy_symm(), nu.toggle_variance(), mu) *clifford_unit(nu, A)-A.trace()*clifford_unit(mu, A), mu); - e = clifford_unit(nu, G) * clifford_unit(nu.toggle_variance(), G) - * clifford_unit(mu, G) * clifford_unit(mu.toggle_variance(), G); + e = clifford_unit(nu, A) * clifford_unit(nu.toggle_variance(), A) + * clifford_unit(mu, A) * clifford_unit(mu.toggle_variance(), A); result += check_equal_simplify(e, pow(A.trace(), 2) * dirac_ONE()); - e = clifford_unit(mu, G) * clifford_unit(nu, G) - * clifford_unit(nu.toggle_variance(), G) * clifford_unit(mu.toggle_variance(), G); + e = clifford_unit(mu, A) * clifford_unit(nu, A) + * clifford_unit(nu.toggle_variance(), A) * clifford_unit(mu.toggle_variance(), A); result += check_equal_simplify(e, pow(A.trace(), 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*A2.trace()*dirac_ONE() - pow(A.trace(), 2)*dirac_ONE()); + e = clifford_unit(mu, A) * clifford_unit(nu, A) + * clifford_unit(mu.toggle_variance(), A) * clifford_unit(nu.toggle_variance(), A); + if (anticommuting) + result += check_equal_simplify(e, 2*A2.trace()*dirac_ONE() - pow(A.trace(), 2)*dirac_ONE()); - e = clifford_unit(mu.toggle_variance(), G) * clifford_unit(nu, G) - * clifford_unit(mu, G) * clifford_unit(nu.toggle_variance(), G); - result += check_equal_simplify(e, 2*A2.trace()*dirac_ONE() - pow(A.trace(), 2)*dirac_ONE()); + result += check_equal_simplify_term2(e, 2*indexed(A_symm, sy_symm(), nu.toggle_variance(), mu.toggle_variance()) * clifford_unit(mu, A) * clifford_unit(nu, A) - pow(A.trace(), 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(A.trace(), 2)+4-4*A.trace()*indexed(A, mu, mu)) * clifford_unit(mu, G)); + e = clifford_unit(mu.toggle_variance(), A) * clifford_unit(nu, A) + * clifford_unit(mu, A) * clifford_unit(nu.toggle_variance(), A); + if (anticommuting) { + result += check_equal_simplify(e, 2*A2.trace()*dirac_ONE() - pow(A.trace(), 2)*dirac_ONE()); + e1 = remove_dirac_ONE(simplify_indexed(e)); + result += check_equal(e1, 2*A2.trace() - pow(A.trace(), 2)); + } - e = clifford_unit(nu.toggle_variance(), G) * clifford_unit(rho, G) - * clifford_unit(mu, G) * clifford_unit(rho.toggle_variance(), G) * clifford_unit(nu, G); - e = e.simplify_indexed().collect(clifford_unit(mu, G)); - result += check_equal(e, (pow(A.trace(), 2)+4-4*A.trace()*indexed(A, mu, mu))* clifford_unit(mu, G)); + result += check_equal_simplify_term2(e, 2*indexed(A_symm, nu, mu) * clifford_unit(mu.toggle_variance(), A) * clifford_unit(nu.toggle_variance(), A) - pow(A.trace(), 2)*dirac_ONE()); - // canonicalize_clifford() checks - 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()*indexed(G, sy_symm(), mu, nu)); - - 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 - + indexed(G, sy_symm(), mu, nu) * clifford_unit(lam, G) - - indexed(G, sy_symm(), mu, lam) * clifford_unit(nu, G) - + indexed(G, sy_symm(), nu, lam) * clifford_unit(mu, G) - - clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(lam, G); + e = clifford_unit(nu.toggle_variance(), A) * clifford_unit(rho.toggle_variance(), A) + * clifford_unit(mu, A) * clifford_unit(rho, A) * clifford_unit(nu, A); + e = e.simplify_indexed().collect(clifford_unit(mu, A)); + if (anticommuting) + result += check_equal(e, (4*indexed(A2, sy_symm(), mu, mu) - 4 * indexed(A_symm, sy_symm(), mu, mu)*A.trace() +pow(A.trace(), 2)) * clifford_unit(mu, A)); + + result += check_equal_simplify_term(e, 4* indexed(A_symm, sy_symm(), nu.toggle_variance(), rho)*indexed(A_symm, sy_symm(), rho.toggle_variance(), mu) *clifford_unit(nu, A) + - 2*A.trace() * (clifford_unit(rho, A) * indexed(A_symm, sy_symm(), rho.toggle_variance(), mu) + + clifford_unit(nu, A) * indexed(A_symm, sy_symm(), nu.toggle_variance(), mu)) + pow(A.trace(),2)* clifford_unit(mu, A), mu); + + e = clifford_unit(nu.toggle_variance(), A) * clifford_unit(rho, A) + * clifford_unit(mu, A) * clifford_unit(rho.toggle_variance(), A) * clifford_unit(nu, A); + e = e.simplify_indexed().collect(clifford_unit(mu, A)); + if (anticommuting) + result += check_equal(e, (4*indexed(A2, sy_symm(), mu, mu) - 4*indexed(A_symm, sy_symm(), mu, mu)*A.trace() +pow(A.trace(), 2))* clifford_unit(mu, A)); + + result += check_equal_simplify_term(e, 4* indexed(A_symm, sy_symm(), nu.toggle_variance(), rho)*indexed(A_symm, sy_symm(), rho.toggle_variance(), mu) *clifford_unit(nu, A) + - 2*A.trace() * (clifford_unit(rho, A) * indexed(A_symm, sy_symm(), rho.toggle_variance(), mu) + + clifford_unit(nu, A) * indexed(A_symm, sy_symm(), nu.toggle_variance(), 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 - symbol x("x"), y("y"), t("t"), z("z"); + realsymbol x("x"), y("y"), t("t"), z("z"); - e = lst_to_clifford(lst(t, x, y, z), mu, G) * lst_to_clifford(lst(1, 2, 3, 4), nu, G); + 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((e*e1).simplify_indexed().normal(), dirac_ONE()); + result += check_equal((e*e1).simplify_indexed(), dirac_ONE(1)); + + // 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() + +static unsigned clifford_check7(const ex & G, const symbol & dim) { // checks general identities and contractions unsigned result = 0; - symbol dim("D"); 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; + ex e, G_base; - ex G = minkmetric(); + if (is_a(G)) + G_base = G.op(0); + else + G_base = G; e = dirac_ONE() * dirac_ONE(); result += check_equal(e, dirac_ONE()); @@ -385,22 +487,38 @@ static unsigned clifford_check7() e = e.simplify_indexed().collect(clifford_unit(mu, G)); result += check_equal(e, pow(2 - dim, 2).expand() * clifford_unit(mu, G)); - // canonicalize_clifford() checks - 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()*indexed(G, sy_symm(), mu, nu)); - - 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 - + indexed(G, sy_symm(), mu, nu) * clifford_unit(lam, G) - - indexed(G, sy_symm(), mu, lam) * clifford_unit(nu, G) - + indexed(G, sy_symm(), 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); - + // 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()*indexed(G_base, sy_symm(), 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 + + indexed(G_base, sy_symm(), mu, nu) * clifford_unit(lam, G) + - indexed(G_base, sy_symm(), mu, lam) * clifford_unit(nu, G) + + indexed(G_base, sy_symm(), 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()*(indexed(G_base, mu, nu) + indexed(G_base, 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 * (indexed(G_base, mu, nu) + indexed(G_base, nu, mu)) * clifford_unit(lam, G) + - half * (indexed(G_base, mu, lam) + indexed(G_base, lam, mu)) * clifford_unit(nu, G) + + half * (indexed(G_base, nu, lam) + indexed(G_base, 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; } @@ -417,32 +535,52 @@ unsigned exam_clifford() 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)))); cout << '.' << flush; + 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)))); cout << '.' << flush; + result += 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)))); cout << '.' << flush; + + realsymbol s("s"), t("t"); // symbolic entries in matric + result += clifford_check6(ex_to(diag_matrix(lst(-1, 1, s, t)))); cout << '.' << flush; + matrix A(4, 4); - A = -1, 0, 0, 0, - 0, 1, 0, 0, - 0, 0, 1, 0, - 0, 0, 0, 1; + 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); cout << '.' << flush; - A = -1, 0, 0, 0, - 0,-1, 0, 0, - 0, 0,-1, 0, - 0, 0, 0,-1; + 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); cout << '.' << flush; - - A = -1, 0, 0, 0, - 0, 1, 0, 0, - 0, 0, 1, 0, - 0, 0, 0,-1; + + 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); cout << '.' << flush; - A = -1, 0, 0, 0, - 0, 0, 0, 0, - 0, 0, 1, 0, - 0, 0, 0,-1; + 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); cout << '.' << flush; - result += clifford_check7(); 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); cout << '.' << flush; + + symbol dim("D"); + result += clifford_check7(minkmetric(), dim); cout << '.' << flush; + + varidx chi(symbol("chi"), dim), xi(symbol("xi"), dim); + result += clifford_check7(lorentz_g(xi, chi), dim); cout << '.' << flush; if (!result) { cout << " passed " << endl;