X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=check%2Fexam_clifford.cpp;h=b067438bb1edbd310c0b942ed0cf84da4d6debd5;hp=8eee701337931fd94fab5e3f79c4d77812f0d034;hb=8cffcdf13d817a47f217f1a1043317d95969e070;hpb=d81773c9ebc817764485203f4892607a98667d48 diff --git a/check/exam_clifford.cpp b/check/exam_clifford.cpp index 8eee7013..b067438b 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) 1999-2001 Johannes Gutenberg University Mainz, Germany + * GiNaC Copyright (C) 1999-2019 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 +17,22 @@ * * 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" +#include "ginac.h" +using namespace GiNaC; + +#include +using namespace std; + +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; } @@ -35,23 +41,70 @@ 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 clifford_check1(void) +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); + varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), rho(symbol("rho"), dim); ex e; e = dirac_ONE() * dirac_ONE(); @@ -72,10 +125,15 @@ static unsigned clifford_check1(void) 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(void) +static unsigned clifford_check2() { // checks identities relating to gamma5 @@ -94,7 +152,7 @@ static unsigned clifford_check2(void) return result; } -static unsigned clifford_check3(void) +static unsigned clifford_check3() { // checks traces @@ -102,7 +160,7 @@ static unsigned clifford_check3(void) 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"), 4); + sig(symbol("sig"), dim), kap(symbol("kap"), dim), lam(symbol("lam"), dim); ex e; e = dirac_gamma(mu); @@ -149,45 +207,460 @@ static unsigned clifford_check3(void) // 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); + e = dirac_trace(e).simplify_indexed(); e = (e / (dim - 4)).normal(); - result += check_equal(e, 8 * eps0123(nu, rho, sig, kap)); + 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, dim) + m * dirac_ONE()) * + (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, dim) * - (dirac_slash(l, dim) + dirac_slash(q, dim) + m * dirac_ONE()) * - dirac_slash(q, dim) * + 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 gamma 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; } -unsigned exam_clifford(void) +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); - cout << "examining clifford objects" << flush; - clog << "----------clifford objects:" << endl; + 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 += clifford_check1(); cout << '.' << flush; - result += clifford_check2(); cout << '.' << flush; - result += clifford_check3(); cout << '.' << flush; + 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)); - if (!result) { - cout << " passed " << endl; - clog << "(no output)" << endl; + 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 */ + 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*/ + matrix 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*/ + matrix 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 (is_a(ex_to(clifford_unit(mu, G)).get_metric()) && + 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 { - cout << " failed " << endl; + 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; +} + +static unsigned clifford_check8() +{ + unsigned result = 0; + + realsymbol a("a"), b("b"), x("x"); + varidx mu(symbol("mu", "\\mu"), 1); + + ex e = clifford_unit(mu, diag_matrix({-1})), e0 = e.subs(mu==0); + result += ( exp(a*e0)*e0*e0 == -exp(e0*a) ) ? 0 : 1; + + ex P = color_T(idx(a,8))*color_T(idx(b,8))*(x*dirac_ONE()+sqrt(x-1)*e0); + ex P_prime = color_T(idx(a,8))*color_T(idx(b,8))*(x*dirac_ONE()-sqrt(x-1)*e0); + + result += check_equal(clifford_prime(P), P_prime); + result += check_equal(clifford_star(P), P); + result += check_equal(clifford_bar(P), P_prime); + + return result; +} + +static unsigned clifford_check9() +{ + unsigned result = 0; + + realsymbol a("a"), b("b"), x("x");; + varidx mu(symbol("mu", "\\mu"), 4), nu(symbol("nu", "\\nu"), 4); + + ex e = clifford_unit(mu, lorentz_g(mu, nu)); + ex e0 = e.subs(mu==0); + ex e1 = e.subs(mu==1); + ex e2 = e.subs(mu==2); + ex e3 = e.subs(mu==3); + ex one = dirac_ONE(); + + ex P = color_T(idx(a,8))*color_T(idx(b,8)) + *(x*one+sqrt(x-1)*e0+sqrt(x-2)*e0*e1 +sqrt(x-3)*e0*e1*e2 +sqrt(x-4)*e0*e1*e2*e3); + ex P_prime = color_T(idx(a,8))*color_T(idx(b,8)) + *(x*one-sqrt(x-1)*e0+sqrt(x-2)*e0*e1 -sqrt(x-3)*e0*e1*e2 +sqrt(x-4)*e0*e1*e2*e3); + ex P_star = color_T(idx(a,8))*color_T(idx(b,8)) + *(x*one+sqrt(x-1)*e0+sqrt(x-2)*e1*e0 +sqrt(x-3)*e2*e1*e0 +sqrt(x-4)*e3*e2*e1*e0); + ex P_bar = color_T(idx(a,8))*color_T(idx(b,8)) + *(x*one-sqrt(x-1)*e0+sqrt(x-2)*e1*e0 -sqrt(x-3)*e2*e1*e0 +sqrt(x-4)*e3*e2*e1*e0); + + result += check_equal(clifford_prime(P), P_prime); + result += check_equal(clifford_star(P), P_star); + result += check_equal(clifford_bar(P), P_bar); + + 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({-1, 1, 1, 1}))); + result += clifford_check6(ex_to(diag_matrix({-1, 1, 1, 1})));; cout << '.' << flush; + result += clifford_check6(ex_to(diag_matrix({-1, -1, -1, -1})))+clifford_check6(ex_to(diag_matrix({-1, -1, -1, -1})));; cout << '.' << flush; + result += clifford_check6(ex_to(diag_matrix({-1, 1, 1, -1})))+clifford_check6(ex_to(diag_matrix({-1, 1, 1, -1})));; cout << '.' << flush; + result += clifford_check6(ex_to(diag_matrix({-1, 0, 1, -1})))+clifford_check6(ex_to(diag_matrix({-1, 0, 1, -1})));; cout << '.' << flush; + result += clifford_check6(ex_to(diag_matrix({-3, 0, 2, -1})))+clifford_check6(ex_to(diag_matrix({-3, 0, 2, -1})));; cout << '.' << flush; + + realsymbol s("s"), t("t"); // symbolic entries in matrix + result += clifford_check6(ex_to(diag_matrix({-1, 1, s, t})))+clifford_check6(ex_to(diag_matrix({-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; + + result += clifford_check8(); cout << '.' << flush; + + result += clifford_check9(); cout << '.' << flush; + return result; } + +int main(int argc, char** argv) +{ + return exam_clifford(); +}