/** @file exam_clifford.cpp * * Here we test GiNaC's Clifford algebra objects. */ /* * GiNaC Copyright (C) 1999-2004 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., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA */ #include "exams.h" static unsigned check_equal(const ex &e1, const ex &e2) { ex e = 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 = 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 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)); result += check_equal_simplify(dirac_trace(e, 2), e); 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; } 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); ex e, e1; 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 = clifford_unit(varidx(2, 4), G) * clifford_unit(varidx(1, 4), G) * clifford_unit(varidx(1, 4), G) * clifford_unit(varidx(2, 4), G); result += check_equal(e, A(1, 1) * A(2, 2) * dirac_ONE()); e = clifford_unit(nu, G) * clifford_unit(nu.toggle_variance(), G); 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, 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, 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, G) * clifford_unit(nu.toggle_variance(), G) * clifford_unit(mu, G) * clifford_unit(mu.toggle_variance(), G); 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); 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.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()); 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(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)); // 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); // lst_to_clifford() and clifford_inverse() check symbol 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); e1 = clifford_inverse(e); result += check_equal((e*e1).simplify_indexed().normal(), dirac_ONE()); return result; } static unsigned clifford_check7() { // 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 G = minkmetric(); 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()); 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, 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*dirac_ONE() - pow(dim, 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(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); return result; } unsigned exam_clifford() { unsigned result = 0; cout << "examining clifford objects" << flush; clog << "----------clifford objects:" << endl; 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; matrix A(4, 4); A = -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1; result += clifford_check6(A); cout << '.' << flush; A = -1, 0, 0, 0, 0,-1, 0, 0, 0, 0,-1, 0, 0, 0, 0,-1; result += clifford_check6(A); cout << '.' << flush; A = -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0,-1; result += clifford_check6(A); cout << '.' << flush; A = -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,-1; result += clifford_check6(A); cout << '.' << flush; result += clifford_check7(); cout << '.' << flush; if (!result) { cout << " passed " << endl; clog << "(no output)" << endl; } else { cout << " failed " << endl; } return result; }