X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=check%2Fexam_clifford.cpp;h=06396e1a0fb5dd8b09011a85ad48c7a587b357b4;hp=2b28be4c5030636af3e55b83998f4d1fd513de0e;hb=4b70436fd4b5d1059636b34844a5c3737af628b2;hpb=2e13bfeb92ce55b3ee7e24cc6041f70c7a0eff4e diff --git a/check/exam_clifford.cpp b/check/exam_clifford.cpp index 2b28be4c..06396e1a 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) 1999-2004 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 @@ -44,14 +44,14 @@ static unsigned check_equal_simplify(const ex &e1, const ex &e2) return 0; } -static unsigned clifford_check1(void) +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 +72,15 @@ 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 @@ -94,22 +99,357 @@ static unsigned clifford_check2(void) return result; } -unsigned exam_clifford(void) +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_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; }