From: Jens Vollinga Date: Fri, 3 Jun 2005 18:32:05 +0000 (+0000) Subject: * Now two different simplification paths in clifford::contract_with(). X-Git-Tag: release_1-4-0~172 X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=commitdiff_plain;h=6dfb8aee92f97422e9c0e2b7aa4706ecf13cac84 * Now two different simplification paths in clifford::contract_with(). * Clifford now works with non-symmetric metric as well. * Several small corrections and update of tutorial and automatic checks. [V.Kisil] --- diff --git a/check/exam_clifford.cpp b/check/exam_clifford.cpp index 3a9c65db..75110bfa 100644 --- a/check/exam_clifford.cpp +++ b/check/exam_clifford.cpp @@ -22,11 +22,13 @@ #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(int i = 0; i++; i < e1.nops()) { + ex e = e1.op(i) - e2.op(i); + if (!e.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 @@ -262,106 +304,163 @@ 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 A_symm(4,4), A2(4, 4); + A_symm = A.add(A.transpose()).mul(half); + A2 = A_symm.mul(A_symm); - matrix A2(4, 4); - A2 = A.mul(A); 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(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(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, G) * clifford_unit(nu.toggle_variance(), 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, 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"); - ex c = clifford_unit(nu, G, 1); - e = lst_to_clifford(lst(t, x, y, z), mu, G, 1) * lst_to_clifford(lst(1, 2, 3, 4), c); + 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(1)); + result += check_equal_lst((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()); @@ -386,22 +485,38 @@ 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; } @@ -418,32 +533,52 @@ 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; diff --git a/doc/tutorial/ginac.texi b/doc/tutorial/ginac.texi index 232f883e..02d9e508 100644 --- a/doc/tutorial/ginac.texi +++ b/doc/tutorial/ginac.texi @@ -2567,6 +2567,47 @@ of a sum are consistent: @} @end example +@cindex @code{expand_dummy_sum()} +A dummy index summation like +@tex +$ a_i b^i$ +@end tex +@ifnottex +a.i b~i +@end ifnottex +can be expanded for indices with numeric +dimensions (e.g. 3) into the explicit sum like +@tex +$a_1b^1+a_2b^2+a_3b^3 $. +@end tex +@ifnottex +a.1 b~1 + a.2 b~2 + a.3 b~3. +@end ifnottex +This is performed by the function + +@example + ex expand_dummy_sum(const ex & e, bool subs_idx = false); +@end example + +which takes an expression @code{e} and returns the expanded sum for all +dummy indices with numeric dimensions. If the parameter @code{subs_idx} +is set to @code{true} then all substitutions are made by @code{idx} class +indices, i.e. without variance. In this case the above sum +@tex +$ a_i b^i$ +@end tex +@ifnottex +a.i b~i +@end ifnottex +will be expanded to +@tex +$a_1b_1+a_2b_2+a_3b_3 $. +@end tex +@ifnottex +a.1 b.1 + a.2 b.2 + a.3 b.3. +@end ifnottex + + @cindex @code{simplify_indexed()} @subsection Simplifying indexed expressions @@ -3189,27 +3230,51 @@ generators @end tex satisfying the identities @tex -$e_i e_j + e_j e_i = M(i, j) $ +$e_i e_j + e_j e_i = M(i, j) + M(j, i) $ @end tex @ifnottex -e~i e~j + e~j e~i = M(i, j) +e~i e~j + e~j e~i = M(i, j) + M(j, i) @end ifnottex -for some matrix (@code{metric}) -@math{M(i, j)}, which may be non-symmetric and containing symbolic -entries. Such generators are created by the function +for some bilinear form (@code{metric}) +@math{M(i, j)}, which may be non-symmetric (see arXiv:math.QA/9911180) +and contain symbolic entries. Such generators are created by the +function @example - ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0); + ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0, + bool anticommuting = false); @end example where @code{mu} should be a @code{varidx} class object indexing the -generators, @code{metr} defines the metric @math{M(i, j)} and can be +generators, an index @code{mu} with a numeric value may be of type +@code{idx} as well. +Parameter @code{metr} defines the metric @math{M(i, j)} and can be represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class -object, optional parameter @code{rl} allows to distinguish different -Clifford algebras (which will commute with each other). Note that the call -@code{clifford_unit(mu, minkmetric())} creates something very close to -@code{dirac_gamma(mu)}. The method @code{clifford::get_metric()} returns a -metric defining this Clifford number. +object. Optional parameter @code{rl} allows to distinguish different +Clifford algebras, which will commute with each other. The last +optional parameter @code{anticommuting} defines if the anticommuting +assumption (i.e. +@tex +$e_i e_j + e_j e_i = 0$) +@end tex +@ifnottex +e~i e~j + e~j e~i = 0) +@end ifnottex +will be used for contraction of Clifford units. If the @code{metric} is +supplied by a @code{matrix} object, then the value of +@code{anticommuting} is calculated automatically and the supplied one +will be ignored. One can overcome this by giving @code{metric} through +matrix wrapped into an @code{indexed} object. + +Note that the call @code{clifford_unit(mu, minkmetric())} creates +something very close to @code{dirac_gamma(mu)}, although +@code{dirac_gamma} have more efficient simplification mechanism. +@cindex @code{clifford::get_metric()} +The method @code{clifford::get_metric()} returns a metric defining this +Clifford number. +@cindex @code{clifford::is_anticommuting()} +The method @code{clifford::is_anticommuting()} returns the +@code{anticommuting} property of a unit. If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create the Clifford algebra units with a call like that @@ -3218,7 +3283,9 @@ the Clifford algebra units with a call like that ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j)); @end example -since this may yield some further automatic simplifications. +since this may yield some further automatic simplifications. Again, for a +metric defined through a @code{matrix} such a symmetry is detected +automatically. Individual generators of a Clifford algebra can be accessed in several ways. For example @@ -3243,7 +3310,8 @@ will produce four anti-commuting generators of a Clifford algebra with propertie $e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$. @end tex @ifnottex -@code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and @code{pow(e3, 2) = s}. +@code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and +@code{pow(e3, 2) = s}. @end ifnottex @cindex @code{lst_to_clifford()} @@ -3251,7 +3319,7 @@ A similar effect can be achieved from the function @example ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, - unsigned char rl = 0); + unsigned char rl = 0, bool anticommuting = false); ex lst_to_clifford(const ex & v, const ex & e); @end example @@ -3273,7 +3341,7 @@ $v^0 e_0 + v^1 e_1 + ... + v^n e_n$ with @samp{e.k} directly supplied in the second form of the procedure. In the first form the Clifford unit @samp{e.k} is generated by the call of -@code{clifford_unit(mu, metr, rl)}. The previous code may be rewritten +@code{clifford_unit(mu, metr, rl, anticommuting)}. The previous code may be rewritten with the help of @code{lst_to_clifford()} as follows @example @@ -3321,7 +3389,7 @@ $(e c_k + c_k e)/c_k^2$. If $c_k^2$ @ifnottex @samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)} @end ifnottex -is zero or is not a @code{numeric} for some @samp{k} +is zero or is not @code{numeric} for some @samp{k} then the method will be automatically changed to symbolic. The same effect is obtained by the assignment (@code{algebraic = false}) in the procedure call. @@ -3410,30 +3478,44 @@ expression by the function The function @code{canonicalize_clifford()} works for a generic Clifford algebra in a similar way as for Dirac gammas. -The last provided function is +The next provided function is @cindex @code{clifford_moebius_map()} @example ex clifford_moebius_map(const ex & a, const ex & b, const ex & c, const ex & d, const ex & v, const ex & G, - unsigned char rl = 0); + unsigned char rl = 0, bool anticommuting = false); ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, - unsigned char rl = 0); + unsigned char rl = 0, bool anticommuting = false); @end example It takes a list or vector @code{v} and makes the Moebius (conformal or linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines -the metric of the surrounding (pseudo-)Euclidean space. This can be a -matrix or a Clifford unit, in the later case the parameter @code{rl} is -ignored even if supplied. The returned value of this function is a list -of components of the resulting vector. - -LaTeX output for Clifford units looks like @code{\clifford[1]@{e@}^@{@{\nu@}@}}, -where @code{1} is the @code{representation_label} and @code{\nu} is the -index of the corresponding unit. This provides a flexible typesetting -with a suitable defintion of the @code{\clifford} command. For example, the -definition +the metric of the surrounding (pseudo-)Euclidean space. This can be an +indexed object, tensormetric, matrix or a Clifford unit, in the later +case the optional parameters @code{rl} and @code{anticommuting} are ignored +even if supplied. The returned value of this function is a list of +components of the resulting vector. + +@cindex @code{clifford_max_label()} +Finally the function + +@example +char clifford_max_label(const ex & e, bool ignore_ONE = false); +@end example + +can detect a presence of Clifford objects in the expression @code{e}: if +such objects are found it returns the maximal +@code{representation_label} of them, otherwise @code{-1}. The optional +parameter @code{ignore_ONE} indicates if @code{dirac_ONE} objects should +be ignored during the search. + +LaTeX output for Clifford units looks like +@code{\clifford[1]@{e@}^@{@{\nu@}@}}, where @code{1} is the +@code{representation_label} and @code{\nu} is the index of the +corresponding unit. This provides a flexible typesetting with a suitable +defintion of the @code{\clifford} command. For example, the definition @example \newcommand@{\clifford@}[1][]@{@} @end example diff --git a/ginac/clifford.cpp b/ginac/clifford.cpp index 99c5a65d..61007389 100644 --- a/ginac/clifford.cpp +++ b/ginac/clifford.cpp @@ -20,6 +20,8 @@ * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA */ +#include + #include "clifford.h" #include "ex.h" @@ -78,7 +80,7 @@ static ex default_metric() return m; } -clifford::clifford() : representation_label(0), metric(default_metric()) +clifford::clifford() : representation_label(0), metric(default_metric()), anticommuting(false) { tinfo_key = TINFO_clifford; } @@ -97,7 +99,7 @@ DEFAULT_CTOR(diracgammaR) /** Construct object without any indices. This constructor is for internal * use only. Use the dirac_ONE() function instead. * @see dirac_ONE */ -clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl), metric(0) +clifford::clifford(const ex & b, unsigned char rl, bool anticommut) : inherited(b), representation_label(rl), metric(0), anticommuting(anticommut) { tinfo_key = TINFO_clifford; } @@ -106,18 +108,18 @@ clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representatio * use only. Use the clifford_unit() or dirac_gamma() functions instead. * @see clifford_unit * @see dirac_gamma */ -clifford::clifford(const ex & b, const ex & mu, const ex & metr, unsigned char rl) : inherited(b, mu), representation_label(rl), metric(metr) +clifford::clifford(const ex & b, const ex & mu, const ex & metr, unsigned char rl, bool anticommut) : inherited(b, mu), representation_label(rl), metric(metr), anticommuting(anticommut) { GINAC_ASSERT(is_a(mu)); tinfo_key = TINFO_clifford; } -clifford::clifford(unsigned char rl, const ex & metr, const exvector & v, bool discardable) : inherited(not_symmetric(), v, discardable), representation_label(rl), metric(metr) +clifford::clifford(unsigned char rl, const ex & metr, bool anticommut, const exvector & v, bool discardable) : inherited(not_symmetric(), v, discardable), representation_label(rl), metric(metr), anticommuting(anticommut) { tinfo_key = TINFO_clifford; } -clifford::clifford(unsigned char rl, const ex & metr, std::auto_ptr vp) : inherited(not_symmetric(), vp), representation_label(rl), metric(metr) +clifford::clifford(unsigned char rl, const ex & metr, bool anticommut, std::auto_ptr vp) : inherited(not_symmetric(), vp), representation_label(rl), metric(metr), anticommuting(anticommut) { tinfo_key = TINFO_clifford; } @@ -132,6 +134,7 @@ clifford::clifford(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst n.find_unsigned("label", rl); representation_label = rl; n.find_ex("metric", metric, sym_lst); + n.find_bool("anticommuting", anticommuting); } void clifford::archive(archive_node & n) const @@ -139,6 +142,7 @@ void clifford::archive(archive_node & n) const inherited::archive(n); n.add_unsigned("label", representation_label); n.add_ex("metric", metric); + n.add_bool("anticommuting", anticommuting); } DEFAULT_UNARCHIVE(clifford) @@ -153,15 +157,29 @@ DEFAULT_ARCHIVING(diracgammaR) // functions overriding virtual functions from base classes ////////// -ex clifford::get_metric(const ex & i, const ex & j) const +ex clifford::get_metric(const ex & i, const ex & j, bool symmetrised) const { - return indexed(metric, symmetric2(), i, j); + if (is_a(metric)) { + if (symmetrised && !(ex_to(ex_to(metric).get_symmetry()).has_symmetry())) { + if (is_a(metric.op(0))) { + return indexed((ex_to(metric.op(0)).add(ex_to(metric.op(0)).transpose())).mul(numeric(1,2)), + symmetric2(), i, j); + } else { + return simplify_indexed(indexed(metric.op(0)*_ex1_2, i, j) + indexed(metric.op(0)*_ex1_2, j, i)); + } + } else { + return indexed(metric.op(0), ex_to(ex_to(metric).get_symmetry()), i, j); + } + } else { + // should not really happen since all constructors but clifford() make the metric an indexed object + return indexed(metric, i, j); + } } bool clifford::same_metric(const ex & other) const { if (is_a(other)) { - return get_metric().is_equal(ex_to(other).get_metric()); + return same_metric(ex_to(other).get_metric()); } else if (is_a(other)) { return get_metric(other.op(1), other.op(2)).is_equal(other); } else @@ -375,15 +393,16 @@ bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other * used in cliffordunit::contract_with(). */ static int find_same_metric(exvector & v, ex & c) { - for (int i=0; i(v[i]) && is_a(v[i]) - && ex_to(c).same_metric(v[i]) - && (ex_to(c.op(1)) == ex_to(v[i]).get_indices()[0] - || ex_to(c.op(1)).toggle_variance() == ex_to(v[i]).get_indices()[0])) { - return ++i; // next to found + for (size_t i=0; i(v[i]) && !is_a(v[i]) + && ((ex_to(c.op(1)) == ex_to(v[i]).get_indices()[0] + && ex_to(c.op(1)) == ex_to(v[i]).get_indices()[1]) + || (ex_to(c.op(1)).toggle_variance() == ex_to(v[i]).get_indices()[0] + && ex_to(c.op(1)).toggle_variance() == ex_to(v[i]).get_indices()[1]))) { + return i; // the index of the found } } - return 0; //nothing found + return -1; //nothing found } /** Contraction of a Clifford unit with something else. */ @@ -403,70 +422,88 @@ bool cliffordunit::contract_with(exvector::iterator self, exvector::iterator oth return false; // Find if a previous contraction produces the square of self - int prev_square = find_same_metric(v, self[0]); - varidx d((new symbol)->setflag(status_flags::dynallocated), ex_to(self->op(1)).get_dim()); - ex squared_metric = unit.get_metric(self->op(1), d) * unit.get_metric(d.toggle_variance(), other->op(1)); + int prev_square = find_same_metric(v, *self); + const varidx d((new symbol)->setflag(status_flags::dynallocated), ex_to(self->op(1)).get_dim()), + in1((new symbol)->setflag(status_flags::dynallocated), ex_to(self->op(1)).get_dim()), + in2((new symbol)->setflag(status_flags::dynallocated), ex_to(self->op(1)).get_dim()); + ex squared_metric; + if (prev_square > -1) + squared_metric = simplify_indexed(indexed(v[prev_square].op(0), in1, d) + * unit.get_metric(d.toggle_variance(), in2, true)).op(0); + + exvector::iterator before_other = other - 1; + const varidx & mu = ex_to(self->op(1)); + const varidx & mu_toggle = ex_to(other->op(1)); + const varidx & alpha = ex_to(before_other->op(1)); // e~mu e.mu = Tr ONE if (other - self == 1) { - if (prev_square != 0) { - *self = squared_metric; - v[prev_square-1] = _ex1; - } else - *self = unit.get_metric(self->op(1), other->op(1)); + if (prev_square > -1) { + *self = indexed(squared_metric, mu, mu_toggle); + v[prev_square] = _ex1; + } else { + *self = unit.get_metric(mu, mu_toggle, true); + } *other = dirac_ONE(rl); return true; - // e~mu e~alpha e.mu = (2e~alpha^2-Tr) e~alpha - } else if (other - self == 2 - && is_a(self[1])) { - - const ex & ia = self[1].op(1); - const ex & ib = self[1].op(1); - if (is_a(unit.get_metric())) - *self = 2 - unit.get_metric(self->op(1), other->op(1)); - else if (prev_square != 0) { - *self = 2-squared_metric; - v[prev_square-1] = _ex1; - } else - *self = 2*unit.get_metric(ia, ib) - unit.get_metric(self->op(1), other->op(1)); - *other = _ex1; - return true; - - // e~mu S e~alpha e.mu = 2 e~alpha^3 S - e~mu S e.mu e~alpha + } else if (other - self == 2) { + if (is_a(*before_other) && ex_to(*before_other).get_representation_label() == rl) { + if (ex_to(*self).is_anticommuting()) { + // e~mu e~alpha e.mu = (2*pow(e~alpha, 2) -Tr(B)) e~alpha + if (prev_square > -1) { + *self = 2 * indexed(squared_metric, alpha, alpha) + - indexed(squared_metric, mu, mu_toggle); + v[prev_square] = _ex1; + } else { + *self = 2 * unit.get_metric(alpha, alpha, true) - unit.get_metric(mu, mu_toggle, true); + } + *other = _ex1; + return true; + + } else { + // e~mu e~alpha e.mu = 2*e~mu B(alpha, mu.toggle_variance())-Tr(B) e~alpha + *self = 2 * (*self) * unit.get_metric(alpha, mu_toggle, true) - unit.get_metric(mu, mu_toggle, true) * (*before_other); + *before_other = _ex1; + *other = _ex1; + return true; + } + } else { + // e~mu S e.mu = Tr S ONE + *self = unit.get_metric(mu, mu_toggle, true); + *other = dirac_ONE(rl); + return true; + } + } else { + // e~mu S e~alpha e.mu = 2 e~mu S B(alpha, mu.toggle_variance()) - e~mu S e.mu e~alpha // (commutate contracted indices towards each other, simplify_indexed() // will re-expand and re-run the simplification) - } else { - exvector::iterator it = self + 1, next_to_last = other - 1; - while (it != other) { - if (!is_a(*it)) - return false; - ++it; + if (std::find_if(self + 1, other, is_not_a_clifford()) != other) { + return false; } - - it = self + 1; - ex S = _ex1; - while (it != next_to_last) { - S *= *it; - *it++ = _ex1; + + ex S = ncmul(exvector(self + 1, before_other), true); + + if (is_a(*before_other) && ex_to(*before_other).get_representation_label() == rl) { + if (ex_to(*self).is_anticommuting()) { + if (prev_square > -1) { + *self = 2 * (*before_other) * S * indexed(squared_metric, alpha, alpha) + - (*self) * S * (*other) * (*before_other); + } else { + *self = 2 * (*before_other) * S * unit.get_metric(alpha, alpha, true) - (*self) * S * (*other) * (*before_other); + } + } else { + *self = 2 * (*self) * S * unit.get_metric(alpha, mu_toggle, true) - (*self) * S * (*other) * (*before_other); + } + } else { + // simply commutes + *self = (*self) * S * (*other) * (*before_other); } - - const ex & ia = next_to_last->op(1); - const ex & ib = next_to_last->op(1); - if (is_a(unit.get_metric())) - *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last); - else if (prev_square != 0) { - *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last)*unit.get_metric(self->op(1),self->op(1)); - v[prev_square-1] = _ex1; - } else - *self = 2 * (*next_to_last) * S* unit.get_metric(ia,ib) - (*self) * S * (*other) * (*next_to_last); - *next_to_last = _ex1; - *other = _ex1; + + std::fill(self + 1, other + 1, _ex1); return true; } - - } - + } return false; } @@ -580,7 +617,7 @@ ex clifford::eval_ncmul(const exvector & v) const const ex & ia = a.op(1); const ex & ib = b.op(1); if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha - a = ex_to(a).get_metric(ia, ib); + a = ex_to(a).get_metric(ia, ib, true); b = dirac_ONE(representation_label); something_changed = true; } @@ -630,7 +667,7 @@ ex clifford::eval_ncmul(const exvector & v) const } if (s.empty()) - return clifford(diracone(), representation_label) * sign; + return dirac_ONE(representation_label) * sign; if (something_changed) return reeval_ncmul(s) * sign; else @@ -639,12 +676,12 @@ ex clifford::eval_ncmul(const exvector & v) const ex clifford::thiscontainer(const exvector & v) const { - return clifford(representation_label, get_metric(), v); + return clifford(representation_label, get_metric(), is_anticommuting(), v); } ex clifford::thiscontainer(std::auto_ptr vp) const { - return clifford(representation_label, get_metric(), vp); + return clifford(representation_label, get_metric(), is_anticommuting(), vp); } ex diracgamma5::conjugate() const @@ -669,22 +706,56 @@ ex diracgammaR::conjugate() const ex dirac_ONE(unsigned char rl) { static ex ONE = (new diracone)->setflag(status_flags::dynallocated); - return clifford(ONE, rl); + return clifford(ONE, rl, false); } -ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl) +ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl, bool anticommuting) { static ex unit = (new cliffordunit)->setflag(status_flags::dynallocated); - if (!is_a(mu)) - throw(std::invalid_argument("index of Clifford unit must be of type varidx")); + if (!is_a(mu)) + throw(std::invalid_argument("clifford_unit(): index of Clifford unit must be of type idx or varidx")); - if (is_a(metr)) - return clifford(unit, mu, metr.op(0), rl); - else if(is_a(metr) || is_a(metr)) - return clifford(unit, mu, metr, rl); - else - throw(std::invalid_argument("metric for Clifford unit must be of type indexed, tensormetric or matrix")); + if (ex_to(mu).is_symbolic() && !is_a(mu)) + throw(std::invalid_argument("clifford_unit(): symbolic index of Clifford unit must be of type varidx (not idx)")); + + if (is_a(metr)) { + exvector indices = ex_to(metr).get_indices(); + if ((indices.size() == 2) && is_a(indices[0]) && is_a(indices[1])) { + return clifford(unit, mu, metr, rl, anticommuting); + } else { + throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be indexed exactly by two indices of same type as the given index")); + } + } else if (is_a(metr)) { + static varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to(mu).get_dim()), + chi((new symbol)->setflag(status_flags::dynallocated), ex_to(mu).get_dim()); + return clifford(unit, mu, indexed(metr, xi, chi), rl, anticommuting); + } else if (is_a(metr)) { + matrix M = ex_to(metr); + unsigned n = M.rows(); + bool symmetric = true; + anticommuting = true; + + static varidx xi((new symbol)->setflag(status_flags::dynallocated), n), + chi((new symbol)->setflag(status_flags::dynallocated), n); + if ((n == M.cols()) && (n == ex_to(mu).get_dim())) { + for (unsigned i = 0; i < n; i++) { + for (unsigned j = i+1; j < n; j++) { + if (M(i, j) != M(j, i)) { + symmetric = false; + } + if (M(i, j) != -M(j, i)) { + anticommuting = false; + } + } + } + return clifford(unit, mu, indexed(metr, symmetric?symmetric2():not_symmetric(), xi, chi), rl, anticommuting); + } else { + throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be a square matrix with the same dimensions as index")); + } + } else { + throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be of type indexed, tensormetric or matrix")); + } } ex dirac_gamma(const ex & mu, unsigned char rl) @@ -692,9 +763,11 @@ ex dirac_gamma(const ex & mu, unsigned char rl) static ex gamma = (new diracgamma)->setflag(status_flags::dynallocated); if (!is_a(mu)) - throw(std::invalid_argument("index of Dirac gamma must be of type varidx")); + throw(std::invalid_argument("dirac_gamma(): index of Dirac gamma must be of type varidx")); - return clifford(gamma, mu, default_metric(), rl); + static varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to(mu).get_dim()), + chi((new symbol)->setflag(status_flags::dynallocated), ex_to(mu).get_dim()); + return clifford(gamma, mu, indexed(default_metric(), symmetric2(), xi, chi), rl, true); } ex dirac_gamma5(unsigned char rl) @@ -984,7 +1057,7 @@ ex canonicalize_clifford(const ex & e_) ex b1, i1, b2, i2; base_and_index(it[0], b1, i1); base_and_index(it[1], b2, i2); - it[0] = (ex_to(save0).get_metric(i1, i2) * b1 * b2).simplify_indexed(); + it[0] = (ex_to(save0).get_metric(i1, i2, true) * b1 * b2).simplify_indexed(); it[1] = v.size() == 2 ? _ex2 * dirac_ONE(ex_to(it[1]).get_representation_label()) : _ex2; ex sum = ncmul(v); it[0] = save1; @@ -1016,42 +1089,81 @@ ex clifford_prime(const ex & e) return e; } -ex remove_dirac_ONE(const ex & e, unsigned char rl) +ex remove_dirac_ONE(const ex & e, unsigned char rl, unsigned options) { - pointer_to_map_function_1arg fcn(remove_dirac_ONE, rl); - if (is_a(e) && ex_to(e).get_representation_label() >= rl) { - if (is_a(e.op(0))) + pointer_to_map_function_2args fcn(remove_dirac_ONE, rl, options | 1); + bool need_reevaluation = false; + ex e1 = e; + if (! (options & 1) ) { // is not a child + if (options & 2) + e1 = expand_dummy_sum(e, true); + e1 = canonicalize_clifford(e1); + } + + if (is_a(e1) && ex_to(e1).get_representation_label() >= rl) { + if (is_a(e1.op(0))) return 1; + else + throw(std::invalid_argument("remove_dirac_ONE(): expression is a non-scalar Clifford number!")); + } else if (is_a(e1) || is_a(e1) || is_a(e1) + || is_a(e1) || is_a(e1)) { + if (options & 3) // is a child or was already expanded + return e1.map(fcn); else - throw(std::invalid_argument("Expression is a non-scalar Clifford number!")); - } else if (is_a(e) || is_a(e) || is_a(e) // || is_a(e) || is_a(e) - || is_a(e) || is_a(e)) { - return e.map(fcn); - } else if (is_a(e)) { - return pow(remove_dirac_ONE(e.op(0)), e.op(1)); - } else - return e; + try { + return e1.map(fcn); + } catch (std::exception &p) { + need_reevaluation = true; + } + } else if (is_a(e1)) { + if (options & 3) // is a child or was already expanded + return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1)); + else + try { + return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1)); + } catch (std::exception &p) { + need_reevaluation = true; + } + } + if (need_reevaluation) + return remove_dirac_ONE(e, rl, options | 2); + return e1; } -ex clifford_norm(const ex & e) +char clifford_max_label(const ex & e, bool ignore_ONE) { - return sqrt(remove_dirac_ONE(canonicalize_clifford(e * clifford_bar(e)).simplify_indexed())); + if (is_a(e)) + if (ignore_ONE && is_a(e.op(0))) + return -1; + else + return ex_to(e).get_representation_label(); + else { + char rl = -1; + for (size_t i=0; i < e.nops(); i++) + rl = (rl > clifford_max_label(e.op(i), ignore_ONE)) ? rl : clifford_max_label(e.op(i), ignore_ONE); + return rl; + } } +ex clifford_norm(const ex & e) +{ + return sqrt(remove_dirac_ONE(e * clifford_bar(e))); +} + ex clifford_inverse(const ex & e) { ex norm = clifford_norm(e); if (!norm.is_zero()) return clifford_bar(e) / pow(norm, 2); else - throw(std::invalid_argument("Cannot find inverse of Clifford number with zero norm!")); + throw(std::invalid_argument("clifford_inverse(): cannot find inverse of Clifford number with zero norm!")); } -ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl) +ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl, bool anticommuting) { if (!ex_to(mu).is_dim_numeric()) - throw(std::invalid_argument("Index should have a numeric dimension")); - ex e = clifford_unit(mu, metr, rl); + throw(std::invalid_argument("lst_to_clifford(): Index should have a numeric dimension")); + ex e = clifford_unit(mu, metr, rl, anticommuting); return lst_to_clifford(v, e); } @@ -1074,20 +1186,20 @@ ex lst_to_clifford(const ex & v, const ex & e) { if (dim == max) return indexed(v, ex_to(mu).toggle_variance()) * e; else - throw(std::invalid_argument("Dimensions of vector and clifford unit mismatch")); + throw(std::invalid_argument("lst_to_clifford(): dimensions of vector and clifford unit mismatch")); } else - throw(std::invalid_argument("First argument should be a vector vector")); + throw(std::invalid_argument("lst_to_clifford(): first argument should be a vector vector")); } else if (is_a(v)) { if (dim == ex_to(v).nops()) return indexed(matrix(dim, 1, ex_to(v)), ex_to(mu).toggle_variance()) * e; else - throw(std::invalid_argument("List length and dimension of clifford unit mismatch")); + throw(std::invalid_argument("lst_to_clifford(): list length and dimension of clifford unit mismatch")); } else - throw(std::invalid_argument("Cannot construct from anything but list or vector")); + throw(std::invalid_argument("lst_to_clifford(): cannot construct from anything but list or vector")); } else - throw(std::invalid_argument("The second argument should be a Clifford unit")); + throw(std::invalid_argument("lst_to_clifford(): the second argument should be a Clifford unit")); } - + /** Auxiliary structure to define a function for striping one Clifford unit * from vectors. Used in clifford_to_lst(). */ static ex get_clifford_comp(const ex & e, const ex & c) @@ -1106,7 +1218,7 @@ static ex get_clifford_comp(const ex & e, const ex & c) if (ind > e.nops()) ind = j; else - throw(std::invalid_argument("Expression is a Clifford multi-vector")); + throw(std::invalid_argument("get_clifford_comp(): expression is a Clifford multi-vector")); if (ind < e.nops()) { ex S = 1; bool same_value_index, found_dummy; @@ -1131,7 +1243,7 @@ static ex get_clifford_comp(const ex & e, const ex & c) } return (found_dummy ? S : 0); } else - throw(std::invalid_argument("Expression is not a Clifford vector to the given units")); + throw(std::invalid_argument("get_clifford_comp(): expression is not a Clifford vector to the given units")); } else if (e.is_zero()) return e; else if (is_a(e) && ex_to(e).same_metric(c)) @@ -1141,7 +1253,7 @@ static ex get_clifford_comp(const ex & e, const ex & c) else return 1; else - throw(std::invalid_argument("Expression is not usable as a Clifford vector")); + throw(std::invalid_argument("get_clifford_comp(): expression is not usable as a Clifford vector")); } @@ -1150,7 +1262,7 @@ lst clifford_to_lst(const ex & e, const ex & c, bool algebraic) GINAC_ASSERT(is_a(c)); varidx mu = ex_to(c.op(1)); if (! mu.is_dim_numeric()) - throw(std::invalid_argument("Index should have a numeric dimension")); + throw(std::invalid_argument("clifford_to_lst(): index should have a numeric dimension")); unsigned int D = ex_to(mu.get_dim()).to_int(); if (algebraic) // check if algebraic method is applicable @@ -1159,26 +1271,33 @@ lst clifford_to_lst(const ex & e, const ex & c, bool algebraic) or (not is_a(pow(c.subs(mu == i), 2)))) algebraic = false; lst V; - if (algebraic) + if (algebraic) { for (unsigned int i = 0; i < D; i++) V.append(remove_dirac_ONE( simplify_indexed(canonicalize_clifford(e * c.subs(mu == i) + c.subs(mu == i) * e)) / (2*pow(c.subs(mu == i), 2)))); - else { + } else { ex e1 = canonicalize_clifford(e); - for (unsigned int i = 0; i < D; i++) - V.append(get_clifford_comp(e1, c.subs(c.op(1) == i))); + try { + for (unsigned int i = 0; i < D; i++) + V.append(get_clifford_comp(e1, c.subs(c.op(1) == i))); + } catch (std::exception &p) { + /* Try to expand dummy summations to simplify the expression*/ + e1 = canonicalize_clifford(expand_dummy_sum(e1, true)); + for (unsigned int i = 0; i < D; i++) + V.append(get_clifford_comp(e1, c.subs(c.op(1) == i))); + } } return V; } -ex clifford_moebius_map(const ex & a, const ex & b, const ex & c, const ex & d, const ex & v, const ex & G, unsigned char rl) +ex clifford_moebius_map(const ex & a, const ex & b, const ex & c, const ex & d, const ex & v, const ex & G, unsigned char rl, bool anticommuting) { ex x, D, cu; if (! is_a(v) && ! is_a(v)) - throw(std::invalid_argument("parameter v should be either vector or list")); + throw(std::invalid_argument("clifford_moebius_map(): parameter v should be either vector or list")); if (is_a(G)) { cu = G; @@ -1187,24 +1306,24 @@ ex clifford_moebius_map(const ex & a, const ex & b, const ex & c, const ex & d, D = ex_to(G.op(1)).get_dim(); else if (is_a(G)) D = ex_to(G).rows(); - else throw(std::invalid_argument("metric should be an indexed object, matrix, or a Clifford unit")); + else throw(std::invalid_argument("clifford_moebius_map(): metric should be an indexed object, matrix, or a Clifford unit")); varidx mu((new symbol)->setflag(status_flags::dynallocated), D); - cu = clifford_unit(mu, G, rl); + cu = clifford_unit(mu, G, rl, anticommuting); } - + x = lst_to_clifford(v, cu); ex e = simplify_indexed(canonicalize_clifford((a * x + b) * clifford_inverse(c * x + d))); return clifford_to_lst(e, cu, false); } -ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl) +ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl, bool anticommuting) { if (is_a(M)) - return clifford_moebius_map(ex_to(M)(0,0), ex_to(M)(0,1), - ex_to(M)(1,0), ex_to(M)(1,1), v, G, rl); + return clifford_moebius_map(ex_to(M)(0,0), ex_to(M)(0,1), + ex_to(M)(1,0), ex_to(M)(1,1), v, G, rl, anticommuting); else - throw(std::invalid_argument("parameter M should be a matrix")); + throw(std::invalid_argument("clifford_moebius_map(): parameter M should be a matrix")); } } // namespace GiNaC diff --git a/ginac/clifford.h b/ginac/clifford.h index c9b17a94..7b120734 100644 --- a/ginac/clifford.h +++ b/ginac/clifford.h @@ -44,12 +44,12 @@ class clifford : public indexed // other constructors public: - clifford(const ex & b, unsigned char rl = 0); - clifford(const ex & b, const ex & mu, const ex & metr, unsigned char rl = 0); + clifford(const ex & b, unsigned char rl = 0, bool anticommut = false); + clifford(const ex & b, const ex & mu, const ex & metr, unsigned char rl = 0, bool anticommut = false); // internal constructors - clifford(unsigned char rl, const ex & metr, const exvector & v, bool discardable = false); - clifford(unsigned char rl, const ex & metr, std::auto_ptr vp); + clifford(unsigned char rl, const ex & metr, bool anticommut, const exvector & v, bool discardable = false); + clifford(unsigned char rl, const ex & metr, bool anticommut, std::auto_ptr vp); // functions overriding virtual functions from base classes public: @@ -66,8 +66,9 @@ protected: public: unsigned char get_representation_label() const { return representation_label; } ex get_metric() const { return metric; } - ex get_metric(const ex & i, const ex & j) const; + ex get_metric(const ex & i, const ex & j, bool symmetrised = false) const; bool same_metric(const ex & other) const; + bool is_anticommuting() const { return anticommuting; } //**< See the member variable anticommuting */ protected: void do_print_dflt(const print_dflt & c, unsigned level) const; @@ -76,7 +77,8 @@ protected: // member variables private: unsigned char representation_label; /**< Representation label to distinguish independent spin lines */ - ex metric; + ex metric; /**< Metric of the space, all constructors make it an indexed object */ + bool anticommuting; /**< Simplifications for anticommuting units is much simpler and we need this info readily available */ }; @@ -193,10 +195,10 @@ ex dirac_ONE(unsigned char rl = 0); /** Create a Clifford unit object. * * @param mu Index (must be of class varidx or a derived class) - * @param metr Metric (should be of class tensmetric or a derived class, or a matrix) + * @param metr Metric (should be indexed, tensmetric or a derived class, or a matrix) * @param rl Representation label * @return newly constructed Clifford unit object */ -ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0); +ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0, bool anticommuting = false); /** Create a Dirac gamma object. * @@ -280,10 +282,18 @@ inline ex clifford_star(const ex & e) { return e.conjugate(); } /** Replaces dirac_ONE's (with a representation_label no less than rl) in e with 1. * For the default value rl = 0 remove all of them. Aborts if e contains any * clifford_unit with representation_label to be removed. - * + * + * @param e Expression to be processed + * @param rl Value of representation label + * @param options Defines some internal use */ +ex remove_dirac_ONE(const ex & e, unsigned char rl = 0, unsigned options = 0); + +/** Returns the maximal representation label of a clifford object + * if e contains at least one, otherwise returns -1 + * * @param e Expression to be processed - * @param rl Value of representation label */ -ex remove_dirac_ONE(const ex & e, unsigned char rl = 0); + * @ignore_ONE defines if clifford_ONE should be ignored in the search*/ +char clifford_max_label(const ex & e, bool ignore_ONE = false); /** Calculation of the norm in the Clifford algebra. */ ex clifford_norm(const ex & e); @@ -295,11 +305,11 @@ ex clifford_inverse(const ex & e); * * @param v List or vector of coordinates * @param mu Index (must be of class varidx or a derived class) - * @param metr Metric (should be of class tensmetric or a derived class, or a matrix) + * @param metr Metric (should be indexed, tensmetric or a derived class, or a matrix) * @param rl Representation label * @param e Clifford unit object * @return Clifford vector with given components */ -ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl = 0); +ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl = 0, bool anticommuting = false); ex lst_to_clifford(const ex & v, const ex & e); /** An inverse function to lst_to_clifford(). For given Clifford vector extracts @@ -327,8 +337,9 @@ lst clifford_to_lst(const ex & e, const ex & c, bool algebraic=true); * @param v Vector to be transformed * @param G Metric of the surrounding space, may be a Clifford unit then the next parameter is ignored * @param rl Representation label + * @param anticommuting indicates if Clifford units anticommutes * @return List of components of the transformed vector*/ -ex clifford_moebius_map(const ex & a, const ex & b, const ex & c, const ex & d, const ex & v, const ex & G, unsigned char rl = 0); +ex clifford_moebius_map(const ex & a, const ex & b, const ex & c, const ex & d, const ex & v, const ex & G, unsigned char rl = 0, bool anticommuting = false); /** The second form of Moebius transformations defined by a 2x2 Clifford matrix M * This function takes the transformation matrix M as a single entity. @@ -337,8 +348,9 @@ ex clifford_moebius_map(const ex & a, const ex & b, const ex & c, const ex & d, * @param v Vector to be transformed * @param G Metric of the surrounding space, may be a Clifford unit then the next parameter is ignored * @param rl Representation label + * @param anticommuting indicates if Clifford units anticommutes * @return List of components of the transformed vector*/ -ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl = 0); +ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl = 0, bool anticommuting = false); } // namespace GiNaC