* Here we test GiNaC's Clifford algebra objects. */
/*
- * GiNaC Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2010 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
*
* 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 <iostream>
+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;
}
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<varidx>(mu)
+ ? lst (
+ mu == idx(j, mu.get_dim()),
+ ex_to<varidx>(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
return result;
}
-static unsigned clifford_check2(void)
+static unsigned clifford_check2()
{
// checks identities relating to gamma5
return result;
}
-static unsigned clifford_check3(void)
+static unsigned clifford_check3()
{
// checks traces
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 gamme matrices with different rl depends on luck.
+ // TODO: better check.
+ //result += check_equal_simplify(dirac_trace(e, 2), canonicalize_clifford(e)); // e will be canonicalized by the calculation of the trace
+ result += check_equal_simplify(dirac_trace(e, lst(0, 1)), 16 * dim);
+
return result;
}
-static unsigned clifford_check4(void)
+static unsigned clifford_check4()
{
// simplify_indexed()/dirac_trace() cross-checks
return result;
}
-static unsigned clifford_check5(void)
+static unsigned clifford_check5()
{
// canonicalize_clifford() checks
ex e;
e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
- result += check_equal(canonicalize_clifford(e), 2*lorentz_g(mu, nu));
+ 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)
return result;
}
-unsigned exam_clifford(void)
+/* 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 <typename IDX> 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<varidx>(mu) ? ex_to<varidx>(mu).toggle_variance() : mu;
+ ex nu_TOGGLE = is_a<varidx>(nu) ? ex_to<varidx>(nu).toggle_variance() : nu;
+ ex rho_TOGGLE
+ = is_a<varidx>(rho) ? ex_to<varidx>(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());
- 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;
+ e = clifford_unit(mu_TOGGLE, A) * clifford_unit(nu, A)
+ * clifford_unit(mu, A) * clifford_unit(nu_TOGGLE, A);
- if (!result) {
- cout << " passed " << endl;
- clog << "(no output)" << endl;
+ 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));
+
+ 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 */
+ 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<lst>(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>(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 (ex_to<symmetry>(ex_to<indexed>(ex_to<clifford>(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");
+ varidx mu(symbol("mu", "\\mu"), 1);
+
+ ex e = clifford_unit(mu, diag_matrix(lst(-1))), e0 = e.subs(mu==0);
+ result += ( exp(a*e0)*e0*e0 == -exp(e0*a) ) ? 0 : 1;
+
+ 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<varidx>(ex_to<matrix>(diag_matrix(lst(-1, 1, 1, 1))));
+ result += clifford_check6<idx>(ex_to<matrix>(diag_matrix(lst(-1, 1, 1, 1))));; cout << '.' << flush;
+ result += clifford_check6<varidx>(ex_to<matrix>(diag_matrix(lst(-1, -1, -1, -1))))+clifford_check6<idx>(ex_to<matrix>(diag_matrix(lst(-1, -1, -1, -1))));; cout << '.' << flush;
+ result += clifford_check6<idx>(ex_to<matrix>(diag_matrix(lst(-1, 1, 1, -1))))+clifford_check6<idx>(ex_to<matrix>(diag_matrix(lst(-1, 1, 1, -1))));; cout << '.' << flush;
+ result += clifford_check6<varidx>(ex_to<matrix>(diag_matrix(lst(-1, 0, 1, -1))))+clifford_check6<idx>(ex_to<matrix>(diag_matrix(lst(-1, 0, 1, -1))));; cout << '.' << flush;
+ result += clifford_check6<varidx>(ex_to<matrix>(diag_matrix(lst(-3, 0, 2, -1))))+clifford_check6<idx>(ex_to<matrix>(diag_matrix(lst(-3, 0, 2, -1))));; cout << '.' << flush;
+
+ realsymbol s("s"), t("t"); // symbolic entries in matric
+ result += clifford_check6<varidx>(ex_to<matrix>(diag_matrix(lst(-1, 1, s, t))))+clifford_check6<idx>(ex_to<matrix>(diag_matrix(lst(-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<varidx>(A)+clifford_check6<idx>(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<varidx>(A)+clifford_check6<idx>(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<varidx>(A)+clifford_check6<idx>(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<varidx>(A)+clifford_check6<idx>(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<varidx>(A)+clifford_check6<idx>(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;
+
return result;
}
+
+int main(int argc, char** argv)
+{
+ return exam_clifford();
+}
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