* 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
return 0;
}
-static unsigned clifford_check1(void)
+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
symbol dim("D"), m("m"), q("q"), l("l"), ldotq("ldotq");
varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), rho(symbol("rho"), dim),
- sig(symbol("sig"), dim), kap(symbol("kap"), dim), lam(symbol("lam"), 4);
+ sig(symbol("sig"), dim), kap(symbol("kap"), dim), lam(symbol("lam"), dim);
ex e;
e = dirac_gamma(mu);
+ 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 * eps0123(nu, rho, sig, kap));
+ result += check_equal(e, 8 * I * lorentz_eps(nu.replace_dim(4), rho.replace_dim(4), sig.replace_dim(4), kap.replace_dim(4)));
// one-loop vacuum polarization in QED
e = dirac_gamma(mu) *
- (dirac_slash(l, dim) + dirac_slash(q, dim) + m * dirac_ONE()) *
+ (dirac_slash(l, dim) + dirac_slash(q, 4) + m * dirac_ONE()) *
dirac_gamma(mu.toggle_variance()) *
(dirac_slash(l, dim) + m * dirac_ONE());
e = dirac_trace(e).simplify_indexed(sp);
result += check_equal(e, 4*((2-dim)*l*l + (2-dim)*ldotq + dim*m*m).expand());
- e = dirac_slash(q, dim) *
- (dirac_slash(l, dim) + dirac_slash(q, dim) + m * dirac_ONE()) *
- dirac_slash(q, dim) *
+ e = dirac_slash(q, 4) *
+ (dirac_slash(l, dim) + dirac_slash(q, 4) + m * dirac_ONE()) *
+ dirac_slash(q, 4) *
(dirac_slash(l, dim) + m * dirac_ONE());
e = dirac_trace(e).simplify_indexed(sp);
result += check_equal(e, 4*(2*ldotq*ldotq + q*q*ldotq - q*q*l*l + q*q*m*m).expand());
+ // stuff that had problems in the past
+ ex prop = dirac_slash(q, dim) - m * dirac_ONE();
+ e = dirac_slash(l, dim) * dirac_gamma5() * dirac_slash(l, dim) * prop;
+ e = dirac_trace(dirac_slash(q, dim) * e) - dirac_trace(m * e)
+ - dirac_trace(prop * e);
+ result += check_equal(e, 0);
+
+ e = (dirac_gamma5() + dirac_ONE()) * dirac_gamma5();
+ e = dirac_trace(e);
+ result += check_equal(e, 4);
+
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)
+static unsigned clifford_check6(const matrix & A)
+{
+ varidx v(symbol("v"), 4), nu(symbol("nu"), 4), mu(symbol("mu"), 4),
+ psi(symbol("psi"),4), lam(symbol("lambda"), 4),
+ xi(symbol("xi"), 4), rho(symbol("rho"),4);
+
+ ex G = A;
+
+ matrix A2(4, 4);
+ A2 = A.mul(A);
+ ex e, e1;
+
+ int result = 0;
+
+ // checks general identities and contractions for clifford_unit
+ e = dirac_ONE() * clifford_unit(mu, G) * dirac_ONE();
+ result += check_equal(e, clifford_unit(mu, G));
+
+ e = clifford_unit(varidx(2, 4), G) * clifford_unit(varidx(1, 4), G)
+ * clifford_unit(varidx(1, 4), G) * clifford_unit(varidx(2, 4), G);
+ result += check_equal(e, A(1, 1) * A(2, 2) * dirac_ONE());
+
+ e = clifford_unit(nu, G) * clifford_unit(nu.toggle_variance(), G);
+ result += check_equal_simplify(e, A.trace() * dirac_ONE());
+
+ e = clifford_unit(nu, G) * clifford_unit(nu, G);
+ result += check_equal_simplify(e, indexed(G, sy_symm(), nu, nu) * dirac_ONE());
+
+ e = clifford_unit(nu, G) * clifford_unit(nu.toggle_variance(), G) * clifford_unit(mu, G);
+ result += check_equal_simplify(e, A.trace() * clifford_unit(mu, G));
+
+ e = clifford_unit(nu, G) * clifford_unit(mu, G) * clifford_unit(nu.toggle_variance(), G);
+ result += check_equal_simplify(e, 2*indexed(G, sy_symm(), mu, mu)*clifford_unit(mu, G) - A.trace()*clifford_unit(mu, G));
+
+ e = clifford_unit(nu, G) * clifford_unit(nu.toggle_variance(), G)
+ * clifford_unit(mu, G) * clifford_unit(mu.toggle_variance(), G);
+ result += check_equal_simplify(e, pow(A.trace(), 2) * dirac_ONE());
+
+ e = clifford_unit(mu, G) * clifford_unit(nu, G)
+ * clifford_unit(nu.toggle_variance(), G) * clifford_unit(mu.toggle_variance(), G);
+ result += check_equal_simplify(e, pow(A.trace(), 2) * dirac_ONE());
+
+ e = clifford_unit(mu, G) * clifford_unit(nu, G)
+ * clifford_unit(mu.toggle_variance(), G) * clifford_unit(nu.toggle_variance(), G);
+ result += check_equal_simplify(e, 2*A2.trace()*dirac_ONE() - pow(A.trace(), 2)*dirac_ONE());
+
+ e = clifford_unit(mu.toggle_variance(), G) * clifford_unit(nu, G)
+ * clifford_unit(mu, G) * clifford_unit(nu.toggle_variance(), G);
+ result += check_equal_simplify(e, 2*A2.trace()*dirac_ONE() - pow(A.trace(), 2)*dirac_ONE());
+
+ e = clifford_unit(nu.toggle_variance(), G) * clifford_unit(rho.toggle_variance(), G)
+ * clifford_unit(mu, G) * clifford_unit(rho, G) * clifford_unit(nu, G);
+ e = e.simplify_indexed().collect(clifford_unit(mu, G));
+ result += check_equal(e, (pow(A.trace(), 2)+4-4*A.trace()*indexed(A, mu, mu)) * clifford_unit(mu, G));
+
+ e = clifford_unit(nu.toggle_variance(), G) * clifford_unit(rho, G)
+ * clifford_unit(mu, G) * clifford_unit(rho.toggle_variance(), G) * clifford_unit(nu, G);
+ e = e.simplify_indexed().collect(clifford_unit(mu, G));
+ result += check_equal(e, (pow(A.trace(), 2)+4-4*A.trace()*indexed(A, mu, mu))* clifford_unit(mu, G));
+
+ // canonicalize_clifford() checks
+ e = clifford_unit(mu, G) * clifford_unit(nu, G) + clifford_unit(nu, G) * clifford_unit(mu, G);
+ result += check_equal(canonicalize_clifford(e), 2*dirac_ONE()*indexed(G, sy_symm(), mu, nu));
+
+ e = (clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(lam, G)
+ + clifford_unit(nu, G) * clifford_unit(lam, G) * clifford_unit(mu, G)
+ + clifford_unit(lam, G) * clifford_unit(mu, G) * clifford_unit(nu, G)
+ - clifford_unit(nu, G) * clifford_unit(mu, G) * clifford_unit(lam, G)
+ - clifford_unit(lam, G) * clifford_unit(nu, G) * clifford_unit(mu, G)
+ - clifford_unit(mu, G) * clifford_unit(lam, G) * clifford_unit(nu, G)) / 6
+ + indexed(G, sy_symm(), mu, nu) * clifford_unit(lam, G)
+ - indexed(G, sy_symm(), mu, lam) * clifford_unit(nu, G)
+ + indexed(G, sy_symm(), nu, lam) * clifford_unit(mu, G)
+ - clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(lam, G);
+ result += check_equal(canonicalize_clifford(e), 0);
+
+ // lst_to_clifford() and clifford_inverse() check
+ symbol x("x"), y("y"), t("t"), z("z");
+
+ e = lst_to_clifford(lst(t, x, y, z), mu, G) * lst_to_clifford(lst(1, 2, 3, 4), nu, G);
+ e1 = clifford_inverse(e);
+ result += check_equal((e*e1).simplify_indexed().normal(), dirac_ONE());
+
+ return result;
+}
+
+static unsigned clifford_check7()
+{
+ // checks general identities and contractions
+
+ unsigned result = 0;
+
+ symbol dim("D");
+ varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), rho(symbol("rho"), dim),
+ psi(symbol("psi"),dim), lam(symbol("lambda"), dim), xi(symbol("xi"), dim);
+
+ ex e;
+
+ ex G = minkmetric();
+
+ e = dirac_ONE() * dirac_ONE();
+ result += check_equal(e, dirac_ONE());
+
+ e = dirac_ONE() * clifford_unit(mu, G) * dirac_ONE();
+ result += check_equal(e, clifford_unit(mu, G));
+
+ e = clifford_unit(varidx(2, dim), G) * clifford_unit(varidx(1, dim), G)
+ * clifford_unit(varidx(1, dim), G) * clifford_unit(varidx(2, dim), G);
+ result += check_equal(e, dirac_ONE());
+
+ e = clifford_unit(mu, G) * clifford_unit(nu, G)
+ * clifford_unit(nu.toggle_variance(), G) * clifford_unit(mu.toggle_variance(), G);
+ result += check_equal_simplify(e, pow(dim, 2) * dirac_ONE());
+
+ e = clifford_unit(mu, G) * clifford_unit(nu, G)
+ * clifford_unit(mu.toggle_variance(), G) * clifford_unit(nu.toggle_variance(), G);
+ result += check_equal_simplify(e, 2*dim*dirac_ONE() - pow(dim, 2)*dirac_ONE());
+
+ e = clifford_unit(nu.toggle_variance(), G) * clifford_unit(rho.toggle_variance(), G)
+ * clifford_unit(mu, G) * clifford_unit(rho, G) * clifford_unit(nu, G);
+ e = e.simplify_indexed().collect(clifford_unit(mu, G));
+ result += check_equal(e, pow(2 - dim, 2).expand() * clifford_unit(mu, G));
+
+ // canonicalize_clifford() checks
+ e = clifford_unit(mu, G) * clifford_unit(nu, G) + clifford_unit(nu, G) * clifford_unit(mu, G);
+ result += check_equal(canonicalize_clifford(e), 2*dirac_ONE()*indexed(G, sy_symm(), mu, nu));
+
+ e = (clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(lam, G)
+ + clifford_unit(nu, G) * clifford_unit(lam, G) * clifford_unit(mu, G)
+ + clifford_unit(lam, G) * clifford_unit(mu, G) * clifford_unit(nu, G)
+ - clifford_unit(nu, G) * clifford_unit(mu, G) * clifford_unit(lam, G)
+ - clifford_unit(lam, G) * clifford_unit(nu, G) * clifford_unit(mu, G)
+ - clifford_unit(mu, G) * clifford_unit(lam, G) * clifford_unit(nu, G)) / 6
+ + indexed(G, sy_symm(), mu, nu) * clifford_unit(lam, G)
+ - indexed(G, sy_symm(), mu, lam) * clifford_unit(nu, G)
+ + indexed(G, sy_symm(), nu, lam) * clifford_unit(mu, G)
+ - clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(lam, G);
+ result += check_equal(canonicalize_clifford(e), 0);
+
+ return result;
+}
+
+unsigned exam_clifford()
{
unsigned result = 0;
cout << "examining clifford objects" << flush;
clog << "----------clifford objects:" << endl;
- result += clifford_check1(); cout << '.' << flush;
- result += clifford_check2(); cout << '.' << flush;
- result += clifford_check3(); cout << '.' << flush;
- result += clifford_check4(); cout << '.' << flush;
- result += clifford_check5(); cout << '.' << flush;
+ 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;
}
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