* Here we test GiNaC's Clifford algebra objects. */
/*
- * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2003 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
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();
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
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);
+
+ 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*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;
+}
+
+unsigned exam_clifford()
{
unsigned result = 0;
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;
+
if (!result) {
cout << " passed " << endl;
clog << "(no output)" << endl;
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