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;
}
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);
// 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);
+ e = dirac_trace(e).simplify_indexed();
e = (e / (dim - 4)).normal();
- result += check_equal(e, 8 * eps0123(nu, rho, sig, kap));
+ result += check_equal(e, 8 * I * eps0123(nu, rho, sig, kap));
// one-loop vacuum polarization in QED
e = dirac_gamma(mu) *
return result;
}
+static unsigned clifford_check4(void)
+{
+ // 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(void)
+{
+ // 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(void)
{
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;