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);
- scalar_products sp;
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) * indexed(q, mu) * indexed(q, nu)
- * dirac_gamma(mu.toggle_variance()) * dirac_gamma(nu.toggle_variance());
+ 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
+ // 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.expand(), 0);
+ 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.expand(), 0);
+ 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);
+ e = (e / (dim - 4)).normal();
+ result += check_equal(e, 8 * eps0123(nu, rho, sig, kap));
// one-loop vacuum polarization in QED
e = dirac_gamma(mu) *
- (dirac_gamma(nu) * indexed(l, nu.toggle_variance()) + dirac_gamma(nu) * indexed(q, nu.toggle_variance()) + m * dirac_ONE()) *
+ (dirac_slash(l, dim) + dirac_slash(q, dim) + m * dirac_ONE()) *
dirac_gamma(mu.toggle_variance()) *
- (dirac_gamma(rho) * indexed(l, rho.toggle_variance()) + m * dirac_ONE());
- e = dirac_trace(e.expand()).simplify_indexed(sp);
+ (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_gamma(mu) * indexed(q, mu.toggle_variance()) *
- (dirac_gamma(nu) * indexed(l, nu.toggle_variance()) + dirac_gamma(nu) * indexed(q, nu.toggle_variance()) + m * dirac_ONE()) *
- dirac_gamma(sig) * indexed(q, sig.toggle_variance()) *
- (dirac_gamma(rho) * indexed(l, rho.toggle_variance()) + m * dirac_ONE());
- e = dirac_trace(e.expand()).simplify_indexed(sp);
+ e = dirac_slash(q, dim) *
+ (dirac_slash(l, dim) + dirac_slash(q, dim) + m * dirac_ONE()) *
+ dirac_slash(q, dim) *
+ (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());
return result;