unsigned result = 0;
- symbol dim("D"), m("m"), q("q");
- varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim);
+ 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;
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 = dirac_trace(e).simplify_indexed(sp);
result += check_equal(e, 4*pow(m, 2)*pow(q, 2));
+ // cyclicity
+ 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);
+
+ 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);
+ result += check_equal(e.expand(), 0);
+
+ // 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_gamma(mu.toggle_variance()) *
+ (dirac_gamma(rho) * indexed(l, rho.toggle_variance()) + m * dirac_ONE());
+ e = dirac_trace(e.expand()).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);
+ result += check_equal(e, 4*(2*ldotq*ldotq + q*q*ldotq - q*q*l*l + q*q*m*m).expand());
+
return result;
}