added missing factor I in gamma5 trace

[ginac.git] / check / exam_clifford.cpp

diff --git a/check/exam_clifford.cpp b/check/exam_clifford.cpp
index d4491191acfce830ca8409969a7fa86d4c1aee79..2d538354400d759aa594ebe5fdf29dd8ac737419 100644 (file)
--- a/check/exam_clifford.cpp
+++ b/check/exam_clifford.cpp
@@ -51,16 +51,161 @@ static unsigned clifford_check1(void)
        unsigned result = 0;
 
        symbol dim("D");
        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;
 
        ex e;
 

+       e = dirac_ONE() * dirac_ONE();
+       result += check_equal(e, dirac_ONE());
+
+       e = dirac_ONE() * dirac_gamma(mu) * dirac_ONE();
+       result += check_equal(e, dirac_gamma(mu));
+
+       e = dirac_gamma(varidx(2, dim)) * dirac_gamma(varidx(1, dim)) *
+           dirac_gamma(varidx(1, dim)) * dirac_gamma(varidx(2, dim));
+       result += check_equal(e, dirac_ONE());
+

        e = dirac_gamma(mu) * dirac_gamma(nu) *
            dirac_gamma(nu.toggle_variance()) * dirac_gamma(mu.toggle_variance());
        e = dirac_gamma(mu) * dirac_gamma(nu) *
            dirac_gamma(nu.toggle_variance()) * dirac_gamma(mu.toggle_variance());

-       result += check_equal_simplify(e, pow(dim, 2) * dirac_one());
+       result += check_equal_simplify(e, pow(dim, 2) * dirac_ONE());

 
        e = dirac_gamma(mu) * dirac_gamma(nu) *
            dirac_gamma(mu.toggle_variance()) * dirac_gamma(nu.toggle_variance());
 
        e = dirac_gamma(mu) * dirac_gamma(nu) *
            dirac_gamma(mu.toggle_variance()) * dirac_gamma(nu.toggle_variance());

-       result += check_equal_simplify(e, 2*dim*dirac_one()-pow(dim, 2)*dirac_one());
+       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)
+{
+       // checks identities relating to gamma5
+
+       unsigned result = 0;
+
+       symbol dim("D");
+       varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim);
+       ex e;
+
+       e = dirac_gamma(mu) * dirac_gamma5() + dirac_gamma5() * dirac_gamma(mu);
+       result += check_equal(e, 0);
+
+       e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma5() + dirac_gamma(mu);
+       result += check_equal(e, 0);
+
+       return result;
+}
+
+static unsigned clifford_check3(void)
+{
+       // 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"), 4);
+       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);
+       e = (e / (dim - 4)).normal();
+       result += check_equal(e, 8 * I * eps0123(nu, rho, sig, kap));
+
+       // one-loop vacuum polarization in QED
+       e = dirac_gamma(mu) *
+           (dirac_slash(l, dim) + dirac_slash(q, dim) + 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) *
+           (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;
+}
+
+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;
 }
 
        return result;
 }


@@ -73,6 +218,9 @@ unsigned exam_clifford(void)
        clog << "----------clifford objects:" << endl;
 
        result += clifford_check1();  cout << '.' << 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;

        
        if (!result) {
                cout << " passed " << endl;
        
        if (!result) {
                cout << " passed " << endl;