X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=check%2Fexam_clifford.cpp;h=2d538354400d759aa594ebe5fdf29dd8ac737419;hp=d4491191acfce830ca8409969a7fa86d4c1aee79;hb=8b3f3cb2d72aa91b9a635f63d5a4c9a24afa5b00;hpb=aef64ef1f63f7b4cb9236c5c189873e48c6c999b;ds=sidebyside

diff --git a/check/exam_clifford.cpp b/check/exam_clifford.cpp
index d4491191..2d538354 100644
--- 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");
-	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();
+	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());
-	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());
-	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;
 }
@@ -73,6 +218,9 @@ unsigned exam_clifford(void)
 	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;