1 /** @file exam_clifford.cpp
3 * Here we test GiNaC's Clifford algebra objects. */
6 * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
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20 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
25 static unsigned check_equal(const ex &e1, const ex &e2)
29 clog << e1 << "-" << e2 << " erroneously returned "
30 << e << " instead of 0" << endl;
36 static unsigned check_equal_simplify(const ex &e1, const ex &e2)
38 ex e = simplify_indexed(e1) - e2;
40 clog << "simplify_indexed(" << e1 << ")-" << e2 << " erroneously returned "
41 << e << " instead of 0" << endl;
47 static unsigned clifford_check1(void)
49 // checks general identities and contractions
54 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim);
57 e = dirac_ONE() * dirac_ONE();
58 result += check_equal(e, dirac_ONE());
60 e = dirac_ONE() * dirac_gamma(mu) * dirac_ONE();
61 result += check_equal(e, dirac_gamma(mu));
63 e = dirac_gamma(varidx(2, dim)) * dirac_gamma(varidx(1, dim)) *
64 dirac_gamma(varidx(1, dim)) * dirac_gamma(varidx(2, dim));
65 result += check_equal(e, dirac_ONE());
67 e = dirac_gamma(mu) * dirac_gamma(nu) *
68 dirac_gamma(nu.toggle_variance()) * dirac_gamma(mu.toggle_variance());
69 result += check_equal_simplify(e, pow(dim, 2) * dirac_ONE());
71 e = dirac_gamma(mu) * dirac_gamma(nu) *
72 dirac_gamma(mu.toggle_variance()) * dirac_gamma(nu.toggle_variance());
73 result += check_equal_simplify(e, 2*dim*dirac_ONE()-pow(dim, 2)*dirac_ONE());
78 static unsigned clifford_check2(void)
80 // checks identities relating to gamma5
85 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim);
88 e = dirac_gamma(mu) * dirac_gamma5() + dirac_gamma5() * dirac_gamma(mu);
89 result += check_equal(e, 0);
91 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma5() + dirac_gamma(mu);
92 result += check_equal(e, 0);
97 static unsigned clifford_check3(void)
103 symbol dim("D"), m("m"), q("q"), l("l"), ldotq("ldotq");
104 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), rho(symbol("rho"), dim),
105 sig(symbol("sig"), dim), kap(symbol("kap"), dim), lam(symbol("lam"), 4);
109 result += check_equal(dirac_trace(e), 0);
111 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho);
112 result += check_equal(dirac_trace(e), 0);
114 e = dirac_gamma5() * dirac_gamma(mu);
115 result += check_equal(dirac_trace(e), 0);
117 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu);
118 result += check_equal(dirac_trace(e), 0);
120 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho);
121 result += check_equal(dirac_trace(e), 0);
124 sp.add(q, q, pow(q, 2));
125 sp.add(l, l, pow(l, 2));
128 e = pow(m, 2) * dirac_slash(q, dim) * dirac_slash(q, dim);
129 e = dirac_trace(e).simplify_indexed(sp);
130 result += check_equal(e, 4*pow(m, 2)*pow(q, 2));
132 // cyclicity without gamma5
133 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig)
134 - dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(mu);
136 result += check_equal(e, 0);
138 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap) * dirac_gamma(lam)
139 - dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap) * dirac_gamma(lam) * dirac_gamma(mu);
140 e = dirac_trace(e).expand();
141 result += check_equal(e, 0);
143 // cyclicity of gamma5 * S_4
144 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig)
145 - dirac_gamma(sig) * dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho);
147 result += check_equal(e, 0);
149 // non-cyclicity of order D-4 of gamma5 * S_6
150 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap) * dirac_gamma(mu.toggle_variance())
151 + dim * dirac_gamma5() * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap);
153 e = (e / (dim - 4)).normal();
154 result += check_equal(e, 8 * eps0123(nu, rho, sig, kap));
156 // one-loop vacuum polarization in QED
157 e = dirac_gamma(mu) *
158 (dirac_slash(l, dim) + dirac_slash(q, dim) + m * dirac_ONE()) *
159 dirac_gamma(mu.toggle_variance()) *
160 (dirac_slash(l, dim) + m * dirac_ONE());
161 e = dirac_trace(e).simplify_indexed(sp);
162 result += check_equal(e, 4*((2-dim)*l*l + (2-dim)*ldotq + dim*m*m).expand());
164 e = dirac_slash(q, dim) *
165 (dirac_slash(l, dim) + dirac_slash(q, dim) + m * dirac_ONE()) *
166 dirac_slash(q, dim) *
167 (dirac_slash(l, dim) + m * dirac_ONE());
168 e = dirac_trace(e).simplify_indexed(sp);
169 result += check_equal(e, 4*(2*ldotq*ldotq + q*q*ldotq - q*q*l*l + q*q*m*m).expand());
174 unsigned exam_clifford(void)
178 cout << "examining clifford objects" << flush;
179 clog << "----------clifford objects:" << endl;
181 result += clifford_check1(); cout << '.' << flush;
182 result += clifford_check2(); cout << '.' << flush;
183 result += clifford_check3(); cout << '.' << flush;
186 cout << " passed " << endl;
187 clog << "(no output)" << endl;
189 cout << " failed " << endl;