1 /** @file exam_clifford.cpp
3 * Here we test GiNaC's Clifford algebra objects. */
6 * GiNaC Copyright (C) 1999-2007 Johannes Gutenberg University Mainz, Germany
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14 * but WITHOUT ANY WARRANTY; without even the implied warranty of
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16 * GNU General Public License for more details.
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20 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 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()
49 // checks general identities and contractions
54 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), rho(symbol("rho"), 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());
75 e = dirac_gamma(nu.toggle_variance()) * dirac_gamma(rho.toggle_variance()) *
76 dirac_gamma(mu) * dirac_gamma(rho) * dirac_gamma(nu);
77 e = e.simplify_indexed().collect(dirac_gamma(mu));
78 result += check_equal(e, pow(2 - dim, 2).expand() * dirac_gamma(mu));
83 static unsigned clifford_check2()
85 // checks identities relating to gamma5
90 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim);
93 e = dirac_gamma(mu) * dirac_gamma5() + dirac_gamma5() * dirac_gamma(mu);
94 result += check_equal(e, 0);
96 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma5() + dirac_gamma(mu);
97 result += check_equal(e, 0);
102 static unsigned clifford_check3()
108 symbol dim("D"), m("m"), q("q"), l("l"), ldotq("ldotq");
109 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), rho(symbol("rho"), dim),
110 sig(symbol("sig"), dim), kap(symbol("kap"), dim), lam(symbol("lam"), dim);
114 result += check_equal(dirac_trace(e), 0);
116 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho);
117 result += check_equal(dirac_trace(e), 0);
119 e = dirac_gamma5() * dirac_gamma(mu);
120 result += check_equal(dirac_trace(e), 0);
122 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu);
123 result += check_equal(dirac_trace(e), 0);
125 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho);
126 result += check_equal(dirac_trace(e), 0);
129 sp.add(q, q, pow(q, 2));
130 sp.add(l, l, pow(l, 2));
133 e = pow(m, 2) * dirac_slash(q, dim) * dirac_slash(q, dim);
134 e = dirac_trace(e).simplify_indexed(sp);
135 result += check_equal(e, 4*pow(m, 2)*pow(q, 2));
137 // cyclicity without gamma5
138 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig)
139 - dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(mu);
141 result += check_equal(e, 0);
143 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap) * dirac_gamma(lam)
144 - dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap) * dirac_gamma(lam) * dirac_gamma(mu);
145 e = dirac_trace(e).expand();
146 result += check_equal(e, 0);
148 // cyclicity of gamma5 * S_4
149 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig)
150 - dirac_gamma(sig) * dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho);
152 result += check_equal(e, 0);
154 // non-cyclicity of order D-4 of gamma5 * S_6
155 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap) * dirac_gamma(mu.toggle_variance())
156 + dim * dirac_gamma5() * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap);
157 e = dirac_trace(e).simplify_indexed();
158 e = (e / (dim - 4)).normal();
159 result += check_equal(e, 8 * I * lorentz_eps(nu.replace_dim(4), rho.replace_dim(4), sig.replace_dim(4), kap.replace_dim(4)));
161 // one-loop vacuum polarization in QED
162 e = dirac_gamma(mu) *
163 (dirac_slash(l, dim) + dirac_slash(q, 4) + m * dirac_ONE()) *
164 dirac_gamma(mu.toggle_variance()) *
165 (dirac_slash(l, dim) + m * dirac_ONE());
166 e = dirac_trace(e).simplify_indexed(sp);
167 result += check_equal(e, 4*((2-dim)*l*l + (2-dim)*ldotq + dim*m*m).expand());
169 e = dirac_slash(q, 4) *
170 (dirac_slash(l, dim) + dirac_slash(q, 4) + m * dirac_ONE()) *
172 (dirac_slash(l, dim) + m * dirac_ONE());
173 e = dirac_trace(e).simplify_indexed(sp);
174 result += check_equal(e, 4*(2*ldotq*ldotq + q*q*ldotq - q*q*l*l + q*q*m*m).expand());
176 // stuff that had problems in the past
177 ex prop = dirac_slash(q, dim) - m * dirac_ONE();
178 e = dirac_slash(l, dim) * dirac_gamma5() * dirac_slash(l, dim) * prop;
179 e = dirac_trace(dirac_slash(q, dim) * e) - dirac_trace(m * e)
180 - dirac_trace(prop * e);
181 result += check_equal(e, 0);
183 e = (dirac_gamma5() + dirac_ONE()) * dirac_gamma5();
185 result += check_equal(e, 4);
187 // traces with multiple representation labels
188 e = dirac_ONE(0) * dirac_ONE(1) / 16;
189 result += check_equal(dirac_trace(e, 0), dirac_ONE(1) / 4);
190 result += check_equal(dirac_trace(e, 1), dirac_ONE(0) / 4);
191 result += check_equal(dirac_trace(e, 2), e);
192 result += check_equal(dirac_trace(e, lst(0, 1)), 1);
194 e = dirac_gamma(mu, 0) * dirac_gamma(mu.toggle_variance(), 1) * dirac_gamma(nu, 0) * dirac_gamma(nu.toggle_variance(), 1);
195 result += check_equal_simplify(dirac_trace(e, 0), 4 * dim * dirac_ONE(1));
196 result += check_equal_simplify(dirac_trace(e, 1), 4 * dim * dirac_ONE(0));
197 result += check_equal_simplify(dirac_trace(e, 2), e);
198 result += check_equal_simplify(dirac_trace(e, lst(0, 1)), 16 * dim);
203 static unsigned clifford_check4()
205 // simplify_indexed()/dirac_trace() cross-checks
210 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), rho(symbol("rho"), dim),
211 sig(symbol("sig"), dim), lam(symbol("lam"), dim);
214 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(mu.toggle_variance());
215 t1 = dirac_trace(e).simplify_indexed();
216 t2 = dirac_trace(e.simplify_indexed());
217 result += check_equal((t1 - t2).expand(), 0);
219 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(mu.toggle_variance()) * dirac_gamma(lam);
220 t1 = dirac_trace(e).simplify_indexed();
221 t2 = dirac_trace(e.simplify_indexed());
222 result += check_equal((t1 - t2).expand(), 0);
224 e = dirac_gamma(sig) * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(nu.toggle_variance()) * dirac_gamma(mu.toggle_variance());
225 t1 = dirac_trace(e).simplify_indexed();
226 t2 = dirac_trace(e.simplify_indexed());
227 result += check_equal((t1 - t2).expand(), 0);
229 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(mu.toggle_variance()) * dirac_gamma(sig) * dirac_gamma(nu.toggle_variance());
230 t1 = dirac_trace(e).simplify_indexed();
231 t2 = dirac_trace(e.simplify_indexed());
232 result += check_equal((t1 - t2).expand(), 0);
237 static unsigned clifford_check5()
239 // canonicalize_clifford() checks
244 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), lam(symbol("lam"), dim);
247 e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
248 result += check_equal(canonicalize_clifford(e), 2*dirac_ONE()*lorentz_g(mu, nu));
250 e = (dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(lam)
251 + dirac_gamma(nu) * dirac_gamma(lam) * dirac_gamma(mu)
252 + dirac_gamma(lam) * dirac_gamma(mu) * dirac_gamma(nu)
253 - dirac_gamma(nu) * dirac_gamma(mu) * dirac_gamma(lam)
254 - dirac_gamma(lam) * dirac_gamma(nu) * dirac_gamma(mu)
255 - dirac_gamma(mu) * dirac_gamma(lam) * dirac_gamma(nu)) / 6
256 + lorentz_g(mu, nu) * dirac_gamma(lam)
257 - lorentz_g(mu, lam) * dirac_gamma(nu)
258 + lorentz_g(nu, lam) * dirac_gamma(mu)
259 - dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(lam);
260 result += check_equal(canonicalize_clifford(e), 0);
265 static unsigned clifford_check6(const matrix & A)
267 varidx v(symbol("v"), 4), nu(symbol("nu"), 4), mu(symbol("mu"), 4),
268 psi(symbol("psi"),4), lam(symbol("lambda"), 4),
269 xi(symbol("xi"), 4), rho(symbol("rho"),4);
279 // checks general identities and contractions for clifford_unit
280 e = dirac_ONE() * clifford_unit(mu, G) * dirac_ONE();
281 result += check_equal(e, clifford_unit(mu, G));
283 e = clifford_unit(varidx(2, 4), G) * clifford_unit(varidx(1, 4), G)
284 * clifford_unit(varidx(1, 4), G) * clifford_unit(varidx(2, 4), G);
285 result += check_equal(e, A(1, 1) * A(2, 2) * dirac_ONE());
287 e = clifford_unit(nu, G) * clifford_unit(nu.toggle_variance(), G);
288 result += check_equal_simplify(e, A.trace() * dirac_ONE());
290 e = clifford_unit(nu, G) * clifford_unit(nu, G);
291 result += check_equal_simplify(e, indexed(G, sy_symm(), nu, nu) * dirac_ONE());
293 e = clifford_unit(nu, G) * clifford_unit(nu.toggle_variance(), G) * clifford_unit(mu, G);
294 result += check_equal_simplify(e, A.trace() * clifford_unit(mu, G));
296 e = clifford_unit(nu, G) * clifford_unit(mu, G) * clifford_unit(nu.toggle_variance(), G);
297 result += check_equal_simplify(e, 2*indexed(G, sy_symm(), mu, mu)*clifford_unit(mu, G) - A.trace()*clifford_unit(mu, G));
299 e = clifford_unit(nu, G) * clifford_unit(nu.toggle_variance(), G)
300 * clifford_unit(mu, G) * clifford_unit(mu.toggle_variance(), G);
301 result += check_equal_simplify(e, pow(A.trace(), 2) * dirac_ONE());
303 e = clifford_unit(mu, G) * clifford_unit(nu, G)
304 * clifford_unit(nu.toggle_variance(), G) * clifford_unit(mu.toggle_variance(), G);
305 result += check_equal_simplify(e, pow(A.trace(), 2) * dirac_ONE());
307 e = clifford_unit(mu, G) * clifford_unit(nu, G)
308 * clifford_unit(mu.toggle_variance(), G) * clifford_unit(nu.toggle_variance(), G);
309 result += check_equal_simplify(e, 2*A2.trace()*dirac_ONE() - pow(A.trace(), 2)*dirac_ONE());
311 e = clifford_unit(mu.toggle_variance(), G) * clifford_unit(nu, G)
312 * clifford_unit(mu, G) * clifford_unit(nu.toggle_variance(), G);
313 result += check_equal_simplify(e, 2*A2.trace()*dirac_ONE() - pow(A.trace(), 2)*dirac_ONE());
315 e = clifford_unit(nu.toggle_variance(), G) * clifford_unit(rho.toggle_variance(), G)
316 * clifford_unit(mu, G) * clifford_unit(rho, G) * clifford_unit(nu, G);
317 e = e.simplify_indexed().collect(clifford_unit(mu, G));
318 result += check_equal(e, (pow(A.trace(), 2)+4-4*A.trace()*indexed(A, mu, mu)) * clifford_unit(mu, G));
320 e = clifford_unit(nu.toggle_variance(), G) * clifford_unit(rho, G)
321 * clifford_unit(mu, G) * clifford_unit(rho.toggle_variance(), G) * clifford_unit(nu, G);
322 e = e.simplify_indexed().collect(clifford_unit(mu, G));
323 result += check_equal(e, (pow(A.trace(), 2)+4-4*A.trace()*indexed(A, mu, mu))* clifford_unit(mu, G));
325 // canonicalize_clifford() checks
326 e = clifford_unit(mu, G) * clifford_unit(nu, G) + clifford_unit(nu, G) * clifford_unit(mu, G);
327 result += check_equal(canonicalize_clifford(e), 2*dirac_ONE()*indexed(G, sy_symm(), mu, nu));
329 e = (clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(lam, G)
330 + clifford_unit(nu, G) * clifford_unit(lam, G) * clifford_unit(mu, G)
331 + clifford_unit(lam, G) * clifford_unit(mu, G) * clifford_unit(nu, G)
332 - clifford_unit(nu, G) * clifford_unit(mu, G) * clifford_unit(lam, G)
333 - clifford_unit(lam, G) * clifford_unit(nu, G) * clifford_unit(mu, G)
334 - clifford_unit(mu, G) * clifford_unit(lam, G) * clifford_unit(nu, G)) / 6
335 + indexed(G, sy_symm(), mu, nu) * clifford_unit(lam, G)
336 - indexed(G, sy_symm(), mu, lam) * clifford_unit(nu, G)
337 + indexed(G, sy_symm(), nu, lam) * clifford_unit(mu, G)
338 - clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(lam, G);
339 result += check_equal(canonicalize_clifford(e), 0);
341 // lst_to_clifford() and clifford_inverse() check
342 symbol x("x"), y("y"), t("t"), z("z");
344 ex c = clifford_unit(nu, G, 1);
345 e = lst_to_clifford(lst(t, x, y, z), mu, G, 1) * lst_to_clifford(lst(1, 2, 3, 4), c);
346 e1 = clifford_inverse(e);
347 result += check_equal((e*e1).simplify_indexed().normal(), dirac_ONE(1));
352 static unsigned clifford_check7()
354 // checks general identities and contractions
359 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), rho(symbol("rho"), dim),
360 psi(symbol("psi"),dim), lam(symbol("lambda"), dim), xi(symbol("xi"), dim);
366 e = dirac_ONE() * dirac_ONE();
367 result += check_equal(e, dirac_ONE());
369 e = dirac_ONE() * clifford_unit(mu, G) * dirac_ONE();
370 result += check_equal(e, clifford_unit(mu, G));
372 e = clifford_unit(varidx(2, dim), G) * clifford_unit(varidx(1, dim), G)
373 * clifford_unit(varidx(1, dim), G) * clifford_unit(varidx(2, dim), G);
374 result += check_equal(e, dirac_ONE());
376 e = clifford_unit(mu, G) * clifford_unit(nu, G)
377 * clifford_unit(nu.toggle_variance(), G) * clifford_unit(mu.toggle_variance(), G);
378 result += check_equal_simplify(e, pow(dim, 2) * dirac_ONE());
380 e = clifford_unit(mu, G) * clifford_unit(nu, G)
381 * clifford_unit(mu.toggle_variance(), G) * clifford_unit(nu.toggle_variance(), G);
382 result += check_equal_simplify(e, 2*dim*dirac_ONE() - pow(dim, 2)*dirac_ONE());
384 e = clifford_unit(nu.toggle_variance(), G) * clifford_unit(rho.toggle_variance(), G)
385 * clifford_unit(mu, G) * clifford_unit(rho, G) * clifford_unit(nu, G);
386 e = e.simplify_indexed().collect(clifford_unit(mu, G));
387 result += check_equal(e, pow(2 - dim, 2).expand() * clifford_unit(mu, G));
389 // canonicalize_clifford() checks
390 e = clifford_unit(mu, G) * clifford_unit(nu, G) + clifford_unit(nu, G) * clifford_unit(mu, G);
391 result += check_equal(canonicalize_clifford(e), 2*dirac_ONE()*indexed(G, sy_symm(), mu, nu));
393 e = (clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(lam, G)
394 + clifford_unit(nu, G) * clifford_unit(lam, G) * clifford_unit(mu, G)
395 + clifford_unit(lam, G) * clifford_unit(mu, G) * clifford_unit(nu, G)
396 - clifford_unit(nu, G) * clifford_unit(mu, G) * clifford_unit(lam, G)
397 - clifford_unit(lam, G) * clifford_unit(nu, G) * clifford_unit(mu, G)
398 - clifford_unit(mu, G) * clifford_unit(lam, G) * clifford_unit(nu, G)) / 6
399 + indexed(G, sy_symm(), mu, nu) * clifford_unit(lam, G)
400 - indexed(G, sy_symm(), mu, lam) * clifford_unit(nu, G)
401 + indexed(G, sy_symm(), nu, lam) * clifford_unit(mu, G)
402 - clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(lam, G);
403 result += check_equal(canonicalize_clifford(e), 0);
408 unsigned exam_clifford()
412 cout << "examining clifford objects" << flush;
413 clog << "----------clifford objects:" << endl;
415 result += clifford_check1(); cout << '.' << flush;
416 result += clifford_check2(); cout << '.' << flush;
417 result += clifford_check3(); cout << '.' << flush;
418 result += clifford_check4(); cout << '.' << flush;
419 result += clifford_check5(); cout << '.' << flush;
426 result += clifford_check6(A); cout << '.' << flush;
432 result += clifford_check6(A); cout << '.' << flush;
438 result += clifford_check6(A); cout << '.' << flush;
444 result += clifford_check6(A); cout << '.' << flush;
446 result += clifford_check7(); cout << '.' << flush;
449 cout << " passed " << endl;
450 clog << "(no output)" << endl;
452 cout << " failed " << endl;