1 /** @file exam_clifford.cpp
3 * Here we test GiNaC's Clifford algebra objects. */
6 * GiNaC Copyright (C) 1999-2005 Johannes Gutenberg University Mainz, Germany
8 * This program is free software; you can redistribute it and/or modify
9 * it under the terms of the GNU General Public License as published by
10 * the Free Software Foundation; either version 2 of the License, or
11 * (at your option) any later version.
13 * This program is distributed in the hope that it will be useful,
14 * but WITHOUT ANY WARRANTY; without even the implied warranty of
15 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
16 * GNU General Public License for more details.
18 * You should have received a copy of the GNU General Public License
19 * along with this program; if not, write to the Free Software
20 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
25 const numeric half(1, 2);
27 static unsigned check_equal(const ex &e1, const ex &e2)
29 ex e = normal(e1 - e2);
31 clog << "(" << e1 << ") - (" << e2 << ") erroneously returned "
32 << e << " instead of 0" << endl;
38 static unsigned check_equal_simplify(const ex &e1, const ex &e2)
40 ex e = normal(simplify_indexed(e1) - e2);
42 clog << "simplify_indexed(" << e1 << ") - (" << e2 << ") erroneously returned "
43 << e << " instead of 0" << endl;
49 static unsigned check_equal_lst(const ex & e1, const ex & e2)
51 for (unsigned int i = 0; i < e1.nops(); i++) {
52 ex e = e1.op(i) - e2.op(i);
53 if (!e.normal().is_zero()) {
54 clog << "(" << e1 << ") - (" << e2 << ") erroneously returned "
55 << e << " instead of 0 (in the entry " << i << ")" << endl;
62 static unsigned check_equal_simplify_term(const ex & e1, const ex & e2, varidx & mu)
64 ex e = expand_dummy_sum(normal(simplify_indexed(e1) - e2), true);
66 for (int j=0; j<4; j++) {
67 ex esub = e.subs(lst(mu == idx(j, mu.get_dim()), mu.toggle_variance() == idx(j, mu.get_dim())));
68 if (!(canonicalize_clifford(esub).is_zero())) {
69 clog << "simplify_indexed(" << e1 << ") - (" << e2 << ") erroneously returned "
70 << canonicalize_clifford(esub) << " instead of 0 for mu=" << j << endl;
77 static unsigned check_equal_simplify_term2(const ex & e1, const ex & e2)
79 ex e = expand_dummy_sum(normal(simplify_indexed(e1) - e2), true);
80 if (!(canonicalize_clifford(e).is_zero())) {
81 clog << "simplify_indexed(" << e1 << ") - (" << e2 << ") erroneously returned "
82 << canonicalize_clifford(e) << " instead of 0" << endl;
89 static unsigned clifford_check1()
91 // checks general identities and contractions
96 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), rho(symbol("rho"), dim);
99 e = dirac_ONE() * dirac_ONE();
100 result += check_equal(e, dirac_ONE());
102 e = dirac_ONE() * dirac_gamma(mu) * dirac_ONE();
103 result += check_equal(e, dirac_gamma(mu));
105 e = dirac_gamma(varidx(2, dim)) * dirac_gamma(varidx(1, dim)) *
106 dirac_gamma(varidx(1, dim)) * dirac_gamma(varidx(2, dim));
107 result += check_equal(e, dirac_ONE());
109 e = dirac_gamma(mu) * dirac_gamma(nu) *
110 dirac_gamma(nu.toggle_variance()) * dirac_gamma(mu.toggle_variance());
111 result += check_equal_simplify(e, pow(dim, 2) * dirac_ONE());
113 e = dirac_gamma(mu) * dirac_gamma(nu) *
114 dirac_gamma(mu.toggle_variance()) * dirac_gamma(nu.toggle_variance());
115 result += check_equal_simplify(e, 2*dim*dirac_ONE()-pow(dim, 2)*dirac_ONE());
117 e = dirac_gamma(nu.toggle_variance()) * dirac_gamma(rho.toggle_variance()) *
118 dirac_gamma(mu) * dirac_gamma(rho) * dirac_gamma(nu);
119 e = e.simplify_indexed().collect(dirac_gamma(mu));
120 result += check_equal(e, pow(2 - dim, 2).expand() * dirac_gamma(mu));
125 static unsigned clifford_check2()
127 // checks identities relating to gamma5
132 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim);
135 e = dirac_gamma(mu) * dirac_gamma5() + dirac_gamma5() * dirac_gamma(mu);
136 result += check_equal(e, 0);
138 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma5() + dirac_gamma(mu);
139 result += check_equal(e, 0);
144 static unsigned clifford_check3()
150 symbol dim("D"), m("m"), q("q"), l("l"), ldotq("ldotq");
151 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), rho(symbol("rho"), dim),
152 sig(symbol("sig"), dim), kap(symbol("kap"), dim), lam(symbol("lam"), dim);
156 result += check_equal(dirac_trace(e), 0);
158 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho);
159 result += check_equal(dirac_trace(e), 0);
161 e = dirac_gamma5() * dirac_gamma(mu);
162 result += check_equal(dirac_trace(e), 0);
164 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu);
165 result += check_equal(dirac_trace(e), 0);
167 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho);
168 result += check_equal(dirac_trace(e), 0);
171 sp.add(q, q, pow(q, 2));
172 sp.add(l, l, pow(l, 2));
175 e = pow(m, 2) * dirac_slash(q, dim) * dirac_slash(q, dim);
176 e = dirac_trace(e).simplify_indexed(sp);
177 result += check_equal(e, 4*pow(m, 2)*pow(q, 2));
179 // cyclicity without gamma5
180 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig)
181 - dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(mu);
183 result += check_equal(e, 0);
185 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap) * dirac_gamma(lam)
186 - dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap) * dirac_gamma(lam) * dirac_gamma(mu);
187 e = dirac_trace(e).expand();
188 result += check_equal(e, 0);
190 // cyclicity of gamma5 * S_4
191 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig)
192 - dirac_gamma(sig) * dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho);
194 result += check_equal(e, 0);
196 // non-cyclicity of order D-4 of gamma5 * S_6
197 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap) * dirac_gamma(mu.toggle_variance())
198 + dim * dirac_gamma5() * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap);
199 e = dirac_trace(e).simplify_indexed();
200 e = (e / (dim - 4)).normal();
201 result += check_equal(e, 8 * I * lorentz_eps(nu.replace_dim(4), rho.replace_dim(4), sig.replace_dim(4), kap.replace_dim(4)));
203 // one-loop vacuum polarization in QED
204 e = dirac_gamma(mu) *
205 (dirac_slash(l, dim) + dirac_slash(q, 4) + m * dirac_ONE()) *
206 dirac_gamma(mu.toggle_variance()) *
207 (dirac_slash(l, dim) + m * dirac_ONE());
208 e = dirac_trace(e).simplify_indexed(sp);
209 result += check_equal(e, 4*((2-dim)*l*l + (2-dim)*ldotq + dim*m*m).expand());
211 e = dirac_slash(q, 4) *
212 (dirac_slash(l, dim) + dirac_slash(q, 4) + m * dirac_ONE()) *
214 (dirac_slash(l, dim) + m * dirac_ONE());
215 e = dirac_trace(e).simplify_indexed(sp);
216 result += check_equal(e, 4*(2*ldotq*ldotq + q*q*ldotq - q*q*l*l + q*q*m*m).expand());
218 // stuff that had problems in the past
219 ex prop = dirac_slash(q, dim) - m * dirac_ONE();
220 e = dirac_slash(l, dim) * dirac_gamma5() * dirac_slash(l, dim) * prop;
221 e = dirac_trace(dirac_slash(q, dim) * e) - dirac_trace(m * e)
222 - dirac_trace(prop * e);
223 result += check_equal(e, 0);
225 e = (dirac_gamma5() + dirac_ONE()) * dirac_gamma5();
227 result += check_equal(e, 4);
229 // traces with multiple representation labels
230 e = dirac_ONE(0) * dirac_ONE(1) / 16;
231 result += check_equal(dirac_trace(e, 0), dirac_ONE(1) / 4);
232 result += check_equal(dirac_trace(e, 1), dirac_ONE(0) / 4);
233 result += check_equal(dirac_trace(e, 2), e);
234 result += check_equal(dirac_trace(e, lst(0, 1)), 1);
236 e = dirac_gamma(mu, 0) * dirac_gamma(mu.toggle_variance(), 1) * dirac_gamma(nu, 0) * dirac_gamma(nu.toggle_variance(), 1);
237 result += check_equal_simplify(dirac_trace(e, 0), 4 * dim * dirac_ONE(1));
238 result += check_equal_simplify(dirac_trace(e, 1), 4 * dim * dirac_ONE(0));
239 // Fails with new tinfo mechanism because the order of gamme matrices with different rl depends on luck.
240 // TODO: better check.
241 //result += check_equal_simplify(dirac_trace(e, 2), canonicalize_clifford(e)); // e will be canonicalized by the calculation of the trace
242 result += check_equal_simplify(dirac_trace(e, lst(0, 1)), 16 * dim);
247 static unsigned clifford_check4()
249 // simplify_indexed()/dirac_trace() cross-checks
254 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), rho(symbol("rho"), dim),
255 sig(symbol("sig"), dim), lam(symbol("lam"), dim);
258 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(mu.toggle_variance());
259 t1 = dirac_trace(e).simplify_indexed();
260 t2 = dirac_trace(e.simplify_indexed());
261 result += check_equal((t1 - t2).expand(), 0);
263 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(mu.toggle_variance()) * dirac_gamma(lam);
264 t1 = dirac_trace(e).simplify_indexed();
265 t2 = dirac_trace(e.simplify_indexed());
266 result += check_equal((t1 - t2).expand(), 0);
268 e = dirac_gamma(sig) * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(nu.toggle_variance()) * dirac_gamma(mu.toggle_variance());
269 t1 = dirac_trace(e).simplify_indexed();
270 t2 = dirac_trace(e.simplify_indexed());
271 result += check_equal((t1 - t2).expand(), 0);
273 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(mu.toggle_variance()) * dirac_gamma(sig) * dirac_gamma(nu.toggle_variance());
274 t1 = dirac_trace(e).simplify_indexed();
275 t2 = dirac_trace(e.simplify_indexed());
276 result += check_equal((t1 - t2).expand(), 0);
281 static unsigned clifford_check5()
283 // canonicalize_clifford() checks
288 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), lam(symbol("lam"), dim);
291 e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
292 result += check_equal(canonicalize_clifford(e), 2*dirac_ONE()*lorentz_g(mu, nu));
294 e = (dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(lam)
295 + dirac_gamma(nu) * dirac_gamma(lam) * dirac_gamma(mu)
296 + dirac_gamma(lam) * dirac_gamma(mu) * dirac_gamma(nu)
297 - dirac_gamma(nu) * dirac_gamma(mu) * dirac_gamma(lam)
298 - dirac_gamma(lam) * dirac_gamma(nu) * dirac_gamma(mu)
299 - dirac_gamma(mu) * dirac_gamma(lam) * dirac_gamma(nu)) / 6
300 + lorentz_g(mu, nu) * dirac_gamma(lam)
301 - lorentz_g(mu, lam) * dirac_gamma(nu)
302 + lorentz_g(nu, lam) * dirac_gamma(mu)
303 - dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(lam);
304 result += check_equal(canonicalize_clifford(e), 0);
310 static unsigned clifford_check6(const matrix & A)
312 varidx v(symbol("v"), 4), nu(symbol("nu"), 4), mu(symbol("mu"), 4),
313 psi(symbol("psi"),4), lam(symbol("lambda"), 4),
314 xi(symbol("xi"), 4), rho(symbol("rho"),4);
316 matrix A_symm(4,4), A2(4, 4);
317 A_symm = A.add(A.transpose()).mul(half);
318 A2 = A_symm.mul(A_symm);
321 bool anticommuting = ex_to<clifford>(clifford_unit(nu, A)).is_anticommuting();
324 // checks general identities and contractions for clifford_unit
325 e = dirac_ONE(2) * clifford_unit(mu, A, 2) * dirac_ONE(2);
326 result += check_equal(e, clifford_unit(mu, A, 2));
328 e = clifford_unit(idx(2, 4), A) * clifford_unit(idx(1, 4), A)
329 * clifford_unit(idx(1, 4), A) * clifford_unit(idx(2, 4), A);
330 result += check_equal(e, A(1, 1) * A(2, 2) * dirac_ONE());
332 e = clifford_unit(varidx(2, 4), A) * clifford_unit(varidx(1, 4), A)
333 * clifford_unit(varidx(1, 4), A) * clifford_unit(varidx(2, 4), A);
334 result += check_equal(e, A(1, 1) * A(2, 2) * dirac_ONE());
336 e = clifford_unit(nu, A) * clifford_unit(nu.toggle_variance(), A);
337 result += check_equal_simplify(e, A.trace() * dirac_ONE());
339 e = clifford_unit(nu, A) * clifford_unit(nu, A);
340 result += check_equal_simplify(e, indexed(A_symm, sy_symm(), nu, nu) * dirac_ONE());
342 e = clifford_unit(nu, A) * clifford_unit(nu.toggle_variance(), A) * clifford_unit(mu, A);
343 result += check_equal_simplify(e, A.trace() * clifford_unit(mu, A));
345 e = clifford_unit(nu, A) * clifford_unit(mu, A) * clifford_unit(nu.toggle_variance(), A);
347 result += check_equal_simplify(e, 2*indexed(A_symm, sy_symm(), mu, mu)*clifford_unit(mu, A) - A.trace()*clifford_unit(mu, A));
349 result += check_equal_simplify_term(e, 2 * indexed(A_symm, sy_symm(), nu.toggle_variance(), mu) *clifford_unit(nu, A)-A.trace()*clifford_unit(mu, A), mu);
351 e = clifford_unit(nu, A) * clifford_unit(nu.toggle_variance(), A)
352 * clifford_unit(mu, A) * clifford_unit(mu.toggle_variance(), A);
353 result += check_equal_simplify(e, pow(A.trace(), 2) * dirac_ONE());
355 e = clifford_unit(mu, A) * clifford_unit(nu, A)
356 * clifford_unit(nu.toggle_variance(), A) * clifford_unit(mu.toggle_variance(), A);
357 result += check_equal_simplify(e, pow(A.trace(), 2) * dirac_ONE());
359 e = clifford_unit(mu, A) * clifford_unit(nu, A)
360 * clifford_unit(mu.toggle_variance(), A) * clifford_unit(nu.toggle_variance(), A);
362 result += check_equal_simplify(e, 2*A2.trace()*dirac_ONE() - pow(A.trace(), 2)*dirac_ONE());
364 result += check_equal_simplify_term2(e, 2*indexed(A_symm, sy_symm(), nu.toggle_variance(), mu.toggle_variance()) * clifford_unit(mu, A) * clifford_unit(nu, A) - pow(A.trace(), 2)*dirac_ONE());
366 e = clifford_unit(mu.toggle_variance(), A) * clifford_unit(nu, A)
367 * clifford_unit(mu, A) * clifford_unit(nu.toggle_variance(), A);
369 result += check_equal_simplify(e, 2*A2.trace()*dirac_ONE() - pow(A.trace(), 2)*dirac_ONE());
370 e1 = remove_dirac_ONE(simplify_indexed(e));
371 result += check_equal(e1, 2*A2.trace() - pow(A.trace(), 2));
374 result += check_equal_simplify_term2(e, 2*indexed(A_symm, nu, mu) * clifford_unit(mu.toggle_variance(), A) * clifford_unit(nu.toggle_variance(), A) - pow(A.trace(), 2)*dirac_ONE());
376 e = clifford_unit(nu.toggle_variance(), A) * clifford_unit(rho.toggle_variance(), A)
377 * clifford_unit(mu, A) * clifford_unit(rho, A) * clifford_unit(nu, A);
378 e = e.simplify_indexed().collect(clifford_unit(mu, A));
380 result += check_equal(e, (4*indexed(A2, sy_symm(), mu, mu) - 4 * indexed(A_symm, sy_symm(), mu, mu)*A.trace() +pow(A.trace(), 2)) * clifford_unit(mu, A));
382 result += check_equal_simplify_term(e, 4* indexed(A_symm, sy_symm(), nu.toggle_variance(), rho)*indexed(A_symm, sy_symm(), rho.toggle_variance(), mu) *clifford_unit(nu, A)
383 - 2*A.trace() * (clifford_unit(rho, A) * indexed(A_symm, sy_symm(), rho.toggle_variance(), mu)
384 + clifford_unit(nu, A) * indexed(A_symm, sy_symm(), nu.toggle_variance(), mu)) + pow(A.trace(),2)* clifford_unit(mu, A), mu);
386 e = clifford_unit(nu.toggle_variance(), A) * clifford_unit(rho, A)
387 * clifford_unit(mu, A) * clifford_unit(rho.toggle_variance(), A) * clifford_unit(nu, A);
388 e = e.simplify_indexed().collect(clifford_unit(mu, A));
390 result += check_equal(e, (4*indexed(A2, sy_symm(), mu, mu) - 4*indexed(A_symm, sy_symm(), mu, mu)*A.trace() +pow(A.trace(), 2))* clifford_unit(mu, A));
392 result += check_equal_simplify_term(e, 4* indexed(A_symm, sy_symm(), nu.toggle_variance(), rho)*indexed(A_symm, sy_symm(), rho.toggle_variance(), mu) *clifford_unit(nu, A)
393 - 2*A.trace() * (clifford_unit(rho, A) * indexed(A_symm, sy_symm(), rho.toggle_variance(), mu)
394 + clifford_unit(nu, A) * indexed(A_symm, sy_symm(), nu.toggle_variance(), mu)) + pow(A.trace(),2)* clifford_unit(mu, A), mu);
396 e = clifford_unit(mu, A) * clifford_unit(nu, A) + clifford_unit(nu, A) * clifford_unit(mu, A);
397 result += check_equal(canonicalize_clifford(e), 2*dirac_ONE()*indexed(A_symm, sy_symm(), mu, nu));
399 e = (clifford_unit(mu, A) * clifford_unit(nu, A) * clifford_unit(lam, A)
400 + clifford_unit(nu, A) * clifford_unit(lam, A) * clifford_unit(mu, A)
401 + clifford_unit(lam, A) * clifford_unit(mu, A) * clifford_unit(nu, A)
402 - clifford_unit(nu, A) * clifford_unit(mu, A) * clifford_unit(lam, A)
403 - clifford_unit(lam, A) * clifford_unit(nu, A) * clifford_unit(mu, A)
404 - clifford_unit(mu, A) * clifford_unit(lam, A) * clifford_unit(nu, A)) / 6
405 + indexed(A_symm, sy_symm(), mu, nu) * clifford_unit(lam, A)
406 - indexed(A_symm, sy_symm(), mu, lam) * clifford_unit(nu, A)
407 + indexed(A_symm, sy_symm(), nu, lam) * clifford_unit(mu, A)
408 - clifford_unit(mu, A) * clifford_unit(nu, A) * clifford_unit(lam, A);
409 result += check_equal(canonicalize_clifford(e), 0);
411 // lst_to_clifford() and clifford_inverse() check
412 realsymbol x("x"), y("y"), t("t"), z("z");
414 ex c = clifford_unit(nu, A, 1);
415 e = lst_to_clifford(lst(t, x, y, z), mu, A, 1) * lst_to_clifford(lst(1, 2, 3, 4), c);
416 e1 = clifford_inverse(e);
417 result += check_equal((e*e1).simplify_indexed(), dirac_ONE(1));
419 // Moebius map (both forms) checks for symmetric metrics only
420 matrix M1(2, 2), M2(2, 2);
421 c = clifford_unit(nu, A);
423 e = clifford_moebius_map(0, dirac_ONE(),
424 dirac_ONE(), 0, lst(t, x, y, z), A); // this is just the inversion
427 e1 = clifford_moebius_map(M1, lst(t, x, y, z), A); // the inversion again
428 result += check_equal_lst(e, e1);
430 e1 = clifford_to_lst(clifford_inverse(lst_to_clifford(lst(t, x, y, z), mu, A)), c);
431 result += check_equal_lst(e, e1);
433 e = clifford_moebius_map(dirac_ONE(), lst_to_clifford(lst(1, 2, 3, 4), nu, A),
434 0, dirac_ONE(), lst(t, x, y, z), A); //this is just a shift
435 M2 = dirac_ONE(), lst_to_clifford(lst(1, 2, 3, 4), c),
437 e1 = clifford_moebius_map(M2, lst(t, x, y, z), c); // the same shift
438 result += check_equal_lst(e, e1);
440 result += check_equal(e, lst(t+1, x+2, y+3, z+4));
442 // Check the group law for Moebius maps
443 e = clifford_moebius_map(M1, ex_to<lst>(e1), c); //composition of M1 and M2
444 e1 = clifford_moebius_map(M1.mul(M2), lst(t, x, y, z), c); // the product M1*M2
445 result += check_equal_lst(e, e1);
451 static unsigned clifford_check7(const ex & G, const symbol & dim)
453 // checks general identities and contractions
457 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), rho(symbol("rho"), dim),
458 psi(symbol("psi"),dim), lam(symbol("lambda"), dim), xi(symbol("xi"), dim);
462 if (is_a<indexed>(G))
467 e = dirac_ONE() * dirac_ONE();
468 result += check_equal(e, dirac_ONE());
470 e = dirac_ONE() * clifford_unit(mu, G) * dirac_ONE();
471 result += check_equal(e, clifford_unit(mu, G));
473 e = clifford_unit(varidx(2, dim), G) * clifford_unit(varidx(1, dim), G)
474 * clifford_unit(varidx(1, dim), G) * clifford_unit(varidx(2, dim), G);
475 result += check_equal(e, dirac_ONE());
477 e = clifford_unit(mu, G) * clifford_unit(nu, G)
478 * clifford_unit(nu.toggle_variance(), G) * clifford_unit(mu.toggle_variance(), G);
479 result += check_equal_simplify(e, pow(dim, 2) * dirac_ONE());
481 e = clifford_unit(mu, G) * clifford_unit(nu, G)
482 * clifford_unit(mu.toggle_variance(), G) * clifford_unit(nu.toggle_variance(), G);
483 result += check_equal_simplify(e, 2*dim*dirac_ONE() - pow(dim, 2)*dirac_ONE());
485 e = clifford_unit(nu.toggle_variance(), G) * clifford_unit(rho.toggle_variance(), G)
486 * clifford_unit(mu, G) * clifford_unit(rho, G) * clifford_unit(nu, G);
487 e = e.simplify_indexed().collect(clifford_unit(mu, G));
488 result += check_equal(e, pow(2 - dim, 2).expand() * clifford_unit(mu, G));
490 // canonicalize_clifford() checks, only for symmetric metrics
491 if (ex_to<symmetry>(ex_to<indexed>(ex_to<clifford>(clifford_unit(mu, G)).get_metric()).get_symmetry()).has_symmetry()) {
492 e = clifford_unit(mu, G) * clifford_unit(nu, G) + clifford_unit(nu, G) * clifford_unit(mu, G);
493 result += check_equal(canonicalize_clifford(e), 2*dirac_ONE()*indexed(G_base, sy_symm(), nu, mu));
495 e = (clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(lam, G)
496 + clifford_unit(nu, G) * clifford_unit(lam, G) * clifford_unit(mu, G)
497 + clifford_unit(lam, G) * clifford_unit(mu, G) * clifford_unit(nu, G)
498 - clifford_unit(nu, G) * clifford_unit(mu, G) * clifford_unit(lam, G)
499 - clifford_unit(lam, G) * clifford_unit(nu, G) * clifford_unit(mu, G)
500 - clifford_unit(mu, G) * clifford_unit(lam, G) * clifford_unit(nu, G)) / 6
501 + indexed(G_base, sy_symm(), mu, nu) * clifford_unit(lam, G)
502 - indexed(G_base, sy_symm(), mu, lam) * clifford_unit(nu, G)
503 + indexed(G_base, sy_symm(), nu, lam) * clifford_unit(mu, G)
504 - clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(lam, G);
505 result += check_equal(canonicalize_clifford(e), 0);
507 e = clifford_unit(mu, G) * clifford_unit(nu, G) + clifford_unit(nu, G) * clifford_unit(mu, G);
508 result += check_equal(canonicalize_clifford(e), dirac_ONE()*(indexed(G_base, mu, nu) + indexed(G_base, nu, mu)));
510 e = (clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(lam, G)
511 + clifford_unit(nu, G) * clifford_unit(lam, G) * clifford_unit(mu, G)
512 + clifford_unit(lam, G) * clifford_unit(mu, G) * clifford_unit(nu, G)
513 - clifford_unit(nu, G) * clifford_unit(mu, G) * clifford_unit(lam, G)
514 - clifford_unit(lam, G) * clifford_unit(nu, G) * clifford_unit(mu, G)
515 - clifford_unit(mu, G) * clifford_unit(lam, G) * clifford_unit(nu, G)) / 6
516 + half * (indexed(G_base, mu, nu) + indexed(G_base, nu, mu)) * clifford_unit(lam, G)
517 - half * (indexed(G_base, mu, lam) + indexed(G_base, lam, mu)) * clifford_unit(nu, G)
518 + half * (indexed(G_base, nu, lam) + indexed(G_base, lam, nu)) * clifford_unit(mu, G)
519 - clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(lam, G);
520 result += check_equal(canonicalize_clifford(e), 0);
525 unsigned exam_clifford()
529 cout << "examining clifford objects" << flush;
530 clog << "----------clifford objects:" << endl;
532 result += clifford_check1(); cout << '.' << flush;
533 result += clifford_check2(); cout << '.' << flush;
534 result += clifford_check3(); cout << '.' << flush;
535 result += clifford_check4(); cout << '.' << flush;
536 result += clifford_check5(); cout << '.' << flush;
538 // anticommuting, symmetric examples
539 result += clifford_check6(ex_to<matrix>(diag_matrix(lst(-1, 1, 1, 1)))); cout << '.' << flush;
540 result += clifford_check6(ex_to<matrix>(diag_matrix(lst(-1, -1, -1, -1)))); cout << '.' << flush;
541 result += clifford_check6(ex_to<matrix>(diag_matrix(lst(-1, 1, 1, -1)))); cout << '.' << flush;
542 result += clifford_check6(ex_to<matrix>(diag_matrix(lst(-1, 0, 1, -1)))); cout << '.' << flush;
543 result += clifford_check6(ex_to<matrix>(diag_matrix(lst(-3, 0, 2, -1)))); cout << '.' << flush;
545 realsymbol s("s"), t("t"); // symbolic entries in matric
546 result += clifford_check6(ex_to<matrix>(diag_matrix(lst(-1, 1, s, t)))); cout << '.' << flush;
549 A = 1, 0, 0, 0, // anticommuting, not symmetric, Tr=0
553 result += clifford_check6(A); cout << '.' << flush;
555 A = 1, 0, 0, 0, // anticommuting, not symmetric, Tr=2
559 result += clifford_check6(A); cout << '.' << flush;
561 A = 1, 0, 0, 0, // not anticommuting, symmetric, Tr=0
565 result += clifford_check6(A); cout << '.' << flush;
567 A = 1, 0, 0, 0, // not anticommuting, symmetric, Tr=2
571 result += clifford_check6(A); cout << '.' << flush;
573 A = 1, 1, 0, 0, // not anticommuting, not symmetric, Tr=4
577 result += clifford_check6(A); cout << '.' << flush;
580 result += clifford_check7(minkmetric(), dim); cout << '.' << flush;
582 varidx chi(symbol("chi"), dim), xi(symbol("xi"), dim);
583 result += clifford_check7(lorentz_g(xi, chi), dim); cout << '.' << flush;
586 cout << " passed " << endl;
587 clog << "(no output)" << endl;
589 cout << " failed " << endl;