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
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, idx & mu)
64 ex e = expand_dummy_sum(normal(simplify_indexed(e1) - e2), true);
66 for (int j=0; j<4; j++) {
70 mu == idx(j, mu.get_dim()),
71 ex_to<varidx>(mu).toggle_variance() == idx(j, mu.get_dim())
73 : lst(mu == idx(j, mu.get_dim()))
75 if (!(canonicalize_clifford(esub).is_zero())) {
76 clog << "simplify_indexed(" << e1 << ") - (" << e2 << ") erroneously returned "
77 << canonicalize_clifford(esub) << " instead of 0 for mu=" << j << endl;
84 static unsigned check_equal_simplify_term2(const ex & e1, const ex & e2)
86 ex e = expand_dummy_sum(normal(simplify_indexed(e1) - e2), true);
87 if (!(canonicalize_clifford(e).is_zero())) {
88 clog << "simplify_indexed(" << e1 << ") - (" << e2 << ") erroneously returned "
89 << canonicalize_clifford(e) << " instead of 0" << endl;
96 static unsigned clifford_check1()
98 // checks general identities and contractions
103 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), rho(symbol("rho"), dim);
106 e = dirac_ONE() * dirac_ONE();
107 result += check_equal(e, dirac_ONE());
109 e = dirac_ONE() * dirac_gamma(mu) * dirac_ONE();
110 result += check_equal(e, dirac_gamma(mu));
112 e = dirac_gamma(varidx(2, dim)) * dirac_gamma(varidx(1, dim)) *
113 dirac_gamma(varidx(1, dim)) * dirac_gamma(varidx(2, dim));
114 result += check_equal(e, dirac_ONE());
116 e = dirac_gamma(mu) * dirac_gamma(nu) *
117 dirac_gamma(nu.toggle_variance()) * dirac_gamma(mu.toggle_variance());
118 result += check_equal_simplify(e, pow(dim, 2) * dirac_ONE());
120 e = dirac_gamma(mu) * dirac_gamma(nu) *
121 dirac_gamma(mu.toggle_variance()) * dirac_gamma(nu.toggle_variance());
122 result += check_equal_simplify(e, 2*dim*dirac_ONE()-pow(dim, 2)*dirac_ONE());
124 e = dirac_gamma(nu.toggle_variance()) * dirac_gamma(rho.toggle_variance()) *
125 dirac_gamma(mu) * dirac_gamma(rho) * dirac_gamma(nu);
126 e = e.simplify_indexed().collect(dirac_gamma(mu));
127 result += check_equal(e, pow(2 - dim, 2).expand() * dirac_gamma(mu));
132 static unsigned clifford_check2()
134 // checks identities relating to gamma5
139 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim);
142 e = dirac_gamma(mu) * dirac_gamma5() + dirac_gamma5() * dirac_gamma(mu);
143 result += check_equal(e, 0);
145 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma5() + dirac_gamma(mu);
146 result += check_equal(e, 0);
151 static unsigned clifford_check3()
157 symbol dim("D"), m("m"), q("q"), l("l"), ldotq("ldotq");
158 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), rho(symbol("rho"), dim),
159 sig(symbol("sig"), dim), kap(symbol("kap"), dim), lam(symbol("lam"), dim);
163 result += check_equal(dirac_trace(e), 0);
165 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho);
166 result += check_equal(dirac_trace(e), 0);
168 e = dirac_gamma5() * dirac_gamma(mu);
169 result += check_equal(dirac_trace(e), 0);
171 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu);
172 result += check_equal(dirac_trace(e), 0);
174 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho);
175 result += check_equal(dirac_trace(e), 0);
178 sp.add(q, q, pow(q, 2));
179 sp.add(l, l, pow(l, 2));
182 e = pow(m, 2) * dirac_slash(q, dim) * dirac_slash(q, dim);
183 e = dirac_trace(e).simplify_indexed(sp);
184 result += check_equal(e, 4*pow(m, 2)*pow(q, 2));
186 // cyclicity without gamma5
187 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig)
188 - dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(mu);
190 result += check_equal(e, 0);
192 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap) * dirac_gamma(lam)
193 - dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap) * dirac_gamma(lam) * dirac_gamma(mu);
194 e = dirac_trace(e).expand();
195 result += check_equal(e, 0);
197 // cyclicity of gamma5 * S_4
198 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig)
199 - dirac_gamma(sig) * dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho);
201 result += check_equal(e, 0);
203 // non-cyclicity of order D-4 of gamma5 * S_6
204 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap) * dirac_gamma(mu.toggle_variance())
205 + dim * dirac_gamma5() * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap);
206 e = dirac_trace(e).simplify_indexed();
207 e = (e / (dim - 4)).normal();
208 result += check_equal(e, 8 * I * lorentz_eps(nu.replace_dim(4), rho.replace_dim(4), sig.replace_dim(4), kap.replace_dim(4)));
210 // one-loop vacuum polarization in QED
211 e = dirac_gamma(mu) *
212 (dirac_slash(l, dim) + dirac_slash(q, 4) + m * dirac_ONE()) *
213 dirac_gamma(mu.toggle_variance()) *
214 (dirac_slash(l, dim) + m * dirac_ONE());
215 e = dirac_trace(e).simplify_indexed(sp);
216 result += check_equal(e, 4*((2-dim)*l*l + (2-dim)*ldotq + dim*m*m).expand());
218 e = dirac_slash(q, 4) *
219 (dirac_slash(l, dim) + dirac_slash(q, 4) + m * dirac_ONE()) *
221 (dirac_slash(l, dim) + m * dirac_ONE());
222 e = dirac_trace(e).simplify_indexed(sp);
223 result += check_equal(e, 4*(2*ldotq*ldotq + q*q*ldotq - q*q*l*l + q*q*m*m).expand());
225 // stuff that had problems in the past
226 ex prop = dirac_slash(q, dim) - m * dirac_ONE();
227 e = dirac_slash(l, dim) * dirac_gamma5() * dirac_slash(l, dim) * prop;
228 e = dirac_trace(dirac_slash(q, dim) * e) - dirac_trace(m * e)
229 - dirac_trace(prop * e);
230 result += check_equal(e, 0);
232 e = (dirac_gamma5() + dirac_ONE()) * dirac_gamma5();
234 result += check_equal(e, 4);
236 // traces with multiple representation labels
237 e = dirac_ONE(0) * dirac_ONE(1) / 16;
238 result += check_equal(dirac_trace(e, 0), dirac_ONE(1) / 4);
239 result += check_equal(dirac_trace(e, 1), dirac_ONE(0) / 4);
240 result += check_equal(dirac_trace(e, 2), e);
241 result += check_equal(dirac_trace(e, lst(0, 1)), 1);
243 e = dirac_gamma(mu, 0) * dirac_gamma(mu.toggle_variance(), 1) * dirac_gamma(nu, 0) * dirac_gamma(nu.toggle_variance(), 1);
244 result += check_equal_simplify(dirac_trace(e, 0), 4 * dim * dirac_ONE(1));
245 result += check_equal_simplify(dirac_trace(e, 1), 4 * dim * dirac_ONE(0));
246 // Fails with new tinfo mechanism because the order of gamme matrices with different rl depends on luck.
247 // TODO: better check.
248 //result += check_equal_simplify(dirac_trace(e, 2), canonicalize_clifford(e)); // e will be canonicalized by the calculation of the trace
249 result += check_equal_simplify(dirac_trace(e, lst(0, 1)), 16 * dim);
254 static unsigned clifford_check4()
256 // simplify_indexed()/dirac_trace() cross-checks
261 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), rho(symbol("rho"), dim),
262 sig(symbol("sig"), dim), lam(symbol("lam"), dim);
265 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(mu.toggle_variance());
266 t1 = dirac_trace(e).simplify_indexed();
267 t2 = dirac_trace(e.simplify_indexed());
268 result += check_equal((t1 - t2).expand(), 0);
270 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(mu.toggle_variance()) * dirac_gamma(lam);
271 t1 = dirac_trace(e).simplify_indexed();
272 t2 = dirac_trace(e.simplify_indexed());
273 result += check_equal((t1 - t2).expand(), 0);
275 e = dirac_gamma(sig) * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(nu.toggle_variance()) * dirac_gamma(mu.toggle_variance());
276 t1 = dirac_trace(e).simplify_indexed();
277 t2 = dirac_trace(e.simplify_indexed());
278 result += check_equal((t1 - t2).expand(), 0);
280 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(mu.toggle_variance()) * dirac_gamma(sig) * dirac_gamma(nu.toggle_variance());
281 t1 = dirac_trace(e).simplify_indexed();
282 t2 = dirac_trace(e.simplify_indexed());
283 result += check_equal((t1 - t2).expand(), 0);
288 static unsigned clifford_check5()
290 // canonicalize_clifford() checks
295 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), lam(symbol("lam"), dim);
298 e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
299 result += check_equal(canonicalize_clifford(e), 2*dirac_ONE()*lorentz_g(mu, nu));
301 e = (dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(lam)
302 + dirac_gamma(nu) * dirac_gamma(lam) * dirac_gamma(mu)
303 + dirac_gamma(lam) * dirac_gamma(mu) * dirac_gamma(nu)
304 - dirac_gamma(nu) * dirac_gamma(mu) * dirac_gamma(lam)
305 - dirac_gamma(lam) * dirac_gamma(nu) * dirac_gamma(mu)
306 - dirac_gamma(mu) * dirac_gamma(lam) * dirac_gamma(nu)) / 6
307 + lorentz_g(mu, nu) * dirac_gamma(lam)
308 - lorentz_g(mu, lam) * dirac_gamma(nu)
309 + lorentz_g(nu, lam) * dirac_gamma(mu)
310 - dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(lam);
311 result += check_equal(canonicalize_clifford(e), 0);
316 /* We make two identical checks with metrics defined through a matrix in
317 * the cases when used indexes have or have not variance.
318 * To this end we recycle the code through the following macros */
320 template <typename IDX> unsigned clifford_check6(const matrix &A)
324 matrix A_symm(4,4), A2(4, 4);
325 A_symm = A.add(A.transpose()).mul(half);
326 A2 = A_symm.mul(A_symm);
328 IDX v(symbol("v"), 4), nu(symbol("nu"), 4), mu(symbol("mu"), 4),
329 psi(symbol("psi"),4), lam(symbol("lambda"), 4),
330 xi(symbol("xi"), 4), rho(symbol("rho"),4);
331 ex mu_TOGGLE = is_a<varidx>(mu) ? ex_to<varidx>(mu).toggle_variance() : mu;
332 ex nu_TOGGLE = is_a<varidx>(nu) ? ex_to<varidx>(nu).toggle_variance() : nu;
334 = is_a<varidx>(rho) ? ex_to<varidx>(rho).toggle_variance() : rho;
338 /* checks general identities and contractions for clifford_unit*/
339 e = dirac_ONE(2) * clifford_unit(mu, A, 2) * dirac_ONE(2);
340 result += check_equal(e, clifford_unit(mu, A, 2));
342 e = clifford_unit(IDX(2, 4), A) * clifford_unit(IDX(1, 4), A)
343 * clifford_unit(IDX(1, 4), A) * clifford_unit(IDX(2, 4), A);
344 result += check_equal(e, A(1, 1) * A(2, 2) * dirac_ONE());
346 e = clifford_unit(IDX(2, 4), A) * clifford_unit(IDX(1, 4), A)
347 * clifford_unit(IDX(1, 4), A) * clifford_unit(IDX(2, 4), A);
348 result += check_equal(e, A(1, 1) * A(2, 2) * dirac_ONE());
350 e = clifford_unit(nu, A) * clifford_unit(nu_TOGGLE, A);
351 result += check_equal_simplify(e, A.trace() * dirac_ONE());
353 e = clifford_unit(nu, A) * clifford_unit(nu, A);
354 result += check_equal_simplify(e, indexed(A_symm, sy_symm(), nu, nu) * dirac_ONE());
356 e = clifford_unit(nu, A) * clifford_unit(nu_TOGGLE, A) * clifford_unit(mu, A);
357 result += check_equal_simplify(e, A.trace() * clifford_unit(mu, A));
359 e = clifford_unit(nu, A) * clifford_unit(mu, A) * clifford_unit(nu_TOGGLE, A);
361 result += check_equal_simplify_term(e, 2 * indexed(A_symm, sy_symm(), nu_TOGGLE, mu) *clifford_unit(nu, A)-A.trace()*clifford_unit(mu, A), mu);
363 e = clifford_unit(nu, A) * clifford_unit(nu_TOGGLE, A)
364 * clifford_unit(mu, A) * clifford_unit(mu_TOGGLE, A);
365 result += check_equal_simplify(e, pow(A.trace(), 2) * dirac_ONE());
367 e = clifford_unit(mu, A) * clifford_unit(nu, A)
368 * clifford_unit(nu_TOGGLE, A) * clifford_unit(mu_TOGGLE, A);
369 result += check_equal_simplify(e, pow(A.trace(), 2) * dirac_ONE());
371 e = clifford_unit(mu, A) * clifford_unit(nu, A)
372 * clifford_unit(mu_TOGGLE, A) * clifford_unit(nu_TOGGLE, A);
374 result += check_equal_simplify_term2(e, 2*indexed(A_symm, sy_symm(), nu_TOGGLE, mu_TOGGLE) * clifford_unit(mu, A) * clifford_unit(nu, A) - pow(A.trace(), 2)*dirac_ONE());
376 e = clifford_unit(mu_TOGGLE, A) * clifford_unit(nu, A)
377 * clifford_unit(mu, A) * clifford_unit(nu_TOGGLE, A);
379 result += check_equal_simplify_term2(e, 2*indexed(A_symm, nu, mu) * clifford_unit(mu_TOGGLE, A) * clifford_unit(nu_TOGGLE, A) - pow(A.trace(), 2)*dirac_ONE());
381 e = clifford_unit(nu_TOGGLE, A) * clifford_unit(rho_TOGGLE, A)
382 * clifford_unit(mu, A) * clifford_unit(rho, A) * clifford_unit(nu, A);
383 e = e.simplify_indexed().collect(clifford_unit(mu, A));
385 result += check_equal_simplify_term(e, 4* indexed(A_symm, sy_symm(), nu_TOGGLE, rho)*indexed(A_symm, sy_symm(), rho_TOGGLE, mu) *clifford_unit(nu, A)
386 - 2*A.trace() * (clifford_unit(rho, A) * indexed(A_symm, sy_symm(), rho_TOGGLE, mu)
387 + clifford_unit(nu, A) * indexed(A_symm, sy_symm(), nu_TOGGLE, mu)) + pow(A.trace(),2)* clifford_unit(mu, A), mu);
389 e = clifford_unit(nu_TOGGLE, A) * clifford_unit(rho, A)
390 * clifford_unit(mu, A) * clifford_unit(rho_TOGGLE, A) * clifford_unit(nu, A);
391 e = e.simplify_indexed().collect(clifford_unit(mu, A));
393 result += check_equal_simplify_term(e, 4* indexed(A_symm, sy_symm(), nu_TOGGLE, rho)*indexed(A_symm, sy_symm(), rho_TOGGLE, mu) *clifford_unit(nu, A)
394 - 2*A.trace() * (clifford_unit(rho, A) * indexed(A_symm, sy_symm(), rho_TOGGLE, mu)
395 + clifford_unit(nu, A) * indexed(A_symm, sy_symm(), nu_TOGGLE, mu)) + pow(A.trace(),2)* clifford_unit(mu, A), mu);
397 e = clifford_unit(mu, A) * clifford_unit(nu, A) + clifford_unit(nu, A) * clifford_unit(mu, A);
398 result += check_equal(canonicalize_clifford(e), 2*dirac_ONE()*indexed(A_symm, sy_symm(), mu, nu));
400 e = (clifford_unit(mu, A) * clifford_unit(nu, A) * clifford_unit(lam, A)
401 + clifford_unit(nu, A) * clifford_unit(lam, A) * clifford_unit(mu, A)
402 + clifford_unit(lam, A) * clifford_unit(mu, A) * clifford_unit(nu, A)
403 - clifford_unit(nu, A) * clifford_unit(mu, A) * clifford_unit(lam, A)
404 - clifford_unit(lam, A) * clifford_unit(nu, A) * clifford_unit(mu, A)
405 - clifford_unit(mu, A) * clifford_unit(lam, A) * clifford_unit(nu, A)) / 6
406 + indexed(A_symm, sy_symm(), mu, nu) * clifford_unit(lam, A)
407 - indexed(A_symm, sy_symm(), mu, lam) * clifford_unit(nu, A)
408 + indexed(A_symm, sy_symm(), nu, lam) * clifford_unit(mu, A)
409 - clifford_unit(mu, A) * clifford_unit(nu, A) * clifford_unit(lam, A);
410 result += check_equal(canonicalize_clifford(e), 0);
412 /* lst_to_clifford() and clifford_inverse() check*/
413 realsymbol s("s"), t("t"), x("x"), y("y"), z("z");
415 ex c = clifford_unit(nu, A, 1);
416 e = lst_to_clifford(lst(t, x, y, z), mu, A, 1) * lst_to_clifford(lst(1, 2, 3, 4), c);
417 e1 = clifford_inverse(e);
418 result += check_equal_simplify_term2((e*e1).simplify_indexed(), dirac_ONE(1));
420 /* lst_to_clifford() and clifford_to_lst() check for vectors*/
422 result += check_equal_lst(clifford_to_lst(lst_to_clifford(e, c), c, false), e);
423 result += check_equal_lst(clifford_to_lst(lst_to_clifford(e, c), c, true), e);
425 /* lst_to_clifford() and clifford_to_lst() check for pseudovectors*/
426 e = lst(s, t, x, y, z);
427 result += check_equal_lst(clifford_to_lst(lst_to_clifford(e, c), c, false), e);
428 result += check_equal_lst(clifford_to_lst(lst_to_clifford(e, c), c, true), e);
430 /* Moebius map (both forms) checks for symmetric metrics only */
431 matrix M1(2, 2), M2(2, 2);
432 c = clifford_unit(nu, A);
434 e = clifford_moebius_map(0, dirac_ONE(),
435 dirac_ONE(), 0, lst(t, x, y, z), A);
436 /* this is just the inversion*/
439 e1 = clifford_moebius_map(M1, lst(t, x, y, z), A);
440 /* the inversion again*/
441 result += check_equal_lst(e, e1);
443 e1 = clifford_to_lst(clifford_inverse(lst_to_clifford(lst(t, x, y, z), mu, A)), c);
444 result += check_equal_lst(e, e1);
446 e = clifford_moebius_map(dirac_ONE(), lst_to_clifford(lst(1, 2, 3, 4), nu, A),
447 0, dirac_ONE(), lst(t, x, y, z), A);
448 /*this is just a shift*/
449 M2 = dirac_ONE(), lst_to_clifford(lst(1, 2, 3, 4), c),
451 e1 = clifford_moebius_map(M2, lst(t, x, y, z), c);
453 result += check_equal_lst(e, e1);
455 result += check_equal(e, lst(t+1, x+2, y+3, z+4));
457 /* Check the group law for Moebius maps */
458 e = clifford_moebius_map(M1, ex_to<lst>(e1), c);
459 /*composition of M1 and M2*/
460 e1 = clifford_moebius_map(M1.mul(M2), lst(t, x, y, z), c);
461 /* the product M1*M2*/
462 result += check_equal_lst(e, e1);
466 static unsigned clifford_check7(const ex & G, const symbol & dim)
468 // checks general identities and contractions
472 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), rho(symbol("rho"), dim),
473 psi(symbol("psi"),dim), lam(symbol("lambda"), dim), xi(symbol("xi"), dim);
476 clifford unit = ex_to<clifford>(clifford_unit(mu, G));
477 ex scalar = unit.get_metric(varidx(0, dim), varidx(0, dim));
479 e = dirac_ONE() * dirac_ONE();
480 result += check_equal(e, dirac_ONE());
482 e = dirac_ONE() * clifford_unit(mu, G) * dirac_ONE();
483 result += check_equal(e, clifford_unit(mu, G));
485 e = clifford_unit(varidx(2, dim), G) * clifford_unit(varidx(1, dim), G)
486 * clifford_unit(varidx(1, dim), G) * clifford_unit(varidx(2, dim), G);
487 result += check_equal(e, dirac_ONE()*pow(scalar, 2));
489 e = clifford_unit(mu, G) * clifford_unit(nu, G)
490 * clifford_unit(nu.toggle_variance(), G) * clifford_unit(mu.toggle_variance(), G);
491 result += check_equal_simplify(e, pow(dim*scalar, 2) * dirac_ONE());
493 e = clifford_unit(mu, G) * clifford_unit(nu, G)
494 * clifford_unit(mu.toggle_variance(), G) * clifford_unit(nu.toggle_variance(), G);
495 result += check_equal_simplify(e, (2*dim - pow(dim, 2))*pow(scalar,2)*dirac_ONE());
497 e = clifford_unit(nu.toggle_variance(), G) * clifford_unit(rho.toggle_variance(), G)
498 * clifford_unit(mu, G) * clifford_unit(rho, G) * clifford_unit(nu, G);
499 e = e.simplify_indexed().collect(clifford_unit(mu, G));
500 result += check_equal(e, pow(scalar*(dim-2), 2).expand() * clifford_unit(mu, G));
502 // canonicalize_clifford() checks, only for symmetric metrics
503 if (ex_to<symmetry>(ex_to<indexed>(ex_to<clifford>(clifford_unit(mu, G)).get_metric()).get_symmetry()).has_symmetry()) {
504 e = clifford_unit(mu, G) * clifford_unit(nu, G) + clifford_unit(nu, G) * clifford_unit(mu, G);
505 result += check_equal(canonicalize_clifford(e), 2*dirac_ONE()*unit.get_metric(nu, mu));
507 e = (clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(lam, G)
508 + clifford_unit(nu, G) * clifford_unit(lam, G) * clifford_unit(mu, G)
509 + clifford_unit(lam, G) * clifford_unit(mu, G) * clifford_unit(nu, G)
510 - clifford_unit(nu, G) * clifford_unit(mu, G) * clifford_unit(lam, G)
511 - clifford_unit(lam, G) * clifford_unit(nu, G) * clifford_unit(mu, G)
512 - clifford_unit(mu, G) * clifford_unit(lam, G) * clifford_unit(nu, G)) / 6
513 + unit.get_metric(mu, nu) * clifford_unit(lam, G)
514 - unit.get_metric(mu, lam) * clifford_unit(nu, G)
515 + unit.get_metric(nu, lam) * clifford_unit(mu, G)
516 - clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(lam, G);
517 result += check_equal(canonicalize_clifford(e), 0);
519 e = clifford_unit(mu, G) * clifford_unit(nu, G) + clifford_unit(nu, G) * clifford_unit(mu, G);
520 result += check_equal(canonicalize_clifford(e), dirac_ONE()*(unit.get_metric(mu, nu) + unit.get_metric(nu, mu)));
522 e = (clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(lam, G)
523 + clifford_unit(nu, G) * clifford_unit(lam, G) * clifford_unit(mu, G)
524 + clifford_unit(lam, G) * clifford_unit(mu, G) * clifford_unit(nu, G)
525 - clifford_unit(nu, G) * clifford_unit(mu, G) * clifford_unit(lam, G)
526 - clifford_unit(lam, G) * clifford_unit(nu, G) * clifford_unit(mu, G)
527 - clifford_unit(mu, G) * clifford_unit(lam, G) * clifford_unit(nu, G)) / 6
528 + half * (unit.get_metric(mu, nu) + unit.get_metric(nu, mu)) * clifford_unit(lam, G)
529 - half * (unit.get_metric(mu, lam) + unit.get_metric(lam, mu)) * clifford_unit(nu, G)
530 + half * (unit.get_metric(nu, lam) + unit.get_metric(lam, nu)) * clifford_unit(mu, G)
531 - clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(lam, G);
532 result += check_equal(canonicalize_clifford(e), 0);
537 unsigned exam_clifford()
541 cout << "examining clifford objects" << flush;
542 clog << "----------clifford objects:" << endl;
544 result += clifford_check1(); cout << '.' << flush;
545 result += clifford_check2(); cout << '.' << flush;
546 result += clifford_check3(); cout << '.' << flush;
547 result += clifford_check4(); cout << '.' << flush;
548 result += clifford_check5(); cout << '.' << flush;
550 // anticommuting, symmetric examples
551 result += clifford_check6<varidx>(ex_to<matrix>(diag_matrix(lst(-1, 1, 1, 1))));
552 result += clifford_check6<idx>(ex_to<matrix>(diag_matrix(lst(-1, 1, 1, 1))));; cout << '.' << flush;
553 result += clifford_check6<varidx>(ex_to<matrix>(diag_matrix(lst(-1, -1, -1, -1))))+clifford_check6<idx>(ex_to<matrix>(diag_matrix(lst(-1, -1, -1, -1))));; cout << '.' << flush;
554 result += clifford_check6<idx>(ex_to<matrix>(diag_matrix(lst(-1, 1, 1, -1))))+clifford_check6<idx>(ex_to<matrix>(diag_matrix(lst(-1, 1, 1, -1))));; cout << '.' << flush;
555 result += clifford_check6<varidx>(ex_to<matrix>(diag_matrix(lst(-1, 0, 1, -1))))+clifford_check6<idx>(ex_to<matrix>(diag_matrix(lst(-1, 0, 1, -1))));; cout << '.' << flush;
556 result += clifford_check6<varidx>(ex_to<matrix>(diag_matrix(lst(-3, 0, 2, -1))))+clifford_check6<idx>(ex_to<matrix>(diag_matrix(lst(-3, 0, 2, -1))));; cout << '.' << flush;
558 realsymbol s("s"), t("t"); // symbolic entries in matric
559 result += clifford_check6<varidx>(ex_to<matrix>(diag_matrix(lst(-1, 1, s, t))))+clifford_check6<idx>(ex_to<matrix>(diag_matrix(lst(-1, 1, s, t))));; cout << '.' << flush;
562 A = 1, 0, 0, 0, // anticommuting, not symmetric, Tr=0
566 result += clifford_check6<varidx>(A)+clifford_check6<idx>(A);; cout << '.' << flush;
568 A = 1, 0, 0, 0, // anticommuting, not symmetric, Tr=2
572 result += clifford_check6<varidx>(A)+clifford_check6<idx>(A);; cout << '.' << flush;
574 A = 1, 0, 0, 0, // not anticommuting, symmetric, Tr=0
578 result += clifford_check6<varidx>(A)+clifford_check6<idx>(A);; cout << '.' << flush;
580 A = 1, 0, 0, 0, // not anticommuting, symmetric, Tr=2
584 result += clifford_check6<varidx>(A)+clifford_check6<idx>(A);; cout << '.' << flush;
586 A = 1, 1, 0, 0, // not anticommuting, not symmetric, Tr=4
590 result += clifford_check6<varidx>(A)+clifford_check6<idx>(A);; cout << '.' << flush;
593 result += clifford_check7(minkmetric(), dim); cout << '.' << flush;
595 varidx chi(symbol("chi"), dim), xi(symbol("xi"), dim);
596 result += clifford_check7(delta_tensor(xi, chi), dim); cout << '.' << flush;
598 result += clifford_check7(lorentz_g(xi, chi), dim); cout << '.' << flush;
600 result += clifford_check7(indexed(-2*minkmetric(), sy_symm(), xi, chi), dim); cout << '.' << flush;
601 result += clifford_check7(-2*delta_tensor(xi, chi), dim); cout << '.' << flush;
604 cout << " passed " << endl;
605 clog << "(no output)" << endl;
607 cout << " failed " << endl;