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
26 using namespace GiNaC;
28 const numeric half(1, 2);
30 static unsigned check_equal(const ex &e1, const ex &e2)
32 ex e = normal(e1 - e2);
34 clog << "(" << e1 << ") - (" << e2 << ") erroneously returned "
35 << e << " instead of 0" << endl;
41 static unsigned check_equal_simplify(const ex &e1, const ex &e2)
43 ex e = normal(simplify_indexed(e1) - e2);
45 clog << "simplify_indexed(" << e1 << ") - (" << e2 << ") erroneously returned "
46 << e << " instead of 0" << endl;
52 static unsigned check_equal_lst(const ex & e1, const ex & e2)
54 for (unsigned int i = 0; i < e1.nops(); i++) {
55 ex e = e1.op(i) - e2.op(i);
56 if (!e.normal().is_zero()) {
57 clog << "(" << e1 << ") - (" << e2 << ") erroneously returned "
58 << e << " instead of 0 (in the entry " << i << ")" << endl;
65 static unsigned check_equal_simplify_term(const ex & e1, const ex & e2, idx & mu)
67 ex e = expand_dummy_sum(normal(simplify_indexed(e1) - e2), true);
69 for (int j=0; j<4; j++) {
73 mu == idx(j, mu.get_dim()),
74 ex_to<varidx>(mu).toggle_variance() == idx(j, mu.get_dim())
76 : lst(mu == idx(j, mu.get_dim()))
78 if (!(canonicalize_clifford(esub).is_zero())) {
79 clog << "simplify_indexed(" << e1 << ") - (" << e2 << ") erroneously returned "
80 << canonicalize_clifford(esub) << " instead of 0 for mu=" << j << endl;
87 static unsigned check_equal_simplify_term2(const ex & e1, const ex & e2)
89 ex e = expand_dummy_sum(normal(simplify_indexed(e1) - e2), true);
90 if (!(canonicalize_clifford(e).is_zero())) {
91 clog << "simplify_indexed(" << e1 << ") - (" << e2 << ") erroneously returned "
92 << canonicalize_clifford(e) << " instead of 0" << endl;
99 static unsigned clifford_check1()
101 // checks general identities and contractions
106 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), rho(symbol("rho"), dim);
109 e = dirac_ONE() * dirac_ONE();
110 result += check_equal(e, dirac_ONE());
112 e = dirac_ONE() * dirac_gamma(mu) * dirac_ONE();
113 result += check_equal(e, dirac_gamma(mu));
115 e = dirac_gamma(varidx(2, dim)) * dirac_gamma(varidx(1, dim)) *
116 dirac_gamma(varidx(1, dim)) * dirac_gamma(varidx(2, dim));
117 result += check_equal(e, dirac_ONE());
119 e = dirac_gamma(mu) * dirac_gamma(nu) *
120 dirac_gamma(nu.toggle_variance()) * dirac_gamma(mu.toggle_variance());
121 result += check_equal_simplify(e, pow(dim, 2) * dirac_ONE());
123 e = dirac_gamma(mu) * dirac_gamma(nu) *
124 dirac_gamma(mu.toggle_variance()) * dirac_gamma(nu.toggle_variance());
125 result += check_equal_simplify(e, 2*dim*dirac_ONE()-pow(dim, 2)*dirac_ONE());
127 e = dirac_gamma(nu.toggle_variance()) * dirac_gamma(rho.toggle_variance()) *
128 dirac_gamma(mu) * dirac_gamma(rho) * dirac_gamma(nu);
129 e = e.simplify_indexed().collect(dirac_gamma(mu));
130 result += check_equal(e, pow(2 - dim, 2).expand() * dirac_gamma(mu));
135 static unsigned clifford_check2()
137 // checks identities relating to gamma5
142 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim);
145 e = dirac_gamma(mu) * dirac_gamma5() + dirac_gamma5() * dirac_gamma(mu);
146 result += check_equal(e, 0);
148 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma5() + dirac_gamma(mu);
149 result += check_equal(e, 0);
154 static unsigned clifford_check3()
160 symbol dim("D"), m("m"), q("q"), l("l"), ldotq("ldotq");
161 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), rho(symbol("rho"), dim),
162 sig(symbol("sig"), dim), kap(symbol("kap"), dim), lam(symbol("lam"), dim);
166 result += check_equal(dirac_trace(e), 0);
168 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho);
169 result += check_equal(dirac_trace(e), 0);
171 e = dirac_gamma5() * dirac_gamma(mu);
172 result += check_equal(dirac_trace(e), 0);
174 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu);
175 result += check_equal(dirac_trace(e), 0);
177 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho);
178 result += check_equal(dirac_trace(e), 0);
181 sp.add(q, q, pow(q, 2));
182 sp.add(l, l, pow(l, 2));
185 e = pow(m, 2) * dirac_slash(q, dim) * dirac_slash(q, dim);
186 e = dirac_trace(e).simplify_indexed(sp);
187 result += check_equal(e, 4*pow(m, 2)*pow(q, 2));
189 // cyclicity without gamma5
190 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig)
191 - dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(mu);
193 result += check_equal(e, 0);
195 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap) * dirac_gamma(lam)
196 - dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap) * dirac_gamma(lam) * dirac_gamma(mu);
197 e = dirac_trace(e).expand();
198 result += check_equal(e, 0);
200 // cyclicity of gamma5 * S_4
201 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig)
202 - dirac_gamma(sig) * dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho);
204 result += check_equal(e, 0);
206 // non-cyclicity of order D-4 of gamma5 * S_6
207 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap) * dirac_gamma(mu.toggle_variance())
208 + dim * dirac_gamma5() * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap);
209 e = dirac_trace(e).simplify_indexed();
210 e = (e / (dim - 4)).normal();
211 result += check_equal(e, 8 * I * lorentz_eps(nu.replace_dim(4), rho.replace_dim(4), sig.replace_dim(4), kap.replace_dim(4)));
213 // one-loop vacuum polarization in QED
214 e = dirac_gamma(mu) *
215 (dirac_slash(l, dim) + dirac_slash(q, 4) + m * dirac_ONE()) *
216 dirac_gamma(mu.toggle_variance()) *
217 (dirac_slash(l, dim) + m * dirac_ONE());
218 e = dirac_trace(e).simplify_indexed(sp);
219 result += check_equal(e, 4*((2-dim)*l*l + (2-dim)*ldotq + dim*m*m).expand());
221 e = dirac_slash(q, 4) *
222 (dirac_slash(l, dim) + dirac_slash(q, 4) + m * dirac_ONE()) *
224 (dirac_slash(l, dim) + m * dirac_ONE());
225 e = dirac_trace(e).simplify_indexed(sp);
226 result += check_equal(e, 4*(2*ldotq*ldotq + q*q*ldotq - q*q*l*l + q*q*m*m).expand());
228 // stuff that had problems in the past
229 ex prop = dirac_slash(q, dim) - m * dirac_ONE();
230 e = dirac_slash(l, dim) * dirac_gamma5() * dirac_slash(l, dim) * prop;
231 e = dirac_trace(dirac_slash(q, dim) * e) - dirac_trace(m * e)
232 - dirac_trace(prop * e);
233 result += check_equal(e, 0);
235 e = (dirac_gamma5() + dirac_ONE()) * dirac_gamma5();
237 result += check_equal(e, 4);
239 // traces with multiple representation labels
240 e = dirac_ONE(0) * dirac_ONE(1) / 16;
241 result += check_equal(dirac_trace(e, 0), dirac_ONE(1) / 4);
242 result += check_equal(dirac_trace(e, 1), dirac_ONE(0) / 4);
243 result += check_equal(dirac_trace(e, 2), e);
244 result += check_equal(dirac_trace(e, lst(0, 1)), 1);
246 e = dirac_gamma(mu, 0) * dirac_gamma(mu.toggle_variance(), 1) * dirac_gamma(nu, 0) * dirac_gamma(nu.toggle_variance(), 1);
247 result += check_equal_simplify(dirac_trace(e, 0), 4 * dim * dirac_ONE(1));
248 result += check_equal_simplify(dirac_trace(e, 1), 4 * dim * dirac_ONE(0));
249 // Fails with new tinfo mechanism because the order of gamme matrices with different rl depends on luck.
250 // TODO: better check.
251 //result += check_equal_simplify(dirac_trace(e, 2), canonicalize_clifford(e)); // e will be canonicalized by the calculation of the trace
252 result += check_equal_simplify(dirac_trace(e, lst(0, 1)), 16 * dim);
257 static unsigned clifford_check4()
259 // simplify_indexed()/dirac_trace() cross-checks
264 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), rho(symbol("rho"), dim),
265 sig(symbol("sig"), dim), lam(symbol("lam"), dim);
268 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * 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(sig) * dirac_gamma(mu.toggle_variance()) * dirac_gamma(lam);
274 t1 = dirac_trace(e).simplify_indexed();
275 t2 = dirac_trace(e.simplify_indexed());
276 result += check_equal((t1 - t2).expand(), 0);
278 e = dirac_gamma(sig) * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(nu.toggle_variance()) * dirac_gamma(mu.toggle_variance());
279 t1 = dirac_trace(e).simplify_indexed();
280 t2 = dirac_trace(e.simplify_indexed());
281 result += check_equal((t1 - t2).expand(), 0);
283 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(mu.toggle_variance()) * dirac_gamma(sig) * dirac_gamma(nu.toggle_variance());
284 t1 = dirac_trace(e).simplify_indexed();
285 t2 = dirac_trace(e.simplify_indexed());
286 result += check_equal((t1 - t2).expand(), 0);
291 static unsigned clifford_check5()
293 // canonicalize_clifford() checks
298 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), lam(symbol("lam"), dim);
301 e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
302 result += check_equal(canonicalize_clifford(e), 2*dirac_ONE()*lorentz_g(mu, nu));
304 e = (dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(lam)
305 + dirac_gamma(nu) * dirac_gamma(lam) * dirac_gamma(mu)
306 + dirac_gamma(lam) * dirac_gamma(mu) * dirac_gamma(nu)
307 - dirac_gamma(nu) * dirac_gamma(mu) * dirac_gamma(lam)
308 - dirac_gamma(lam) * dirac_gamma(nu) * dirac_gamma(mu)
309 - dirac_gamma(mu) * dirac_gamma(lam) * dirac_gamma(nu)) / 6
310 + lorentz_g(mu, nu) * dirac_gamma(lam)
311 - lorentz_g(mu, lam) * dirac_gamma(nu)
312 + lorentz_g(nu, lam) * dirac_gamma(mu)
313 - dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(lam);
314 result += check_equal(canonicalize_clifford(e), 0);
319 /* We make two identical checks with metrics defined through a matrix in
320 * the cases when used indexes have or have not variance.
321 * To this end we recycle the code through the following macros */
323 template <typename IDX> unsigned clifford_check6(const matrix &A)
327 matrix A_symm(4,4), A2(4, 4);
328 A_symm = A.add(A.transpose()).mul(half);
329 A2 = A_symm.mul(A_symm);
331 IDX v(symbol("v"), 4), nu(symbol("nu"), 4), mu(symbol("mu"), 4),
332 psi(symbol("psi"),4), lam(symbol("lambda"), 4),
333 xi(symbol("xi"), 4), rho(symbol("rho"),4);
334 ex mu_TOGGLE = is_a<varidx>(mu) ? ex_to<varidx>(mu).toggle_variance() : mu;
335 ex nu_TOGGLE = is_a<varidx>(nu) ? ex_to<varidx>(nu).toggle_variance() : nu;
337 = is_a<varidx>(rho) ? ex_to<varidx>(rho).toggle_variance() : rho;
341 /* checks general identities and contractions for clifford_unit*/
342 e = dirac_ONE(2) * clifford_unit(mu, A, 2) * dirac_ONE(2);
343 result += check_equal(e, clifford_unit(mu, A, 2));
345 e = clifford_unit(IDX(2, 4), A) * clifford_unit(IDX(1, 4), A)
346 * clifford_unit(IDX(1, 4), A) * clifford_unit(IDX(2, 4), A);
347 result += check_equal(e, A(1, 1) * A(2, 2) * dirac_ONE());
349 e = clifford_unit(IDX(2, 4), A) * clifford_unit(IDX(1, 4), A)
350 * clifford_unit(IDX(1, 4), A) * clifford_unit(IDX(2, 4), A);
351 result += check_equal(e, A(1, 1) * A(2, 2) * dirac_ONE());
353 e = clifford_unit(nu, A) * clifford_unit(nu_TOGGLE, A);
354 result += check_equal_simplify(e, A.trace() * dirac_ONE());
356 e = clifford_unit(nu, A) * clifford_unit(nu, A);
357 result += check_equal_simplify(e, indexed(A_symm, sy_symm(), nu, nu) * dirac_ONE());
359 e = clifford_unit(nu, A) * clifford_unit(nu_TOGGLE, A) * clifford_unit(mu, A);
360 result += check_equal_simplify(e, A.trace() * clifford_unit(mu, A));
362 e = clifford_unit(nu, A) * clifford_unit(mu, A) * clifford_unit(nu_TOGGLE, A);
364 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);
366 e = clifford_unit(nu, A) * clifford_unit(nu_TOGGLE, A)
367 * clifford_unit(mu, A) * clifford_unit(mu_TOGGLE, A);
368 result += check_equal_simplify(e, pow(A.trace(), 2) * dirac_ONE());
370 e = clifford_unit(mu, A) * clifford_unit(nu, A)
371 * clifford_unit(nu_TOGGLE, A) * clifford_unit(mu_TOGGLE, A);
372 result += check_equal_simplify(e, pow(A.trace(), 2) * dirac_ONE());
374 e = clifford_unit(mu, A) * clifford_unit(nu, A)
375 * clifford_unit(mu_TOGGLE, A) * clifford_unit(nu_TOGGLE, A);
377 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());
379 e = clifford_unit(mu_TOGGLE, A) * clifford_unit(nu, A)
380 * clifford_unit(mu, A) * clifford_unit(nu_TOGGLE, A);
382 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());
384 e = clifford_unit(nu_TOGGLE, A) * clifford_unit(rho_TOGGLE, A)
385 * clifford_unit(mu, A) * clifford_unit(rho, A) * clifford_unit(nu, A);
386 e = e.simplify_indexed().collect(clifford_unit(mu, A));
388 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)
389 - 2*A.trace() * (clifford_unit(rho, A) * indexed(A_symm, sy_symm(), rho_TOGGLE, mu)
390 + clifford_unit(nu, A) * indexed(A_symm, sy_symm(), nu_TOGGLE, mu)) + pow(A.trace(),2)* clifford_unit(mu, A), mu);
392 e = clifford_unit(nu_TOGGLE, A) * clifford_unit(rho, A)
393 * clifford_unit(mu, A) * clifford_unit(rho_TOGGLE, A) * clifford_unit(nu, A);
394 e = e.simplify_indexed().collect(clifford_unit(mu, A));
396 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)
397 - 2*A.trace() * (clifford_unit(rho, A) * indexed(A_symm, sy_symm(), rho_TOGGLE, mu)
398 + clifford_unit(nu, A) * indexed(A_symm, sy_symm(), nu_TOGGLE, mu)) + pow(A.trace(),2)* clifford_unit(mu, A), mu);
400 e = clifford_unit(mu, A) * clifford_unit(nu, A) + clifford_unit(nu, A) * clifford_unit(mu, A);
401 result += check_equal(canonicalize_clifford(e), 2*dirac_ONE()*indexed(A_symm, sy_symm(), mu, nu));
403 e = (clifford_unit(mu, A) * clifford_unit(nu, A) * clifford_unit(lam, A)
404 + clifford_unit(nu, A) * clifford_unit(lam, A) * clifford_unit(mu, A)
405 + clifford_unit(lam, A) * clifford_unit(mu, A) * clifford_unit(nu, A)
406 - clifford_unit(nu, A) * clifford_unit(mu, A) * clifford_unit(lam, A)
407 - clifford_unit(lam, A) * clifford_unit(nu, A) * clifford_unit(mu, A)
408 - clifford_unit(mu, A) * clifford_unit(lam, A) * clifford_unit(nu, A)) / 6
409 + indexed(A_symm, sy_symm(), mu, nu) * clifford_unit(lam, A)
410 - indexed(A_symm, sy_symm(), mu, lam) * clifford_unit(nu, A)
411 + indexed(A_symm, sy_symm(), nu, lam) * clifford_unit(mu, A)
412 - clifford_unit(mu, A) * clifford_unit(nu, A) * clifford_unit(lam, A);
413 result += check_equal(canonicalize_clifford(e), 0);
415 /* lst_to_clifford() and clifford_inverse() check*/
416 realsymbol s("s"), t("t"), x("x"), y("y"), z("z");
418 ex c = clifford_unit(nu, A, 1);
419 e = lst_to_clifford(lst(t, x, y, z), mu, A, 1) * lst_to_clifford(lst(1, 2, 3, 4), c);
420 e1 = clifford_inverse(e);
421 result += check_equal_simplify_term2((e*e1).simplify_indexed(), dirac_ONE(1));
423 /* lst_to_clifford() and clifford_to_lst() check for vectors*/
425 result += check_equal_lst(clifford_to_lst(lst_to_clifford(e, c), c, false), e);
426 result += check_equal_lst(clifford_to_lst(lst_to_clifford(e, c), c, true), e);
428 /* lst_to_clifford() and clifford_to_lst() check for pseudovectors*/
429 e = lst(s, t, x, y, z);
430 result += check_equal_lst(clifford_to_lst(lst_to_clifford(e, c), c, false), e);
431 result += check_equal_lst(clifford_to_lst(lst_to_clifford(e, c), c, true), e);
433 /* Moebius map (both forms) checks for symmetric metrics only */
434 matrix M1(2, 2), M2(2, 2);
435 c = clifford_unit(nu, A);
437 e = clifford_moebius_map(0, dirac_ONE(),
438 dirac_ONE(), 0, lst(t, x, y, z), A);
439 /* this is just the inversion*/
442 e1 = clifford_moebius_map(M1, lst(t, x, y, z), A);
443 /* the inversion again*/
444 result += check_equal_lst(e, e1);
446 e1 = clifford_to_lst(clifford_inverse(lst_to_clifford(lst(t, x, y, z), mu, A)), c);
447 result += check_equal_lst(e, e1);
449 e = clifford_moebius_map(dirac_ONE(), lst_to_clifford(lst(1, 2, 3, 4), nu, A),
450 0, dirac_ONE(), lst(t, x, y, z), A);
451 /*this is just a shift*/
452 M2 = dirac_ONE(), lst_to_clifford(lst(1, 2, 3, 4), c),
454 e1 = clifford_moebius_map(M2, lst(t, x, y, z), c);
456 result += check_equal_lst(e, e1);
458 result += check_equal(e, lst(t+1, x+2, y+3, z+4));
460 /* Check the group law for Moebius maps */
461 e = clifford_moebius_map(M1, ex_to<lst>(e1), c);
462 /*composition of M1 and M2*/
463 e1 = clifford_moebius_map(M1.mul(M2), lst(t, x, y, z), c);
464 /* the product M1*M2*/
465 result += check_equal_lst(e, e1);
469 static unsigned clifford_check7(const ex & G, const symbol & dim)
471 // checks general identities and contractions
475 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), rho(symbol("rho"), dim),
476 psi(symbol("psi"),dim), lam(symbol("lambda"), dim), xi(symbol("xi"), dim);
479 clifford unit = ex_to<clifford>(clifford_unit(mu, G));
480 ex scalar = unit.get_metric(varidx(0, dim), varidx(0, dim));
482 e = dirac_ONE() * dirac_ONE();
483 result += check_equal(e, dirac_ONE());
485 e = dirac_ONE() * clifford_unit(mu, G) * dirac_ONE();
486 result += check_equal(e, clifford_unit(mu, G));
488 e = clifford_unit(varidx(2, dim), G) * clifford_unit(varidx(1, dim), G)
489 * clifford_unit(varidx(1, dim), G) * clifford_unit(varidx(2, dim), G);
490 result += check_equal(e, dirac_ONE()*pow(scalar, 2));
492 e = clifford_unit(mu, G) * clifford_unit(nu, G)
493 * clifford_unit(nu.toggle_variance(), G) * clifford_unit(mu.toggle_variance(), G);
494 result += check_equal_simplify(e, pow(dim*scalar, 2) * dirac_ONE());
496 e = clifford_unit(mu, G) * clifford_unit(nu, G)
497 * clifford_unit(mu.toggle_variance(), G) * clifford_unit(nu.toggle_variance(), G);
498 result += check_equal_simplify(e, (2*dim - pow(dim, 2))*pow(scalar,2)*dirac_ONE());
500 e = clifford_unit(nu.toggle_variance(), G) * clifford_unit(rho.toggle_variance(), G)
501 * clifford_unit(mu, G) * clifford_unit(rho, G) * clifford_unit(nu, G);
502 e = e.simplify_indexed().collect(clifford_unit(mu, G));
503 result += check_equal(e, pow(scalar*(dim-2), 2).expand() * clifford_unit(mu, G));
505 // canonicalize_clifford() checks, only for symmetric metrics
506 if (ex_to<symmetry>(ex_to<indexed>(ex_to<clifford>(clifford_unit(mu, G)).get_metric()).get_symmetry()).has_symmetry()) {
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), 2*dirac_ONE()*unit.get_metric(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 + unit.get_metric(mu, nu) * clifford_unit(lam, G)
517 - unit.get_metric(mu, lam) * clifford_unit(nu, G)
518 + unit.get_metric(nu, lam) * 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);
522 e = clifford_unit(mu, G) * clifford_unit(nu, G) + clifford_unit(nu, G) * clifford_unit(mu, G);
523 result += check_equal(canonicalize_clifford(e), dirac_ONE()*(unit.get_metric(mu, nu) + unit.get_metric(nu, mu)));
525 e = (clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(lam, G)
526 + clifford_unit(nu, G) * clifford_unit(lam, G) * clifford_unit(mu, G)
527 + clifford_unit(lam, G) * clifford_unit(mu, G) * clifford_unit(nu, G)
528 - clifford_unit(nu, G) * clifford_unit(mu, G) * clifford_unit(lam, G)
529 - clifford_unit(lam, G) * clifford_unit(nu, G) * clifford_unit(mu, G)
530 - clifford_unit(mu, G) * clifford_unit(lam, G) * clifford_unit(nu, G)) / 6
531 + half * (unit.get_metric(mu, nu) + unit.get_metric(nu, mu)) * clifford_unit(lam, G)
532 - half * (unit.get_metric(mu, lam) + unit.get_metric(lam, mu)) * clifford_unit(nu, G)
533 + half * (unit.get_metric(nu, lam) + unit.get_metric(lam, nu)) * clifford_unit(mu, G)
534 - clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(lam, G);
535 result += check_equal(canonicalize_clifford(e), 0);
540 unsigned exam_clifford()
544 cout << "examining clifford objects" << flush;
546 result += clifford_check1(); cout << '.' << flush;
547 result += clifford_check2(); cout << '.' << flush;
548 result += clifford_check3(); cout << '.' << flush;
549 result += clifford_check4(); cout << '.' << flush;
550 result += clifford_check5(); cout << '.' << flush;
552 // anticommuting, symmetric examples
553 result += clifford_check6<varidx>(ex_to<matrix>(diag_matrix(lst(-1, 1, 1, 1))));
554 result += 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, -1, -1, -1))))+clifford_check6<idx>(ex_to<matrix>(diag_matrix(lst(-1, -1, -1, -1))));; cout << '.' << flush;
556 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;
557 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;
558 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;
560 realsymbol s("s"), t("t"); // symbolic entries in matric
561 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;
564 A = 1, 0, 0, 0, // anticommuting, not symmetric, Tr=0
568 result += clifford_check6<varidx>(A)+clifford_check6<idx>(A);; cout << '.' << flush;
570 A = 1, 0, 0, 0, // anticommuting, not symmetric, Tr=2
574 result += clifford_check6<varidx>(A)+clifford_check6<idx>(A);; cout << '.' << flush;
576 A = 1, 0, 0, 0, // not anticommuting, symmetric, Tr=0
580 result += clifford_check6<varidx>(A)+clifford_check6<idx>(A);; cout << '.' << flush;
582 A = 1, 0, 0, 0, // not anticommuting, symmetric, Tr=2
586 result += clifford_check6<varidx>(A)+clifford_check6<idx>(A);; cout << '.' << flush;
588 A = 1, 1, 0, 0, // not anticommuting, not symmetric, Tr=4
592 result += clifford_check6<varidx>(A)+clifford_check6<idx>(A);; cout << '.' << flush;
595 result += clifford_check7(minkmetric(), dim); cout << '.' << flush;
597 varidx chi(symbol("chi"), dim), xi(symbol("xi"), dim);
598 result += clifford_check7(delta_tensor(xi, chi), dim); cout << '.' << flush;
600 result += clifford_check7(lorentz_g(xi, chi), dim); cout << '.' << flush;
602 result += clifford_check7(indexed(-2*minkmetric(), sy_symm(), xi, chi), dim); cout << '.' << flush;
603 result += clifford_check7(-2*delta_tensor(xi, chi), dim); cout << '.' << flush;
608 int main(int argc, char** argv)
610 return exam_clifford();