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, idx & 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(is_a<varidx>(mu) ?
68 mu == idx(j, mu.get_dim()), ex_to<varidx>(mu).toggle_variance() == idx(j, mu.get_dim())
69 : mu == idx(j, mu.get_dim())));
70 if (!(canonicalize_clifford(esub).is_zero())) {
71 clog << "simplify_indexed(" << e1 << ") - (" << e2 << ") erroneously returned "
72 << canonicalize_clifford(esub) << " instead of 0 for mu=" << j << endl;
79 static unsigned check_equal_simplify_term2(const ex & e1, const ex & e2)
81 ex e = expand_dummy_sum(normal(simplify_indexed(e1) - e2), true);
82 if (!(canonicalize_clifford(e).is_zero())) {
83 clog << "simplify_indexed(" << e1 << ") - (" << e2 << ") erroneously returned "
84 << canonicalize_clifford(e) << " instead of 0" << endl;
91 static unsigned clifford_check1()
93 // checks general identities and contractions
98 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), rho(symbol("rho"), dim);
101 e = dirac_ONE() * dirac_ONE();
102 result += check_equal(e, dirac_ONE());
104 e = dirac_ONE() * dirac_gamma(mu) * dirac_ONE();
105 result += check_equal(e, dirac_gamma(mu));
107 e = dirac_gamma(varidx(2, dim)) * dirac_gamma(varidx(1, dim)) *
108 dirac_gamma(varidx(1, dim)) * dirac_gamma(varidx(2, dim));
109 result += check_equal(e, dirac_ONE());
111 e = dirac_gamma(mu) * dirac_gamma(nu) *
112 dirac_gamma(nu.toggle_variance()) * dirac_gamma(mu.toggle_variance());
113 result += check_equal_simplify(e, pow(dim, 2) * dirac_ONE());
115 e = dirac_gamma(mu) * dirac_gamma(nu) *
116 dirac_gamma(mu.toggle_variance()) * dirac_gamma(nu.toggle_variance());
117 result += check_equal_simplify(e, 2*dim*dirac_ONE()-pow(dim, 2)*dirac_ONE());
119 e = dirac_gamma(nu.toggle_variance()) * dirac_gamma(rho.toggle_variance()) *
120 dirac_gamma(mu) * dirac_gamma(rho) * dirac_gamma(nu);
121 e = e.simplify_indexed().collect(dirac_gamma(mu));
122 result += check_equal(e, pow(2 - dim, 2).expand() * dirac_gamma(mu));
127 static unsigned clifford_check2()
129 // checks identities relating to gamma5
134 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim);
137 e = dirac_gamma(mu) * dirac_gamma5() + dirac_gamma5() * dirac_gamma(mu);
138 result += check_equal(e, 0);
140 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma5() + dirac_gamma(mu);
141 result += check_equal(e, 0);
146 static unsigned clifford_check3()
152 symbol dim("D"), m("m"), q("q"), l("l"), ldotq("ldotq");
153 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), rho(symbol("rho"), dim),
154 sig(symbol("sig"), dim), kap(symbol("kap"), dim), lam(symbol("lam"), dim);
158 result += check_equal(dirac_trace(e), 0);
160 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho);
161 result += check_equal(dirac_trace(e), 0);
163 e = dirac_gamma5() * dirac_gamma(mu);
164 result += check_equal(dirac_trace(e), 0);
166 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu);
167 result += check_equal(dirac_trace(e), 0);
169 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho);
170 result += check_equal(dirac_trace(e), 0);
173 sp.add(q, q, pow(q, 2));
174 sp.add(l, l, pow(l, 2));
177 e = pow(m, 2) * dirac_slash(q, dim) * dirac_slash(q, dim);
178 e = dirac_trace(e).simplify_indexed(sp);
179 result += check_equal(e, 4*pow(m, 2)*pow(q, 2));
181 // cyclicity without gamma5
182 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig)
183 - dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(mu);
185 result += check_equal(e, 0);
187 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap) * dirac_gamma(lam)
188 - dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap) * dirac_gamma(lam) * dirac_gamma(mu);
189 e = dirac_trace(e).expand();
190 result += check_equal(e, 0);
192 // cyclicity of gamma5 * S_4
193 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig)
194 - dirac_gamma(sig) * dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho);
196 result += check_equal(e, 0);
198 // non-cyclicity of order D-4 of gamma5 * S_6
199 e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap) * dirac_gamma(mu.toggle_variance())
200 + dim * dirac_gamma5() * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap);
201 e = dirac_trace(e).simplify_indexed();
202 e = (e / (dim - 4)).normal();
203 result += check_equal(e, 8 * I * lorentz_eps(nu.replace_dim(4), rho.replace_dim(4), sig.replace_dim(4), kap.replace_dim(4)));
205 // one-loop vacuum polarization in QED
206 e = dirac_gamma(mu) *
207 (dirac_slash(l, dim) + dirac_slash(q, 4) + m * dirac_ONE()) *
208 dirac_gamma(mu.toggle_variance()) *
209 (dirac_slash(l, dim) + m * dirac_ONE());
210 e = dirac_trace(e).simplify_indexed(sp);
211 result += check_equal(e, 4*((2-dim)*l*l + (2-dim)*ldotq + dim*m*m).expand());
213 e = dirac_slash(q, 4) *
214 (dirac_slash(l, dim) + dirac_slash(q, 4) + m * dirac_ONE()) *
216 (dirac_slash(l, dim) + m * dirac_ONE());
217 e = dirac_trace(e).simplify_indexed(sp);
218 result += check_equal(e, 4*(2*ldotq*ldotq + q*q*ldotq - q*q*l*l + q*q*m*m).expand());
220 // stuff that had problems in the past
221 ex prop = dirac_slash(q, dim) - m * dirac_ONE();
222 e = dirac_slash(l, dim) * dirac_gamma5() * dirac_slash(l, dim) * prop;
223 e = dirac_trace(dirac_slash(q, dim) * e) - dirac_trace(m * e)
224 - dirac_trace(prop * e);
225 result += check_equal(e, 0);
227 e = (dirac_gamma5() + dirac_ONE()) * dirac_gamma5();
229 result += check_equal(e, 4);
231 // traces with multiple representation labels
232 e = dirac_ONE(0) * dirac_ONE(1) / 16;
233 result += check_equal(dirac_trace(e, 0), dirac_ONE(1) / 4);
234 result += check_equal(dirac_trace(e, 1), dirac_ONE(0) / 4);
235 result += check_equal(dirac_trace(e, 2), e);
236 result += check_equal(dirac_trace(e, lst(0, 1)), 1);
238 e = dirac_gamma(mu, 0) * dirac_gamma(mu.toggle_variance(), 1) * dirac_gamma(nu, 0) * dirac_gamma(nu.toggle_variance(), 1);
239 result += check_equal_simplify(dirac_trace(e, 0), 4 * dim * dirac_ONE(1));
240 result += check_equal_simplify(dirac_trace(e, 1), 4 * dim * dirac_ONE(0));
241 // Fails with new tinfo mechanism because the order of gamme matrices with different rl depends on luck.
242 // TODO: better check.
243 //result += check_equal_simplify(dirac_trace(e, 2), canonicalize_clifford(e)); // e will be canonicalized by the calculation of the trace
244 result += check_equal_simplify(dirac_trace(e, lst(0, 1)), 16 * dim);
249 static unsigned clifford_check4()
251 // simplify_indexed()/dirac_trace() cross-checks
256 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), rho(symbol("rho"), dim),
257 sig(symbol("sig"), dim), lam(symbol("lam"), dim);
260 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(mu.toggle_variance());
261 t1 = dirac_trace(e).simplify_indexed();
262 t2 = dirac_trace(e.simplify_indexed());
263 result += check_equal((t1 - t2).expand(), 0);
265 e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(mu.toggle_variance()) * dirac_gamma(lam);
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(sig) * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(nu.toggle_variance()) * dirac_gamma(mu.toggle_variance());
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(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(mu.toggle_variance()) * dirac_gamma(sig) * dirac_gamma(nu.toggle_variance());
276 t1 = dirac_trace(e).simplify_indexed();
277 t2 = dirac_trace(e.simplify_indexed());
278 result += check_equal((t1 - t2).expand(), 0);
283 static unsigned clifford_check5()
285 // canonicalize_clifford() checks
290 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), lam(symbol("lam"), dim);
293 e = dirac_gamma(mu) * dirac_gamma(nu) + dirac_gamma(nu) * dirac_gamma(mu);
294 result += check_equal(canonicalize_clifford(e), 2*dirac_ONE()*lorentz_g(mu, nu));
296 e = (dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(lam)
297 + dirac_gamma(nu) * dirac_gamma(lam) * dirac_gamma(mu)
298 + dirac_gamma(lam) * dirac_gamma(mu) * dirac_gamma(nu)
299 - dirac_gamma(nu) * dirac_gamma(mu) * dirac_gamma(lam)
300 - dirac_gamma(lam) * dirac_gamma(nu) * dirac_gamma(mu)
301 - dirac_gamma(mu) * dirac_gamma(lam) * dirac_gamma(nu)) / 6
302 + lorentz_g(mu, nu) * dirac_gamma(lam)
303 - lorentz_g(mu, lam) * dirac_gamma(nu)
304 + lorentz_g(nu, lam) * dirac_gamma(mu)
305 - dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(lam);
306 result += check_equal(canonicalize_clifford(e), 0);
311 /* We make two identical checks with metrics defined through a matrix in
312 * the cases when used indexes have or have not variance.
313 * To this end we recycle the code through the following macros */
315 #define CHECK6(IDX,TOGGLE) {IDX v(symbol("v"), 4), nu(symbol("nu"), 4), mu(symbol("mu"), 4), \
316 psi(symbol("psi"),4), lam(symbol("lambda"), 4),\
317 xi(symbol("xi"), 4), rho(symbol("rho"),4);\
319 /* checks general identities and contractions for clifford_unit*/\
320 e = dirac_ONE(2) * clifford_unit(mu, A, 2) * dirac_ONE(2);\
321 result += check_equal(e, clifford_unit(mu, A, 2));\
323 e = clifford_unit(idx(2, 4), A) * clifford_unit(idx(1, 4), A)\
324 * clifford_unit(idx(1, 4), A) * clifford_unit(idx(2, 4), A);\
325 result += check_equal(e, A(1, 1) * A(2, 2) * dirac_ONE());\
327 e = clifford_unit(IDX(2, 4), A) * clifford_unit(IDX(1, 4), A)\
328 * clifford_unit(IDX(1, 4), A) * clifford_unit(IDX(2, 4), A);\
329 result += check_equal(e, A(1, 1) * A(2, 2) * dirac_ONE());\
331 e = clifford_unit(nu, A) * clifford_unit(nu TOGGLE, A);\
332 result += check_equal_simplify(e, A.trace() * dirac_ONE());\
334 e = clifford_unit(nu, A) * clifford_unit(nu, A);\
335 result += check_equal_simplify(e, indexed(A_symm, sy_symm(), nu, nu) * dirac_ONE());\
337 e = clifford_unit(nu, A) * clifford_unit(nu TOGGLE, A) * clifford_unit(mu, A);\
338 result += check_equal_simplify(e, A.trace() * clifford_unit(mu, A));\
340 e = clifford_unit(nu, A) * clifford_unit(mu, A) * clifford_unit(nu TOGGLE, A);\
342 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);\
344 e = clifford_unit(nu, A) * clifford_unit(nu TOGGLE, A)\
345 * clifford_unit(mu, A) * clifford_unit(mu TOGGLE, A);\
346 result += check_equal_simplify(e, pow(A.trace(), 2) * dirac_ONE());\
348 e = clifford_unit(mu, A) * clifford_unit(nu, A)\
349 * clifford_unit(nu TOGGLE, A) * clifford_unit(mu TOGGLE, A);\
350 result += check_equal_simplify(e, pow(A.trace(), 2) * dirac_ONE());\
352 e = clifford_unit(mu, A) * clifford_unit(nu, A)\
353 * clifford_unit(mu TOGGLE, A) * clifford_unit(nu TOGGLE, A);\
355 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());\
357 e = clifford_unit(mu TOGGLE, A) * clifford_unit(nu, A)\
358 * clifford_unit(mu, A) * clifford_unit(nu TOGGLE, A);\
360 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());\
362 e = clifford_unit(nu TOGGLE, A) * clifford_unit(rho TOGGLE, A)\
363 * clifford_unit(mu, A) * clifford_unit(rho, A) * clifford_unit(nu, A);\
364 e = e.simplify_indexed().collect(clifford_unit(mu, A));\
366 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) \
367 - 2*A.trace() * (clifford_unit(rho, A) * indexed(A_symm, sy_symm(), rho TOGGLE, mu) \
368 + clifford_unit(nu, A) * indexed(A_symm, sy_symm(), nu TOGGLE, mu)) + pow(A.trace(),2)* clifford_unit(mu, A), mu);\
370 e = clifford_unit(nu TOGGLE, A) * clifford_unit(rho, A)\
371 * clifford_unit(mu, A) * clifford_unit(rho TOGGLE, A) * clifford_unit(nu, A);\
372 e = e.simplify_indexed().collect(clifford_unit(mu, A));\
374 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) \
375 - 2*A.trace() * (clifford_unit(rho, A) * indexed(A_symm, sy_symm(), rho TOGGLE, mu) \
376 + clifford_unit(nu, A) * indexed(A_symm, sy_symm(), nu TOGGLE, mu)) + pow(A.trace(),2)* clifford_unit(mu, A), mu);\
378 e = clifford_unit(mu, A) * clifford_unit(nu, A) + clifford_unit(nu, A) * clifford_unit(mu, A);\
379 result += check_equal(canonicalize_clifford(e), 2*dirac_ONE()*indexed(A_symm, sy_symm(), mu, nu));\
381 e = (clifford_unit(mu, A) * clifford_unit(nu, A) * clifford_unit(lam, A)\
382 + clifford_unit(nu, A) * clifford_unit(lam, A) * clifford_unit(mu, A)\
383 + clifford_unit(lam, A) * clifford_unit(mu, A) * clifford_unit(nu, A)\
384 - clifford_unit(nu, A) * clifford_unit(mu, A) * clifford_unit(lam, A)\
385 - clifford_unit(lam, A) * clifford_unit(nu, A) * clifford_unit(mu, A)\
386 - clifford_unit(mu, A) * clifford_unit(lam, A) * clifford_unit(nu, A)) / 6\
387 + indexed(A_symm, sy_symm(), mu, nu) * clifford_unit(lam, A)\
388 - indexed(A_symm, sy_symm(), mu, lam) * clifford_unit(nu, A)\
389 + indexed(A_symm, sy_symm(), nu, lam) * clifford_unit(mu, A)\
390 - clifford_unit(mu, A) * clifford_unit(nu, A) * clifford_unit(lam, A);\
391 result += check_equal(canonicalize_clifford(e), 0);\
393 /* lst_to_clifford() and clifford_inverse() check*/\
394 realsymbol x("x"), y("y"), t("t"), z("z");\
396 ex c = clifford_unit(nu, A, 1);\
397 e = lst_to_clifford(lst(t, x, y, z), mu, A, 1) * lst_to_clifford(lst(1, 2, 3, 4), c);\
398 e1 = clifford_inverse(e);\
399 result += check_equal_simplify_term2((e*e1).simplify_indexed(), dirac_ONE(1));\
401 /* Moebius map (both forms) checks for symmetric metrics only */\
402 matrix M1(2, 2), M2(2, 2);\
403 c = clifford_unit(nu, A);\
405 e = clifford_moebius_map(0, dirac_ONE(), \
406 dirac_ONE(), 0, lst(t, x, y, z), A); \
407 /* this is just the inversion*/\
408 M1 = 0, dirac_ONE(),\
410 e1 = clifford_moebius_map(M1, lst(t, x, y, z), A); \
411 /* the inversion again*/\
412 result += check_equal_lst(e, e1);\
414 e1 = clifford_to_lst(clifford_inverse(lst_to_clifford(lst(t, x, y, z), mu, A)), c);\
415 result += check_equal_lst(e, e1);\
417 e = clifford_moebius_map(dirac_ONE(), lst_to_clifford(lst(1, 2, 3, 4), nu, A), \
418 0, dirac_ONE(), lst(t, x, y, z), A); \
419 /*this is just a shift*/\
420 M2 = dirac_ONE(), lst_to_clifford(lst(1, 2, 3, 4), c),\
422 e1 = clifford_moebius_map(M2, lst(t, x, y, z), c); \
424 result += check_equal_lst(e, e1);\
426 result += check_equal(e, lst(t+1, x+2, y+3, z+4));\
428 /* Check the group law for Moebius maps */\
429 e = clifford_moebius_map(M1, ex_to<lst>(e1), c); \
430 /*composition of M1 and M2*/\
431 e1 = clifford_moebius_map(M1.mul(M2), lst(t, x, y, z), c); \
432 /* the product M1*M2*/\
433 result += check_equal_lst(e, e1);}
435 static unsigned clifford_check6(const matrix & A)
437 matrix A_symm(4,4), A2(4, 4);
438 A_symm = A.add(A.transpose()).mul(half);
439 A2 = A_symm.mul(A_symm);
444 CHECK6(varidx,.toggle_variance())
449 static unsigned clifford_check6a(const matrix & A)
451 matrix A_symm(4,4), A2(4, 4);
452 A_symm = A.add(A.transpose()).mul(half);
453 A2 = A_symm.mul(A_symm);
463 static unsigned clifford_check7(const ex & G, const symbol & dim)
465 // checks general identities and contractions
469 varidx mu(symbol("mu"), dim), nu(symbol("nu"), dim), rho(symbol("rho"), dim),
470 psi(symbol("psi"),dim), lam(symbol("lambda"), dim), xi(symbol("xi"), dim);
473 clifford unit = ex_to<clifford>(clifford_unit(mu, G));
474 ex scalar = unit.get_metric(varidx(0, dim), varidx(0, dim));
476 e = dirac_ONE() * dirac_ONE();
477 result += check_equal(e, dirac_ONE());
479 e = dirac_ONE() * clifford_unit(mu, G) * dirac_ONE();
480 result += check_equal(e, clifford_unit(mu, G));
482 e = clifford_unit(varidx(2, dim), G) * clifford_unit(varidx(1, dim), G)
483 * clifford_unit(varidx(1, dim), G) * clifford_unit(varidx(2, dim), G);
484 result += check_equal(e, dirac_ONE()*pow(scalar, 2));
486 e = clifford_unit(mu, G) * clifford_unit(nu, G)
487 * clifford_unit(nu.toggle_variance(), G) * clifford_unit(mu.toggle_variance(), G);
488 result += check_equal_simplify(e, pow(dim*scalar, 2) * dirac_ONE());
490 e = clifford_unit(mu, G) * clifford_unit(nu, G)
491 * clifford_unit(mu.toggle_variance(), G) * clifford_unit(nu.toggle_variance(), G);
492 result += check_equal_simplify(e, (2*dim - pow(dim, 2))*pow(scalar,2)*dirac_ONE());
494 e = clifford_unit(nu.toggle_variance(), G) * clifford_unit(rho.toggle_variance(), G)
495 * clifford_unit(mu, G) * clifford_unit(rho, G) * clifford_unit(nu, G);
496 e = e.simplify_indexed().collect(clifford_unit(mu, G));
497 result += check_equal(e, pow(scalar*(dim-2), 2).expand() * clifford_unit(mu, G));
499 // canonicalize_clifford() checks, only for symmetric metrics
500 if (ex_to<symmetry>(ex_to<indexed>(ex_to<clifford>(clifford_unit(mu, G)).get_metric()).get_symmetry()).has_symmetry()) {
501 e = clifford_unit(mu, G) * clifford_unit(nu, G) + clifford_unit(nu, G) * clifford_unit(mu, G);
502 result += check_equal(canonicalize_clifford(e), 2*dirac_ONE()*unit.get_metric(nu, mu));
504 e = (clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(lam, G)
505 + clifford_unit(nu, G) * clifford_unit(lam, G) * clifford_unit(mu, G)
506 + clifford_unit(lam, G) * clifford_unit(mu, G) * clifford_unit(nu, G)
507 - clifford_unit(nu, G) * clifford_unit(mu, G) * clifford_unit(lam, G)
508 - clifford_unit(lam, G) * clifford_unit(nu, G) * clifford_unit(mu, G)
509 - clifford_unit(mu, G) * clifford_unit(lam, G) * clifford_unit(nu, G)) / 6
510 + unit.get_metric(mu, nu) * clifford_unit(lam, G)
511 - unit.get_metric(mu, lam) * clifford_unit(nu, G)
512 + unit.get_metric(nu, lam) * clifford_unit(mu, G)
513 - clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(lam, G);
514 result += check_equal(canonicalize_clifford(e), 0);
516 e = clifford_unit(mu, G) * clifford_unit(nu, G) + clifford_unit(nu, G) * clifford_unit(mu, G);
517 result += check_equal(canonicalize_clifford(e), dirac_ONE()*(unit.get_metric(mu, nu) + unit.get_metric(nu, mu)));
519 e = (clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(lam, G)
520 + clifford_unit(nu, G) * clifford_unit(lam, G) * clifford_unit(mu, G)
521 + clifford_unit(lam, G) * clifford_unit(mu, G) * clifford_unit(nu, G)
522 - clifford_unit(nu, G) * clifford_unit(mu, G) * clifford_unit(lam, G)
523 - clifford_unit(lam, G) * clifford_unit(nu, G) * clifford_unit(mu, G)
524 - clifford_unit(mu, G) * clifford_unit(lam, G) * clifford_unit(nu, G)) / 6
525 + half * (unit.get_metric(mu, nu) + unit.get_metric(nu, mu)) * clifford_unit(lam, G)
526 - half * (unit.get_metric(mu, lam) + unit.get_metric(lam, mu)) * clifford_unit(nu, G)
527 + half * (unit.get_metric(nu, lam) + unit.get_metric(lam, nu)) * clifford_unit(mu, G)
528 - clifford_unit(mu, G) * clifford_unit(nu, G) * clifford_unit(lam, G);
529 result += check_equal(canonicalize_clifford(e), 0);
534 unsigned exam_clifford()
538 cout << "examining clifford objects" << flush;
539 clog << "----------clifford objects:" << endl;
541 result += clifford_check1(); cout << '.' << flush;
542 result += clifford_check2(); cout << '.' << flush;
543 result += clifford_check3(); cout << '.' << flush;
544 result += clifford_check4(); cout << '.' << flush;
545 result += clifford_check5(); cout << '.' << flush;
547 // anticommuting, symmetric examples
548 result += clifford_check6(ex_to<matrix>(diag_matrix(lst(-1, 1, 1, 1))))+clifford_check6a(ex_to<matrix>(diag_matrix(lst(-1, 1, 1, 1))));; cout << '.' << flush;
549 result += clifford_check6(ex_to<matrix>(diag_matrix(lst(-1, -1, -1, -1))))+clifford_check6a(ex_to<matrix>(diag_matrix(lst(-1, -1, -1, -1))));; cout << '.' << flush;
550 result += clifford_check6(ex_to<matrix>(diag_matrix(lst(-1, 1, 1, -1))))+clifford_check6a(ex_to<matrix>(diag_matrix(lst(-1, 1, 1, -1))));; cout << '.' << flush;
551 result += clifford_check6(ex_to<matrix>(diag_matrix(lst(-1, 0, 1, -1))))+clifford_check6a(ex_to<matrix>(diag_matrix(lst(-1, 0, 1, -1))));; cout << '.' << flush;
552 result += clifford_check6(ex_to<matrix>(diag_matrix(lst(-3, 0, 2, -1))))+clifford_check6a(ex_to<matrix>(diag_matrix(lst(-3, 0, 2, -1))));; cout << '.' << flush;
554 realsymbol s("s"), t("t"); // symbolic entries in matric
555 result += clifford_check6(ex_to<matrix>(diag_matrix(lst(-1, 1, s, t))))+clifford_check6a(ex_to<matrix>(diag_matrix(lst(-1, 1, s, t))));; cout << '.' << flush;
558 A = 1, 0, 0, 0, // anticommuting, not symmetric, Tr=0
562 result += clifford_check6(A)+clifford_check6a(A);; cout << '.' << flush;
564 A = 1, 0, 0, 0, // anticommuting, not symmetric, Tr=2
568 result += clifford_check6(A)+clifford_check6a(A);; cout << '.' << flush;
570 A = 1, 0, 0, 0, // not anticommuting, symmetric, Tr=0
574 result += clifford_check6(A)+clifford_check6a(A);; cout << '.' << flush;
576 A = 1, 0, 0, 0, // not anticommuting, symmetric, Tr=2
580 result += clifford_check6(A)+clifford_check6a(A);; cout << '.' << flush;
582 A = 1, 1, 0, 0, // not anticommuting, not symmetric, Tr=4
586 result += clifford_check6(A)+clifford_check6a(A);; cout << '.' << flush;
589 result += clifford_check7(minkmetric(), dim); cout << '.' << flush;
591 varidx chi(symbol("chi"), dim), xi(symbol("xi"), dim);
592 result += clifford_check7(delta_tensor(xi, chi), dim); cout << '.' << flush;
594 result += clifford_check7(lorentz_g(xi, chi), dim); cout << '.' << flush;
596 result += clifford_check7(indexed(-2*minkmetric(), sy_symm(), xi, chi), dim); cout << '.' << flush;
597 result += clifford_check7(-2*delta_tensor(xi, chi), dim); cout << '.' << flush;
600 cout << " passed " << endl;
601 clog << "(no output)" << endl;
603 cout << " failed " << endl;