1 /** @file exam_indexed.cpp
3 * Here we test manipulations on GiNaC's indexed objects. */
6 * GiNaC Copyright (C) 1999-2004 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
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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.
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20 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
25 static unsigned check_equal(const ex &e1, const ex &e2)
29 clog << e1 << "-" << e2 << " erroneously returned "
30 << e << " instead of 0" << endl;
36 static unsigned check_equal_simplify(const ex &e1, const ex &e2)
38 ex e = simplify_indexed(e1) - e2;
40 clog << "simplify_indexed(" << e1 << ")-" << e2 << " erroneously returned "
41 << e << " instead of 0" << endl;
47 static unsigned check_equal_simplify(const ex &e1, const ex &e2, const scalar_products &sp)
49 ex e = simplify_indexed(e1, sp) - e2;
51 clog << "simplify_indexed(" << e1 << ")-" << e2 << " erroneously returned "
52 << e << " instead of 0" << endl;
58 static unsigned delta_check()
60 // checks identities of the delta tensor
64 symbol s_i("i"), s_j("j"), s_k("k");
65 idx i(s_i, 3), j(s_j, 3), k(s_k, 3);
69 result += check_equal(delta_tensor(i, j), delta_tensor(j, i));
71 // trace = dimension of index space
72 result += check_equal(delta_tensor(i, i), 3);
73 result += check_equal_simplify(delta_tensor(i, j) * delta_tensor(i, j), 3);
75 // contraction with delta tensor
76 result += check_equal_simplify(delta_tensor(i, j) * indexed(A, k), delta_tensor(i, j) * indexed(A, k));
77 result += check_equal_simplify(delta_tensor(i, j) * indexed(A, j), indexed(A, i));
78 result += check_equal_simplify(delta_tensor(i, j) * indexed(A, i), indexed(A, j));
79 result += check_equal_simplify(delta_tensor(i, j) * delta_tensor(j, k) * indexed(A, i), indexed(A, k));
84 static unsigned metric_check()
86 // checks identities of the metric tensor
90 symbol s_mu("mu"), s_nu("nu"), s_rho("rho"), s_sigma("sigma");
91 varidx mu(s_mu, 4), nu(s_nu, 4), rho(s_rho, 4), sigma(s_sigma, 4);
94 // becomes delta tensor if indices have opposite variance
95 result += check_equal(metric_tensor(mu, nu.toggle_variance()), delta_tensor(mu, nu.toggle_variance()));
97 // scalar contraction = dimension of index space
98 result += check_equal(metric_tensor(mu, mu.toggle_variance()), 4);
99 result += check_equal_simplify(metric_tensor(mu, nu) * metric_tensor(mu.toggle_variance(), nu.toggle_variance()), 4);
101 // contraction with metric tensor
102 result += check_equal_simplify(metric_tensor(mu, nu) * indexed(A, nu), metric_tensor(mu, nu) * indexed(A, nu));
103 result += check_equal_simplify(metric_tensor(mu, nu) * indexed(A, nu.toggle_variance()), indexed(A, mu));
104 result += check_equal_simplify(metric_tensor(mu, nu) * indexed(A, mu.toggle_variance()), indexed(A, nu));
105 result += check_equal_simplify(metric_tensor(mu, nu) * metric_tensor(mu.toggle_variance(), rho.toggle_variance()) * indexed(A, nu.toggle_variance()), indexed(A, rho.toggle_variance()));
106 result += check_equal_simplify(metric_tensor(mu, rho) * metric_tensor(nu, sigma) * indexed(A, rho.toggle_variance(), sigma.toggle_variance()), indexed(A, mu, nu));
107 result += check_equal_simplify(indexed(A, mu.toggle_variance()) * metric_tensor(mu, nu) - indexed(A, mu.toggle_variance()) * metric_tensor(nu, mu), 0);
108 result += check_equal_simplify(indexed(A, mu.toggle_variance(), nu.toggle_variance()) * metric_tensor(nu, rho), indexed(A, mu.toggle_variance(), rho));
110 // contraction with delta tensor yields a metric tensor
111 result += check_equal_simplify(delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho), metric_tensor(mu, rho));
112 result += check_equal_simplify(metric_tensor(mu, nu) * indexed(A, nu.toggle_variance()) * delta_tensor(mu.toggle_variance(), rho), indexed(A, rho));
117 static unsigned epsilon_check()
119 // checks identities of the epsilon tensor
123 symbol s_mu("mu"), s_nu("nu"), s_rho("rho"), s_sigma("sigma"), s_tau("tau");
125 varidx mu(s_mu, 4), nu(s_nu, 4), rho(s_rho, 4), sigma(s_sigma, 4), tau(s_tau, 4);
126 varidx mu_co(s_mu, 4, true), nu_co(s_nu, 4, true), rho_co(s_rho, 4, true), sigma_co(s_sigma, 4, true), tau_co(s_tau, 4, true);
129 result += check_equal(lorentz_eps(mu, nu, rho, sigma) + lorentz_eps(sigma, rho, mu, nu), 0);
131 // convolution is zero
132 result += check_equal(lorentz_eps(mu, nu, rho, nu_co), 0);
133 result += check_equal(lorentz_eps(mu, nu, mu_co, nu_co), 0);
134 result += check_equal_simplify(lorentz_g(mu_co, nu_co) * lorentz_eps(mu, nu, rho, sigma), 0);
136 // contraction with symmetric tensor is zero
137 result += check_equal_simplify(lorentz_eps(mu, nu, rho, sigma) * indexed(d, sy_symm(), mu_co, nu_co), 0);
138 result += check_equal_simplify(lorentz_eps(mu, nu, rho, sigma) * indexed(d, sy_symm(), nu_co, sigma_co, rho_co), 0);
139 result += check_equal_simplify(lorentz_eps(mu, nu, rho, sigma) * indexed(d, mu_co) * indexed(d, nu_co), 0);
140 result += check_equal_simplify(lorentz_eps(mu_co, nu, rho, sigma) * indexed(d, mu) * indexed(d, nu_co), 0);
141 ex e = lorentz_eps(mu, nu, rho, sigma) * indexed(d, mu_co) - lorentz_eps(mu_co, nu, rho, sigma) * indexed(d, mu);
142 result += check_equal_simplify(e, 0);
144 // contractions of epsilon tensors
145 result += check_equal_simplify(lorentz_eps(mu, nu, rho, sigma) * lorentz_eps(mu_co, nu_co, rho_co, sigma_co), -24);
146 result += check_equal_simplify(lorentz_eps(tau, nu, rho, sigma) * lorentz_eps(mu_co, nu_co, rho_co, sigma_co), -6 * delta_tensor(tau, mu_co));
151 DECLARE_FUNCTION_2P(symm_fcn)
152 REGISTER_FUNCTION(symm_fcn, set_symmetry(sy_symm(0, 1)));
153 DECLARE_FUNCTION_2P(anti_fcn)
154 REGISTER_FUNCTION(anti_fcn, set_symmetry(sy_anti(0, 1)));
156 static unsigned symmetry_check()
158 // check symmetric/antisymmetric objects
162 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3), l(symbol("l"), 3);
163 symbol A("A"), B("B"), C("C");
166 result += check_equal(indexed(A, sy_symm(), i, j), indexed(A, sy_symm(), j, i));
167 result += check_equal(indexed(A, sy_anti(), i, j) + indexed(A, sy_anti(), j, i), 0);
168 result += check_equal(indexed(A, sy_anti(), i, j, k) - indexed(A, sy_anti(), j, k, i), 0);
169 e = indexed(A, sy_symm(), i, j, k) *
170 indexed(B, sy_anti(), l, k, i);
171 result += check_equal_simplify(e, 0);
172 e = indexed(A, sy_symm(), i, i, j, j) *
173 indexed(B, sy_anti(), k, l); // GiNaC 0.8.0 had a bug here
174 result += check_equal_simplify(e, e);
176 symmetry R = sy_symm(sy_anti(0, 1), sy_anti(2, 3));
177 e = indexed(A, R, i, j, k, l) + indexed(A, R, j, i, k, l);
178 result += check_equal(e, 0);
179 e = indexed(A, R, i, j, k, l) + indexed(A, R, i, j, l, k);
180 result += check_equal(e, 0);
181 e = indexed(A, R, i, j, k, l) - indexed(A, R, j, i, l, k);
182 result += check_equal(e, 0);
183 e = indexed(A, R, i, j, k, l) + indexed(A, R, k, l, j, i);
184 result += check_equal(e, 0);
186 e = indexed(A, i, j);
187 result += check_equal(symmetrize(e) + antisymmetrize(e), e);
188 e = indexed(A, sy_symm(), i, j, k, l);
189 result += check_equal(symmetrize(e), e);
190 result += check_equal(antisymmetrize(e), 0);
191 e = indexed(A, sy_anti(), i, j, k, l);
192 result += check_equal(symmetrize(e), 0);
193 result += check_equal(antisymmetrize(e), e);
195 e = (indexed(A, sy_anti(), i, j, k, l) * (indexed(B, j) * indexed(C, k) + indexed(B, k) * indexed(C, j)) + indexed(B, i, l)).expand();
196 result += check_equal_simplify(e, indexed(B, i, l));
198 result += check_equal(symm_fcn(0, 1) + symm_fcn(1, 0), 2*symm_fcn(0, 1));
199 result += check_equal(anti_fcn(0, 1) + anti_fcn(1, 0), 0);
200 result += check_equal(anti_fcn(0, 0), 0);
205 static unsigned scalar_product_check()
207 // check scalar product replacement
211 idx i(symbol("i"), 3), j(symbol("j"), 3);
212 symbol A("A"), B("B"), C("C");
216 sp.add(A, B, 0); // A and B are orthogonal
217 sp.add(A, C, 0); // A and C are orthogonal
218 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
220 e = (indexed(A + B, i) * indexed(A + C, i)).expand(expand_options::expand_indexed);
221 result += check_equal_simplify(e, indexed(B, i) * indexed(C, i) + 4, sp);
222 e = indexed(A, i, i) * indexed(B, j, j); // GiNaC 0.8.0 had a bug here
223 result += check_equal_simplify(e, e, sp);
228 static unsigned edyn_check()
230 // Relativistic electrodynamics
232 // Test 1: check transformation laws of electric and magnetic fields by
233 // applying a Lorentz boost to the field tensor
238 ex gamma = 1 / sqrt(1 - pow(beta, 2));
239 symbol Ex("Ex"), Ey("Ey"), Ez("Ez");
240 symbol Bx("Bx"), By("By"), Bz("Bz");
242 // Lorentz transformation matrix (boost along x axis)
244 L = gamma, -beta*gamma, 0, 0,
245 -beta*gamma, gamma, 0, 0,
249 // Electromagnetic field tensor
251 F = 0, -Ex, -Ey, -Ez,
257 symbol s_mu("mu"), s_nu("nu"), s_rho("rho"), s_sigma("sigma");
258 varidx mu(s_mu, 4), nu(s_nu, 4), rho(s_rho, 4), sigma(s_sigma, 4);
260 // Apply transformation law of second rank tensor
261 ex e = (indexed(L, mu, rho.toggle_variance())
262 * indexed(L, nu, sigma.toggle_variance())
263 * indexed(F, rho, sigma)).simplify_indexed();
265 // Extract transformed electric and magnetic fields
266 ex Ex_p = e.subs(lst(mu == 1, nu == 0)).normal();
267 ex Ey_p = e.subs(lst(mu == 2, nu == 0)).normal();
268 ex Ez_p = e.subs(lst(mu == 3, nu == 0)).normal();
269 ex Bx_p = e.subs(lst(mu == 3, nu == 2)).normal();
270 ex By_p = e.subs(lst(mu == 1, nu == 3)).normal();
271 ex Bz_p = e.subs(lst(mu == 2, nu == 1)).normal();
274 result += check_equal(Ex_p, Ex);
275 result += check_equal(Ey_p, gamma * (Ey - beta * Bz));
276 result += check_equal(Ez_p, gamma * (Ez + beta * By));
277 result += check_equal(Bx_p, Bx);
278 result += check_equal(By_p, gamma * (By + beta * Ez));
279 result += check_equal(Bz_p, gamma * (Bz - beta * Ey));
281 // Test 2: check energy density and Poynting vector of electromagnetic field
284 ex eta = diag_matrix(lst(1, -1, -1, -1));
286 // Covariant field tensor
287 ex F_mu_nu = (indexed(eta, mu.toggle_variance(), rho.toggle_variance())
288 * indexed(eta, nu.toggle_variance(), sigma.toggle_variance())
289 * indexed(F, rho, sigma)).simplify_indexed();
291 // Energy-momentum tensor
292 ex T = (-indexed(eta, rho, sigma) * F_mu_nu.subs(s_nu == s_rho)
293 * F_mu_nu.subs(lst(s_mu == s_nu, s_nu == s_sigma))
294 + indexed(eta, mu.toggle_variance(), nu.toggle_variance())
295 * F_mu_nu.subs(lst(s_mu == s_rho, s_nu == s_sigma))
296 * indexed(F, rho, sigma) / 4).simplify_indexed() / (4 * Pi);
298 // Extract energy density and Poynting vector
299 ex E = T.subs(lst(s_mu == 0, s_nu == 0)).normal();
300 ex Px = T.subs(lst(s_mu == 0, s_nu == 1));
301 ex Py = T.subs(lst(s_mu == 0, s_nu == 2));
302 ex Pz = T.subs(lst(s_mu == 0, s_nu == 3));
305 result += check_equal(E, (Ex*Ex+Ey*Ey+Ez*Ez+Bx*Bx+By*By+Bz*Bz) / (8 * Pi));
306 result += check_equal(Px, (Ez*By-Ey*Bz) / (4 * Pi));
307 result += check_equal(Py, (Ex*Bz-Ez*Bx) / (4 * Pi));
308 result += check_equal(Pz, (Ey*Bx-Ex*By) / (4 * Pi));
313 static unsigned spinor_check()
315 // check identities of the spinor metric
320 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C")), D(symbol("D"));
321 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
324 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
325 result += check_equal_simplify(e, 2);
326 e = spinor_metric(A_co, B_co) * spinor_metric(B, A);
327 result += check_equal_simplify(e, -2);
328 e = spinor_metric(A_co, B_co) * spinor_metric(A, C);
329 result += check_equal_simplify(e, delta_tensor(B_co, C));
330 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
331 result += check_equal_simplify(e, -delta_tensor(A_co, C));
332 e = spinor_metric(A_co, B_co) * spinor_metric(C, A);
333 result += check_equal_simplify(e, -delta_tensor(B_co, C));
334 e = spinor_metric(A, B) * indexed(psi, B_co);
335 result += check_equal_simplify(e, indexed(psi, A));
336 e = spinor_metric(A, B) * indexed(psi, A_co);
337 result += check_equal_simplify(e, -indexed(psi, B));
338 e = spinor_metric(A_co, B_co) * indexed(psi, B);
339 result += check_equal_simplify(e, -indexed(psi, A_co));
340 e = spinor_metric(A_co, B_co) * indexed(psi, A);
341 result += check_equal_simplify(e, indexed(psi, B_co));
342 e = spinor_metric(D, A) * spinor_metric(A_co, B_co) * spinor_metric(B, C) - spinor_metric(D, A_co) * spinor_metric(A, B_co) * spinor_metric(B, C);
343 result += check_equal_simplify(e, 0);
348 static unsigned dummy_check()
350 // check dummy index renaming/repositioning
354 symbol p("p"), q("q");
355 idx i(symbol("i"), 3), j(symbol("j"), 3), n(symbol("n"), 3);
356 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
359 e = indexed(p, i) * indexed(q, i) - indexed(p, j) * indexed(q, j);
360 result += check_equal_simplify(e, 0);
362 e = indexed(p, i) * indexed(p, i) * indexed(q, j) * indexed(q, j)
363 - indexed(p, n) * indexed(p, n) * indexed(q, j) * indexed(q, j);
364 result += check_equal_simplify(e, 0);
366 e = indexed(p, mu, mu.toggle_variance()) - indexed(p, nu, nu.toggle_variance());
367 result += check_equal_simplify(e, 0);
369 e = indexed(p, mu.toggle_variance(), nu, mu) * indexed(q, i)
370 - indexed(p, mu, nu, mu.toggle_variance()) * indexed(q, i);
371 result += check_equal_simplify(e, 0);
373 e = indexed(p, mu, mu.toggle_variance()) - indexed(p, nu.toggle_variance(), nu);
374 result += check_equal_simplify(e, 0);
375 e = indexed(p, mu.toggle_variance(), mu) - indexed(p, nu, nu.toggle_variance());
376 result += check_equal_simplify(e, 0);
381 unsigned exam_indexed()
385 cout << "examining indexed objects" << flush;
386 clog << "----------indexed objects:" << endl;
388 result += delta_check(); cout << '.' << flush;
389 result += metric_check(); cout << '.' << flush;
390 result += epsilon_check(); cout << '.' << flush;
391 result += symmetry_check(); cout << '.' << flush;
392 result += scalar_product_check(); cout << '.' << flush;
393 result += edyn_check(); cout << '.' << flush;
394 result += spinor_check(); cout << '.' << flush;
395 result += dummy_check(); cout << '.' << flush;
398 cout << " passed " << endl;
399 clog << "(no output)" << endl;
401 cout << " failed " << endl;