1 /** @file exam_indexed.cpp
3 * Here we test manipulations on GiNaC's indexed objects. */
6 * GiNaC Copyright (C) 1999-2002 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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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(void)
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(void)
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(void)
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 static unsigned symmetry_check(void)
153 // check symmetric/antisymmetric objects
157 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3), l(symbol("l"), 3);
158 symbol A("A"), B("B");
161 result += check_equal(indexed(A, sy_symm(), i, j), indexed(A, sy_symm(), j, i));
162 result += check_equal(indexed(A, sy_anti(), i, j) + indexed(A, sy_anti(), j, i), 0);
163 result += check_equal(indexed(A, sy_anti(), i, j, k) - indexed(A, sy_anti(), j, k, i), 0);
164 e = indexed(A, sy_symm(), i, j, k) *
165 indexed(B, sy_anti(), l, k, i);
166 result += check_equal_simplify(e, 0);
167 e = indexed(A, sy_symm(), i, i, j, j) *
168 indexed(B, sy_anti(), k, l); // GiNaC 0.8.0 had a bug here
169 result += check_equal_simplify(e, e);
171 symmetry R = sy_symm(sy_anti(0, 1), sy_anti(2, 3));
172 e = indexed(A, R, i, j, k, l) + indexed(A, R, j, i, k, l);
173 result += check_equal(e, 0);
174 e = indexed(A, R, i, j, k, l) + indexed(A, R, i, j, l, k);
175 result += check_equal(e, 0);
176 e = indexed(A, R, i, j, k, l) - indexed(A, R, j, i, l, k);
177 result += check_equal(e, 0);
178 e = indexed(A, R, i, j, k, l) + indexed(A, R, k, l, j, i);
179 result += check_equal(e, 0);
181 e = indexed(A, i, j);
182 result += check_equal(symmetrize(e) + antisymmetrize(e), e);
183 e = indexed(A, sy_symm(), i, j, k, l);
184 result += check_equal(symmetrize(e), e);
185 result += check_equal(antisymmetrize(e), 0);
186 e = indexed(A, sy_anti(), i, j, k, l);
187 result += check_equal(symmetrize(e), 0);
188 result += check_equal(antisymmetrize(e), e);
193 static unsigned scalar_product_check(void)
195 // check scalar product replacement
199 idx i(symbol("i"), 3), j(symbol("j"), 3);
200 symbol A("A"), B("B"), C("C");
204 sp.add(A, B, 0); // A and B are orthogonal
205 sp.add(A, C, 0); // A and C are orthogonal
206 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
208 e = (indexed(A + B, i) * indexed(A + C, i)).expand(expand_options::expand_indexed);
209 result += check_equal_simplify(e, indexed(B, i) * indexed(C, i) + 4, sp);
210 e = indexed(A, i, i) * indexed(B, j, j); // GiNaC 0.8.0 had a bug here
211 result += check_equal_simplify(e, e, sp);
216 static unsigned edyn_check(void)
218 // Relativistic electrodynamics
220 // Test 1: check transformation laws of electric and magnetic fields by
221 // applying a Lorentz boost to the field tensor
226 ex gamma = 1 / sqrt(1 - pow(beta, 2));
227 symbol Ex("Ex"), Ey("Ey"), Ez("Ez");
228 symbol Bx("Bx"), By("By"), Bz("Bz");
230 // Lorentz transformation matrix (boost along x axis)
233 L(0, 1) = -beta*gamma;
234 L(1, 0) = -beta*gamma;
236 L(2, 2) = 1; L(3, 3) = 1;
238 // Electromagnetic field tensor
247 symbol s_mu("mu"), s_nu("nu"), s_rho("rho"), s_sigma("sigma");
248 varidx mu(s_mu, 4), nu(s_nu, 4), rho(s_rho, 4), sigma(s_sigma, 4);
250 // Apply transformation law of second rank tensor
251 ex e = (indexed(L, mu, rho.toggle_variance())
252 * indexed(L, nu, sigma.toggle_variance())
253 * indexed(F, rho, sigma)).simplify_indexed();
255 // Extract transformed electric and magnetic fields
256 ex Ex_p = e.subs(lst(mu == 1, nu == 0)).normal();
257 ex Ey_p = e.subs(lst(mu == 2, nu == 0)).normal();
258 ex Ez_p = e.subs(lst(mu == 3, nu == 0)).normal();
259 ex Bx_p = e.subs(lst(mu == 3, nu == 2)).normal();
260 ex By_p = e.subs(lst(mu == 1, nu == 3)).normal();
261 ex Bz_p = e.subs(lst(mu == 2, nu == 1)).normal();
264 result += check_equal(Ex_p, Ex);
265 result += check_equal(Ey_p, gamma * (Ey - beta * Bz));
266 result += check_equal(Ez_p, gamma * (Ez + beta * By));
267 result += check_equal(Bx_p, Bx);
268 result += check_equal(By_p, gamma * (By + beta * Ez));
269 result += check_equal(Bz_p, gamma * (Bz - beta * Ey));
271 // Test 2: check energy density and Poynting vector of electromagnetic field
274 ex eta = diag_matrix(lst(1, -1, -1, -1));
276 // Covariant field tensor
277 ex F_mu_nu = (indexed(eta, mu.toggle_variance(), rho.toggle_variance())
278 * indexed(eta, nu.toggle_variance(), sigma.toggle_variance())
279 * indexed(F, rho, sigma)).simplify_indexed();
281 // Energy-momentum tensor
282 ex T = (-indexed(eta, rho, sigma) * F_mu_nu.subs(s_nu == s_rho)
283 * F_mu_nu.subs(lst(s_mu == s_nu, s_nu == s_sigma))
284 + indexed(eta, mu.toggle_variance(), nu.toggle_variance())
285 * F_mu_nu.subs(lst(s_mu == s_rho, s_nu == s_sigma))
286 * indexed(F, rho, sigma) / 4).simplify_indexed() / (4 * Pi);
288 // Extract energy density and Poynting vector
289 ex E = T.subs(lst(s_mu == 0, s_nu == 0)).normal();
290 ex Px = T.subs(lst(s_mu == 0, s_nu == 1));
291 ex Py = T.subs(lst(s_mu == 0, s_nu == 2));
292 ex Pz = T.subs(lst(s_mu == 0, s_nu == 3));
295 result += check_equal(E, (Ex*Ex+Ey*Ey+Ez*Ez+Bx*Bx+By*By+Bz*Bz) / (8 * Pi));
296 result += check_equal(Px, (Ez*By-Ey*Bz) / (4 * Pi));
297 result += check_equal(Py, (Ex*Bz-Ez*Bx) / (4 * Pi));
298 result += check_equal(Pz, (Ey*Bx-Ex*By) / (4 * Pi));
303 static unsigned spinor_check(void)
305 // check identities of the spinor metric
310 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C")), D(symbol("D"));
311 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
314 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
315 result += check_equal_simplify(e, 2);
316 e = spinor_metric(A_co, B_co) * spinor_metric(B, A);
317 result += check_equal_simplify(e, -2);
318 e = spinor_metric(A_co, B_co) * spinor_metric(A, C);
319 result += check_equal_simplify(e, delta_tensor(B_co, C));
320 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
321 result += check_equal_simplify(e, -delta_tensor(A_co, C));
322 e = spinor_metric(A_co, B_co) * spinor_metric(C, A);
323 result += check_equal_simplify(e, -delta_tensor(B_co, C));
324 e = spinor_metric(A, B) * indexed(psi, B_co);
325 result += check_equal_simplify(e, indexed(psi, A));
326 e = spinor_metric(A, B) * indexed(psi, A_co);
327 result += check_equal_simplify(e, -indexed(psi, B));
328 e = spinor_metric(A_co, B_co) * indexed(psi, B);
329 result += check_equal_simplify(e, -indexed(psi, A_co));
330 e = spinor_metric(A_co, B_co) * indexed(psi, A);
331 result += check_equal_simplify(e, indexed(psi, B_co));
332 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);
333 result += check_equal_simplify(e, 0);
338 static unsigned dummy_check(void)
340 // check dummy index renaming/repositioning
344 symbol p("p"), q("q");
345 idx i(symbol("i"), 3), j(symbol("j"), 3), n(symbol("n"), 3);
346 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
349 e = indexed(p, i) * indexed(q, i) - indexed(p, j) * indexed(q, j);
350 result += check_equal_simplify(e, 0);
352 e = indexed(p, i) * indexed(p, i) * indexed(q, j) * indexed(q, j)
353 - indexed(p, n) * indexed(p, n) * indexed(q, j) * indexed(q, j);
354 result += check_equal_simplify(e, 0);
356 e = indexed(p, mu, mu.toggle_variance()) - indexed(p, nu, nu.toggle_variance());
357 result += check_equal_simplify(e, 0);
359 e = indexed(p, mu.toggle_variance(), nu, mu) * indexed(q, i)
360 - indexed(p, mu, nu, mu.toggle_variance()) * indexed(q, i);
361 result += check_equal_simplify(e, 0);
363 e = indexed(p, mu, mu.toggle_variance()) - indexed(p, nu.toggle_variance(), nu);
364 result += check_equal_simplify(e, 0);
365 e = indexed(p, mu.toggle_variance(), mu) - indexed(p, nu, nu.toggle_variance());
366 result += check_equal_simplify(e, 0);
371 unsigned exam_indexed(void)
375 cout << "examining indexed objects" << flush;
376 clog << "----------indexed objects:" << endl;
378 result += delta_check(); cout << '.' << flush;
379 result += metric_check(); cout << '.' << flush;
380 result += epsilon_check(); cout << '.' << flush;
381 result += symmetry_check(); cout << '.' << flush;
382 result += scalar_product_check(); cout << '.' << flush;
383 result += edyn_check(); cout << '.' << flush;
384 result += spinor_check(); cout << '.' << flush;
385 result += dummy_check(); cout << '.' << flush;
388 cout << " passed " << endl;
389 clog << "(no output)" << endl;
391 cout << " failed " << endl;