1 /** @file exam_indexed.cpp
3 * Here we test manipulations on GiNaC's indexed objects. */
6 * GiNaC Copyright (C) 1999-2019 Johannes Gutenberg University Mainz, Germany
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20 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
24 using namespace GiNaC;
29 static unsigned check_equal(const ex &e1, const ex &e2)
33 clog << e1 << "-" << e2 << " erroneously returned "
34 << e << " instead of 0" << endl;
40 static unsigned check_equal_simplify(const ex &e1, const ex &e2)
42 ex e = simplify_indexed(e1) - e2;
44 clog << "simplify_indexed(" << e1 << ")-" << e2 << " erroneously returned "
45 << e << " instead of 0" << endl;
51 static unsigned check_equal_simplify(const ex &e1, const ex &e2, const scalar_products &sp)
53 ex e = simplify_indexed(e1, sp) - e2;
55 clog << "simplify_indexed(" << e1 << ")-" << e2 << " erroneously returned "
56 << e << " instead of 0" << endl;
62 static unsigned delta_check()
64 // checks identities of the delta tensor
68 symbol s_i("i"), s_j("j"), s_k("k");
69 idx i(s_i, 3), j(s_j, 3), k(s_k, 3);
73 result += check_equal(delta_tensor(i, j), delta_tensor(j, i));
75 // trace = dimension of index space
76 result += check_equal(delta_tensor(i, i), 3);
77 result += check_equal_simplify(delta_tensor(i, j) * delta_tensor(i, j), 3);
79 // contraction with delta tensor
80 result += check_equal_simplify(delta_tensor(i, j) * indexed(A, k), delta_tensor(i, j) * indexed(A, k));
81 result += check_equal_simplify(delta_tensor(i, j) * indexed(A, j), indexed(A, i));
82 result += check_equal_simplify(delta_tensor(i, j) * indexed(A, i), indexed(A, j));
83 result += check_equal_simplify(delta_tensor(i, j) * delta_tensor(j, k) * indexed(A, i), indexed(A, k));
88 static unsigned metric_check()
90 // checks identities of the metric tensor
94 symbol s_mu("mu"), s_nu("nu"), s_rho("rho"), s_sigma("sigma");
95 varidx mu(s_mu, 4), nu(s_nu, 4), rho(s_rho, 4), sigma(s_sigma, 4);
98 // becomes delta tensor if indices have opposite variance
99 result += check_equal(metric_tensor(mu, nu.toggle_variance()), delta_tensor(mu, nu.toggle_variance()));
101 // scalar contraction = dimension of index space
102 result += check_equal(metric_tensor(mu, mu.toggle_variance()), 4);
103 result += check_equal_simplify(metric_tensor(mu, nu) * metric_tensor(mu.toggle_variance(), nu.toggle_variance()), 4);
105 // contraction with metric tensor
106 result += check_equal_simplify(metric_tensor(mu, nu) * indexed(A, nu), metric_tensor(mu, nu) * indexed(A, nu));
107 result += check_equal_simplify(metric_tensor(mu, nu) * indexed(A, nu.toggle_variance()), indexed(A, mu));
108 result += check_equal_simplify(metric_tensor(mu, nu) * indexed(A, mu.toggle_variance()), indexed(A, nu));
109 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()));
110 result += check_equal_simplify(metric_tensor(mu, rho) * metric_tensor(nu, sigma) * indexed(A, rho.toggle_variance(), sigma.toggle_variance()), indexed(A, mu, nu));
111 result += check_equal_simplify(indexed(A, mu.toggle_variance()) * metric_tensor(mu, nu) - indexed(A, mu.toggle_variance()) * metric_tensor(nu, mu), 0);
112 result += check_equal_simplify(indexed(A, mu.toggle_variance(), nu.toggle_variance()) * metric_tensor(nu, rho), indexed(A, mu.toggle_variance(), rho));
114 // contraction with delta tensor yields a metric tensor
115 result += check_equal_simplify(delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho), metric_tensor(mu, rho));
116 result += check_equal_simplify(metric_tensor(mu, nu) * indexed(A, nu.toggle_variance()) * delta_tensor(mu.toggle_variance(), rho), indexed(A, rho));
121 static unsigned epsilon_check()
123 // checks identities of the epsilon tensor
127 symbol s_mu("mu"), s_nu("nu"), s_rho("rho"), s_sigma("sigma"), s_tau("tau");
129 varidx mu(s_mu, 4), nu(s_nu, 4), rho(s_rho, 4), sigma(s_sigma, 4), tau(s_tau, 4);
130 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);
133 result += check_equal(lorentz_eps(mu, nu, rho, sigma) + lorentz_eps(sigma, rho, mu, nu), 0);
135 // convolution is zero
136 result += check_equal(lorentz_eps(mu, nu, rho, nu_co), 0);
137 result += check_equal(lorentz_eps(mu, nu, mu_co, nu_co), 0);
138 result += check_equal_simplify(lorentz_g(mu_co, nu_co) * lorentz_eps(mu, nu, rho, sigma), 0);
140 // contraction with symmetric tensor is zero
141 result += check_equal_simplify(lorentz_eps(mu, nu, rho, sigma) * indexed(d, sy_symm(), mu_co, nu_co), 0);
142 result += check_equal_simplify(lorentz_eps(mu, nu, rho, sigma) * indexed(d, sy_symm(), nu_co, sigma_co, rho_co), 0);
143 result += check_equal_simplify(lorentz_eps(mu, nu, rho, sigma) * indexed(d, mu_co) * indexed(d, nu_co), 0);
144 result += check_equal_simplify(lorentz_eps(mu_co, nu, rho, sigma) * indexed(d, mu) * indexed(d, nu_co), 0);
145 ex e = lorentz_eps(mu, nu, rho, sigma) * indexed(d, mu_co) - lorentz_eps(mu_co, nu, rho, sigma) * indexed(d, mu);
146 result += check_equal_simplify(e, 0);
148 // contractions of epsilon tensors
149 result += check_equal_simplify(lorentz_eps(mu, nu, rho, sigma) * lorentz_eps(mu_co, nu_co, rho_co, sigma_co), -24);
150 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));
155 DECLARE_FUNCTION_2P(symm_fcn)
156 REGISTER_FUNCTION(symm_fcn, set_symmetry(sy_symm(0, 1)));
157 DECLARE_FUNCTION_2P(anti_fcn)
158 REGISTER_FUNCTION(anti_fcn, set_symmetry(sy_anti(0, 1)));
160 static unsigned symmetry_check()
162 // check symmetric/antisymmetric objects
166 idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3), l(symbol("l"), 3);
167 symbol A("A"), B("B"), C("C");
170 result += check_equal(indexed(A, sy_symm(), i, j), indexed(A, sy_symm(), j, i));
171 result += check_equal(indexed(A, sy_anti(), i, j) + indexed(A, sy_anti(), j, i), 0);
172 result += check_equal(indexed(A, sy_anti(), i, j, k) - indexed(A, sy_anti(), j, k, i), 0);
173 e = indexed(A, sy_symm(), i, j, k) *
174 indexed(B, sy_anti(), l, k, i);
175 result += check_equal_simplify(e, 0);
176 e = indexed(A, sy_symm(), i, i, j, j) *
177 indexed(B, sy_anti(), k, l); // GiNaC 0.8.0 had a bug here
178 result += check_equal_simplify(e, e);
180 symmetry R = sy_symm(sy_anti(0, 1), sy_anti(2, 3));
181 e = indexed(A, R, i, j, k, l) + indexed(A, R, j, i, k, l);
182 result += check_equal(e, 0);
183 e = indexed(A, R, i, j, k, l) + indexed(A, R, i, j, l, k);
184 result += check_equal(e, 0);
185 e = indexed(A, R, i, j, k, l) - indexed(A, R, j, i, l, k);
186 result += check_equal(e, 0);
187 e = indexed(A, R, i, j, k, l) + indexed(A, R, k, l, j, i);
188 result += check_equal(e, 0);
190 e = indexed(A, i, j);
191 result += check_equal(symmetrize(e) + antisymmetrize(e), e);
192 e = indexed(A, sy_symm(), i, j, k, l);
193 result += check_equal(symmetrize(e), e);
194 result += check_equal(antisymmetrize(e), 0);
195 e = indexed(A, sy_anti(), i, j, k, l);
196 result += check_equal(symmetrize(e), 0);
197 result += check_equal(antisymmetrize(e), e);
199 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();
200 result += check_equal_simplify(e, indexed(B, i, l));
202 result += check_equal(symm_fcn(0, 1) + symm_fcn(1, 0), 2*symm_fcn(0, 1));
203 result += check_equal(anti_fcn(0, 1) + anti_fcn(1, 0), 0);
204 result += check_equal(anti_fcn(0, 0), 0);
209 static unsigned scalar_product_check()
211 // check scalar product replacement
215 idx i(symbol("i"), 3), j(symbol("j"), 3);
216 symbol A("A"), B("B"), C("C");
220 sp.add(A, B, 0); // A and B are orthogonal
221 sp.add(A, C, 0); // A and C are orthogonal
222 sp.add(A, A, 4); // A^2 = 4 (A has length 2)
224 e = (indexed(A + B, i) * indexed(A + C, i)).expand(expand_options::expand_indexed);
225 result += check_equal_simplify(e, indexed(B, i) * indexed(C, i) + 4, sp);
226 e = indexed(A, i, i) * indexed(B, j, j); // GiNaC 0.8.0 had a bug here
227 result += check_equal_simplify(e, e, sp);
232 static unsigned edyn_check()
234 // Relativistic electrodynamics
236 // Test 1: check transformation laws of electric and magnetic fields by
237 // applying a Lorentz boost to the field tensor
242 ex gamma = 1 / sqrt(1 - pow(beta, 2));
243 symbol Ex("Ex"), Ey("Ey"), Ez("Ez");
244 symbol Bx("Bx"), By("By"), Bz("Bz");
246 // Lorentz transformation matrix (boost along x axis)
247 matrix L = {{ gamma, -beta*gamma, 0, 0},
248 {-beta*gamma, gamma, 0, 0},
252 // Electromagnetic field tensor
253 matrix F = {{ 0, -Ex, -Ey, -Ez},
259 symbol s_mu("mu"), s_nu("nu"), s_rho("rho"), s_sigma("sigma");
260 varidx mu(s_mu, 4), nu(s_nu, 4), rho(s_rho, 4), sigma(s_sigma, 4);
262 // Apply transformation law of second rank tensor
263 ex e = (indexed(L, mu, rho.toggle_variance())
264 * indexed(L, nu, sigma.toggle_variance())
265 * indexed(F, rho, sigma)).simplify_indexed();
267 // Extract transformed electric and magnetic fields
268 ex Ex_p = e.subs(lst{mu == 1, nu == 0}).normal();
269 ex Ey_p = e.subs(lst{mu == 2, nu == 0}).normal();
270 ex Ez_p = e.subs(lst{mu == 3, nu == 0}).normal();
271 ex Bx_p = e.subs(lst{mu == 3, nu == 2}).normal();
272 ex By_p = e.subs(lst{mu == 1, nu == 3}).normal();
273 ex Bz_p = e.subs(lst{mu == 2, nu == 1}).normal();
276 result += check_equal(Ex_p, Ex);
277 result += check_equal(Ey_p, gamma * (Ey - beta * Bz));
278 result += check_equal(Ez_p, gamma * (Ez + beta * By));
279 result += check_equal(Bx_p, Bx);
280 result += check_equal(By_p, gamma * (By + beta * Ez));
281 result += check_equal(Bz_p, gamma * (Bz - beta * Ey));
283 // Test 2: check energy density and Poynting vector of electromagnetic field
286 ex eta = diag_matrix(lst{1, -1, -1, -1});
288 // Covariant field tensor
289 ex F_mu_nu = (indexed(eta, mu.toggle_variance(), rho.toggle_variance())
290 * indexed(eta, nu.toggle_variance(), sigma.toggle_variance())
291 * indexed(F, rho, sigma)).simplify_indexed();
293 // Energy-momentum tensor
294 ex T = (-indexed(eta, rho, sigma) * F_mu_nu.subs(s_nu == s_rho)
295 * F_mu_nu.subs(lst{s_mu == s_nu, s_nu == s_sigma})
296 + indexed(eta, mu.toggle_variance(), nu.toggle_variance())
297 * F_mu_nu.subs(lst{s_mu == s_rho, s_nu == s_sigma})
298 * indexed(F, rho, sigma) / 4).simplify_indexed() / (4 * Pi);
300 // Extract energy density and Poynting vector
301 ex E = T.subs(lst{s_mu == 0, s_nu == 0}).normal();
302 ex Px = T.subs(lst{s_mu == 0, s_nu == 1});
303 ex Py = T.subs(lst{s_mu == 0, s_nu == 2});
304 ex Pz = T.subs(lst{s_mu == 0, s_nu == 3});
307 result += check_equal(E, (Ex*Ex+Ey*Ey+Ez*Ez+Bx*Bx+By*By+Bz*Bz) / (8 * Pi));
308 result += check_equal(Px, (Ez*By-Ey*Bz) / (4 * Pi));
309 result += check_equal(Py, (Ex*Bz-Ez*Bx) / (4 * Pi));
310 result += check_equal(Pz, (Ey*Bx-Ex*By) / (4 * Pi));
315 static unsigned spinor_check()
317 // check identities of the spinor metric
322 spinidx A(symbol("A")), B(symbol("B")), C(symbol("C")), D(symbol("D"));
323 ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
326 e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
327 result += check_equal_simplify(e, 2);
328 e = spinor_metric(A_co, B_co) * spinor_metric(B, A);
329 result += check_equal_simplify(e, -2);
330 e = spinor_metric(A_co, B_co) * spinor_metric(A, C);
331 result += check_equal_simplify(e, delta_tensor(B_co, C));
332 e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
333 result += check_equal_simplify(e, -delta_tensor(A_co, C));
334 e = spinor_metric(A_co, B_co) * spinor_metric(C, A);
335 result += check_equal_simplify(e, -delta_tensor(B_co, C));
336 e = spinor_metric(A, B) * indexed(psi, B_co);
337 result += check_equal_simplify(e, indexed(psi, A));
338 e = spinor_metric(A, B) * indexed(psi, A_co);
339 result += check_equal_simplify(e, -indexed(psi, B));
340 e = spinor_metric(A_co, B_co) * indexed(psi, B);
341 result += check_equal_simplify(e, -indexed(psi, A_co));
342 e = spinor_metric(A_co, B_co) * indexed(psi, A);
343 result += check_equal_simplify(e, indexed(psi, B_co));
344 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);
345 result += check_equal_simplify(e, 0);
350 static unsigned dummy_check()
352 // check dummy index renaming/repositioning
356 symbol p("p"), q("q");
357 idx i(symbol("i"), 3), j(symbol("j"), 3), n(symbol("n"), 3);
358 varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
361 e = indexed(p, i) * indexed(q, i) - indexed(p, j) * indexed(q, j);
362 result += check_equal_simplify(e, 0);
364 e = indexed(p, i) * indexed(p, i) * indexed(q, j) * indexed(q, j)
365 - indexed(p, n) * indexed(p, n) * indexed(q, j) * indexed(q, j);
366 result += check_equal_simplify(e, 0);
368 e = indexed(p, mu, mu.toggle_variance()) - indexed(p, nu, nu.toggle_variance());
369 result += check_equal_simplify(e, 0);
371 e = indexed(p, mu.toggle_variance(), nu, mu) * indexed(q, i)
372 - indexed(p, mu, nu, mu.toggle_variance()) * indexed(q, i);
373 result += check_equal_simplify(e, 0);
375 e = indexed(p, mu, mu.toggle_variance()) - indexed(p, nu.toggle_variance(), nu);
376 result += check_equal_simplify(e, 0);
377 e = indexed(p, mu.toggle_variance(), mu) - indexed(p, nu, nu.toggle_variance());
378 result += check_equal_simplify(e, 0);
380 // GiNaC 1.2.1 had a bug here because p.i*p.i -> (p.i)^2
381 e = indexed(p, i) * indexed(p, i) * indexed(p, j) + indexed(p, j);
382 ex fi = exprseq(e.get_free_indices());
383 if (!fi.is_equal(exprseq{j})) {
384 clog << "get_free_indices(" << e << ") erroneously returned "
385 << fi << " instead of (.j)" << endl;
392 unsigned exam_indexed()
396 cout << "examining indexed objects" << flush;
398 result += delta_check(); cout << '.' << flush;
399 result += metric_check(); cout << '.' << flush;
400 result += epsilon_check(); cout << '.' << flush;
401 result += symmetry_check(); cout << '.' << flush;
402 result += scalar_product_check(); cout << '.' << flush;
403 result += edyn_check(); cout << '.' << flush;
404 result += spinor_check(); cout << '.' << flush;
405 result += dummy_check(); cout << '.' << flush;
410 int main(int argc, char** argv)
412 return exam_indexed();