/** @file exam_indexed.cpp
*
* Here we test manipulations on GiNaC's indexed objects. */
/*
* GiNaC Copyright (C) 1999-2007 Johannes Gutenberg University Mainz, Germany
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
*/
#include "exams.h"
static unsigned check_equal(const ex &e1, const ex &e2)
{
ex e = e1 - e2;
if (!e.is_zero()) {
clog << e1 << "-" << e2 << " erroneously returned "
<< e << " instead of 0" << endl;
return 1;
}
return 0;
}
static unsigned check_equal_simplify(const ex &e1, const ex &e2)
{
ex e = simplify_indexed(e1) - e2;
if (!e.is_zero()) {
clog << "simplify_indexed(" << e1 << ")-" << e2 << " erroneously returned "
<< e << " instead of 0" << endl;
return 1;
}
return 0;
}
static unsigned check_equal_simplify(const ex &e1, const ex &e2, const scalar_products &sp)
{
ex e = simplify_indexed(e1, sp) - e2;
if (!e.is_zero()) {
clog << "simplify_indexed(" << e1 << ")-" << e2 << " erroneously returned "
<< e << " instead of 0" << endl;
return 1;
}
return 0;
}
static unsigned delta_check()
{
// checks identities of the delta tensor
unsigned result = 0;
symbol s_i("i"), s_j("j"), s_k("k");
idx i(s_i, 3), j(s_j, 3), k(s_k, 3);
symbol A("A");
// symmetry
result += check_equal(delta_tensor(i, j), delta_tensor(j, i));
// trace = dimension of index space
result += check_equal(delta_tensor(i, i), 3);
result += check_equal_simplify(delta_tensor(i, j) * delta_tensor(i, j), 3);
// contraction with delta tensor
result += check_equal_simplify(delta_tensor(i, j) * indexed(A, k), delta_tensor(i, j) * indexed(A, k));
result += check_equal_simplify(delta_tensor(i, j) * indexed(A, j), indexed(A, i));
result += check_equal_simplify(delta_tensor(i, j) * indexed(A, i), indexed(A, j));
result += check_equal_simplify(delta_tensor(i, j) * delta_tensor(j, k) * indexed(A, i), indexed(A, k));
return result;
}
static unsigned metric_check()
{
// checks identities of the metric tensor
unsigned result = 0;
symbol s_mu("mu"), s_nu("nu"), s_rho("rho"), s_sigma("sigma");
varidx mu(s_mu, 4), nu(s_nu, 4), rho(s_rho, 4), sigma(s_sigma, 4);
symbol A("A");
// becomes delta tensor if indices have opposite variance
result += check_equal(metric_tensor(mu, nu.toggle_variance()), delta_tensor(mu, nu.toggle_variance()));
// scalar contraction = dimension of index space
result += check_equal(metric_tensor(mu, mu.toggle_variance()), 4);
result += check_equal_simplify(metric_tensor(mu, nu) * metric_tensor(mu.toggle_variance(), nu.toggle_variance()), 4);
// contraction with metric tensor
result += check_equal_simplify(metric_tensor(mu, nu) * indexed(A, nu), metric_tensor(mu, nu) * indexed(A, nu));
result += check_equal_simplify(metric_tensor(mu, nu) * indexed(A, nu.toggle_variance()), indexed(A, mu));
result += check_equal_simplify(metric_tensor(mu, nu) * indexed(A, mu.toggle_variance()), indexed(A, nu));
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()));
result += check_equal_simplify(metric_tensor(mu, rho) * metric_tensor(nu, sigma) * indexed(A, rho.toggle_variance(), sigma.toggle_variance()), indexed(A, mu, nu));
result += check_equal_simplify(indexed(A, mu.toggle_variance()) * metric_tensor(mu, nu) - indexed(A, mu.toggle_variance()) * metric_tensor(nu, mu), 0);
result += check_equal_simplify(indexed(A, mu.toggle_variance(), nu.toggle_variance()) * metric_tensor(nu, rho), indexed(A, mu.toggle_variance(), rho));
// contraction with delta tensor yields a metric tensor
result += check_equal_simplify(delta_tensor(mu, nu.toggle_variance()) * metric_tensor(nu, rho), metric_tensor(mu, rho));
result += check_equal_simplify(metric_tensor(mu, nu) * indexed(A, nu.toggle_variance()) * delta_tensor(mu.toggle_variance(), rho), indexed(A, rho));
return result;
}
static unsigned epsilon_check()
{
// checks identities of the epsilon tensor
unsigned result = 0;
symbol s_mu("mu"), s_nu("nu"), s_rho("rho"), s_sigma("sigma"), s_tau("tau");
symbol d("d");
varidx mu(s_mu, 4), nu(s_nu, 4), rho(s_rho, 4), sigma(s_sigma, 4), tau(s_tau, 4);
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);
// antisymmetry
result += check_equal(lorentz_eps(mu, nu, rho, sigma) + lorentz_eps(sigma, rho, mu, nu), 0);
// convolution is zero
result += check_equal(lorentz_eps(mu, nu, rho, nu_co), 0);
result += check_equal(lorentz_eps(mu, nu, mu_co, nu_co), 0);
result += check_equal_simplify(lorentz_g(mu_co, nu_co) * lorentz_eps(mu, nu, rho, sigma), 0);
// contraction with symmetric tensor is zero
result += check_equal_simplify(lorentz_eps(mu, nu, rho, sigma) * indexed(d, sy_symm(), mu_co, nu_co), 0);
result += check_equal_simplify(lorentz_eps(mu, nu, rho, sigma) * indexed(d, sy_symm(), nu_co, sigma_co, rho_co), 0);
result += check_equal_simplify(lorentz_eps(mu, nu, rho, sigma) * indexed(d, mu_co) * indexed(d, nu_co), 0);
result += check_equal_simplify(lorentz_eps(mu_co, nu, rho, sigma) * indexed(d, mu) * indexed(d, nu_co), 0);
ex e = lorentz_eps(mu, nu, rho, sigma) * indexed(d, mu_co) - lorentz_eps(mu_co, nu, rho, sigma) * indexed(d, mu);
result += check_equal_simplify(e, 0);
// contractions of epsilon tensors
result += check_equal_simplify(lorentz_eps(mu, nu, rho, sigma) * lorentz_eps(mu_co, nu_co, rho_co, sigma_co), -24);
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));
return result;
}
DECLARE_FUNCTION_2P(symm_fcn)
REGISTER_FUNCTION(symm_fcn, set_symmetry(sy_symm(0, 1)));
DECLARE_FUNCTION_2P(anti_fcn)
REGISTER_FUNCTION(anti_fcn, set_symmetry(sy_anti(0, 1)));
static unsigned symmetry_check()
{
// check symmetric/antisymmetric objects
unsigned result = 0;
idx i(symbol("i"), 3), j(symbol("j"), 3), k(symbol("k"), 3), l(symbol("l"), 3);
symbol A("A"), B("B"), C("C");
ex e;
result += check_equal(indexed(A, sy_symm(), i, j), indexed(A, sy_symm(), j, i));
result += check_equal(indexed(A, sy_anti(), i, j) + indexed(A, sy_anti(), j, i), 0);
result += check_equal(indexed(A, sy_anti(), i, j, k) - indexed(A, sy_anti(), j, k, i), 0);
e = indexed(A, sy_symm(), i, j, k) *
indexed(B, sy_anti(), l, k, i);
result += check_equal_simplify(e, 0);
e = indexed(A, sy_symm(), i, i, j, j) *
indexed(B, sy_anti(), k, l); // GiNaC 0.8.0 had a bug here
result += check_equal_simplify(e, e);
symmetry R = sy_symm(sy_anti(0, 1), sy_anti(2, 3));
e = indexed(A, R, i, j, k, l) + indexed(A, R, j, i, k, l);
result += check_equal(e, 0);
e = indexed(A, R, i, j, k, l) + indexed(A, R, i, j, l, k);
result += check_equal(e, 0);
e = indexed(A, R, i, j, k, l) - indexed(A, R, j, i, l, k);
result += check_equal(e, 0);
e = indexed(A, R, i, j, k, l) + indexed(A, R, k, l, j, i);
result += check_equal(e, 0);
e = indexed(A, i, j);
result += check_equal(symmetrize(e) + antisymmetrize(e), e);
e = indexed(A, sy_symm(), i, j, k, l);
result += check_equal(symmetrize(e), e);
result += check_equal(antisymmetrize(e), 0);
e = indexed(A, sy_anti(), i, j, k, l);
result += check_equal(symmetrize(e), 0);
result += check_equal(antisymmetrize(e), e);
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();
result += check_equal_simplify(e, indexed(B, i, l));
result += check_equal(symm_fcn(0, 1) + symm_fcn(1, 0), 2*symm_fcn(0, 1));
result += check_equal(anti_fcn(0, 1) + anti_fcn(1, 0), 0);
result += check_equal(anti_fcn(0, 0), 0);
return result;
}
static unsigned scalar_product_check()
{
// check scalar product replacement
unsigned result = 0;
idx i(symbol("i"), 3), j(symbol("j"), 3);
symbol A("A"), B("B"), C("C");
ex e;
scalar_products sp;
sp.add(A, B, 0); // A and B are orthogonal
sp.add(A, C, 0); // A and C are orthogonal
sp.add(A, A, 4); // A^2 = 4 (A has length 2)
e = (indexed(A + B, i) * indexed(A + C, i)).expand(expand_options::expand_indexed);
result += check_equal_simplify(e, indexed(B, i) * indexed(C, i) + 4, sp);
e = indexed(A, i, i) * indexed(B, j, j); // GiNaC 0.8.0 had a bug here
result += check_equal_simplify(e, e, sp);
return result;
}
static unsigned edyn_check()
{
// Relativistic electrodynamics
// Test 1: check transformation laws of electric and magnetic fields by
// applying a Lorentz boost to the field tensor
unsigned result = 0;
symbol beta("beta");
ex gamma = 1 / sqrt(1 - pow(beta, 2));
symbol Ex("Ex"), Ey("Ey"), Ez("Ez");
symbol Bx("Bx"), By("By"), Bz("Bz");
// Lorentz transformation matrix (boost along x axis)
matrix L(4, 4);
L = gamma, -beta*gamma, 0, 0,
-beta*gamma, gamma, 0, 0,
0, 0, 1, 0,
0, 0, 0, 1;
// Electromagnetic field tensor
matrix F(4, 4);
F = 0, -Ex, -Ey, -Ez,
Ex, 0, -Bz, By,
Ey, Bz, 0, -Bx,
Ez, -By, Bx, 0;
// Indices
symbol s_mu("mu"), s_nu("nu"), s_rho("rho"), s_sigma("sigma");
varidx mu(s_mu, 4), nu(s_nu, 4), rho(s_rho, 4), sigma(s_sigma, 4);
// Apply transformation law of second rank tensor
ex e = (indexed(L, mu, rho.toggle_variance())
* indexed(L, nu, sigma.toggle_variance())
* indexed(F, rho, sigma)).simplify_indexed();
// Extract transformed electric and magnetic fields
ex Ex_p = e.subs(lst(mu == 1, nu == 0)).normal();
ex Ey_p = e.subs(lst(mu == 2, nu == 0)).normal();
ex Ez_p = e.subs(lst(mu == 3, nu == 0)).normal();
ex Bx_p = e.subs(lst(mu == 3, nu == 2)).normal();
ex By_p = e.subs(lst(mu == 1, nu == 3)).normal();
ex Bz_p = e.subs(lst(mu == 2, nu == 1)).normal();
// Check results
result += check_equal(Ex_p, Ex);
result += check_equal(Ey_p, gamma * (Ey - beta * Bz));
result += check_equal(Ez_p, gamma * (Ez + beta * By));
result += check_equal(Bx_p, Bx);
result += check_equal(By_p, gamma * (By + beta * Ez));
result += check_equal(Bz_p, gamma * (Bz - beta * Ey));
// Test 2: check energy density and Poynting vector of electromagnetic field
// Minkowski metric
ex eta = diag_matrix(lst(1, -1, -1, -1));
// Covariant field tensor
ex F_mu_nu = (indexed(eta, mu.toggle_variance(), rho.toggle_variance())
* indexed(eta, nu.toggle_variance(), sigma.toggle_variance())
* indexed(F, rho, sigma)).simplify_indexed();
// Energy-momentum tensor
ex T = (-indexed(eta, rho, sigma) * F_mu_nu.subs(s_nu == s_rho)
* F_mu_nu.subs(lst(s_mu == s_nu, s_nu == s_sigma))
+ indexed(eta, mu.toggle_variance(), nu.toggle_variance())
* F_mu_nu.subs(lst(s_mu == s_rho, s_nu == s_sigma))
* indexed(F, rho, sigma) / 4).simplify_indexed() / (4 * Pi);
// Extract energy density and Poynting vector
ex E = T.subs(lst(s_mu == 0, s_nu == 0)).normal();
ex Px = T.subs(lst(s_mu == 0, s_nu == 1));
ex Py = T.subs(lst(s_mu == 0, s_nu == 2));
ex Pz = T.subs(lst(s_mu == 0, s_nu == 3));
// Check results
result += check_equal(E, (Ex*Ex+Ey*Ey+Ez*Ez+Bx*Bx+By*By+Bz*Bz) / (8 * Pi));
result += check_equal(Px, (Ez*By-Ey*Bz) / (4 * Pi));
result += check_equal(Py, (Ex*Bz-Ez*Bx) / (4 * Pi));
result += check_equal(Pz, (Ey*Bx-Ex*By) / (4 * Pi));
return result;
}
static unsigned spinor_check()
{
// check identities of the spinor metric
unsigned result = 0;
symbol psi("psi");
spinidx A(symbol("A")), B(symbol("B")), C(symbol("C")), D(symbol("D"));
ex A_co = A.toggle_variance(), B_co = B.toggle_variance();
ex e;
e = spinor_metric(A_co, B_co) * spinor_metric(A, B);
result += check_equal_simplify(e, 2);
e = spinor_metric(A_co, B_co) * spinor_metric(B, A);
result += check_equal_simplify(e, -2);
e = spinor_metric(A_co, B_co) * spinor_metric(A, C);
result += check_equal_simplify(e, delta_tensor(B_co, C));
e = spinor_metric(A_co, B_co) * spinor_metric(B, C);
result += check_equal_simplify(e, -delta_tensor(A_co, C));
e = spinor_metric(A_co, B_co) * spinor_metric(C, A);
result += check_equal_simplify(e, -delta_tensor(B_co, C));
e = spinor_metric(A, B) * indexed(psi, B_co);
result += check_equal_simplify(e, indexed(psi, A));
e = spinor_metric(A, B) * indexed(psi, A_co);
result += check_equal_simplify(e, -indexed(psi, B));
e = spinor_metric(A_co, B_co) * indexed(psi, B);
result += check_equal_simplify(e, -indexed(psi, A_co));
e = spinor_metric(A_co, B_co) * indexed(psi, A);
result += check_equal_simplify(e, indexed(psi, B_co));
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);
result += check_equal_simplify(e, 0);
return result;
}
static unsigned dummy_check()
{
// check dummy index renaming/repositioning
unsigned result = 0;
symbol p("p"), q("q");
idx i(symbol("i"), 3), j(symbol("j"), 3), n(symbol("n"), 3);
varidx mu(symbol("mu"), 4), nu(symbol("nu"), 4);
ex e;
e = indexed(p, i) * indexed(q, i) - indexed(p, j) * indexed(q, j);
result += check_equal_simplify(e, 0);
e = indexed(p, i) * indexed(p, i) * indexed(q, j) * indexed(q, j)
- indexed(p, n) * indexed(p, n) * indexed(q, j) * indexed(q, j);
result += check_equal_simplify(e, 0);
e = indexed(p, mu, mu.toggle_variance()) - indexed(p, nu, nu.toggle_variance());
result += check_equal_simplify(e, 0);
e = indexed(p, mu.toggle_variance(), nu, mu) * indexed(q, i)
- indexed(p, mu, nu, mu.toggle_variance()) * indexed(q, i);
result += check_equal_simplify(e, 0);
e = indexed(p, mu, mu.toggle_variance()) - indexed(p, nu.toggle_variance(), nu);
result += check_equal_simplify(e, 0);
e = indexed(p, mu.toggle_variance(), mu) - indexed(p, nu, nu.toggle_variance());
result += check_equal_simplify(e, 0);
// GiNaC 1.2.1 had a bug here because p.i*p.i -> (p.i)^2
e = indexed(p, i) * indexed(p, i) * indexed(p, j) + indexed(p, j);
ex fi = exprseq(e.get_free_indices());
if (!fi.is_equal(exprseq(j))) {
clog << "get_free_indices(" << e << ") erroneously returned "
<< fi << " instead of (.j)" << endl;
++result;
}
return result;
}
unsigned exam_indexed()
{
unsigned result = 0;
cout << "examining indexed objects" << flush;
clog << "----------indexed objects:" << endl;
result += delta_check(); cout << '.' << flush;
result += metric_check(); cout << '.' << flush;
result += epsilon_check(); cout << '.' << flush;
result += symmetry_check(); cout << '.' << flush;
result += scalar_product_check(); cout << '.' << flush;
result += edyn_check(); cout << '.' << flush;
result += spinor_check(); cout << '.' << flush;
result += dummy_check(); cout << '.' << flush;
if (!result) {
cout << " passed " << endl;
clog << "(no output)" << endl;
} else {
cout << " failed " << endl;
}
return result;
}
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