ginac/tensor.cpp

   1 /** @file tensor.cpp
   2  *
   3  *  Implementation of GiNaC's special tensors. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2004 Johannes Gutenberg University Mainz, Germany
   7  *
   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.
  12  *
  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.
  17  *
  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., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  21  */
  22
  23 #include <iostream>
  24 #include <stdexcept>
  25 #include <vector>
  26
  27 #include "tensor.h"
  28 #include "idx.h"
  29 #include "indexed.h"
  30 #include "symmetry.h"
  31 #include "relational.h"
  32 #include "operators.h"
  33 #include "lst.h"
  34 #include "numeric.h"
  35 #include "matrix.h"
  36 #include "archive.h"
  37 #include "utils.h"
  38
  39 namespace GiNaC {
  40
  41 GINAC_IMPLEMENT_REGISTERED_CLASS(tensor, basic)
  42
  43 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(tensdelta, tensor,
  44   print_func<print_dflt>(&tensdelta::do_print).
  45   print_func<print_latex>(&tensdelta::do_print_latex))
  46
  47 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(tensmetric, tensor,
  48   print_func<print_dflt>(&tensmetric::do_print).
  49   print_func<print_latex>(&tensmetric::do_print))
  50
  51 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(minkmetric, tensmetric,
  52   print_func<print_dflt>(&minkmetric::do_print).
  53   print_func<print_latex>(&minkmetric::do_print_latex))
  54
  55 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(spinmetric, tensmetric,
  56   print_func<print_dflt>(&spinmetric::do_print).
  57   print_func<print_latex>(&spinmetric::do_print_latex))
  58
  59 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(tensepsilon, tensor,
  60   print_func<print_dflt>(&tensepsilon::do_print).
  61   print_func<print_latex>(&tensepsilon::do_print_latex))
  62
  63 //////////
  64 // constructors
  65 //////////
  66
  67 tensor::tensor() : inherited(TINFO_tensor)
  68 {
  69         setflag(status_flags::evaluated | status_flags::expanded);
  70 }
  71
  72 DEFAULT_CTOR(tensdelta)
  73 DEFAULT_CTOR(tensmetric)
  74
  75 minkmetric::minkmetric() : pos_sig(false)
  76 {
  77         tinfo_key = TINFO_minkmetric;
  78 }
  79
  80 spinmetric::spinmetric()
  81 {
  82         tinfo_key = TINFO_spinmetric;
  83 }
  84
  85 minkmetric::minkmetric(bool ps) : pos_sig(ps)
  86 {
  87         tinfo_key = TINFO_minkmetric;
  88 }
  89
  90 tensepsilon::tensepsilon() : minkowski(false), pos_sig(false)
  91 {
  92         tinfo_key = TINFO_tensepsilon;
  93 }
  94
  95 tensepsilon::tensepsilon(bool mink, bool ps) : minkowski(mink), pos_sig(ps)
  96 {
  97         tinfo_key = TINFO_tensepsilon;
  98 }
  99
 100 //////////
 101 // archiving
 102 //////////
 103
 104 DEFAULT_ARCHIVING(tensor)
 105 DEFAULT_ARCHIVING(tensdelta)
 106 DEFAULT_ARCHIVING(tensmetric)
 107 DEFAULT_ARCHIVING(spinmetric)
 108 DEFAULT_UNARCHIVE(minkmetric)
 109 DEFAULT_UNARCHIVE(tensepsilon)
 110
 111 minkmetric::minkmetric(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
 112 {
 113         n.find_bool("pos_sig", pos_sig);
 114 }
 115
 116 void minkmetric::archive(archive_node &n) const
 117 {
 118         inherited::archive(n);
 119         n.add_bool("pos_sig", pos_sig);
 120 }
 121
 122 tensepsilon::tensepsilon(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
 123 {
 124         n.find_bool("minkowski", minkowski);
 125         n.find_bool("pos_sig", pos_sig);
 126 }
 127
 128 void tensepsilon::archive(archive_node &n) const
 129 {
 130         inherited::archive(n);
 131         n.add_bool("minkowski", minkowski);
 132         n.add_bool("pos_sig", pos_sig);
 133 }
 134
 135 //////////
 136 // functions overriding virtual functions from base classes
 137 //////////
 138
 139 DEFAULT_COMPARE(tensor)
 140 DEFAULT_COMPARE(tensdelta)
 141 DEFAULT_COMPARE(tensmetric)
 142 DEFAULT_COMPARE(spinmetric)
 143
 144 int minkmetric::compare_same_type(const basic & other) const
 145 {
 146         GINAC_ASSERT(is_a<minkmetric>(other));
 147         const minkmetric &o = static_cast<const minkmetric &>(other);
 148
 149         if (pos_sig != o.pos_sig)
 150                 return pos_sig ? -1 : 1;
 151         else
 152                 return inherited::compare_same_type(other);
 153 }
 154
 155 int tensepsilon::compare_same_type(const basic & other) const
 156 {
 157         GINAC_ASSERT(is_a<tensepsilon>(other));
 158         const tensepsilon &o = static_cast<const tensepsilon &>(other);
 159
 160         if (minkowski != o.minkowski)
 161                 return minkowski ? -1 : 1;
 162         else if (pos_sig != o.pos_sig)
 163                 return pos_sig ? -1 : 1;
 164         else
 165                 return inherited::compare_same_type(other);
 166 }
 167
 168 DEFAULT_PRINT_LATEX(tensdelta, "delta", "\\delta")
 169 DEFAULT_PRINT(tensmetric, "g")
 170 DEFAULT_PRINT_LATEX(minkmetric, "eta", "\\eta")
 171 DEFAULT_PRINT_LATEX(spinmetric, "eps", "\\varepsilon")
 172 DEFAULT_PRINT_LATEX(tensepsilon, "eps", "\\varepsilon")
 173
 174 /** Automatic symbolic evaluation of an indexed delta tensor. */
 175 ex tensdelta::eval_indexed(const basic & i) const
 176 {
 177         GINAC_ASSERT(is_a<indexed>(i));
 178         GINAC_ASSERT(i.nops() == 3);
 179         GINAC_ASSERT(is_a<tensdelta>(i.op(0)));
 180
 181         const idx & i1 = ex_to<idx>(i.op(1));
 182         const idx & i2 = ex_to<idx>(i.op(2));
 183
 184         // The dimension of the indices must be equal, otherwise we use the minimal
 185         // dimension
 186         if (!i1.get_dim().is_equal(i2.get_dim())) {
 187                 ex min_dim = i1.minimal_dim(i2);
 188                 exmap m;
 189                 m[i1] = i1.replace_dim(min_dim);
 190                 m[i2] = i2.replace_dim(min_dim);
 191                 return i.subs(m, subs_options::no_pattern);
 192         }
 193
 194         // Trace of delta tensor is the (effective) dimension of the space
 195         if (is_dummy_pair(i1, i2)) {
 196                 try {
 197                         return i1.minimal_dim(i2);
 198                 } catch (std::exception &e) {
 199                         return i.hold();
 200                 }
 201         }
 202
 203         // Numeric evaluation
 204         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::integer)) {
 205                 int n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
 206                 if (n1 == n2)
 207                         return _ex1;
 208                 else
 209                         return _ex0;
 210         }
 211
 212         // No further simplifications
 213         return i.hold();
 214 }
 215
 216 /** Automatic symbolic evaluation of an indexed metric tensor. */
 217 ex tensmetric::eval_indexed(const basic & i) const
 218 {
 219         GINAC_ASSERT(is_a<indexed>(i));
 220         GINAC_ASSERT(i.nops() == 3);
 221         GINAC_ASSERT(is_a<tensmetric>(i.op(0)));
 222         GINAC_ASSERT(is_a<varidx>(i.op(1)));
 223         GINAC_ASSERT(is_a<varidx>(i.op(2)));
 224
 225         const varidx & i1 = ex_to<varidx>(i.op(1));
 226         const varidx & i2 = ex_to<varidx>(i.op(2));
 227
 228         // The dimension of the indices must be equal, otherwise we use the minimal
 229         // dimension
 230         if (!i1.get_dim().is_equal(i2.get_dim())) {
 231                 ex min_dim = i1.minimal_dim(i2);
 232                 exmap m;
 233                 m[i1] = i1.replace_dim(min_dim);
 234                 m[i2] = i2.replace_dim(min_dim);
 235                 return i.subs(m, subs_options::no_pattern);
 236         }
 237
 238         // A metric tensor with one covariant and one contravariant index gets
 239         // replaced by a delta tensor
 240         if (i1.is_covariant() != i2.is_covariant())
 241                 return delta_tensor(i1, i2);
 242
 243         // No further simplifications
 244         return i.hold();
 245 }
 246
 247 /** Automatic symbolic evaluation of an indexed Lorentz metric tensor. */
 248 ex minkmetric::eval_indexed(const basic & i) const
 249 {
 250         GINAC_ASSERT(is_a<indexed>(i));
 251         GINAC_ASSERT(i.nops() == 3);
 252         GINAC_ASSERT(is_a<minkmetric>(i.op(0)));
 253         GINAC_ASSERT(is_a<varidx>(i.op(1)));
 254         GINAC_ASSERT(is_a<varidx>(i.op(2)));
 255
 256         const varidx & i1 = ex_to<varidx>(i.op(1));
 257         const varidx & i2 = ex_to<varidx>(i.op(2));
 258
 259         // Numeric evaluation
 260         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 261                 int n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
 262                 if (n1 != n2)
 263                         return _ex0;
 264                 else if (n1 == 0)
 265                         return pos_sig ? _ex_1 : _ex1;
 266                 else
 267                         return pos_sig ? _ex1 : _ex_1;
 268         }
 269
 270         // Perform the usual evaluations of a metric tensor
 271         return inherited::eval_indexed(i);
 272 }
 273
 274 /** Automatic symbolic evaluation of an indexed metric tensor. */
 275 ex spinmetric::eval_indexed(const basic & i) const
 276 {
 277         GINAC_ASSERT(is_a<indexed>(i));
 278         GINAC_ASSERT(i.nops() == 3);
 279         GINAC_ASSERT(is_a<spinmetric>(i.op(0)));
 280         GINAC_ASSERT(is_a<spinidx>(i.op(1)));
 281         GINAC_ASSERT(is_a<spinidx>(i.op(2)));
 282
 283         const spinidx & i1 = ex_to<spinidx>(i.op(1));
 284         const spinidx & i2 = ex_to<spinidx>(i.op(2));
 285
 286         // Convolutions are zero
 287         if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
 288                 return _ex0;
 289
 290         // Numeric evaluation
 291         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 292                 int n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
 293                 if (n1 == n2)
 294                         return _ex0;
 295                 else if (n1 < n2)
 296                         return _ex1;
 297                 else
 298                         return _ex_1;
 299         }
 300
 301         // No further simplifications
 302         return i.hold();
 303 }
 304
 305 /** Automatic symbolic evaluation of an indexed epsilon tensor. */
 306 ex tensepsilon::eval_indexed(const basic & i) const
 307 {
 308         GINAC_ASSERT(is_a<indexed>(i));
 309         GINAC_ASSERT(i.nops() > 1);
 310         GINAC_ASSERT(is_a<tensepsilon>(i.op(0)));
 311
 312         // Convolutions are zero
 313         if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
 314                 return _ex0;
 315
 316         // Numeric evaluation
 317         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 318
 319                 // Get sign of index permutation (the indices should already be in
 320                 // a canonic order but we can't assume what exactly that order is)
 321                 std::vector<int> v;
 322                 v.reserve(i.nops() - 1);
 323                 for (size_t j=1; j<i.nops(); j++)
 324                         v.push_back(ex_to<numeric>(ex_to<idx>(i.op(j)).get_value()).to_int());
 325                 int sign = permutation_sign(v.begin(), v.end());
 326
 327                 // In a Minkowski space, check for covariant indices
 328                 if (minkowski) {
 329                         for (size_t j=1; j<i.nops(); j++) {
 330                                 const ex & x = i.op(j);
 331                                 if (!is_a<varidx>(x))
 332                                         throw(std::runtime_error("indices of epsilon tensor in Minkowski space must be of type varidx"));
 333                                 if (ex_to<varidx>(x).is_covariant())
 334                                         if (ex_to<idx>(x).get_value().is_zero())
 335                                                 sign = (pos_sig ? -sign : sign);
 336                                         else
 337                                                 sign = (pos_sig ? sign : -sign);
 338                         }
 339                 }
 340
 341                 return sign;
 342         }
 343
 344         // No further simplifications
 345         return i.hold();
 346 }
 347
 348 bool tensor::replace_contr_index(exvector::iterator self, exvector::iterator other) const
 349 {
 350         // Try to contract the first index
 351         const idx *self_idx = &ex_to<idx>(self->op(1));
 352         const idx *free_idx = &ex_to<idx>(self->op(2));
 353         bool first_index_tried = false;
 354
 355 again:
 356         if (self_idx->is_symbolic()) {
 357                 for (size_t i=1; i<other->nops(); i++) {
 358                         const idx &other_idx = ex_to<idx>(other->op(i));
 359                         if (is_dummy_pair(*self_idx, other_idx)) {
 360
 361                                 // Contraction found, remove this tensor and substitute the
 362                                 // index in the second object
 363                                 try {
 364                                         // minimal_dim() throws an exception when index dimensions are not comparable
 365                                         ex min_dim = self_idx->minimal_dim(other_idx);
 366                                         *other = other->subs(other_idx == free_idx->replace_dim(min_dim));
 367                                         *self = _ex1; // *other is assigned first because assigning *self invalidates free_idx
 368                                         return true;
 369                                 } catch (std::exception &e) {
 370                                         return false;
 371                                 }
 372                         }
 373                 }
 374         }
 375
 376         if (!first_index_tried) {
 377
 378                 // No contraction with the first index found, try the second index
 379                 self_idx = &ex_to<idx>(self->op(2));
 380                 free_idx = &ex_to<idx>(self->op(1));
 381                 first_index_tried = true;
 382                 goto again;
 383         }
 384
 385         return false;
 386 }
 387
 388 /** Contraction of an indexed delta tensor with something else. */
 389 bool tensdelta::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 390 {
 391         GINAC_ASSERT(is_a<indexed>(*self));
 392         GINAC_ASSERT(is_a<indexed>(*other));
 393         GINAC_ASSERT(self->nops() == 3);
 394         GINAC_ASSERT(is_a<tensdelta>(self->op(0)));
 395
 396         // Replace the dummy index with this tensor's other index and remove
 397         // the tensor (this is valid for contractions with all other tensors)
 398         return replace_contr_index(self, other);
 399 }
 400
 401 /** Contraction of an indexed metric tensor with something else. */
 402 bool tensmetric::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 403 {
 404         GINAC_ASSERT(is_a<indexed>(*self));
 405         GINAC_ASSERT(is_a<indexed>(*other));
 406         GINAC_ASSERT(self->nops() == 3);
 407         GINAC_ASSERT(is_a<tensmetric>(self->op(0)));
 408
 409         // If contracting with the delta tensor, let the delta do it
 410         // (don't raise/lower delta indices)
 411         if (is_a<tensdelta>(other->op(0)))
 412                 return false;
 413
 414         // Replace the dummy index with this tensor's other index and remove
 415         // the tensor
 416         return replace_contr_index(self, other);
 417 }
 418
 419 /** Contraction of an indexed spinor metric with something else. */
 420 bool spinmetric::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 421 {
 422         GINAC_ASSERT(is_a<indexed>(*self));
 423         GINAC_ASSERT(is_a<indexed>(*other));
 424         GINAC_ASSERT(self->nops() == 3);
 425         GINAC_ASSERT(is_a<spinmetric>(self->op(0)));
 426
 427         // Contractions between spinor metrics
 428         if (is_a<spinmetric>(other->op(0))) {
 429                 const idx &self_i1 = ex_to<idx>(self->op(1));
 430                 const idx &self_i2 = ex_to<idx>(self->op(2));
 431                 const idx &other_i1 = ex_to<idx>(other->op(1));
 432                 const idx &other_i2 = ex_to<idx>(other->op(2));
 433
 434                 if (is_dummy_pair(self_i1, other_i1)) {
 435                         if (is_dummy_pair(self_i2, other_i2))
 436                                 *self = _ex2;
 437                         else
 438                                 *self = delta_tensor(self_i2, other_i2);
 439                         *other = _ex1;
 440                         return true;
 441                 } else if (is_dummy_pair(self_i1, other_i2)) {
 442                         if (is_dummy_pair(self_i2, other_i1))
 443                                 *self = _ex_2;
 444                         else
 445                                 *self = -delta_tensor(self_i2, other_i1);
 446                         *other = _ex1;
 447                         return true;
 448                 } else if (is_dummy_pair(self_i2, other_i1)) {
 449                         *self = -delta_tensor(self_i1, other_i2);
 450                         *other = _ex1;
 451                         return true;
 452                 } else if (is_dummy_pair(self_i2, other_i2)) {
 453                         *self = delta_tensor(self_i1, other_i1);
 454                         *other = _ex1;
 455                         return true;
 456                 }
 457         }
 458
 459         // If contracting with the delta tensor, let the delta do it
 460         // (don't raise/lower delta indices)
 461         if (is_a<tensdelta>(other->op(0)))
 462                 return false;
 463
 464         // Try to contract first index
 465         const idx *self_idx = &ex_to<idx>(self->op(1));
 466         const idx *free_idx = &ex_to<idx>(self->op(2));
 467         bool first_index_tried = false;
 468         int sign = 1;
 469
 470 again:
 471         if (self_idx->is_symbolic()) {
 472                 for (size_t i=1; i<other->nops(); i++) {
 473                         const idx &other_idx = ex_to<idx>(other->op(i));
 474                         if (is_dummy_pair(*self_idx, other_idx)) {
 475
 476                                 // Contraction found, remove metric tensor and substitute
 477                                 // index in second object (assign *self last because this
 478                                 // invalidates free_idx)
 479                                 *other = other->subs(other_idx == *free_idx);
 480                                 *self = (static_cast<const spinidx *>(self_idx)->is_covariant() ? sign : -sign);
 481                                 return true;
 482                         }
 483                 }
 484         }
 485
 486         if (!first_index_tried) {
 487
 488                 // No contraction with first index found, try second index
 489                 self_idx = &ex_to<idx>(self->op(2));
 490                 free_idx = &ex_to<idx>(self->op(1));
 491                 first_index_tried = true;
 492                 sign = -sign;
 493                 goto again;
 494         }
 495
 496         return false;
 497 }
 498
 499 /** Contraction of epsilon tensor with something else. */
 500 bool tensepsilon::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 501 {
 502         GINAC_ASSERT(is_a<indexed>(*self));
 503         GINAC_ASSERT(is_a<indexed>(*other));
 504         GINAC_ASSERT(is_a<tensepsilon>(self->op(0)));
 505         size_t num = self->nops() - 1;
 506
 507         if (is_exactly_a<tensepsilon>(other->op(0)) && num+1 == other->nops()) {
 508
 509                 // Contraction of two epsilon tensors is a determinant
 510                 bool variance = is_a<varidx>(self->op(1));
 511                 matrix M(num, num);
 512                 for (size_t i=0; i<num; i++) {
 513                         for (size_t j=0; j<num; j++) {
 514                                 if (minkowski)
 515                                         M(i, j) = lorentz_g(self->op(i+1), other->op(j+1), pos_sig);
 516                                 else if (variance)
 517                                         M(i, j) = metric_tensor(self->op(i+1), other->op(j+1));
 518                                 else
 519                                         M(i, j) = delta_tensor(self->op(i+1), other->op(j+1));
 520                         }
 521                 }
 522                 int sign = minkowski ? -1 : 1;
 523                 *self = sign * M.determinant().simplify_indexed();
 524                 *other = _ex1;
 525                 return true;
 526         }
 527
 528         return false;
 529 }
 530
 531 //////////
 532 // global functions
 533 //////////
 534
 535 ex delta_tensor(const ex & i1, const ex & i2)
 536 {
 537         if (!is_a<idx>(i1) || !is_a<idx>(i2))
 538                 throw(std::invalid_argument("indices of delta tensor must be of type idx"));
 539
 540         return indexed(tensdelta(), sy_symm(), i1, i2);
 541 }
 542
 543 ex metric_tensor(const ex & i1, const ex & i2)
 544 {
 545         if (!is_a<varidx>(i1) || !is_a<varidx>(i2))
 546                 throw(std::invalid_argument("indices of metric tensor must be of type varidx"));
 547
 548         return indexed(tensmetric(), sy_symm(), i1, i2);
 549 }
 550
 551 ex lorentz_g(const ex & i1, const ex & i2, bool pos_sig)
 552 {
 553         if (!is_a<varidx>(i1) || !is_a<varidx>(i2))
 554                 throw(std::invalid_argument("indices of metric tensor must be of type varidx"));
 555
 556         return indexed(minkmetric(pos_sig), sy_symm(), i1, i2);
 557 }
 558
 559 ex spinor_metric(const ex & i1, const ex & i2)
 560 {
 561         if (!is_a<spinidx>(i1) || !is_a<spinidx>(i2))
 562                 throw(std::invalid_argument("indices of spinor metric must be of type spinidx"));
 563         if (!ex_to<idx>(i1).get_dim().is_equal(2) || !ex_to<idx>(i2).get_dim().is_equal(2))
 564                 throw(std::runtime_error("index dimension for spinor metric must be 2"));
 565
 566         return indexed(spinmetric(), sy_anti(), i1, i2);
 567 }
 568
 569 ex epsilon_tensor(const ex & i1, const ex & i2)
 570 {
 571         if (!is_a<idx>(i1) || !is_a<idx>(i2))
 572                 throw(std::invalid_argument("indices of epsilon tensor must be of type idx"));
 573
 574         ex dim = ex_to<idx>(i1).get_dim();
 575         if (!dim.is_equal(ex_to<idx>(i2).get_dim()))
 576                 throw(std::invalid_argument("all indices of epsilon tensor must have the same dimension"));
 577         if (!ex_to<idx>(i1).get_dim().is_equal(_ex2))
 578                 throw(std::runtime_error("index dimension of epsilon tensor must match number of indices"));
 579
 580         return indexed(tensepsilon(), sy_anti(), i1, i2);
 581 }
 582
 583 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3)
 584 {
 585         if (!is_a<idx>(i1) || !is_a<idx>(i2) || !is_a<idx>(i3))
 586                 throw(std::invalid_argument("indices of epsilon tensor must be of type idx"));
 587
 588         ex dim = ex_to<idx>(i1).get_dim();
 589         if (!dim.is_equal(ex_to<idx>(i2).get_dim()) || !dim.is_equal(ex_to<idx>(i3).get_dim()))
 590                 throw(std::invalid_argument("all indices of epsilon tensor must have the same dimension"));
 591         if (!ex_to<idx>(i1).get_dim().is_equal(_ex3))
 592                 throw(std::runtime_error("index dimension of epsilon tensor must match number of indices"));
 593
 594         return indexed(tensepsilon(), sy_anti(), i1, i2, i3);
 595 }
 596
 597 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig)
 598 {
 599         if (!is_a<varidx>(i1) || !is_a<varidx>(i2) || !is_a<varidx>(i3) || !is_a<varidx>(i4))
 600                 throw(std::invalid_argument("indices of Lorentz epsilon tensor must be of type varidx"));
 601
 602         ex dim = ex_to<idx>(i1).get_dim();
 603         if (!dim.is_equal(ex_to<idx>(i2).get_dim()) || !dim.is_equal(ex_to<idx>(i3).get_dim()) || !dim.is_equal(ex_to<idx>(i4).get_dim()))
 604                 throw(std::invalid_argument("all indices of epsilon tensor must have the same dimension"));
 605         if (!ex_to<idx>(i1).get_dim().is_equal(_ex4))
 606                 throw(std::runtime_error("index dimension of epsilon tensor must match number of indices"));
 607
 608         return indexed(tensepsilon(true, pos_sig), sy_anti(), i1, i2, i3, i4);
 609 }
 610
 611 } // namespace GiNaC