ginac/tensor.cpp

   1 /** @file tensor.cpp
   2  *
   3  *  Implementation of GiNaC's special tensors. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2003 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 "print.h"
  37 #include "archive.h"
  38 #include "utils.h"
  39
  40 namespace GiNaC {
  41
  42 GINAC_IMPLEMENT_REGISTERED_CLASS(tensor, basic)
  43 GINAC_IMPLEMENT_REGISTERED_CLASS(tensdelta, tensor)
  44 GINAC_IMPLEMENT_REGISTERED_CLASS(tensmetric, tensor)
  45 GINAC_IMPLEMENT_REGISTERED_CLASS(minkmetric, tensmetric)
  46 GINAC_IMPLEMENT_REGISTERED_CLASS(spinmetric, tensmetric)
  47 GINAC_IMPLEMENT_REGISTERED_CLASS(tensepsilon, tensor)
  48
  49 //////////
  50 // default ctor, dtor, copy ctor, assignment operator and helpers
  51 //////////
  52
  53 DEFAULT_CTORS(tensor)
  54 DEFAULT_CTORS(tensdelta)
  55 DEFAULT_CTORS(tensmetric)
  56 DEFAULT_COPY(spinmetric)
  57 DEFAULT_DESTROY(spinmetric)
  58 DEFAULT_DESTROY(minkmetric)
  59 DEFAULT_DESTROY(tensepsilon)
  60
  61 minkmetric::minkmetric() : pos_sig(false)
  62 {
  63         tinfo_key = TINFO_minkmetric;
  64 }
  65
  66 spinmetric::spinmetric()
  67 {
  68         tinfo_key = TINFO_spinmetric;
  69 }
  70
  71 minkmetric::minkmetric(bool ps) : pos_sig(ps)
  72 {
  73         tinfo_key = TINFO_minkmetric;
  74 }
  75
  76 void minkmetric::copy(const minkmetric & other)
  77 {
  78         inherited::copy(other);
  79         pos_sig = other.pos_sig;
  80 }
  81
  82 tensepsilon::tensepsilon() : minkowski(false), pos_sig(false)
  83 {
  84         tinfo_key = TINFO_tensepsilon;
  85 }
  86
  87 tensepsilon::tensepsilon(bool mink, bool ps) : minkowski(mink), pos_sig(ps)
  88 {
  89         tinfo_key = TINFO_tensepsilon;
  90 }
  91
  92 void tensepsilon::copy(const tensepsilon & other)
  93 {
  94         inherited::copy(other);
  95         minkowski = other.minkowski;
  96         pos_sig = other.pos_sig;
  97 }
  98
  99 //////////
 100 // archiving
 101 //////////
 102
 103 DEFAULT_ARCHIVING(tensor)
 104 DEFAULT_ARCHIVING(tensdelta)
 105 DEFAULT_ARCHIVING(tensmetric)
 106 DEFAULT_ARCHIVING(spinmetric)
 107 DEFAULT_UNARCHIVE(minkmetric)
 108 DEFAULT_UNARCHIVE(tensepsilon)
 109
 110 minkmetric::minkmetric(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
 111 {
 112         n.find_bool("pos_sig", pos_sig);
 113 }
 114
 115 void minkmetric::archive(archive_node &n) const
 116 {
 117         inherited::archive(n);
 118         n.add_bool("pos_sig", pos_sig);
 119 }
 120
 121 tensepsilon::tensepsilon(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
 122 {
 123         n.find_bool("minkowski", minkowski);
 124         n.find_bool("pos_sig", pos_sig);
 125 }
 126
 127 void tensepsilon::archive(archive_node &n) const
 128 {
 129         inherited::archive(n);
 130         n.add_bool("minkowski", minkowski);
 131         n.add_bool("pos_sig", pos_sig);
 132 }
 133
 134 //////////
 135 // functions overriding virtual functions from base classes
 136 //////////
 137
 138 DEFAULT_COMPARE(tensor)
 139 DEFAULT_COMPARE(tensdelta)
 140 DEFAULT_COMPARE(tensmetric)
 141 DEFAULT_COMPARE(spinmetric)
 142
 143 int minkmetric::compare_same_type(const basic & other) const
 144 {
 145         GINAC_ASSERT(is_a<minkmetric>(other));
 146         const minkmetric &o = static_cast<const minkmetric &>(other);
 147
 148         if (pos_sig != o.pos_sig)
 149                 return pos_sig ? -1 : 1;
 150         else
 151                 return inherited::compare_same_type(other);
 152 }
 153
 154 int tensepsilon::compare_same_type(const basic & other) const
 155 {
 156         GINAC_ASSERT(is_a<tensepsilon>(other));
 157         const tensepsilon &o = static_cast<const tensepsilon &>(other);
 158
 159         if (minkowski != o.minkowski)
 160                 return minkowski ? -1 : 1;
 161         else if (pos_sig != o.pos_sig)
 162                 return pos_sig ? -1 : 1;
 163         else
 164                 return inherited::compare_same_type(other);
 165 }
 166
 167 DEFAULT_PRINT_LATEX(tensdelta, "delta", "\\delta")
 168 DEFAULT_PRINT(tensmetric, "g")
 169 DEFAULT_PRINT_LATEX(minkmetric, "eta", "\\eta")
 170 DEFAULT_PRINT_LATEX(spinmetric, "eps", "\\varepsilon")
 171 DEFAULT_PRINT_LATEX(tensepsilon, "eps", "\\varepsilon")
 172
 173 /** Automatic symbolic evaluation of an indexed delta tensor. */
 174 ex tensdelta::eval_indexed(const basic & i) const
 175 {
 176         GINAC_ASSERT(is_a<indexed>(i));
 177         GINAC_ASSERT(i.nops() == 3);
 178         GINAC_ASSERT(is_a<tensdelta>(i.op(0)));
 179
 180         const idx & i1 = ex_to<idx>(i.op(1));
 181         const idx & i2 = ex_to<idx>(i.op(2));
 182
 183         // The dimension of the indices must be equal, otherwise we use the minimal
 184         // dimension
 185         if (!i1.get_dim().is_equal(i2.get_dim())) {
 186                 ex min_dim = i1.minimal_dim(i2);
 187                 return i.subs(lst(i1 == i1.replace_dim(min_dim), i2 == i2.replace_dim(min_dim)));
 188         }
 189
 190         // Trace of delta tensor is the (effective) dimension of the space
 191         if (is_dummy_pair(i1, i2)) {
 192                 try {
 193                         return i1.minimal_dim(i2);
 194                 } catch (std::exception &e) {
 195                         return i.hold();
 196                 }
 197         }
 198
 199         // Numeric evaluation
 200         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::integer)) {
 201                 int n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
 202                 if (n1 == n2)
 203                         return _ex1;
 204                 else
 205                         return _ex0;
 206         }
 207
 208         // No further simplifications
 209         return i.hold();
 210 }
 211
 212 /** Automatic symbolic evaluation of an indexed metric tensor. */
 213 ex tensmetric::eval_indexed(const basic & i) const
 214 {
 215         GINAC_ASSERT(is_a<indexed>(i));
 216         GINAC_ASSERT(i.nops() == 3);
 217         GINAC_ASSERT(is_a<tensmetric>(i.op(0)));
 218         GINAC_ASSERT(is_a<varidx>(i.op(1)));
 219         GINAC_ASSERT(is_a<varidx>(i.op(2)));
 220
 221         const varidx & i1 = ex_to<varidx>(i.op(1));
 222         const varidx & i2 = ex_to<varidx>(i.op(2));
 223
 224         // The dimension of the indices must be equal, otherwise we use the minimal
 225         // dimension
 226         if (!i1.get_dim().is_equal(i2.get_dim())) {
 227                 ex min_dim = i1.minimal_dim(i2);
 228                 return i.subs(lst(i1 == i1.replace_dim(min_dim), i2 == i2.replace_dim(min_dim)));
 229         }
 230
 231         // A metric tensor with one covariant and one contravariant index gets
 232         // replaced by a delta tensor
 233         if (i1.is_covariant() != i2.is_covariant())
 234                 return delta_tensor(i1, i2);
 235
 236         // No further simplifications
 237         return i.hold();
 238 }
 239
 240 /** Automatic symbolic evaluation of an indexed Lorentz metric tensor. */
 241 ex minkmetric::eval_indexed(const basic & i) const
 242 {
 243         GINAC_ASSERT(is_a<indexed>(i));
 244         GINAC_ASSERT(i.nops() == 3);
 245         GINAC_ASSERT(is_a<minkmetric>(i.op(0)));
 246         GINAC_ASSERT(is_a<varidx>(i.op(1)));
 247         GINAC_ASSERT(is_a<varidx>(i.op(2)));
 248
 249         const varidx & i1 = ex_to<varidx>(i.op(1));
 250         const varidx & i2 = ex_to<varidx>(i.op(2));
 251
 252         // Numeric evaluation
 253         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 254                 int n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
 255                 if (n1 != n2)
 256                         return _ex0;
 257                 else if (n1 == 0)
 258                         return pos_sig ? _ex_1 : _ex1;
 259                 else
 260                         return pos_sig ? _ex1 : _ex_1;
 261         }
 262
 263         // Perform the usual evaluations of a metric tensor
 264         return inherited::eval_indexed(i);
 265 }
 266
 267 /** Automatic symbolic evaluation of an indexed metric tensor. */
 268 ex spinmetric::eval_indexed(const basic & i) const
 269 {
 270         GINAC_ASSERT(is_a<indexed>(i));
 271         GINAC_ASSERT(i.nops() == 3);
 272         GINAC_ASSERT(is_a<spinmetric>(i.op(0)));
 273         GINAC_ASSERT(is_a<spinidx>(i.op(1)));
 274         GINAC_ASSERT(is_a<spinidx>(i.op(2)));
 275
 276         const spinidx & i1 = ex_to<spinidx>(i.op(1));
 277         const spinidx & i2 = ex_to<spinidx>(i.op(2));
 278
 279         // Convolutions are zero
 280         if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
 281                 return _ex0;
 282
 283         // Numeric evaluation
 284         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 285                 int n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
 286                 if (n1 == n2)
 287                         return _ex0;
 288                 else if (n1 < n2)
 289                         return _ex1;
 290                 else
 291                         return _ex_1;
 292         }
 293
 294         // No further simplifications
 295         return i.hold();
 296 }
 297
 298 /** Automatic symbolic evaluation of an indexed epsilon tensor. */
 299 ex tensepsilon::eval_indexed(const basic & i) const
 300 {
 301         GINAC_ASSERT(is_a<indexed>(i));
 302         GINAC_ASSERT(i.nops() > 1);
 303         GINAC_ASSERT(is_a<tensepsilon>(i.op(0)));
 304
 305         // Convolutions are zero
 306         if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
 307                 return _ex0;
 308
 309         // Numeric evaluation
 310         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 311
 312                 // Get sign of index permutation (the indices should already be in
 313                 // a canonic order but we can't assume what exactly that order is)
 314                 std::vector<int> v;
 315                 v.reserve(i.nops() - 1);
 316                 for (size_t j=1; j<i.nops(); j++)
 317                         v.push_back(ex_to<numeric>(ex_to<idx>(i.op(j)).get_value()).to_int());
 318                 int sign = permutation_sign(v.begin(), v.end());
 319
 320                 // In a Minkowski space, check for covariant indices
 321                 if (minkowski) {
 322                         for (size_t j=1; j<i.nops(); j++) {
 323                                 const ex & x = i.op(j);
 324                                 if (!is_a<varidx>(x))
 325                                         throw(std::runtime_error("indices of epsilon tensor in Minkowski space must be of type varidx"));
 326                                 if (ex_to<varidx>(x).is_covariant())
 327                                         if (ex_to<idx>(x).get_value().is_zero())
 328                                                 sign = (pos_sig ? -sign : sign);
 329                                         else
 330                                                 sign = (pos_sig ? sign : -sign);
 331                         }
 332                 }
 333
 334                 return sign;
 335         }
 336
 337         // No further simplifications
 338         return i.hold();
 339 }
 340
 341 bool tensor::replace_contr_index(exvector::iterator self, exvector::iterator other) const
 342 {
 343         // Try to contract the first index
 344         const idx *self_idx = &ex_to<idx>(self->op(1));
 345         const idx *free_idx = &ex_to<idx>(self->op(2));
 346         bool first_index_tried = false;
 347
 348 again:
 349         if (self_idx->is_symbolic()) {
 350                 for (size_t i=1; i<other->nops(); i++) {
 351                         const idx &other_idx = ex_to<idx>(other->op(i));
 352                         if (is_dummy_pair(*self_idx, other_idx)) {
 353
 354                                 // Contraction found, remove this tensor and substitute the
 355                                 // index in the second object
 356                                 try {
 357                                         // minimal_dim() throws an exception when index dimensions are not comparable
 358                                         ex min_dim = self_idx->minimal_dim(other_idx);
 359                                         *other = other->subs(other_idx == free_idx->replace_dim(min_dim));
 360                                         *self = _ex1; // *other is assigned first because assigning *self invalidates free_idx
 361                                         return true;
 362                                 } catch (std::exception &e) {
 363                                         return false;
 364                                 }
 365                         }
 366                 }
 367         }
 368
 369         if (!first_index_tried) {
 370
 371                 // No contraction with the first index found, try the second index
 372                 self_idx = &ex_to<idx>(self->op(2));
 373                 free_idx = &ex_to<idx>(self->op(1));
 374                 first_index_tried = true;
 375                 goto again;
 376         }
 377
 378         return false;
 379 }
 380
 381 /** Contraction of an indexed delta tensor with something else. */
 382 bool tensdelta::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 383 {
 384         GINAC_ASSERT(is_a<indexed>(*self));
 385         GINAC_ASSERT(is_a<indexed>(*other));
 386         GINAC_ASSERT(self->nops() == 3);
 387         GINAC_ASSERT(is_a<tensdelta>(self->op(0)));
 388
 389         // Replace the dummy index with this tensor's other index and remove
 390         // the tensor (this is valid for contractions with all other tensors)
 391         return replace_contr_index(self, other);
 392 }
 393
 394 /** Contraction of an indexed metric tensor with something else. */
 395 bool tensmetric::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 396 {
 397         GINAC_ASSERT(is_a<indexed>(*self));
 398         GINAC_ASSERT(is_a<indexed>(*other));
 399         GINAC_ASSERT(self->nops() == 3);
 400         GINAC_ASSERT(is_a<tensmetric>(self->op(0)));
 401
 402         // If contracting with the delta tensor, let the delta do it
 403         // (don't raise/lower delta indices)
 404         if (is_a<tensdelta>(other->op(0)))
 405                 return false;
 406
 407         // Replace the dummy index with this tensor's other index and remove
 408         // the tensor
 409         return replace_contr_index(self, other);
 410 }
 411
 412 /** Contraction of an indexed spinor metric with something else. */
 413 bool spinmetric::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 414 {
 415         GINAC_ASSERT(is_a<indexed>(*self));
 416         GINAC_ASSERT(is_a<indexed>(*other));
 417         GINAC_ASSERT(self->nops() == 3);
 418         GINAC_ASSERT(is_a<spinmetric>(self->op(0)));
 419
 420         // Contractions between spinor metrics
 421         if (is_a<spinmetric>(other->op(0))) {
 422                 const idx &self_i1 = ex_to<idx>(self->op(1));
 423                 const idx &self_i2 = ex_to<idx>(self->op(2));
 424                 const idx &other_i1 = ex_to<idx>(other->op(1));
 425                 const idx &other_i2 = ex_to<idx>(other->op(2));
 426
 427                 if (is_dummy_pair(self_i1, other_i1)) {
 428                         if (is_dummy_pair(self_i2, other_i2))
 429                                 *self = _ex2;
 430                         else
 431                                 *self = delta_tensor(self_i2, other_i2);
 432                         *other = _ex1;
 433                         return true;
 434                 } else if (is_dummy_pair(self_i1, other_i2)) {
 435                         if (is_dummy_pair(self_i2, other_i1))
 436                                 *self = _ex_2;
 437                         else
 438                                 *self = -delta_tensor(self_i2, other_i1);
 439                         *other = _ex1;
 440                         return true;
 441                 } else if (is_dummy_pair(self_i2, other_i1)) {
 442                         *self = -delta_tensor(self_i1, other_i2);
 443                         *other = _ex1;
 444                         return true;
 445                 } else if (is_dummy_pair(self_i2, other_i2)) {
 446                         *self = delta_tensor(self_i1, other_i1);
 447                         *other = _ex1;
 448                         return true;
 449                 }
 450         }
 451
 452         // If contracting with the delta tensor, let the delta do it
 453         // (don't raise/lower delta indices)
 454         if (is_a<tensdelta>(other->op(0)))
 455                 return false;
 456
 457         // Try to contract first index
 458         const idx *self_idx = &ex_to<idx>(self->op(1));
 459         const idx *free_idx = &ex_to<idx>(self->op(2));
 460         bool first_index_tried = false;
 461         int sign = 1;
 462
 463 again:
 464         if (self_idx->is_symbolic()) {
 465                 for (size_t i=1; i<other->nops(); i++) {
 466                         const idx &other_idx = ex_to<idx>(other->op(i));
 467                         if (is_dummy_pair(*self_idx, other_idx)) {
 468
 469                                 // Contraction found, remove metric tensor and substitute
 470                                 // index in second object (assign *self last because this
 471                                 // invalidates free_idx)
 472                                 *other = other->subs(other_idx == *free_idx);
 473                                 *self = (static_cast<const spinidx *>(self_idx)->is_covariant() ? sign : -sign);
 474                                 return true;
 475                         }
 476                 }
 477         }
 478
 479         if (!first_index_tried) {
 480
 481                 // No contraction with first index found, try second index
 482                 self_idx = &ex_to<idx>(self->op(2));
 483                 free_idx = &ex_to<idx>(self->op(1));
 484                 first_index_tried = true;
 485                 sign = -sign;
 486                 goto again;
 487         }
 488
 489         return false;
 490 }
 491
 492 /** Contraction of epsilon tensor with something else. */
 493 bool tensepsilon::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 494 {
 495         GINAC_ASSERT(is_a<indexed>(*self));
 496         GINAC_ASSERT(is_a<indexed>(*other));
 497         GINAC_ASSERT(is_a<tensepsilon>(self->op(0)));
 498         size_t num = self->nops() - 1;
 499
 500         if (is_exactly_a<tensepsilon>(other->op(0)) && num+1 == other->nops()) {
 501
 502                 // Contraction of two epsilon tensors is a determinant
 503                 bool variance = is_a<varidx>(self->op(1));
 504                 matrix M(num, num);
 505                 for (size_t i=0; i<num; i++) {
 506                         for (size_t j=0; j<num; j++) {
 507                                 if (minkowski)
 508                                         M(i, j) = lorentz_g(self->op(i+1), other->op(j+1), pos_sig);
 509                                 else if (variance)
 510                                         M(i, j) = metric_tensor(self->op(i+1), other->op(j+1));
 511                                 else
 512                                         M(i, j) = delta_tensor(self->op(i+1), other->op(j+1));
 513                         }
 514                 }
 515                 int sign = minkowski ? -1 : 1;
 516                 *self = sign * M.determinant().simplify_indexed();
 517                 *other = _ex1;
 518                 return true;
 519         }
 520
 521         return false;
 522 }
 523
 524 //////////
 525 // global functions
 526 //////////
 527
 528 ex delta_tensor(const ex & i1, const ex & i2)
 529 {
 530         if (!is_a<idx>(i1) || !is_a<idx>(i2))
 531                 throw(std::invalid_argument("indices of delta tensor must be of type idx"));
 532
 533         return indexed(tensdelta(), sy_symm(), i1, i2);
 534 }
 535
 536 ex metric_tensor(const ex & i1, const ex & i2)
 537 {
 538         if (!is_a<varidx>(i1) || !is_a<varidx>(i2))
 539                 throw(std::invalid_argument("indices of metric tensor must be of type varidx"));
 540
 541         return indexed(tensmetric(), sy_symm(), i1, i2);
 542 }
 543
 544 ex lorentz_g(const ex & i1, const ex & i2, bool pos_sig)
 545 {
 546         if (!is_a<varidx>(i1) || !is_a<varidx>(i2))
 547                 throw(std::invalid_argument("indices of metric tensor must be of type varidx"));
 548
 549         return indexed(minkmetric(pos_sig), sy_symm(), i1, i2);
 550 }
 551
 552 ex spinor_metric(const ex & i1, const ex & i2)
 553 {
 554         if (!is_a<spinidx>(i1) || !is_a<spinidx>(i2))
 555                 throw(std::invalid_argument("indices of spinor metric must be of type spinidx"));
 556         if (!ex_to<idx>(i1).get_dim().is_equal(2) || !ex_to<idx>(i2).get_dim().is_equal(2))
 557                 throw(std::runtime_error("index dimension for spinor metric must be 2"));
 558
 559         return indexed(spinmetric(), sy_anti(), i1, i2);
 560 }
 561
 562 ex epsilon_tensor(const ex & i1, const ex & i2)
 563 {
 564         if (!is_a<idx>(i1) || !is_a<idx>(i2))
 565                 throw(std::invalid_argument("indices of epsilon tensor must be of type idx"));
 566
 567         ex dim = ex_to<idx>(i1).get_dim();
 568         if (!dim.is_equal(ex_to<idx>(i2).get_dim()))
 569                 throw(std::invalid_argument("all indices of epsilon tensor must have the same dimension"));
 570         if (!ex_to<idx>(i1).get_dim().is_equal(_ex2))
 571                 throw(std::runtime_error("index dimension of epsilon tensor must match number of indices"));
 572
 573         return indexed(tensepsilon(), sy_anti(), i1, i2);
 574 }
 575
 576 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3)
 577 {
 578         if (!is_a<idx>(i1) || !is_a<idx>(i2) || !is_a<idx>(i3))
 579                 throw(std::invalid_argument("indices of epsilon tensor must be of type idx"));
 580
 581         ex dim = ex_to<idx>(i1).get_dim();
 582         if (!dim.is_equal(ex_to<idx>(i2).get_dim()) || !dim.is_equal(ex_to<idx>(i3).get_dim()))
 583                 throw(std::invalid_argument("all indices of epsilon tensor must have the same dimension"));
 584         if (!ex_to<idx>(i1).get_dim().is_equal(_ex3))
 585                 throw(std::runtime_error("index dimension of epsilon tensor must match number of indices"));
 586
 587         return indexed(tensepsilon(), sy_anti(), i1, i2, i3);
 588 }
 589
 590 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig)
 591 {
 592         if (!is_a<varidx>(i1) || !is_a<varidx>(i2) || !is_a<varidx>(i3) || !is_a<varidx>(i4))
 593                 throw(std::invalid_argument("indices of Lorentz epsilon tensor must be of type varidx"));
 594
 595         ex dim = ex_to<idx>(i1).get_dim();
 596         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()))
 597                 throw(std::invalid_argument("all indices of epsilon tensor must have the same dimension"));
 598         if (!ex_to<idx>(i1).get_dim().is_equal(_ex4))
 599                 throw(std::runtime_error("index dimension of epsilon tensor must match number of indices"));
 600
 601         return indexed(tensepsilon(true, pos_sig), sy_anti(), i1, i2, i3, i4);
 602 }
 603
 604 } // namespace GiNaC