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