ginac/tensor.cpp

   1 /** @file tensor.cpp
   2  *
   3  *  Implementation of GiNaC's special tensors. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2011 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., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  21  */
  22
  23 #include "tensor.h"
  24 #include "idx.h"
  25 #include "indexed.h"
  26 #include "symmetry.h"
  27 #include "relational.h"
  28 #include "operators.h"
  29 #include "lst.h"
  30 #include "numeric.h"
  31 #include "matrix.h"
  32 #include "archive.h"
  33 #include "utils.h"
  34
  35 #include <iostream>
  36 #include <stdexcept>
  37 #include <vector>
  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()
  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 }
  78
  79 spinmetric::spinmetric()
  80 {
  81 }
  82
  83 minkmetric::minkmetric(bool ps) : pos_sig(ps)
  84 {
  85 }
  86
  87 tensepsilon::tensepsilon() : minkowski(false), pos_sig(false)
  88 {
  89 }
  90
  91 tensepsilon::tensepsilon(bool mink, bool ps) : minkowski(mink), pos_sig(ps)
  92 {
  93 }
  94
  95 //////////
  96 // archiving
  97 //////////
  98
  99 void minkmetric::read_archive(const archive_node& n, lst& sym_lst)
 100 {
 101         inherited::read_archive(n, sym_lst);
 102         n.find_bool("pos_sig", pos_sig);
 103 }
 104 GINAC_BIND_UNARCHIVER(minkmetric);
 105
 106 void minkmetric::archive(archive_node &n) const
 107 {
 108         inherited::archive(n);
 109         n.add_bool("pos_sig", pos_sig);
 110 }
 111
 112 void tensepsilon::read_archive(const archive_node& n, lst& sym_lst)
 113 {
 114         inherited::read_archive(n, sym_lst);
 115         n.find_bool("minkowski", minkowski);
 116         n.find_bool("pos_sig", pos_sig);
 117 }
 118 GINAC_BIND_UNARCHIVER(tensepsilon);
 119
 120 void tensepsilon::archive(archive_node &n) const
 121 {
 122         inherited::archive(n);
 123         n.add_bool("minkowski", minkowski);
 124         n.add_bool("pos_sig", pos_sig);
 125 }
 126
 127 GINAC_BIND_UNARCHIVER(tensdelta);
 128 GINAC_BIND_UNARCHIVER(tensmetric);
 129 GINAC_BIND_UNARCHIVER(spinmetric);
 130
 131 //////////
 132 // functions overriding virtual functions from base classes
 133 //////////
 134
 135 DEFAULT_COMPARE(tensor)
 136 DEFAULT_COMPARE(tensdelta)
 137 DEFAULT_COMPARE(tensmetric)
 138 DEFAULT_COMPARE(spinmetric)
 139
 140 bool tensdelta::info(unsigned inf) const
 141 {
 142         if(inf == info_flags::real)
 143                 return true;
 144
 145         return false;
 146 }
 147
 148 bool tensmetric::info(unsigned inf) const
 149 {
 150         if(inf == info_flags::real)
 151                 return true;
 152
 153         return false;
 154 }
 155
 156 int minkmetric::compare_same_type(const basic & other) const
 157 {
 158         GINAC_ASSERT(is_a<minkmetric>(other));
 159         const minkmetric &o = static_cast<const minkmetric &>(other);
 160
 161         if (pos_sig != o.pos_sig)
 162                 return pos_sig ? -1 : 1;
 163         else
 164                 return inherited::compare_same_type(other);
 165 }
 166
 167 bool minkmetric::info(unsigned inf) const
 168 {
 169         if(inf == info_flags::real)
 170                 return true;
 171
 172         return false;
 173 }
 174
 175 int tensepsilon::compare_same_type(const basic & other) const
 176 {
 177         GINAC_ASSERT(is_a<tensepsilon>(other));
 178         const tensepsilon &o = static_cast<const tensepsilon &>(other);
 179
 180         if (minkowski != o.minkowski)
 181                 return minkowski ? -1 : 1;
 182         else if (pos_sig != o.pos_sig)
 183                 return pos_sig ? -1 : 1;
 184         else
 185                 return inherited::compare_same_type(other);
 186 }
 187
 188 bool tensepsilon::info(unsigned inf) const
 189 {
 190         if(inf == info_flags::real)
 191                 return true;
 192
 193         return false;
 194 }
 195
 196 bool spinmetric::info(unsigned inf) const
 197 {
 198         if(inf == info_flags::real)
 199                 return true;
 200
 201         return false;
 202 }
 203
 204 DEFAULT_PRINT_LATEX(tensdelta, "delta", "\\delta")
 205 DEFAULT_PRINT(tensmetric, "g")
 206 DEFAULT_PRINT_LATEX(minkmetric, "eta", "\\eta")
 207 DEFAULT_PRINT_LATEX(spinmetric, "eps", "\\varepsilon")
 208 DEFAULT_PRINT_LATEX(tensepsilon, "eps", "\\varepsilon")
 209
 210 /** Automatic symbolic evaluation of an indexed delta tensor. */
 211 ex tensdelta::eval_indexed(const basic & i) const
 212 {
 213         GINAC_ASSERT(is_a<indexed>(i));
 214         GINAC_ASSERT(i.nops() == 3);
 215         GINAC_ASSERT(is_a<tensdelta>(i.op(0)));
 216
 217         const idx & i1 = ex_to<idx>(i.op(1));
 218         const idx & i2 = ex_to<idx>(i.op(2));
 219
 220         // The dimension of the indices must be equal, otherwise we use the minimal
 221         // dimension
 222         if (!i1.get_dim().is_equal(i2.get_dim())) {
 223                 ex min_dim = i1.minimal_dim(i2);
 224                 exmap m;
 225                 m[i1] = i1.replace_dim(min_dim);
 226                 m[i2] = i2.replace_dim(min_dim);
 227                 return i.subs(m, subs_options::no_pattern);
 228         }
 229
 230         // Trace of delta tensor is the (effective) dimension of the space
 231         if (is_dummy_pair(i1, i2)) {
 232                 try {
 233                         return i1.minimal_dim(i2);
 234                 } catch (std::exception &e) {
 235                         return i.hold();
 236                 }
 237         }
 238
 239         // Numeric evaluation
 240         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::integer)) {
 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 _ex1;
 244                 else
 245                         return _ex0;
 246         }
 247
 248         // No further simplifications
 249         return i.hold();
 250 }
 251
 252 /** Automatic symbolic evaluation of an indexed metric tensor. */
 253 ex tensmetric::eval_indexed(const basic & i) const
 254 {
 255         GINAC_ASSERT(is_a<indexed>(i));
 256         GINAC_ASSERT(i.nops() == 3);
 257         GINAC_ASSERT(is_a<tensmetric>(i.op(0)));
 258         GINAC_ASSERT(is_a<varidx>(i.op(1)));
 259         GINAC_ASSERT(is_a<varidx>(i.op(2)));
 260
 261         const varidx & i1 = ex_to<varidx>(i.op(1));
 262         const varidx & i2 = ex_to<varidx>(i.op(2));
 263
 264         // The dimension of the indices must be equal, otherwise we use the minimal
 265         // dimension
 266         if (!i1.get_dim().is_equal(i2.get_dim())) {
 267                 ex min_dim = i1.minimal_dim(i2);
 268                 exmap m;
 269                 m[i1] = i1.replace_dim(min_dim);
 270                 m[i2] = i2.replace_dim(min_dim);
 271                 return i.subs(m, subs_options::no_pattern);
 272         }
 273
 274         // A metric tensor with one covariant and one contravariant index gets
 275         // replaced by a delta tensor
 276         if (i1.is_covariant() != i2.is_covariant())
 277                 return delta_tensor(i1, i2);
 278
 279         // No further simplifications
 280         return i.hold();
 281 }
 282
 283 /** Automatic symbolic evaluation of an indexed Lorentz metric tensor. */
 284 ex minkmetric::eval_indexed(const basic & i) const
 285 {
 286         GINAC_ASSERT(is_a<indexed>(i));
 287         GINAC_ASSERT(i.nops() == 3);
 288         GINAC_ASSERT(is_a<minkmetric>(i.op(0)));
 289         GINAC_ASSERT(is_a<varidx>(i.op(1)));
 290         GINAC_ASSERT(is_a<varidx>(i.op(2)));
 291
 292         const varidx & i1 = ex_to<varidx>(i.op(1));
 293         const varidx & i2 = ex_to<varidx>(i.op(2));
 294
 295         // Numeric evaluation
 296         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 297                 int n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
 298                 if (n1 != n2)
 299                         return _ex0;
 300                 else if (n1 == 0)
 301                         return pos_sig ? _ex_1 : _ex1;
 302                 else
 303                         return pos_sig ? _ex1 : _ex_1;
 304         }
 305
 306         // Perform the usual evaluations of a metric tensor
 307         return inherited::eval_indexed(i);
 308 }
 309
 310 /** Automatic symbolic evaluation of an indexed metric tensor. */
 311 ex spinmetric::eval_indexed(const basic & i) const
 312 {
 313         GINAC_ASSERT(is_a<indexed>(i));
 314         GINAC_ASSERT(i.nops() == 3);
 315         GINAC_ASSERT(is_a<spinmetric>(i.op(0)));
 316         GINAC_ASSERT(is_a<spinidx>(i.op(1)));
 317         GINAC_ASSERT(is_a<spinidx>(i.op(2)));
 318
 319         const spinidx & i1 = ex_to<spinidx>(i.op(1));
 320         const spinidx & i2 = ex_to<spinidx>(i.op(2));
 321
 322         // Convolutions are zero
 323         if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
 324                 return _ex0;
 325
 326         // Numeric evaluation
 327         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 328                 int n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
 329                 if (n1 == n2)
 330                         return _ex0;
 331                 else if (n1 < n2)
 332                         return _ex1;
 333                 else
 334                         return _ex_1;
 335         }
 336
 337         // No further simplifications
 338         return i.hold();
 339 }
 340
 341 /** Automatic symbolic evaluation of an indexed epsilon tensor. */
 342 ex tensepsilon::eval_indexed(const basic & i) const
 343 {
 344         GINAC_ASSERT(is_a<indexed>(i));
 345         GINAC_ASSERT(i.nops() > 1);
 346         GINAC_ASSERT(is_a<tensepsilon>(i.op(0)));
 347
 348         // Convolutions are zero
 349         if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
 350                 return _ex0;
 351
 352         // Numeric evaluation
 353         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 354
 355                 // Get sign of index permutation (the indices should already be in
 356                 // a canonic order but we can't assume what exactly that order is)
 357                 std::vector<int> v;
 358                 v.reserve(i.nops() - 1);
 359                 for (size_t j=1; j<i.nops(); j++)
 360                         v.push_back(ex_to<numeric>(ex_to<idx>(i.op(j)).get_value()).to_int());
 361                 int sign = permutation_sign(v.begin(), v.end());
 362
 363                 // In a Minkowski space, check for covariant indices
 364                 if (minkowski) {
 365                         for (size_t j=1; j<i.nops(); j++) {
 366                                 const ex & x = i.op(j);
 367                                 if (!is_a<varidx>(x)) {
 368                                         throw(std::runtime_error("indices of epsilon tensor in Minkowski space must be of type varidx"));
 369                                 }
 370                                 if (ex_to<varidx>(x).is_covariant()) {
 371                                         if (ex_to<idx>(x).get_value().is_zero()) {
 372                                                 sign = (pos_sig ? -sign : sign);
 373                                         }
 374                                         else {
 375                                                 sign = (pos_sig ? sign : -sign);
 376                                         }
 377                                 }
 378                         }
 379                 }
 380
 381                 return sign;
 382         }
 383
 384         // No further simplifications
 385         return i.hold();
 386 }
 387
 388 bool tensor::replace_contr_index(exvector::iterator self, exvector::iterator other) const
 389 {
 390         // Try to contract the first index
 391         const idx *self_idx = &ex_to<idx>(self->op(1));
 392         const idx *free_idx = &ex_to<idx>(self->op(2));
 393         bool first_index_tried = false;
 394
 395 again:
 396         if (self_idx->is_symbolic()) {
 397                 for (size_t i=1; i<other->nops(); i++) {
 398                         if (! is_a<idx>(other->op(i)))
 399                                 continue;
 400                         const idx &other_idx = ex_to<idx>(other->op(i));
 401                         if (is_dummy_pair(*self_idx, other_idx)) {
 402
 403                                 // Contraction found, remove this tensor and substitute the
 404                                 // index in the second object
 405                                 try {
 406                                         // minimal_dim() throws an exception when index dimensions are not comparable
 407                                         ex min_dim = self_idx->minimal_dim(other_idx);
 408                                         *other = other->subs(other_idx == free_idx->replace_dim(min_dim));
 409                                         *self = _ex1; // *other is assigned first because assigning *self invalidates free_idx
 410                                         return true;
 411                                 } catch (std::exception &e) {
 412                                         return false;
 413                                 }
 414                         }
 415                 }
 416         }
 417
 418         if (!first_index_tried) {
 419
 420                 // No contraction with the first index found, try the second index
 421                 self_idx = &ex_to<idx>(self->op(2));
 422                 free_idx = &ex_to<idx>(self->op(1));
 423                 first_index_tried = true;
 424                 goto again;
 425         }
 426
 427         return false;
 428 }
 429
 430 /** Contraction of an indexed delta tensor with something else. */
 431 bool tensdelta::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 432 {
 433         GINAC_ASSERT(is_a<indexed>(*self));
 434         GINAC_ASSERT(is_a<indexed>(*other));
 435         GINAC_ASSERT(self->nops() == 3);
 436         GINAC_ASSERT(is_a<tensdelta>(self->op(0)));
 437
 438         // Replace the dummy index with this tensor's other index and remove
 439         // the tensor (this is valid for contractions with all other tensors)
 440         return replace_contr_index(self, other);
 441 }
 442
 443 /** Contraction of an indexed metric tensor with something else. */
 444 bool tensmetric::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 445 {
 446         GINAC_ASSERT(is_a<indexed>(*self));
 447         GINAC_ASSERT(is_a<indexed>(*other));
 448         GINAC_ASSERT(self->nops() == 3);
 449         GINAC_ASSERT(is_a<tensmetric>(self->op(0)));
 450
 451         // If contracting with the delta tensor, let the delta do it
 452         // (don't raise/lower delta indices)
 453         if (is_a<tensdelta>(other->op(0)))
 454                 return false;
 455
 456         // Replace the dummy index with this tensor's other index and remove
 457         // the tensor
 458         return replace_contr_index(self, other);
 459 }
 460
 461 /** Contraction of an indexed spinor metric with something else. */
 462 bool spinmetric::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 463 {
 464         GINAC_ASSERT(is_a<indexed>(*self));
 465         GINAC_ASSERT(is_a<indexed>(*other));
 466         GINAC_ASSERT(self->nops() == 3);
 467         GINAC_ASSERT(is_a<spinmetric>(self->op(0)));
 468
 469         // Contractions between spinor metrics
 470         if (is_a<spinmetric>(other->op(0))) {
 471                 const idx &self_i1 = ex_to<idx>(self->op(1));
 472                 const idx &self_i2 = ex_to<idx>(self->op(2));
 473                 const idx &other_i1 = ex_to<idx>(other->op(1));
 474                 const idx &other_i2 = ex_to<idx>(other->op(2));
 475
 476                 if (is_dummy_pair(self_i1, other_i1)) {
 477                         if (is_dummy_pair(self_i2, other_i2))
 478                                 *self = _ex2;
 479                         else
 480                                 *self = delta_tensor(self_i2, other_i2);
 481                         *other = _ex1;
 482                         return true;
 483                 } else if (is_dummy_pair(self_i1, other_i2)) {
 484                         if (is_dummy_pair(self_i2, other_i1))
 485                                 *self = _ex_2;
 486                         else
 487                                 *self = -delta_tensor(self_i2, other_i1);
 488                         *other = _ex1;
 489                         return true;
 490                 } else if (is_dummy_pair(self_i2, other_i1)) {
 491                         *self = -delta_tensor(self_i1, other_i2);
 492                         *other = _ex1;
 493                         return true;
 494                 } else if (is_dummy_pair(self_i2, other_i2)) {
 495                         *self = delta_tensor(self_i1, other_i1);
 496                         *other = _ex1;
 497                         return true;
 498                 }
 499         }
 500
 501         // If contracting with the delta tensor, let the delta do it
 502         // (don't raise/lower delta indices)
 503         if (is_a<tensdelta>(other->op(0)))
 504                 return false;
 505
 506         // Try to contract first index
 507         const idx *self_idx = &ex_to<idx>(self->op(1));
 508         const idx *free_idx = &ex_to<idx>(self->op(2));
 509         bool first_index_tried = false;
 510         int sign = 1;
 511
 512 again:
 513         if (self_idx->is_symbolic()) {
 514                 for (size_t i=1; i<other->nops(); i++) {
 515                         const idx &other_idx = ex_to<idx>(other->op(i));
 516                         if (is_dummy_pair(*self_idx, other_idx)) {
 517
 518                                 // Contraction found, remove metric tensor and substitute
 519                                 // index in second object (assign *self last because this
 520                                 // invalidates free_idx)
 521                                 *other = other->subs(other_idx == *free_idx);
 522                                 *self = (static_cast<const spinidx *>(self_idx)->is_covariant() ? sign : -sign);
 523                                 return true;
 524                         }
 525                 }
 526         }
 527
 528         if (!first_index_tried) {
 529
 530                 // No contraction with first index found, try second index
 531                 self_idx = &ex_to<idx>(self->op(2));
 532                 free_idx = &ex_to<idx>(self->op(1));
 533                 first_index_tried = true;
 534                 sign = -sign;
 535                 goto again;
 536         }
 537
 538         return false;
 539 }
 540
 541 /** Contraction of epsilon tensor with something else. */
 542 bool tensepsilon::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 543 {
 544         GINAC_ASSERT(is_a<indexed>(*self));
 545         GINAC_ASSERT(is_a<indexed>(*other));
 546         GINAC_ASSERT(is_a<tensepsilon>(self->op(0)));
 547         size_t num = self->nops() - 1;
 548
 549         if (is_exactly_a<tensepsilon>(other->op(0)) && num+1 == other->nops()) {
 550
 551                 // Contraction of two epsilon tensors is a determinant
 552                 bool variance = is_a<varidx>(self->op(1));
 553                 matrix M(num, num);
 554                 for (size_t i=0; i<num; i++) {
 555                         for (size_t j=0; j<num; j++) {
 556                                 if (minkowski)
 557                                         M(i, j) = lorentz_g(self->op(i+1), other->op(j+1), pos_sig);
 558                                 else if (variance)
 559                                         M(i, j) = metric_tensor(self->op(i+1), other->op(j+1));
 560                                 else
 561                                         M(i, j) = delta_tensor(self->op(i+1), other->op(j+1));
 562                         }
 563                 }
 564                 int sign = minkowski ? -1 : 1;
 565                 *self = sign * M.determinant().simplify_indexed();
 566                 *other = _ex1;
 567                 return true;
 568         }
 569
 570         return false;
 571 }
 572
 573 //////////
 574 // global functions
 575 //////////
 576
 577 ex delta_tensor(const ex & i1, const ex & i2)
 578 {
 579         static ex delta = (new tensdelta)->setflag(status_flags::dynallocated);
 580
 581         if (!is_a<idx>(i1) || !is_a<idx>(i2))
 582                 throw(std::invalid_argument("indices of delta tensor must be of type idx"));
 583
 584         return indexed(delta, symmetric2(), i1, i2);
 585 }
 586
 587 ex metric_tensor(const ex & i1, const ex & i2)
 588 {
 589         static ex metric = (new tensmetric)->setflag(status_flags::dynallocated);
 590
 591         if (!is_a<varidx>(i1) || !is_a<varidx>(i2))
 592                 throw(std::invalid_argument("indices of metric tensor must be of type varidx"));
 593
 594         return indexed(metric, symmetric2(), i1, i2);
 595 }
 596
 597 ex lorentz_g(const ex & i1, const ex & i2, bool pos_sig)
 598 {
 599         static ex metric_neg = (new minkmetric(false))->setflag(status_flags::dynallocated);
 600         static ex metric_pos = (new minkmetric(true))->setflag(status_flags::dynallocated);
 601
 602         if (!is_a<varidx>(i1) || !is_a<varidx>(i2))
 603                 throw(std::invalid_argument("indices of metric tensor must be of type varidx"));
 604
 605         return indexed(pos_sig ? metric_pos : metric_neg, symmetric2(), i1, i2);
 606 }
 607
 608 ex spinor_metric(const ex & i1, const ex & i2)
 609 {
 610         static ex metric = (new spinmetric)->setflag(status_flags::dynallocated);
 611
 612         if (!is_a<spinidx>(i1) || !is_a<spinidx>(i2))
 613                 throw(std::invalid_argument("indices of spinor metric must be of type spinidx"));
 614         if (!ex_to<idx>(i1).get_dim().is_equal(2) || !ex_to<idx>(i2).get_dim().is_equal(2))
 615                 throw(std::runtime_error("index dimension for spinor metric must be 2"));
 616
 617         return indexed(metric, antisymmetric2(), i1, i2);
 618 }
 619
 620 ex epsilon_tensor(const ex & i1, const ex & i2)
 621 {
 622         static ex epsilon = (new tensepsilon)->setflag(status_flags::dynallocated);
 623
 624         if (!is_a<idx>(i1) || !is_a<idx>(i2))
 625                 throw(std::invalid_argument("indices of epsilon tensor must be of type idx"));
 626
 627         ex dim = ex_to<idx>(i1).get_dim();
 628         if (!dim.is_equal(ex_to<idx>(i2).get_dim()))
 629                 throw(std::invalid_argument("all indices of epsilon tensor must have the same dimension"));
 630         if (!ex_to<idx>(i1).get_dim().is_equal(_ex2))
 631                 throw(std::runtime_error("index dimension of epsilon tensor must match number of indices"));
 632
 633         if(is_a<wildcard>(i1.op(0))||is_a<wildcard>(i2.op(0)))
 634                 return indexed(epsilon, antisymmetric2(), i1, i2).hold();
 635
 636         return indexed(epsilon, antisymmetric2(), i1, i2);
 637 }
 638
 639 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3)
 640 {
 641         static ex epsilon = (new tensepsilon)->setflag(status_flags::dynallocated);
 642
 643         if (!is_a<idx>(i1) || !is_a<idx>(i2) || !is_a<idx>(i3))
 644                 throw(std::invalid_argument("indices of epsilon tensor must be of type idx"));
 645
 646         ex dim = ex_to<idx>(i1).get_dim();
 647         if (!dim.is_equal(ex_to<idx>(i2).get_dim()) || !dim.is_equal(ex_to<idx>(i3).get_dim()))
 648                 throw(std::invalid_argument("all indices of epsilon tensor must have the same dimension"));
 649         if (!ex_to<idx>(i1).get_dim().is_equal(_ex3))
 650                 throw(std::runtime_error("index dimension of epsilon tensor must match number of indices"));
 651
 652         if(is_a<wildcard>(i1.op(0))||is_a<wildcard>(i2.op(0))||is_a<wildcard>(i3.op(0)))
 653                 return indexed(epsilon, antisymmetric3(), i1, i2, i3).hold();
 654
 655         return indexed(epsilon, antisymmetric3(), i1, i2, i3);
 656 }
 657
 658 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig)
 659 {
 660         static ex epsilon_neg = (new tensepsilon(true, false))->setflag(status_flags::dynallocated);
 661         static ex epsilon_pos = (new tensepsilon(true, true))->setflag(status_flags::dynallocated);
 662
 663         if (!is_a<varidx>(i1) || !is_a<varidx>(i2) || !is_a<varidx>(i3) || !is_a<varidx>(i4))
 664                 throw(std::invalid_argument("indices of Lorentz epsilon tensor must be of type varidx"));
 665
 666         ex dim = ex_to<idx>(i1).get_dim();
 667         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()))
 668                 throw(std::invalid_argument("all indices of epsilon tensor must have the same dimension"));
 669         if (!ex_to<idx>(i1).get_dim().is_equal(_ex4))
 670                 throw(std::runtime_error("index dimension of epsilon tensor must match number of indices"));
 671
 672         if(is_a<wildcard>(i1.op(0))||is_a<wildcard>(i2.op(0))||is_a<wildcard>(i3.op(0))||is_a<wildcard>(i4.op(0)))
 673                 return indexed(pos_sig ? epsilon_pos : epsilon_neg, antisymmetric4(), i1, i2, i3, i4).hold();
 674
 675         return indexed(pos_sig ? epsilon_pos : epsilon_neg, antisymmetric4(), i1, i2, i3, i4);
 676 }
 677
 678 } // namespace GiNaC