ginac/tensor.cpp

   1 /** @file tensor.cpp
   2  *
   3  *  Implementation of GiNaC's special tensors. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2001 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 <stdexcept>
  24 #include <vector>
  25
  26 #include "tensor.h"
  27 #include "idx.h"
  28 #include "indexed.h"
  29 #include "symmetry.h"
  30 #include "relational.h"
  31 #include "lst.h"
  32 #include "numeric.h"
  33 #include "matrix.h"
  34 #include "print.h"
  35 #include "archive.h"
  36 #include "utils.h"
  37 #include "debugmsg.h"
  38
  39 namespace GiNaC {
  40
  41 GINAC_IMPLEMENT_REGISTERED_CLASS(tensor, basic)
  42 GINAC_IMPLEMENT_REGISTERED_CLASS(tensdelta, tensor)
  43 GINAC_IMPLEMENT_REGISTERED_CLASS(tensmetric, tensor)
  44 GINAC_IMPLEMENT_REGISTERED_CLASS(minkmetric, tensmetric)
  45 GINAC_IMPLEMENT_REGISTERED_CLASS(spinmetric, tensmetric)
  46 GINAC_IMPLEMENT_REGISTERED_CLASS(tensepsilon, tensor)
  47
  48 //////////
  49 // default constructor, destructor, copy constructor assignment operator and helpers
  50 //////////
  51
  52 tensor::tensor(unsigned ti) : inherited(ti)
  53 {
  54         debugmsg("tensor constructor from unsigned", LOGLEVEL_CONSTRUCT); \
  55 }
  56
  57 DEFAULT_CTORS(tensor)
  58 DEFAULT_CTORS(tensdelta)
  59 DEFAULT_CTORS(tensmetric)
  60 DEFAULT_COPY(spinmetric)
  61 DEFAULT_DESTROY(spinmetric)
  62 DEFAULT_DESTROY(minkmetric)
  63 DEFAULT_DESTROY(tensepsilon)
  64
  65 minkmetric::minkmetric() : pos_sig(false)
  66 {
  67         debugmsg("minkmetric default constructor", LOGLEVEL_CONSTRUCT);
  68         tinfo_key = TINFO_minkmetric;
  69 }
  70
  71 spinmetric::spinmetric()
  72 {
  73         debugmsg("spinmetric default constructor", LOGLEVEL_CONSTRUCT);
  74         tinfo_key = TINFO_spinmetric;
  75 }
  76
  77 minkmetric::minkmetric(bool ps) : pos_sig(ps)
  78 {
  79         debugmsg("minkmetric constructor from bool", LOGLEVEL_CONSTRUCT);
  80         tinfo_key = TINFO_minkmetric;
  81 }
  82
  83 void minkmetric::copy(const minkmetric & other)
  84 {
  85         inherited::copy(other);
  86         pos_sig = other.pos_sig;
  87 }
  88
  89 tensepsilon::tensepsilon() : minkowski(false), pos_sig(false)
  90 {
  91         debugmsg("tensepsilon default constructor", LOGLEVEL_CONSTRUCT);
  92         tinfo_key = TINFO_tensepsilon;
  93 }
  94
  95 tensepsilon::tensepsilon(bool mink, bool ps) : minkowski(mink), pos_sig(ps)
  96 {
  97         debugmsg("tensepsilon constructor from bool,bool", LOGLEVEL_CONSTRUCT);
  98         tinfo_key = TINFO_tensepsilon;
  99 }
 100
 101 void tensepsilon::copy(const tensepsilon & other)
 102 {
 103         inherited::copy(other);
 104         minkowski = other.minkowski;
 105         pos_sig = other.pos_sig;
 106 }
 107
 108 //////////
 109 // archiving
 110 //////////
 111
 112 DEFAULT_ARCHIVING(tensor)
 113 DEFAULT_ARCHIVING(tensdelta)
 114 DEFAULT_ARCHIVING(tensmetric)
 115 DEFAULT_ARCHIVING(spinmetric)
 116 DEFAULT_UNARCHIVE(minkmetric)
 117 DEFAULT_UNARCHIVE(tensepsilon)
 118
 119 minkmetric::minkmetric(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 120 {
 121         debugmsg("minkmetric constructor from archive_node", LOGLEVEL_CONSTRUCT);
 122         n.find_bool("pos_sig", pos_sig);
 123 }
 124
 125 void minkmetric::archive(archive_node &n) const
 126 {
 127         inherited::archive(n);
 128         n.add_bool("pos_sig", pos_sig);
 129 }
 130
 131 tensepsilon::tensepsilon(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 132 {
 133         debugmsg("tensepsilon constructor from archive_node", LOGLEVEL_CONSTRUCT);
 134         n.find_bool("minkowski", minkowski);
 135         n.find_bool("pos_sig", pos_sig);
 136 }
 137
 138 void tensepsilon::archive(archive_node &n) const
 139 {
 140         inherited::archive(n);
 141         n.add_bool("minkowski", minkowski);
 142         n.add_bool("pos_sig", pos_sig);
 143 }
 144
 145 //////////
 146 // functions overriding virtual functions from base classes
 147 //////////
 148
 149 DEFAULT_COMPARE(tensor)
 150 DEFAULT_COMPARE(tensdelta)
 151 DEFAULT_COMPARE(tensmetric)
 152 DEFAULT_COMPARE(spinmetric)
 153
 154 int minkmetric::compare_same_type(const basic & other) const
 155 {
 156         GINAC_ASSERT(is_a<minkmetric>(other));
 157         const minkmetric &o = static_cast<const minkmetric &>(other);
 158
 159         if (pos_sig != o.pos_sig)
 160                 return pos_sig ? -1 : 1;
 161         else
 162                 return inherited::compare_same_type(other);
 163 }
 164
 165 int tensepsilon::compare_same_type(const basic & other) const
 166 {
 167         GINAC_ASSERT(is_a<tensepsilon>(other));
 168         const tensepsilon &o = static_cast<const tensepsilon &>(other);
 169
 170         if (minkowski != o.minkowski)
 171                 return minkowski ? -1 : 1;
 172         else if (pos_sig != o.pos_sig)
 173                 return pos_sig ? -1 : 1;
 174         else
 175                 return inherited::compare_same_type(other);
 176 }
 177
 178 DEFAULT_PRINT_LATEX(tensdelta, "delta", "\\delta")
 179 DEFAULT_PRINT(tensmetric, "g")
 180 DEFAULT_PRINT_LATEX(minkmetric, "eta", "\\eta")
 181 DEFAULT_PRINT_LATEX(spinmetric, "eps", "\\varepsilon")
 182 DEFAULT_PRINT_LATEX(tensepsilon, "eps", "\\varepsilon")
 183
 184 /** Automatic symbolic evaluation of an indexed delta tensor. */
 185 ex tensdelta::eval_indexed(const basic & i) const
 186 {
 187         GINAC_ASSERT(is_a<indexed>(i));
 188         GINAC_ASSERT(i.nops() == 3);
 189         GINAC_ASSERT(is_a<tensdelta>(i.op(0)));
 190
 191         const idx & i1 = ex_to<idx>(i.op(1));
 192         const idx & i2 = ex_to<idx>(i.op(2));
 193
 194         // Trace of delta tensor is the dimension of the space
 195         if (is_dummy_pair(i1, i2))
 196                 return i1.get_dim();
 197
 198         // Numeric evaluation
 199         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::integer)) {
 200                 int n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
 201                 if (n1 == n2)
 202                         return _ex1();
 203                 else
 204                         return _ex0();
 205         }
 206
 207         // No further simplifications
 208         return i.hold();
 209 }
 210
 211 /** Automatic symbolic evaluation of an indexed metric tensor. */
 212 ex tensmetric::eval_indexed(const basic & i) const
 213 {
 214         GINAC_ASSERT(is_a<indexed>(i));
 215         GINAC_ASSERT(i.nops() == 3);
 216         GINAC_ASSERT(is_a<tensmetric>(i.op(0)));
 217         GINAC_ASSERT(is_a<varidx>(i.op(1)));
 218         GINAC_ASSERT(is_a<varidx>(i.op(2)));
 219
 220         const varidx & i1 = ex_to<varidx>(i.op(1));
 221         const varidx & i2 = ex_to<varidx>(i.op(2));
 222
 223         // A metric tensor with one covariant and one contravariant index gets
 224         // replaced by a delta tensor
 225         if (i1.is_covariant() != i2.is_covariant())
 226                 return delta_tensor(i1, i2);
 227
 228         // No further simplifications
 229         return i.hold();
 230 }
 231
 232 /** Automatic symbolic evaluation of an indexed Lorentz metric tensor. */
 233 ex minkmetric::eval_indexed(const basic & i) const
 234 {
 235         GINAC_ASSERT(is_a<indexed>(i));
 236         GINAC_ASSERT(i.nops() == 3);
 237         GINAC_ASSERT(is_a<minkmetric>(i.op(0)));
 238         GINAC_ASSERT(is_a<varidx>(i.op(1)));
 239         GINAC_ASSERT(is_a<varidx>(i.op(2)));
 240
 241         const varidx & i1 = ex_to<varidx>(i.op(1));
 242         const varidx & i2 = ex_to<varidx>(i.op(2));
 243
 244         // Numeric evaluation
 245         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 246                 int n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
 247                 if (n1 != n2)
 248                         return _ex0();
 249                 else if (n1 == 0)
 250                         return pos_sig ? _ex_1() : _ex1();
 251                 else
 252                         return pos_sig ? _ex1() : _ex_1();
 253         }
 254
 255         // Perform the usual evaluations of a metric tensor
 256         return inherited::eval_indexed(i);
 257 }
 258
 259 /** Automatic symbolic evaluation of an indexed metric tensor. */
 260 ex spinmetric::eval_indexed(const basic & i) const
 261 {
 262         GINAC_ASSERT(is_a<indexed>(i));
 263         GINAC_ASSERT(i.nops() == 3);
 264         GINAC_ASSERT(is_a<spinmetric>(i.op(0)));
 265         GINAC_ASSERT(is_a<spinidx>(i.op(1)));
 266         GINAC_ASSERT(is_a<spinidx>(i.op(2)));
 267
 268         const spinidx & i1 = ex_to<spinidx>(i.op(1));
 269         const spinidx & i2 = ex_to<spinidx>(i.op(2));
 270
 271         // Convolutions are zero
 272         if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
 273                 return _ex0();
 274
 275         // Numeric evaluation
 276         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 277                 int n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
 278                 if (n1 == n2)
 279                         return _ex0();
 280                 else if (n1 < n2)
 281                         return _ex1();
 282                 else
 283                         return _ex_1();
 284         }
 285
 286         // No further simplifications
 287         return i.hold();
 288 }
 289
 290 /** Automatic symbolic evaluation of an indexed epsilon tensor. */
 291 ex tensepsilon::eval_indexed(const basic & i) const
 292 {
 293         GINAC_ASSERT(is_a<indexed>(i));
 294         GINAC_ASSERT(i.nops() > 1);
 295         GINAC_ASSERT(is_a<tensepsilon>(i.op(0)));
 296
 297         // Convolutions are zero
 298         if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
 299                 return _ex0();
 300
 301         // Numeric evaluation
 302         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 303
 304                 // Get sign of index permutation (the indices should already be in
 305                 // a canonic order but we can't assume what exactly that order is)
 306                 std::vector<int> v;
 307                 v.reserve(i.nops() - 1);
 308                 for (unsigned j=1; j<i.nops(); j++)
 309                         v.push_back(ex_to<numeric>(ex_to<idx>(i.op(j)).get_value()).to_int());
 310                 int sign = permutation_sign(v.begin(), v.end());
 311
 312                 // In a Minkowski space, check for covariant indices
 313                 if (minkowski) {
 314                         for (unsigned j=1; j<i.nops(); j++) {
 315                                 const ex & x = i.op(j);
 316                                 if (!is_ex_of_type(x, varidx))
 317                                         throw(std::runtime_error("indices of epsilon tensor in Minkowski space must be of type varidx"));
 318                                 if (ex_to<varidx>(x).is_covariant())
 319                                         if (ex_to<idx>(x).get_value().is_zero())
 320                                                 sign = (pos_sig ? -sign : sign);
 321                                         else
 322                                                 sign = (pos_sig ? sign : -sign);
 323                         }
 324                 }
 325
 326                 return sign;
 327         }
 328
 329         // No further simplifications
 330         return i.hold();
 331 }
 332
 333 /** Contraction of an indexed delta tensor with something else. */
 334 bool tensdelta::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 335 {
 336         GINAC_ASSERT(is_a<indexed>(*self));
 337         GINAC_ASSERT(is_a<indexed>(*other));
 338         GINAC_ASSERT(self->nops() == 3);
 339         GINAC_ASSERT(is_a<tensdelta>(self->op(0)));
 340
 341         // Try to contract first index
 342         const idx *self_idx = &ex_to<idx>(self->op(1));
 343         const idx *free_idx = &ex_to<idx>(self->op(2));
 344         bool first_index_tried = false;
 345
 346 again:
 347         if (self_idx->is_symbolic()) {
 348                 for (unsigned i=1; i<other->nops(); i++) {
 349                         const idx &other_idx = ex_to<idx>(other->op(i));
 350                         if (is_dummy_pair(*self_idx, other_idx)) {
 351
 352                                 // Contraction found, remove delta tensor and substitute
 353                                 // index in second object
 354                                 *self = _ex1();
 355                                 *other = other->subs(other_idx == *free_idx);
 356                                 return true;
 357                         }
 358                 }
 359         }
 360
 361         if (!first_index_tried) {
 362
 363                 // No contraction with first index found, try second index
 364                 self_idx = &ex_to<idx>(self->op(2));
 365                 free_idx = &ex_to<idx>(self->op(1));
 366                 first_index_tried = true;
 367                 goto again;
 368         }
 369
 370         return false;
 371 }
 372
 373 /** Contraction of an indexed metric tensor with something else. */
 374 bool tensmetric::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 375 {
 376         GINAC_ASSERT(is_a<indexed>(*self));
 377         GINAC_ASSERT(is_a<indexed>(*other));
 378         GINAC_ASSERT(self->nops() == 3);
 379         GINAC_ASSERT(is_a<tensmetric>(self->op(0)));
 380
 381         // If contracting with the delta tensor, let the delta do it
 382         // (don't raise/lower delta indices)
 383         if (is_ex_of_type(other->op(0), tensdelta))
 384                 return false;
 385
 386         // Try to contract first index
 387         const idx *self_idx = &ex_to<idx>(self->op(1));
 388         const idx *free_idx = &ex_to<idx>(self->op(2));
 389         bool first_index_tried = false;
 390
 391 again:
 392         if (self_idx->is_symbolic()) {
 393                 for (unsigned i=1; i<other->nops(); i++) {
 394                         const idx &other_idx = ex_to<idx>(other->op(i));
 395                         if (is_dummy_pair(*self_idx, other_idx)) {
 396
 397                                 // Contraction found, remove metric tensor and substitute
 398                                 // index in second object
 399                                 *self = _ex1();
 400                                 *other = other->subs(other_idx == *free_idx);
 401                                 return true;
 402                         }
 403                 }
 404         }
 405
 406         if (!first_index_tried) {
 407
 408                 // No contraction with first index found, try second index
 409                 self_idx = &ex_to<idx>(self->op(2));
 410                 free_idx = &ex_to<idx>(self->op(1));
 411                 first_index_tried = true;
 412                 goto again;
 413         }
 414
 415         return false;
 416 }
 417
 418 /** Contraction of an indexed spinor metric with something else. */
 419 bool spinmetric::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 420 {
 421         GINAC_ASSERT(is_a<indexed>(*self));
 422         GINAC_ASSERT(is_a<indexed>(*other));
 423         GINAC_ASSERT(self->nops() == 3);
 424         GINAC_ASSERT(is_a<spinmetric>(self->op(0)));
 425
 426         // Contractions between spinor metrics
 427         if (is_ex_of_type(other->op(0), spinmetric)) {
 428                 const idx &self_i1 = ex_to<idx>(self->op(1));
 429                 const idx &self_i2 = ex_to<idx>(self->op(2));
 430                 const idx &other_i1 = ex_to<idx>(other->op(1));
 431                 const idx &other_i2 = ex_to<idx>(other->op(2));
 432
 433                 if (is_dummy_pair(self_i1, other_i1)) {
 434                         if (is_dummy_pair(self_i2, other_i2))
 435                                 *self = _ex2();
 436                         else
 437                                 *self = delta_tensor(self_i2, other_i2);
 438                         *other = _ex1();
 439                         return true;
 440                 } else if (is_dummy_pair(self_i1, other_i2)) {
 441                         if (is_dummy_pair(self_i2, other_i1))
 442                                 *self = _ex_2();
 443                         else
 444                                 *self = -delta_tensor(self_i2, other_i1);
 445                         *other = _ex1();
 446                         return true;
 447                 } else if (is_dummy_pair(self_i2, other_i1)) {
 448                         *self = -delta_tensor(self_i1, other_i2);
 449                         *other = _ex1();
 450                         return true;
 451                 } else if (is_dummy_pair(self_i2, other_i2)) {
 452                         *self = delta_tensor(self_i1, other_i1);
 453                         *other = _ex1();
 454                         return true;
 455                 }
 456         }
 457
 458         // If contracting with the delta tensor, let the delta do it
 459         // (don't raise/lower delta indices)
 460         if (is_ex_of_type(other->op(0), tensdelta))
 461                 return false;
 462
 463         // Try to contract first index
 464         const idx *self_idx = &ex_to<idx>(self->op(1));
 465         const idx *free_idx = &ex_to<idx>(self->op(2));
 466         bool first_index_tried = false;
 467         int sign = 1;
 468
 469 again:
 470         if (self_idx->is_symbolic()) {
 471                 for (unsigned i=1; i<other->nops(); i++) {
 472                         const idx &other_idx = ex_to<idx>(other->op(i));
 473                         if (is_dummy_pair(*self_idx, other_idx)) {
 474
 475                                 // Contraction found, remove metric tensor and substitute
 476                                 // index in second object
 477                                 *self = (static_cast<const spinidx *>(self_idx)->is_covariant() ? sign : -sign);
 478                                 *other = other->subs(other_idx == *free_idx);
 479                                 return true;
 480                         }
 481                 }
 482         }
 483
 484         if (!first_index_tried) {
 485
 486                 // No contraction with first index found, try second index
 487                 self_idx = &ex_to<idx>(self->op(2));
 488                 free_idx = &ex_to<idx>(self->op(1));
 489                 first_index_tried = true;
 490                 sign = -sign;
 491                 goto again;
 492         }
 493
 494         return false;
 495 }
 496
 497 /** Contraction of epsilon tensor with something else. */
 498 bool tensepsilon::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 499 {
 500         GINAC_ASSERT(is_a<indexed>(*self));
 501         GINAC_ASSERT(is_a<indexed>(*other));
 502         GINAC_ASSERT(is_a<tensepsilon>(self->op(0)));
 503         unsigned num = self->nops() - 1;
 504
 505         if (is_ex_exactly_of_type(other->op(0), tensepsilon) && num+1 == other->nops()) {
 506
 507                 // Contraction of two epsilon tensors is a determinant
 508                 ex dim = ex_to<idx>(self->op(1)).get_dim();
 509                 matrix M(num, num);
 510                 for (int i=0; i<num; i++) {
 511                         for (int j=0; j<num; j++) {
 512                                 if (minkowski)
 513                                         M(i, j) = lorentz_g(self->op(i+1), other->op(j+1), pos_sig);
 514                                 else
 515                                         M(i, j) = metric_tensor(self->op(i+1), other->op(j+1));
 516                         }
 517                 }
 518                 int sign = minkowski ? -1 : 1;
 519                 *self = sign * M.determinant().simplify_indexed();
 520                 *other = _ex1();
 521                 return true;
 522
 523         } else if (other->return_type() == return_types::commutative) {
 524
 525 #if 0
 526                 // This handles eps.i.j.k * p.j * p.k = 0
 527                 // Maybe something like this should go to simplify_indexed() because
 528                 // such relations are true for any antisymmetric tensors...
 529                 exvector c;
 530
 531                 // Handle all indices of the epsilon tensor
 532                 for (int i=0; i<num; i++) {
 533                         ex idx = self->op(i+1);
 534
 535                         // Look whether there's a contraction with this index
 536                         exvector::const_iterator ait, aitend = v.end();
 537                         for (ait = v.begin(); ait != aitend; ait++) {
 538                                 if (ait == self)
 539                                         continue;
 540                                 if (is_a<indexed>(*ait) && ait->return_type() == return_types::commutative && ex_to<indexed>(*ait).has_dummy_index_for(idx) && ait->nops() == 2) {
 541
 542                                         // Yes, did we already have another contraction with the same base expression?
 543                                         ex base = ait->op(0);
 544                                         if (std::find_if(c.begin(), c.end(), bind2nd(ex_is_equal(), base)) == c.end()) {
 545
 546                                                 // No, add the base expression to the list
 547                                                 c.push_back(base);
 548
 549                                         } else {
 550
 551                                                 // Yes, the contraction is zero
 552                                                 *self = _ex0();
 553                                                 *other = _ex0();
 554                                                 return true;
 555                                         }
 556                                 }
 557                         }
 558                 }
 559 #endif
 560         }
 561
 562         return false;
 563 }
 564
 565 //////////
 566 // global functions
 567 //////////
 568
 569 ex delta_tensor(const ex & i1, const ex & i2)
 570 {
 571         if (!is_ex_of_type(i1, idx) || !is_ex_of_type(i2, idx))
 572                 throw(std::invalid_argument("indices of delta tensor must be of type idx"));
 573
 574         return indexed(tensdelta(), sy_symm(), i1, i2);
 575 }
 576
 577 ex metric_tensor(const ex & i1, const ex & i2)
 578 {
 579         if (!is_ex_of_type(i1, varidx) || !is_ex_of_type(i2, varidx))
 580                 throw(std::invalid_argument("indices of metric tensor must be of type varidx"));
 581
 582         return indexed(tensmetric(), sy_symm(), i1, i2);
 583 }
 584
 585 ex lorentz_g(const ex & i1, const ex & i2, bool pos_sig)
 586 {
 587         if (!is_ex_of_type(i1, varidx) || !is_ex_of_type(i2, varidx))
 588                 throw(std::invalid_argument("indices of metric tensor must be of type varidx"));
 589
 590         return indexed(minkmetric(pos_sig), sy_symm(), i1, i2);
 591 }
 592
 593 ex spinor_metric(const ex & i1, const ex & i2)
 594 {
 595         if (!is_ex_of_type(i1, spinidx) || !is_ex_of_type(i2, spinidx))
 596                 throw(std::invalid_argument("indices of spinor metric must be of type spinidx"));
 597         if (!ex_to<idx>(i1).get_dim().is_equal(2) || !ex_to<idx>(i2).get_dim().is_equal(2))
 598                 throw(std::runtime_error("index dimension for spinor metric must be 2"));
 599
 600         return indexed(spinmetric(), sy_anti(), i1, i2);
 601 }
 602
 603 ex epsilon_tensor(const ex & i1, const ex & i2)
 604 {
 605         if (!is_ex_of_type(i1, idx) || !is_ex_of_type(i2, idx))
 606                 throw(std::invalid_argument("indices of epsilon tensor must be of type idx"));
 607
 608         ex dim = ex_to<idx>(i1).get_dim();
 609         if (!dim.is_equal(ex_to<idx>(i2).get_dim()))
 610                 throw(std::invalid_argument("all indices of epsilon tensor must have the same dimension"));
 611         if (!ex_to<idx>(i1).get_dim().is_equal(_ex2()))
 612                 throw(std::runtime_error("index dimension of epsilon tensor must match number of indices"));
 613
 614         return indexed(tensepsilon(), sy_anti(), i1, i2);
 615 }
 616
 617 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3)
 618 {
 619         if (!is_ex_of_type(i1, idx) || !is_ex_of_type(i2, idx) || !is_ex_of_type(i3, idx))
 620                 throw(std::invalid_argument("indices of epsilon tensor must be of type idx"));
 621
 622         ex dim = ex_to<idx>(i1).get_dim();
 623         if (!dim.is_equal(ex_to<idx>(i2).get_dim()) || !dim.is_equal(ex_to<idx>(i3).get_dim()))
 624                 throw(std::invalid_argument("all indices of epsilon tensor must have the same dimension"));
 625         if (!ex_to<idx>(i1).get_dim().is_equal(_ex3()))
 626                 throw(std::runtime_error("index dimension of epsilon tensor must match number of indices"));
 627
 628         return indexed(tensepsilon(), sy_anti(), i1, i2, i3);
 629 }
 630
 631 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig)
 632 {
 633         if (!is_ex_of_type(i1, varidx) || !is_ex_of_type(i2, varidx) || !is_ex_of_type(i3, varidx) || !is_ex_of_type(i4, varidx))
 634                 throw(std::invalid_argument("indices of Lorentz epsilon tensor must be of type varidx"));
 635
 636         ex dim = ex_to<idx>(i1).get_dim();
 637         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()))
 638                 throw(std::invalid_argument("all indices of epsilon tensor must have the same dimension"));
 639         if (!ex_to<idx>(i1).get_dim().is_equal(_ex4()))
 640                 throw(std::runtime_error("index dimension of epsilon tensor must match number of indices"));
 641
 642         return indexed(tensepsilon(true, pos_sig), sy_anti(), i1, i2, i3, i4);
 643 }
 644
 645 ex eps0123(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig)
 646 {
 647         if (!is_ex_of_type(i1, varidx) || !is_ex_of_type(i2, varidx) || !is_ex_of_type(i3, varidx) || !is_ex_of_type(i4, varidx))
 648                 throw(std::invalid_argument("indices of epsilon tensor must be of type varidx"));
 649
 650         ex dim = ex_to<idx>(i1).get_dim();
 651         if (dim.is_equal(4))
 652                 return lorentz_eps(i1, i2, i3, i4, pos_sig);
 653         else
 654                 return indexed(tensepsilon(true, pos_sig), sy_anti(), i1, i2, i3, i4);
 655 }
 656
 657 } // namespace GiNaC