Remove 'level' argument of normal().
[ginac.git] / ginac / tensor.cpp
1 /** @file tensor.cpp
2  *
3  *  Implementation of GiNaC's special tensors. */
4
5 /*
6  *  GiNaC Copyright (C) 1999-2016 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 = dynallocate<tensdelta>();
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 = dynallocate<tensmetric>();
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 = dynallocate<minkmetric>(false);
600         static ex metric_pos = dynallocate<minkmetric>(true);
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 = dynallocate<spinmetric>();
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 = dynallocate<tensepsilon>();
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 = dynallocate<tensepsilon>();
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 = dynallocate<tensepsilon>(true, false);
661         static ex epsilon_pos = dynallocate<tensepsilon>(true, true);
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