3 * Implementation of GiNaC's special tensors. */
6 * GiNaC Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany
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.
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.
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
31 #include "relational.h"
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)
49 // default ctor, dtor, copy ctor, assignment operator and helpers
53 DEFAULT_CTORS(tensdelta)
54 DEFAULT_CTORS(tensmetric)
55 DEFAULT_COPY(spinmetric)
56 DEFAULT_DESTROY(spinmetric)
57 DEFAULT_DESTROY(minkmetric)
58 DEFAULT_DESTROY(tensepsilon)
60 minkmetric::minkmetric() : pos_sig(false)
62 tinfo_key = TINFO_minkmetric;
65 spinmetric::spinmetric()
67 tinfo_key = TINFO_spinmetric;
70 minkmetric::minkmetric(bool ps) : pos_sig(ps)
72 tinfo_key = TINFO_minkmetric;
75 void minkmetric::copy(const minkmetric & other)
77 inherited::copy(other);
78 pos_sig = other.pos_sig;
81 tensepsilon::tensepsilon() : minkowski(false), pos_sig(false)
83 tinfo_key = TINFO_tensepsilon;
86 tensepsilon::tensepsilon(bool mink, bool ps) : minkowski(mink), pos_sig(ps)
88 tinfo_key = TINFO_tensepsilon;
91 void tensepsilon::copy(const tensepsilon & other)
93 inherited::copy(other);
94 minkowski = other.minkowski;
95 pos_sig = other.pos_sig;
102 DEFAULT_ARCHIVING(tensor)
103 DEFAULT_ARCHIVING(tensdelta)
104 DEFAULT_ARCHIVING(tensmetric)
105 DEFAULT_ARCHIVING(spinmetric)
106 DEFAULT_UNARCHIVE(minkmetric)
107 DEFAULT_UNARCHIVE(tensepsilon)
109 minkmetric::minkmetric(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
111 n.find_bool("pos_sig", pos_sig);
114 void minkmetric::archive(archive_node &n) const
116 inherited::archive(n);
117 n.add_bool("pos_sig", pos_sig);
120 tensepsilon::tensepsilon(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
122 n.find_bool("minkowski", minkowski);
123 n.find_bool("pos_sig", pos_sig);
126 void tensepsilon::archive(archive_node &n) const
128 inherited::archive(n);
129 n.add_bool("minkowski", minkowski);
130 n.add_bool("pos_sig", pos_sig);
134 // functions overriding virtual functions from base classes
137 DEFAULT_COMPARE(tensor)
138 DEFAULT_COMPARE(tensdelta)
139 DEFAULT_COMPARE(tensmetric)
140 DEFAULT_COMPARE(spinmetric)
142 int minkmetric::compare_same_type(const basic & other) const
144 GINAC_ASSERT(is_a<minkmetric>(other));
145 const minkmetric &o = static_cast<const minkmetric &>(other);
147 if (pos_sig != o.pos_sig)
148 return pos_sig ? -1 : 1;
150 return inherited::compare_same_type(other);
153 int tensepsilon::compare_same_type(const basic & other) const
155 GINAC_ASSERT(is_a<tensepsilon>(other));
156 const tensepsilon &o = static_cast<const tensepsilon &>(other);
158 if (minkowski != o.minkowski)
159 return minkowski ? -1 : 1;
160 else if (pos_sig != o.pos_sig)
161 return pos_sig ? -1 : 1;
163 return inherited::compare_same_type(other);
166 DEFAULT_PRINT_LATEX(tensdelta, "delta", "\\delta")
167 DEFAULT_PRINT(tensmetric, "g")
168 DEFAULT_PRINT_LATEX(minkmetric, "eta", "\\eta")
169 DEFAULT_PRINT_LATEX(spinmetric, "eps", "\\varepsilon")
170 DEFAULT_PRINT_LATEX(tensepsilon, "eps", "\\varepsilon")
172 /** Automatic symbolic evaluation of an indexed delta tensor. */
173 ex tensdelta::eval_indexed(const basic & i) const
175 GINAC_ASSERT(is_a<indexed>(i));
176 GINAC_ASSERT(i.nops() == 3);
177 GINAC_ASSERT(is_a<tensdelta>(i.op(0)));
179 const idx & i1 = ex_to<idx>(i.op(1));
180 const idx & i2 = ex_to<idx>(i.op(2));
182 // The dimension of the indices must be equal, otherwise we use the minimal
184 if (!i1.get_dim().is_equal(i2.get_dim())) {
185 ex min_dim = i1.minimal_dim(i2);
186 return i.subs(lst(i1 == i1.replace_dim(min_dim), i2 == i2.replace_dim(min_dim)));
189 // Trace of delta tensor is the (effective) dimension of the space
190 if (is_dummy_pair(i1, i2)) {
192 return i1.minimal_dim(i2);
193 } catch (std::exception &e) {
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();
207 // No further simplifications
211 /** Automatic symbolic evaluation of an indexed metric tensor. */
212 ex tensmetric::eval_indexed(const basic & i) const
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)));
220 const varidx & i1 = ex_to<varidx>(i.op(1));
221 const varidx & i2 = ex_to<varidx>(i.op(2));
223 // The dimension of the indices must be equal, otherwise we use the minimal
225 if (!i1.get_dim().is_equal(i2.get_dim())) {
226 ex min_dim = i1.minimal_dim(i2);
227 return i.subs(lst(i1 == i1.replace_dim(min_dim), i2 == i2.replace_dim(min_dim)));
230 // A metric tensor with one covariant and one contravariant index gets
231 // replaced by a delta tensor
232 if (i1.is_covariant() != i2.is_covariant())
233 return delta_tensor(i1, i2);
235 // No further simplifications
239 /** Automatic symbolic evaluation of an indexed Lorentz metric tensor. */
240 ex minkmetric::eval_indexed(const basic & i) const
242 GINAC_ASSERT(is_a<indexed>(i));
243 GINAC_ASSERT(i.nops() == 3);
244 GINAC_ASSERT(is_a<minkmetric>(i.op(0)));
245 GINAC_ASSERT(is_a<varidx>(i.op(1)));
246 GINAC_ASSERT(is_a<varidx>(i.op(2)));
248 const varidx & i1 = ex_to<varidx>(i.op(1));
249 const varidx & i2 = ex_to<varidx>(i.op(2));
251 // Numeric evaluation
252 if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
253 int n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
257 return pos_sig ? _ex_1 : _ex1;
259 return pos_sig ? _ex1 : _ex_1;
262 // Perform the usual evaluations of a metric tensor
263 return inherited::eval_indexed(i);
266 /** Automatic symbolic evaluation of an indexed metric tensor. */
267 ex spinmetric::eval_indexed(const basic & i) const
269 GINAC_ASSERT(is_a<indexed>(i));
270 GINAC_ASSERT(i.nops() == 3);
271 GINAC_ASSERT(is_a<spinmetric>(i.op(0)));
272 GINAC_ASSERT(is_a<spinidx>(i.op(1)));
273 GINAC_ASSERT(is_a<spinidx>(i.op(2)));
275 const spinidx & i1 = ex_to<spinidx>(i.op(1));
276 const spinidx & i2 = ex_to<spinidx>(i.op(2));
278 // Convolutions are zero
279 if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
282 // Numeric evaluation
283 if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
284 int n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
293 // No further simplifications
297 /** Automatic symbolic evaluation of an indexed epsilon tensor. */
298 ex tensepsilon::eval_indexed(const basic & i) const
300 GINAC_ASSERT(is_a<indexed>(i));
301 GINAC_ASSERT(i.nops() > 1);
302 GINAC_ASSERT(is_a<tensepsilon>(i.op(0)));
304 // Convolutions are zero
305 if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
308 // Numeric evaluation
309 if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
311 // Get sign of index permutation (the indices should already be in
312 // a canonic order but we can't assume what exactly that order is)
314 v.reserve(i.nops() - 1);
315 for (unsigned j=1; j<i.nops(); j++)
316 v.push_back(ex_to<numeric>(ex_to<idx>(i.op(j)).get_value()).to_int());
317 int sign = permutation_sign(v.begin(), v.end());
319 // In a Minkowski space, check for covariant indices
321 for (unsigned j=1; j<i.nops(); j++) {
322 const ex & x = i.op(j);
323 if (!is_ex_of_type(x, varidx))
324 throw(std::runtime_error("indices of epsilon tensor in Minkowski space must be of type varidx"));
325 if (ex_to<varidx>(x).is_covariant())
326 if (ex_to<idx>(x).get_value().is_zero())
327 sign = (pos_sig ? -sign : sign);
329 sign = (pos_sig ? sign : -sign);
336 // No further simplifications
340 bool tensor::replace_contr_index(exvector::iterator self, exvector::iterator other) const
342 // Try to contract the first index
343 const idx *self_idx = &ex_to<idx>(self->op(1));
344 const idx *free_idx = &ex_to<idx>(self->op(2));
345 bool first_index_tried = false;
348 if (self_idx->is_symbolic()) {
349 for (unsigned i=1; i<other->nops(); i++) {
350 const idx &other_idx = ex_to<idx>(other->op(i));
351 if (is_dummy_pair(*self_idx, other_idx)) {
353 // Contraction found, remove this tensor and substitute the
354 // index in the second object
356 // minimal_dim() throws an exception when index dimensions are not comparable
357 ex min_dim = self_idx->minimal_dim(other_idx);
358 *other = other->subs(other_idx == free_idx->replace_dim(min_dim));
359 *self = _ex1; // *other is assigned first because assigning *self invalidates free_idx
361 } catch (std::exception &e) {
368 if (!first_index_tried) {
370 // No contraction with the first index found, try the second index
371 self_idx = &ex_to<idx>(self->op(2));
372 free_idx = &ex_to<idx>(self->op(1));
373 first_index_tried = true;
380 /** Contraction of an indexed delta tensor with something else. */
381 bool tensdelta::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
383 GINAC_ASSERT(is_a<indexed>(*self));
384 GINAC_ASSERT(is_a<indexed>(*other));
385 GINAC_ASSERT(self->nops() == 3);
386 GINAC_ASSERT(is_a<tensdelta>(self->op(0)));
388 // Replace the dummy index with this tensor's other index and remove
389 // the tensor (this is valid for contractions with all other tensors)
390 return replace_contr_index(self, other);
393 /** Contraction of an indexed metric tensor with something else. */
394 bool tensmetric::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
396 GINAC_ASSERT(is_a<indexed>(*self));
397 GINAC_ASSERT(is_a<indexed>(*other));
398 GINAC_ASSERT(self->nops() == 3);
399 GINAC_ASSERT(is_a<tensmetric>(self->op(0)));
401 // If contracting with the delta tensor, let the delta do it
402 // (don't raise/lower delta indices)
403 if (is_ex_of_type(other->op(0), tensdelta))
406 // Replace the dummy index with this tensor's other index and remove
407 // the tensor (this is valid for contractions with all other tensors)
408 return replace_contr_index(self, other);
411 /** Contraction of an indexed spinor metric with something else. */
412 bool spinmetric::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
414 GINAC_ASSERT(is_a<indexed>(*self));
415 GINAC_ASSERT(is_a<indexed>(*other));
416 GINAC_ASSERT(self->nops() == 3);
417 GINAC_ASSERT(is_a<spinmetric>(self->op(0)));
419 // Contractions between spinor metrics
420 if (is_ex_of_type(other->op(0), spinmetric)) {
421 const idx &self_i1 = ex_to<idx>(self->op(1));
422 const idx &self_i2 = ex_to<idx>(self->op(2));
423 const idx &other_i1 = ex_to<idx>(other->op(1));
424 const idx &other_i2 = ex_to<idx>(other->op(2));
426 if (is_dummy_pair(self_i1, other_i1)) {
427 if (is_dummy_pair(self_i2, other_i2))
430 *self = delta_tensor(self_i2, other_i2);
433 } else if (is_dummy_pair(self_i1, other_i2)) {
434 if (is_dummy_pair(self_i2, other_i1))
437 *self = -delta_tensor(self_i2, other_i1);
440 } else if (is_dummy_pair(self_i2, other_i1)) {
441 *self = -delta_tensor(self_i1, other_i2);
444 } else if (is_dummy_pair(self_i2, other_i2)) {
445 *self = delta_tensor(self_i1, other_i1);
451 // If contracting with the delta tensor, let the delta do it
452 // (don't raise/lower delta indices)
453 if (is_ex_of_type(other->op(0), tensdelta))
456 // Try to contract first index
457 const idx *self_idx = &ex_to<idx>(self->op(1));
458 const idx *free_idx = &ex_to<idx>(self->op(2));
459 bool first_index_tried = false;
463 if (self_idx->is_symbolic()) {
464 for (unsigned i=1; i<other->nops(); i++) {
465 const idx &other_idx = ex_to<idx>(other->op(i));
466 if (is_dummy_pair(*self_idx, other_idx)) {
468 // Contraction found, remove metric tensor and substitute
469 // index in second object (assign *self last because this
470 // invalidates free_idx)
471 *other = other->subs(other_idx == *free_idx);
472 *self = (static_cast<const spinidx *>(self_idx)->is_covariant() ? sign : -sign);
478 if (!first_index_tried) {
480 // No contraction with first index found, try second index
481 self_idx = &ex_to<idx>(self->op(2));
482 free_idx = &ex_to<idx>(self->op(1));
483 first_index_tried = true;
491 /** Contraction of epsilon tensor with something else. */
492 bool tensepsilon::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
494 GINAC_ASSERT(is_a<indexed>(*self));
495 GINAC_ASSERT(is_a<indexed>(*other));
496 GINAC_ASSERT(is_a<tensepsilon>(self->op(0)));
497 unsigned num = self->nops() - 1;
499 if (is_ex_exactly_of_type(other->op(0), tensepsilon) && num+1 == other->nops()) {
501 // Contraction of two epsilon tensors is a determinant
502 bool variance = is_a<varidx>(self->op(1));
504 for (int i=0; i<num; i++) {
505 for (int j=0; j<num; j++) {
507 M(i, j) = lorentz_g(self->op(i+1), other->op(j+1), pos_sig);
509 M(i, j) = metric_tensor(self->op(i+1), other->op(j+1));
511 M(i, j) = delta_tensor(self->op(i+1), other->op(j+1));
514 int sign = minkowski ? -1 : 1;
515 *self = sign * M.determinant().simplify_indexed();
527 ex delta_tensor(const ex & i1, const ex & i2)
529 if (!is_ex_of_type(i1, idx) || !is_ex_of_type(i2, idx))
530 throw(std::invalid_argument("indices of delta tensor must be of type idx"));
532 return indexed(tensdelta(), sy_symm(), i1, i2);
535 ex metric_tensor(const ex & i1, const ex & i2)
537 if (!is_ex_of_type(i1, varidx) || !is_ex_of_type(i2, varidx))
538 throw(std::invalid_argument("indices of metric tensor must be of type varidx"));
540 return indexed(tensmetric(), sy_symm(), i1, i2);
543 ex lorentz_g(const ex & i1, const ex & i2, bool pos_sig)
545 if (!is_ex_of_type(i1, varidx) || !is_ex_of_type(i2, varidx))
546 throw(std::invalid_argument("indices of metric tensor must be of type varidx"));
548 return indexed(minkmetric(pos_sig), sy_symm(), i1, i2);
551 ex spinor_metric(const ex & i1, const ex & i2)
553 if (!is_ex_of_type(i1, spinidx) || !is_ex_of_type(i2, spinidx))
554 throw(std::invalid_argument("indices of spinor metric must be of type spinidx"));
555 if (!ex_to<idx>(i1).get_dim().is_equal(2) || !ex_to<idx>(i2).get_dim().is_equal(2))
556 throw(std::runtime_error("index dimension for spinor metric must be 2"));
558 return indexed(spinmetric(), sy_anti(), i1, i2);
561 ex epsilon_tensor(const ex & i1, const ex & i2)
563 if (!is_ex_of_type(i1, idx) || !is_ex_of_type(i2, idx))
564 throw(std::invalid_argument("indices of epsilon tensor must be of type idx"));
566 ex dim = ex_to<idx>(i1).get_dim();
567 if (!dim.is_equal(ex_to<idx>(i2).get_dim()))
568 throw(std::invalid_argument("all indices of epsilon tensor must have the same dimension"));
569 if (!ex_to<idx>(i1).get_dim().is_equal(_ex2))
570 throw(std::runtime_error("index dimension of epsilon tensor must match number of indices"));
572 return indexed(tensepsilon(), sy_anti(), i1, i2);
575 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3)
577 if (!is_ex_of_type(i1, idx) || !is_ex_of_type(i2, idx) || !is_ex_of_type(i3, idx))
578 throw(std::invalid_argument("indices of epsilon tensor must be of type idx"));
580 ex dim = ex_to<idx>(i1).get_dim();
581 if (!dim.is_equal(ex_to<idx>(i2).get_dim()) || !dim.is_equal(ex_to<idx>(i3).get_dim()))
582 throw(std::invalid_argument("all indices of epsilon tensor must have the same dimension"));
583 if (!ex_to<idx>(i1).get_dim().is_equal(_ex3))
584 throw(std::runtime_error("index dimension of epsilon tensor must match number of indices"));
586 return indexed(tensepsilon(), sy_anti(), i1, i2, i3);
589 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig)
591 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))
592 throw(std::invalid_argument("indices of Lorentz epsilon tensor must be of type varidx"));
594 ex dim = ex_to<idx>(i1).get_dim();
595 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()))
596 throw(std::invalid_argument("all indices of epsilon tensor must have the same dimension"));
597 if (!ex_to<idx>(i1).get_dim().is_equal(_ex4))
598 throw(std::runtime_error("index dimension of epsilon tensor must match number of indices"));
600 return indexed(tensepsilon(true, pos_sig), sy_anti(), i1, i2, i3, i4);