3 * Implementation of GiNaC's special tensors. */
6 * GiNaC Copyright (C) 1999-2008 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., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
31 #include "relational.h"
32 #include "operators.h"
41 GINAC_IMPLEMENT_REGISTERED_CLASS(tensor, basic)
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))
47 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(tensmetric, tensor,
48 print_func<print_dflt>(&tensmetric::do_print).
49 print_func<print_latex>(&tensmetric::do_print))
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))
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))
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))
69 setflag(status_flags::evaluated | status_flags::expanded);
72 DEFAULT_CTOR(tensdelta)
73 DEFAULT_CTOR(tensmetric)
75 minkmetric::minkmetric() : pos_sig(false)
79 spinmetric::spinmetric()
83 minkmetric::minkmetric(bool ps) : pos_sig(ps)
87 tensepsilon::tensepsilon() : minkowski(false), pos_sig(false)
91 tensepsilon::tensepsilon(bool mink, bool ps) : minkowski(mink), pos_sig(ps)
99 void minkmetric::read_archive(const archive_node& n, lst& sym_lst)
101 inherited::read_archive(n, sym_lst);
102 n.find_bool("pos_sig", pos_sig);
104 GINAC_BIND_UNARCHIVER(minkmetric);
106 void minkmetric::archive(archive_node &n) const
108 inherited::archive(n);
109 n.add_bool("pos_sig", pos_sig);
112 void tensepsilon::read_archive(const archive_node& n, lst& sym_lst)
114 inherited::read_archive(n, sym_lst);
115 n.find_bool("minkowski", minkowski);
116 n.find_bool("pos_sig", pos_sig);
118 GINAC_BIND_UNARCHIVER(tensepsilon);
120 void tensepsilon::archive(archive_node &n) const
122 inherited::archive(n);
123 n.add_bool("minkowski", minkowski);
124 n.add_bool("pos_sig", pos_sig);
127 GINAC_BIND_UNARCHIVER(tensdelta);
128 GINAC_BIND_UNARCHIVER(tensmetric);
129 GINAC_BIND_UNARCHIVER(spinmetric);
132 // functions overriding virtual functions from base classes
135 DEFAULT_COMPARE(tensor)
136 DEFAULT_COMPARE(tensdelta)
137 DEFAULT_COMPARE(tensmetric)
138 DEFAULT_COMPARE(spinmetric)
140 bool tensdelta::info(unsigned inf) const
142 if(inf == info_flags::real)
148 bool tensmetric::info(unsigned inf) const
150 if(inf == info_flags::real)
156 int minkmetric::compare_same_type(const basic & other) const
158 GINAC_ASSERT(is_a<minkmetric>(other));
159 const minkmetric &o = static_cast<const minkmetric &>(other);
161 if (pos_sig != o.pos_sig)
162 return pos_sig ? -1 : 1;
164 return inherited::compare_same_type(other);
167 bool minkmetric::info(unsigned inf) const
169 if(inf == info_flags::real)
175 int tensepsilon::compare_same_type(const basic & other) const
177 GINAC_ASSERT(is_a<tensepsilon>(other));
178 const tensepsilon &o = static_cast<const tensepsilon &>(other);
180 if (minkowski != o.minkowski)
181 return minkowski ? -1 : 1;
182 else if (pos_sig != o.pos_sig)
183 return pos_sig ? -1 : 1;
185 return inherited::compare_same_type(other);
188 bool tensepsilon::info(unsigned inf) const
190 if(inf == info_flags::real)
196 bool spinmetric::info(unsigned inf) const
198 if(inf == info_flags::real)
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")
210 /** Automatic symbolic evaluation of an indexed delta tensor. */
211 ex tensdelta::eval_indexed(const basic & i) const
213 GINAC_ASSERT(is_a<indexed>(i));
214 GINAC_ASSERT(i.nops() == 3);
215 GINAC_ASSERT(is_a<tensdelta>(i.op(0)));
217 const idx & i1 = ex_to<idx>(i.op(1));
218 const idx & i2 = ex_to<idx>(i.op(2));
220 // The dimension of the indices must be equal, otherwise we use the minimal
222 if (!i1.get_dim().is_equal(i2.get_dim())) {
223 ex min_dim = i1.minimal_dim(i2);
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);
230 // Trace of delta tensor is the (effective) dimension of the space
231 if (is_dummy_pair(i1, i2)) {
233 return i1.minimal_dim(i2);
234 } catch (std::exception &e) {
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();
248 // No further simplifications
252 /** Automatic symbolic evaluation of an indexed metric tensor. */
253 ex tensmetric::eval_indexed(const basic & i) const
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)));
261 const varidx & i1 = ex_to<varidx>(i.op(1));
262 const varidx & i2 = ex_to<varidx>(i.op(2));
264 // The dimension of the indices must be equal, otherwise we use the minimal
266 if (!i1.get_dim().is_equal(i2.get_dim())) {
267 ex min_dim = i1.minimal_dim(i2);
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);
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);
279 // No further simplifications
283 /** Automatic symbolic evaluation of an indexed Lorentz metric tensor. */
284 ex minkmetric::eval_indexed(const basic & i) const
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)));
292 const varidx & i1 = ex_to<varidx>(i.op(1));
293 const varidx & i2 = ex_to<varidx>(i.op(2));
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();
301 return pos_sig ? _ex_1 : _ex1;
303 return pos_sig ? _ex1 : _ex_1;
306 // Perform the usual evaluations of a metric tensor
307 return inherited::eval_indexed(i);
310 /** Automatic symbolic evaluation of an indexed metric tensor. */
311 ex spinmetric::eval_indexed(const basic & i) const
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)));
319 const spinidx & i1 = ex_to<spinidx>(i.op(1));
320 const spinidx & i2 = ex_to<spinidx>(i.op(2));
322 // Convolutions are zero
323 if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
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();
337 // No further simplifications
341 /** Automatic symbolic evaluation of an indexed epsilon tensor. */
342 ex tensepsilon::eval_indexed(const basic & i) const
344 GINAC_ASSERT(is_a<indexed>(i));
345 GINAC_ASSERT(i.nops() > 1);
346 GINAC_ASSERT(is_a<tensepsilon>(i.op(0)));
348 // Convolutions are zero
349 if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
352 // Numeric evaluation
353 if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
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)
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());
363 // In a Minkowski space, check for covariant indices
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 if (ex_to<varidx>(x).is_covariant())
370 if (ex_to<idx>(x).get_value().is_zero())
371 sign = (pos_sig ? -sign : sign);
373 sign = (pos_sig ? sign : -sign);
380 // No further simplifications
384 bool tensor::replace_contr_index(exvector::iterator self, exvector::iterator other) const
386 // Try to contract the 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;
392 if (self_idx->is_symbolic()) {
393 for (size_t i=1; i<other->nops(); i++) {
394 if (! is_a<idx>(other->op(i)))
396 const idx &other_idx = ex_to<idx>(other->op(i));
397 if (is_dummy_pair(*self_idx, other_idx)) {
399 // Contraction found, remove this tensor and substitute the
400 // index in the second object
402 // minimal_dim() throws an exception when index dimensions are not comparable
403 ex min_dim = self_idx->minimal_dim(other_idx);
404 *other = other->subs(other_idx == free_idx->replace_dim(min_dim));
405 *self = _ex1; // *other is assigned first because assigning *self invalidates free_idx
407 } catch (std::exception &e) {
414 if (!first_index_tried) {
416 // No contraction with the first index found, try the second index
417 self_idx = &ex_to<idx>(self->op(2));
418 free_idx = &ex_to<idx>(self->op(1));
419 first_index_tried = true;
426 /** Contraction of an indexed delta tensor with something else. */
427 bool tensdelta::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
429 GINAC_ASSERT(is_a<indexed>(*self));
430 GINAC_ASSERT(is_a<indexed>(*other));
431 GINAC_ASSERT(self->nops() == 3);
432 GINAC_ASSERT(is_a<tensdelta>(self->op(0)));
434 // Replace the dummy index with this tensor's other index and remove
435 // the tensor (this is valid for contractions with all other tensors)
436 return replace_contr_index(self, other);
439 /** Contraction of an indexed metric tensor with something else. */
440 bool tensmetric::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
442 GINAC_ASSERT(is_a<indexed>(*self));
443 GINAC_ASSERT(is_a<indexed>(*other));
444 GINAC_ASSERT(self->nops() == 3);
445 GINAC_ASSERT(is_a<tensmetric>(self->op(0)));
447 // If contracting with the delta tensor, let the delta do it
448 // (don't raise/lower delta indices)
449 if (is_a<tensdelta>(other->op(0)))
452 // Replace the dummy index with this tensor's other index and remove
454 return replace_contr_index(self, other);
457 /** Contraction of an indexed spinor metric with something else. */
458 bool spinmetric::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
460 GINAC_ASSERT(is_a<indexed>(*self));
461 GINAC_ASSERT(is_a<indexed>(*other));
462 GINAC_ASSERT(self->nops() == 3);
463 GINAC_ASSERT(is_a<spinmetric>(self->op(0)));
465 // Contractions between spinor metrics
466 if (is_a<spinmetric>(other->op(0))) {
467 const idx &self_i1 = ex_to<idx>(self->op(1));
468 const idx &self_i2 = ex_to<idx>(self->op(2));
469 const idx &other_i1 = ex_to<idx>(other->op(1));
470 const idx &other_i2 = ex_to<idx>(other->op(2));
472 if (is_dummy_pair(self_i1, other_i1)) {
473 if (is_dummy_pair(self_i2, other_i2))
476 *self = delta_tensor(self_i2, other_i2);
479 } else if (is_dummy_pair(self_i1, other_i2)) {
480 if (is_dummy_pair(self_i2, other_i1))
483 *self = -delta_tensor(self_i2, other_i1);
486 } else if (is_dummy_pair(self_i2, other_i1)) {
487 *self = -delta_tensor(self_i1, other_i2);
490 } else if (is_dummy_pair(self_i2, other_i2)) {
491 *self = delta_tensor(self_i1, other_i1);
497 // If contracting with the delta tensor, let the delta do it
498 // (don't raise/lower delta indices)
499 if (is_a<tensdelta>(other->op(0)))
502 // Try to contract first index
503 const idx *self_idx = &ex_to<idx>(self->op(1));
504 const idx *free_idx = &ex_to<idx>(self->op(2));
505 bool first_index_tried = false;
509 if (self_idx->is_symbolic()) {
510 for (size_t i=1; i<other->nops(); i++) {
511 const idx &other_idx = ex_to<idx>(other->op(i));
512 if (is_dummy_pair(*self_idx, other_idx)) {
514 // Contraction found, remove metric tensor and substitute
515 // index in second object (assign *self last because this
516 // invalidates free_idx)
517 *other = other->subs(other_idx == *free_idx);
518 *self = (static_cast<const spinidx *>(self_idx)->is_covariant() ? sign : -sign);
524 if (!first_index_tried) {
526 // No contraction with first index found, try second index
527 self_idx = &ex_to<idx>(self->op(2));
528 free_idx = &ex_to<idx>(self->op(1));
529 first_index_tried = true;
537 /** Contraction of epsilon tensor with something else. */
538 bool tensepsilon::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
540 GINAC_ASSERT(is_a<indexed>(*self));
541 GINAC_ASSERT(is_a<indexed>(*other));
542 GINAC_ASSERT(is_a<tensepsilon>(self->op(0)));
543 size_t num = self->nops() - 1;
545 if (is_exactly_a<tensepsilon>(other->op(0)) && num+1 == other->nops()) {
547 // Contraction of two epsilon tensors is a determinant
548 bool variance = is_a<varidx>(self->op(1));
550 for (size_t i=0; i<num; i++) {
551 for (size_t j=0; j<num; j++) {
553 M(i, j) = lorentz_g(self->op(i+1), other->op(j+1), pos_sig);
555 M(i, j) = metric_tensor(self->op(i+1), other->op(j+1));
557 M(i, j) = delta_tensor(self->op(i+1), other->op(j+1));
560 int sign = minkowski ? -1 : 1;
561 *self = sign * M.determinant().simplify_indexed();
573 ex delta_tensor(const ex & i1, const ex & i2)
575 static ex delta = (new tensdelta)->setflag(status_flags::dynallocated);
577 if (!is_a<idx>(i1) || !is_a<idx>(i2))
578 throw(std::invalid_argument("indices of delta tensor must be of type idx"));
580 return indexed(delta, symmetric2(), i1, i2);
583 ex metric_tensor(const ex & i1, const ex & i2)
585 static ex metric = (new tensmetric)->setflag(status_flags::dynallocated);
587 if (!is_a<varidx>(i1) || !is_a<varidx>(i2))
588 throw(std::invalid_argument("indices of metric tensor must be of type varidx"));
590 return indexed(metric, symmetric2(), i1, i2);
593 ex lorentz_g(const ex & i1, const ex & i2, bool pos_sig)
595 static ex metric_neg = (new minkmetric(false))->setflag(status_flags::dynallocated);
596 static ex metric_pos = (new minkmetric(true))->setflag(status_flags::dynallocated);
598 if (!is_a<varidx>(i1) || !is_a<varidx>(i2))
599 throw(std::invalid_argument("indices of metric tensor must be of type varidx"));
601 return indexed(pos_sig ? metric_pos : metric_neg, symmetric2(), i1, i2);
604 ex spinor_metric(const ex & i1, const ex & i2)
606 static ex metric = (new spinmetric)->setflag(status_flags::dynallocated);
608 if (!is_a<spinidx>(i1) || !is_a<spinidx>(i2))
609 throw(std::invalid_argument("indices of spinor metric must be of type spinidx"));
610 if (!ex_to<idx>(i1).get_dim().is_equal(2) || !ex_to<idx>(i2).get_dim().is_equal(2))
611 throw(std::runtime_error("index dimension for spinor metric must be 2"));
613 return indexed(metric, antisymmetric2(), i1, i2);
616 ex epsilon_tensor(const ex & i1, const ex & i2)
618 static ex epsilon = (new tensepsilon)->setflag(status_flags::dynallocated);
620 if (!is_a<idx>(i1) || !is_a<idx>(i2))
621 throw(std::invalid_argument("indices of epsilon tensor must be of type idx"));
623 ex dim = ex_to<idx>(i1).get_dim();
624 if (!dim.is_equal(ex_to<idx>(i2).get_dim()))
625 throw(std::invalid_argument("all indices of epsilon tensor must have the same dimension"));
626 if (!ex_to<idx>(i1).get_dim().is_equal(_ex2))
627 throw(std::runtime_error("index dimension of epsilon tensor must match number of indices"));
629 if(is_a<wildcard>(i1.op(0))||is_a<wildcard>(i2.op(0)))
630 return indexed(epsilon, antisymmetric2(), i1, i2).hold();
632 return indexed(epsilon, antisymmetric2(), i1, i2);
635 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3)
637 static ex epsilon = (new tensepsilon)->setflag(status_flags::dynallocated);
639 if (!is_a<idx>(i1) || !is_a<idx>(i2) || !is_a<idx>(i3))
640 throw(std::invalid_argument("indices of epsilon tensor must be of type idx"));
642 ex dim = ex_to<idx>(i1).get_dim();
643 if (!dim.is_equal(ex_to<idx>(i2).get_dim()) || !dim.is_equal(ex_to<idx>(i3).get_dim()))
644 throw(std::invalid_argument("all indices of epsilon tensor must have the same dimension"));
645 if (!ex_to<idx>(i1).get_dim().is_equal(_ex3))
646 throw(std::runtime_error("index dimension of epsilon tensor must match number of indices"));
648 if(is_a<wildcard>(i1.op(0))||is_a<wildcard>(i2.op(0))||is_a<wildcard>(i3.op(0)))
649 return indexed(epsilon, antisymmetric3(), i1, i2, i3).hold();
651 return indexed(epsilon, antisymmetric3(), i1, i2, i3);
654 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig)
656 static ex epsilon_neg = (new tensepsilon(true, false))->setflag(status_flags::dynallocated);
657 static ex epsilon_pos = (new tensepsilon(true, true))->setflag(status_flags::dynallocated);
659 if (!is_a<varidx>(i1) || !is_a<varidx>(i2) || !is_a<varidx>(i3) || !is_a<varidx>(i4))
660 throw(std::invalid_argument("indices of Lorentz epsilon tensor must be of type varidx"));
662 ex dim = ex_to<idx>(i1).get_dim();
663 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()))
664 throw(std::invalid_argument("all indices of epsilon tensor must have the same dimension"));
665 if (!ex_to<idx>(i1).get_dim().is_equal(_ex4))
666 throw(std::runtime_error("index dimension of epsilon tensor must match number of indices"));
668 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)))
669 return indexed(pos_sig ? epsilon_pos : epsilon_neg, antisymmetric4(), i1, i2, i3, i4).hold();
671 return indexed(pos_sig ? epsilon_pos : epsilon_neg, antisymmetric4(), i1, i2, i3, i4);