3 * Implementation of GiNaC's special tensors. */
6 * GiNaC Copyright (C) 1999-2001 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
30 #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 constructor, destructor, copy constructor assignment operator and helpers
52 tensor::tensor(unsigned ti) : inherited(ti)
54 debugmsg("tensor constructor from unsigned", LOGLEVEL_CONSTRUCT); \
58 DEFAULT_CTORS(tensdelta)
59 DEFAULT_CTORS(tensmetric)
60 DEFAULT_COPY(spinmetric)
61 DEFAULT_DESTROY(spinmetric)
62 DEFAULT_DESTROY(minkmetric)
63 DEFAULT_DESTROY(tensepsilon)
65 minkmetric::minkmetric() : pos_sig(false)
67 debugmsg("minkmetric default constructor", LOGLEVEL_CONSTRUCT);
68 tinfo_key = TINFO_minkmetric;
71 spinmetric::spinmetric()
73 debugmsg("spinmetric default constructor", LOGLEVEL_CONSTRUCT);
74 tinfo_key = TINFO_spinmetric;
77 minkmetric::minkmetric(bool ps) : pos_sig(ps)
79 debugmsg("minkmetric constructor from bool", LOGLEVEL_CONSTRUCT);
80 tinfo_key = TINFO_minkmetric;
83 void minkmetric::copy(const minkmetric & other)
85 inherited::copy(other);
86 pos_sig = other.pos_sig;
89 tensepsilon::tensepsilon() : minkowski(false), pos_sig(false)
91 debugmsg("tensepsilon default constructor", LOGLEVEL_CONSTRUCT);
92 tinfo_key = TINFO_tensepsilon;
95 tensepsilon::tensepsilon(bool mink, bool ps) : minkowski(mink), pos_sig(ps)
97 debugmsg("tensepsilon constructor from bool,bool", LOGLEVEL_CONSTRUCT);
98 tinfo_key = TINFO_tensepsilon;
101 void tensepsilon::copy(const tensepsilon & other)
103 inherited::copy(other);
104 minkowski = other.minkowski;
105 pos_sig = other.pos_sig;
112 DEFAULT_ARCHIVING(tensor)
113 DEFAULT_ARCHIVING(tensdelta)
114 DEFAULT_ARCHIVING(tensmetric)
115 DEFAULT_ARCHIVING(spinmetric)
116 DEFAULT_UNARCHIVE(minkmetric)
117 DEFAULT_UNARCHIVE(tensepsilon)
119 minkmetric::minkmetric(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
121 debugmsg("minkmetric constructor from archive_node", LOGLEVEL_CONSTRUCT);
122 n.find_bool("pos_sig", pos_sig);
125 void minkmetric::archive(archive_node &n) const
127 inherited::archive(n);
128 n.add_bool("pos_sig", pos_sig);
131 tensepsilon::tensepsilon(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
133 debugmsg("tensepsilon constructor from archive_node", LOGLEVEL_CONSTRUCT);
134 n.find_bool("minkowski", minkowski);
135 n.find_bool("pos_sig", pos_sig);
138 void tensepsilon::archive(archive_node &n) const
140 inherited::archive(n);
141 n.add_bool("minkowski", minkowski);
142 n.add_bool("pos_sig", pos_sig);
146 // functions overriding virtual functions from base classes
149 DEFAULT_COMPARE(tensor)
150 DEFAULT_COMPARE(tensdelta)
151 DEFAULT_COMPARE(tensmetric)
152 DEFAULT_COMPARE(spinmetric)
154 int minkmetric::compare_same_type(const basic & other) const
156 GINAC_ASSERT(is_of_type(other, minkmetric));
157 const minkmetric &o = static_cast<const minkmetric &>(other);
159 if (pos_sig != o.pos_sig)
160 return pos_sig ? -1 : 1;
162 return inherited::compare_same_type(other);
165 int tensepsilon::compare_same_type(const basic & other) const
167 GINAC_ASSERT(is_of_type(other, tensepsilon));
168 const tensepsilon &o = static_cast<const tensepsilon &>(other);
170 if (minkowski != o.minkowski)
171 return minkowski ? -1 : 1;
172 else if (pos_sig != o.pos_sig)
173 return pos_sig ? -1 : 1;
175 return inherited::compare_same_type(other);
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")
184 /** Automatic symbolic evaluation of an indexed delta tensor. */
185 ex tensdelta::eval_indexed(const basic & i) const
187 GINAC_ASSERT(is_of_type(i, indexed));
188 GINAC_ASSERT(i.nops() == 3);
189 GINAC_ASSERT(is_ex_of_type(i.op(0), tensdelta));
191 const idx & i1 = ex_to<idx>(i.op(1));
192 const idx & i2 = ex_to<idx>(i.op(2));
194 // Trace of delta tensor is the dimension of the space
195 if (is_dummy_pair(i1, i2))
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_of_type(i, indexed));
215 GINAC_ASSERT(i.nops() == 3);
216 GINAC_ASSERT(is_ex_of_type(i.op(0), tensmetric));
217 GINAC_ASSERT(is_ex_of_type(i.op(1), varidx));
218 GINAC_ASSERT(is_ex_of_type(i.op(2), varidx));
220 const varidx & i1 = ex_to<varidx>(i.op(1));
221 const varidx & i2 = ex_to<varidx>(i.op(2));
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);
228 // No further simplifications
232 /** Automatic symbolic evaluation of an indexed Lorentz metric tensor. */
233 ex minkmetric::eval_indexed(const basic & i) const
235 GINAC_ASSERT(is_of_type(i, indexed));
236 GINAC_ASSERT(i.nops() == 3);
237 GINAC_ASSERT(is_ex_of_type(i.op(0), minkmetric));
238 GINAC_ASSERT(is_ex_of_type(i.op(1), varidx));
239 GINAC_ASSERT(is_ex_of_type(i.op(2), varidx));
241 const varidx & i1 = ex_to<varidx>(i.op(1));
242 const varidx & i2 = ex_to<varidx>(i.op(2));
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();
250 return pos_sig ? _ex_1() : _ex1();
252 return pos_sig ? _ex1() : _ex_1();
255 // Perform the usual evaluations of a metric tensor
256 return inherited::eval_indexed(i);
259 /** Automatic symbolic evaluation of an indexed metric tensor. */
260 ex spinmetric::eval_indexed(const basic & i) const
262 GINAC_ASSERT(is_of_type(i, indexed));
263 GINAC_ASSERT(i.nops() == 3);
264 GINAC_ASSERT(is_ex_of_type(i.op(0), spinmetric));
265 GINAC_ASSERT(is_ex_of_type(i.op(1), spinidx));
266 GINAC_ASSERT(is_ex_of_type(i.op(2), spinidx));
268 const spinidx & i1 = ex_to<spinidx>(i.op(1));
269 const spinidx & i2 = ex_to<spinidx>(i.op(2));
271 // Convolutions are zero
272 if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
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();
286 // No further simplifications
290 /** Automatic symbolic evaluation of an indexed epsilon tensor. */
291 ex tensepsilon::eval_indexed(const basic & i) const
293 GINAC_ASSERT(is_of_type(i, indexed));
294 GINAC_ASSERT(i.nops() > 1);
295 GINAC_ASSERT(is_ex_of_type(i.op(0), tensepsilon));
297 // Convolutions are zero
298 if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
301 // Numeric evaluation
302 if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
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)
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());
312 // In a Minkowski space, check for covariant indices
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);
322 sign = (pos_sig ? sign : -sign);
329 // No further simplifications
333 /** Contraction of an indexed delta tensor with something else. */
334 bool tensdelta::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
336 GINAC_ASSERT(is_ex_of_type(*self, indexed));
337 GINAC_ASSERT(is_ex_of_type(*other, indexed));
338 GINAC_ASSERT(self->nops() == 3);
339 GINAC_ASSERT(is_ex_of_type(self->op(0), tensdelta));
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;
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)) {
352 // Contraction found, remove delta tensor and substitute
353 // index in second object
355 *other = other->subs(other_idx == *free_idx);
361 if (!first_index_tried) {
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;
373 /** Contraction of an indexed metric tensor with something else. */
374 bool tensmetric::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
376 GINAC_ASSERT(is_ex_of_type(*self, indexed));
377 GINAC_ASSERT(is_ex_of_type(*other, indexed));
378 GINAC_ASSERT(self->nops() == 3);
379 GINAC_ASSERT(is_ex_of_type(self->op(0), tensmetric));
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))
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;
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)) {
397 // Contraction found, remove metric tensor and substitute
398 // index in second object
400 *other = other->subs(other_idx == *free_idx);
406 if (!first_index_tried) {
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;
418 /** Contraction of an indexed spinor metric with something else. */
419 bool spinmetric::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
421 GINAC_ASSERT(is_ex_of_type(*self, indexed));
422 GINAC_ASSERT(is_ex_of_type(*other, indexed));
423 GINAC_ASSERT(self->nops() == 3);
424 GINAC_ASSERT(is_ex_of_type(self->op(0), spinmetric));
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));
433 if (is_dummy_pair(self_i1, other_i1)) {
434 if (is_dummy_pair(self_i2, other_i2))
437 *self = delta_tensor(self_i2, other_i2);
440 } else if (is_dummy_pair(self_i1, other_i2)) {
441 if (is_dummy_pair(self_i2, other_i1))
444 *self = -delta_tensor(self_i2, other_i1);
447 } else if (is_dummy_pair(self_i2, other_i1)) {
448 *self = -delta_tensor(self_i1, other_i2);
451 } else if (is_dummy_pair(self_i2, other_i2)) {
452 *self = delta_tensor(self_i1, other_i1);
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))
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;
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)) {
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);
484 if (!first_index_tried) {
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;
497 /** Contraction of epsilon tensor with something else. */
498 bool tensepsilon::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
500 GINAC_ASSERT(is_ex_of_type(*self, indexed));
501 GINAC_ASSERT(is_ex_of_type(*other, indexed));
502 GINAC_ASSERT(is_ex_of_type(self->op(0), tensepsilon));
503 unsigned num = self->nops() - 1;
505 if (is_ex_exactly_of_type(other->op(0), tensepsilon) && num+1 == other->nops()) {
507 // Contraction of two epsilon tensors is a determinant
508 ex dim = ex_to<idx>(self->op(1)).get_dim();
510 for (int i=0; i<num; i++) {
511 for (int j=0; j<num; j++) {
513 M(i, j) = lorentz_g(self->op(i+1), other->op(j+1), pos_sig);
515 M(i, j) = metric_tensor(self->op(i+1), other->op(j+1));
518 int sign = minkowski ? -1 : 1;
519 *self = sign * M.determinant().simplify_indexed();
523 } else if (other->return_type() == return_types::commutative) {
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...
531 // Handle all indices of the epsilon tensor
532 for (int i=0; i<num; i++) {
533 ex idx = self->op(i+1);
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++) {
540 if (is_a<indexed>(*ait) && ait->return_type() == return_types::commutative && ex_to<indexed>(*ait).has_dummy_index_for(idx) && ait->nops() == 2) {
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()) {
546 // No, add the base expression to the list
551 // Yes, the contraction is zero
569 ex delta_tensor(const ex & i1, const ex & i2)
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"));
574 return indexed(tensdelta(), sy_symm(), i1, i2);
577 ex metric_tensor(const ex & i1, const ex & i2)
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"));
582 return indexed(tensmetric(), sy_symm(), i1, i2);
585 ex lorentz_g(const ex & i1, const ex & i2, bool pos_sig)
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"));
590 return indexed(minkmetric(pos_sig), sy_symm(), i1, i2);
593 ex spinor_metric(const ex & i1, const ex & i2)
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"));
600 return indexed(spinmetric(), sy_anti(), i1, i2);
603 ex epsilon_tensor(const ex & i1, const ex & i2)
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"));
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"));
614 return indexed(tensepsilon(), sy_anti(), i1, i2);
617 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3)
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"));
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"));
628 return indexed(tensepsilon(), sy_anti(), i1, i2, i3);
631 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig)
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"));
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"));
642 return indexed(tensepsilon(true, pos_sig), sy_anti(), i1, i2, i3, i4);
645 ex eps0123(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig)
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"));
650 ex dim = ex_to<idx>(i1).get_dim();
652 return lorentz_eps(i1, i2, i3, i4, pos_sig);
654 return indexed(tensepsilon(true, pos_sig), sy_anti(), i1, i2, i3, i4);