3 * Implementation of GiNaC's special tensors. */
6 * GiNaC Copyright (C) 1999-2002 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 // Trace of delta tensor is the dimension of the space
183 if (is_dummy_pair(i1, i2))
186 // Numeric evaluation
187 if (static_cast<const indexed &>(i).all_index_values_are(info_flags::integer)) {
188 int n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
195 // No further simplifications
199 /** Automatic symbolic evaluation of an indexed metric tensor. */
200 ex tensmetric::eval_indexed(const basic & i) const
202 GINAC_ASSERT(is_a<indexed>(i));
203 GINAC_ASSERT(i.nops() == 3);
204 GINAC_ASSERT(is_a<tensmetric>(i.op(0)));
205 GINAC_ASSERT(is_a<varidx>(i.op(1)));
206 GINAC_ASSERT(is_a<varidx>(i.op(2)));
208 const varidx & i1 = ex_to<varidx>(i.op(1));
209 const varidx & i2 = ex_to<varidx>(i.op(2));
211 // A metric tensor with one covariant and one contravariant index gets
212 // replaced by a delta tensor
213 if (i1.is_covariant() != i2.is_covariant())
214 return delta_tensor(i1, i2);
216 // No further simplifications
220 /** Automatic symbolic evaluation of an indexed Lorentz metric tensor. */
221 ex minkmetric::eval_indexed(const basic & i) const
223 GINAC_ASSERT(is_a<indexed>(i));
224 GINAC_ASSERT(i.nops() == 3);
225 GINAC_ASSERT(is_a<minkmetric>(i.op(0)));
226 GINAC_ASSERT(is_a<varidx>(i.op(1)));
227 GINAC_ASSERT(is_a<varidx>(i.op(2)));
229 const varidx & i1 = ex_to<varidx>(i.op(1));
230 const varidx & i2 = ex_to<varidx>(i.op(2));
232 // Numeric evaluation
233 if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
234 int n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
238 return pos_sig ? _ex_1 : _ex1;
240 return pos_sig ? _ex1 : _ex_1;
243 // Perform the usual evaluations of a metric tensor
244 return inherited::eval_indexed(i);
247 /** Automatic symbolic evaluation of an indexed metric tensor. */
248 ex spinmetric::eval_indexed(const basic & i) const
250 GINAC_ASSERT(is_a<indexed>(i));
251 GINAC_ASSERT(i.nops() == 3);
252 GINAC_ASSERT(is_a<spinmetric>(i.op(0)));
253 GINAC_ASSERT(is_a<spinidx>(i.op(1)));
254 GINAC_ASSERT(is_a<spinidx>(i.op(2)));
256 const spinidx & i1 = ex_to<spinidx>(i.op(1));
257 const spinidx & i2 = ex_to<spinidx>(i.op(2));
259 // Convolutions are zero
260 if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
263 // Numeric evaluation
264 if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
265 int n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
274 // No further simplifications
278 /** Automatic symbolic evaluation of an indexed epsilon tensor. */
279 ex tensepsilon::eval_indexed(const basic & i) const
281 GINAC_ASSERT(is_a<indexed>(i));
282 GINAC_ASSERT(i.nops() > 1);
283 GINAC_ASSERT(is_a<tensepsilon>(i.op(0)));
285 // Convolutions are zero
286 if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
289 // Numeric evaluation
290 if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
292 // Get sign of index permutation (the indices should already be in
293 // a canonic order but we can't assume what exactly that order is)
295 v.reserve(i.nops() - 1);
296 for (unsigned j=1; j<i.nops(); j++)
297 v.push_back(ex_to<numeric>(ex_to<idx>(i.op(j)).get_value()).to_int());
298 int sign = permutation_sign(v.begin(), v.end());
300 // In a Minkowski space, check for covariant indices
302 for (unsigned j=1; j<i.nops(); j++) {
303 const ex & x = i.op(j);
304 if (!is_ex_of_type(x, varidx))
305 throw(std::runtime_error("indices of epsilon tensor in Minkowski space must be of type varidx"));
306 if (ex_to<varidx>(x).is_covariant())
307 if (ex_to<idx>(x).get_value().is_zero())
308 sign = (pos_sig ? -sign : sign);
310 sign = (pos_sig ? sign : -sign);
317 // No further simplifications
321 /** Contraction of an indexed delta tensor with something else. */
322 bool tensdelta::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
324 GINAC_ASSERT(is_a<indexed>(*self));
325 GINAC_ASSERT(is_a<indexed>(*other));
326 GINAC_ASSERT(self->nops() == 3);
327 GINAC_ASSERT(is_a<tensdelta>(self->op(0)));
329 // Try to contract first index
330 const idx *self_idx = &ex_to<idx>(self->op(1));
331 const idx *free_idx = &ex_to<idx>(self->op(2));
332 bool first_index_tried = false;
335 if (self_idx->is_symbolic()) {
336 for (unsigned i=1; i<other->nops(); i++) {
337 const idx &other_idx = ex_to<idx>(other->op(i));
338 if (is_dummy_pair(*self_idx, other_idx)) {
340 // Contraction found, remove delta tensor and substitute
341 // index in second object
343 *other = other->subs(other_idx == *free_idx);
349 if (!first_index_tried) {
351 // No contraction with first index found, try second index
352 self_idx = &ex_to<idx>(self->op(2));
353 free_idx = &ex_to<idx>(self->op(1));
354 first_index_tried = true;
361 /** Contraction of an indexed metric tensor with something else. */
362 bool tensmetric::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
364 GINAC_ASSERT(is_a<indexed>(*self));
365 GINAC_ASSERT(is_a<indexed>(*other));
366 GINAC_ASSERT(self->nops() == 3);
367 GINAC_ASSERT(is_a<tensmetric>(self->op(0)));
369 // If contracting with the delta tensor, let the delta do it
370 // (don't raise/lower delta indices)
371 if (is_ex_of_type(other->op(0), tensdelta))
374 // Try to contract first index
375 const idx *self_idx = &ex_to<idx>(self->op(1));
376 const idx *free_idx = &ex_to<idx>(self->op(2));
377 bool first_index_tried = false;
380 if (self_idx->is_symbolic()) {
381 for (unsigned i=1; i<other->nops(); i++) {
382 const idx &other_idx = ex_to<idx>(other->op(i));
383 if (is_dummy_pair(*self_idx, other_idx)) {
385 // Contraction found, remove metric tensor and substitute
386 // index in second object
388 *other = other->subs(other_idx == *free_idx);
394 if (!first_index_tried) {
396 // No contraction with first index found, try second index
397 self_idx = &ex_to<idx>(self->op(2));
398 free_idx = &ex_to<idx>(self->op(1));
399 first_index_tried = true;
406 /** Contraction of an indexed spinor metric with something else. */
407 bool spinmetric::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
409 GINAC_ASSERT(is_a<indexed>(*self));
410 GINAC_ASSERT(is_a<indexed>(*other));
411 GINAC_ASSERT(self->nops() == 3);
412 GINAC_ASSERT(is_a<spinmetric>(self->op(0)));
414 // Contractions between spinor metrics
415 if (is_ex_of_type(other->op(0), spinmetric)) {
416 const idx &self_i1 = ex_to<idx>(self->op(1));
417 const idx &self_i2 = ex_to<idx>(self->op(2));
418 const idx &other_i1 = ex_to<idx>(other->op(1));
419 const idx &other_i2 = ex_to<idx>(other->op(2));
421 if (is_dummy_pair(self_i1, other_i1)) {
422 if (is_dummy_pair(self_i2, other_i2))
425 *self = delta_tensor(self_i2, other_i2);
428 } else if (is_dummy_pair(self_i1, other_i2)) {
429 if (is_dummy_pair(self_i2, other_i1))
432 *self = -delta_tensor(self_i2, other_i1);
435 } else if (is_dummy_pair(self_i2, other_i1)) {
436 *self = -delta_tensor(self_i1, other_i2);
439 } else if (is_dummy_pair(self_i2, other_i2)) {
440 *self = delta_tensor(self_i1, other_i1);
446 // If contracting with the delta tensor, let the delta do it
447 // (don't raise/lower delta indices)
448 if (is_ex_of_type(other->op(0), tensdelta))
451 // Try to contract first index
452 const idx *self_idx = &ex_to<idx>(self->op(1));
453 const idx *free_idx = &ex_to<idx>(self->op(2));
454 bool first_index_tried = false;
458 if (self_idx->is_symbolic()) {
459 for (unsigned i=1; i<other->nops(); i++) {
460 const idx &other_idx = ex_to<idx>(other->op(i));
461 if (is_dummy_pair(*self_idx, other_idx)) {
463 // Contraction found, remove metric tensor and substitute
464 // index in second object
465 *self = (static_cast<const spinidx *>(self_idx)->is_covariant() ? sign : -sign);
466 *other = other->subs(other_idx == *free_idx);
472 if (!first_index_tried) {
474 // No contraction with first index found, try second index
475 self_idx = &ex_to<idx>(self->op(2));
476 free_idx = &ex_to<idx>(self->op(1));
477 first_index_tried = true;
485 /** Contraction of epsilon tensor with something else. */
486 bool tensepsilon::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
488 GINAC_ASSERT(is_a<indexed>(*self));
489 GINAC_ASSERT(is_a<indexed>(*other));
490 GINAC_ASSERT(is_a<tensepsilon>(self->op(0)));
491 unsigned num = self->nops() - 1;
493 if (is_ex_exactly_of_type(other->op(0), tensepsilon) && num+1 == other->nops()) {
495 // Contraction of two epsilon tensors is a determinant
496 ex dim = ex_to<idx>(self->op(1)).get_dim();
498 for (int i=0; i<num; i++) {
499 for (int j=0; j<num; j++) {
501 M(i, j) = lorentz_g(self->op(i+1), other->op(j+1), pos_sig);
503 M(i, j) = metric_tensor(self->op(i+1), other->op(j+1));
506 int sign = minkowski ? -1 : 1;
507 *self = sign * M.determinant().simplify_indexed();
511 } else if (other->return_type() == return_types::commutative) {
514 // This handles eps.i.j.k * p.j * p.k = 0
515 // Maybe something like this should go to simplify_indexed() because
516 // such relations are true for any antisymmetric tensors...
519 // Handle all indices of the epsilon tensor
520 for (int i=0; i<num; i++) {
521 ex idx = self->op(i+1);
523 // Look whether there's a contraction with this index
524 exvector::const_iterator ait, aitend = v.end();
525 for (ait = v.begin(); ait != aitend; ait++) {
528 if (is_a<indexed>(*ait) && ait->return_type() == return_types::commutative && ex_to<indexed>(*ait).has_dummy_index_for(idx) && ait->nops() == 2) {
530 // Yes, did we already have another contraction with the same base expression?
531 ex base = ait->op(0);
532 if (std::find_if(c.begin(), c.end(), bind2nd(ex_is_equal(), base)) == c.end()) {
534 // No, add the base expression to the list
539 // Yes, the contraction is zero
557 ex delta_tensor(const ex & i1, const ex & i2)
559 if (!is_ex_of_type(i1, idx) || !is_ex_of_type(i2, idx))
560 throw(std::invalid_argument("indices of delta tensor must be of type idx"));
562 return indexed(tensdelta(), sy_symm(), i1, i2);
565 ex metric_tensor(const ex & i1, const ex & i2)
567 if (!is_ex_of_type(i1, varidx) || !is_ex_of_type(i2, varidx))
568 throw(std::invalid_argument("indices of metric tensor must be of type varidx"));
570 return indexed(tensmetric(), sy_symm(), i1, i2);
573 ex lorentz_g(const ex & i1, const ex & i2, bool pos_sig)
575 if (!is_ex_of_type(i1, varidx) || !is_ex_of_type(i2, varidx))
576 throw(std::invalid_argument("indices of metric tensor must be of type varidx"));
578 return indexed(minkmetric(pos_sig), sy_symm(), i1, i2);
581 ex spinor_metric(const ex & i1, const ex & i2)
583 if (!is_ex_of_type(i1, spinidx) || !is_ex_of_type(i2, spinidx))
584 throw(std::invalid_argument("indices of spinor metric must be of type spinidx"));
585 if (!ex_to<idx>(i1).get_dim().is_equal(2) || !ex_to<idx>(i2).get_dim().is_equal(2))
586 throw(std::runtime_error("index dimension for spinor metric must be 2"));
588 return indexed(spinmetric(), sy_anti(), i1, i2);
591 ex epsilon_tensor(const ex & i1, const ex & i2)
593 if (!is_ex_of_type(i1, idx) || !is_ex_of_type(i2, idx))
594 throw(std::invalid_argument("indices of epsilon tensor must be of type idx"));
596 ex dim = ex_to<idx>(i1).get_dim();
597 if (!dim.is_equal(ex_to<idx>(i2).get_dim()))
598 throw(std::invalid_argument("all indices of epsilon tensor must have the same dimension"));
599 if (!ex_to<idx>(i1).get_dim().is_equal(_ex2))
600 throw(std::runtime_error("index dimension of epsilon tensor must match number of indices"));
602 return indexed(tensepsilon(), sy_anti(), i1, i2);
605 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3)
607 if (!is_ex_of_type(i1, idx) || !is_ex_of_type(i2, idx) || !is_ex_of_type(i3, idx))
608 throw(std::invalid_argument("indices of epsilon tensor must be of type idx"));
610 ex dim = ex_to<idx>(i1).get_dim();
611 if (!dim.is_equal(ex_to<idx>(i2).get_dim()) || !dim.is_equal(ex_to<idx>(i3).get_dim()))
612 throw(std::invalid_argument("all indices of epsilon tensor must have the same dimension"));
613 if (!ex_to<idx>(i1).get_dim().is_equal(_ex3))
614 throw(std::runtime_error("index dimension of epsilon tensor must match number of indices"));
616 return indexed(tensepsilon(), sy_anti(), i1, i2, i3);
619 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig)
621 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))
622 throw(std::invalid_argument("indices of Lorentz epsilon tensor must be of type varidx"));
624 ex dim = ex_to<idx>(i1).get_dim();
625 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()))
626 throw(std::invalid_argument("all indices of epsilon tensor must have the same dimension"));
627 if (!ex_to<idx>(i1).get_dim().is_equal(_ex4))
628 throw(std::runtime_error("index dimension of epsilon tensor must match number of indices"));
630 return indexed(tensepsilon(true, pos_sig), sy_anti(), i1, i2, i3, i4);
633 ex eps0123(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig)
635 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))
636 throw(std::invalid_argument("indices of epsilon tensor must be of type varidx"));
638 ex dim = ex_to<idx>(i1).get_dim();
640 return lorentz_eps(i1, i2, i3, i4, pos_sig);
642 return indexed(tensepsilon(true, pos_sig), sy_anti(), i1, i2, i3, i4);