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 (effective) dimension of the space
183 if (is_dummy_pair(i1, i2)) {
185 return i1.minimal_dim(i2);
186 } catch (std::exception &e) {
191 // Numeric evaluation
192 if (static_cast<const indexed &>(i).all_index_values_are(info_flags::integer)) {
193 int n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
200 // No further simplifications
204 /** Automatic symbolic evaluation of an indexed metric tensor. */
205 ex tensmetric::eval_indexed(const basic & i) const
207 GINAC_ASSERT(is_a<indexed>(i));
208 GINAC_ASSERT(i.nops() == 3);
209 GINAC_ASSERT(is_a<tensmetric>(i.op(0)));
210 GINAC_ASSERT(is_a<varidx>(i.op(1)));
211 GINAC_ASSERT(is_a<varidx>(i.op(2)));
213 const varidx & i1 = ex_to<varidx>(i.op(1));
214 const varidx & i2 = ex_to<varidx>(i.op(2));
216 // A metric tensor with one covariant and one contravariant index gets
217 // replaced by a delta tensor
218 if (i1.is_covariant() != i2.is_covariant())
219 return delta_tensor(i1, i2);
221 // No further simplifications
225 /** Automatic symbolic evaluation of an indexed Lorentz metric tensor. */
226 ex minkmetric::eval_indexed(const basic & i) const
228 GINAC_ASSERT(is_a<indexed>(i));
229 GINAC_ASSERT(i.nops() == 3);
230 GINAC_ASSERT(is_a<minkmetric>(i.op(0)));
231 GINAC_ASSERT(is_a<varidx>(i.op(1)));
232 GINAC_ASSERT(is_a<varidx>(i.op(2)));
234 const varidx & i1 = ex_to<varidx>(i.op(1));
235 const varidx & i2 = ex_to<varidx>(i.op(2));
237 // Numeric evaluation
238 if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
239 int n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
243 return pos_sig ? _ex_1 : _ex1;
245 return pos_sig ? _ex1 : _ex_1;
248 // Perform the usual evaluations of a metric tensor
249 return inherited::eval_indexed(i);
252 /** Automatic symbolic evaluation of an indexed metric tensor. */
253 ex spinmetric::eval_indexed(const basic & i) const
255 GINAC_ASSERT(is_a<indexed>(i));
256 GINAC_ASSERT(i.nops() == 3);
257 GINAC_ASSERT(is_a<spinmetric>(i.op(0)));
258 GINAC_ASSERT(is_a<spinidx>(i.op(1)));
259 GINAC_ASSERT(is_a<spinidx>(i.op(2)));
261 const spinidx & i1 = ex_to<spinidx>(i.op(1));
262 const spinidx & i2 = ex_to<spinidx>(i.op(2));
264 // Convolutions are zero
265 if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
268 // Numeric evaluation
269 if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
270 int n1 = ex_to<numeric>(i1.get_value()).to_int(), n2 = ex_to<numeric>(i2.get_value()).to_int();
279 // No further simplifications
283 /** Automatic symbolic evaluation of an indexed epsilon tensor. */
284 ex tensepsilon::eval_indexed(const basic & i) const
286 GINAC_ASSERT(is_a<indexed>(i));
287 GINAC_ASSERT(i.nops() > 1);
288 GINAC_ASSERT(is_a<tensepsilon>(i.op(0)));
290 // Convolutions are zero
291 if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
294 // Numeric evaluation
295 if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
297 // Get sign of index permutation (the indices should already be in
298 // a canonic order but we can't assume what exactly that order is)
300 v.reserve(i.nops() - 1);
301 for (unsigned j=1; j<i.nops(); j++)
302 v.push_back(ex_to<numeric>(ex_to<idx>(i.op(j)).get_value()).to_int());
303 int sign = permutation_sign(v.begin(), v.end());
305 // In a Minkowski space, check for covariant indices
307 for (unsigned j=1; j<i.nops(); j++) {
308 const ex & x = i.op(j);
309 if (!is_ex_of_type(x, varidx))
310 throw(std::runtime_error("indices of epsilon tensor in Minkowski space must be of type varidx"));
311 if (ex_to<varidx>(x).is_covariant())
312 if (ex_to<idx>(x).get_value().is_zero())
313 sign = (pos_sig ? -sign : sign);
315 sign = (pos_sig ? sign : -sign);
322 // No further simplifications
326 bool tensor::replace_contr_index(exvector::iterator self, exvector::iterator other) const
328 // Try to contract the first index
329 const idx *self_idx = &ex_to<idx>(self->op(1));
330 const idx *free_idx = &ex_to<idx>(self->op(2));
331 bool first_index_tried = false;
334 if (self_idx->is_symbolic()) {
335 for (unsigned i=1; i<other->nops(); i++) {
336 const idx &other_idx = ex_to<idx>(other->op(i));
337 if (is_dummy_pair(*self_idx, other_idx)) {
339 // Contraction found, remove this tensor and substitute the
340 // index in the second object
342 // minimal_dim() throws an exception when index dimensions are not comparable
343 ex min_dim = self_idx->minimal_dim(other_idx);
345 *other = other->subs(other_idx == free_idx->replace_dim(min_dim));
347 } catch (std::exception &e) {
354 if (!first_index_tried) {
356 // No contraction with the first index found, try the second index
357 self_idx = &ex_to<idx>(self->op(2));
358 free_idx = &ex_to<idx>(self->op(1));
359 first_index_tried = true;
366 /** Contraction of an indexed delta tensor with something else. */
367 bool tensdelta::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
369 GINAC_ASSERT(is_a<indexed>(*self));
370 GINAC_ASSERT(is_a<indexed>(*other));
371 GINAC_ASSERT(self->nops() == 3);
372 GINAC_ASSERT(is_a<tensdelta>(self->op(0)));
374 // Replace the dummy index with this tensor's other index and remove
375 // the tensor (this is valid for contractions with all other tensors)
376 return replace_contr_index(self, other);
379 /** Contraction of an indexed metric tensor with something else. */
380 bool tensmetric::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
382 GINAC_ASSERT(is_a<indexed>(*self));
383 GINAC_ASSERT(is_a<indexed>(*other));
384 GINAC_ASSERT(self->nops() == 3);
385 GINAC_ASSERT(is_a<tensmetric>(self->op(0)));
387 // If contracting with the delta tensor, let the delta do it
388 // (don't raise/lower delta indices)
389 if (is_ex_of_type(other->op(0), tensdelta))
392 // Replace the dummy index with this tensor's other index and remove
393 // the tensor (this is valid for contractions with all other tensors)
394 return replace_contr_index(self, other);
397 /** Contraction of an indexed spinor metric with something else. */
398 bool spinmetric::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
400 GINAC_ASSERT(is_a<indexed>(*self));
401 GINAC_ASSERT(is_a<indexed>(*other));
402 GINAC_ASSERT(self->nops() == 3);
403 GINAC_ASSERT(is_a<spinmetric>(self->op(0)));
405 // Contractions between spinor metrics
406 if (is_ex_of_type(other->op(0), spinmetric)) {
407 const idx &self_i1 = ex_to<idx>(self->op(1));
408 const idx &self_i2 = ex_to<idx>(self->op(2));
409 const idx &other_i1 = ex_to<idx>(other->op(1));
410 const idx &other_i2 = ex_to<idx>(other->op(2));
412 if (is_dummy_pair(self_i1, other_i1)) {
413 if (is_dummy_pair(self_i2, other_i2))
416 *self = delta_tensor(self_i2, other_i2);
419 } else if (is_dummy_pair(self_i1, other_i2)) {
420 if (is_dummy_pair(self_i2, other_i1))
423 *self = -delta_tensor(self_i2, other_i1);
426 } else if (is_dummy_pair(self_i2, other_i1)) {
427 *self = -delta_tensor(self_i1, other_i2);
430 } else if (is_dummy_pair(self_i2, other_i2)) {
431 *self = delta_tensor(self_i1, other_i1);
437 // If contracting with the delta tensor, let the delta do it
438 // (don't raise/lower delta indices)
439 if (is_ex_of_type(other->op(0), tensdelta))
442 // Try to contract first index
443 const idx *self_idx = &ex_to<idx>(self->op(1));
444 const idx *free_idx = &ex_to<idx>(self->op(2));
445 bool first_index_tried = false;
449 if (self_idx->is_symbolic()) {
450 for (unsigned i=1; i<other->nops(); i++) {
451 const idx &other_idx = ex_to<idx>(other->op(i));
452 if (is_dummy_pair(*self_idx, other_idx)) {
454 // Contraction found, remove metric tensor and substitute
455 // index in second object
456 *self = (static_cast<const spinidx *>(self_idx)->is_covariant() ? sign : -sign);
457 *other = other->subs(other_idx == *free_idx);
463 if (!first_index_tried) {
465 // No contraction with first index found, try second index
466 self_idx = &ex_to<idx>(self->op(2));
467 free_idx = &ex_to<idx>(self->op(1));
468 first_index_tried = true;
476 /** Contraction of epsilon tensor with something else. */
477 bool tensepsilon::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
479 GINAC_ASSERT(is_a<indexed>(*self));
480 GINAC_ASSERT(is_a<indexed>(*other));
481 GINAC_ASSERT(is_a<tensepsilon>(self->op(0)));
482 unsigned num = self->nops() - 1;
484 if (is_ex_exactly_of_type(other->op(0), tensepsilon) && num+1 == other->nops()) {
486 // Contraction of two epsilon tensors is a determinant
487 ex dim = ex_to<idx>(self->op(1)).get_dim();
489 for (int i=0; i<num; i++) {
490 for (int j=0; j<num; j++) {
492 M(i, j) = lorentz_g(self->op(i+1), other->op(j+1), pos_sig);
494 M(i, j) = metric_tensor(self->op(i+1), other->op(j+1));
497 int sign = minkowski ? -1 : 1;
498 *self = sign * M.determinant().simplify_indexed();
510 ex delta_tensor(const ex & i1, const ex & i2)
512 if (!is_ex_of_type(i1, idx) || !is_ex_of_type(i2, idx))
513 throw(std::invalid_argument("indices of delta tensor must be of type idx"));
515 return indexed(tensdelta(), sy_symm(), i1, i2);
518 ex metric_tensor(const ex & i1, const ex & i2)
520 if (!is_ex_of_type(i1, varidx) || !is_ex_of_type(i2, varidx))
521 throw(std::invalid_argument("indices of metric tensor must be of type varidx"));
522 ex dim = ex_to<idx>(i1).get_dim();
523 if (!dim.is_equal(ex_to<idx>(i2).get_dim()))
524 throw(std::invalid_argument("all indices of metric tensor must have the same dimension"));
526 return indexed(tensmetric(), sy_symm(), i1, i2);
529 ex lorentz_g(const ex & i1, const ex & i2, bool pos_sig)
531 if (!is_ex_of_type(i1, varidx) || !is_ex_of_type(i2, varidx))
532 throw(std::invalid_argument("indices of metric tensor must be of type varidx"));
533 ex dim = ex_to<idx>(i1).get_dim();
534 if (!dim.is_equal(ex_to<idx>(i2).get_dim()))
535 throw(std::invalid_argument("all indices of metric tensor must have the same dimension"));
537 return indexed(minkmetric(pos_sig), sy_symm(), i1, i2);
540 ex spinor_metric(const ex & i1, const ex & i2)
542 if (!is_ex_of_type(i1, spinidx) || !is_ex_of_type(i2, spinidx))
543 throw(std::invalid_argument("indices of spinor metric must be of type spinidx"));
544 if (!ex_to<idx>(i1).get_dim().is_equal(2) || !ex_to<idx>(i2).get_dim().is_equal(2))
545 throw(std::runtime_error("index dimension for spinor metric must be 2"));
547 return indexed(spinmetric(), sy_anti(), i1, i2);
550 ex epsilon_tensor(const ex & i1, const ex & i2)
552 if (!is_ex_of_type(i1, idx) || !is_ex_of_type(i2, idx))
553 throw(std::invalid_argument("indices of epsilon tensor must be of type idx"));
555 ex dim = ex_to<idx>(i1).get_dim();
556 if (!dim.is_equal(ex_to<idx>(i2).get_dim()))
557 throw(std::invalid_argument("all indices of epsilon tensor must have the same dimension"));
558 if (!ex_to<idx>(i1).get_dim().is_equal(_ex2))
559 throw(std::runtime_error("index dimension of epsilon tensor must match number of indices"));
561 return indexed(tensepsilon(), sy_anti(), i1, i2);
564 ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3)
566 if (!is_ex_of_type(i1, idx) || !is_ex_of_type(i2, idx) || !is_ex_of_type(i3, idx))
567 throw(std::invalid_argument("indices of epsilon tensor must be of type idx"));
569 ex dim = ex_to<idx>(i1).get_dim();
570 if (!dim.is_equal(ex_to<idx>(i2).get_dim()) || !dim.is_equal(ex_to<idx>(i3).get_dim()))
571 throw(std::invalid_argument("all indices of epsilon tensor must have the same dimension"));
572 if (!ex_to<idx>(i1).get_dim().is_equal(_ex3))
573 throw(std::runtime_error("index dimension of epsilon tensor must match number of indices"));
575 return indexed(tensepsilon(), sy_anti(), i1, i2, i3);
578 ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig)
580 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))
581 throw(std::invalid_argument("indices of Lorentz epsilon tensor must be of type varidx"));
583 ex dim = ex_to<idx>(i1).get_dim();
584 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()))
585 throw(std::invalid_argument("all indices of epsilon tensor must have the same dimension"));
586 if (!ex_to<idx>(i1).get_dim().is_equal(_ex4))
587 throw(std::runtime_error("index dimension of epsilon tensor must match number of indices"));
589 return indexed(tensepsilon(true, pos_sig), sy_anti(), i1, i2, i3, i4);