Finalize 1.7.6 release.
[ginac.git] / ginac / clifford.cpp
1 /** @file clifford.cpp
2  *
3  *  Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
4
5 /*
6  *  GiNaC Copyright (C) 1999-2019 Johannes Gutenberg University Mainz, Germany
7  *
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.
12  *
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.
17  *
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
21  */
22
23 #include "clifford.h"
24
25 #include "ex.h"
26 #include "idx.h"
27 #include "ncmul.h"
28 #include "symbol.h"
29 #include "numeric.h" // for I
30 #include "symmetry.h"
31 #include "lst.h"
32 #include "relational.h"
33 #include "operators.h"
34 #include "add.h"
35 #include "mul.h"
36 #include "power.h"
37 #include "matrix.h"
38 #include "archive.h"
39 #include "utils.h"
40
41 #include <stdexcept>
42
43 namespace GiNaC {
44
45 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(clifford, indexed,
46   print_func<print_dflt>(&clifford::do_print_dflt).
47   print_func<print_latex>(&clifford::do_print_latex).
48   print_func<print_tree>(&clifford::do_print_tree))
49
50 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracone, tensor,
51   print_func<print_dflt>(&diracone::do_print).
52   print_func<print_latex>(&diracone::do_print_latex))
53
54 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(cliffordunit, tensor,
55   print_func<print_dflt>(&cliffordunit::do_print).
56   print_func<print_latex>(&cliffordunit::do_print_latex))
57
58 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma, cliffordunit,
59   print_func<print_dflt>(&diracgamma::do_print).
60   print_func<print_latex>(&diracgamma::do_print_latex))
61
62 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma5, tensor,
63   print_func<print_dflt>(&diracgamma5::do_print).
64   print_func<print_latex>(&diracgamma5::do_print_latex))
65
66 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaL, tensor,
67   print_func<print_context>(&diracgammaL::do_print).
68   print_func<print_latex>(&diracgammaL::do_print_latex))
69
70 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaR, tensor,
71   print_func<print_context>(&diracgammaR::do_print).
72   print_func<print_latex>(&diracgammaR::do_print_latex))
73
74 //////////
75 // default constructors
76 //////////
77
78 clifford::clifford() : representation_label(0), metric(0), commutator_sign(-1)
79 {
80 }
81
82 DEFAULT_CTOR(diracone)
83 DEFAULT_CTOR(cliffordunit)
84 DEFAULT_CTOR(diracgamma)
85 DEFAULT_CTOR(diracgamma5)
86 DEFAULT_CTOR(diracgammaL)
87 DEFAULT_CTOR(diracgammaR)
88
89 //////////
90 // other constructors
91 //////////
92
93 /** Construct object without any indices. This constructor is for internal
94  *  use only. Use the dirac_ONE() function instead.
95  *  @see dirac_ONE */
96 clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl), metric(0), commutator_sign(-1)
97 {
98 }
99
100 /** Construct object with one Lorentz index. This constructor is for internal
101  *  use only. Use the clifford_unit() or dirac_gamma() functions instead.
102  *  @see clifford_unit
103  *  @see dirac_gamma */
104 clifford::clifford(const ex & b, const ex & mu, const ex & metr, unsigned char rl, int comm_sign) : inherited(b, mu), representation_label(rl), metric(metr), commutator_sign(comm_sign)
105 {
106         GINAC_ASSERT(is_a<idx>(mu));
107 }
108
109 clifford::clifford(unsigned char rl, const ex & metr, int comm_sign, const exvector & v) : inherited(not_symmetric(), v), representation_label(rl), metric(metr), commutator_sign(comm_sign)
110 {
111 }
112
113 clifford::clifford(unsigned char rl, const ex & metr, int comm_sign, exvector && v) : inherited(not_symmetric(), std::move(v)), representation_label(rl), metric(metr), commutator_sign(comm_sign)
114 {
115 }
116
117 return_type_t clifford::return_type_tinfo() const
118 {
119         return make_return_type_t<clifford>(representation_label);
120 }
121
122 //////////
123 // archiving
124 //////////
125
126 void clifford::read_archive(const archive_node& n, lst& sym_lst)
127 {
128         inherited::read_archive(n, sym_lst);
129         unsigned rl;
130         n.find_unsigned("label", rl);
131         representation_label = rl;
132         n.find_ex("metric", metric, sym_lst);
133         n.find_unsigned("commutator_sign+1", rl);
134         commutator_sign = rl - 1;
135 }
136
137 void clifford::archive(archive_node & n) const
138 {
139         inherited::archive(n);
140         n.add_unsigned("label", representation_label);
141         n.add_ex("metric", metric);
142         n.add_unsigned("commutator_sign+1", commutator_sign+1);
143 }
144
145 GINAC_BIND_UNARCHIVER(clifford);
146 GINAC_BIND_UNARCHIVER(cliffordunit);
147 GINAC_BIND_UNARCHIVER(diracone);
148 GINAC_BIND_UNARCHIVER(diracgamma);
149 GINAC_BIND_UNARCHIVER(diracgamma5);
150 GINAC_BIND_UNARCHIVER(diracgammaL);
151 GINAC_BIND_UNARCHIVER(diracgammaR);
152
153
154 ex clifford::get_metric(const ex & i, const ex & j, bool symmetrised) const
155 {
156         if (is_a<indexed>(metric)) {
157                 if (symmetrised && !(ex_to<symmetry>(ex_to<indexed>(metric).get_symmetry()).has_symmetry())) {
158                         if (is_a<matrix>(metric.op(0))) {
159                                 return indexed((ex_to<matrix>(metric.op(0)).add(ex_to<matrix>(metric.op(0)).transpose())).mul(numeric(1, 2)),
160                                                symmetric2(), i, j);
161                         } else {
162                                 return simplify_indexed(indexed(metric.op(0)*_ex1_2, i, j) + indexed(metric.op(0)*_ex1_2, j, i));
163                         }
164                 } else {
165                         return metric.subs(lst{metric.op(1) == i, metric.op(2) == j}, subs_options::no_pattern);
166                 }
167         } else {
168                 exvector indices = metric.get_free_indices();
169                 if (symmetrised)
170                         return _ex1_2*simplify_indexed(metric.subs(lst{indices[0] == i, indices[1] == j}, subs_options::no_pattern)
171                                                      + metric.subs(lst{indices[0] == j, indices[1] == i}, subs_options::no_pattern));
172                 else
173                         return metric.subs(lst{indices[0] == i, indices[1] == j}, subs_options::no_pattern);
174         }
175 }
176
177 bool clifford::same_metric(const ex & other) const
178 {
179         ex metr;
180         if (is_a<clifford>(other)) 
181                 metr = ex_to<clifford>(other).get_metric();
182         else 
183                 metr = other;
184
185         if (is_a<indexed>(metr))
186                 return metr.op(0).is_equal(get_metric().op(0));
187         else {
188                 exvector indices = metr.get_free_indices();
189                 return  (indices.size() == 2) 
190                         && simplify_indexed(get_metric(indices[0], indices[1])-metr).is_zero();
191         }
192 }
193
194 //////////
195 // functions overriding virtual functions from base classes
196 //////////
197
198 ex clifford::op(size_t i) const
199 {
200         GINAC_ASSERT(i<nops());
201         if (nops()-i == 1)
202                 return representation_label;
203         else 
204                 return inherited::op(i);
205 }
206
207 ex & clifford::let_op(size_t i)
208 {
209         GINAC_ASSERT(i<nops());
210
211         static ex rl = numeric(representation_label);
212         ensure_if_modifiable();
213         if (nops()-i == 1)
214                 return rl;
215         else 
216                 return inherited::let_op(i);
217 }
218
219 ex clifford::subs(const exmap & m, unsigned options) const
220 {
221         ex subsed = inherited::subs(m, options);
222         if(is_a<clifford>(subsed)) {
223                 ex prevmetric = ex_to<clifford>(subsed).metric;
224                 ex newmetric = prevmetric.subs(m, options);
225                 if(!are_ex_trivially_equal(prevmetric, newmetric)) {
226                         clifford c = ex_to<clifford>(subsed);
227                         c.metric = newmetric;
228                         subsed = c;
229                 }
230         }
231         return subsed;
232 }
233
234 int clifford::compare_same_type(const basic & other) const
235 {
236         GINAC_ASSERT(is_a<clifford>(other));
237         const clifford &o = static_cast<const clifford &>(other);
238
239         if (representation_label != o.representation_label) {
240                 // different representation label
241                 return representation_label < o.representation_label ? -1 : 1;
242         }
243
244         return inherited::compare_same_type(other);
245 }
246
247 bool clifford::match_same_type(const basic & other) const
248 {
249         GINAC_ASSERT(is_a<clifford>(other));
250         const clifford &o = static_cast<const clifford &>(other);
251
252         return ((representation_label == o.representation_label) && (commutator_sign == o.get_commutator_sign()) && same_metric(o));
253 }
254
255 static bool is_dirac_slash(const ex & seq0)
256 {
257         return !is_a<diracgamma5>(seq0) && !is_a<diracgammaL>(seq0) &&
258                !is_a<diracgammaR>(seq0) && !is_a<cliffordunit>(seq0) &&
259                !is_a<diracone>(seq0);
260 }
261
262 void clifford::do_print_dflt(const print_dflt & c, unsigned level) const
263 {
264         // dirac_slash() object is printed differently
265         if (is_dirac_slash(seq[0])) {
266                 seq[0].print(c, precedence());
267                 c.s << "\\";
268         } else { // We do not print representation label if it is 0
269                 if (representation_label == 0) {
270                         this->print_dispatch<inherited>(c, level);
271                 } else { // otherwise we put it before indices in square brackets; the code is borrowed from indexed.cpp 
272                         if (precedence() <= level) {
273                                 c.s << '(';
274                         }
275                         seq[0].print(c, precedence());
276                         c.s << '[' << int(representation_label) << ']';
277                         printindices(c, level);
278                         if (precedence() <= level) {
279                                 c.s << ')';
280                         }
281                 }
282         }
283 }
284
285 void clifford::do_print_latex(const print_latex & c, unsigned level) const
286 {
287         // dirac_slash() object is printed differently
288         if (is_dirac_slash(seq[0])) {
289                 c.s << "{";
290                 seq[0].print(c, precedence());
291                 c.s << "\\hspace{-1.0ex}/}";
292         } else {
293                 c.s << "\\clifford[" << int(representation_label) << "]";
294                 this->print_dispatch<inherited>(c, level);
295         }
296 }
297
298 void clifford::do_print_tree(const print_tree & c, unsigned level) const
299 {
300         c.s << std::string(level, ' ') << class_name() << " @" << this
301             << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
302             << ", " << seq.size()-1 << " indices"
303             << ", symmetry=" << symtree << std::endl;
304         metric.print(c, level + c.delta_indent);
305         seq[0].print(c, level + c.delta_indent);
306         printindices(c, level + c.delta_indent);
307 }
308
309 DEFAULT_COMPARE(diracone)
310 DEFAULT_COMPARE(cliffordunit)
311 DEFAULT_COMPARE(diracgamma)
312 DEFAULT_COMPARE(diracgamma5)
313 DEFAULT_COMPARE(diracgammaL)
314 DEFAULT_COMPARE(diracgammaR)
315
316 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbf{1}")
317 DEFAULT_PRINT_LATEX(cliffordunit, "e", "e")
318 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
319 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
320 DEFAULT_PRINT_LATEX(diracgammaL, "gammaL", "{\\gamma_L}")
321 DEFAULT_PRINT_LATEX(diracgammaR, "gammaR", "{\\gamma_R}")
322
323 /** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
324 static void base_and_index(const ex & c, ex & b, ex & i)
325 {
326         GINAC_ASSERT(is_a<clifford>(c));
327         GINAC_ASSERT(c.nops() == 2+1);
328
329         if (is_a<cliffordunit>(c.op(0))) { // proper dirac gamma object or clifford unit
330                 i = c.op(1);
331                 b = _ex1;
332         } else if (is_a<diracgamma5>(c.op(0)) || is_a<diracgammaL>(c.op(0)) || is_a<diracgammaR>(c.op(0))) { // gamma5/L/R
333                 i = _ex0;
334                 b = _ex1;
335         } else { // slash object, generate new dummy index
336                 varidx ix(dynallocate<symbol>(), ex_to<idx>(c.op(1)).get_dim());
337                 b = indexed(c.op(0), ix.toggle_variance());
338                 i = ix;
339         }
340 }
341
342 /** Predicate for finding non-clifford objects. */
343 struct is_not_a_clifford {
344         bool operator()(const ex & e)
345         {
346                 return !is_a<clifford>(e);
347         }
348 };
349
350 /** Contraction of a gamma matrix with something else. */
351 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
352 {
353         GINAC_ASSERT(is_a<clifford>(*self));
354         GINAC_ASSERT(is_a<indexed>(*other));
355         GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
356         unsigned char rl = ex_to<clifford>(*self).get_representation_label();
357
358         ex dim = ex_to<idx>(self->op(1)).get_dim();
359         if (other->nops() > 1)
360                 dim = minimal_dim(dim, ex_to<idx>(other->op(1)).get_dim());
361
362         if (is_a<clifford>(*other)) {
363
364                 // Contraction only makes sense if the representation labels are equal
365                 if (ex_to<clifford>(*other).get_representation_label() != rl)
366                         return false;
367
368                 size_t num = other - self;
369
370                 // gamma~mu gamma.mu = dim ONE
371                 if (num == 1) {
372                         *self = dim;
373                         *other = dirac_ONE(rl);
374                         return true;
375
376                 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
377                 } else if (num == 2
378                         && is_a<clifford>(self[1])) {
379                         *self = 2 - dim;
380                         *other = _ex1;
381                         return true;
382
383                 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
384                 } else if (num == 3
385                         && is_a<clifford>(self[1])
386                         && is_a<clifford>(self[2])) {
387                         ex b1, i1, b2, i2;
388                         base_and_index(self[1], b1, i1);
389                         base_and_index(self[2], b2, i2);
390                         *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
391                         self[1] = _ex1;
392                         self[2] = _ex1;
393                         *other = _ex1;
394                         return true;
395
396                 // gamma~mu gamma~alpha gamma~beta gamma~delta gamma.mu = -2 gamma~delta gamma~beta gamma~alpha - (dim-4) gamam~alpha gamma~beta gamma~delta
397                 } else if (num == 4
398                         && is_a<clifford>(self[1])
399                         && is_a<clifford>(self[2])
400                         && is_a<clifford>(self[3])) {
401                         *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3];
402                         self[1] = _ex1;
403                         self[2] = _ex1;
404                         self[3] = _ex1;
405                         *other = _ex1;
406                         return true;
407
408                 // gamma~mu Sodd gamma.mu = -2 Sodd_R
409                 // (Chisholm identity in 4 dimensions)
410                 } else if (!((other - self) & 1) && dim.is_equal(4)) {
411                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
412                                 return false;
413
414                         *self = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(other), std::reverse_iterator<exvector::const_iterator>(self + 1)));
415                         std::fill(self + 1, other, _ex1);
416                         *other = _ex_2;
417                         return true;
418
419                 // gamma~mu Sodd gamma~alpha gamma.mu = 2 gamma~alpha Sodd + 2 Sodd_R gamma~alpha
420                 // (commutate contracted indices towards each other, then use
421                 // Chisholm identity in 4 dimensions)
422                 } else if (((other - self) & 1) && dim.is_equal(4)) {
423                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
424                                 return false;
425
426                         auto next_to_last = other - 1;
427                         ex S = ncmul(exvector(self + 1, next_to_last));
428                         ex SR = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(next_to_last), std::reverse_iterator<exvector::const_iterator>(self + 1)));
429
430                         *self = (*next_to_last) * S + SR * (*next_to_last);
431                         std::fill(self + 1, other, _ex1);
432                         *other = _ex2;
433                         return true;
434
435                 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
436                 // (commutate contracted indices towards each other, simplify_indexed()
437                 // will re-expand and re-run the simplification)
438                 } else {
439                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
440                                 return false;
441
442                         auto next_to_last = other - 1;
443                         ex S = ncmul(exvector(self + 1, next_to_last));
444
445                         *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
446                         std::fill(self + 1, other + 1, _ex1);
447                         return true;
448                 }
449
450         } else if (is_a<symbol>(other->op(0)) && other->nops() == 2) {
451
452                 // x.mu gamma~mu -> x-slash
453                 *self = dirac_slash(other->op(0), dim, rl);
454                 *other = _ex1;
455                 return true;
456         }
457
458         return false;
459 }
460
461 /** Contraction of a Clifford unit with something else. */
462 bool cliffordunit::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
463 {
464         GINAC_ASSERT(is_a<clifford>(*self));
465         GINAC_ASSERT(is_a<indexed>(*other));
466         GINAC_ASSERT(is_a<cliffordunit>(self->op(0)));
467         clifford unit = ex_to<clifford>(*self);
468         unsigned char rl = unit.get_representation_label();
469
470         if (is_a<clifford>(*other)) {
471                 // Contraction only makes sense if the representation labels are equal
472                 // and the metrics are the same
473                 if ((ex_to<clifford>(*other).get_representation_label() != rl) 
474                     && unit.same_metric(*other))
475                         return false;
476
477                 auto before_other = other - 1;
478                 ex mu = self->op(1);
479                 ex mu_toggle = other->op(1);
480                 ex alpha = before_other->op(1);
481
482                 // e~mu e.mu = Tr ONE
483                 if (other - self == 1) {
484                         *self = unit.get_metric(mu, mu_toggle, true);
485                         *other = dirac_ONE(rl);
486                         return true;
487
488                 } else if (other - self == 2) {
489                         if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
490                                 // e~mu e~alpha e.mu = 2*e~mu B(alpha, mu.toggle_variance())-Tr(B) e~alpha
491                                 *self = 2 * (*self) * unit.get_metric(alpha, mu_toggle, true) - unit.get_metric(mu, mu_toggle, true) * (*before_other);
492                                 *before_other = _ex1;
493                                 *other = _ex1;
494                                 return true;
495
496                         } else {
497                                 // e~mu S e.mu = Tr S ONE
498                                 *self = unit.get_metric(mu, mu_toggle, true);
499                                 *other = dirac_ONE(rl);
500                                 return true;
501                         }
502                 } else {
503                 // e~mu S e~alpha e.mu = 2 e~mu S B(alpha, mu.toggle_variance()) - e~mu S e.mu e~alpha
504                 // (commutate contracted indices towards each other, simplify_indexed()
505                 // will re-expand and re-run the simplification)
506                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other) {
507                                 return false;
508                         }
509                         
510                         ex S = ncmul(exvector(self + 1, before_other));
511
512                         if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
513                                 *self = 2 * (*self) * S * unit.get_metric(alpha, mu_toggle, true) - (*self) * S * (*other) * (*before_other);
514                         } else {
515                                 // simply commutes
516                                 *self = (*self) * S * (*other) * (*before_other);
517                         }
518                                 
519                         std::fill(self + 1, other + 1, _ex1);
520                         return true;
521                 }
522         }
523         return false;
524 }
525
526 /** Perform automatic simplification on noncommutative product of clifford
527  *  objects. This removes superfluous ONEs, permutes gamma5/L/R's to the front
528  *  and removes squares of gamma objects. */
529 ex clifford::eval_ncmul(const exvector & v) const
530 {
531         exvector s;
532         s.reserve(v.size());
533
534         // Remove superfluous ONEs
535         for (auto & it : v) {
536                 if (!is_a<clifford>(it) || !is_a<diracone>(it.op(0)))
537                         s.push_back(it);
538         }
539
540         bool something_changed = false;
541         int sign = 1;
542
543         // Anticommutate gamma5/L/R's to the front
544         if (s.size() >= 2) {
545                 auto first = s.begin(), next_to_last = s.end() - 2;
546                 while (true) {
547                         auto it = next_to_last;
548                         while (true) {
549                                 auto it2 = it + 1;
550                                 if (is_a<clifford>(*it) && is_a<clifford>(*it2)) {
551                                         ex e1 = it->op(0), e2 = it2->op(0);
552
553                                         if (is_a<diracgamma5>(e2)) {
554
555                                                 if (is_a<diracgammaL>(e1) || is_a<diracgammaR>(e1)) {
556
557                                                         // gammaL/R gamma5 -> gamma5 gammaL/R
558                                                         it->swap(*it2);
559                                                         something_changed = true;
560
561                                                 } else if (!is_a<diracgamma5>(e1)) {
562
563                                                         // gamma5 gamma5 -> gamma5 gamma5 (do nothing)
564                                                         // x gamma5 -> -gamma5 x
565                                                         it->swap(*it2);
566                                                         sign = -sign;
567                                                         something_changed = true;
568                                                 }
569
570                                         } else if (is_a<diracgammaL>(e2)) {
571
572                                                 if (is_a<diracgammaR>(e1)) {
573
574                                                         // gammaR gammaL -> 0
575                                                         return _ex0;
576
577                                                 } else if (!is_a<diracgammaL>(e1) && !is_a<diracgamma5>(e1)) {
578
579                                                         // gammaL gammaL -> gammaL gammaL (do nothing)
580                                                         // gamma5 gammaL -> gamma5 gammaL (do nothing)
581                                                         // x gammaL -> gammaR x
582                                                         it->swap(*it2);
583                                                         *it = clifford(diracgammaR(), ex_to<clifford>(*it).get_representation_label());
584                                                         something_changed = true;
585                                                 }
586
587                                         } else if (is_a<diracgammaR>(e2)) {
588
589                                                 if (is_a<diracgammaL>(e1)) {
590
591                                                         // gammaL gammaR -> 0
592                                                         return _ex0;
593
594                                                 } else if (!is_a<diracgammaR>(e1) && !is_a<diracgamma5>(e1)) {
595
596                                                         // gammaR gammaR -> gammaR gammaR (do nothing)
597                                                         // gamma5 gammaR -> gamma5 gammaR (do nothing)
598                                                         // x gammaR -> gammaL x
599                                                         it->swap(*it2);
600                                                         *it = clifford(diracgammaL(), ex_to<clifford>(*it).get_representation_label());
601                                                         something_changed = true;
602                                                 }
603                                         }
604                                 }
605                                 if (it == first)
606                                         break;
607                                 --it;
608                         }
609                         if (next_to_last == first)
610                                 break;
611                         --next_to_last;
612                 }
613         }
614
615         // Remove equal adjacent gammas
616         if (s.size() >= 2) {
617                 exvector::iterator it, itend = s.end() - 1;
618                 for (it = s.begin(); it != itend; ++it) {
619                         ex & a = it[0];
620                         ex & b = it[1];
621                         if (!is_a<clifford>(a) || !is_a<clifford>(b))
622                                 continue;
623
624                         const ex & ag = a.op(0);
625                         const ex & bg = b.op(0);
626                         bool a_is_cliffordunit = is_a<cliffordunit>(ag);
627                         bool b_is_cliffordunit =  is_a<cliffordunit>(bg);
628
629                         if (a_is_cliffordunit && b_is_cliffordunit && ex_to<clifford>(a).same_metric(b)
630                                 && (ex_to<clifford>(a).get_commutator_sign() == -1)) {
631                                 // This is done only for Clifford algebras 
632                                 
633                                 const ex & ia = a.op(1);
634                                 const ex & ib = b.op(1);
635                                 if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
636                                         a = ex_to<clifford>(a).get_metric(ia, ib, true);
637                                         b = dirac_ONE(representation_label);
638                                         something_changed = true;
639                                 }
640
641                         } else if ((is_a<diracgamma5>(ag) && is_a<diracgamma5>(bg))) {
642
643                                 // Remove squares of gamma5
644                                 a = dirac_ONE(representation_label);
645                                 b = dirac_ONE(representation_label);
646                                 something_changed = true;
647
648                         } else if ((is_a<diracgammaL>(ag) && is_a<diracgammaL>(bg))
649                                 || (is_a<diracgammaR>(ag) && is_a<diracgammaR>(bg))) {
650
651                                 // Remove squares of gammaL/R
652                                 b = dirac_ONE(representation_label);
653                                 something_changed = true;
654
655                         } else if (is_a<diracgammaL>(ag) && is_a<diracgammaR>(bg)) {
656
657                                 // gammaL and gammaR are orthogonal
658                                 return _ex0;
659
660                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaL>(bg)) {
661
662                                 // gamma5 gammaL -> -gammaL
663                                 a = dirac_ONE(representation_label);
664                                 sign = -sign;
665                                 something_changed = true;
666
667                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaR>(bg)) {
668
669                                 // gamma5 gammaR -> gammaR
670                                 a = dirac_ONE(representation_label);
671                                 something_changed = true;
672
673                         } else if (!a_is_cliffordunit && !b_is_cliffordunit && ag.is_equal(bg)) {
674
675                                 // a\ a\ -> a^2
676                                 varidx ix(dynallocate<symbol>(), ex_to<idx>(a.op(1)).minimal_dim(ex_to<idx>(b.op(1))));
677                                 
678                                 a = indexed(ag, ix) * indexed(ag, ix.toggle_variance());
679                                 b = dirac_ONE(representation_label);
680                                 something_changed = true;
681                         }
682                 }
683         }
684
685         if (s.empty())
686                 return dirac_ONE(representation_label) * sign;
687         if (something_changed)
688                 return reeval_ncmul(s) * sign;
689         else
690                 return hold_ncmul(s) * sign;
691 }
692
693 ex clifford::thiscontainer(const exvector & v) const
694 {
695         return clifford(representation_label, metric, commutator_sign, v);
696 }
697
698 ex clifford::thiscontainer(exvector && v) const
699 {
700         return clifford(representation_label, metric, commutator_sign, std::move(v));
701 }
702
703 ex diracgamma5::conjugate() const
704 {       
705         return _ex_1 * (*this);
706 }
707
708 ex diracgammaL::conjugate() const
709 {
710         return dynallocate<diracgammaR>();
711 }
712
713 ex diracgammaR::conjugate() const
714 {
715         return dynallocate<diracgammaL>();
716 }
717
718 //////////
719 // global functions
720 //////////
721
722 ex dirac_ONE(unsigned char rl)
723 {
724         static ex ONE = dynallocate<diracone>();
725         return clifford(ONE, rl);
726 }
727
728 static unsigned get_dim_uint(const ex& e)
729 {
730         if (!is_a<idx>(e))
731                 throw std::invalid_argument("get_dim_uint: argument is not an index");
732         ex dim = ex_to<idx>(e).get_dim();
733         if (!dim.info(info_flags::posint))
734                 throw std::invalid_argument("get_dim_uint: dimension of index should be a positive integer");
735         unsigned d = ex_to<numeric>(dim).to_int();
736         return d;
737 }
738
739 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl)
740 {
741         ex unit = dynallocate<cliffordunit>();
742
743         if (!is_a<idx>(mu))
744                 throw(std::invalid_argument("clifford_unit(): index of Clifford unit must be of type idx or varidx"));
745
746         exvector indices = metr.get_free_indices();
747
748         if (indices.size() == 2) {
749                 return clifford(unit, mu, metr, rl);
750         } else if (is_a<matrix>(metr)) {
751                 matrix M = ex_to<matrix>(metr);
752                 unsigned n = M.rows();
753                 bool symmetric = true;
754
755                 //static idx xi(dynallocate<symbol>(), n),
756                 //           chi(dynallocate<symbol>(), n);
757                 idx xi(dynallocate<symbol>(), n),
758                     chi(dynallocate<symbol>(), n);
759                 if ((n ==  M.cols()) && (n == get_dim_uint(mu))) {
760                         for (unsigned i = 0; i < n; i++) {
761                                 for (unsigned j = i+1; j < n; j++) {
762                                         if (!M(i, j).is_equal(M(j, i))) {
763                                                 symmetric = false;
764                                         }
765                                 }
766                         }
767                         return clifford(unit, mu, indexed(metr, symmetric?symmetric2():not_symmetric(), xi, chi), rl);
768                 } else {
769                         throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be a square matrix with the same dimensions as index"));
770                 }
771         } else if (indices.size() == 0) { // a tensor or other expression without indices
772                 //static varidx xi(dynallocate<symbol>(), ex_to<idx>(mu).get_dim()),
773                 //              chi(dynallocate<symbol>(), ex_to<idx>(mu).get_dim());
774                 varidx xi(dynallocate<symbol>(), ex_to<idx>(mu).get_dim()),
775                        chi(dynallocate<symbol>(), ex_to<idx>(mu).get_dim());
776                 return clifford(unit, mu, indexed(metr, xi, chi), rl);
777         }  else 
778                 throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be of type tensor, matrix or an expression with two free indices"));
779 }
780
781 ex dirac_gamma(const ex & mu, unsigned char rl)
782 {
783         static ex gamma = dynallocate<diracgamma>();
784
785         if (!is_a<varidx>(mu))
786                 throw(std::invalid_argument("dirac_gamma(): index of Dirac gamma must be of type varidx"));
787
788         static varidx xi(dynallocate<symbol>(), ex_to<varidx>(mu).get_dim()),
789                       chi(dynallocate<symbol>(), ex_to<varidx>(mu).get_dim());
790         return clifford(gamma, mu, indexed(dynallocate<minkmetric>(), symmetric2(), xi, chi), rl);
791 }
792
793 ex dirac_gamma5(unsigned char rl)
794 {
795         static ex gamma5 = dynallocate<diracgamma5>();
796         return clifford(gamma5, rl);
797 }
798
799 ex dirac_gammaL(unsigned char rl)
800 {
801         static ex gammaL = dynallocate<diracgammaL>();
802         return clifford(gammaL, rl);
803 }
804
805 ex dirac_gammaR(unsigned char rl)
806 {
807         static ex gammaR = dynallocate<diracgammaR>();
808         return clifford(gammaR, rl);
809 }
810
811 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
812 {
813         // Slashed vectors are actually stored as a clifford object with the
814         // vector as its base expression and a (dummy) index that just serves
815         // for storing the space dimensionality
816
817         static varidx xi(dynallocate<symbol>(), dim),
818                       chi(dynallocate<symbol>(), dim);
819         return clifford(e, varidx(0, dim), indexed(dynallocate<minkmetric>(), symmetric2(), xi, chi), rl);
820 }
821
822 /** Extract representation label from tinfo key (as returned by
823  *  return_type_tinfo()). */
824 static unsigned char get_representation_label(const return_type_t& ti)
825 {
826         return (unsigned char)ti.rl;
827 }
828
829 /** Take trace of a string of an even number of Dirac gammas given a vector
830  *  of indices. */
831 static ex trace_string(exvector::const_iterator ix, size_t num)
832 {
833         // Tr gamma.mu gamma.nu = 4 g.mu.nu
834         if (num == 2)
835                 return lorentz_g(ix[0], ix[1]);
836
837         // Tr gamma.mu gamma.nu gamma.rho gamma.sig = 4 (g.mu.nu g.rho.sig + g.nu.rho g.mu.sig - g.mu.rho g.nu.sig )
838         else if (num == 4)
839                 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
840                      + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
841                      - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
842
843         // Traces of 6 or more gammas are computed recursively:
844         // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
845         //   + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
846         //   - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
847         //   + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
848         //   - ...
849         //   + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
850         exvector v(num - 2);
851         int sign = 1;
852         ex result;
853         for (size_t i=1; i<num; i++) {
854                 for (size_t n=1, j=0; n<num; n++) {
855                         if (n == i)
856                                 continue;
857                         v[j++] = ix[n];
858                 }
859                 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
860                 sign = -sign;
861         }
862         return result;
863 }
864
865 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE)
866 {
867         if (is_a<clifford>(e)) {
868
869                 unsigned char rl = ex_to<clifford>(e).get_representation_label();
870
871                 // Are we taking the trace over this object's representation label?
872                 if (rls.find(rl) == rls.end())
873                         return e;
874
875                 // Yes, all elements are traceless, except for dirac_ONE and dirac_L/R
876                 const ex & g = e.op(0);
877                 if (is_a<diracone>(g))
878                         return trONE;
879                 else if (is_a<diracgammaL>(g) || is_a<diracgammaR>(g))
880                         return trONE/2;
881                 else
882                         return _ex0;
883
884         } else if (is_exactly_a<mul>(e)) {
885
886                 // Trace of product: pull out non-clifford factors
887                 ex prod = _ex1;
888                 for (size_t i=0; i<e.nops(); i++) {
889                         const ex &o = e.op(i);
890                         if (is_clifford_tinfo(o.return_type_tinfo()))
891                                 prod *= dirac_trace(o, rls, trONE);
892                         else
893                                 prod *= o;
894                 }
895                 return prod;
896
897         } else if (is_exactly_a<ncmul>(e)) {
898
899                 unsigned char rl = get_representation_label(e.return_type_tinfo());
900
901                 // Are we taking the trace over this string's representation label?
902                 if (rls.find(rl) == rls.end())
903                         return e;
904
905                 // Substitute gammaL/R and expand product, if necessary
906                 ex e_expanded = e.subs(lst{
907                         dirac_gammaL(rl) == (dirac_ONE(rl)-dirac_gamma5(rl))/2,
908                         dirac_gammaR(rl) == (dirac_ONE(rl)+dirac_gamma5(rl))/2
909                 }, subs_options::no_pattern).expand();
910                 if (!is_a<ncmul>(e_expanded))
911                         return dirac_trace(e_expanded, rls, trONE);
912
913                 // gamma5 gets moved to the front so this check is enough
914                 bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
915                 size_t num = e.nops();
916
917                 if (has_gamma5) {
918
919                         // Trace of gamma5 * odd number of gammas and trace of
920                         // gamma5 * gamma.mu * gamma.nu are zero
921                         if ((num & 1) == 0 || num == 3)
922                                 return _ex0;
923
924                         // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
925                         // (the epsilon is always 4-dimensional)
926                         if (num == 5) {
927                                 ex b1, i1, b2, i2, b3, i3, b4, i4;
928                                 base_and_index(e.op(1), b1, i1);
929                                 base_and_index(e.op(2), b2, i2);
930                                 base_and_index(e.op(3), b3, i3);
931                                 base_and_index(e.op(4), b4, i4);
932                                 return trONE * I * (lorentz_eps(ex_to<idx>(i1).replace_dim(_ex4), ex_to<idx>(i2).replace_dim(_ex4), ex_to<idx>(i3).replace_dim(_ex4), ex_to<idx>(i4).replace_dim(_ex4)) * b1 * b2 * b3 * b4).simplify_indexed();
933                         }
934
935                         // Tr gamma5 S_2k =
936                         //   I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
937                         // (the epsilon is always 4-dimensional)
938                         exvector ix(num-1), bv(num-1);
939                         for (size_t i=1; i<num; i++)
940                                 base_and_index(e.op(i), bv[i-1], ix[i-1]);
941                         num--;
942                         int *iv = new int[num];
943                         ex result;
944                         for (size_t i=0; i<num-3; i++) {
945                                 ex idx1 = ix[i];
946                                 for (size_t j=i+1; j<num-2; j++) {
947                                         ex idx2 = ix[j];
948                                         for (size_t k=j+1; k<num-1; k++) {
949                                                 ex idx3 = ix[k];
950                                                 for (size_t l=k+1; l<num; l++) {
951                                                         ex idx4 = ix[l];
952                                                         iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
953                                                         exvector v;
954                                                         v.reserve(num - 4);
955                                                         for (size_t n=0, t=4; n<num; n++) {
956                                                                 if (n == i || n == j || n == k || n == l)
957                                                                         continue;
958                                                                 iv[t++] = n;
959                                                                 v.push_back(ix[n]);
960                                                         }
961                                                         int sign = permutation_sign(iv, iv + num);
962                                                         result += sign * lorentz_eps(ex_to<idx>(idx1).replace_dim(_ex4), ex_to<idx>(idx2).replace_dim(_ex4), ex_to<idx>(idx3).replace_dim(_ex4), ex_to<idx>(idx4).replace_dim(_ex4))
963                                                                 * trace_string(v.begin(), num - 4);
964                                                 }
965                                         }
966                                 }
967                         }
968                         delete[] iv;
969                         return trONE * I * result * mul(bv);
970
971                 } else { // no gamma5
972
973                         // Trace of odd number of gammas is zero
974                         if ((num & 1) == 1)
975                                 return _ex0;
976
977                         // Tr gamma.mu gamma.nu = 4 g.mu.nu
978                         if (num == 2) {
979                                 ex b1, i1, b2, i2;
980                                 base_and_index(e.op(0), b1, i1);
981                                 base_and_index(e.op(1), b2, i2);
982                                 return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
983                         }
984
985                         exvector iv(num), bv(num);
986                         for (size_t i=0; i<num; i++)
987                                 base_and_index(e.op(i), bv[i], iv[i]);
988
989                         return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
990                 }
991
992         } else if (e.nops() > 0) {
993
994                 // Trace maps to all other container classes (this includes sums)
995                 pointer_to_map_function_2args<const std::set<unsigned char> &, const ex &> fcn(dirac_trace, rls, trONE);
996                 return e.map(fcn);
997
998         } else
999                 return _ex0;
1000 }
1001
1002 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE)
1003 {
1004         // Convert list to set
1005         std::set<unsigned char> rls;
1006         for (const auto & i : rll) {
1007                 if (i.info(info_flags::nonnegint))
1008                         rls.insert(ex_to<numeric>(i).to_int());
1009         }
1010
1011         return dirac_trace(e, rls, trONE);
1012 }
1013
1014 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
1015 {
1016         // Convert label to set
1017         std::set<unsigned char> rls;
1018         rls.insert(rl);
1019
1020         return dirac_trace(e, rls, trONE);
1021 }
1022
1023
1024 ex canonicalize_clifford(const ex & e_)
1025 {
1026         pointer_to_map_function fcn(canonicalize_clifford);
1027
1028         if (is_a<matrix>(e_)    // || is_a<pseries>(e) || is_a<integral>(e)
1029                 || e_.info(info_flags::list)) {
1030                 return e_.map(fcn);
1031         } else {
1032                 ex e=simplify_indexed(e_);
1033                 // Scan for any ncmul objects
1034                 exmap srl;
1035                 ex aux = e.to_rational(srl);
1036                 for (auto & i : srl) {
1037
1038                         ex lhs = i.first;
1039                         ex rhs = i.second;
1040
1041                         if (is_exactly_a<ncmul>(rhs)
1042                                         && rhs.return_type() == return_types::noncommutative
1043                                         && is_clifford_tinfo(rhs.return_type_tinfo())) {
1044
1045                                 // Expand product, if necessary
1046                                 ex rhs_expanded = rhs.expand();
1047                                 if (!is_a<ncmul>(rhs_expanded)) {
1048                                         i.second = canonicalize_clifford(rhs_expanded);
1049                                         continue;
1050
1051                                 } else if (!is_a<clifford>(rhs.op(0)))
1052                                         continue;
1053
1054                                 exvector v;
1055                                 v.reserve(rhs.nops());
1056                                 for (size_t j=0; j<rhs.nops(); j++)
1057                                         v.push_back(rhs.op(j));
1058
1059                                 // Stupid recursive bubble sort because we only want to swap adjacent gammas
1060                                 auto it = v.begin(), next_to_last = v.end() - 1;
1061                                 if (is_a<diracgamma5>(it->op(0)) || is_a<diracgammaL>(it->op(0)) || is_a<diracgammaR>(it->op(0)))
1062                                         ++it;
1063
1064                                 while (it != next_to_last) {
1065                                         if (it[0].compare(it[1]) > 0) {
1066
1067                                                 ex save0 = it[0], save1 = it[1];
1068                                                 ex b1, i1, b2, i2;
1069                                                 base_and_index(it[0], b1, i1);
1070                                                 base_and_index(it[1], b2, i2);
1071                                                 // for Clifford algebras (commutator_sign == -1) metric should be symmetrised
1072                                                 it[0] = (ex_to<clifford>(save0).get_metric(i1, i2, ex_to<clifford>(save0).get_commutator_sign() == -1) * b1 * b2).simplify_indexed();
1073                                                 it[1] = v.size() ? _ex2 * dirac_ONE(ex_to<clifford>(save0).get_representation_label()) : _ex2;
1074                                                 ex sum = ncmul(v);
1075                                                 it[0] = save1;
1076                                                 it[1] = save0;
1077                                                 sum += ex_to<clifford>(save0).get_commutator_sign() * ncmul(std::move(v));
1078                                                 i.second = canonicalize_clifford(sum);
1079                                                 goto next_sym;
1080                                         }
1081                                         ++it;
1082                                 }
1083 next_sym:       ;
1084                         }
1085                 }
1086                 return aux.subs(srl, subs_options::no_pattern).simplify_indexed();
1087         }
1088 }
1089
1090 ex clifford_star_bar(const ex & e, bool do_bar, unsigned options)
1091 {
1092         pointer_to_map_function_2args<bool, unsigned> fcn(clifford_star_bar, do_bar, options | 1);
1093
1094         // is a child, no need to expand
1095         ex e1= (options & 1 ? e : e.expand());
1096
1097         if (is_a<ncmul>(e1) ) { // reversing order of clifford units
1098                 exvector ev, cv;
1099                 ev.reserve(e1.nops());
1100                 cv.reserve(e1.nops());
1101                 // separate clifford and non-clifford entries
1102                 for (int i= 0; i < e1.nops(); ++i) {
1103                         if (is_a<clifford>(e1.op(i)) && is_a<cliffordunit>(e1.op(i).op(0)))
1104                                 cv.push_back(e1.op(i));
1105                         else
1106                                 ev.push_back(e1.op(i));
1107                 }
1108                 for (auto i=cv.rbegin(); i!=cv.rend(); ++i) { // reverse order of Clifford units
1109                         ev.push_back(i->conjugate());
1110                 }
1111                 // For clifford_bar an odd number of clifford units reverts the sign
1112                 if (do_bar && (cv.size() % 2 == 1))
1113                         return -dynallocate<ncmul>(std::move(ev));
1114                 else
1115                         return dynallocate<ncmul>(std::move(ev));
1116         } else if (is_a<clifford>(e1) && is_a<cliffordunit>(e1.op(0))) {
1117                 if (do_bar)
1118                         return -e;
1119                 else
1120                         return e;
1121         } else if (is_a<power>(e1)) {
1122                 // apply the procedure to the base of a power
1123                 return pow(clifford_star_bar(e1.op(0), do_bar, 0), e1.op(1));
1124         } else if (is_a<add>(e1) || is_a<mul>(e1) || e.info(info_flags::list)) {
1125                 // recurse into subexpressions
1126                 return e1.map(fcn);
1127         } else  // nothing meaningful can be done
1128                 return e;
1129 }
1130
1131 ex clifford_prime(const ex & e)
1132 {
1133         pointer_to_map_function fcn(clifford_prime);
1134         if (is_a<clifford>(e) && is_a<cliffordunit>(e.op(0))) {
1135                 return -e;
1136         } else if (is_a<add>(e) || is_a<ncmul>(e) || is_a<mul>(e) //|| is_a<pseries>(e) || is_a<integral>(e)
1137                            || is_a<matrix>(e) || e.info(info_flags::list)) {
1138                 return e.map(fcn);
1139         } else if (is_a<power>(e)) {
1140                 return pow(clifford_prime(e.op(0)), e.op(1));
1141         } else
1142                 return e;
1143 }
1144
1145 ex remove_dirac_ONE(const ex & e, unsigned char rl, unsigned options)
1146 {
1147         pointer_to_map_function_2args<unsigned char, unsigned> fcn(remove_dirac_ONE, rl, options | 1);
1148         bool need_reevaluation = false;
1149         ex e1 = e;
1150         if (! (options & 1) )  { // is not a child
1151                 if (options & 2)
1152                         e1 = expand_dummy_sum(e, true);
1153                 e1 = canonicalize_clifford(e1);
1154         }
1155         
1156         if (is_a<clifford>(e1) && ex_to<clifford>(e1).get_representation_label() >= rl) {
1157                 if (is_a<diracone>(e1.op(0)))
1158                         return 1;
1159                 else 
1160                         throw(std::invalid_argument("remove_dirac_ONE(): expression is a non-scalar Clifford number!"));
1161         } else if (is_a<add>(e1) || is_a<ncmul>(e1) || is_a<mul>(e1)  
1162                            || is_a<matrix>(e1) || e1.info(info_flags::list)) {
1163                 if (options & 3) // is a child or was already expanded
1164                         return e1.map(fcn);
1165                 else
1166                         try {
1167                                 return e1.map(fcn);
1168                         } catch (std::exception &p) {
1169                                 need_reevaluation = true;
1170                         }
1171         } else if (is_a<power>(e1)) {
1172                 if (options & 3) // is a child or was already expanded
1173                         return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
1174                 else
1175                         try {
1176                                 return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
1177                         } catch (std::exception &p) {
1178                                 need_reevaluation = true;
1179                         }
1180         } 
1181         if (need_reevaluation)
1182                 return remove_dirac_ONE(e, rl, options | 2);
1183         return e1;
1184 }
1185
1186 int clifford_max_label(const ex & e, bool ignore_ONE)
1187 {
1188         if (is_a<clifford>(e))
1189                 if (ignore_ONE && is_a<diracone>(e.op(0)))
1190                         return -1;
1191                 else
1192                         return ex_to<clifford>(e).get_representation_label();
1193         else {
1194                 int rl = -1;
1195                 for (size_t i=0; i < e.nops(); i++) 
1196                         rl = (rl > clifford_max_label(e.op(i), ignore_ONE)) ? rl : clifford_max_label(e.op(i), ignore_ONE);
1197                 return rl;
1198         }
1199 }
1200
1201 ex clifford_norm(const ex & e)
1202 {
1203         return sqrt(remove_dirac_ONE(e * clifford_bar(e)));
1204 }
1205         
1206 ex clifford_inverse(const ex & e)
1207 {
1208         ex norm = clifford_norm(e);
1209         if (!norm.is_zero())
1210                 return clifford_bar(e) / pow(norm, 2);
1211         else 
1212                 throw(std::invalid_argument("clifford_inverse(): cannot find inverse of Clifford number with zero norm!"));
1213 }
1214
1215 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl)
1216 {
1217         if (!ex_to<idx>(mu).is_dim_numeric())
1218                 throw(std::invalid_argument("lst_to_clifford(): Index should have a numeric dimension"));
1219         ex e = clifford_unit(mu, metr, rl);
1220         return lst_to_clifford(v, e);
1221 }
1222
1223 ex lst_to_clifford(const ex & v, const ex & e) {
1224         unsigned min, max;
1225
1226         if (is_a<clifford>(e)) {
1227                 ex mu = e.op(1);
1228                 ex mu_toggle
1229                         = is_a<varidx>(mu) ? ex_to<varidx>(mu).toggle_variance() : mu;
1230                 unsigned dim = get_dim_uint(mu);
1231
1232                 if (is_a<matrix>(v)) {
1233                         if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows()) {
1234                                 min = ex_to<matrix>(v).rows();
1235                                 max = ex_to<matrix>(v).cols();
1236                         } else {
1237                                 min = ex_to<matrix>(v).cols();
1238                                 max = ex_to<matrix>(v).rows();
1239                         }
1240                         if (min == 1) {
1241                                 if (dim == max)
1242                                         return indexed(v, mu_toggle) * e;
1243                                 else if (max - dim == 1) {
1244                                         if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows())
1245                                                 return v.op(0) * dirac_ONE(ex_to<clifford>(e).get_representation_label()) + indexed(sub_matrix(ex_to<matrix>(v), 0, 1, 1, dim), mu_toggle) * e;
1246                                         else 
1247                                                 return v.op(0) * dirac_ONE(ex_to<clifford>(e).get_representation_label()) + indexed(sub_matrix(ex_to<matrix>(v), 1, dim, 0, 1), mu_toggle) * e;
1248                                 } else
1249                                         throw(std::invalid_argument("lst_to_clifford(): dimensions of vector and clifford unit mismatch"));
1250                         } else
1251                                 throw(std::invalid_argument("lst_to_clifford(): first argument should be a vector (nx1 or 1xn matrix)"));
1252                 } else if (v.info(info_flags::list)) {
1253                         if (dim == ex_to<lst>(v).nops())
1254                                 return indexed(matrix(dim, 1, ex_to<lst>(v)), mu_toggle) * e;
1255                         else if (ex_to<lst>(v).nops() - dim == 1)
1256                                 return v.op(0) * dirac_ONE(ex_to<clifford>(e).get_representation_label()) + indexed(sub_matrix(matrix(dim+1, 1, ex_to<lst>(v)), 1, dim, 0, 1), mu_toggle) * e;
1257                         else
1258                                 throw(std::invalid_argument("lst_to_clifford(): list length and dimension of clifford unit mismatch"));
1259                 } else
1260                         throw(std::invalid_argument("lst_to_clifford(): cannot construct from anything but list or vector"));
1261         } else
1262                 throw(std::invalid_argument("lst_to_clifford(): the second argument should be a Clifford unit"));
1263 }
1264
1265 /** Auxiliary structure to define a function for striping one Clifford unit
1266  * from vectors. Used in  clifford_to_lst(). */
1267 static ex get_clifford_comp(const ex & e, const ex & c, bool root=true)
1268 {
1269         // make expansion on the top-level call only
1270         ex e1=(root? e.expand() : e);
1271
1272         pointer_to_map_function_2args<const ex &, bool> fcn(get_clifford_comp, c, false);
1273         int ival = ex_to<numeric>(ex_to<idx>(c.op(1)).get_value()).to_int();
1274         int rl=ex_to<clifford>(c).get_representation_label();
1275
1276         if ( (is_a<add>(e1) || e1.info(info_flags::list) || is_a<matrix>(e1))) {
1277                 return e1.map(fcn);
1278         } else if (is_a<ncmul>(e1) || is_a<mul>(e1)) {
1279                 // searches are done within products only
1280                 exvector ev, all_dummy=get_all_dummy_indices(e1);
1281                 bool found=false, same_value_found=false;
1282                 ex dummy_ind=0;
1283                 ev.reserve(e1.nops());
1284                 for (int i=0; i < e1.nops();++i) {
1285                         // look for a Clifford unit with the same metric and representation label,
1286                         // if found remember its index
1287                         if (is_a<clifford>(e1.op(i)) && ex_to<clifford>(e1.op(i)).get_representation_label() == rl
1288                                 && is_a<cliffordunit>(e1.op(i).op(0)) && ex_to<clifford>(e1.op(i)).same_metric(c)) { // same Clifford unit
1289                                 if (found)
1290                                         throw(std::invalid_argument("get_clifford_comp(): expression is a Clifford multi-vector"));
1291                                 found=true;
1292                                 if (ex_to<idx>(e1.op(i).op(1)).is_numeric() &&
1293                                     (ival == ex_to<numeric>(ex_to<idx>(e1.op(i).op(1)).get_value()).to_int())) {
1294                                         same_value_found = true; // desired index value is found
1295                                 } else if ((std::find(all_dummy.begin(), all_dummy.end(), e1.op(i).op(1)) != all_dummy.end())
1296                                            || (is_a<varidx>(e1.op(i).op(1))
1297                                                && std::find(all_dummy.begin(), all_dummy.end(),
1298                                                             ex_to<varidx>(e1.op(i).op(1)).toggle_variance()) != all_dummy.end())) {
1299                                         dummy_ind=(e1.op(i).op(1)); // suitable dummy index found
1300                                 } else
1301                                         ev.push_back(e.op(i)); // another index value
1302                         } else
1303                                 ev.push_back(e1.op(i));
1304                 }
1305
1306                 if (! found) // no Clifford units found at all
1307                         throw(std::invalid_argument("get_clifford_comp(): expression is not a Clifford vector to the given units"));
1308
1309                 ex res=dynallocate<ncmul>(std::move(ev));
1310                 if (same_value_found) {
1311                         return  res;
1312                 } else if (! dummy_ind.is_zero()) { // a dummy index was found
1313                         if (is_a<varidx>(dummy_ind))
1314                                 dummy_ind = ex_to<varidx>(dummy_ind).toggle_variance();
1315                         return res.subs(dummy_ind==ival, subs_options::no_pattern);
1316                 } else // found a Clifford unit with another index
1317                         return 0;
1318         } else if (e1.is_zero()) {
1319                 return 0;
1320         } else if (is_a<clifford>(e1) && is_a<cliffordunit>(e1.op(0)) && ex_to<clifford>(e1).same_metric(c)) {
1321                 if (ex_to<idx>(e1.op(1)).is_numeric() &&
1322                     (ival == ex_to<numeric>(ex_to<idx>(e1.op(1)).get_value()).to_int()) )
1323                         return 1;
1324                 else
1325                         return 0;
1326         } else
1327                 throw(std::invalid_argument("get_clifford_comp(): expression is not usable as a Clifford vector"));
1328 }
1329
1330 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic)
1331 {
1332         GINAC_ASSERT(is_a<clifford>(c));
1333         ex mu = c.op(1);
1334         if (! ex_to<idx>(mu).is_dim_numeric())
1335                 throw(std::invalid_argument("clifford_to_lst(): index should have a numeric dimension"));
1336         unsigned int D = ex_to<numeric>(ex_to<idx>(mu).get_dim()).to_int();
1337
1338         if (algebraic) // check if algebraic method is applicable
1339                 for (unsigned int i = 0; i < D; i++) 
1340                         if (pow(c.subs(mu == i, subs_options::no_pattern), 2).is_zero() 
1341                                 || (! is_a<numeric>(pow(c.subs(mu == i, subs_options::no_pattern), 2))))
1342                                 algebraic = false;
1343         lst V; 
1344         ex v0 = remove_dirac_ONE(canonicalize_clifford(e+clifford_prime(e)))/2;
1345         if (! v0.is_zero())
1346                 V.append(v0);
1347         ex e1 = canonicalize_clifford(e - v0 * dirac_ONE(ex_to<clifford>(c).get_representation_label())); 
1348         if (algebraic) {
1349                 for (unsigned int i = 0; i < D; i++) 
1350                         V.append(remove_dirac_ONE(
1351                                                 simplify_indexed(canonicalize_clifford(e1 * c.subs(mu == i, subs_options::no_pattern) +  c.subs(mu == i, subs_options::no_pattern) * e1))
1352                                                 / (2*pow(c.subs(mu == i, subs_options::no_pattern), 2))));
1353         } else {
1354                 try {
1355                         for (unsigned int i = 0; i < D; i++)
1356                                 V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
1357                 } catch  (std::exception &p) {
1358                         /* Try to expand dummy summations to simplify the expression*/
1359                         e1 = canonicalize_clifford(expand_dummy_sum(e, true));
1360                         V.remove_all();
1361                         v0 = remove_dirac_ONE(canonicalize_clifford(e1+clifford_prime(e1)))/2;
1362                         if (! v0.is_zero()) {
1363                                 V.append(v0);
1364                                 e1 = canonicalize_clifford(e1 - v0 * dirac_ONE(ex_to<clifford>(c).get_representation_label())); 
1365                         }
1366                         for (unsigned int i = 0; i < D; i++) 
1367                                 V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
1368                 }
1369         }
1370         return V;
1371 }
1372
1373
1374 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c, const ex & d, const ex & v, const ex & G, unsigned char rl)
1375 {
1376         ex x, D, cu;
1377         
1378         if (! is_a<matrix>(v) && ! v.info(info_flags::list))
1379                 throw(std::invalid_argument("clifford_moebius_map(): parameter v should be either vector or list"));
1380         
1381         if (is_a<clifford>(G)) {
1382                 cu = G;
1383         } else {
1384                 if (is_a<indexed>(G)) {
1385                         D = ex_to<idx>(G.op(1)).get_dim();
1386                         varidx mu(dynallocate<symbol>(), D);
1387                         cu = clifford_unit(mu, G, rl);
1388                 } else if (is_a<matrix>(G)) {
1389                         D = ex_to<matrix>(G).rows(); 
1390                         idx mu(dynallocate<symbol>(), D);
1391                         cu = clifford_unit(mu, G, rl);
1392                 } else throw(std::invalid_argument("clifford_moebius_map(): metric should be an indexed object, matrix, or a Clifford unit"));
1393                 
1394         }
1395         
1396         x = lst_to_clifford(v, cu); 
1397         ex e = clifford_to_lst(simplify_indexed(canonicalize_clifford((a * x + b) * clifford_inverse(c * x + d))), cu, false);
1398         return (is_a<matrix>(v) ? matrix(ex_to<matrix>(v).rows(), ex_to<matrix>(v).cols(), ex_to<lst>(e)) : e);
1399 }
1400
1401 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl)
1402 {
1403         if (is_a<matrix>(M) && (ex_to<matrix>(M).rows() == 2) && (ex_to<matrix>(M).cols() == 2)) 
1404                 return clifford_moebius_map(M.op(0), M.op(1), M.op(2), M.op(3), v, G, rl);
1405         else
1406                 throw(std::invalid_argument("clifford_moebius_map(): parameter M should be a 2x2 matrix"));
1407 }
1408
1409 } // namespace GiNaC