ginac/clifford.cpp

   1 /** @file clifford.cpp
   2  *
   3  *  Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2015 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
  49 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracone, tensor,
  50   print_func<print_dflt>(&diracone::do_print).
  51   print_func<print_latex>(&diracone::do_print_latex))
  52
  53 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(cliffordunit, tensor,
  54   print_func<print_dflt>(&cliffordunit::do_print).
  55   print_func<print_latex>(&cliffordunit::do_print_latex))
  56
  57 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma, cliffordunit,
  58   print_func<print_dflt>(&diracgamma::do_print).
  59   print_func<print_latex>(&diracgamma::do_print_latex))
  60
  61 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma5, tensor,
  62   print_func<print_dflt>(&diracgamma5::do_print).
  63   print_func<print_latex>(&diracgamma5::do_print_latex))
  64
  65 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaL, tensor,
  66   print_func<print_context>(&diracgammaL::do_print).
  67   print_func<print_latex>(&diracgammaL::do_print_latex))
  68
  69 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaR, tensor,
  70   print_func<print_context>(&diracgammaR::do_print).
  71   print_func<print_latex>(&diracgammaR::do_print_latex))
  72
  73 //////////
  74 // default constructors
  75 //////////
  76
  77 clifford::clifford() : representation_label(0), metric(0), commutator_sign(-1)
  78 {
  79 }
  80
  81 DEFAULT_CTOR(diracone)
  82 DEFAULT_CTOR(cliffordunit)
  83 DEFAULT_CTOR(diracgamma)
  84 DEFAULT_CTOR(diracgamma5)
  85 DEFAULT_CTOR(diracgammaL)
  86 DEFAULT_CTOR(diracgammaR)
  87
  88 //////////
  89 // other constructors
  90 //////////
  91
  92 /** Construct object without any indices. This constructor is for internal
  93  *  use only. Use the dirac_ONE() function instead.
  94  *  @see dirac_ONE */
  95 clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl), metric(0), commutator_sign(-1)
  96 {
  97 }
  98
  99 /** Construct object with one Lorentz index. This constructor is for internal
 100  *  use only. Use the clifford_unit() or dirac_gamma() functions instead.
 101  *  @see clifford_unit
 102  *  @see dirac_gamma */
 103 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)
 104 {
 105         GINAC_ASSERT(is_a<idx>(mu));
 106 }
 107
 108 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)
 109 {
 110 }
 111
 112 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)
 113 {
 114 }
 115
 116 return_type_t clifford::return_type_tinfo() const
 117 {
 118         return make_return_type_t<clifford>(representation_label);
 119 }
 120
 121 //////////
 122 // archiving
 123 //////////
 124
 125 void clifford::read_archive(const archive_node& n, lst& sym_lst)
 126 {
 127         inherited::read_archive(n, sym_lst);
 128         unsigned rl;
 129         n.find_unsigned("label", rl);
 130         representation_label = rl;
 131         n.find_ex("metric", metric, sym_lst);
 132         n.find_unsigned("commutator_sign+1", rl);
 133         commutator_sign = rl - 1;
 134 }
 135
 136 void clifford::archive(archive_node & n) const
 137 {
 138         inherited::archive(n);
 139         n.add_unsigned("label", representation_label);
 140         n.add_ex("metric", metric);
 141         n.add_unsigned("commutator_sign+1", commutator_sign+1);
 142 }
 143
 144 GINAC_BIND_UNARCHIVER(clifford);
 145 GINAC_BIND_UNARCHIVER(cliffordunit);
 146 GINAC_BIND_UNARCHIVER(diracone);
 147 GINAC_BIND_UNARCHIVER(diracgamma);
 148 GINAC_BIND_UNARCHIVER(diracgamma5);
 149 GINAC_BIND_UNARCHIVER(diracgammaL);
 150 GINAC_BIND_UNARCHIVER(diracgammaR);
 151
 152
 153 ex clifford::get_metric(const ex & i, const ex & j, bool symmetrised) const
 154 {
 155         if (is_a<indexed>(metric)) {
 156                 if (symmetrised && !(ex_to<symmetry>(ex_to<indexed>(metric).get_symmetry()).has_symmetry())) {
 157                         if (is_a<matrix>(metric.op(0))) {
 158                                 return indexed((ex_to<matrix>(metric.op(0)).add(ex_to<matrix>(metric.op(0)).transpose())).mul(numeric(1, 2)),
 159                                                symmetric2(), i, j);
 160                         } else {
 161                                 return simplify_indexed(indexed(metric.op(0)*_ex1_2, i, j) + indexed(metric.op(0)*_ex1_2, j, i));
 162                         }
 163                 } else {
 164                         return metric.subs(lst{metric.op(1) == i, metric.op(2) == j}, subs_options::no_pattern);
 165                 }
 166         } else {
 167                 exvector indices = metric.get_free_indices();
 168                 if (symmetrised)
 169                         return _ex1_2*simplify_indexed(metric.subs(lst{indices[0] == i, indices[1] == j}, subs_options::no_pattern)
 170                                                      + metric.subs(lst{indices[0] == j, indices[1] == i}, subs_options::no_pattern));
 171                 else
 172                         return metric.subs(lst{indices[0] == i, indices[1] == j}, subs_options::no_pattern);
 173         }
 174 }
 175
 176 bool clifford::same_metric(const ex & other) const
 177 {
 178         ex metr;
 179         if (is_a<clifford>(other))
 180                 metr = ex_to<clifford>(other).get_metric();
 181         else
 182                 metr = other;
 183
 184         if (is_a<indexed>(metr))
 185                 return metr.op(0).is_equal(get_metric().op(0));
 186         else {
 187                 exvector indices = metr.get_free_indices();
 188                 return  (indices.size() == 2)
 189                         && simplify_indexed(get_metric(indices[0], indices[1])-metr).is_zero();
 190         }
 191 }
 192
 193 //////////
 194 // functions overriding virtual functions from base classes
 195 //////////
 196
 197 ex clifford::op(size_t i) const
 198 {
 199         GINAC_ASSERT(i<nops());
 200         if (nops()-i == 1)
 201                 return representation_label;
 202         else
 203                 return inherited::op(i);
 204 }
 205
 206 ex & clifford::let_op(size_t i)
 207 {
 208         GINAC_ASSERT(i<nops());
 209
 210         static ex rl = numeric(representation_label);
 211         ensure_if_modifiable();
 212         if (nops()-i == 1)
 213                 return rl;
 214         else
 215                 return inherited::let_op(i);
 216 }
 217
 218 ex clifford::subs(const exmap & m, unsigned options) const
 219 {
 220         ex subsed = inherited::subs(m, options);
 221         if(is_a<clifford>(subsed)) {
 222                 ex prevmetric = ex_to<clifford>(subsed).metric;
 223                 ex newmetric = prevmetric.subs(m, options);
 224                 if(!are_ex_trivially_equal(prevmetric, newmetric)) {
 225                         clifford c = ex_to<clifford>(subsed);
 226                         c.metric = newmetric;
 227                         subsed = c;
 228                 }
 229         }
 230         return subsed;
 231 }
 232
 233 int clifford::compare_same_type(const basic & other) const
 234 {
 235         GINAC_ASSERT(is_a<clifford>(other));
 236         const clifford &o = static_cast<const clifford &>(other);
 237
 238         if (representation_label != o.representation_label) {
 239                 // different representation label
 240                 return representation_label < o.representation_label ? -1 : 1;
 241         }
 242
 243         return inherited::compare_same_type(other);
 244 }
 245
 246 bool clifford::match_same_type(const basic & other) const
 247 {
 248         GINAC_ASSERT(is_a<clifford>(other));
 249         const clifford &o = static_cast<const clifford &>(other);
 250
 251         return ((representation_label == o.representation_label) && (commutator_sign == o.get_commutator_sign()) && same_metric(o));
 252 }
 253
 254 static bool is_dirac_slash(const ex & seq0)
 255 {
 256         return !is_a<diracgamma5>(seq0) && !is_a<diracgammaL>(seq0) &&
 257                !is_a<diracgammaR>(seq0) && !is_a<cliffordunit>(seq0) &&
 258                !is_a<diracone>(seq0);
 259 }
 260
 261 void clifford::do_print_dflt(const print_dflt & c, unsigned level) const
 262 {
 263         // dirac_slash() object is printed differently
 264         if (is_dirac_slash(seq[0])) {
 265                 seq[0].print(c, precedence());
 266                 c.s << "\\";
 267         } else { // We do not print representation label if it is 0
 268                 if (representation_label == 0) {
 269                         this->print_dispatch<inherited>(c, level);
 270                 } else { // otherwise we put it before indices in square brackets; the code is borrowed from indexed.cpp
 271                         if (precedence() <= level) {
 272                                 c.s << '(';
 273                         }
 274                         seq[0].print(c, precedence());
 275                         c.s << '[' << int(representation_label) << ']';
 276                         printindices(c, level);
 277                         if (precedence() <= level) {
 278                                 c.s << ')';
 279                         }
 280                 }
 281         }
 282 }
 283
 284 void clifford::do_print_latex(const print_latex & c, unsigned level) const
 285 {
 286         // dirac_slash() object is printed differently
 287         if (is_dirac_slash(seq[0])) {
 288                 c.s << "{";
 289                 seq[0].print(c, precedence());
 290                 c.s << "\\hspace{-1.0ex}/}";
 291         } else {
 292                 c.s << "\\clifford[" << int(representation_label) << "]";
 293                 this->print_dispatch<inherited>(c, level);
 294         }
 295 }
 296
 297 DEFAULT_COMPARE(diracone)
 298 DEFAULT_COMPARE(cliffordunit)
 299 DEFAULT_COMPARE(diracgamma)
 300 DEFAULT_COMPARE(diracgamma5)
 301 DEFAULT_COMPARE(diracgammaL)
 302 DEFAULT_COMPARE(diracgammaR)
 303
 304 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbf{1}")
 305 DEFAULT_PRINT_LATEX(cliffordunit, "e", "e")
 306 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
 307 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
 308 DEFAULT_PRINT_LATEX(diracgammaL, "gammaL", "{\\gamma_L}")
 309 DEFAULT_PRINT_LATEX(diracgammaR, "gammaR", "{\\gamma_R}")
 310
 311 /** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
 312 static void base_and_index(const ex & c, ex & b, ex & i)
 313 {
 314         GINAC_ASSERT(is_a<clifford>(c));
 315         GINAC_ASSERT(c.nops() == 2+1);
 316
 317         if (is_a<cliffordunit>(c.op(0))) { // proper dirac gamma object or clifford unit
 318                 i = c.op(1);
 319                 b = _ex1;
 320         } else if (is_a<diracgamma5>(c.op(0)) || is_a<diracgammaL>(c.op(0)) || is_a<diracgammaR>(c.op(0))) { // gamma5/L/R
 321                 i = _ex0;
 322                 b = _ex1;
 323         } else { // slash object, generate new dummy index
 324                 varidx ix(dynallocate<symbol>(), ex_to<idx>(c.op(1)).get_dim());
 325                 b = indexed(c.op(0), ix.toggle_variance());
 326                 i = ix;
 327         }
 328 }
 329
 330 /** Predicate for finding non-clifford objects. */
 331 struct is_not_a_clifford : public std::unary_function<ex, bool> {
 332         bool operator()(const ex & e)
 333         {
 334                 return !is_a<clifford>(e);
 335         }
 336 };
 337
 338 /** Contraction of a gamma matrix with something else. */
 339 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 340 {
 341         GINAC_ASSERT(is_a<clifford>(*self));
 342         GINAC_ASSERT(is_a<indexed>(*other));
 343         GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
 344         unsigned char rl = ex_to<clifford>(*self).get_representation_label();
 345
 346         ex dim = ex_to<idx>(self->op(1)).get_dim();
 347         if (other->nops() > 1)
 348                 dim = minimal_dim(dim, ex_to<idx>(other->op(1)).get_dim());
 349
 350         if (is_a<clifford>(*other)) {
 351
 352                 // Contraction only makes sense if the representation labels are equal
 353                 if (ex_to<clifford>(*other).get_representation_label() != rl)
 354                         return false;
 355
 356                 size_t num = other - self;
 357
 358                 // gamma~mu gamma.mu = dim ONE
 359                 if (num == 1) {
 360                         *self = dim;
 361                         *other = dirac_ONE(rl);
 362                         return true;
 363
 364                 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
 365                 } else if (num == 2
 366                         && is_a<clifford>(self[1])) {
 367                         *self = 2 - dim;
 368                         *other = _ex1;
 369                         return true;
 370
 371                 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
 372                 } else if (num == 3
 373                         && is_a<clifford>(self[1])
 374                         && is_a<clifford>(self[2])) {
 375                         ex b1, i1, b2, i2;
 376                         base_and_index(self[1], b1, i1);
 377                         base_and_index(self[2], b2, i2);
 378                         *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
 379                         self[1] = _ex1;
 380                         self[2] = _ex1;
 381                         *other = _ex1;
 382                         return true;
 383
 384                 // 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
 385                 } else if (num == 4
 386                         && is_a<clifford>(self[1])
 387                         && is_a<clifford>(self[2])
 388                         && is_a<clifford>(self[3])) {
 389                         *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3];
 390                         self[1] = _ex1;
 391                         self[2] = _ex1;
 392                         self[3] = _ex1;
 393                         *other = _ex1;
 394                         return true;
 395
 396                 // gamma~mu Sodd gamma.mu = -2 Sodd_R
 397                 // (Chisholm identity in 4 dimensions)
 398                 } else if (!((other - self) & 1) && dim.is_equal(4)) {
 399                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
 400                                 return false;
 401
 402                         *self = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(other), std::reverse_iterator<exvector::const_iterator>(self + 1)));
 403                         std::fill(self + 1, other, _ex1);
 404                         *other = _ex_2;
 405                         return true;
 406
 407                 // gamma~mu Sodd gamma~alpha gamma.mu = 2 gamma~alpha Sodd + 2 Sodd_R gamma~alpha
 408                 // (commutate contracted indices towards each other, then use
 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                         exvector::iterator next_to_last = other - 1;
 415                         ex S = ncmul(exvector(self + 1, next_to_last));
 416                         ex SR = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(next_to_last), std::reverse_iterator<exvector::const_iterator>(self + 1)));
 417
 418                         *self = (*next_to_last) * S + SR * (*next_to_last);
 419                         std::fill(self + 1, other, _ex1);
 420                         *other = _ex2;
 421                         return true;
 422
 423                 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
 424                 // (commutate contracted indices towards each other, simplify_indexed()
 425                 // will re-expand and re-run the simplification)
 426                 } else {
 427                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
 428                                 return false;
 429
 430                         exvector::iterator next_to_last = other - 1;
 431                         ex S = ncmul(exvector(self + 1, next_to_last));
 432
 433                         *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
 434                         std::fill(self + 1, other + 1, _ex1);
 435                         return true;
 436                 }
 437
 438         } else if (is_a<symbol>(other->op(0)) && other->nops() == 2) {
 439
 440                 // x.mu gamma~mu -> x-slash
 441                 *self = dirac_slash(other->op(0), dim, rl);
 442                 *other = _ex1;
 443                 return true;
 444         }
 445
 446         return false;
 447 }
 448
 449 /** Contraction of a Clifford unit with something else. */
 450 bool cliffordunit::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 451 {
 452         GINAC_ASSERT(is_a<clifford>(*self));
 453         GINAC_ASSERT(is_a<indexed>(*other));
 454         GINAC_ASSERT(is_a<cliffordunit>(self->op(0)));
 455         clifford unit = ex_to<clifford>(*self);
 456         unsigned char rl = unit.get_representation_label();
 457
 458         if (is_a<clifford>(*other)) {
 459                 // Contraction only makes sense if the representation labels are equal
 460                 // and the metrics are the same
 461                 if ((ex_to<clifford>(*other).get_representation_label() != rl)
 462                     && unit.same_metric(*other))
 463                         return false;
 464
 465                 exvector::iterator before_other = other - 1;
 466                 ex mu = self->op(1);
 467                 ex mu_toggle = other->op(1);
 468                 ex alpha = before_other->op(1);
 469
 470                 // e~mu e.mu = Tr ONE
 471                 if (other - self == 1) {
 472                         *self = unit.get_metric(mu, mu_toggle, true);
 473                         *other = dirac_ONE(rl);
 474                         return true;
 475
 476                 } else if (other - self == 2) {
 477                         if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
 478                                 // e~mu e~alpha e.mu = 2*e~mu B(alpha, mu.toggle_variance())-Tr(B) e~alpha
 479                                 *self = 2 * (*self) * unit.get_metric(alpha, mu_toggle, true) - unit.get_metric(mu, mu_toggle, true) * (*before_other);
 480                                 *before_other = _ex1;
 481                                 *other = _ex1;
 482                                 return true;
 483
 484                         } else {
 485                                 // e~mu S e.mu = Tr S ONE
 486                                 *self = unit.get_metric(mu, mu_toggle, true);
 487                                 *other = dirac_ONE(rl);
 488                                 return true;
 489                         }
 490                 } else {
 491                 // e~mu S e~alpha e.mu = 2 e~mu S B(alpha, mu.toggle_variance()) - e~mu S e.mu e~alpha
 492                 // (commutate contracted indices towards each other, simplify_indexed()
 493                 // will re-expand and re-run the simplification)
 494                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other) {
 495                                 return false;
 496                         }
 497
 498                         ex S = ncmul(exvector(self + 1, before_other));
 499
 500                         if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
 501                                 *self = 2 * (*self) * S * unit.get_metric(alpha, mu_toggle, true) - (*self) * S * (*other) * (*before_other);
 502                         } else {
 503                                 // simply commutes
 504                                 *self = (*self) * S * (*other) * (*before_other);
 505                         }
 506
 507                         std::fill(self + 1, other + 1, _ex1);
 508                         return true;
 509                 }
 510         }
 511         return false;
 512 }
 513
 514 /** Perform automatic simplification on noncommutative product of clifford
 515  *  objects. This removes superfluous ONEs, permutes gamma5/L/R's to the front
 516  *  and removes squares of gamma objects. */
 517 ex clifford::eval_ncmul(const exvector & v) const
 518 {
 519         exvector s;
 520         s.reserve(v.size());
 521
 522         // Remove superfluous ONEs
 523         for (auto & it : v) {
 524                 if (!is_a<clifford>(it) || !is_a<diracone>(it.op(0)))
 525                         s.push_back(it);
 526         }
 527
 528         bool something_changed = false;
 529         int sign = 1;
 530
 531         // Anticommutate gamma5/L/R's to the front
 532         if (s.size() >= 2) {
 533                 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
 534                 while (true) {
 535                         exvector::iterator it = next_to_last;
 536                         while (true) {
 537                                 exvector::iterator it2 = it + 1;
 538                                 if (is_a<clifford>(*it) && is_a<clifford>(*it2)) {
 539                                         ex e1 = it->op(0), e2 = it2->op(0);
 540
 541                                         if (is_a<diracgamma5>(e2)) {
 542
 543                                                 if (is_a<diracgammaL>(e1) || is_a<diracgammaR>(e1)) {
 544
 545                                                         // gammaL/R gamma5 -> gamma5 gammaL/R
 546                                                         it->swap(*it2);
 547                                                         something_changed = true;
 548
 549                                                 } else if (!is_a<diracgamma5>(e1)) {
 550
 551                                                         // gamma5 gamma5 -> gamma5 gamma5 (do nothing)
 552                                                         // x gamma5 -> -gamma5 x
 553                                                         it->swap(*it2);
 554                                                         sign = -sign;
 555                                                         something_changed = true;
 556                                                 }
 557
 558                                         } else if (is_a<diracgammaL>(e2)) {
 559
 560                                                 if (is_a<diracgammaR>(e1)) {
 561
 562                                                         // gammaR gammaL -> 0
 563                                                         return _ex0;
 564
 565                                                 } else if (!is_a<diracgammaL>(e1) && !is_a<diracgamma5>(e1)) {
 566
 567                                                         // gammaL gammaL -> gammaL gammaL (do nothing)
 568                                                         // gamma5 gammaL -> gamma5 gammaL (do nothing)
 569                                                         // x gammaL -> gammaR x
 570                                                         it->swap(*it2);
 571                                                         *it = clifford(diracgammaR(), ex_to<clifford>(*it).get_representation_label());
 572                                                         something_changed = true;
 573                                                 }
 574
 575                                         } else if (is_a<diracgammaR>(e2)) {
 576
 577                                                 if (is_a<diracgammaL>(e1)) {
 578
 579                                                         // gammaL gammaR -> 0
 580                                                         return _ex0;
 581
 582                                                 } else if (!is_a<diracgammaR>(e1) && !is_a<diracgamma5>(e1)) {
 583
 584                                                         // gammaR gammaR -> gammaR gammaR (do nothing)
 585                                                         // gamma5 gammaR -> gamma5 gammaR (do nothing)
 586                                                         // x gammaR -> gammaL x
 587                                                         it->swap(*it2);
 588                                                         *it = clifford(diracgammaL(), ex_to<clifford>(*it).get_representation_label());
 589                                                         something_changed = true;
 590                                                 }
 591                                         }
 592                                 }
 593                                 if (it == first)
 594                                         break;
 595                                 --it;
 596                         }
 597                         if (next_to_last == first)
 598                                 break;
 599                         --next_to_last;
 600                 }
 601         }
 602
 603         // Remove equal adjacent gammas
 604         if (s.size() >= 2) {
 605                 exvector::iterator it, itend = s.end() - 1;
 606                 for (it = s.begin(); it != itend; ++it) {
 607                         ex & a = it[0];
 608                         ex & b = it[1];
 609                         if (!is_a<clifford>(a) || !is_a<clifford>(b))
 610                                 continue;
 611
 612                         const ex & ag = a.op(0);
 613                         const ex & bg = b.op(0);
 614                         bool a_is_cliffordunit = is_a<cliffordunit>(ag);
 615                         bool b_is_cliffordunit =  is_a<cliffordunit>(bg);
 616
 617                         if (a_is_cliffordunit && b_is_cliffordunit && ex_to<clifford>(a).same_metric(b)
 618                                 && (ex_to<clifford>(a).get_commutator_sign() == -1)) {
 619                                 // This is done only for Clifford algebras
 620
 621                                 const ex & ia = a.op(1);
 622                                 const ex & ib = b.op(1);
 623                                 if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
 624                                         a = ex_to<clifford>(a).get_metric(ia, ib, true);
 625                                         b = dirac_ONE(representation_label);
 626                                         something_changed = true;
 627                                 }
 628
 629                         } else if ((is_a<diracgamma5>(ag) && is_a<diracgamma5>(bg))) {
 630
 631                                 // Remove squares of gamma5
 632                                 a = dirac_ONE(representation_label);
 633                                 b = dirac_ONE(representation_label);
 634                                 something_changed = true;
 635
 636                         } else if ((is_a<diracgammaL>(ag) && is_a<diracgammaL>(bg))
 637                                 || (is_a<diracgammaR>(ag) && is_a<diracgammaR>(bg))) {
 638
 639                                 // Remove squares of gammaL/R
 640                                 b = dirac_ONE(representation_label);
 641                                 something_changed = true;
 642
 643                         } else if (is_a<diracgammaL>(ag) && is_a<diracgammaR>(bg)) {
 644
 645                                 // gammaL and gammaR are orthogonal
 646                                 return _ex0;
 647
 648                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaL>(bg)) {
 649
 650                                 // gamma5 gammaL -> -gammaL
 651                                 a = dirac_ONE(representation_label);
 652                                 sign = -sign;
 653                                 something_changed = true;
 654
 655                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaR>(bg)) {
 656
 657                                 // gamma5 gammaR -> gammaR
 658                                 a = dirac_ONE(representation_label);
 659                                 something_changed = true;
 660
 661                         } else if (!a_is_cliffordunit && !b_is_cliffordunit && ag.is_equal(bg)) {
 662
 663                                 // a\ a\ -> a^2
 664                                 varidx ix(dynallocate<symbol>(), ex_to<idx>(a.op(1)).minimal_dim(ex_to<idx>(b.op(1))));
 665
 666                                 a = indexed(ag, ix) * indexed(ag, ix.toggle_variance());
 667                                 b = dirac_ONE(representation_label);
 668                                 something_changed = true;
 669                         }
 670                 }
 671         }
 672
 673         if (s.empty())
 674                 return dirac_ONE(representation_label) * sign;
 675         if (something_changed)
 676                 return reeval_ncmul(s) * sign;
 677         else
 678                 return hold_ncmul(s) * sign;
 679 }
 680
 681 ex clifford::thiscontainer(const exvector & v) const
 682 {
 683         return clifford(representation_label, metric, commutator_sign, v);
 684 }
 685
 686 ex clifford::thiscontainer(exvector && v) const
 687 {
 688         return clifford(representation_label, metric, commutator_sign, std::move(v));
 689 }
 690
 691 ex diracgamma5::conjugate() const
 692 {
 693         return _ex_1 * (*this);
 694 }
 695
 696 ex diracgammaL::conjugate() const
 697 {
 698         return dynallocate<diracgammaR>();
 699 }
 700
 701 ex diracgammaR::conjugate() const
 702 {
 703         return dynallocate<diracgammaL>();
 704 }
 705
 706 //////////
 707 // global functions
 708 //////////
 709
 710 ex dirac_ONE(unsigned char rl)
 711 {
 712         static ex ONE = dynallocate<diracone>();
 713         return clifford(ONE, rl);
 714 }
 715
 716 static unsigned get_dim_uint(const ex& e)
 717 {
 718         if (!is_a<idx>(e))
 719                 throw std::invalid_argument("get_dim_uint: argument is not an index");
 720         ex dim = ex_to<idx>(e).get_dim();
 721         if (!dim.info(info_flags::posint))
 722                 throw std::invalid_argument("get_dim_uint: dimension of index should be a positive integer");
 723         unsigned d = ex_to<numeric>(dim).to_int();
 724         return d;
 725 }
 726
 727 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl)
 728 {
 729         ex unit = dynallocate<cliffordunit>();
 730
 731         if (!is_a<idx>(mu))
 732                 throw(std::invalid_argument("clifford_unit(): index of Clifford unit must be of type idx or varidx"));
 733
 734         exvector indices = metr.get_free_indices();
 735
 736         if (indices.size() == 2) {
 737                 return clifford(unit, mu, metr, rl);
 738         } else if (is_a<matrix>(metr)) {
 739                 matrix M = ex_to<matrix>(metr);
 740                 unsigned n = M.rows();
 741                 bool symmetric = true;
 742
 743                 //static idx xi(dynallocate<symbol>(), n),
 744                 //           chi(dynallocate<symbol>(), n);
 745                 idx xi(dynallocate<symbol>(), n),
 746                     chi(dynallocate<symbol>(), n);
 747                 if ((n ==  M.cols()) && (n == get_dim_uint(mu))) {
 748                         for (unsigned i = 0; i < n; i++) {
 749                                 for (unsigned j = i+1; j < n; j++) {
 750                                         if (!M(i, j).is_equal(M(j, i))) {
 751                                                 symmetric = false;
 752                                         }
 753                                 }
 754                         }
 755                         return clifford(unit, mu, indexed(metr, symmetric?symmetric2():not_symmetric(), xi, chi), rl);
 756                 } else {
 757                         throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be a square matrix with the same dimensions as index"));
 758                 }
 759         } else if (indices.size() == 0) { // a tensor or other expression without indices
 760                 //static varidx xi(dynallocate<symbol>(), ex_to<idx>(mu).get_dim()),
 761                 //              chi(dynallocate<symbol>(), ex_to<idx>(mu).get_dim());
 762                 varidx xi(dynallocate<symbol>(), ex_to<idx>(mu).get_dim()),
 763                        chi(dynallocate<symbol>(), ex_to<idx>(mu).get_dim());
 764                 return clifford(unit, mu, indexed(metr, xi, chi), rl);
 765         }  else
 766                 throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be of type tensor, matrix or an expression with two free indices"));
 767 }
 768
 769 ex dirac_gamma(const ex & mu, unsigned char rl)
 770 {
 771         static ex gamma = dynallocate<diracgamma>();
 772
 773         if (!is_a<varidx>(mu))
 774                 throw(std::invalid_argument("dirac_gamma(): index of Dirac gamma must be of type varidx"));
 775
 776         static varidx xi(dynallocate<symbol>(), ex_to<varidx>(mu).get_dim()),
 777                       chi(dynallocate<symbol>(), ex_to<varidx>(mu).get_dim());
 778         return clifford(gamma, mu, indexed(dynallocate<minkmetric>(), symmetric2(), xi, chi), rl);
 779 }
 780
 781 ex dirac_gamma5(unsigned char rl)
 782 {
 783         static ex gamma5 = dynallocate<diracgamma5>();
 784         return clifford(gamma5, rl);
 785 }
 786
 787 ex dirac_gammaL(unsigned char rl)
 788 {
 789         static ex gammaL = dynallocate<diracgammaL>();
 790         return clifford(gammaL, rl);
 791 }
 792
 793 ex dirac_gammaR(unsigned char rl)
 794 {
 795         static ex gammaR = dynallocate<diracgammaR>();
 796         return clifford(gammaR, rl);
 797 }
 798
 799 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
 800 {
 801         // Slashed vectors are actually stored as a clifford object with the
 802         // vector as its base expression and a (dummy) index that just serves
 803         // for storing the space dimensionality
 804
 805         static varidx xi(dynallocate<symbol>(), dim),
 806                       chi(dynallocate<symbol>(), dim);
 807         return clifford(e, varidx(0, dim), indexed(dynallocate<minkmetric>(), symmetric2(), xi, chi), rl);
 808 }
 809
 810 /** Extract representation label from tinfo key (as returned by
 811  *  return_type_tinfo()). */
 812 static unsigned char get_representation_label(const return_type_t& ti)
 813 {
 814         return (unsigned char)ti.rl;
 815 }
 816
 817 /** Take trace of a string of an even number of Dirac gammas given a vector
 818  *  of indices. */
 819 static ex trace_string(exvector::const_iterator ix, size_t num)
 820 {
 821         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 822         if (num == 2)
 823                 return lorentz_g(ix[0], ix[1]);
 824
 825         // 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 )
 826         else if (num == 4)
 827                 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
 828                      + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
 829                      - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
 830
 831         // Traces of 6 or more gammas are computed recursively:
 832         // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
 833         //   + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
 834         //   - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
 835         //   + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
 836         //   - ...
 837         //   + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
 838         exvector v(num - 2);
 839         int sign = 1;
 840         ex result;
 841         for (size_t i=1; i<num; i++) {
 842                 for (size_t n=1, j=0; n<num; n++) {
 843                         if (n == i)
 844                                 continue;
 845                         v[j++] = ix[n];
 846                 }
 847                 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
 848                 sign = -sign;
 849         }
 850         return result;
 851 }
 852
 853 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE)
 854 {
 855         if (is_a<clifford>(e)) {
 856
 857                 unsigned char rl = ex_to<clifford>(e).get_representation_label();
 858
 859                 // Are we taking the trace over this object's representation label?
 860                 if (rls.find(rl) == rls.end())
 861                         return e;
 862
 863                 // Yes, all elements are traceless, except for dirac_ONE and dirac_L/R
 864                 const ex & g = e.op(0);
 865                 if (is_a<diracone>(g))
 866                         return trONE;
 867                 else if (is_a<diracgammaL>(g) || is_a<diracgammaR>(g))
 868                         return trONE/2;
 869                 else
 870                         return _ex0;
 871
 872         } else if (is_exactly_a<mul>(e)) {
 873
 874                 // Trace of product: pull out non-clifford factors
 875                 ex prod = _ex1;
 876                 for (size_t i=0; i<e.nops(); i++) {
 877                         const ex &o = e.op(i);
 878                         if (is_clifford_tinfo(o.return_type_tinfo()))
 879                                 prod *= dirac_trace(o, rls, trONE);
 880                         else
 881                                 prod *= o;
 882                 }
 883                 return prod;
 884
 885         } else if (is_exactly_a<ncmul>(e)) {
 886
 887                 unsigned char rl = get_representation_label(e.return_type_tinfo());
 888
 889                 // Are we taking the trace over this string's representation label?
 890                 if (rls.find(rl) == rls.end())
 891                         return e;
 892
 893                 // Substitute gammaL/R and expand product, if necessary
 894                 ex e_expanded = e.subs(lst{
 895                         dirac_gammaL(rl) == (dirac_ONE(rl)-dirac_gamma5(rl))/2,
 896                         dirac_gammaR(rl) == (dirac_ONE(rl)+dirac_gamma5(rl))/2
 897                 }, subs_options::no_pattern).expand();
 898                 if (!is_a<ncmul>(e_expanded))
 899                         return dirac_trace(e_expanded, rls, trONE);
 900
 901                 // gamma5 gets moved to the front so this check is enough
 902                 bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
 903                 size_t num = e.nops();
 904
 905                 if (has_gamma5) {
 906
 907                         // Trace of gamma5 * odd number of gammas and trace of
 908                         // gamma5 * gamma.mu * gamma.nu are zero
 909                         if ((num & 1) == 0 || num == 3)
 910                                 return _ex0;
 911
 912                         // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
 913                         // (the epsilon is always 4-dimensional)
 914                         if (num == 5) {
 915                                 ex b1, i1, b2, i2, b3, i3, b4, i4;
 916                                 base_and_index(e.op(1), b1, i1);
 917                                 base_and_index(e.op(2), b2, i2);
 918                                 base_and_index(e.op(3), b3, i3);
 919                                 base_and_index(e.op(4), b4, i4);
 920                                 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();
 921                         }
 922
 923                         // Tr gamma5 S_2k =
 924                         //   I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
 925                         // (the epsilon is always 4-dimensional)
 926                         exvector ix(num-1), bv(num-1);
 927                         for (size_t i=1; i<num; i++)
 928                                 base_and_index(e.op(i), bv[i-1], ix[i-1]);
 929                         num--;
 930                         int *iv = new int[num];
 931                         ex result;
 932                         for (size_t i=0; i<num-3; i++) {
 933                                 ex idx1 = ix[i];
 934                                 for (size_t j=i+1; j<num-2; j++) {
 935                                         ex idx2 = ix[j];
 936                                         for (size_t k=j+1; k<num-1; k++) {
 937                                                 ex idx3 = ix[k];
 938                                                 for (size_t l=k+1; l<num; l++) {
 939                                                         ex idx4 = ix[l];
 940                                                         iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
 941                                                         exvector v;
 942                                                         v.reserve(num - 4);
 943                                                         for (size_t n=0, t=4; n<num; n++) {
 944                                                                 if (n == i || n == j || n == k || n == l)
 945                                                                         continue;
 946                                                                 iv[t++] = n;
 947                                                                 v.push_back(ix[n]);
 948                                                         }
 949                                                         int sign = permutation_sign(iv, iv + num);
 950                                                         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))
 951                                                                 * trace_string(v.begin(), num - 4);
 952                                                 }
 953                                         }
 954                                 }
 955                         }
 956                         delete[] iv;
 957                         return trONE * I * result * mul(bv);
 958
 959                 } else { // no gamma5
 960
 961                         // Trace of odd number of gammas is zero
 962                         if ((num & 1) == 1)
 963                                 return _ex0;
 964
 965                         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 966                         if (num == 2) {
 967                                 ex b1, i1, b2, i2;
 968                                 base_and_index(e.op(0), b1, i1);
 969                                 base_and_index(e.op(1), b2, i2);
 970                                 return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
 971                         }
 972
 973                         exvector iv(num), bv(num);
 974                         for (size_t i=0; i<num; i++)
 975                                 base_and_index(e.op(i), bv[i], iv[i]);
 976
 977                         return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
 978                 }
 979
 980         } else if (e.nops() > 0) {
 981
 982                 // Trace maps to all other container classes (this includes sums)
 983                 pointer_to_map_function_2args<const std::set<unsigned char> &, const ex &> fcn(dirac_trace, rls, trONE);
 984                 return e.map(fcn);
 985
 986         } else
 987                 return _ex0;
 988 }
 989
 990 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE)
 991 {
 992         // Convert list to set
 993         std::set<unsigned char> rls;
 994         for (lst::const_iterator i = rll.begin(); i != rll.end(); ++i) {
 995                 if (i->info(info_flags::nonnegint))
 996                         rls.insert(ex_to<numeric>(*i).to_int());
 997         }
 998
 999         return dirac_trace(e, rls, trONE);
1000 }
1001
1002 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
1003 {
1004         // Convert label to set
1005         std::set<unsigned char> rls;
1006         rls.insert(rl);
1007
1008         return dirac_trace(e, rls, trONE);
1009 }
1010
1011
1012 ex canonicalize_clifford(const ex & e_)
1013 {
1014         pointer_to_map_function fcn(canonicalize_clifford);
1015
1016         if (is_a<matrix>(e_)    // || is_a<pseries>(e) || is_a<integral>(e)
1017                 || e_.info(info_flags::list)) {
1018                 return e_.map(fcn);
1019         } else {
1020                 ex e=simplify_indexed(e_);
1021                 // Scan for any ncmul objects
1022                 exmap srl;
1023                 ex aux = e.to_rational(srl);
1024                 for (exmap::iterator i = srl.begin(); i != srl.end(); ++i) {
1025
1026                         ex lhs = i->first;
1027                         ex rhs = i->second;
1028
1029                         if (is_exactly_a<ncmul>(rhs)
1030                                         && rhs.return_type() == return_types::noncommutative
1031                                         && is_clifford_tinfo(rhs.return_type_tinfo())) {
1032
1033                                 // Expand product, if necessary
1034                                 ex rhs_expanded = rhs.expand();
1035                                 if (!is_a<ncmul>(rhs_expanded)) {
1036                                         i->second = canonicalize_clifford(rhs_expanded);
1037                                         continue;
1038
1039                                 } else if (!is_a<clifford>(rhs.op(0)))
1040                                         continue;
1041
1042                                 exvector v;
1043                                 v.reserve(rhs.nops());
1044                                 for (size_t j=0; j<rhs.nops(); j++)
1045                                         v.push_back(rhs.op(j));
1046
1047                                 // Stupid recursive bubble sort because we only want to swap adjacent gammas
1048                                 exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
1049                                 if (is_a<diracgamma5>(it->op(0)) || is_a<diracgammaL>(it->op(0)) || is_a<diracgammaR>(it->op(0)))
1050                                         ++it;
1051
1052                                 while (it != next_to_last) {
1053                                         if (it[0].compare(it[1]) > 0) {
1054
1055                                                 ex save0 = it[0], save1 = it[1];
1056                                                 ex b1, i1, b2, i2;
1057                                                 base_and_index(it[0], b1, i1);
1058                                                 base_and_index(it[1], b2, i2);
1059                                                 // for Clifford algebras (commutator_sign == -1) metric should be symmetrised
1060                                                 it[0] = (ex_to<clifford>(save0).get_metric(i1, i2, ex_to<clifford>(save0).get_commutator_sign() == -1) * b1 * b2).simplify_indexed();
1061                                                 it[1] = v.size() ? _ex2 * dirac_ONE(ex_to<clifford>(save0).get_representation_label()) : _ex2;
1062                                                 ex sum = ncmul(v);
1063                                                 it[0] = save1;
1064                                                 it[1] = save0;
1065                                                 sum += ex_to<clifford>(save0).get_commutator_sign() * ncmul(std::move(v));
1066                                                 i->second = canonicalize_clifford(sum);
1067                                                 goto next_sym;
1068                                         }
1069                                         ++it;
1070                                 }
1071 next_sym:       ;
1072                         }
1073                 }
1074                 return aux.subs(srl, subs_options::no_pattern).simplify_indexed();
1075         }
1076 }
1077
1078 ex clifford_prime(const ex & e)
1079 {
1080         pointer_to_map_function fcn(clifford_prime);
1081         if (is_a<clifford>(e) && is_a<cliffordunit>(e.op(0))) {
1082                 return -e;
1083         } else if (is_a<add>(e) || is_a<ncmul>(e) || is_a<mul>(e) //|| is_a<pseries>(e) || is_a<integral>(e)
1084                            || is_a<matrix>(e) || e.info(info_flags::list)) {
1085                 return e.map(fcn);
1086         } else if (is_a<power>(e)) {
1087                 return pow(clifford_prime(e.op(0)), e.op(1));
1088         } else
1089                 return e;
1090 }
1091
1092 ex remove_dirac_ONE(const ex & e, unsigned char rl, unsigned options)
1093 {
1094         pointer_to_map_function_2args<unsigned char, unsigned> fcn(remove_dirac_ONE, rl, options | 1);
1095         bool need_reevaluation = false;
1096         ex e1 = e;
1097         if (! (options & 1) )  { // is not a child
1098                 if (options & 2)
1099                         e1 = expand_dummy_sum(e, true);
1100                 e1 = canonicalize_clifford(e1);
1101         }
1102
1103         if (is_a<clifford>(e1) && ex_to<clifford>(e1).get_representation_label() >= rl) {
1104                 if (is_a<diracone>(e1.op(0)))
1105                         return 1;
1106                 else
1107                         throw(std::invalid_argument("remove_dirac_ONE(): expression is a non-scalar Clifford number!"));
1108         } else if (is_a<add>(e1) || is_a<ncmul>(e1) || is_a<mul>(e1)
1109                            || is_a<matrix>(e1) || e1.info(info_flags::list)) {
1110                 if (options & 3) // is a child or was already expanded
1111                         return e1.map(fcn);
1112                 else
1113                         try {
1114                                 return e1.map(fcn);
1115                         } catch (std::exception &p) {
1116                                 need_reevaluation = true;
1117                         }
1118         } else if (is_a<power>(e1)) {
1119                 if (options & 3) // is a child or was already expanded
1120                         return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
1121                 else
1122                         try {
1123                                 return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
1124                         } catch (std::exception &p) {
1125                                 need_reevaluation = true;
1126                         }
1127         }
1128         if (need_reevaluation)
1129                 return remove_dirac_ONE(e, rl, options | 2);
1130         return e1;
1131 }
1132
1133 int clifford_max_label(const ex & e, bool ignore_ONE)
1134 {
1135         if (is_a<clifford>(e))
1136                 if (ignore_ONE && is_a<diracone>(e.op(0)))
1137                         return -1;
1138                 else
1139                         return ex_to<clifford>(e).get_representation_label();
1140         else {
1141                 int rl = -1;
1142                 for (size_t i=0; i < e.nops(); i++)
1143                         rl = (rl > clifford_max_label(e.op(i), ignore_ONE)) ? rl : clifford_max_label(e.op(i), ignore_ONE);
1144                 return rl;
1145         }
1146 }
1147
1148 ex clifford_norm(const ex & e)
1149 {
1150         return sqrt(remove_dirac_ONE(e * clifford_bar(e)));
1151 }
1152
1153 ex clifford_inverse(const ex & e)
1154 {
1155         ex norm = clifford_norm(e);
1156         if (!norm.is_zero())
1157                 return clifford_bar(e) / pow(norm, 2);
1158         else
1159                 throw(std::invalid_argument("clifford_inverse(): cannot find inverse of Clifford number with zero norm!"));
1160 }
1161
1162 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl)
1163 {
1164         if (!ex_to<idx>(mu).is_dim_numeric())
1165                 throw(std::invalid_argument("lst_to_clifford(): Index should have a numeric dimension"));
1166         ex e = clifford_unit(mu, metr, rl);
1167         return lst_to_clifford(v, e);
1168 }
1169
1170 ex lst_to_clifford(const ex & v, const ex & e) {
1171         unsigned min, max;
1172
1173         if (is_a<clifford>(e)) {
1174                 ex mu = e.op(1);
1175                 ex mu_toggle
1176                         = is_a<varidx>(mu) ? ex_to<varidx>(mu).toggle_variance() : mu;
1177                 unsigned dim = get_dim_uint(mu);
1178
1179                 if (is_a<matrix>(v)) {
1180                         if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows()) {
1181                                 min = ex_to<matrix>(v).rows();
1182                                 max = ex_to<matrix>(v).cols();
1183                         } else {
1184                                 min = ex_to<matrix>(v).cols();
1185                                 max = ex_to<matrix>(v).rows();
1186                         }
1187                         if (min == 1) {
1188                                 if (dim == max)
1189                                         return indexed(v, mu_toggle) * e;
1190                                 else if (max - dim == 1) {
1191                                         if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows())
1192                                                 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;
1193                                         else
1194                                                 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;
1195                                 } else
1196                                         throw(std::invalid_argument("lst_to_clifford(): dimensions of vector and clifford unit mismatch"));
1197                         } else
1198                                 throw(std::invalid_argument("lst_to_clifford(): first argument should be a vector (nx1 or 1xn matrix)"));
1199                 } else if (v.info(info_flags::list)) {
1200                         if (dim == ex_to<lst>(v).nops())
1201                                 return indexed(matrix(dim, 1, ex_to<lst>(v)), mu_toggle) * e;
1202                         else if (ex_to<lst>(v).nops() - dim == 1)
1203                                 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;
1204                         else
1205                                 throw(std::invalid_argument("lst_to_clifford(): list length and dimension of clifford unit mismatch"));
1206                 } else
1207                         throw(std::invalid_argument("lst_to_clifford(): cannot construct from anything but list or vector"));
1208         } else
1209                 throw(std::invalid_argument("lst_to_clifford(): the second argument should be a Clifford unit"));
1210 }
1211
1212 /** Auxiliary structure to define a function for striping one Clifford unit
1213  * from vectors. Used in  clifford_to_lst(). */
1214 static ex get_clifford_comp(const ex & e, const ex & c)
1215 {
1216         pointer_to_map_function_1arg<const ex &> fcn(get_clifford_comp, c);
1217         int ival = ex_to<numeric>(ex_to<idx>(c.op(1)).get_value()).to_int();
1218
1219         if (is_a<add>(e) || e.info(info_flags::list) // || is_a<pseries>(e) || is_a<integral>(e)
1220                 || is_a<matrix>(e))
1221                 return e.map(fcn);
1222         else if (is_a<ncmul>(e) || is_a<mul>(e)) {
1223                 // find a Clifford unit with the same metric, delete it and substitute its index
1224                 size_t ind = e.nops() + 1;
1225                 for (size_t j = 0; j < e.nops(); j++) {
1226                         if (is_a<clifford>(e.op(j)) && ex_to<clifford>(c).same_metric(e.op(j))) {
1227                                 if (ind > e.nops()) {
1228                                         ind = j;
1229                                 } else {
1230                                         throw(std::invalid_argument("get_clifford_comp(): expression is a Clifford multi-vector"));
1231                                 }
1232                         }
1233                 }
1234                 if (ind < e.nops()) {
1235                         ex S = 1;
1236                         bool same_value_index, found_dummy;
1237                         same_value_index = ( ex_to<idx>(e.op(ind).op(1)).is_numeric()
1238                                                                  &&  (ival == ex_to<numeric>(ex_to<idx>(e.op(ind).op(1)).get_value()).to_int()) );
1239                         found_dummy = same_value_index;
1240                         // Run through the expression collecting all non-clifford factors
1241                         for (size_t j=0; j < e.nops(); j++) {
1242                                 if (j != ind) {
1243                                         if (same_value_index) {
1244                                                 S = S * e.op(j);
1245                                         } else {
1246                                                 exvector ind_vec;
1247                                                 if (is_a<indexed>(e.op(j)))
1248                                                         ind_vec = ex_to<indexed>(e.op(j)).get_dummy_indices(ex_to<indexed>(e.op(ind)));
1249
1250                                                 if (ind_vec.size() > 0) {
1251                                                         found_dummy = true;
1252                                                         for (auto & it : ind_vec) {
1253                                                                 ex curridx = it;
1254                                                                 ex curridx_toggle = is_a<varidx>(curridx)
1255                                                                         ? ex_to<varidx>(curridx).toggle_variance()
1256                                                                         : curridx;
1257                                                                 S = S * e.op(j).subs(lst{curridx == ival, curridx_toggle == ival},
1258                                                                                      subs_options::no_pattern);
1259                                                         }
1260                                                 } else
1261                                                         S = S * e.op(j);
1262                                         }
1263                                 }
1264                         }
1265                         return (found_dummy ? S : 0);
1266                 } else
1267                         throw(std::invalid_argument("get_clifford_comp(): expression is not a Clifford vector to the given units"));
1268         } else if (e.is_zero())
1269                 return e;
1270         else if (is_a<clifford>(e) && ex_to<clifford>(e).same_metric(c))
1271                 if ( ex_to<idx>(e.op(1)).is_numeric() &&
1272                          (ival != ex_to<numeric>(ex_to<idx>(e.op(1)).get_value()).to_int()) )
1273                         return 0;
1274                 else
1275                         return 1;
1276         else
1277                 throw(std::invalid_argument("get_clifford_comp(): expression is not usable as a Clifford vector"));
1278 }
1279
1280
1281 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic)
1282 {
1283         GINAC_ASSERT(is_a<clifford>(c));
1284         ex mu = c.op(1);
1285         if (! ex_to<idx>(mu).is_dim_numeric())
1286                 throw(std::invalid_argument("clifford_to_lst(): index should have a numeric dimension"));
1287         unsigned int D = ex_to<numeric>(ex_to<idx>(mu).get_dim()).to_int();
1288
1289         if (algebraic) // check if algebraic method is applicable
1290                 for (unsigned int i = 0; i < D; i++)
1291                         if (pow(c.subs(mu == i, subs_options::no_pattern), 2).is_zero()
1292                                 || (! is_a<numeric>(pow(c.subs(mu == i, subs_options::no_pattern), 2))))
1293                                 algebraic = false;
1294         lst V;
1295         ex v0 = remove_dirac_ONE(canonicalize_clifford(e+clifford_prime(e)).normal())/2;
1296         if (! v0.is_zero())
1297                 V.append(v0);
1298         ex e1 = canonicalize_clifford(e - v0 * dirac_ONE(ex_to<clifford>(c).get_representation_label()));
1299         if (algebraic) {
1300                 for (unsigned int i = 0; i < D; i++)
1301                         V.append(remove_dirac_ONE(
1302                                                 simplify_indexed(canonicalize_clifford(e1 * c.subs(mu == i, subs_options::no_pattern) +  c.subs(mu == i, subs_options::no_pattern) * e1))
1303                                                 / (2*pow(c.subs(mu == i, subs_options::no_pattern), 2))));
1304         } else {
1305                 try {
1306                         for (unsigned int i = 0; i < D; i++)
1307                                 V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
1308                 } catch  (std::exception &p) {
1309                         /* Try to expand dummy summations to simplify the expression*/
1310                         e1 = canonicalize_clifford(expand_dummy_sum(e, true));
1311                         V.remove_all();
1312                         v0 = remove_dirac_ONE(canonicalize_clifford(e1+clifford_prime(e1)).normal())/2;
1313                         if (! v0.is_zero()) {
1314                                 V.append(v0);
1315                                 e1 = canonicalize_clifford(e1 - v0 * dirac_ONE(ex_to<clifford>(c).get_representation_label()));
1316                         }
1317                         for (unsigned int i = 0; i < D; i++)
1318                                 V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
1319                 }
1320         }
1321         return V;
1322 }
1323
1324
1325 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)
1326 {
1327         ex x, D, cu;
1328
1329         if (! is_a<matrix>(v) && ! v.info(info_flags::list))
1330                 throw(std::invalid_argument("clifford_moebius_map(): parameter v should be either vector or list"));
1331
1332         if (is_a<clifford>(G)) {
1333                 cu = G;
1334         } else {
1335                 if (is_a<indexed>(G)) {
1336                         D = ex_to<idx>(G.op(1)).get_dim();
1337                         varidx mu(dynallocate<symbol>(), D);
1338                         cu = clifford_unit(mu, G, rl);
1339                 } else if (is_a<matrix>(G)) {
1340                         D = ex_to<matrix>(G).rows();
1341                         idx mu(dynallocate<symbol>(), D);
1342                         cu = clifford_unit(mu, G, rl);
1343                 } else throw(std::invalid_argument("clifford_moebius_map(): metric should be an indexed object, matrix, or a Clifford unit"));
1344
1345         }
1346
1347         x = lst_to_clifford(v, cu);
1348         ex e = clifford_to_lst(simplify_indexed(canonicalize_clifford((a * x + b) * clifford_inverse(c * x + d))), cu, false);
1349         return (is_a<matrix>(v) ? matrix(ex_to<matrix>(v).rows(), ex_to<matrix>(v).cols(), ex_to<lst>(e)) : e);
1350 }
1351
1352 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl)
1353 {
1354         if (is_a<matrix>(M) && (ex_to<matrix>(M).rows() == 2) && (ex_to<matrix>(M).cols() == 2))
1355                 return clifford_moebius_map(M.op(0), M.op(1), M.op(2), M.op(3), v, G, rl);
1356         else
1357                 throw(std::invalid_argument("clifford_moebius_map(): parameter M should be a 2x2 matrix"));
1358 }
1359
1360 } // namespace GiNaC