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, bool discardable) : inherited(not_symmetric(), v, discardable), representation_label(rl), metric(metr), commutator_sign(comm_sign)
 109 {
 110 }
 111
 112 clifford::clifford(unsigned char rl, const ex & metr, int comm_sign, std::auto_ptr<exvector> vp) : inherited(not_symmetric(), vp), 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((new symbol)->setflag(status_flags::dynallocated), 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)), true);
 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), true);
 416                         ex SR = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(next_to_last), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
 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), true);
 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), true);
 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         exvector::const_iterator cit = v.begin(), citend = v.end();
 524         while (cit != citend) {
 525                 if (!is_a<clifford>(*cit) || !is_a<diracone>(cit->op(0)))
 526                         s.push_back(*cit);
 527                 cit++;
 528         }
 529
 530         bool something_changed = false;
 531         int sign = 1;
 532
 533         // Anticommutate gamma5/L/R's to the front
 534         if (s.size() >= 2) {
 535                 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
 536                 while (true) {
 537                         exvector::iterator it = next_to_last;
 538                         while (true) {
 539                                 exvector::iterator it2 = it + 1;
 540                                 if (is_a<clifford>(*it) && is_a<clifford>(*it2)) {
 541                                         ex e1 = it->op(0), e2 = it2->op(0);
 542
 543                                         if (is_a<diracgamma5>(e2)) {
 544
 545                                                 if (is_a<diracgammaL>(e1) || is_a<diracgammaR>(e1)) {
 546
 547                                                         // gammaL/R gamma5 -> gamma5 gammaL/R
 548                                                         it->swap(*it2);
 549                                                         something_changed = true;
 550
 551                                                 } else if (!is_a<diracgamma5>(e1)) {
 552
 553                                                         // gamma5 gamma5 -> gamma5 gamma5 (do nothing)
 554                                                         // x gamma5 -> -gamma5 x
 555                                                         it->swap(*it2);
 556                                                         sign = -sign;
 557                                                         something_changed = true;
 558                                                 }
 559
 560                                         } else if (is_a<diracgammaL>(e2)) {
 561
 562                                                 if (is_a<diracgammaR>(e1)) {
 563
 564                                                         // gammaR gammaL -> 0
 565                                                         return _ex0;
 566
 567                                                 } else if (!is_a<diracgammaL>(e1) && !is_a<diracgamma5>(e1)) {
 568
 569                                                         // gammaL gammaL -> gammaL gammaL (do nothing)
 570                                                         // gamma5 gammaL -> gamma5 gammaL (do nothing)
 571                                                         // x gammaL -> gammaR x
 572                                                         it->swap(*it2);
 573                                                         *it = clifford(diracgammaR(), ex_to<clifford>(*it).get_representation_label());
 574                                                         something_changed = true;
 575                                                 }
 576
 577                                         } else if (is_a<diracgammaR>(e2)) {
 578
 579                                                 if (is_a<diracgammaL>(e1)) {
 580
 581                                                         // gammaL gammaR -> 0
 582                                                         return _ex0;
 583
 584                                                 } else if (!is_a<diracgammaR>(e1) && !is_a<diracgamma5>(e1)) {
 585
 586                                                         // gammaR gammaR -> gammaR gammaR (do nothing)
 587                                                         // gamma5 gammaR -> gamma5 gammaR (do nothing)
 588                                                         // x gammaR -> gammaL x
 589                                                         it->swap(*it2);
 590                                                         *it = clifford(diracgammaL(), ex_to<clifford>(*it).get_representation_label());
 591                                                         something_changed = true;
 592                                                 }
 593                                         }
 594                                 }
 595                                 if (it == first)
 596                                         break;
 597                                 --it;
 598                         }
 599                         if (next_to_last == first)
 600                                 break;
 601                         --next_to_last;
 602                 }
 603         }
 604
 605         // Remove equal adjacent gammas
 606         if (s.size() >= 2) {
 607                 exvector::iterator it, itend = s.end() - 1;
 608                 for (it = s.begin(); it != itend; ++it) {
 609                         ex & a = it[0];
 610                         ex & b = it[1];
 611                         if (!is_a<clifford>(a) || !is_a<clifford>(b))
 612                                 continue;
 613
 614                         const ex & ag = a.op(0);
 615                         const ex & bg = b.op(0);
 616                         bool a_is_cliffordunit = is_a<cliffordunit>(ag);
 617                         bool b_is_cliffordunit =  is_a<cliffordunit>(bg);
 618
 619                         if (a_is_cliffordunit && b_is_cliffordunit && ex_to<clifford>(a).same_metric(b)
 620                                 && (ex_to<clifford>(a).get_commutator_sign() == -1)) {
 621                                 // This is done only for Clifford algebras
 622
 623                                 const ex & ia = a.op(1);
 624                                 const ex & ib = b.op(1);
 625                                 if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
 626                                         a = ex_to<clifford>(a).get_metric(ia, ib, true);
 627                                         b = dirac_ONE(representation_label);
 628                                         something_changed = true;
 629                                 }
 630
 631                         } else if ((is_a<diracgamma5>(ag) && is_a<diracgamma5>(bg))) {
 632
 633                                 // Remove squares of gamma5
 634                                 a = dirac_ONE(representation_label);
 635                                 b = dirac_ONE(representation_label);
 636                                 something_changed = true;
 637
 638                         } else if ((is_a<diracgammaL>(ag) && is_a<diracgammaL>(bg))
 639                                 || (is_a<diracgammaR>(ag) && is_a<diracgammaR>(bg))) {
 640
 641                                 // Remove squares of gammaL/R
 642                                 b = dirac_ONE(representation_label);
 643                                 something_changed = true;
 644
 645                         } else if (is_a<diracgammaL>(ag) && is_a<diracgammaR>(bg)) {
 646
 647                                 // gammaL and gammaR are orthogonal
 648                                 return _ex0;
 649
 650                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaL>(bg)) {
 651
 652                                 // gamma5 gammaL -> -gammaL
 653                                 a = dirac_ONE(representation_label);
 654                                 sign = -sign;
 655                                 something_changed = true;
 656
 657                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaR>(bg)) {
 658
 659                                 // gamma5 gammaR -> gammaR
 660                                 a = dirac_ONE(representation_label);
 661                                 something_changed = true;
 662
 663                         } else if (!a_is_cliffordunit && !b_is_cliffordunit && ag.is_equal(bg)) {
 664
 665                                 // a\ a\ -> a^2
 666                                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).minimal_dim(ex_to<idx>(b.op(1))));
 667
 668                                 a = indexed(ag, ix) * indexed(ag, ix.toggle_variance());
 669                                 b = dirac_ONE(representation_label);
 670                                 something_changed = true;
 671                         }
 672                 }
 673         }
 674
 675         if (s.empty())
 676                 return dirac_ONE(representation_label) * sign;
 677         if (something_changed)
 678                 return reeval_ncmul(s) * sign;
 679         else
 680                 return hold_ncmul(s) * sign;
 681 }
 682
 683 ex clifford::thiscontainer(const exvector & v) const
 684 {
 685         return clifford(representation_label, metric, commutator_sign, v);
 686 }
 687
 688 ex clifford::thiscontainer(std::auto_ptr<exvector> vp) const
 689 {
 690         return clifford(representation_label, metric, commutator_sign, vp);
 691 }
 692
 693 ex diracgamma5::conjugate() const
 694 {
 695         return _ex_1 * (*this);
 696 }
 697
 698 ex diracgammaL::conjugate() const
 699 {
 700         return (new diracgammaR)->setflag(status_flags::dynallocated);
 701 }
 702
 703 ex diracgammaR::conjugate() const
 704 {
 705         return (new diracgammaL)->setflag(status_flags::dynallocated);
 706 }
 707
 708 //////////
 709 // global functions
 710 //////////
 711
 712 ex dirac_ONE(unsigned char rl)
 713 {
 714         static ex ONE = (new diracone)->setflag(status_flags::dynallocated);
 715         return clifford(ONE, rl);
 716 }
 717
 718 static unsigned get_dim_uint(const ex& e)
 719 {
 720         if (!is_a<idx>(e))
 721                 throw std::invalid_argument("get_dim_uint: argument is not an index");
 722         ex dim = ex_to<idx>(e).get_dim();
 723         if (!dim.info(info_flags::posint))
 724                 throw std::invalid_argument("get_dim_uint: dimension of index should be a positive integer");
 725         unsigned d = ex_to<numeric>(dim).to_int();
 726         return d;
 727 }
 728
 729 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl)
 730 {
 731         //static ex unit = (new cliffordunit)->setflag(status_flags::dynallocated);
 732         ex unit = (new cliffordunit)->setflag(status_flags::dynallocated);
 733
 734         if (!is_a<idx>(mu))
 735                 throw(std::invalid_argument("clifford_unit(): index of Clifford unit must be of type idx or varidx"));
 736
 737         exvector indices = metr.get_free_indices();
 738
 739         if (indices.size() == 2) {
 740                 return clifford(unit, mu, metr, rl);
 741         } else if (is_a<matrix>(metr)) {
 742                 matrix M = ex_to<matrix>(metr);
 743                 unsigned n = M.rows();
 744                 bool symmetric = true;
 745
 746                 //static idx xi((new symbol)->setflag(status_flags::dynallocated), n),
 747                 //      chi((new symbol)->setflag(status_flags::dynallocated), n);
 748                 idx xi((new symbol)->setflag(status_flags::dynallocated), n),
 749                         chi((new symbol)->setflag(status_flags::dynallocated), n);
 750                 if ((n ==  M.cols()) && (n == get_dim_uint(mu))) {
 751                         for (unsigned i = 0; i < n; i++) {
 752                                 for (unsigned j = i+1; j < n; j++) {
 753                                         if (!M(i, j).is_equal(M(j, i))) {
 754                                                 symmetric = false;
 755                                         }
 756                                 }
 757                         }
 758                         return clifford(unit, mu, indexed(metr, symmetric?symmetric2():not_symmetric(), xi, chi), rl);
 759                 } else {
 760                         throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be a square matrix with the same dimensions as index"));
 761                 }
 762         } else if (indices.size() == 0) { // a tensor or other expression without indices
 763                 //static varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim()),
 764                 //      chi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim());
 765                 varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim()),
 766                         chi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim());
 767                 return clifford(unit, mu, indexed(metr, xi, chi), rl);
 768         }  else
 769                 throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be of type tensor, matrix or an expression with two free indices"));
 770 }
 771
 772 ex dirac_gamma(const ex & mu, unsigned char rl)
 773 {
 774         static ex gamma = (new diracgamma)->setflag(status_flags::dynallocated);
 775
 776         if (!is_a<varidx>(mu))
 777                 throw(std::invalid_argument("dirac_gamma(): index of Dirac gamma must be of type varidx"));
 778
 779         static varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim()),
 780                 chi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim());
 781         return clifford(gamma, mu, indexed((new minkmetric)->setflag(status_flags::dynallocated), symmetric2(), xi, chi), rl);
 782 }
 783
 784 ex dirac_gamma5(unsigned char rl)
 785 {
 786         static ex gamma5 = (new diracgamma5)->setflag(status_flags::dynallocated);
 787         return clifford(gamma5, rl);
 788 }
 789
 790 ex dirac_gammaL(unsigned char rl)
 791 {
 792         static ex gammaL = (new diracgammaL)->setflag(status_flags::dynallocated);
 793         return clifford(gammaL, rl);
 794 }
 795
 796 ex dirac_gammaR(unsigned char rl)
 797 {
 798         static ex gammaR = (new diracgammaR)->setflag(status_flags::dynallocated);
 799         return clifford(gammaR, rl);
 800 }
 801
 802 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
 803 {
 804         // Slashed vectors are actually stored as a clifford object with the
 805         // vector as its base expression and a (dummy) index that just serves
 806         // for storing the space dimensionality
 807
 808         static varidx xi((new symbol)->setflag(status_flags::dynallocated), dim),
 809                 chi((new symbol)->setflag(status_flags::dynallocated), dim);
 810    return clifford(e, varidx(0, dim), indexed((new minkmetric)->setflag(status_flags::dynallocated), symmetric2(), xi, chi), rl);
 811 }
 812
 813 /** Extract representation label from tinfo key (as returned by
 814  *  return_type_tinfo()). */
 815 static unsigned char get_representation_label(const return_type_t& ti)
 816 {
 817         return (unsigned char)ti.rl;
 818 }
 819
 820 /** Take trace of a string of an even number of Dirac gammas given a vector
 821  *  of indices. */
 822 static ex trace_string(exvector::const_iterator ix, size_t num)
 823 {
 824         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 825         if (num == 2)
 826                 return lorentz_g(ix[0], ix[1]);
 827
 828         // 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 )
 829         else if (num == 4)
 830                 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
 831                      + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
 832                      - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
 833
 834         // Traces of 6 or more gammas are computed recursively:
 835         // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
 836         //   + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
 837         //   - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
 838         //   + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
 839         //   - ...
 840         //   + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
 841         exvector v(num - 2);
 842         int sign = 1;
 843         ex result;
 844         for (size_t i=1; i<num; i++) {
 845                 for (size_t n=1, j=0; n<num; n++) {
 846                         if (n == i)
 847                                 continue;
 848                         v[j++] = ix[n];
 849                 }
 850                 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
 851                 sign = -sign;
 852         }
 853         return result;
 854 }
 855
 856 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE)
 857 {
 858         if (is_a<clifford>(e)) {
 859
 860                 unsigned char rl = ex_to<clifford>(e).get_representation_label();
 861
 862                 // Are we taking the trace over this object's representation label?
 863                 if (rls.find(rl) == rls.end())
 864                         return e;
 865
 866                 // Yes, all elements are traceless, except for dirac_ONE and dirac_L/R
 867                 const ex & g = e.op(0);
 868                 if (is_a<diracone>(g))
 869                         return trONE;
 870                 else if (is_a<diracgammaL>(g) || is_a<diracgammaR>(g))
 871                         return trONE/2;
 872                 else
 873                         return _ex0;
 874
 875         } else if (is_exactly_a<mul>(e)) {
 876
 877                 // Trace of product: pull out non-clifford factors
 878                 ex prod = _ex1;
 879                 for (size_t i=0; i<e.nops(); i++) {
 880                         const ex &o = e.op(i);
 881                         if (is_clifford_tinfo(o.return_type_tinfo()))
 882                                 prod *= dirac_trace(o, rls, trONE);
 883                         else
 884                                 prod *= o;
 885                 }
 886                 return prod;
 887
 888         } else if (is_exactly_a<ncmul>(e)) {
 889
 890                 unsigned char rl = get_representation_label(e.return_type_tinfo());
 891
 892                 // Are we taking the trace over this string's representation label?
 893                 if (rls.find(rl) == rls.end())
 894                         return e;
 895
 896                 // Substitute gammaL/R and expand product, if necessary
 897                 ex e_expanded = e.subs(lst(
 898                         dirac_gammaL(rl) == (dirac_ONE(rl)-dirac_gamma5(rl))/2,
 899                         dirac_gammaR(rl) == (dirac_ONE(rl)+dirac_gamma5(rl))/2
 900                 ), subs_options::no_pattern).expand();
 901                 if (!is_a<ncmul>(e_expanded))
 902                         return dirac_trace(e_expanded, rls, trONE);
 903
 904                 // gamma5 gets moved to the front so this check is enough
 905                 bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
 906                 size_t num = e.nops();
 907
 908                 if (has_gamma5) {
 909
 910                         // Trace of gamma5 * odd number of gammas and trace of
 911                         // gamma5 * gamma.mu * gamma.nu are zero
 912                         if ((num & 1) == 0 || num == 3)
 913                                 return _ex0;
 914
 915                         // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
 916                         // (the epsilon is always 4-dimensional)
 917                         if (num == 5) {
 918                                 ex b1, i1, b2, i2, b3, i3, b4, i4;
 919                                 base_and_index(e.op(1), b1, i1);
 920                                 base_and_index(e.op(2), b2, i2);
 921                                 base_and_index(e.op(3), b3, i3);
 922                                 base_and_index(e.op(4), b4, i4);
 923                                 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();
 924                         }
 925
 926                         // Tr gamma5 S_2k =
 927                         //   I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
 928                         // (the epsilon is always 4-dimensional)
 929                         exvector ix(num-1), bv(num-1);
 930                         for (size_t i=1; i<num; i++)
 931                                 base_and_index(e.op(i), bv[i-1], ix[i-1]);
 932                         num--;
 933                         int *iv = new int[num];
 934                         ex result;
 935                         for (size_t i=0; i<num-3; i++) {
 936                                 ex idx1 = ix[i];
 937                                 for (size_t j=i+1; j<num-2; j++) {
 938                                         ex idx2 = ix[j];
 939                                         for (size_t k=j+1; k<num-1; k++) {
 940                                                 ex idx3 = ix[k];
 941                                                 for (size_t l=k+1; l<num; l++) {
 942                                                         ex idx4 = ix[l];
 943                                                         iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
 944                                                         exvector v;
 945                                                         v.reserve(num - 4);
 946                                                         for (size_t n=0, t=4; n<num; n++) {
 947                                                                 if (n == i || n == j || n == k || n == l)
 948                                                                         continue;
 949                                                                 iv[t++] = n;
 950                                                                 v.push_back(ix[n]);
 951                                                         }
 952                                                         int sign = permutation_sign(iv, iv + num);
 953                                                         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))
 954                                                                 * trace_string(v.begin(), num - 4);
 955                                                 }
 956                                         }
 957                                 }
 958                         }
 959                         delete[] iv;
 960                         return trONE * I * result * mul(bv);
 961
 962                 } else { // no gamma5
 963
 964                         // Trace of odd number of gammas is zero
 965                         if ((num & 1) == 1)
 966                                 return _ex0;
 967
 968                         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 969                         if (num == 2) {
 970                                 ex b1, i1, b2, i2;
 971                                 base_and_index(e.op(0), b1, i1);
 972                                 base_and_index(e.op(1), b2, i2);
 973                                 return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
 974                         }
 975
 976                         exvector iv(num), bv(num);
 977                         for (size_t i=0; i<num; i++)
 978                                 base_and_index(e.op(i), bv[i], iv[i]);
 979
 980                         return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
 981                 }
 982
 983         } else if (e.nops() > 0) {
 984
 985                 // Trace maps to all other container classes (this includes sums)
 986                 pointer_to_map_function_2args<const std::set<unsigned char> &, const ex &> fcn(dirac_trace, rls, trONE);
 987                 return e.map(fcn);
 988
 989         } else
 990                 return _ex0;
 991 }
 992
 993 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE)
 994 {
 995         // Convert list to set
 996         std::set<unsigned char> rls;
 997         for (lst::const_iterator i = rll.begin(); i != rll.end(); ++i) {
 998                 if (i->info(info_flags::nonnegint))
 999                         rls.insert(ex_to<numeric>(*i).to_int());
1000         }
1001
1002         return dirac_trace(e, rls, trONE);
1003 }
1004
1005 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
1006 {
1007         // Convert label to set
1008         std::set<unsigned char> rls;
1009         rls.insert(rl);
1010
1011         return dirac_trace(e, rls, trONE);
1012 }
1013
1014
1015 ex canonicalize_clifford(const ex & e_)
1016 {
1017         pointer_to_map_function fcn(canonicalize_clifford);
1018
1019         if (is_a<matrix>(e_)    // || is_a<pseries>(e) || is_a<integral>(e)
1020                 || e_.info(info_flags::list)) {
1021                 return e_.map(fcn);
1022         } else {
1023                 ex e=simplify_indexed(e_);
1024                 // Scan for any ncmul objects
1025                 exmap srl;
1026                 ex aux = e.to_rational(srl);
1027                 for (exmap::iterator i = srl.begin(); i != srl.end(); ++i) {
1028
1029                         ex lhs = i->first;
1030                         ex rhs = i->second;
1031
1032                         if (is_exactly_a<ncmul>(rhs)
1033                                         && rhs.return_type() == return_types::noncommutative
1034                                         && is_clifford_tinfo(rhs.return_type_tinfo())) {
1035
1036                                 // Expand product, if necessary
1037                                 ex rhs_expanded = rhs.expand();
1038                                 if (!is_a<ncmul>(rhs_expanded)) {
1039                                         i->second = canonicalize_clifford(rhs_expanded);
1040                                         continue;
1041
1042                                 } else if (!is_a<clifford>(rhs.op(0)))
1043                                         continue;
1044
1045                                 exvector v;
1046                                 v.reserve(rhs.nops());
1047                                 for (size_t j=0; j<rhs.nops(); j++)
1048                                         v.push_back(rhs.op(j));
1049
1050                                 // Stupid recursive bubble sort because we only want to swap adjacent gammas
1051                                 exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
1052                                 if (is_a<diracgamma5>(it->op(0)) || is_a<diracgammaL>(it->op(0)) || is_a<diracgammaR>(it->op(0)))
1053                                         ++it;
1054
1055                                 while (it != next_to_last) {
1056                                         if (it[0].compare(it[1]) > 0) {
1057
1058                                                 ex save0 = it[0], save1 = it[1];
1059                                                 ex b1, i1, b2, i2;
1060                                                 base_and_index(it[0], b1, i1);
1061                                                 base_and_index(it[1], b2, i2);
1062                                                 // for Clifford algebras (commutator_sign == -1) metric should be symmetrised
1063                                                 it[0] = (ex_to<clifford>(save0).get_metric(i1, i2, ex_to<clifford>(save0).get_commutator_sign() == -1) * b1 * b2).simplify_indexed();
1064                                                 it[1] = v.size() ? _ex2 * dirac_ONE(ex_to<clifford>(save0).get_representation_label()) : _ex2;
1065                                                 ex sum = ncmul(v);
1066                                                 it[0] = save1;
1067                                                 it[1] = save0;
1068                                                 sum += ex_to<clifford>(save0).get_commutator_sign() * ncmul(v, true);
1069                                                 i->second = canonicalize_clifford(sum);
1070                                                 goto next_sym;
1071                                         }
1072                                         ++it;
1073                                 }
1074 next_sym:       ;
1075                         }
1076                 }
1077                 return aux.subs(srl, subs_options::no_pattern).simplify_indexed();
1078         }
1079 }
1080
1081 ex clifford_prime(const ex & e)
1082 {
1083         pointer_to_map_function fcn(clifford_prime);
1084         if (is_a<clifford>(e) && is_a<cliffordunit>(e.op(0))) {
1085                 return -e;
1086         } else if (is_a<add>(e) || is_a<ncmul>(e) || is_a<mul>(e) //|| is_a<pseries>(e) || is_a<integral>(e)
1087                            || is_a<matrix>(e) || e.info(info_flags::list)) {
1088                 return e.map(fcn);
1089         } else if (is_a<power>(e)) {
1090                 return pow(clifford_prime(e.op(0)), e.op(1));
1091         } else
1092                 return e;
1093 }
1094
1095 ex remove_dirac_ONE(const ex & e, unsigned char rl, unsigned options)
1096 {
1097         pointer_to_map_function_2args<unsigned char, unsigned> fcn(remove_dirac_ONE, rl, options | 1);
1098         bool need_reevaluation = false;
1099         ex e1 = e;
1100         if (! (options & 1) )  { // is not a child
1101                 if (options & 2)
1102                         e1 = expand_dummy_sum(e, true);
1103                 e1 = canonicalize_clifford(e1);
1104         }
1105
1106         if (is_a<clifford>(e1) && ex_to<clifford>(e1).get_representation_label() >= rl) {
1107                 if (is_a<diracone>(e1.op(0)))
1108                         return 1;
1109                 else
1110                         throw(std::invalid_argument("remove_dirac_ONE(): expression is a non-scalar Clifford number!"));
1111         } else if (is_a<add>(e1) || is_a<ncmul>(e1) || is_a<mul>(e1)
1112                            || is_a<matrix>(e1) || e1.info(info_flags::list)) {
1113                 if (options & 3) // is a child or was already expanded
1114                         return e1.map(fcn);
1115                 else
1116                         try {
1117                                 return e1.map(fcn);
1118                         } catch (std::exception &p) {
1119                                 need_reevaluation = true;
1120                         }
1121         } else if (is_a<power>(e1)) {
1122                 if (options & 3) // is a child or was already expanded
1123                         return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
1124                 else
1125                         try {
1126                                 return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
1127                         } catch (std::exception &p) {
1128                                 need_reevaluation = true;
1129                         }
1130         }
1131         if (need_reevaluation)
1132                 return remove_dirac_ONE(e, rl, options | 2);
1133         return e1;
1134 }
1135
1136 int clifford_max_label(const ex & e, bool ignore_ONE)
1137 {
1138         if (is_a<clifford>(e))
1139                 if (ignore_ONE && is_a<diracone>(e.op(0)))
1140                         return -1;
1141                 else
1142                         return ex_to<clifford>(e).get_representation_label();
1143         else {
1144                 int rl = -1;
1145                 for (size_t i=0; i < e.nops(); i++)
1146                         rl = (rl > clifford_max_label(e.op(i), ignore_ONE)) ? rl : clifford_max_label(e.op(i), ignore_ONE);
1147                 return rl;
1148         }
1149 }
1150
1151 ex clifford_norm(const ex & e)
1152 {
1153         return sqrt(remove_dirac_ONE(e * clifford_bar(e)));
1154 }
1155
1156 ex clifford_inverse(const ex & e)
1157 {
1158         ex norm = clifford_norm(e);
1159         if (!norm.is_zero())
1160                 return clifford_bar(e) / pow(norm, 2);
1161         else
1162                 throw(std::invalid_argument("clifford_inverse(): cannot find inverse of Clifford number with zero norm!"));
1163 }
1164
1165 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl)
1166 {
1167         if (!ex_to<idx>(mu).is_dim_numeric())
1168                 throw(std::invalid_argument("lst_to_clifford(): Index should have a numeric dimension"));
1169         ex e = clifford_unit(mu, metr, rl);
1170         return lst_to_clifford(v, e);
1171 }
1172
1173 ex lst_to_clifford(const ex & v, const ex & e) {
1174         unsigned min, max;
1175
1176         if (is_a<clifford>(e)) {
1177                 ex mu = e.op(1);
1178                 ex mu_toggle
1179                         = is_a<varidx>(mu) ? ex_to<varidx>(mu).toggle_variance() : mu;
1180                 unsigned dim = get_dim_uint(mu);
1181
1182                 if (is_a<matrix>(v)) {
1183                         if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows()) {
1184                                 min = ex_to<matrix>(v).rows();
1185                                 max = ex_to<matrix>(v).cols();
1186                         } else {
1187                                 min = ex_to<matrix>(v).cols();
1188                                 max = ex_to<matrix>(v).rows();
1189                         }
1190                         if (min == 1) {
1191                                 if (dim == max)
1192                                         return indexed(v, mu_toggle) * e;
1193                                 else if (max - dim == 1) {
1194                                         if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows())
1195                                                 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;
1196                                         else
1197                                                 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;
1198                                 } else
1199                                         throw(std::invalid_argument("lst_to_clifford(): dimensions of vector and clifford unit mismatch"));
1200                         } else
1201                                 throw(std::invalid_argument("lst_to_clifford(): first argument should be a vector (nx1 or 1xn matrix)"));
1202                 } else if (v.info(info_flags::list)) {
1203                         if (dim == ex_to<lst>(v).nops())
1204                                 return indexed(matrix(dim, 1, ex_to<lst>(v)), mu_toggle) * e;
1205                         else if (ex_to<lst>(v).nops() - dim == 1)
1206                                 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;
1207                         else
1208                                 throw(std::invalid_argument("lst_to_clifford(): list length and dimension of clifford unit mismatch"));
1209                 } else
1210                         throw(std::invalid_argument("lst_to_clifford(): cannot construct from anything but list or vector"));
1211         } else
1212                 throw(std::invalid_argument("lst_to_clifford(): the second argument should be a Clifford unit"));
1213 }
1214
1215 /** Auxiliary structure to define a function for striping one Clifford unit
1216  * from vectors. Used in  clifford_to_lst(). */
1217 static ex get_clifford_comp(const ex & e, const ex & c)
1218 {
1219         pointer_to_map_function_1arg<const ex &> fcn(get_clifford_comp, c);
1220         int ival = ex_to<numeric>(ex_to<idx>(c.op(1)).get_value()).to_int();
1221
1222         if (is_a<add>(e) || e.info(info_flags::list) // || is_a<pseries>(e) || is_a<integral>(e)
1223                 || is_a<matrix>(e))
1224                 return e.map(fcn);
1225         else if (is_a<ncmul>(e) || is_a<mul>(e)) {
1226                 // find a Clifford unit with the same metric, delete it and substitute its index
1227                 size_t ind = e.nops() + 1;
1228                 for (size_t j = 0; j < e.nops(); j++) {
1229                         if (is_a<clifford>(e.op(j)) && ex_to<clifford>(c).same_metric(e.op(j))) {
1230                                 if (ind > e.nops()) {
1231                                         ind = j;
1232                                 } else {
1233                                         throw(std::invalid_argument("get_clifford_comp(): expression is a Clifford multi-vector"));
1234                                 }
1235                         }
1236                 }
1237                 if (ind < e.nops()) {
1238                         ex S = 1;
1239                         bool same_value_index, found_dummy;
1240                         same_value_index = ( ex_to<idx>(e.op(ind).op(1)).is_numeric()
1241                                                                  &&  (ival == ex_to<numeric>(ex_to<idx>(e.op(ind).op(1)).get_value()).to_int()) );
1242                         found_dummy = same_value_index;
1243                         // Run through the expression collecting all non-clifford factors
1244                         for (size_t j=0; j < e.nops(); j++) {
1245                                 if (j != ind) {
1246                                         if (same_value_index) {
1247                                                 S = S * e.op(j);
1248                                         } else {
1249                                                 exvector ind_vec;
1250                                                 if (is_a<indexed>(e.op(j)))
1251                                                         ind_vec = ex_to<indexed>(e.op(j)).get_dummy_indices(ex_to<indexed>(e.op(ind)));
1252
1253                                                 if (ind_vec.size() > 0) {
1254                                                         found_dummy = true;
1255                                                         exvector::const_iterator it = ind_vec.begin(), itend = ind_vec.end();
1256                                                         while (it != itend) {
1257                                                                 ex curridx = *it;
1258                                                                 ex curridx_toggle = is_a<varidx>(curridx)
1259                                                                         ? ex_to<varidx>(curridx).toggle_variance()
1260                                                                         : curridx;
1261                                                                 S = S * e.op(j).subs(lst(curridx == ival,
1262                                                                         curridx_toggle == ival), subs_options::no_pattern);
1263                                                                 ++it;
1264                                                         }
1265                                                 } else
1266                                                         S = S * e.op(j);
1267                                         }
1268                                 }
1269                         }
1270                         return (found_dummy ? S : 0);
1271                 } else
1272                         throw(std::invalid_argument("get_clifford_comp(): expression is not a Clifford vector to the given units"));
1273         } else if (e.is_zero())
1274                 return e;
1275         else if (is_a<clifford>(e) && ex_to<clifford>(e).same_metric(c))
1276                 if ( ex_to<idx>(e.op(1)).is_numeric() &&
1277                          (ival != ex_to<numeric>(ex_to<idx>(e.op(1)).get_value()).to_int()) )
1278                         return 0;
1279                 else
1280                         return 1;
1281         else
1282                 throw(std::invalid_argument("get_clifford_comp(): expression is not usable as a Clifford vector"));
1283 }
1284
1285
1286 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic)
1287 {
1288         GINAC_ASSERT(is_a<clifford>(c));
1289         ex mu = c.op(1);
1290         if (! ex_to<idx>(mu).is_dim_numeric())
1291                 throw(std::invalid_argument("clifford_to_lst(): index should have a numeric dimension"));
1292         unsigned int D = ex_to<numeric>(ex_to<idx>(mu).get_dim()).to_int();
1293
1294         if (algebraic) // check if algebraic method is applicable
1295                 for (unsigned int i = 0; i < D; i++)
1296                         if (pow(c.subs(mu == i, subs_options::no_pattern), 2).is_zero()
1297                                 || (! is_a<numeric>(pow(c.subs(mu == i, subs_options::no_pattern), 2))))
1298                                 algebraic = false;
1299         lst V;
1300         ex v0 = remove_dirac_ONE(canonicalize_clifford(e+clifford_prime(e)).normal())/2;
1301         if (! v0.is_zero())
1302                 V.append(v0);
1303         ex e1 = canonicalize_clifford(e - v0 * dirac_ONE(ex_to<clifford>(c).get_representation_label()));
1304         if (algebraic) {
1305                 for (unsigned int i = 0; i < D; i++)
1306                         V.append(remove_dirac_ONE(
1307                                                 simplify_indexed(canonicalize_clifford(e1 * c.subs(mu == i, subs_options::no_pattern) +  c.subs(mu == i, subs_options::no_pattern) * e1))
1308                                                 / (2*pow(c.subs(mu == i, subs_options::no_pattern), 2))));
1309         } else {
1310                 try {
1311                         for (unsigned int i = 0; i < D; i++)
1312                                 V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
1313                 } catch  (std::exception &p) {
1314                         /* Try to expand dummy summations to simplify the expression*/
1315                         e1 = canonicalize_clifford(expand_dummy_sum(e, true));
1316                         V.remove_all();
1317                         v0 = remove_dirac_ONE(canonicalize_clifford(e1+clifford_prime(e1)).normal())/2;
1318                         if (! v0.is_zero()) {
1319                                 V.append(v0);
1320                                 e1 = canonicalize_clifford(e1 - v0 * dirac_ONE(ex_to<clifford>(c).get_representation_label()));
1321                         }
1322                         for (unsigned int i = 0; i < D; i++)
1323                                 V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
1324                 }
1325         }
1326         return V;
1327 }
1328
1329
1330 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)
1331 {
1332         ex x, D, cu;
1333
1334         if (! is_a<matrix>(v) && ! v.info(info_flags::list))
1335                 throw(std::invalid_argument("clifford_moebius_map(): parameter v should be either vector or list"));
1336
1337         if (is_a<clifford>(G)) {
1338                 cu = G;
1339         } else {
1340                 if (is_a<indexed>(G)) {
1341                         D = ex_to<idx>(G.op(1)).get_dim();
1342                         varidx mu((new symbol)->setflag(status_flags::dynallocated), D);
1343                         cu = clifford_unit(mu, G, rl);
1344                 } else if (is_a<matrix>(G)) {
1345                         D = ex_to<matrix>(G).rows();
1346                         idx mu((new symbol)->setflag(status_flags::dynallocated), D);
1347                         cu = clifford_unit(mu, G, rl);
1348                 } else throw(std::invalid_argument("clifford_moebius_map(): metric should be an indexed object, matrix, or a Clifford unit"));
1349
1350         }
1351
1352         x = lst_to_clifford(v, cu);
1353         ex e = clifford_to_lst(simplify_indexed(canonicalize_clifford((a * x + b) * clifford_inverse(c * x + d))), cu, false);
1354         return (is_a<matrix>(v) ? matrix(ex_to<matrix>(v).rows(), ex_to<matrix>(v).cols(), ex_to<lst>(e)) : e);
1355 }
1356
1357 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl)
1358 {
1359         if (is_a<matrix>(M) && (ex_to<matrix>(M).rows() == 2) && (ex_to<matrix>(M).cols() == 2))
1360                 return clifford_moebius_map(M.op(0), M.op(1), M.op(2), M.op(3), v, G, rl);
1361         else
1362                 throw(std::invalid_argument("clifford_moebius_map(): parameter M should be a 2x2 matrix"));
1363 }
1364
1365 } // namespace GiNaC