ginac/clifford.cpp

   1 /** @file clifford.cpp
   2  *
   3  *  Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2005 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 namespace GiNaC {
  42
  43 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(clifford, indexed,
  44   print_func<print_dflt>(&clifford::do_print_dflt).
  45   print_func<print_latex>(&clifford::do_print_latex))
  46
  47 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracone, tensor,
  48   print_func<print_dflt>(&diracone::do_print).
  49   print_func<print_latex>(&diracone::do_print_latex))
  50
  51 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(cliffordunit, tensor,
  52   print_func<print_dflt>(&cliffordunit::do_print).
  53   print_func<print_latex>(&cliffordunit::do_print_latex))
  54
  55 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma, cliffordunit,
  56   print_func<print_dflt>(&diracgamma::do_print).
  57   print_func<print_latex>(&diracgamma::do_print_latex))
  58
  59 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma5, tensor,
  60   print_func<print_dflt>(&diracgamma5::do_print).
  61   print_func<print_latex>(&diracgamma5::do_print_latex))
  62
  63 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaL, tensor,
  64   print_func<print_context>(&diracgammaL::do_print).
  65   print_func<print_latex>(&diracgammaL::do_print_latex))
  66
  67 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaR, tensor,
  68   print_func<print_context>(&diracgammaR::do_print).
  69   print_func<print_latex>(&diracgammaR::do_print_latex))
  70
  71 //////////
  72 // default constructors
  73 //////////
  74
  75 static ex default_metric()
  76 {
  77         static ex m = (new minkmetric)->setflag(status_flags::dynallocated);
  78         return m;
  79 }
  80
  81 clifford::clifford() : representation_label(0), metric(default_metric())
  82 {
  83         tinfo_key = TINFO_clifford;
  84 }
  85
  86 DEFAULT_CTOR(diracone)
  87 DEFAULT_CTOR(cliffordunit)
  88 DEFAULT_CTOR(diracgamma)
  89 DEFAULT_CTOR(diracgamma5)
  90 DEFAULT_CTOR(diracgammaL)
  91 DEFAULT_CTOR(diracgammaR)
  92
  93 //////////
  94 // other constructors
  95 //////////
  96
  97 /** Construct object without any indices. This constructor is for internal
  98  *  use only. Use the dirac_ONE() function instead.
  99  *  @see dirac_ONE */
 100 clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl), metric(0)
 101 {
 102         tinfo_key = TINFO_clifford;
 103 }
 104
 105 /** Construct object with one Lorentz index. This constructor is for internal
 106  *  use only. Use the clifford_unit() or dirac_gamma() functions instead.
 107  *  @see clifford_unit
 108  *  @see dirac_gamma */
 109 clifford::clifford(const ex & b, const ex & mu, const ex & metr, unsigned char rl) : inherited(b, mu), representation_label(rl), metric(metr)
 110 {
 111         GINAC_ASSERT(is_a<varidx>(mu));
 112         tinfo_key = TINFO_clifford;
 113 }
 114
 115 clifford::clifford(unsigned char rl, const ex & metr, const exvector & v, bool discardable) : inherited(not_symmetric(), v, discardable), representation_label(rl), metric(metr)
 116 {
 117         tinfo_key = TINFO_clifford;
 118 }
 119
 120 clifford::clifford(unsigned char rl, const ex & metr, std::auto_ptr<exvector> vp) : inherited(not_symmetric(), vp), representation_label(rl), metric(metr)
 121 {
 122         tinfo_key = TINFO_clifford;
 123 }
 124
 125 //////////
 126 // archiving
 127 //////////
 128
 129 clifford::clifford(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
 130 {
 131         unsigned rl;
 132         n.find_unsigned("label", rl);
 133         representation_label = rl;
 134         n.find_ex("metric", metric, sym_lst);
 135 }
 136
 137 void clifford::archive(archive_node & n) const
 138 {
 139         inherited::archive(n);
 140         n.add_unsigned("label", representation_label);
 141         n.add_ex("metric", metric);
 142 }
 143
 144 DEFAULT_UNARCHIVE(clifford)
 145 DEFAULT_ARCHIVING(diracone)
 146 DEFAULT_ARCHIVING(cliffordunit)
 147 DEFAULT_ARCHIVING(diracgamma)
 148 DEFAULT_ARCHIVING(diracgamma5)
 149 DEFAULT_ARCHIVING(diracgammaL)
 150 DEFAULT_ARCHIVING(diracgammaR)
 151
 152 //////////
 153 // functions overriding virtual functions from base classes
 154 //////////
 155
 156 ex clifford::get_metric(const ex & i, const ex & j) const
 157 {
 158         return indexed(metric, symmetric2(), i, j);
 159 }
 160
 161 bool clifford::same_metric(const ex & other) const
 162 {
 163         if (is_a<clifford>(other)) {
 164                 return get_metric().is_equal(ex_to<clifford>(other).get_metric());
 165         } else if (is_a<indexed>(other)) {
 166                 return get_metric(other.op(1), other.op(2)).is_equal(other);
 167         } else
 168                 return false;
 169 }
 170
 171 int clifford::compare_same_type(const basic & other) const
 172 {
 173         GINAC_ASSERT(is_a<clifford>(other));
 174         const clifford &o = static_cast<const clifford &>(other);
 175
 176         if (representation_label != o.representation_label) {
 177                 // different representation label
 178                 return representation_label < o.representation_label ? -1 : 1;
 179         }
 180
 181         return inherited::compare_same_type(other);
 182 }
 183
 184 bool clifford::match_same_type(const basic & other) const
 185 {
 186         GINAC_ASSERT(is_a<clifford>(other));
 187         const clifford &o = static_cast<const clifford &>(other);
 188
 189         return (representation_label == o.representation_label) && same_metric(o);
 190 }
 191
 192 static bool is_dirac_slash(const ex & seq0)
 193 {
 194         return !is_a<diracgamma5>(seq0) && !is_a<diracgammaL>(seq0) &&
 195                !is_a<diracgammaR>(seq0) && !is_a<cliffordunit>(seq0) &&
 196                !is_a<diracone>(seq0);
 197 }
 198
 199 void clifford::do_print_dflt(const print_dflt & c, unsigned level) const
 200 {
 201         // dirac_slash() object is printed differently
 202         if (is_dirac_slash(seq[0])) {
 203                 seq[0].print(c, precedence());
 204                 c.s << "\\";
 205         } else
 206                 this->print_dispatch<inherited>(c, level);
 207 }
 208
 209 void clifford::do_print_latex(const print_latex & c, unsigned level) const
 210 {
 211         // dirac_slash() object is printed differently
 212         if (is_dirac_slash(seq[0])) {
 213                 c.s << "{";
 214                 seq[0].print(c, precedence());
 215                 c.s << "\\hspace{-1.0ex}/}";
 216         } else {
 217                 c.s << "\\clifford[" << int(representation_label) << "]";
 218                 this->print_dispatch<inherited>(c, level);
 219         }
 220 }
 221
 222 DEFAULT_COMPARE(diracone)
 223 DEFAULT_COMPARE(cliffordunit)
 224 DEFAULT_COMPARE(diracgamma)
 225 DEFAULT_COMPARE(diracgamma5)
 226 DEFAULT_COMPARE(diracgammaL)
 227 DEFAULT_COMPARE(diracgammaR)
 228
 229 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbf{1}")
 230 DEFAULT_PRINT_LATEX(cliffordunit, "e", "e")
 231 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
 232 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
 233 DEFAULT_PRINT_LATEX(diracgammaL, "gammaL", "{\\gamma_L}")
 234 DEFAULT_PRINT_LATEX(diracgammaR, "gammaR", "{\\gamma_R}")
 235
 236 /** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
 237 static void base_and_index(const ex & c, ex & b, ex & i)
 238 {
 239         GINAC_ASSERT(is_a<clifford>(c));
 240         GINAC_ASSERT(c.nops() == 2);
 241
 242         if (is_a<cliffordunit>(c.op(0))) { // proper dirac gamma object or clifford unit
 243                 i = c.op(1);
 244                 b = _ex1;
 245         } else if (is_a<diracgamma5>(c.op(0)) || is_a<diracgammaL>(c.op(0)) || is_a<diracgammaR>(c.op(0))) { // gamma5/L/R
 246                 i = _ex0;
 247                 b = _ex1;
 248         } else { // slash object, generate new dummy index
 249                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(c.op(1)).get_dim());
 250                 b = indexed(c.op(0), ix.toggle_variance());
 251                 i = ix;
 252         }
 253 }
 254
 255 /** Predicate for finding non-clifford objects. */
 256 struct is_not_a_clifford : public std::unary_function<ex, bool> {
 257         bool operator()(const ex & e)
 258         {
 259                 return !is_a<clifford>(e);
 260         }
 261 };
 262
 263 /** Contraction of a gamma matrix with something else. */
 264 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 265 {
 266         GINAC_ASSERT(is_a<clifford>(*self));
 267         GINAC_ASSERT(is_a<indexed>(*other));
 268         GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
 269         unsigned char rl = ex_to<clifford>(*self).get_representation_label();
 270
 271         ex dim = ex_to<idx>(self->op(1)).get_dim();
 272         if (other->nops() > 1)
 273                 dim = minimal_dim(dim, ex_to<idx>(other->op(1)).get_dim());
 274
 275         if (is_a<clifford>(*other)) {
 276
 277                 // Contraction only makes sense if the represenation labels are equal
 278                 if (ex_to<clifford>(*other).get_representation_label() != rl)
 279                         return false;
 280
 281                 size_t num = other - self;
 282
 283                 // gamma~mu gamma.mu = dim ONE
 284                 if (num == 1) {
 285                         *self = dim;
 286                         *other = dirac_ONE(rl);
 287                         return true;
 288
 289                 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
 290                 } else if (num == 2
 291                         && is_a<clifford>(self[1])) {
 292                         *self = 2 - dim;
 293                         *other = _ex1;
 294                         return true;
 295
 296                 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
 297                 } else if (num == 3
 298                         && is_a<clifford>(self[1])
 299                         && is_a<clifford>(self[2])) {
 300                         ex b1, i1, b2, i2;
 301                         base_and_index(self[1], b1, i1);
 302                         base_and_index(self[2], b2, i2);
 303                         *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
 304                         self[1] = _ex1;
 305                         self[2] = _ex1;
 306                         *other = _ex1;
 307                         return true;
 308
 309                 // 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
 310                 } else if (num == 4
 311                         && is_a<clifford>(self[1])
 312                         && is_a<clifford>(self[2])
 313                         && is_a<clifford>(self[3])) {
 314                         *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3];
 315                         self[1] = _ex1;
 316                         self[2] = _ex1;
 317                         self[3] = _ex1;
 318                         *other = _ex1;
 319                         return true;
 320
 321                 // gamma~mu Sodd gamma.mu = -2 Sodd_R
 322                 // (Chisholm identity in 4 dimensions)
 323                 } else if (!((other - self) & 1) && dim.is_equal(4)) {
 324                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
 325                                 return false;
 326
 327                         *self = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(other), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
 328                         std::fill(self + 1, other, _ex1);
 329                         *other = _ex_2;
 330                         return true;
 331
 332                 // gamma~mu Sodd gamma~alpha gamma.mu = 2 gamma~alpha Sodd + 2 Sodd_R gamma~alpha
 333                 // (commutate contracted indices towards each other, then use
 334                 // Chisholm identity in 4 dimensions)
 335                 } else if (((other - self) & 1) && dim.is_equal(4)) {
 336                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
 337                                 return false;
 338
 339                         exvector::iterator next_to_last = other - 1;
 340                         ex S = ncmul(exvector(self + 1, next_to_last), true);
 341                         ex SR = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(next_to_last), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
 342
 343                         *self = (*next_to_last) * S + SR * (*next_to_last);
 344                         std::fill(self + 1, other, _ex1);
 345                         *other = _ex2;
 346                         return true;
 347
 348                 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
 349                 // (commutate contracted indices towards each other, simplify_indexed()
 350                 // will re-expand and re-run the simplification)
 351                 } else {
 352                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
 353                                 return false;
 354
 355                         exvector::iterator next_to_last = other - 1;
 356                         ex S = ncmul(exvector(self + 1, next_to_last), true);
 357
 358                         *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
 359                         std::fill(self + 1, other + 1, _ex1);
 360                         return true;
 361                 }
 362
 363         } else if (is_a<symbol>(other->op(0)) && other->nops() == 2) {
 364
 365                 // x.mu gamma~mu -> x-slash
 366                 *self = dirac_slash(other->op(0), dim, rl);
 367                 *other = _ex1;
 368                 return true;
 369         }
 370
 371         return false;
 372 }
 373
 374 /** An utility function looking for a given metric within an exvector,
 375  *  used in cliffordunit::contract_with(). */
 376 static int find_same_metric(exvector & v, ex & c)
 377 {
 378         for (int i=0; i<v.size();i++) {
 379                 if (!is_a<clifford>(v[i]) && is_a<indexed>(v[i])
 380                     && ex_to<clifford>(c).same_metric(v[i])
 381                     && (ex_to<varidx>(c.op(1)) == ex_to<indexed>(v[i]).get_indices()[0]
 382                         || ex_to<varidx>(c.op(1)).toggle_variance() == ex_to<indexed>(v[i]).get_indices()[0])) {
 383                         return ++i; // next to found
 384                 }
 385         }
 386         return 0; //nothing found
 387 }
 388
 389 /** Contraction of a Clifford unit with something else. */
 390 bool cliffordunit::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 391 {
 392         GINAC_ASSERT(is_a<clifford>(*self));
 393         GINAC_ASSERT(is_a<indexed>(*other));
 394         GINAC_ASSERT(is_a<cliffordunit>(self->op(0)));
 395         clifford unit = ex_to<clifford>(*self);
 396         unsigned char rl = unit.get_representation_label();
 397
 398         if (is_a<clifford>(*other)) {
 399                 // Contraction only makes sense if the represenation labels are equal
 400                 // and the metrics are the same
 401                 if ((ex_to<clifford>(*other).get_representation_label() != rl)
 402                     && unit.same_metric(*other))
 403                         return false;
 404
 405                 // Find if a previous contraction produces the square of self
 406                 int prev_square = find_same_metric(v, self[0]);
 407                 varidx d((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(self->op(1)).get_dim());
 408                 ex squared_metric = unit.get_metric(self->op(1), d) * unit.get_metric(d.toggle_variance(), other->op(1));
 409
 410                 // e~mu e.mu = Tr ONE
 411                 if (other - self == 1) {
 412                         if (prev_square != 0) {
 413                                 *self = squared_metric;
 414                                 v[prev_square-1] = _ex1;
 415                         } else
 416                                 *self = unit.get_metric(self->op(1), other->op(1));
 417                         *other = dirac_ONE(rl);
 418                         return true;
 419
 420                 // e~mu e~alpha e.mu = (2e~alpha^2-Tr) e~alpha
 421                 } else if (other - self == 2
 422                         && is_a<clifford>(self[1])) {
 423
 424                         const ex & ia = self[1].op(1);
 425                         const ex & ib = self[1].op(1);
 426                         if (is_a<tensmetric>(unit.get_metric()))
 427                                 *self = 2 - unit.get_metric(self->op(1), other->op(1));
 428                         else if (prev_square != 0) {
 429                                 *self = 2-squared_metric;
 430                                 v[prev_square-1] = _ex1;
 431                         } else
 432                                 *self = 2*unit.get_metric(ia, ib) - unit.get_metric(self->op(1), other->op(1));
 433                         *other = _ex1;
 434                         return true;
 435
 436                 // e~mu S e~alpha e.mu = 2 e~alpha^3 S - e~mu S e.mu e~alpha
 437                 // (commutate contracted indices towards each other, simplify_indexed()
 438                 // will re-expand and re-run the simplification)
 439                 } else {
 440                         exvector::iterator it = self + 1, next_to_last = other - 1;
 441                         while (it != other) {
 442                                 if (!is_a<clifford>(*it))
 443                                         return false;
 444                                 ++it;
 445                         }
 446
 447                         it = self + 1;
 448                         ex S = _ex1;
 449                         while (it != next_to_last) {
 450                                 S *= *it;
 451                                 *it++ = _ex1;
 452                         }
 453
 454                         const ex & ia = next_to_last->op(1);
 455                         const ex & ib = next_to_last->op(1);
 456                         if (is_a<tensmetric>(unit.get_metric()))
 457                                 *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
 458                         else if (prev_square != 0) {
 459                                 *self = 2 * (*next_to_last) * S  - (*self) * S * (*other) * (*next_to_last)*unit.get_metric(self->op(1),self->op(1));
 460                                 v[prev_square-1] = _ex1;
 461                         } else
 462                                 *self = 2 * (*next_to_last) * S* unit.get_metric(ia,ib) - (*self) * S * (*other) * (*next_to_last);
 463                         *next_to_last = _ex1;
 464                         *other = _ex1;
 465                         return true;
 466                 }
 467
 468         }
 469
 470         return false;
 471 }
 472
 473 /** Perform automatic simplification on noncommutative product of clifford
 474  *  objects. This removes superfluous ONEs, permutes gamma5/L/R's to the front
 475  *  and removes squares of gamma objects. */
 476 ex clifford::eval_ncmul(const exvector & v) const
 477 {
 478         exvector s;
 479         s.reserve(v.size());
 480
 481         // Remove superfluous ONEs
 482         exvector::const_iterator cit = v.begin(), citend = v.end();
 483         while (cit != citend) {
 484                 if (!is_a<clifford>(*cit) || !is_a<diracone>(cit->op(0)))
 485                         s.push_back(*cit);
 486                 cit++;
 487         }
 488
 489         bool something_changed = false;
 490         int sign = 1;
 491
 492         // Anticommutate gamma5/L/R's to the front
 493         if (s.size() >= 2) {
 494                 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
 495                 while (true) {
 496                         exvector::iterator it = next_to_last;
 497                         while (true) {
 498                                 exvector::iterator it2 = it + 1;
 499                                 if (is_a<clifford>(*it) && is_a<clifford>(*it2)) {
 500                                         ex e1 = it->op(0), e2 = it2->op(0);
 501
 502                                         if (is_a<diracgamma5>(e2)) {
 503
 504                                                 if (is_a<diracgammaL>(e1) || is_a<diracgammaR>(e1)) {
 505
 506                                                         // gammaL/R gamma5 -> gamma5 gammaL/R
 507                                                         it->swap(*it2);
 508                                                         something_changed = true;
 509
 510                                                 } else if (!is_a<diracgamma5>(e1)) {
 511
 512                                                         // gamma5 gamma5 -> gamma5 gamma5 (do nothing)
 513                                                         // x gamma5 -> -gamma5 x
 514                                                         it->swap(*it2);
 515                                                         sign = -sign;
 516                                                         something_changed = true;
 517                                                 }
 518
 519                                         } else if (is_a<diracgammaL>(e2)) {
 520
 521                                                 if (is_a<diracgammaR>(e1)) {
 522
 523                                                         // gammaR gammaL -> 0
 524                                                         return _ex0;
 525
 526                                                 } else if (!is_a<diracgammaL>(e1) && !is_a<diracgamma5>(e1)) {
 527
 528                                                         // gammaL gammaL -> gammaL gammaL (do nothing)
 529                                                         // gamma5 gammaL -> gamma5 gammaL (do nothing)
 530                                                         // x gammaL -> gammaR x
 531                                                         it->swap(*it2);
 532                                                         *it = clifford(diracgammaR(), ex_to<clifford>(*it).get_representation_label());
 533                                                         something_changed = true;
 534                                                 }
 535
 536                                         } else if (is_a<diracgammaR>(e2)) {
 537
 538                                                 if (is_a<diracgammaL>(e1)) {
 539
 540                                                         // gammaL gammaR -> 0
 541                                                         return _ex0;
 542
 543                                                 } else if (!is_a<diracgammaR>(e1) && !is_a<diracgamma5>(e1)) {
 544
 545                                                         // gammaR gammaR -> gammaR gammaR (do nothing)
 546                                                         // gamma5 gammaR -> gamma5 gammaR (do nothing)
 547                                                         // x gammaR -> gammaL x
 548                                                         it->swap(*it2);
 549                                                         *it = clifford(diracgammaL(), ex_to<clifford>(*it).get_representation_label());
 550                                                         something_changed = true;
 551                                                 }
 552                                         }
 553                                 }
 554                                 if (it == first)
 555                                         break;
 556                                 --it;
 557                         }
 558                         if (next_to_last == first)
 559                                 break;
 560                         --next_to_last;
 561                 }
 562         }
 563
 564         // Remove equal adjacent gammas
 565         if (s.size() >= 2) {
 566                 exvector::iterator it, itend = s.end() - 1;
 567                 for (it = s.begin(); it != itend; ++it) {
 568                         ex & a = it[0];
 569                         ex & b = it[1];
 570                         if (!is_a<clifford>(a) || !is_a<clifford>(b))
 571                                 continue;
 572
 573                         const ex & ag = a.op(0);
 574                         const ex & bg = b.op(0);
 575                         bool a_is_cliffordunit = is_a<cliffordunit>(ag);
 576                         bool b_is_cliffordunit =  is_a<cliffordunit>(bg);
 577
 578                         if (a_is_cliffordunit && b_is_cliffordunit && ex_to<clifford>(a).same_metric(b)) {
 579
 580                                 const ex & ia = a.op(1);
 581                                 const ex & ib = b.op(1);
 582                                 if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
 583                                         a = ex_to<clifford>(a).get_metric(ia, ib);
 584                                         b = dirac_ONE(representation_label);
 585                                         something_changed = true;
 586                                 }
 587
 588                         } else if ((is_a<diracgamma5>(ag) && is_a<diracgamma5>(bg))) {
 589
 590                                 // Remove squares of gamma5
 591                                 a = dirac_ONE(representation_label);
 592                                 b = dirac_ONE(representation_label);
 593                                 something_changed = true;
 594
 595                         } else if ((is_a<diracgammaL>(ag) && is_a<diracgammaL>(bg))
 596                                 || (is_a<diracgammaR>(ag) && is_a<diracgammaR>(bg))) {
 597
 598                                 // Remove squares of gammaL/R
 599                                 b = dirac_ONE(representation_label);
 600                                 something_changed = true;
 601
 602                         } else if (is_a<diracgammaL>(ag) && is_a<diracgammaR>(bg)) {
 603
 604                                 // gammaL and gammaR are orthogonal
 605                                 return _ex0;
 606
 607                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaL>(bg)) {
 608
 609                                 // gamma5 gammaL -> -gammaL
 610                                 a = dirac_ONE(representation_label);
 611                                 sign = -sign;
 612                                 something_changed = true;
 613
 614                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaR>(bg)) {
 615
 616                                 // gamma5 gammaR -> gammaR
 617                                 a = dirac_ONE(representation_label);
 618                                 something_changed = true;
 619
 620                         } else if (!a_is_cliffordunit && !b_is_cliffordunit && ag.is_equal(bg)) {
 621
 622                                 // a\ a\ -> a^2
 623                                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).minimal_dim(ex_to<idx>(b.op(1))));
 624
 625                                 a = indexed(ag, ix) * indexed(ag, ix.toggle_variance());
 626                                 b = dirac_ONE(representation_label);
 627                                 something_changed = true;
 628                         }
 629                 }
 630         }
 631
 632         if (s.empty())
 633                 return clifford(diracone(), representation_label) * sign;
 634         if (something_changed)
 635                 return reeval_ncmul(s) * sign;
 636         else
 637                 return hold_ncmul(s) * sign;
 638 }
 639
 640 ex clifford::thiscontainer(const exvector & v) const
 641 {
 642         return clifford(representation_label, get_metric(), v);
 643 }
 644
 645 ex clifford::thiscontainer(std::auto_ptr<exvector> vp) const
 646 {
 647         return clifford(representation_label, get_metric(), vp);
 648 }
 649
 650 ex diracgamma5::conjugate() const
 651 {
 652         return _ex_1 * (*this);
 653 }
 654
 655 ex diracgammaL::conjugate() const
 656 {
 657         return (new diracgammaR)->setflag(status_flags::dynallocated);
 658 }
 659
 660 ex diracgammaR::conjugate() const
 661 {
 662         return (new diracgammaL)->setflag(status_flags::dynallocated);
 663 }
 664
 665 //////////
 666 // global functions
 667 //////////
 668
 669 ex dirac_ONE(unsigned char rl)
 670 {
 671         static ex ONE = (new diracone)->setflag(status_flags::dynallocated);
 672         return clifford(ONE, rl);
 673 }
 674
 675 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl)
 676 {
 677         static ex unit = (new cliffordunit)->setflag(status_flags::dynallocated);
 678
 679         if (!is_a<varidx>(mu))
 680                 throw(std::invalid_argument("index of Clifford unit must be of type varidx"));
 681
 682         if (is_a<indexed>(metr))
 683                 return clifford(unit, mu, metr.op(0), rl);
 684         else if(is_a<tensmetric>(metr) || is_a<matrix>(metr))
 685                 return clifford(unit, mu, metr, rl);
 686         else
 687                 throw(std::invalid_argument("metric for Clifford unit must be of type indexed, tensormetric or matrix"));
 688 }
 689
 690 ex dirac_gamma(const ex & mu, unsigned char rl)
 691 {
 692         static ex gamma = (new diracgamma)->setflag(status_flags::dynallocated);
 693
 694         if (!is_a<varidx>(mu))
 695                 throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
 696
 697         return clifford(gamma, mu, default_metric(), rl);
 698 }
 699
 700 ex dirac_gamma5(unsigned char rl)
 701 {
 702         static ex gamma5 = (new diracgamma5)->setflag(status_flags::dynallocated);
 703         return clifford(gamma5, rl);
 704 }
 705
 706 ex dirac_gammaL(unsigned char rl)
 707 {
 708         static ex gammaL = (new diracgammaL)->setflag(status_flags::dynallocated);
 709         return clifford(gammaL, rl);
 710 }
 711
 712 ex dirac_gammaR(unsigned char rl)
 713 {
 714         static ex gammaR = (new diracgammaR)->setflag(status_flags::dynallocated);
 715         return clifford(gammaR, rl);
 716 }
 717
 718 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
 719 {
 720         // Slashed vectors are actually stored as a clifford object with the
 721         // vector as its base expression and a (dummy) index that just serves
 722         // for storing the space dimensionality
 723         return clifford(e, varidx(0, dim), default_metric(), rl);
 724 }
 725
 726 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 727  *  is that of a clifford object with the specified representation label. */
 728 static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
 729 {
 730         return ti == (TINFO_clifford + rl);
 731 }
 732
 733 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 734  *  is that of a clifford object (with an arbitrary representation label). */
 735 static bool is_clifford_tinfo(unsigned ti)
 736 {
 737         return (ti & ~0xff) == TINFO_clifford;
 738 }
 739
 740 /** Extract representation label from tinfo key (as returned by
 741  *  return_type_tinfo()). */
 742 static unsigned char get_representation_label(unsigned ti)
 743 {
 744         return ti & 0xff;
 745 }
 746
 747 /** Take trace of a string of an even number of Dirac gammas given a vector
 748  *  of indices. */
 749 static ex trace_string(exvector::const_iterator ix, size_t num)
 750 {
 751         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 752         if (num == 2)
 753                 return lorentz_g(ix[0], ix[1]);
 754
 755         // 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 )
 756         else if (num == 4)
 757                 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
 758                      + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
 759                      - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
 760
 761         // Traces of 6 or more gammas are computed recursively:
 762         // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
 763         //   + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
 764         //   - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
 765         //   + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
 766         //   - ...
 767         //   + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
 768         exvector v(num - 2);
 769         int sign = 1;
 770         ex result;
 771         for (size_t i=1; i<num; i++) {
 772                 for (size_t n=1, j=0; n<num; n++) {
 773                         if (n == i)
 774                                 continue;
 775                         v[j++] = ix[n];
 776                 }
 777                 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
 778                 sign = -sign;
 779         }
 780         return result;
 781 }
 782
 783 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE)
 784 {
 785         if (is_a<clifford>(e)) {
 786
 787                 unsigned char rl = ex_to<clifford>(e).get_representation_label();
 788
 789                 // Are we taking the trace over this object's representation label?
 790                 if (rls.find(rl) == rls.end())
 791                         return e;
 792
 793                 // Yes, all elements are traceless, except for dirac_ONE and dirac_L/R
 794                 const ex & g = e.op(0);
 795                 if (is_a<diracone>(g))
 796                         return trONE;
 797                 else if (is_a<diracgammaL>(g) || is_a<diracgammaR>(g))
 798                         return trONE/2;
 799                 else
 800                         return _ex0;
 801
 802         } else if (is_exactly_a<mul>(e)) {
 803
 804                 // Trace of product: pull out non-clifford factors
 805                 ex prod = _ex1;
 806                 for (size_t i=0; i<e.nops(); i++) {
 807                         const ex &o = e.op(i);
 808                         if (is_clifford_tinfo(o.return_type_tinfo()))
 809                                 prod *= dirac_trace(o, rls, trONE);
 810                         else
 811                                 prod *= o;
 812                 }
 813                 return prod;
 814
 815         } else if (is_exactly_a<ncmul>(e)) {
 816
 817                 unsigned char rl = get_representation_label(e.return_type_tinfo());
 818
 819                 // Are we taking the trace over this string's representation label?
 820                 if (rls.find(rl) == rls.end())
 821                         return e;
 822
 823                 // Substitute gammaL/R and expand product, if necessary
 824                 ex e_expanded = e.subs(lst(
 825                         dirac_gammaL(rl) == (dirac_ONE(rl)-dirac_gamma5(rl))/2,
 826                         dirac_gammaR(rl) == (dirac_ONE(rl)+dirac_gamma5(rl))/2
 827                 ), subs_options::no_pattern).expand();
 828                 if (!is_a<ncmul>(e_expanded))
 829                         return dirac_trace(e_expanded, rls, trONE);
 830
 831                 // gamma5 gets moved to the front so this check is enough
 832                 bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
 833                 size_t num = e.nops();
 834
 835                 if (has_gamma5) {
 836
 837                         // Trace of gamma5 * odd number of gammas and trace of
 838                         // gamma5 * gamma.mu * gamma.nu are zero
 839                         if ((num & 1) == 0 || num == 3)
 840                                 return _ex0;
 841
 842                         // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
 843                         // (the epsilon is always 4-dimensional)
 844                         if (num == 5) {
 845                                 ex b1, i1, b2, i2, b3, i3, b4, i4;
 846                                 base_and_index(e.op(1), b1, i1);
 847                                 base_and_index(e.op(2), b2, i2);
 848                                 base_and_index(e.op(3), b3, i3);
 849                                 base_and_index(e.op(4), b4, i4);
 850                                 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();
 851                         }
 852
 853                         // Tr gamma5 S_2k =
 854                         //   I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
 855                         // (the epsilon is always 4-dimensional)
 856                         exvector ix(num-1), bv(num-1);
 857                         for (size_t i=1; i<num; i++)
 858                                 base_and_index(e.op(i), bv[i-1], ix[i-1]);
 859                         num--;
 860                         int *iv = new int[num];
 861                         ex result;
 862                         for (size_t i=0; i<num-3; i++) {
 863                                 ex idx1 = ix[i];
 864                                 for (size_t j=i+1; j<num-2; j++) {
 865                                         ex idx2 = ix[j];
 866                                         for (size_t k=j+1; k<num-1; k++) {
 867                                                 ex idx3 = ix[k];
 868                                                 for (size_t l=k+1; l<num; l++) {
 869                                                         ex idx4 = ix[l];
 870                                                         iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
 871                                                         exvector v;
 872                                                         v.reserve(num - 4);
 873                                                         for (size_t n=0, t=4; n<num; n++) {
 874                                                                 if (n == i || n == j || n == k || n == l)
 875                                                                         continue;
 876                                                                 iv[t++] = n;
 877                                                                 v.push_back(ix[n]);
 878                                                         }
 879                                                         int sign = permutation_sign(iv, iv + num);
 880                                                         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))
 881                                                                 * trace_string(v.begin(), num - 4);
 882                                                 }
 883                                         }
 884                                 }
 885                         }
 886                         delete[] iv;
 887                         return trONE * I * result * mul(bv);
 888
 889                 } else { // no gamma5
 890
 891                         // Trace of odd number of gammas is zero
 892                         if ((num & 1) == 1)
 893                                 return _ex0;
 894
 895                         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 896                         if (num == 2) {
 897                                 ex b1, i1, b2, i2;
 898                                 base_and_index(e.op(0), b1, i1);
 899                                 base_and_index(e.op(1), b2, i2);
 900                                 return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
 901                         }
 902
 903                         exvector iv(num), bv(num);
 904                         for (size_t i=0; i<num; i++)
 905                                 base_and_index(e.op(i), bv[i], iv[i]);
 906
 907                         return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
 908                 }
 909
 910         } else if (e.nops() > 0) {
 911
 912                 // Trace maps to all other container classes (this includes sums)
 913                 pointer_to_map_function_2args<const std::set<unsigned char> &, const ex &> fcn(dirac_trace, rls, trONE);
 914                 return e.map(fcn);
 915
 916         } else
 917                 return _ex0;
 918 }
 919
 920 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE)
 921 {
 922         // Convert list to set
 923         std::set<unsigned char> rls;
 924         for (lst::const_iterator i = rll.begin(); i != rll.end(); ++i) {
 925                 if (i->info(info_flags::nonnegint))
 926                         rls.insert(ex_to<numeric>(*i).to_int());
 927         }
 928
 929         return dirac_trace(e, rls, trONE);
 930 }
 931
 932 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
 933 {
 934         // Convert label to set
 935         std::set<unsigned char> rls;
 936         rls.insert(rl);
 937
 938         return dirac_trace(e, rls, trONE);
 939 }
 940
 941
 942 ex canonicalize_clifford(const ex & e_)
 943 {
 944         pointer_to_map_function fcn(canonicalize_clifford);
 945
 946         if (is_a<matrix>(e_)    // || is_a<pseries>(e) || is_a<integral>(e)
 947                 || is_a<lst>(e_)) {
 948                 return e_.map(fcn);
 949         } else {
 950                 ex e=simplify_indexed(e_);
 951                 // Scan for any ncmul objects
 952                 exmap srl;
 953                 ex aux = e.to_rational(srl);
 954                 for (exmap::iterator i = srl.begin(); i != srl.end(); ++i) {
 955
 956                         ex lhs = i->first;
 957                         ex rhs = i->second;
 958
 959                         if (is_exactly_a<ncmul>(rhs)
 960                                         && rhs.return_type() == return_types::noncommutative
 961                                         && is_clifford_tinfo(rhs.return_type_tinfo())) {
 962
 963                                 // Expand product, if necessary
 964                                 ex rhs_expanded = rhs.expand();
 965                                 if (!is_a<ncmul>(rhs_expanded)) {
 966                                         i->second = canonicalize_clifford(rhs_expanded);
 967                                         continue;
 968
 969                                 } else if (!is_a<clifford>(rhs.op(0)))
 970                                         continue;
 971
 972                                 exvector v;
 973                                 v.reserve(rhs.nops());
 974                                 for (size_t j=0; j<rhs.nops(); j++)
 975                                         v.push_back(rhs.op(j));
 976
 977                                 // Stupid recursive bubble sort because we only want to swap adjacent gammas
 978                                 exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
 979                                 if (is_a<diracgamma5>(it->op(0)) || is_a<diracgammaL>(it->op(0)) || is_a<diracgammaR>(it->op(0)))
 980                                         ++it;
 981                                 while (it != next_to_last) {
 982                                         if (it[0].compare(it[1]) > 0) {
 983                                                 ex save0 = it[0], save1 = it[1];
 984                                                 ex b1, i1, b2, i2;
 985                                                 base_and_index(it[0], b1, i1);
 986                                                 base_and_index(it[1], b2, i2);
 987                                                 it[0] = (ex_to<clifford>(save0).get_metric(i1, i2) * b1 * b2).simplify_indexed();
 988                                                 it[1] = v.size() == 2 ? _ex2 * dirac_ONE(ex_to<clifford>(it[1]).get_representation_label()) : _ex2;
 989                                                 ex sum = ncmul(v);
 990                                                 it[0] = save1;
 991                                                 it[1] = save0;
 992                                                 sum -= ncmul(v, true);
 993                                                 i->second = canonicalize_clifford(sum);
 994                                                 goto next_sym;
 995                                         }
 996                                         ++it;
 997                                 }
 998 next_sym:       ;
 999                         }
1000                 }
1001                 return aux.subs(srl, subs_options::no_pattern).simplify_indexed();
1002         }
1003 }
1004
1005 ex clifford_prime(const ex & e)
1006 {
1007         pointer_to_map_function fcn(clifford_prime);
1008         if (is_a<clifford>(e) && is_a<cliffordunit>(e.op(0))) {
1009                 return -e;
1010         } else if (is_a<add>(e) || is_a<ncmul>(e) || is_a<mul>(e) //|| is_a<pseries>(e) || is_a<integral>(e)
1011                            || is_a<matrix>(e) || is_a<lst>(e)) {
1012                 return e.map(fcn);
1013         } else if (is_a<power>(e)) {
1014                 return pow(clifford_prime(e.op(0)), e.op(1));
1015         } else
1016                 return e;
1017 }
1018
1019 ex remove_dirac_ONE(const ex & e,  unsigned char rl)
1020 {
1021         pointer_to_map_function_1arg<unsigned char> fcn(remove_dirac_ONE, rl);
1022         if (is_a<clifford>(e) && ex_to<clifford>(e).get_representation_label() >= rl) {
1023                 if (is_a<diracone>(e.op(0)))
1024                         return 1;
1025                 else
1026                         throw(std::invalid_argument("Expression is a non-scalar Clifford number!"));
1027         } else if (is_a<add>(e) || is_a<ncmul>(e) || is_a<mul>(e)  // || is_a<pseries>(e) || is_a<integral>(e)
1028                            || is_a<matrix>(e) || is_a<lst>(e)) {
1029                 return e.map(fcn);
1030         } else if (is_a<power>(e)) {
1031                 return pow(remove_dirac_ONE(e.op(0)), e.op(1));
1032         } else
1033                 return e;
1034 }
1035
1036 ex clifford_norm(const ex & e)
1037 {
1038         return sqrt(remove_dirac_ONE(canonicalize_clifford(e * clifford_bar(e)).simplify_indexed()));
1039 }
1040
1041 ex clifford_inverse(const ex & e)
1042 {
1043         ex norm = clifford_norm(e);
1044         if (!norm.is_zero())
1045                 return clifford_bar(e) / pow(norm, 2);
1046         else
1047                 throw(std::invalid_argument("Cannot find inverse of Clifford number with zero norm!"));
1048 }
1049
1050 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl)
1051 {
1052         if (!ex_to<idx>(mu).is_dim_numeric())
1053                 throw(std::invalid_argument("Index should have a numeric dimension"));
1054         ex e = clifford_unit(mu, metr, rl);
1055         return lst_to_clifford(v, e);
1056 }
1057
1058 ex lst_to_clifford(const ex & v, const ex & e) {
1059         unsigned min, max;
1060
1061         if (is_a<clifford>(e)) {
1062                 varidx mu = ex_to<varidx>(e.op(1));
1063                 unsigned dim = (ex_to<numeric>(mu.get_dim())).to_int();
1064
1065                 if (is_a<matrix>(v)) {
1066                         if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows()) {
1067                                 min = ex_to<matrix>(v).rows();
1068                                 max = ex_to<matrix>(v).cols();
1069                         } else {
1070                                 min = ex_to<matrix>(v).cols();
1071                                 max = ex_to<matrix>(v).rows();
1072                         }
1073                         if (min == 1) {
1074                                 if (dim == max)
1075                                         return indexed(v, ex_to<varidx>(mu).toggle_variance()) * e;
1076                                 else
1077                                         throw(std::invalid_argument("Dimensions of vector and clifford unit mismatch"));
1078                         } else
1079                                 throw(std::invalid_argument("First argument should be a vector vector"));
1080                 } else if (is_a<lst>(v)) {
1081                         if (dim == ex_to<lst>(v).nops())
1082                                 return indexed(matrix(dim, 1, ex_to<lst>(v)), ex_to<varidx>(mu).toggle_variance()) * e;
1083                         else
1084                                 throw(std::invalid_argument("List length and dimension of clifford unit mismatch"));
1085                 } else
1086                         throw(std::invalid_argument("Cannot construct from anything but list or vector"));
1087         } else
1088                 throw(std::invalid_argument("The second argument should be a Clifford unit"));
1089 }
1090
1091 /** Auxiliary structure to define a function for striping one Clifford unit
1092  * from vectors. Used in  clifford_to_lst(). */
1093 static ex get_clifford_comp(const ex & e, const ex & c)
1094 {
1095         pointer_to_map_function_1arg<const ex &> fcn(get_clifford_comp, c);
1096         int ival = ex_to<numeric>(ex_to<varidx>(c.op(1)).get_value()).to_int();
1097
1098         if (is_a<add>(e) || is_a<lst>(e) // || is_a<pseries>(e) || is_a<integral>(e)
1099                 || is_a<matrix>(e))
1100                 return e.map(fcn);
1101         else if (is_a<ncmul>(e) || is_a<mul>(e)) {
1102                 // find a Clifford unit with the same metric, delete it and substitute its index
1103                 size_t ind = e.nops() + 1;
1104                 for (size_t j = 0; j < e.nops(); j++)
1105                         if (is_a<clifford>(e.op(j)) && ex_to<clifford>(c).same_metric(e.op(j)))
1106                                 if (ind > e.nops())
1107                                         ind = j;
1108                                 else
1109                                         throw(std::invalid_argument("Expression is a Clifford multi-vector"));
1110                 if (ind < e.nops()) {
1111                         ex S = 1;
1112                         bool same_value_index, found_dummy;
1113                         same_value_index = ( ex_to<varidx>(e.op(ind).op(1)).is_numeric()
1114                                                                  &&  (ival == ex_to<numeric>(ex_to<varidx>(e.op(ind).op(1)).get_value()).to_int()) );
1115                         found_dummy = same_value_index;
1116                         for(size_t j=0; j < e.nops(); j++)
1117                                 if (j != ind)
1118                                         if (same_value_index)
1119                                                 S = S * e.op(j);
1120                                         else {
1121                                                 exvector ind_vec = ex_to<indexed>(e.op(j)).get_dummy_indices(ex_to<indexed>(e.op(ind)));
1122                                                 if (ind_vec.size() > 0) {
1123                                                         found_dummy = true;
1124                                                         exvector::const_iterator it = ind_vec.begin(), itend = ind_vec.end();
1125                                                         while (it != itend) {
1126                                                                 S = S * e.op(j).subs(lst(ex_to<varidx>(*it) == ival, ex_to<varidx>(*it).toggle_variance() == ival), subs_options::no_pattern);
1127                                                                 ++it;
1128                                                         }
1129                                                 } else
1130                                                         S = S * e.op(j);
1131                                         }
1132                         return (found_dummy ? S : 0);
1133                 } else
1134                         throw(std::invalid_argument("Expression is not a Clifford vector to the given units"));
1135         } else if (e.is_zero())
1136                 return e;
1137         else if (is_a<clifford>(e) && ex_to<clifford>(e).same_metric(c))
1138                 if ( ex_to<varidx>(e.op(1)).is_numeric() &&
1139                          (ival != ex_to<numeric>(ex_to<varidx>(e.op(1)).get_value()).to_int()) )
1140                         return 0;
1141                 else
1142                         return 1;
1143         else
1144                 throw(std::invalid_argument("Expression is not usable as a Clifford vector"));
1145 }
1146
1147
1148 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic)
1149 {
1150         GINAC_ASSERT(is_a<clifford>(c));
1151         varidx mu = ex_to<varidx>(c.op(1));
1152         if (! mu.is_dim_numeric())
1153                 throw(std::invalid_argument("Index should have a numeric dimension"));
1154         unsigned int D = ex_to<numeric>(mu.get_dim()).to_int();
1155
1156         if (algebraic) // check if algebraic method is applicable
1157                 for (unsigned int i = 0; i < D; i++)
1158                         if (pow(c.subs(mu == i), 2).is_zero()
1159                                 or (not is_a<numeric>(pow(c.subs(mu == i), 2))))
1160                                 algebraic = false;
1161         lst V;
1162         if (algebraic)
1163                 for (unsigned int i = 0; i < D; i++)
1164                         V.append(remove_dirac_ONE(
1165                                                 simplify_indexed(canonicalize_clifford(e * c.subs(mu == i) +  c.subs(mu == i) * e))
1166                                                 / (2*pow(c.subs(mu == i), 2))));
1167         else {
1168                 ex e1 = canonicalize_clifford(e);
1169                 for (unsigned int i = 0; i < D; i++)
1170                         V.append(get_clifford_comp(e1, c.subs(c.op(1) == i)));
1171         }
1172         return V;
1173 }
1174
1175
1176 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)
1177 {
1178         ex x, D, cu;
1179
1180         if (! is_a<matrix>(v) && ! is_a<lst>(v))
1181                 throw(std::invalid_argument("parameter v should be either vector or list"));
1182
1183         if (is_a<clifford>(G)) {
1184                 cu = G;
1185         } else {
1186                 if (is_a<indexed>(G))
1187                         D = ex_to<varidx>(G.op(1)).get_dim();
1188                 else if (is_a<matrix>(G))
1189                         D = ex_to<matrix>(G).rows();
1190                 else throw(std::invalid_argument("metric should be an indexed object, matrix, or a Clifford unit"));
1191
1192                 varidx mu((new symbol)->setflag(status_flags::dynallocated), D);
1193                 cu = clifford_unit(mu, G, rl);
1194         }
1195
1196         x = lst_to_clifford(v, cu);
1197         ex e = simplify_indexed(canonicalize_clifford((a * x + b) * clifford_inverse(c * x + d)));
1198         return clifford_to_lst(e, cu, false);
1199 }
1200
1201 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl)
1202 {
1203         if (is_a<matrix>(M))
1204                 return clifford_moebius_map(ex_to<matrix>(M)(0,0), ex_to<matrix>(M)(0,1),
1205                                                 ex_to<matrix>(M)(1,0), ex_to<matrix>(M)(1,1), v, G, rl);
1206         else
1207                 throw(std::invalid_argument("parameter M should be a matrix"));
1208 }
1209
1210 } // namespace GiNaC