ginac/clifford.cpp

   1 /** @file clifford.cpp
   2  *
   3  *  Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2007 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 { // We do not print representation label if it is 0
 206                 if (representation_label == 0) {
 207                         this->print_dispatch<inherited>(c, level);
 208                 } else { // otherwise we put it before indices in square brackets; the code is borrowed from indexed.cpp
 209                         if (precedence() <= level) {
 210                                 c.s << '(';
 211                         }
 212                         seq[0].print(c, precedence());
 213                         c.s << '[' << int(representation_label) << ']';
 214                         printindices(c, level);
 215                         if (precedence() <= level) {
 216                                 c.s << ')';
 217                         }
 218                 }
 219         }
 220 }
 221
 222 void clifford::do_print_latex(const print_latex & c, unsigned level) const
 223 {
 224         // dirac_slash() object is printed differently
 225         if (is_dirac_slash(seq[0])) {
 226                 c.s << "{";
 227                 seq[0].print(c, precedence());
 228                 c.s << "\\hspace{-1.0ex}/}";
 229         } else {
 230                 c.s << "\\clifford[" << int(representation_label) << "]";
 231                 this->print_dispatch<inherited>(c, level);
 232         }
 233 }
 234
 235 DEFAULT_COMPARE(diracone)
 236 DEFAULT_COMPARE(cliffordunit)
 237 DEFAULT_COMPARE(diracgamma)
 238 DEFAULT_COMPARE(diracgamma5)
 239 DEFAULT_COMPARE(diracgammaL)
 240 DEFAULT_COMPARE(diracgammaR)
 241
 242 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbf{1}")
 243 DEFAULT_PRINT_LATEX(cliffordunit, "e", "e")
 244 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
 245 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
 246 DEFAULT_PRINT_LATEX(diracgammaL, "gammaL", "{\\gamma_L}")
 247 DEFAULT_PRINT_LATEX(diracgammaR, "gammaR", "{\\gamma_R}")
 248
 249 /** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
 250 static void base_and_index(const ex & c, ex & b, ex & i)
 251 {
 252         GINAC_ASSERT(is_a<clifford>(c));
 253         GINAC_ASSERT(c.nops() == 2);
 254
 255         if (is_a<cliffordunit>(c.op(0))) { // proper dirac gamma object or clifford unit
 256                 i = c.op(1);
 257                 b = _ex1;
 258         } else if (is_a<diracgamma5>(c.op(0)) || is_a<diracgammaL>(c.op(0)) || is_a<diracgammaR>(c.op(0))) { // gamma5/L/R
 259                 i = _ex0;
 260                 b = _ex1;
 261         } else { // slash object, generate new dummy index
 262                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(c.op(1)).get_dim());
 263                 b = indexed(c.op(0), ix.toggle_variance());
 264                 i = ix;
 265         }
 266 }
 267
 268 /** Predicate for finding non-clifford objects. */
 269 struct is_not_a_clifford : public std::unary_function<ex, bool> {
 270         bool operator()(const ex & e)
 271         {
 272                 return !is_a<clifford>(e);
 273         }
 274 };
 275
 276 /** Contraction of a gamma matrix with something else. */
 277 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 278 {
 279         GINAC_ASSERT(is_a<clifford>(*self));
 280         GINAC_ASSERT(is_a<indexed>(*other));
 281         GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
 282         unsigned char rl = ex_to<clifford>(*self).get_representation_label();
 283
 284         ex dim = ex_to<idx>(self->op(1)).get_dim();
 285         if (other->nops() > 1)
 286                 dim = minimal_dim(dim, ex_to<idx>(other->op(1)).get_dim());
 287
 288         if (is_a<clifford>(*other)) {
 289
 290                 // Contraction only makes sense if the represenation labels are equal
 291                 if (ex_to<clifford>(*other).get_representation_label() != rl)
 292                         return false;
 293
 294                 size_t num = other - self;
 295
 296                 // gamma~mu gamma.mu = dim ONE
 297                 if (num == 1) {
 298                         *self = dim;
 299                         *other = dirac_ONE(rl);
 300                         return true;
 301
 302                 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
 303                 } else if (num == 2
 304                         && is_a<clifford>(self[1])) {
 305                         *self = 2 - dim;
 306                         *other = _ex1;
 307                         return true;
 308
 309                 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
 310                 } else if (num == 3
 311                         && is_a<clifford>(self[1])
 312                         && is_a<clifford>(self[2])) {
 313                         ex b1, i1, b2, i2;
 314                         base_and_index(self[1], b1, i1);
 315                         base_and_index(self[2], b2, i2);
 316                         *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
 317                         self[1] = _ex1;
 318                         self[2] = _ex1;
 319                         *other = _ex1;
 320                         return true;
 321
 322                 // 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
 323                 } else if (num == 4
 324                         && is_a<clifford>(self[1])
 325                         && is_a<clifford>(self[2])
 326                         && is_a<clifford>(self[3])) {
 327                         *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3];
 328                         self[1] = _ex1;
 329                         self[2] = _ex1;
 330                         self[3] = _ex1;
 331                         *other = _ex1;
 332                         return true;
 333
 334                 // gamma~mu Sodd gamma.mu = -2 Sodd_R
 335                 // (Chisholm identity in 4 dimensions)
 336                 } else if (!((other - self) & 1) && dim.is_equal(4)) {
 337                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
 338                                 return false;
 339
 340                         *self = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(other), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
 341                         std::fill(self + 1, other, _ex1);
 342                         *other = _ex_2;
 343                         return true;
 344
 345                 // gamma~mu Sodd gamma~alpha gamma.mu = 2 gamma~alpha Sodd + 2 Sodd_R gamma~alpha
 346                 // (commutate contracted indices towards each other, then use
 347                 // Chisholm identity in 4 dimensions)
 348                 } else if (((other - self) & 1) && dim.is_equal(4)) {
 349                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
 350                                 return false;
 351
 352                         exvector::iterator next_to_last = other - 1;
 353                         ex S = ncmul(exvector(self + 1, next_to_last), true);
 354                         ex SR = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(next_to_last), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
 355
 356                         *self = (*next_to_last) * S + SR * (*next_to_last);
 357                         std::fill(self + 1, other, _ex1);
 358                         *other = _ex2;
 359                         return true;
 360
 361                 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
 362                 // (commutate contracted indices towards each other, simplify_indexed()
 363                 // will re-expand and re-run the simplification)
 364                 } else {
 365                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
 366                                 return false;
 367
 368                         exvector::iterator next_to_last = other - 1;
 369                         ex S = ncmul(exvector(self + 1, next_to_last), true);
 370
 371                         *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
 372                         std::fill(self + 1, other + 1, _ex1);
 373                         return true;
 374                 }
 375
 376         } else if (is_a<symbol>(other->op(0)) && other->nops() == 2) {
 377
 378                 // x.mu gamma~mu -> x-slash
 379                 *self = dirac_slash(other->op(0), dim, rl);
 380                 *other = _ex1;
 381                 return true;
 382         }
 383
 384         return false;
 385 }
 386
 387 /** An utility function looking for a given metric within an exvector,
 388  *  used in cliffordunit::contract_with(). */
 389 static int find_same_metric(exvector & v, ex & c)
 390 {
 391         for (int i=0; i<v.size();i++) {
 392                 if (!is_a<clifford>(v[i]) && is_a<indexed>(v[i])
 393                     && ex_to<clifford>(c).same_metric(v[i])
 394                     && (ex_to<varidx>(c.op(1)) == ex_to<indexed>(v[i]).get_indices()[0]
 395                         || ex_to<varidx>(c.op(1)).toggle_variance() == ex_to<indexed>(v[i]).get_indices()[0])) {
 396                         return ++i; // next to found
 397                 }
 398         }
 399         return 0; //nothing found
 400 }
 401
 402 /** Contraction of a Clifford unit with something else. */
 403 bool cliffordunit::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 404 {
 405         GINAC_ASSERT(is_a<clifford>(*self));
 406         GINAC_ASSERT(is_a<indexed>(*other));
 407         GINAC_ASSERT(is_a<cliffordunit>(self->op(0)));
 408         clifford unit = ex_to<clifford>(*self);
 409         unsigned char rl = unit.get_representation_label();
 410
 411         if (is_a<clifford>(*other)) {
 412                 // Contraction only makes sense if the represenation labels are equal
 413                 // and the metrics are the same
 414                 if ((ex_to<clifford>(*other).get_representation_label() != rl)
 415                     && unit.same_metric(*other))
 416                         return false;
 417
 418                 // Find if a previous contraction produces the square of self
 419                 int prev_square = find_same_metric(v, self[0]);
 420                 varidx d((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(self->op(1)).get_dim());
 421                 ex squared_metric = unit.get_metric(self->op(1), d) * unit.get_metric(d.toggle_variance(), other->op(1));
 422
 423                 // e~mu e.mu = Tr ONE
 424                 if (other - self == 1) {
 425                         if (prev_square != 0) {
 426                                 *self = squared_metric;
 427                                 v[prev_square-1] = _ex1;
 428                         } else
 429                                 *self = unit.get_metric(self->op(1), other->op(1));
 430                         *other = dirac_ONE(rl);
 431                         return true;
 432
 433                 // e~mu e~alpha e.mu = (2e~alpha^2-Tr) e~alpha
 434                 } else if (other - self == 2
 435                         && is_a<clifford>(self[1])) {
 436
 437                         const ex & ia = self[1].op(1);
 438                         const ex & ib = self[1].op(1);
 439                         if (is_a<tensmetric>(unit.get_metric()))
 440                                 *self = 2 - unit.get_metric(self->op(1), other->op(1));
 441                         else if (prev_square != 0) {
 442                                 *self = 2-squared_metric;
 443                                 v[prev_square-1] = _ex1;
 444                         } else
 445                                 *self = 2*unit.get_metric(ia, ib) - unit.get_metric(self->op(1), other->op(1));
 446                         *other = _ex1;
 447                         return true;
 448
 449                 // e~mu S e~alpha e.mu = 2 e~alpha^3 S - e~mu S e.mu e~alpha
 450                 // (commutate contracted indices towards each other, simplify_indexed()
 451                 // will re-expand and re-run the simplification)
 452                 } else {
 453                         exvector::iterator it = self + 1, next_to_last = other - 1;
 454                         while (it != other) {
 455                                 if (!is_a<clifford>(*it))
 456                                         return false;
 457                                 ++it;
 458                         }
 459
 460                         it = self + 1;
 461                         ex S = _ex1;
 462                         while (it != next_to_last) {
 463                                 S *= *it;
 464                                 *it++ = _ex1;
 465                         }
 466
 467                         const ex & ia = next_to_last->op(1);
 468                         const ex & ib = next_to_last->op(1);
 469                         if (is_a<tensmetric>(unit.get_metric()))
 470                                 *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
 471                         else if (prev_square != 0) {
 472                                 *self = 2 * (*next_to_last) * S  - (*self) * S * (*other) * (*next_to_last)*unit.get_metric(self->op(1),self->op(1));
 473                                 v[prev_square-1] = _ex1;
 474                         } else
 475                                 *self = 2 * (*next_to_last) * S* unit.get_metric(ia,ib) - (*self) * S * (*other) * (*next_to_last);
 476                         *next_to_last = _ex1;
 477                         *other = _ex1;
 478                         return true;
 479                 }
 480
 481         }
 482
 483         return false;
 484 }
 485
 486 /** Perform automatic simplification on noncommutative product of clifford
 487  *  objects. This removes superfluous ONEs, permutes gamma5/L/R's to the front
 488  *  and removes squares of gamma objects. */
 489 ex clifford::eval_ncmul(const exvector & v) const
 490 {
 491         exvector s;
 492         s.reserve(v.size());
 493
 494         // Remove superfluous ONEs
 495         exvector::const_iterator cit = v.begin(), citend = v.end();
 496         while (cit != citend) {
 497                 if (!is_a<clifford>(*cit) || !is_a<diracone>(cit->op(0)))
 498                         s.push_back(*cit);
 499                 cit++;
 500         }
 501
 502         bool something_changed = false;
 503         int sign = 1;
 504
 505         // Anticommutate gamma5/L/R's to the front
 506         if (s.size() >= 2) {
 507                 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
 508                 while (true) {
 509                         exvector::iterator it = next_to_last;
 510                         while (true) {
 511                                 exvector::iterator it2 = it + 1;
 512                                 if (is_a<clifford>(*it) && is_a<clifford>(*it2)) {
 513                                         ex e1 = it->op(0), e2 = it2->op(0);
 514
 515                                         if (is_a<diracgamma5>(e2)) {
 516
 517                                                 if (is_a<diracgammaL>(e1) || is_a<diracgammaR>(e1)) {
 518
 519                                                         // gammaL/R gamma5 -> gamma5 gammaL/R
 520                                                         it->swap(*it2);
 521                                                         something_changed = true;
 522
 523                                                 } else if (!is_a<diracgamma5>(e1)) {
 524
 525                                                         // gamma5 gamma5 -> gamma5 gamma5 (do nothing)
 526                                                         // x gamma5 -> -gamma5 x
 527                                                         it->swap(*it2);
 528                                                         sign = -sign;
 529                                                         something_changed = true;
 530                                                 }
 531
 532                                         } else if (is_a<diracgammaL>(e2)) {
 533
 534                                                 if (is_a<diracgammaR>(e1)) {
 535
 536                                                         // gammaR gammaL -> 0
 537                                                         return _ex0;
 538
 539                                                 } else if (!is_a<diracgammaL>(e1) && !is_a<diracgamma5>(e1)) {
 540
 541                                                         // gammaL gammaL -> gammaL gammaL (do nothing)
 542                                                         // gamma5 gammaL -> gamma5 gammaL (do nothing)
 543                                                         // x gammaL -> gammaR x
 544                                                         it->swap(*it2);
 545                                                         *it = clifford(diracgammaR(), ex_to<clifford>(*it).get_representation_label());
 546                                                         something_changed = true;
 547                                                 }
 548
 549                                         } else if (is_a<diracgammaR>(e2)) {
 550
 551                                                 if (is_a<diracgammaL>(e1)) {
 552
 553                                                         // gammaL gammaR -> 0
 554                                                         return _ex0;
 555
 556                                                 } else if (!is_a<diracgammaR>(e1) && !is_a<diracgamma5>(e1)) {
 557
 558                                                         // gammaR gammaR -> gammaR gammaR (do nothing)
 559                                                         // gamma5 gammaR -> gamma5 gammaR (do nothing)
 560                                                         // x gammaR -> gammaL x
 561                                                         it->swap(*it2);
 562                                                         *it = clifford(diracgammaL(), ex_to<clifford>(*it).get_representation_label());
 563                                                         something_changed = true;
 564                                                 }
 565                                         }
 566                                 }
 567                                 if (it == first)
 568                                         break;
 569                                 --it;
 570                         }
 571                         if (next_to_last == first)
 572                                 break;
 573                         --next_to_last;
 574                 }
 575         }
 576
 577         // Remove equal adjacent gammas
 578         if (s.size() >= 2) {
 579                 exvector::iterator it, itend = s.end() - 1;
 580                 for (it = s.begin(); it != itend; ++it) {
 581                         ex & a = it[0];
 582                         ex & b = it[1];
 583                         if (!is_a<clifford>(a) || !is_a<clifford>(b))
 584                                 continue;
 585
 586                         const ex & ag = a.op(0);
 587                         const ex & bg = b.op(0);
 588                         bool a_is_cliffordunit = is_a<cliffordunit>(ag);
 589                         bool b_is_cliffordunit =  is_a<cliffordunit>(bg);
 590
 591                         if (a_is_cliffordunit && b_is_cliffordunit && ex_to<clifford>(a).same_metric(b)) {
 592
 593                                 const ex & ia = a.op(1);
 594                                 const ex & ib = b.op(1);
 595                                 if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
 596                                         a = ex_to<clifford>(a).get_metric(ia, ib);
 597                                         b = dirac_ONE(representation_label);
 598                                         something_changed = true;
 599                                 }
 600
 601                         } else if ((is_a<diracgamma5>(ag) && is_a<diracgamma5>(bg))) {
 602
 603                                 // Remove squares of gamma5
 604                                 a = dirac_ONE(representation_label);
 605                                 b = dirac_ONE(representation_label);
 606                                 something_changed = true;
 607
 608                         } else if ((is_a<diracgammaL>(ag) && is_a<diracgammaL>(bg))
 609                                 || (is_a<diracgammaR>(ag) && is_a<diracgammaR>(bg))) {
 610
 611                                 // Remove squares of gammaL/R
 612                                 b = dirac_ONE(representation_label);
 613                                 something_changed = true;
 614
 615                         } else if (is_a<diracgammaL>(ag) && is_a<diracgammaR>(bg)) {
 616
 617                                 // gammaL and gammaR are orthogonal
 618                                 return _ex0;
 619
 620                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaL>(bg)) {
 621
 622                                 // gamma5 gammaL -> -gammaL
 623                                 a = dirac_ONE(representation_label);
 624                                 sign = -sign;
 625                                 something_changed = true;
 626
 627                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaR>(bg)) {
 628
 629                                 // gamma5 gammaR -> gammaR
 630                                 a = dirac_ONE(representation_label);
 631                                 something_changed = true;
 632
 633                         } else if (!a_is_cliffordunit && !b_is_cliffordunit && ag.is_equal(bg)) {
 634
 635                                 // a\ a\ -> a^2
 636                                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).minimal_dim(ex_to<idx>(b.op(1))));
 637
 638                                 a = indexed(ag, ix) * indexed(ag, ix.toggle_variance());
 639                                 b = dirac_ONE(representation_label);
 640                                 something_changed = true;
 641                         }
 642                 }
 643         }
 644
 645         if (s.empty())
 646                 return clifford(diracone(), representation_label) * sign;
 647         if (something_changed)
 648                 return reeval_ncmul(s) * sign;
 649         else
 650                 return hold_ncmul(s) * sign;
 651 }
 652
 653 ex clifford::thiscontainer(const exvector & v) const
 654 {
 655         return clifford(representation_label, get_metric(), v);
 656 }
 657
 658 ex clifford::thiscontainer(std::auto_ptr<exvector> vp) const
 659 {
 660         return clifford(representation_label, get_metric(), vp);
 661 }
 662
 663 ex diracgamma5::conjugate() const
 664 {
 665         return _ex_1 * (*this);
 666 }
 667
 668 ex diracgammaL::conjugate() const
 669 {
 670         return (new diracgammaR)->setflag(status_flags::dynallocated);
 671 }
 672
 673 ex diracgammaR::conjugate() const
 674 {
 675         return (new diracgammaL)->setflag(status_flags::dynallocated);
 676 }
 677
 678 //////////
 679 // global functions
 680 //////////
 681
 682 ex dirac_ONE(unsigned char rl)
 683 {
 684         static ex ONE = (new diracone)->setflag(status_flags::dynallocated);
 685         return clifford(ONE, rl);
 686 }
 687
 688 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl)
 689 {
 690         static ex unit = (new cliffordunit)->setflag(status_flags::dynallocated);
 691
 692         if (!is_a<varidx>(mu))
 693                 throw(std::invalid_argument("index of Clifford unit must be of type varidx"));
 694
 695         if (is_a<indexed>(metr))
 696                 return clifford(unit, mu, metr.op(0), rl);
 697         else if(is_a<tensmetric>(metr) || is_a<matrix>(metr))
 698                 return clifford(unit, mu, metr, rl);
 699         else
 700                 throw(std::invalid_argument("metric for Clifford unit must be of type indexed, tensormetric or matrix"));
 701 }
 702
 703 ex dirac_gamma(const ex & mu, unsigned char rl)
 704 {
 705         static ex gamma = (new diracgamma)->setflag(status_flags::dynallocated);
 706
 707         if (!is_a<varidx>(mu))
 708                 throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
 709
 710         return clifford(gamma, mu, default_metric(), rl);
 711 }
 712
 713 ex dirac_gamma5(unsigned char rl)
 714 {
 715         static ex gamma5 = (new diracgamma5)->setflag(status_flags::dynallocated);
 716         return clifford(gamma5, rl);
 717 }
 718
 719 ex dirac_gammaL(unsigned char rl)
 720 {
 721         static ex gammaL = (new diracgammaL)->setflag(status_flags::dynallocated);
 722         return clifford(gammaL, rl);
 723 }
 724
 725 ex dirac_gammaR(unsigned char rl)
 726 {
 727         static ex gammaR = (new diracgammaR)->setflag(status_flags::dynallocated);
 728         return clifford(gammaR, rl);
 729 }
 730
 731 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
 732 {
 733         // Slashed vectors are actually stored as a clifford object with the
 734         // vector as its base expression and a (dummy) index that just serves
 735         // for storing the space dimensionality
 736         return clifford(e, varidx(0, dim), default_metric(), rl);
 737 }
 738
 739 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 740  *  is that of a clifford object with the specified representation label. */
 741 static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
 742 {
 743         return ti == (TINFO_clifford + rl);
 744 }
 745
 746 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 747  *  is that of a clifford object (with an arbitrary representation label). */
 748 static bool is_clifford_tinfo(unsigned ti)
 749 {
 750         return (ti & ~0xff) == TINFO_clifford;
 751 }
 752
 753 /** Extract representation label from tinfo key (as returned by
 754  *  return_type_tinfo()). */
 755 static unsigned char get_representation_label(unsigned ti)
 756 {
 757         return ti & 0xff;
 758 }
 759
 760 /** Take trace of a string of an even number of Dirac gammas given a vector
 761  *  of indices. */
 762 static ex trace_string(exvector::const_iterator ix, size_t num)
 763 {
 764         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 765         if (num == 2)
 766                 return lorentz_g(ix[0], ix[1]);
 767
 768         // 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 )
 769         else if (num == 4)
 770                 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
 771                      + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
 772                      - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
 773
 774         // Traces of 6 or more gammas are computed recursively:
 775         // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
 776         //   + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
 777         //   - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
 778         //   + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
 779         //   - ...
 780         //   + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
 781         exvector v(num - 2);
 782         int sign = 1;
 783         ex result;
 784         for (size_t i=1; i<num; i++) {
 785                 for (size_t n=1, j=0; n<num; n++) {
 786                         if (n == i)
 787                                 continue;
 788                         v[j++] = ix[n];
 789                 }
 790                 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
 791                 sign = -sign;
 792         }
 793         return result;
 794 }
 795
 796 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE)
 797 {
 798         if (is_a<clifford>(e)) {
 799
 800                 unsigned char rl = ex_to<clifford>(e).get_representation_label();
 801
 802                 // Are we taking the trace over this object's representation label?
 803                 if (rls.find(rl) == rls.end())
 804                         return e;
 805
 806                 // Yes, all elements are traceless, except for dirac_ONE and dirac_L/R
 807                 const ex & g = e.op(0);
 808                 if (is_a<diracone>(g))
 809                         return trONE;
 810                 else if (is_a<diracgammaL>(g) || is_a<diracgammaR>(g))
 811                         return trONE/2;
 812                 else
 813                         return _ex0;
 814
 815         } else if (is_exactly_a<mul>(e)) {
 816
 817                 // Trace of product: pull out non-clifford factors
 818                 ex prod = _ex1;
 819                 for (size_t i=0; i<e.nops(); i++) {
 820                         const ex &o = e.op(i);
 821                         if (is_clifford_tinfo(o.return_type_tinfo()))
 822                                 prod *= dirac_trace(o, rls, trONE);
 823                         else
 824                                 prod *= o;
 825                 }
 826                 return prod;
 827
 828         } else if (is_exactly_a<ncmul>(e)) {
 829
 830                 unsigned char rl = get_representation_label(e.return_type_tinfo());
 831
 832                 // Are we taking the trace over this string's representation label?
 833                 if (rls.find(rl) == rls.end())
 834                         return e;
 835
 836                 // Substitute gammaL/R and expand product, if necessary
 837                 ex e_expanded = e.subs(lst(
 838                         dirac_gammaL(rl) == (dirac_ONE(rl)-dirac_gamma5(rl))/2,
 839                         dirac_gammaR(rl) == (dirac_ONE(rl)+dirac_gamma5(rl))/2
 840                 ), subs_options::no_pattern).expand();
 841                 if (!is_a<ncmul>(e_expanded))
 842                         return dirac_trace(e_expanded, rls, trONE);
 843
 844                 // gamma5 gets moved to the front so this check is enough
 845                 bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
 846                 size_t num = e.nops();
 847
 848                 if (has_gamma5) {
 849
 850                         // Trace of gamma5 * odd number of gammas and trace of
 851                         // gamma5 * gamma.mu * gamma.nu are zero
 852                         if ((num & 1) == 0 || num == 3)
 853                                 return _ex0;
 854
 855                         // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
 856                         // (the epsilon is always 4-dimensional)
 857                         if (num == 5) {
 858                                 ex b1, i1, b2, i2, b3, i3, b4, i4;
 859                                 base_and_index(e.op(1), b1, i1);
 860                                 base_and_index(e.op(2), b2, i2);
 861                                 base_and_index(e.op(3), b3, i3);
 862                                 base_and_index(e.op(4), b4, i4);
 863                                 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();
 864                         }
 865
 866                         // Tr gamma5 S_2k =
 867                         //   I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
 868                         // (the epsilon is always 4-dimensional)
 869                         exvector ix(num-1), bv(num-1);
 870                         for (size_t i=1; i<num; i++)
 871                                 base_and_index(e.op(i), bv[i-1], ix[i-1]);
 872                         num--;
 873                         int *iv = new int[num];
 874                         ex result;
 875                         for (size_t i=0; i<num-3; i++) {
 876                                 ex idx1 = ix[i];
 877                                 for (size_t j=i+1; j<num-2; j++) {
 878                                         ex idx2 = ix[j];
 879                                         for (size_t k=j+1; k<num-1; k++) {
 880                                                 ex idx3 = ix[k];
 881                                                 for (size_t l=k+1; l<num; l++) {
 882                                                         ex idx4 = ix[l];
 883                                                         iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
 884                                                         exvector v;
 885                                                         v.reserve(num - 4);
 886                                                         for (size_t n=0, t=4; n<num; n++) {
 887                                                                 if (n == i || n == j || n == k || n == l)
 888                                                                         continue;
 889                                                                 iv[t++] = n;
 890                                                                 v.push_back(ix[n]);
 891                                                         }
 892                                                         int sign = permutation_sign(iv, iv + num);
 893                                                         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))
 894                                                                 * trace_string(v.begin(), num - 4);
 895                                                 }
 896                                         }
 897                                 }
 898                         }
 899                         delete[] iv;
 900                         return trONE * I * result * mul(bv);
 901
 902                 } else { // no gamma5
 903
 904                         // Trace of odd number of gammas is zero
 905                         if ((num & 1) == 1)
 906                                 return _ex0;
 907
 908                         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 909                         if (num == 2) {
 910                                 ex b1, i1, b2, i2;
 911                                 base_and_index(e.op(0), b1, i1);
 912                                 base_and_index(e.op(1), b2, i2);
 913                                 return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
 914                         }
 915
 916                         exvector iv(num), bv(num);
 917                         for (size_t i=0; i<num; i++)
 918                                 base_and_index(e.op(i), bv[i], iv[i]);
 919
 920                         return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
 921                 }
 922
 923         } else if (e.nops() > 0) {
 924
 925                 // Trace maps to all other container classes (this includes sums)
 926                 pointer_to_map_function_2args<const std::set<unsigned char> &, const ex &> fcn(dirac_trace, rls, trONE);
 927                 return e.map(fcn);
 928
 929         } else
 930                 return _ex0;
 931 }
 932
 933 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE)
 934 {
 935         // Convert list to set
 936         std::set<unsigned char> rls;
 937         for (lst::const_iterator i = rll.begin(); i != rll.end(); ++i) {
 938                 if (i->info(info_flags::nonnegint))
 939                         rls.insert(ex_to<numeric>(*i).to_int());
 940         }
 941
 942         return dirac_trace(e, rls, trONE);
 943 }
 944
 945 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
 946 {
 947         // Convert label to set
 948         std::set<unsigned char> rls;
 949         rls.insert(rl);
 950
 951         return dirac_trace(e, rls, trONE);
 952 }
 953
 954
 955 ex canonicalize_clifford(const ex & e_)
 956 {
 957         pointer_to_map_function fcn(canonicalize_clifford);
 958
 959         if (is_a<matrix>(e_)    // || is_a<pseries>(e) || is_a<integral>(e)
 960                 || is_a<lst>(e_)) {
 961                 return e_.map(fcn);
 962         } else {
 963                 ex e=simplify_indexed(e_);
 964                 // Scan for any ncmul objects
 965                 exmap srl;
 966                 ex aux = e.to_rational(srl);
 967                 for (exmap::iterator i = srl.begin(); i != srl.end(); ++i) {
 968
 969                         ex lhs = i->first;
 970                         ex rhs = i->second;
 971
 972                         if (is_exactly_a<ncmul>(rhs)
 973                                         && rhs.return_type() == return_types::noncommutative
 974                                         && is_clifford_tinfo(rhs.return_type_tinfo())) {
 975
 976                                 // Expand product, if necessary
 977                                 ex rhs_expanded = rhs.expand();
 978                                 if (!is_a<ncmul>(rhs_expanded)) {
 979                                         i->second = canonicalize_clifford(rhs_expanded);
 980                                         continue;
 981
 982                                 } else if (!is_a<clifford>(rhs.op(0)))
 983                                         continue;
 984
 985                                 exvector v;
 986                                 v.reserve(rhs.nops());
 987                                 for (size_t j=0; j<rhs.nops(); j++)
 988                                         v.push_back(rhs.op(j));
 989
 990                                 // Stupid recursive bubble sort because we only want to swap adjacent gammas
 991                                 exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
 992                                 if (is_a<diracgamma5>(it->op(0)) || is_a<diracgammaL>(it->op(0)) || is_a<diracgammaR>(it->op(0)))
 993                                         ++it;
 994                                 while (it != next_to_last) {
 995                                         if (it[0].compare(it[1]) > 0) {
 996                                                 ex save0 = it[0], save1 = it[1];
 997                                                 ex b1, i1, b2, i2;
 998                                                 base_and_index(it[0], b1, i1);
 999                                                 base_and_index(it[1], b2, i2);
1000                                                 it[0] = (ex_to<clifford>(save0).get_metric(i1, i2) * b1 * b2).simplify_indexed();
1001                                                 it[1] = v.size() == 2 ? _ex2 * dirac_ONE(ex_to<clifford>(it[1]).get_representation_label()) : _ex2;
1002                                                 ex sum = ncmul(v);
1003                                                 it[0] = save1;
1004                                                 it[1] = save0;
1005                                                 sum -= ncmul(v, true);
1006                                                 i->second = canonicalize_clifford(sum);
1007                                                 goto next_sym;
1008                                         }
1009                                         ++it;
1010                                 }
1011 next_sym:       ;
1012                         }
1013                 }
1014                 return aux.subs(srl, subs_options::no_pattern).simplify_indexed();
1015         }
1016 }
1017
1018 ex clifford_prime(const ex & e)
1019 {
1020         pointer_to_map_function fcn(clifford_prime);
1021         if (is_a<clifford>(e) && is_a<cliffordunit>(e.op(0))) {
1022                 return -e;
1023         } else if (is_a<add>(e) || is_a<ncmul>(e) || is_a<mul>(e) //|| is_a<pseries>(e) || is_a<integral>(e)
1024                            || is_a<matrix>(e) || is_a<lst>(e)) {
1025                 return e.map(fcn);
1026         } else if (is_a<power>(e)) {
1027                 return pow(clifford_prime(e.op(0)), e.op(1));
1028         } else
1029                 return e;
1030 }
1031
1032 ex remove_dirac_ONE(const ex & e)
1033 {
1034         return remove_dirac_ONE(e, 0);
1035 }
1036
1037 ex remove_dirac_ONE(const ex & e,  unsigned char rl)
1038 {
1039         pointer_to_map_function_1arg<unsigned char> fcn(remove_dirac_ONE, rl);
1040         if (is_a<clifford>(e) && ex_to<clifford>(e).get_representation_label() >= rl) {
1041                 if (is_a<diracone>(e.op(0)))
1042                         return 1;
1043                 else
1044                         throw(std::invalid_argument("Expression is a non-scalar Clifford number!"));
1045         } else if (is_a<add>(e) || is_a<ncmul>(e) || is_a<mul>(e)  // || is_a<pseries>(e) || is_a<integral>(e)
1046                            || is_a<matrix>(e) || is_a<lst>(e)) {
1047                 return e.map(fcn);
1048         } else if (is_a<power>(e)) {
1049                 return pow(remove_dirac_ONE(e.op(0)), e.op(1));
1050         } else
1051                 return e;
1052 }
1053
1054 ex clifford_norm(const ex & e)
1055 {
1056         return sqrt(remove_dirac_ONE(canonicalize_clifford(e * clifford_bar(e)).simplify_indexed()));
1057 }
1058
1059 ex clifford_inverse(const ex & e)
1060 {
1061         ex norm = clifford_norm(e);
1062         if (!norm.is_zero())
1063                 return clifford_bar(e) / pow(norm, 2);
1064         else
1065                 throw(std::invalid_argument("Cannot find inverse of Clifford number with zero norm!"));
1066 }
1067
1068 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl)
1069 {
1070         if (!ex_to<idx>(mu).is_dim_numeric())
1071                 throw(std::invalid_argument("Index should have a numeric dimension"));
1072         ex e = clifford_unit(mu, metr, rl);
1073         return lst_to_clifford(v, e);
1074 }
1075
1076 ex lst_to_clifford(const ex & v, const ex & e) {
1077         unsigned min, max;
1078
1079         if (is_a<clifford>(e)) {
1080                 varidx mu = ex_to<varidx>(e.op(1));
1081                 unsigned dim = (ex_to<numeric>(mu.get_dim())).to_int();
1082
1083                 if (is_a<matrix>(v)) {
1084                         if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows()) {
1085                                 min = ex_to<matrix>(v).rows();
1086                                 max = ex_to<matrix>(v).cols();
1087                         } else {
1088                                 min = ex_to<matrix>(v).cols();
1089                                 max = ex_to<matrix>(v).rows();
1090                         }
1091                         if (min == 1) {
1092                                 if (dim == max)
1093                                         return indexed(v, ex_to<varidx>(mu).toggle_variance()) * e;
1094                                 else
1095                                         throw(std::invalid_argument("Dimensions of vector and clifford unit mismatch"));
1096                         } else
1097                                 throw(std::invalid_argument("First argument should be a vector vector"));
1098                 } else if (is_a<lst>(v)) {
1099                         if (dim == ex_to<lst>(v).nops())
1100                                 return indexed(matrix(dim, 1, ex_to<lst>(v)), ex_to<varidx>(mu).toggle_variance()) * e;
1101                         else
1102                                 throw(std::invalid_argument("List length and dimension of clifford unit mismatch"));
1103                 } else
1104                         throw(std::invalid_argument("Cannot construct from anything but list or vector"));
1105         } else
1106                 throw(std::invalid_argument("The second argument should be a Clifford unit"));
1107 }
1108
1109 /** Auxiliary structure to define a function for striping one Clifford unit
1110  * from vectors. Used in  clifford_to_lst(). */
1111 static ex get_clifford_comp(const ex & e, const ex & c)
1112 {
1113         pointer_to_map_function_1arg<const ex &> fcn(get_clifford_comp, c);
1114         int ival = ex_to<numeric>(ex_to<varidx>(c.op(1)).get_value()).to_int();
1115
1116         if (is_a<add>(e) || is_a<lst>(e) // || is_a<pseries>(e) || is_a<integral>(e)
1117                 || is_a<matrix>(e))
1118                 return e.map(fcn);
1119         else if (is_a<ncmul>(e) || is_a<mul>(e)) {
1120                 // find a Clifford unit with the same metric, delete it and substitute its index
1121                 size_t ind = e.nops() + 1;
1122                 for (size_t j = 0; j < e.nops(); j++)
1123                         if (is_a<clifford>(e.op(j)) && ex_to<clifford>(c).same_metric(e.op(j)))
1124                                 if (ind > e.nops())
1125                                         ind = j;
1126                                 else
1127                                         throw(std::invalid_argument("Expression is a Clifford multi-vector"));
1128                 if (ind < e.nops()) {
1129                         ex S = 1;
1130                         bool same_value_index, found_dummy;
1131                         same_value_index = ( ex_to<varidx>(e.op(ind).op(1)).is_numeric()
1132                                                                  &&  (ival == ex_to<numeric>(ex_to<varidx>(e.op(ind).op(1)).get_value()).to_int()) );
1133                         found_dummy = same_value_index;
1134                         for(size_t j=0; j < e.nops(); j++)
1135                                 if (j != ind)
1136                                         if (same_value_index)
1137                                                 S = S * e.op(j);
1138                                         else {
1139                                                 exvector ind_vec = ex_to<indexed>(e.op(j)).get_dummy_indices(ex_to<indexed>(e.op(ind)));
1140                                                 if (ind_vec.size() > 0) {
1141                                                         found_dummy = true;
1142                                                         exvector::const_iterator it = ind_vec.begin(), itend = ind_vec.end();
1143                                                         while (it != itend) {
1144                                                                 S = S * e.op(j).subs(lst(ex_to<varidx>(*it) == ival, ex_to<varidx>(*it).toggle_variance() == ival), subs_options::no_pattern);
1145                                                                 ++it;
1146                                                         }
1147                                                 } else
1148                                                         S = S * e.op(j);
1149                                         }
1150                         return (found_dummy ? S : 0);
1151                 } else
1152                         throw(std::invalid_argument("Expression is not a Clifford vector to the given units"));
1153         } else if (e.is_zero())
1154                 return e;
1155         else if (is_a<clifford>(e) && ex_to<clifford>(e).same_metric(c))
1156                 if ( ex_to<varidx>(e.op(1)).is_numeric() &&
1157                          (ival != ex_to<numeric>(ex_to<varidx>(e.op(1)).get_value()).to_int()) )
1158                         return 0;
1159                 else
1160                         return 1;
1161         else
1162                 throw(std::invalid_argument("Expression is not usable as a Clifford vector"));
1163 }
1164
1165
1166 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic)
1167 {
1168         GINAC_ASSERT(is_a<clifford>(c));
1169         varidx mu = ex_to<varidx>(c.op(1));
1170         if (! mu.is_dim_numeric())
1171                 throw(std::invalid_argument("Index should have a numeric dimension"));
1172         unsigned int D = ex_to<numeric>(mu.get_dim()).to_int();
1173
1174         if (algebraic) // check if algebraic method is applicable
1175                 for (unsigned int i = 0; i < D; i++)
1176                         if (pow(c.subs(mu == i), 2).is_zero()
1177                                 or (not is_a<numeric>(pow(c.subs(mu == i), 2))))
1178                                 algebraic = false;
1179         lst V;
1180         if (algebraic)
1181                 for (unsigned int i = 0; i < D; i++)
1182                         V.append(remove_dirac_ONE(
1183                                                 simplify_indexed(canonicalize_clifford(e * c.subs(mu == i) +  c.subs(mu == i) * e))
1184                                                 / (2*pow(c.subs(mu == i), 2))));
1185         else {
1186                 ex e1 = canonicalize_clifford(e);
1187                 for (unsigned int i = 0; i < D; i++)
1188                         V.append(get_clifford_comp(e1, c.subs(c.op(1) == i)));
1189         }
1190         return V;
1191 }
1192
1193
1194 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)
1195 {
1196         ex x, D, cu;
1197
1198         if (! is_a<matrix>(v) && ! is_a<lst>(v))
1199                 throw(std::invalid_argument("parameter v should be either vector or list"));
1200
1201         if (is_a<clifford>(G)) {
1202                 cu = G;
1203         } else {
1204                 if (is_a<indexed>(G))
1205                         D = ex_to<varidx>(G.op(1)).get_dim();
1206                 else if (is_a<matrix>(G))
1207                         D = ex_to<matrix>(G).rows();
1208                 else throw(std::invalid_argument("metric should be an indexed object, matrix, or a Clifford unit"));
1209
1210                 varidx mu((new symbol)->setflag(status_flags::dynallocated), D);
1211                 cu = clifford_unit(mu, G, rl);
1212         }
1213
1214         x = lst_to_clifford(v, cu);
1215         ex e = simplify_indexed(canonicalize_clifford((a * x + b) * clifford_inverse(c * x + d)));
1216         return clifford_to_lst(e, cu, false);
1217 }
1218
1219 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl)
1220 {
1221         if (is_a<matrix>(M))
1222                 return clifford_moebius_map(ex_to<matrix>(M)(0,0), ex_to<matrix>(M)(0,1),
1223                                                 ex_to<matrix>(M)(1,0), ex_to<matrix>(M)(1,1), v, G, rl);
1224         else
1225                 throw(std::invalid_argument("parameter M should be a matrix"));
1226 }
1227
1228 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G)
1229 {
1230         return clifford_moebius_map(M, v, G, 0);
1231 }
1232
1233 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c, const ex & d, const ex & v, const ex & G)
1234 {
1235         return clifford_moebius_map(a, b, c, d, v, G, 0);
1236 }
1237
1238 } // namespace GiNaC