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 <stdexcept>
  24
  25 #include "clifford.h"
  26
  27 #include "ex.h"
  28 #include "idx.h"
  29 #include "ncmul.h"
  30 #include "symbol.h"
  31 #include "numeric.h" // for I
  32 #include "symmetry.h"
  33 #include "lst.h"
  34 #include "relational.h"
  35 #include "operators.h"
  36 #include "add.h"
  37 #include "mul.h"
  38 #include "power.h"
  39 #include "matrix.h"
  40 #include "archive.h"
  41 #include "utils.h"
  42
  43 namespace GiNaC {
  44
  45 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(clifford, indexed,
  46   print_func<print_dflt>(&clifford::do_print_dflt).
  47   print_func<print_latex>(&clifford::do_print_latex))
  48
  49 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracone, tensor,
  50   print_func<print_dflt>(&diracone::do_print).
  51   print_func<print_latex>(&diracone::do_print_latex))
  52
  53 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(cliffordunit, tensor,
  54   print_func<print_dflt>(&cliffordunit::do_print).
  55   print_func<print_latex>(&cliffordunit::do_print_latex))
  56
  57 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma, cliffordunit,
  58   print_func<print_dflt>(&diracgamma::do_print).
  59   print_func<print_latex>(&diracgamma::do_print_latex))
  60
  61 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma5, tensor,
  62   print_func<print_dflt>(&diracgamma5::do_print).
  63   print_func<print_latex>(&diracgamma5::do_print_latex))
  64
  65 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaL, tensor,
  66   print_func<print_context>(&diracgammaL::do_print).
  67   print_func<print_latex>(&diracgammaL::do_print_latex))
  68
  69 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaR, tensor,
  70   print_func<print_context>(&diracgammaR::do_print).
  71   print_func<print_latex>(&diracgammaR::do_print_latex))
  72
  73 //////////
  74 // default constructors
  75 //////////
  76
  77 static ex default_metric()
  78 {
  79         static ex m = (new minkmetric)->setflag(status_flags::dynallocated);
  80         return m;
  81 }
  82
  83 clifford::clifford() : representation_label(0), metric(default_metric()), anticommuting(false)
  84 {
  85         tinfo_key = TINFO_clifford;
  86 }
  87
  88 DEFAULT_CTOR(diracone)
  89 DEFAULT_CTOR(cliffordunit)
  90 DEFAULT_CTOR(diracgamma)
  91 DEFAULT_CTOR(diracgamma5)
  92 DEFAULT_CTOR(diracgammaL)
  93 DEFAULT_CTOR(diracgammaR)
  94
  95 //////////
  96 // other constructors
  97 //////////
  98
  99 /** Construct object without any indices. This constructor is for internal
 100  *  use only. Use the dirac_ONE() function instead.
 101  *  @see dirac_ONE */
 102 clifford::clifford(const ex & b, unsigned char rl, bool anticommut) : inherited(b), representation_label(rl), metric(0), anticommuting(anticommut)
 103 {
 104         tinfo_key = TINFO_clifford;
 105 }
 106
 107 /** Construct object with one Lorentz index. This constructor is for internal
 108  *  use only. Use the clifford_unit() or dirac_gamma() functions instead.
 109  *  @see clifford_unit
 110  *  @see dirac_gamma */
 111 clifford::clifford(const ex & b, const ex & mu, const ex & metr, unsigned char rl, bool anticommut) : inherited(b, mu), representation_label(rl), metric(metr), anticommuting(anticommut)
 112 {
 113         GINAC_ASSERT(is_a<varidx>(mu));
 114         tinfo_key = TINFO_clifford;
 115 }
 116
 117 clifford::clifford(unsigned char rl, const ex & metr, bool anticommut, const exvector & v, bool discardable) : inherited(not_symmetric(), v, discardable), representation_label(rl), metric(metr), anticommuting(anticommut)
 118 {
 119         tinfo_key = TINFO_clifford;
 120 }
 121
 122 clifford::clifford(unsigned char rl, const ex & metr, bool anticommut, std::auto_ptr<exvector> vp) : inherited(not_symmetric(), vp), representation_label(rl), metric(metr), anticommuting(anticommut)
 123 {
 124         tinfo_key = TINFO_clifford;
 125 }
 126
 127 //////////
 128 // archiving
 129 //////////
 130
 131 clifford::clifford(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
 132 {
 133         unsigned rl;
 134         n.find_unsigned("label", rl);
 135         representation_label = rl;
 136         n.find_ex("metric", metric, sym_lst);
 137         n.find_bool("anticommuting", anticommuting);
 138 }
 139
 140 void clifford::archive(archive_node & n) const
 141 {
 142         inherited::archive(n);
 143         n.add_unsigned("label", representation_label);
 144         n.add_ex("metric", metric);
 145         n.add_bool("anticommuting", anticommuting);
 146 }
 147
 148 DEFAULT_UNARCHIVE(clifford)
 149 DEFAULT_ARCHIVING(diracone)
 150 DEFAULT_ARCHIVING(cliffordunit)
 151 DEFAULT_ARCHIVING(diracgamma)
 152 DEFAULT_ARCHIVING(diracgamma5)
 153 DEFAULT_ARCHIVING(diracgammaL)
 154 DEFAULT_ARCHIVING(diracgammaR)
 155
 156 //////////
 157 // functions overriding virtual functions from base classes
 158 //////////
 159
 160 ex clifford::get_metric(const ex & i, const ex & j, bool symmetrised) const
 161 {
 162         if (is_a<indexed>(metric)) {
 163                 if (symmetrised && !(ex_to<symmetry>(ex_to<indexed>(metric).get_symmetry()).has_symmetry())) {
 164                         if (is_a<matrix>(metric.op(0))) {
 165                                 return indexed((ex_to<matrix>(metric.op(0)).add(ex_to<matrix>(metric.op(0)).transpose())).mul(numeric(1,2)),
 166                                                symmetric2(), i, j);
 167                         } else {
 168                                 return simplify_indexed(indexed(metric.op(0)*_ex1_2, i, j) + indexed(metric.op(0)*_ex1_2, j, i));
 169                         }
 170                 } else {
 171                         return indexed(metric.op(0), ex_to<symmetry>(ex_to<indexed>(metric).get_symmetry()), i, j);
 172                 }
 173         } else {
 174                 // should not really happen since all constructors but clifford() make the metric an indexed object
 175                 return indexed(metric, i, j);
 176         }
 177 }
 178
 179 bool clifford::same_metric(const ex & other) const
 180 {
 181         if (is_a<clifford>(other)) {
 182                 return same_metric(ex_to<clifford>(other).get_metric());
 183         } else if (is_a<indexed>(other)) {
 184                 return get_metric(other.op(1), other.op(2)).is_equal(other);
 185         } else
 186                 return false;
 187 }
 188
 189 int clifford::compare_same_type(const basic & other) const
 190 {
 191         GINAC_ASSERT(is_a<clifford>(other));
 192         const clifford &o = static_cast<const clifford &>(other);
 193
 194         if (representation_label != o.representation_label) {
 195                 // different representation label
 196                 return representation_label < o.representation_label ? -1 : 1;
 197         }
 198
 199         return inherited::compare_same_type(other);
 200 }
 201
 202 bool clifford::match_same_type(const basic & other) const
 203 {
 204         GINAC_ASSERT(is_a<clifford>(other));
 205         const clifford &o = static_cast<const clifford &>(other);
 206
 207         return (representation_label == o.representation_label) && same_metric(o);
 208 }
 209
 210 static bool is_dirac_slash(const ex & seq0)
 211 {
 212         return !is_a<diracgamma5>(seq0) && !is_a<diracgammaL>(seq0) &&
 213                !is_a<diracgammaR>(seq0) && !is_a<cliffordunit>(seq0) &&
 214                !is_a<diracone>(seq0);
 215 }
 216
 217 void clifford::do_print_dflt(const print_dflt & c, unsigned level) const
 218 {
 219         // dirac_slash() object is printed differently
 220         if (is_dirac_slash(seq[0])) {
 221                 seq[0].print(c, precedence());
 222                 c.s << "\\";
 223         } else
 224                 this->print_dispatch<inherited>(c, level);
 225 }
 226
 227 void clifford::do_print_latex(const print_latex & c, unsigned level) const
 228 {
 229         // dirac_slash() object is printed differently
 230         if (is_dirac_slash(seq[0])) {
 231                 c.s << "{";
 232                 seq[0].print(c, precedence());
 233                 c.s << "\\hspace{-1.0ex}/}";
 234         } else {
 235                 c.s << "\\clifford[" << int(representation_label) << "]";
 236                 this->print_dispatch<inherited>(c, level);
 237         }
 238 }
 239
 240 DEFAULT_COMPARE(diracone)
 241 DEFAULT_COMPARE(cliffordunit)
 242 DEFAULT_COMPARE(diracgamma)
 243 DEFAULT_COMPARE(diracgamma5)
 244 DEFAULT_COMPARE(diracgammaL)
 245 DEFAULT_COMPARE(diracgammaR)
 246
 247 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbf{1}")
 248 DEFAULT_PRINT_LATEX(cliffordunit, "e", "e")
 249 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
 250 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
 251 DEFAULT_PRINT_LATEX(diracgammaL, "gammaL", "{\\gamma_L}")
 252 DEFAULT_PRINT_LATEX(diracgammaR, "gammaR", "{\\gamma_R}")
 253
 254 /** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
 255 static void base_and_index(const ex & c, ex & b, ex & i)
 256 {
 257         GINAC_ASSERT(is_a<clifford>(c));
 258         GINAC_ASSERT(c.nops() == 2);
 259
 260         if (is_a<cliffordunit>(c.op(0))) { // proper dirac gamma object or clifford unit
 261                 i = c.op(1);
 262                 b = _ex1;
 263         } else if (is_a<diracgamma5>(c.op(0)) || is_a<diracgammaL>(c.op(0)) || is_a<diracgammaR>(c.op(0))) { // gamma5/L/R
 264                 i = _ex0;
 265                 b = _ex1;
 266         } else { // slash object, generate new dummy index
 267                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(c.op(1)).get_dim());
 268                 b = indexed(c.op(0), ix.toggle_variance());
 269                 i = ix;
 270         }
 271 }
 272
 273 /** Predicate for finding non-clifford objects. */
 274 struct is_not_a_clifford : public std::unary_function<ex, bool> {
 275         bool operator()(const ex & e)
 276         {
 277                 return !is_a<clifford>(e);
 278         }
 279 };
 280
 281 /** Contraction of a gamma matrix with something else. */
 282 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 283 {
 284         GINAC_ASSERT(is_a<clifford>(*self));
 285         GINAC_ASSERT(is_a<indexed>(*other));
 286         GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
 287         unsigned char rl = ex_to<clifford>(*self).get_representation_label();
 288
 289         ex dim = ex_to<idx>(self->op(1)).get_dim();
 290         if (other->nops() > 1)
 291                 dim = minimal_dim(dim, ex_to<idx>(other->op(1)).get_dim());
 292
 293         if (is_a<clifford>(*other)) {
 294
 295                 // Contraction only makes sense if the represenation labels are equal
 296                 if (ex_to<clifford>(*other).get_representation_label() != rl)
 297                         return false;
 298
 299                 size_t num = other - self;
 300
 301                 // gamma~mu gamma.mu = dim ONE
 302                 if (num == 1) {
 303                         *self = dim;
 304                         *other = dirac_ONE(rl);
 305                         return true;
 306
 307                 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
 308                 } else if (num == 2
 309                         && is_a<clifford>(self[1])) {
 310                         *self = 2 - dim;
 311                         *other = _ex1;
 312                         return true;
 313
 314                 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
 315                 } else if (num == 3
 316                         && is_a<clifford>(self[1])
 317                         && is_a<clifford>(self[2])) {
 318                         ex b1, i1, b2, i2;
 319                         base_and_index(self[1], b1, i1);
 320                         base_and_index(self[2], b2, i2);
 321                         *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
 322                         self[1] = _ex1;
 323                         self[2] = _ex1;
 324                         *other = _ex1;
 325                         return true;
 326
 327                 // 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
 328                 } else if (num == 4
 329                         && is_a<clifford>(self[1])
 330                         && is_a<clifford>(self[2])
 331                         && is_a<clifford>(self[3])) {
 332                         *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3];
 333                         self[1] = _ex1;
 334                         self[2] = _ex1;
 335                         self[3] = _ex1;
 336                         *other = _ex1;
 337                         return true;
 338
 339                 // gamma~mu Sodd gamma.mu = -2 Sodd_R
 340                 // (Chisholm identity in 4 dimensions)
 341                 } else if (!((other - self) & 1) && dim.is_equal(4)) {
 342                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
 343                                 return false;
 344
 345                         *self = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(other), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
 346                         std::fill(self + 1, other, _ex1);
 347                         *other = _ex_2;
 348                         return true;
 349
 350                 // gamma~mu Sodd gamma~alpha gamma.mu = 2 gamma~alpha Sodd + 2 Sodd_R gamma~alpha
 351                 // (commutate contracted indices towards each other, then use
 352                 // Chisholm identity in 4 dimensions)
 353                 } else if (((other - self) & 1) && dim.is_equal(4)) {
 354                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
 355                                 return false;
 356
 357                         exvector::iterator next_to_last = other - 1;
 358                         ex S = ncmul(exvector(self + 1, next_to_last), true);
 359                         ex SR = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(next_to_last), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
 360
 361                         *self = (*next_to_last) * S + SR * (*next_to_last);
 362                         std::fill(self + 1, other, _ex1);
 363                         *other = _ex2;
 364                         return true;
 365
 366                 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
 367                 // (commutate contracted indices towards each other, simplify_indexed()
 368                 // will re-expand and re-run the simplification)
 369                 } else {
 370                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
 371                                 return false;
 372
 373                         exvector::iterator next_to_last = other - 1;
 374                         ex S = ncmul(exvector(self + 1, next_to_last), true);
 375
 376                         *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
 377                         std::fill(self + 1, other + 1, _ex1);
 378                         return true;
 379                 }
 380
 381         } else if (is_a<symbol>(other->op(0)) && other->nops() == 2) {
 382
 383                 // x.mu gamma~mu -> x-slash
 384                 *self = dirac_slash(other->op(0), dim, rl);
 385                 *other = _ex1;
 386                 return true;
 387         }
 388
 389         return false;
 390 }
 391
 392 /** An utility function looking for a given metric within an exvector,
 393  *  used in cliffordunit::contract_with(). */
 394 static int find_same_metric(exvector & v, ex & c)
 395 {
 396         for (size_t i=0; i<v.size(); i++) {
 397                 if (is_a<indexed>(v[i]) && !is_a<clifford>(v[i])
 398                     && ((ex_to<varidx>(c.op(1)) == ex_to<indexed>(v[i]).get_indices()[0]
 399                     && ex_to<varidx>(c.op(1)) == ex_to<indexed>(v[i]).get_indices()[1])
 400                     || (ex_to<varidx>(c.op(1)).toggle_variance() == ex_to<indexed>(v[i]).get_indices()[0]
 401                     && ex_to<varidx>(c.op(1)).toggle_variance() == ex_to<indexed>(v[i]).get_indices()[1]))) {
 402                         return i; // the index of the found
 403                 }
 404         }
 405         return -1; //nothing found
 406 }
 407
 408 /** Contraction of a Clifford unit with something else. */
 409 bool cliffordunit::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 410 {
 411         GINAC_ASSERT(is_a<clifford>(*self));
 412         GINAC_ASSERT(is_a<indexed>(*other));
 413         GINAC_ASSERT(is_a<cliffordunit>(self->op(0)));
 414         clifford unit = ex_to<clifford>(*self);
 415         unsigned char rl = unit.get_representation_label();
 416
 417         if (is_a<clifford>(*other)) {
 418                 // Contraction only makes sense if the represenation labels are equal
 419                 // and the metrics are the same
 420                 if ((ex_to<clifford>(*other).get_representation_label() != rl)
 421                     && unit.same_metric(*other))
 422                         return false;
 423
 424                 // Find if a previous contraction produces the square of self
 425                 int prev_square = find_same_metric(v, *self);
 426                 const varidx d((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(self->op(1)).get_dim()),
 427                         in1((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(self->op(1)).get_dim()),
 428                         in2((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(self->op(1)).get_dim());
 429                 ex squared_metric;
 430                 if (prev_square > -1)
 431                         squared_metric = simplify_indexed(indexed(v[prev_square].op(0), in1, d)
 432                                                                                           * unit.get_metric(d.toggle_variance(), in2, true)).op(0);
 433
 434                 exvector::iterator before_other = other - 1;
 435                 const varidx & mu = ex_to<varidx>(self->op(1));
 436                 const varidx & mu_toggle = ex_to<varidx>(other->op(1));
 437                 const varidx & alpha = ex_to<varidx>(before_other->op(1));
 438
 439                 // e~mu e.mu = Tr ONE
 440                 if (other - self == 1) {
 441                         if (prev_square > -1) {
 442                                 *self = indexed(squared_metric, mu, mu_toggle);
 443                                 v[prev_square] = _ex1;
 444                         } else {
 445                                 *self = unit.get_metric(mu, mu_toggle, true);
 446                         }
 447                         *other = dirac_ONE(rl);
 448                         return true;
 449
 450                 } else if (other - self == 2) {
 451                         if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
 452                                 if (ex_to<clifford>(*self).is_anticommuting()) {
 453                                         // e~mu e~alpha e.mu = (2*pow(e~alpha, 2) -Tr(B)) e~alpha
 454                                         if (prev_square > -1) {
 455                                                 *self = 2 * indexed(squared_metric, alpha, alpha)
 456                                                         - indexed(squared_metric, mu, mu_toggle);
 457                                                 v[prev_square] = _ex1;
 458                                         } else {
 459                                                 *self = 2 * unit.get_metric(alpha, alpha, true) - unit.get_metric(mu, mu_toggle, true);
 460                                         }
 461                                         *other = _ex1;
 462                                         return true;
 463
 464                                 } else {
 465                                         // e~mu e~alpha e.mu = 2*e~mu B(alpha, mu.toggle_variance())-Tr(B) e~alpha
 466                                         *self = 2 * (*self) * unit.get_metric(alpha, mu_toggle, true) - unit.get_metric(mu, mu_toggle, true) * (*before_other);
 467                                         *before_other = _ex1;
 468                                         *other = _ex1;
 469                                         return true;
 470                                 }
 471                         } else {
 472                                 // e~mu S e.mu = Tr S ONE
 473                                 *self = unit.get_metric(mu, mu_toggle, true);
 474                                 *other = dirac_ONE(rl);
 475                                 return true;
 476                         }
 477                 } else {
 478                 // e~mu S e~alpha e.mu = 2 e~mu S B(alpha, mu.toggle_variance()) - e~mu S e.mu e~alpha
 479                 // (commutate contracted indices towards each other, simplify_indexed()
 480                 // will re-expand and re-run the simplification)
 481                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other) {
 482                                 return false;
 483                         }
 484
 485                         ex S = ncmul(exvector(self + 1, before_other), true);
 486
 487                         if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
 488                                 if (ex_to<clifford>(*self).is_anticommuting()) {
 489                                         if (prev_square > -1) {
 490                                                 *self = 2 * (*before_other) * S * indexed(squared_metric, alpha, alpha)
 491                                                         - (*self) * S * (*other) * (*before_other);
 492                                         } else {
 493                                                 *self = 2 * (*before_other) * S * unit.get_metric(alpha, alpha, true) - (*self) * S * (*other) * (*before_other);
 494                                         }
 495                                 } else {
 496                                         *self = 2 * (*self) * S * unit.get_metric(alpha, mu_toggle, true) - (*self) * S * (*other) * (*before_other);
 497                                 }
 498                         } else {
 499                                 // simply commutes
 500                                 *self = (*self) * S * (*other) * (*before_other);
 501                         }
 502
 503                         std::fill(self + 1, other + 1, _ex1);
 504                         return true;
 505                 }
 506         }
 507         return false;
 508 }
 509
 510 /** Perform automatic simplification on noncommutative product of clifford
 511  *  objects. This removes superfluous ONEs, permutes gamma5/L/R's to the front
 512  *  and removes squares of gamma objects. */
 513 ex clifford::eval_ncmul(const exvector & v) const
 514 {
 515         exvector s;
 516         s.reserve(v.size());
 517
 518         // Remove superfluous ONEs
 519         exvector::const_iterator cit = v.begin(), citend = v.end();
 520         while (cit != citend) {
 521                 if (!is_a<clifford>(*cit) || !is_a<diracone>(cit->op(0)))
 522                         s.push_back(*cit);
 523                 cit++;
 524         }
 525
 526         bool something_changed = false;
 527         int sign = 1;
 528
 529         // Anticommutate gamma5/L/R's to the front
 530         if (s.size() >= 2) {
 531                 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
 532                 while (true) {
 533                         exvector::iterator it = next_to_last;
 534                         while (true) {
 535                                 exvector::iterator it2 = it + 1;
 536                                 if (is_a<clifford>(*it) && is_a<clifford>(*it2)) {
 537                                         ex e1 = it->op(0), e2 = it2->op(0);
 538
 539                                         if (is_a<diracgamma5>(e2)) {
 540
 541                                                 if (is_a<diracgammaL>(e1) || is_a<diracgammaR>(e1)) {
 542
 543                                                         // gammaL/R gamma5 -> gamma5 gammaL/R
 544                                                         it->swap(*it2);
 545                                                         something_changed = true;
 546
 547                                                 } else if (!is_a<diracgamma5>(e1)) {
 548
 549                                                         // gamma5 gamma5 -> gamma5 gamma5 (do nothing)
 550                                                         // x gamma5 -> -gamma5 x
 551                                                         it->swap(*it2);
 552                                                         sign = -sign;
 553                                                         something_changed = true;
 554                                                 }
 555
 556                                         } else if (is_a<diracgammaL>(e2)) {
 557
 558                                                 if (is_a<diracgammaR>(e1)) {
 559
 560                                                         // gammaR gammaL -> 0
 561                                                         return _ex0;
 562
 563                                                 } else if (!is_a<diracgammaL>(e1) && !is_a<diracgamma5>(e1)) {
 564
 565                                                         // gammaL gammaL -> gammaL gammaL (do nothing)
 566                                                         // gamma5 gammaL -> gamma5 gammaL (do nothing)
 567                                                         // x gammaL -> gammaR x
 568                                                         it->swap(*it2);
 569                                                         *it = clifford(diracgammaR(), ex_to<clifford>(*it).get_representation_label());
 570                                                         something_changed = true;
 571                                                 }
 572
 573                                         } else if (is_a<diracgammaR>(e2)) {
 574
 575                                                 if (is_a<diracgammaL>(e1)) {
 576
 577                                                         // gammaL gammaR -> 0
 578                                                         return _ex0;
 579
 580                                                 } else if (!is_a<diracgammaR>(e1) && !is_a<diracgamma5>(e1)) {
 581
 582                                                         // gammaR gammaR -> gammaR gammaR (do nothing)
 583                                                         // gamma5 gammaR -> gamma5 gammaR (do nothing)
 584                                                         // x gammaR -> gammaL x
 585                                                         it->swap(*it2);
 586                                                         *it = clifford(diracgammaL(), ex_to<clifford>(*it).get_representation_label());
 587                                                         something_changed = true;
 588                                                 }
 589                                         }
 590                                 }
 591                                 if (it == first)
 592                                         break;
 593                                 --it;
 594                         }
 595                         if (next_to_last == first)
 596                                 break;
 597                         --next_to_last;
 598                 }
 599         }
 600
 601         // Remove equal adjacent gammas
 602         if (s.size() >= 2) {
 603                 exvector::iterator it, itend = s.end() - 1;
 604                 for (it = s.begin(); it != itend; ++it) {
 605                         ex & a = it[0];
 606                         ex & b = it[1];
 607                         if (!is_a<clifford>(a) || !is_a<clifford>(b))
 608                                 continue;
 609
 610                         const ex & ag = a.op(0);
 611                         const ex & bg = b.op(0);
 612                         bool a_is_cliffordunit = is_a<cliffordunit>(ag);
 613                         bool b_is_cliffordunit =  is_a<cliffordunit>(bg);
 614
 615                         if (a_is_cliffordunit && b_is_cliffordunit && ex_to<clifford>(a).same_metric(b)) {
 616
 617                                 const ex & ia = a.op(1);
 618                                 const ex & ib = b.op(1);
 619                                 if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
 620                                         a = ex_to<clifford>(a).get_metric(ia, ib, true);
 621                                         b = dirac_ONE(representation_label);
 622                                         something_changed = true;
 623                                 }
 624
 625                         } else if ((is_a<diracgamma5>(ag) && is_a<diracgamma5>(bg))) {
 626
 627                                 // Remove squares of gamma5
 628                                 a = dirac_ONE(representation_label);
 629                                 b = dirac_ONE(representation_label);
 630                                 something_changed = true;
 631
 632                         } else if ((is_a<diracgammaL>(ag) && is_a<diracgammaL>(bg))
 633                                 || (is_a<diracgammaR>(ag) && is_a<diracgammaR>(bg))) {
 634
 635                                 // Remove squares of gammaL/R
 636                                 b = dirac_ONE(representation_label);
 637                                 something_changed = true;
 638
 639                         } else if (is_a<diracgammaL>(ag) && is_a<diracgammaR>(bg)) {
 640
 641                                 // gammaL and gammaR are orthogonal
 642                                 return _ex0;
 643
 644                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaL>(bg)) {
 645
 646                                 // gamma5 gammaL -> -gammaL
 647                                 a = dirac_ONE(representation_label);
 648                                 sign = -sign;
 649                                 something_changed = true;
 650
 651                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaR>(bg)) {
 652
 653                                 // gamma5 gammaR -> gammaR
 654                                 a = dirac_ONE(representation_label);
 655                                 something_changed = true;
 656
 657                         } else if (!a_is_cliffordunit && !b_is_cliffordunit && ag.is_equal(bg)) {
 658
 659                                 // a\ a\ -> a^2
 660                                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).minimal_dim(ex_to<idx>(b.op(1))));
 661
 662                                 a = indexed(ag, ix) * indexed(ag, ix.toggle_variance());
 663                                 b = dirac_ONE(representation_label);
 664                                 something_changed = true;
 665                         }
 666                 }
 667         }
 668
 669         if (s.empty())
 670                 return dirac_ONE(representation_label) * sign;
 671         if (something_changed)
 672                 return reeval_ncmul(s) * sign;
 673         else
 674                 return hold_ncmul(s) * sign;
 675 }
 676
 677 ex clifford::thiscontainer(const exvector & v) const
 678 {
 679         return clifford(representation_label, get_metric(), is_anticommuting(), v);
 680 }
 681
 682 ex clifford::thiscontainer(std::auto_ptr<exvector> vp) const
 683 {
 684         return clifford(representation_label, get_metric(), is_anticommuting(), vp);
 685 }
 686
 687 ex diracgamma5::conjugate() const
 688 {
 689         return _ex_1 * (*this);
 690 }
 691
 692 ex diracgammaL::conjugate() const
 693 {
 694         return (new diracgammaR)->setflag(status_flags::dynallocated);
 695 }
 696
 697 ex diracgammaR::conjugate() const
 698 {
 699         return (new diracgammaL)->setflag(status_flags::dynallocated);
 700 }
 701
 702 //////////
 703 // global functions
 704 //////////
 705
 706 ex dirac_ONE(unsigned char rl)
 707 {
 708         static ex ONE = (new diracone)->setflag(status_flags::dynallocated);
 709         return clifford(ONE, rl, false);
 710 }
 711
 712 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl, bool anticommuting)
 713 {
 714         static ex unit = (new cliffordunit)->setflag(status_flags::dynallocated);
 715
 716         if (!is_a<idx>(mu))
 717                 throw(std::invalid_argument("clifford_unit(): index of Clifford unit must be of type idx or varidx"));
 718
 719         if (ex_to<idx>(mu).is_symbolic() && !is_a<varidx>(mu))
 720                 throw(std::invalid_argument("clifford_unit(): symbolic index of Clifford unit must be of type varidx (not idx)"));
 721
 722         if (is_a<indexed>(metr)) {
 723                 exvector indices = ex_to<indexed>(metr).get_indices();
 724                 if ((indices.size() == 2) && is_a<varidx>(indices[0]) && is_a<varidx>(indices[1])) {
 725                         return clifford(unit, mu, metr, rl, anticommuting);
 726                 } else {
 727                         throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be indexed exactly by two indices of same type as the given index"));
 728                 }
 729         } else if (is_a<tensmetric>(metr)) {
 730                 static varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim()),
 731                         chi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim());
 732                 return clifford(unit, mu, indexed(metr, xi, chi), rl, anticommuting);
 733         } else if (is_a<matrix>(metr)) {
 734                 matrix M = ex_to<matrix>(metr);
 735                 unsigned n = M.rows();
 736                 bool symmetric = true;
 737                 anticommuting = true;
 738
 739                 static varidx xi((new symbol)->setflag(status_flags::dynallocated), n),
 740                         chi((new symbol)->setflag(status_flags::dynallocated), n);
 741                 if ((n ==  M.cols()) && (n == ex_to<varidx>(mu).get_dim())) {
 742                         for (unsigned i = 0; i < n; i++) {
 743                                 for (unsigned j = i+1; j < n; j++) {
 744                                         if (M(i, j) != M(j, i)) {
 745                                                 symmetric = false;
 746                                         }
 747                                         if (M(i, j) != -M(j, i)) {
 748                                                 anticommuting = false;
 749                                         }
 750                                 }
 751                         }
 752                         return clifford(unit, mu, indexed(metr, symmetric?symmetric2():not_symmetric(), xi, chi), rl, anticommuting);
 753                 } else {
 754                         throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be a square matrix with the same dimensions as index"));
 755                 }
 756         } else {
 757                 throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be of type indexed, tensormetric or matrix"));
 758         }
 759 }
 760
 761 ex dirac_gamma(const ex & mu, unsigned char rl)
 762 {
 763         static ex gamma = (new diracgamma)->setflag(status_flags::dynallocated);
 764
 765         if (!is_a<varidx>(mu))
 766                 throw(std::invalid_argument("dirac_gamma(): index of Dirac gamma must be of type varidx"));
 767
 768         static varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim()),
 769                 chi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim());
 770         return clifford(gamma, mu, indexed(default_metric(), symmetric2(), xi, chi), rl, true);
 771 }
 772
 773 ex dirac_gamma5(unsigned char rl)
 774 {
 775         static ex gamma5 = (new diracgamma5)->setflag(status_flags::dynallocated);
 776         return clifford(gamma5, rl);
 777 }
 778
 779 ex dirac_gammaL(unsigned char rl)
 780 {
 781         static ex gammaL = (new diracgammaL)->setflag(status_flags::dynallocated);
 782         return clifford(gammaL, rl);
 783 }
 784
 785 ex dirac_gammaR(unsigned char rl)
 786 {
 787         static ex gammaR = (new diracgammaR)->setflag(status_flags::dynallocated);
 788         return clifford(gammaR, rl);
 789 }
 790
 791 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
 792 {
 793         // Slashed vectors are actually stored as a clifford object with the
 794         // vector as its base expression and a (dummy) index that just serves
 795         // for storing the space dimensionality
 796         return clifford(e, varidx(0, dim), default_metric(), rl);
 797 }
 798
 799 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 800  *  is that of a clifford object with the specified representation label. */
 801 static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
 802 {
 803         return ti == (TINFO_clifford + rl);
 804 }
 805
 806 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 807  *  is that of a clifford object (with an arbitrary representation label). */
 808 static bool is_clifford_tinfo(unsigned ti)
 809 {
 810         return (ti & ~0xff) == TINFO_clifford;
 811 }
 812
 813 /** Extract representation label from tinfo key (as returned by
 814  *  return_type_tinfo()). */
 815 static unsigned char get_representation_label(unsigned ti)
 816 {
 817         return ti & 0xff;
 818 }
 819
 820 /** Take trace of a string of an even number of Dirac gammas given a vector
 821  *  of indices. */
 822 static ex trace_string(exvector::const_iterator ix, size_t num)
 823 {
 824         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 825         if (num == 2)
 826                 return lorentz_g(ix[0], ix[1]);
 827
 828         // Tr gamma.mu gamma.nu gamma.rho gamma.sig = 4 (g.mu.nu g.rho.sig + g.nu.rho g.mu.sig - g.mu.rho g.nu.sig )
 829         else if (num == 4)
 830                 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
 831                      + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
 832                      - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
 833
 834         // Traces of 6 or more gammas are computed recursively:
 835         // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
 836         //   + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
 837         //   - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
 838         //   + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
 839         //   - ...
 840         //   + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
 841         exvector v(num - 2);
 842         int sign = 1;
 843         ex result;
 844         for (size_t i=1; i<num; i++) {
 845                 for (size_t n=1, j=0; n<num; n++) {
 846                         if (n == i)
 847                                 continue;
 848                         v[j++] = ix[n];
 849                 }
 850                 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
 851                 sign = -sign;
 852         }
 853         return result;
 854 }
 855
 856 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE)
 857 {
 858         if (is_a<clifford>(e)) {
 859
 860                 unsigned char rl = ex_to<clifford>(e).get_representation_label();
 861
 862                 // Are we taking the trace over this object's representation label?
 863                 if (rls.find(rl) == rls.end())
 864                         return e;
 865
 866                 // Yes, all elements are traceless, except for dirac_ONE and dirac_L/R
 867                 const ex & g = e.op(0);
 868                 if (is_a<diracone>(g))
 869                         return trONE;
 870                 else if (is_a<diracgammaL>(g) || is_a<diracgammaR>(g))
 871                         return trONE/2;
 872                 else
 873                         return _ex0;
 874
 875         } else if (is_exactly_a<mul>(e)) {
 876
 877                 // Trace of product: pull out non-clifford factors
 878                 ex prod = _ex1;
 879                 for (size_t i=0; i<e.nops(); i++) {
 880                         const ex &o = e.op(i);
 881                         if (is_clifford_tinfo(o.return_type_tinfo()))
 882                                 prod *= dirac_trace(o, rls, trONE);
 883                         else
 884                                 prod *= o;
 885                 }
 886                 return prod;
 887
 888         } else if (is_exactly_a<ncmul>(e)) {
 889
 890                 unsigned char rl = get_representation_label(e.return_type_tinfo());
 891
 892                 // Are we taking the trace over this string's representation label?
 893                 if (rls.find(rl) == rls.end())
 894                         return e;
 895
 896                 // Substitute gammaL/R and expand product, if necessary
 897                 ex e_expanded = e.subs(lst(
 898                         dirac_gammaL(rl) == (dirac_ONE(rl)-dirac_gamma5(rl))/2,
 899                         dirac_gammaR(rl) == (dirac_ONE(rl)+dirac_gamma5(rl))/2
 900                 ), subs_options::no_pattern).expand();
 901                 if (!is_a<ncmul>(e_expanded))
 902                         return dirac_trace(e_expanded, rls, trONE);
 903
 904                 // gamma5 gets moved to the front so this check is enough
 905                 bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
 906                 size_t num = e.nops();
 907
 908                 if (has_gamma5) {
 909
 910                         // Trace of gamma5 * odd number of gammas and trace of
 911                         // gamma5 * gamma.mu * gamma.nu are zero
 912                         if ((num & 1) == 0 || num == 3)
 913                                 return _ex0;
 914
 915                         // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
 916                         // (the epsilon is always 4-dimensional)
 917                         if (num == 5) {
 918                                 ex b1, i1, b2, i2, b3, i3, b4, i4;
 919                                 base_and_index(e.op(1), b1, i1);
 920                                 base_and_index(e.op(2), b2, i2);
 921                                 base_and_index(e.op(3), b3, i3);
 922                                 base_and_index(e.op(4), b4, i4);
 923                                 return trONE * I * (lorentz_eps(ex_to<idx>(i1).replace_dim(_ex4), ex_to<idx>(i2).replace_dim(_ex4), ex_to<idx>(i3).replace_dim(_ex4), ex_to<idx>(i4).replace_dim(_ex4)) * b1 * b2 * b3 * b4).simplify_indexed();
 924                         }
 925
 926                         // Tr gamma5 S_2k =
 927                         //   I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
 928                         // (the epsilon is always 4-dimensional)
 929                         exvector ix(num-1), bv(num-1);
 930                         for (size_t i=1; i<num; i++)
 931                                 base_and_index(e.op(i), bv[i-1], ix[i-1]);
 932                         num--;
 933                         int *iv = new int[num];
 934                         ex result;
 935                         for (size_t i=0; i<num-3; i++) {
 936                                 ex idx1 = ix[i];
 937                                 for (size_t j=i+1; j<num-2; j++) {
 938                                         ex idx2 = ix[j];
 939                                         for (size_t k=j+1; k<num-1; k++) {
 940                                                 ex idx3 = ix[k];
 941                                                 for (size_t l=k+1; l<num; l++) {
 942                                                         ex idx4 = ix[l];
 943                                                         iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
 944                                                         exvector v;
 945                                                         v.reserve(num - 4);
 946                                                         for (size_t n=0, t=4; n<num; n++) {
 947                                                                 if (n == i || n == j || n == k || n == l)
 948                                                                         continue;
 949                                                                 iv[t++] = n;
 950                                                                 v.push_back(ix[n]);
 951                                                         }
 952                                                         int sign = permutation_sign(iv, iv + num);
 953                                                         result += sign * lorentz_eps(ex_to<idx>(idx1).replace_dim(_ex4), ex_to<idx>(idx2).replace_dim(_ex4), ex_to<idx>(idx3).replace_dim(_ex4), ex_to<idx>(idx4).replace_dim(_ex4))
 954                                                                 * trace_string(v.begin(), num - 4);
 955                                                 }
 956                                         }
 957                                 }
 958                         }
 959                         delete[] iv;
 960                         return trONE * I * result * mul(bv);
 961
 962                 } else { // no gamma5
 963
 964                         // Trace of odd number of gammas is zero
 965                         if ((num & 1) == 1)
 966                                 return _ex0;
 967
 968                         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 969                         if (num == 2) {
 970                                 ex b1, i1, b2, i2;
 971                                 base_and_index(e.op(0), b1, i1);
 972                                 base_and_index(e.op(1), b2, i2);
 973                                 return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
 974                         }
 975
 976                         exvector iv(num), bv(num);
 977                         for (size_t i=0; i<num; i++)
 978                                 base_and_index(e.op(i), bv[i], iv[i]);
 979
 980                         return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
 981                 }
 982
 983         } else if (e.nops() > 0) {
 984
 985                 // Trace maps to all other container classes (this includes sums)
 986                 pointer_to_map_function_2args<const std::set<unsigned char> &, const ex &> fcn(dirac_trace, rls, trONE);
 987                 return e.map(fcn);
 988
 989         } else
 990                 return _ex0;
 991 }
 992
 993 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE)
 994 {
 995         // Convert list to set
 996         std::set<unsigned char> rls;
 997         for (lst::const_iterator i = rll.begin(); i != rll.end(); ++i) {
 998                 if (i->info(info_flags::nonnegint))
 999                         rls.insert(ex_to<numeric>(*i).to_int());
1000         }
1001
1002         return dirac_trace(e, rls, trONE);
1003 }
1004
1005 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
1006 {
1007         // Convert label to set
1008         std::set<unsigned char> rls;
1009         rls.insert(rl);
1010
1011         return dirac_trace(e, rls, trONE);
1012 }
1013
1014
1015 ex canonicalize_clifford(const ex & e_)
1016 {
1017         pointer_to_map_function fcn(canonicalize_clifford);
1018
1019         if (is_a<matrix>(e_)    // || is_a<pseries>(e) || is_a<integral>(e)
1020                 || is_a<lst>(e_)) {
1021                 return e_.map(fcn);
1022         } else {
1023                 ex e=simplify_indexed(e_);
1024                 // Scan for any ncmul objects
1025                 exmap srl;
1026                 ex aux = e.to_rational(srl);
1027                 for (exmap::iterator i = srl.begin(); i != srl.end(); ++i) {
1028
1029                         ex lhs = i->first;
1030                         ex rhs = i->second;
1031
1032                         if (is_exactly_a<ncmul>(rhs)
1033                                         && rhs.return_type() == return_types::noncommutative
1034                                         && is_clifford_tinfo(rhs.return_type_tinfo())) {
1035
1036                                 // Expand product, if necessary
1037                                 ex rhs_expanded = rhs.expand();
1038                                 if (!is_a<ncmul>(rhs_expanded)) {
1039                                         i->second = canonicalize_clifford(rhs_expanded);
1040                                         continue;
1041
1042                                 } else if (!is_a<clifford>(rhs.op(0)))
1043                                         continue;
1044
1045                                 exvector v;
1046                                 v.reserve(rhs.nops());
1047                                 for (size_t j=0; j<rhs.nops(); j++)
1048                                         v.push_back(rhs.op(j));
1049
1050                                 // Stupid recursive bubble sort because we only want to swap adjacent gammas
1051                                 exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
1052                                 if (is_a<diracgamma5>(it->op(0)) || is_a<diracgammaL>(it->op(0)) || is_a<diracgammaR>(it->op(0)))
1053                                         ++it;
1054                                 while (it != next_to_last) {
1055                                         if (it[0].compare(it[1]) > 0) {
1056                                                 ex save0 = it[0], save1 = it[1];
1057                                                 ex b1, i1, b2, i2;
1058                                                 base_and_index(it[0], b1, i1);
1059                                                 base_and_index(it[1], b2, i2);
1060                                                 it[0] = (ex_to<clifford>(save0).get_metric(i1, i2, true) * b1 * b2).simplify_indexed();
1061                                                 it[1] = v.size() == 2 ? _ex2 * dirac_ONE(ex_to<clifford>(it[1]).get_representation_label()) : _ex2;
1062                                                 ex sum = ncmul(v);
1063                                                 it[0] = save1;
1064                                                 it[1] = save0;
1065                                                 sum -= ncmul(v, true);
1066                                                 i->second = canonicalize_clifford(sum);
1067                                                 goto next_sym;
1068                                         }
1069                                         ++it;
1070                                 }
1071 next_sym:       ;
1072                         }
1073                 }
1074                 return aux.subs(srl, subs_options::no_pattern).simplify_indexed();
1075         }
1076 }
1077
1078 ex clifford_prime(const ex & e)
1079 {
1080         pointer_to_map_function fcn(clifford_prime);
1081         if (is_a<clifford>(e) && is_a<cliffordunit>(e.op(0))) {
1082                 return -e;
1083         } else if (is_a<add>(e) || is_a<ncmul>(e) || is_a<mul>(e) //|| is_a<pseries>(e) || is_a<integral>(e)
1084                            || is_a<matrix>(e) || is_a<lst>(e)) {
1085                 return e.map(fcn);
1086         } else if (is_a<power>(e)) {
1087                 return pow(clifford_prime(e.op(0)), e.op(1));
1088         } else
1089                 return e;
1090 }
1091
1092 ex remove_dirac_ONE(const ex & e, unsigned char rl, unsigned options)
1093 {
1094         pointer_to_map_function_2args<unsigned char, unsigned> fcn(remove_dirac_ONE, rl, options | 1);
1095         bool need_reevaluation = false;
1096         ex e1 = e;
1097         if (! (options & 1) )  { // is not a child
1098                 if (options & 2)
1099                         e1 = expand_dummy_sum(e, true);
1100                 e1 = canonicalize_clifford(e1);
1101         }
1102
1103         if (is_a<clifford>(e1) && ex_to<clifford>(e1).get_representation_label() >= rl) {
1104                 if (is_a<diracone>(e1.op(0)))
1105                         return 1;
1106                 else
1107                         throw(std::invalid_argument("remove_dirac_ONE(): expression is a non-scalar Clifford number!"));
1108         } else if (is_a<add>(e1) || is_a<ncmul>(e1) || is_a<mul>(e1)
1109                            || is_a<matrix>(e1) || is_a<lst>(e1)) {
1110                 if (options & 3) // is a child or was already expanded
1111                         return e1.map(fcn);
1112                 else
1113                         try {
1114                                 return e1.map(fcn);
1115                         } catch (std::exception &p) {
1116                                 need_reevaluation = true;
1117                         }
1118         } else if (is_a<power>(e1)) {
1119                 if (options & 3) // is a child or was already expanded
1120                         return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
1121                 else
1122                         try {
1123                                 return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
1124                         } catch (std::exception &p) {
1125                                 need_reevaluation = true;
1126                         }
1127         }
1128         if (need_reevaluation)
1129                 return remove_dirac_ONE(e, rl, options | 2);
1130         return e1;
1131 }
1132
1133 char clifford_max_label(const ex & e, bool ignore_ONE)
1134 {
1135         if (is_a<clifford>(e))
1136                 if (ignore_ONE && is_a<diracone>(e.op(0)))
1137                         return -1;
1138                 else
1139                         return ex_to<clifford>(e).get_representation_label();
1140         else {
1141                 char rl = -1;
1142                 for (size_t i=0; i < e.nops(); i++)
1143                         rl = (rl > clifford_max_label(e.op(i), ignore_ONE)) ? rl : clifford_max_label(e.op(i), ignore_ONE);
1144                 return rl;
1145         }
1146 }
1147
1148 ex clifford_norm(const ex & e)
1149 {
1150         return sqrt(remove_dirac_ONE(e * clifford_bar(e)));
1151 }
1152
1153 ex clifford_inverse(const ex & e)
1154 {
1155         ex norm = clifford_norm(e);
1156         if (!norm.is_zero())
1157                 return clifford_bar(e) / pow(norm, 2);
1158         else
1159                 throw(std::invalid_argument("clifford_inverse(): cannot find inverse of Clifford number with zero norm!"));
1160 }
1161
1162 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl, bool anticommuting)
1163 {
1164         if (!ex_to<idx>(mu).is_dim_numeric())
1165                 throw(std::invalid_argument("lst_to_clifford(): Index should have a numeric dimension"));
1166         ex e = clifford_unit(mu, metr, rl, anticommuting);
1167         return lst_to_clifford(v, e);
1168 }
1169
1170 ex lst_to_clifford(const ex & v, const ex & e) {
1171         unsigned min, max;
1172
1173         if (is_a<clifford>(e)) {
1174                 varidx mu = ex_to<varidx>(e.op(1));
1175                 unsigned dim = (ex_to<numeric>(mu.get_dim())).to_int();
1176
1177                 if (is_a<matrix>(v)) {
1178                         if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows()) {
1179                                 min = ex_to<matrix>(v).rows();
1180                                 max = ex_to<matrix>(v).cols();
1181                         } else {
1182                                 min = ex_to<matrix>(v).cols();
1183                                 max = ex_to<matrix>(v).rows();
1184                         }
1185                         if (min == 1) {
1186                                 if (dim == max)
1187                                         return indexed(v, ex_to<varidx>(mu).toggle_variance()) * e;
1188                                 else
1189                                         throw(std::invalid_argument("lst_to_clifford(): dimensions of vector and clifford unit mismatch"));
1190                         } else
1191                                 throw(std::invalid_argument("lst_to_clifford(): first argument should be a vector vector"));
1192                 } else if (is_a<lst>(v)) {
1193                         if (dim == ex_to<lst>(v).nops())
1194                                 return indexed(matrix(dim, 1, ex_to<lst>(v)), ex_to<varidx>(mu).toggle_variance()) * e;
1195                         else
1196                                 throw(std::invalid_argument("lst_to_clifford(): list length and dimension of clifford unit mismatch"));
1197                 } else
1198                         throw(std::invalid_argument("lst_to_clifford(): cannot construct from anything but list or vector"));
1199         } else
1200                 throw(std::invalid_argument("lst_to_clifford(): the second argument should be a Clifford unit"));
1201 }
1202
1203 /** Auxiliary structure to define a function for striping one Clifford unit
1204  * from vectors. Used in  clifford_to_lst(). */
1205 static ex get_clifford_comp(const ex & e, const ex & c)
1206 {
1207         pointer_to_map_function_1arg<const ex &> fcn(get_clifford_comp, c);
1208         int ival = ex_to<numeric>(ex_to<varidx>(c.op(1)).get_value()).to_int();
1209
1210         if (is_a<add>(e) || is_a<lst>(e) // || is_a<pseries>(e) || is_a<integral>(e)
1211                 || is_a<matrix>(e))
1212                 return e.map(fcn);
1213         else if (is_a<ncmul>(e) || is_a<mul>(e)) {
1214                 // find a Clifford unit with the same metric, delete it and substitute its index
1215                 size_t ind = e.nops() + 1;
1216                 for (size_t j = 0; j < e.nops(); j++)
1217                         if (is_a<clifford>(e.op(j)) && ex_to<clifford>(c).same_metric(e.op(j)))
1218                                 if (ind > e.nops())
1219                                         ind = j;
1220                                 else
1221                                         throw(std::invalid_argument("get_clifford_comp(): expression is a Clifford multi-vector"));
1222                 if (ind < e.nops()) {
1223                         ex S = 1;
1224                         bool same_value_index, found_dummy;
1225                         same_value_index = ( ex_to<varidx>(e.op(ind).op(1)).is_numeric()
1226                                                                  &&  (ival == ex_to<numeric>(ex_to<varidx>(e.op(ind).op(1)).get_value()).to_int()) );
1227                         found_dummy = same_value_index;
1228                         for(size_t j=0; j < e.nops(); j++)
1229                                 if (j != ind)
1230                                         if (same_value_index)
1231                                                 S = S * e.op(j);
1232                                         else {
1233                                                 exvector ind_vec = ex_to<indexed>(e.op(j)).get_dummy_indices(ex_to<indexed>(e.op(ind)));
1234                                                 if (ind_vec.size() > 0) {
1235                                                         found_dummy = true;
1236                                                         exvector::const_iterator it = ind_vec.begin(), itend = ind_vec.end();
1237                                                         while (it != itend) {
1238                                                                 S = S * e.op(j).subs(lst(ex_to<varidx>(*it) == ival, ex_to<varidx>(*it).toggle_variance() == ival), subs_options::no_pattern);
1239                                                                 ++it;
1240                                                         }
1241                                                 } else
1242                                                         S = S * e.op(j);
1243                                         }
1244                         return (found_dummy ? S : 0);
1245                 } else
1246                         throw(std::invalid_argument("get_clifford_comp(): expression is not a Clifford vector to the given units"));
1247         } else if (e.is_zero())
1248                 return e;
1249         else if (is_a<clifford>(e) && ex_to<clifford>(e).same_metric(c))
1250                 if ( ex_to<varidx>(e.op(1)).is_numeric() &&
1251                          (ival != ex_to<numeric>(ex_to<varidx>(e.op(1)).get_value()).to_int()) )
1252                         return 0;
1253                 else
1254                         return 1;
1255         else
1256                 throw(std::invalid_argument("get_clifford_comp(): expression is not usable as a Clifford vector"));
1257 }
1258
1259
1260 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic)
1261 {
1262         GINAC_ASSERT(is_a<clifford>(c));
1263         varidx mu = ex_to<varidx>(c.op(1));
1264         if (! mu.is_dim_numeric())
1265                 throw(std::invalid_argument("clifford_to_lst(): index should have a numeric dimension"));
1266         unsigned int D = ex_to<numeric>(mu.get_dim()).to_int();
1267
1268         if (algebraic) // check if algebraic method is applicable
1269                 for (unsigned int i = 0; i < D; i++)
1270                         if (pow(c.subs(mu == i), 2).is_zero()
1271                                 or (not is_a<numeric>(pow(c.subs(mu == i), 2))))
1272                                 algebraic = false;
1273         lst V;
1274         if (algebraic) {
1275                 for (unsigned int i = 0; i < D; i++)
1276                         V.append(remove_dirac_ONE(
1277                                                 simplify_indexed(canonicalize_clifford(e * c.subs(mu == i) +  c.subs(mu == i) * e))
1278                                                 / (2*pow(c.subs(mu == i), 2))));
1279         } else {
1280                 ex e1 = canonicalize_clifford(e);
1281                 try {
1282                         for (unsigned int i = 0; i < D; i++)
1283                                 V.append(get_clifford_comp(e1, c.subs(c.op(1) == i)));
1284                 } catch  (std::exception &p) {
1285                         /* Try to expand dummy summations to simplify the expression*/
1286                         e1 = canonicalize_clifford(expand_dummy_sum(e1, true));
1287                         for (unsigned int i = 0; i < D; i++)
1288                                 V.append(get_clifford_comp(e1, c.subs(c.op(1) == i)));
1289                 }
1290         }
1291         return V;
1292 }
1293
1294
1295 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, bool anticommuting)
1296 {
1297         ex x, D, cu;
1298
1299         if (! is_a<matrix>(v) && ! is_a<lst>(v))
1300                 throw(std::invalid_argument("clifford_moebius_map(): parameter v should be either vector or list"));
1301
1302         if (is_a<clifford>(G)) {
1303                 cu = G;
1304         } else {
1305                 if (is_a<indexed>(G))
1306                         D = ex_to<varidx>(G.op(1)).get_dim();
1307                 else if (is_a<matrix>(G))
1308                         D = ex_to<matrix>(G).rows();
1309                 else throw(std::invalid_argument("clifford_moebius_map(): metric should be an indexed object, matrix, or a Clifford unit"));
1310
1311                 varidx mu((new symbol)->setflag(status_flags::dynallocated), D);
1312                 cu = clifford_unit(mu, G, rl, anticommuting);
1313         }
1314
1315         x = lst_to_clifford(v, cu);
1316         ex e = simplify_indexed(canonicalize_clifford((a * x + b) * clifford_inverse(c * x + d)));
1317         return clifford_to_lst(e, cu, false);
1318 }
1319
1320 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl, bool anticommuting)
1321 {
1322         if (is_a<matrix>(M))
1323                 return clifford_moebius_map(ex_to<matrix>(M)(0,0), ex_to<matrix>(M)(0,1),
1324                                             ex_to<matrix>(M)(1,0), ex_to<matrix>(M)(1,1), v, G, rl, anticommuting);
1325         else
1326                 throw(std::invalid_argument("clifford_moebius_map(): parameter M should be a matrix"));
1327 }
1328
1329 } // namespace GiNaC