ginac/clifford.cpp

   1 /** @file clifford.cpp
   2  *
   3  *  Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2008 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 clifford::clifford() : representation_label(0), metric(0), commutator_sign(-1)
  78 {
  79         tinfo_key = &clifford::tinfo_static;
  80 }
  81
  82 DEFAULT_CTOR(diracone)
  83 DEFAULT_CTOR(cliffordunit)
  84 DEFAULT_CTOR(diracgamma)
  85 DEFAULT_CTOR(diracgamma5)
  86 DEFAULT_CTOR(diracgammaL)
  87 DEFAULT_CTOR(diracgammaR)
  88
  89 //////////
  90 // other constructors
  91 //////////
  92
  93 /** Construct object without any indices. This constructor is for internal
  94  *  use only. Use the dirac_ONE() function instead.
  95  *  @see dirac_ONE */
  96 clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl), metric(0), commutator_sign(-1)
  97 {
  98         tinfo_key = &clifford::tinfo_static;
  99 }
 100
 101 /** Construct object with one Lorentz index. This constructor is for internal
 102  *  use only. Use the clifford_unit() or dirac_gamma() functions instead.
 103  *  @see clifford_unit
 104  *  @see dirac_gamma */
 105 clifford::clifford(const ex & b, const ex & mu, const ex & metr, unsigned char rl, int comm_sign) : inherited(b, mu), representation_label(rl), metric(metr), commutator_sign(comm_sign)
 106 {
 107         GINAC_ASSERT(is_a<varidx>(mu));
 108         tinfo_key = &clifford::tinfo_static;
 109 }
 110
 111 clifford::clifford(unsigned char rl, const ex & metr, int comm_sign, const exvector & v, bool discardable) : inherited(not_symmetric(), v, discardable), representation_label(rl), metric(metr), commutator_sign(comm_sign)
 112 {
 113         tinfo_key = &clifford::tinfo_static;
 114 }
 115
 116 clifford::clifford(unsigned char rl, const ex & metr, int comm_sign, std::auto_ptr<exvector> vp) : inherited(not_symmetric(), vp), representation_label(rl), metric(metr), commutator_sign(comm_sign)
 117 {
 118         tinfo_key = &clifford::tinfo_static;
 119 }
 120
 121 return_type_t clifford::return_type_tinfo() const
 122 {
 123         return make_return_type_t<clifford>(representation_label);
 124 }
 125
 126 //////////
 127 // archiving
 128 //////////
 129
 130 clifford::clifford(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
 131 {
 132         unsigned rl;
 133         n.find_unsigned("label", rl);
 134         representation_label = rl;
 135         n.find_ex("metric", metric, sym_lst);
 136         n.find_unsigned("commutator_sign+1", rl);
 137         commutator_sign = rl - 1;
 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_unsigned("commutator_sign+1", commutator_sign+1);
 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 ex clifford::get_metric(const ex & i, const ex & j, bool symmetrised) const
 158 {
 159         if (is_a<indexed>(metric)) {
 160                 if (symmetrised && !(ex_to<symmetry>(ex_to<indexed>(metric).get_symmetry()).has_symmetry())) {
 161                         if (is_a<matrix>(metric.op(0))) {
 162                                 return indexed((ex_to<matrix>(metric.op(0)).add(ex_to<matrix>(metric.op(0)).transpose())).mul(numeric(1, 2)),
 163                                                symmetric2(), i, j);
 164                         } else {
 165                                 return simplify_indexed(indexed(metric.op(0)*_ex1_2, i, j) + indexed(metric.op(0)*_ex1_2, j, i));
 166                         }
 167                 } else {
 168                         return metric.subs(lst(metric.op(1) == i, metric.op(2) == j), subs_options::no_pattern);
 169                 }
 170         } else {
 171                 exvector indices = metric.get_free_indices();
 172                 if (symmetrised)
 173                         return _ex1_2*simplify_indexed(metric.subs(lst(indices[0] == i, indices[1] == j), subs_options::no_pattern)
 174                                                                         + metric.subs(lst(indices[0] == j, indices[1] == i), subs_options::no_pattern));
 175                 else
 176                         return metric.subs(lst(indices[0] == i, indices[1] == j), subs_options::no_pattern);
 177         }
 178 }
 179
 180 bool clifford::same_metric(const ex & other) const
 181 {
 182         ex metr;
 183         if (is_a<clifford>(other))
 184                 metr = ex_to<clifford>(other).get_metric();
 185         else
 186                 metr = other;
 187
 188         if (is_a<indexed>(metr))
 189                 return metr.op(0).is_equal(get_metric().op(0));
 190         else {
 191                 exvector indices = metr.get_free_indices();
 192                 return  (indices.size() == 2)
 193                         && simplify_indexed(get_metric(indices[0], indices[1])-metr).is_zero();
 194         }
 195 }
 196
 197 //////////
 198 // functions overriding virtual functions from base classes
 199 //////////
 200
 201 ex clifford::op(size_t i) const
 202 {
 203         GINAC_ASSERT(i<nops());
 204         if (nops()-i == 1)
 205                 return representation_label;
 206         else
 207                 return inherited::op(i);
 208 }
 209
 210 ex & clifford::let_op(size_t i)
 211 {
 212         GINAC_ASSERT(i<nops());
 213
 214         static ex rl = numeric(representation_label);
 215         ensure_if_modifiable();
 216         if (nops()-i == 1)
 217                 return rl;
 218         else
 219                 return inherited::let_op(i);
 220 }
 221
 222 ex clifford::subs(const exmap & m, unsigned options) const
 223 {
 224         ex subsed = inherited::subs(m, options);
 225         if(is_a<clifford>(subsed)) {
 226                 ex prevmetric = ex_to<clifford>(subsed).metric;
 227                 ex newmetric = prevmetric.subs(m, options);
 228                 if(!are_ex_trivially_equal(prevmetric, newmetric)) {
 229                         clifford c = ex_to<clifford>(subsed);
 230                         c.metric = newmetric;
 231                         subsed = c;
 232                 }
 233         }
 234         return subsed;
 235 }
 236
 237 int clifford::compare_same_type(const basic & other) const
 238 {
 239         GINAC_ASSERT(is_a<clifford>(other));
 240         const clifford &o = static_cast<const clifford &>(other);
 241
 242         if (representation_label != o.representation_label) {
 243                 // different representation label
 244                 return representation_label < o.representation_label ? -1 : 1;
 245         }
 246
 247         return inherited::compare_same_type(other);
 248 }
 249
 250 bool clifford::match_same_type(const basic & other) const
 251 {
 252         GINAC_ASSERT(is_a<clifford>(other));
 253         const clifford &o = static_cast<const clifford &>(other);
 254
 255         return ((representation_label == o.representation_label) && (commutator_sign == o.get_commutator_sign()) && same_metric(o));
 256 }
 257
 258 static bool is_dirac_slash(const ex & seq0)
 259 {
 260         return !is_a<diracgamma5>(seq0) && !is_a<diracgammaL>(seq0) &&
 261                !is_a<diracgammaR>(seq0) && !is_a<cliffordunit>(seq0) &&
 262                !is_a<diracone>(seq0);
 263 }
 264
 265 void clifford::do_print_dflt(const print_dflt & c, unsigned level) const
 266 {
 267         // dirac_slash() object is printed differently
 268         if (is_dirac_slash(seq[0])) {
 269                 seq[0].print(c, precedence());
 270                 c.s << "\\";
 271         } else { // We do not print representation label if it is 0
 272                 if (representation_label == 0) {
 273                         this->print_dispatch<inherited>(c, level);
 274                 } else { // otherwise we put it before indices in square brackets; the code is borrowed from indexed.cpp
 275                         if (precedence() <= level) {
 276                                 c.s << '(';
 277                         }
 278                         seq[0].print(c, precedence());
 279                         c.s << '[' << int(representation_label) << ']';
 280                         printindices(c, level);
 281                         if (precedence() <= level) {
 282                                 c.s << ')';
 283                         }
 284                 }
 285         }
 286 }
 287
 288 void clifford::do_print_latex(const print_latex & c, unsigned level) const
 289 {
 290         // dirac_slash() object is printed differently
 291         if (is_dirac_slash(seq[0])) {
 292                 c.s << "{";
 293                 seq[0].print(c, precedence());
 294                 c.s << "\\hspace{-1.0ex}/}";
 295         } else {
 296                 c.s << "\\clifford[" << int(representation_label) << "]";
 297                 this->print_dispatch<inherited>(c, level);
 298         }
 299 }
 300
 301 DEFAULT_COMPARE(diracone)
 302 DEFAULT_COMPARE(cliffordunit)
 303 DEFAULT_COMPARE(diracgamma)
 304 DEFAULT_COMPARE(diracgamma5)
 305 DEFAULT_COMPARE(diracgammaL)
 306 DEFAULT_COMPARE(diracgammaR)
 307
 308 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbf{1}")
 309 DEFAULT_PRINT_LATEX(cliffordunit, "e", "e")
 310 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
 311 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
 312 DEFAULT_PRINT_LATEX(diracgammaL, "gammaL", "{\\gamma_L}")
 313 DEFAULT_PRINT_LATEX(diracgammaR, "gammaR", "{\\gamma_R}")
 314
 315 /** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
 316 static void base_and_index(const ex & c, ex & b, ex & i)
 317 {
 318         GINAC_ASSERT(is_a<clifford>(c));
 319         GINAC_ASSERT(c.nops() == 2+1);
 320
 321         if (is_a<cliffordunit>(c.op(0))) { // proper dirac gamma object or clifford unit
 322                 i = c.op(1);
 323                 b = _ex1;
 324         } else if (is_a<diracgamma5>(c.op(0)) || is_a<diracgammaL>(c.op(0)) || is_a<diracgammaR>(c.op(0))) { // gamma5/L/R
 325                 i = _ex0;
 326                 b = _ex1;
 327         } else { // slash object, generate new dummy index
 328                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(c.op(1)).get_dim());
 329                 b = indexed(c.op(0), ix.toggle_variance());
 330                 i = ix;
 331         }
 332 }
 333
 334 /** Predicate for finding non-clifford objects. */
 335 struct is_not_a_clifford : public std::unary_function<ex, bool> {
 336         bool operator()(const ex & e)
 337         {
 338                 return !is_a<clifford>(e);
 339         }
 340 };
 341
 342 /** Contraction of a gamma matrix with something else. */
 343 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 344 {
 345         GINAC_ASSERT(is_a<clifford>(*self));
 346         GINAC_ASSERT(is_a<indexed>(*other));
 347         GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
 348         unsigned char rl = ex_to<clifford>(*self).get_representation_label();
 349
 350         ex dim = ex_to<idx>(self->op(1)).get_dim();
 351         if (other->nops() > 1)
 352                 dim = minimal_dim(dim, ex_to<idx>(other->op(1)).get_dim());
 353
 354         if (is_a<clifford>(*other)) {
 355
 356                 // Contraction only makes sense if the represenation labels are equal
 357                 if (ex_to<clifford>(*other).get_representation_label() != rl)
 358                         return false;
 359
 360                 size_t num = other - self;
 361
 362                 // gamma~mu gamma.mu = dim ONE
 363                 if (num == 1) {
 364                         *self = dim;
 365                         *other = dirac_ONE(rl);
 366                         return true;
 367
 368                 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
 369                 } else if (num == 2
 370                         && is_a<clifford>(self[1])) {
 371                         *self = 2 - dim;
 372                         *other = _ex1;
 373                         return true;
 374
 375                 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
 376                 } else if (num == 3
 377                         && is_a<clifford>(self[1])
 378                         && is_a<clifford>(self[2])) {
 379                         ex b1, i1, b2, i2;
 380                         base_and_index(self[1], b1, i1);
 381                         base_and_index(self[2], b2, i2);
 382                         *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
 383                         self[1] = _ex1;
 384                         self[2] = _ex1;
 385                         *other = _ex1;
 386                         return true;
 387
 388                 // 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
 389                 } else if (num == 4
 390                         && is_a<clifford>(self[1])
 391                         && is_a<clifford>(self[2])
 392                         && is_a<clifford>(self[3])) {
 393                         *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3];
 394                         self[1] = _ex1;
 395                         self[2] = _ex1;
 396                         self[3] = _ex1;
 397                         *other = _ex1;
 398                         return true;
 399
 400                 // gamma~mu Sodd gamma.mu = -2 Sodd_R
 401                 // (Chisholm identity in 4 dimensions)
 402                 } else if (!((other - self) & 1) && dim.is_equal(4)) {
 403                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
 404                                 return false;
 405
 406                         *self = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(other), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
 407                         std::fill(self + 1, other, _ex1);
 408                         *other = _ex_2;
 409                         return true;
 410
 411                 // gamma~mu Sodd gamma~alpha gamma.mu = 2 gamma~alpha Sodd + 2 Sodd_R gamma~alpha
 412                 // (commutate contracted indices towards each other, then use
 413                 // Chisholm identity in 4 dimensions)
 414                 } else if (((other - self) & 1) && dim.is_equal(4)) {
 415                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
 416                                 return false;
 417
 418                         exvector::iterator next_to_last = other - 1;
 419                         ex S = ncmul(exvector(self + 1, next_to_last), true);
 420                         ex SR = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(next_to_last), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
 421
 422                         *self = (*next_to_last) * S + SR * (*next_to_last);
 423                         std::fill(self + 1, other, _ex1);
 424                         *other = _ex2;
 425                         return true;
 426
 427                 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
 428                 // (commutate contracted indices towards each other, simplify_indexed()
 429                 // will re-expand and re-run the simplification)
 430                 } else {
 431                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
 432                                 return false;
 433
 434                         exvector::iterator next_to_last = other - 1;
 435                         ex S = ncmul(exvector(self + 1, next_to_last), true);
 436
 437                         *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
 438                         std::fill(self + 1, other + 1, _ex1);
 439                         return true;
 440                 }
 441
 442         } else if (is_a<symbol>(other->op(0)) && other->nops() == 2) {
 443
 444                 // x.mu gamma~mu -> x-slash
 445                 *self = dirac_slash(other->op(0), dim, rl);
 446                 *other = _ex1;
 447                 return true;
 448         }
 449
 450         return false;
 451 }
 452
 453 /** Contraction of a Clifford unit with something else. */
 454 bool cliffordunit::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 455 {
 456         GINAC_ASSERT(is_a<clifford>(*self));
 457         GINAC_ASSERT(is_a<indexed>(*other));
 458         GINAC_ASSERT(is_a<cliffordunit>(self->op(0)));
 459         clifford unit = ex_to<clifford>(*self);
 460         unsigned char rl = unit.get_representation_label();
 461
 462         if (is_a<clifford>(*other)) {
 463                 // Contraction only makes sense if the represenation labels are equal
 464                 // and the metrics are the same
 465                 if ((ex_to<clifford>(*other).get_representation_label() != rl)
 466                     && unit.same_metric(*other))
 467                         return false;
 468
 469                 exvector::iterator before_other = other - 1;
 470                 ex mu = self->op(1);
 471                 ex mu_toggle = other->op(1);
 472                 ex alpha = before_other->op(1);
 473
 474                 // e~mu e.mu = Tr ONE
 475                 if (other - self == 1) {
 476                         *self = unit.get_metric(mu, mu_toggle, true);
 477                         *other = dirac_ONE(rl);
 478                         return true;
 479
 480                 } else if (other - self == 2) {
 481                         if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
 482                                 // e~mu e~alpha e.mu = 2*e~mu B(alpha, mu.toggle_variance())-Tr(B) e~alpha
 483                                 *self = 2 * (*self) * unit.get_metric(alpha, mu_toggle, true) - unit.get_metric(mu, mu_toggle, true) * (*before_other);
 484                                 *before_other = _ex1;
 485                                 *other = _ex1;
 486                                 return true;
 487
 488                         } else {
 489                                 // e~mu S e.mu = Tr S ONE
 490                                 *self = unit.get_metric(mu, mu_toggle, true);
 491                                 *other = dirac_ONE(rl);
 492                                 return true;
 493                         }
 494                 } else {
 495                 // e~mu S e~alpha e.mu = 2 e~mu S B(alpha, mu.toggle_variance()) - e~mu S e.mu e~alpha
 496                 // (commutate contracted indices towards each other, simplify_indexed()
 497                 // will re-expand and re-run the simplification)
 498                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other) {
 499                                 return false;
 500                         }
 501
 502                         ex S = ncmul(exvector(self + 1, before_other), true);
 503
 504                         if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
 505                                 *self = 2 * (*self) * S * unit.get_metric(alpha, mu_toggle, true) - (*self) * S * (*other) * (*before_other);
 506                         } else {
 507                                 // simply commutes
 508                                 *self = (*self) * S * (*other) * (*before_other);
 509                         }
 510
 511                         std::fill(self + 1, other + 1, _ex1);
 512                         return true;
 513                 }
 514         }
 515         return false;
 516 }
 517
 518 /** Perform automatic simplification on noncommutative product of clifford
 519  *  objects. This removes superfluous ONEs, permutes gamma5/L/R's to the front
 520  *  and removes squares of gamma objects. */
 521 ex clifford::eval_ncmul(const exvector & v) const
 522 {
 523         exvector s;
 524         s.reserve(v.size());
 525
 526         // Remove superfluous ONEs
 527         exvector::const_iterator cit = v.begin(), citend = v.end();
 528         while (cit != citend) {
 529                 if (!is_a<clifford>(*cit) || !is_a<diracone>(cit->op(0)))
 530                         s.push_back(*cit);
 531                 cit++;
 532         }
 533
 534         bool something_changed = false;
 535         int sign = 1;
 536
 537         // Anticommutate gamma5/L/R's to the front
 538         if (s.size() >= 2) {
 539                 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
 540                 while (true) {
 541                         exvector::iterator it = next_to_last;
 542                         while (true) {
 543                                 exvector::iterator it2 = it + 1;
 544                                 if (is_a<clifford>(*it) && is_a<clifford>(*it2)) {
 545                                         ex e1 = it->op(0), e2 = it2->op(0);
 546
 547                                         if (is_a<diracgamma5>(e2)) {
 548
 549                                                 if (is_a<diracgammaL>(e1) || is_a<diracgammaR>(e1)) {
 550
 551                                                         // gammaL/R gamma5 -> gamma5 gammaL/R
 552                                                         it->swap(*it2);
 553                                                         something_changed = true;
 554
 555                                                 } else if (!is_a<diracgamma5>(e1)) {
 556
 557                                                         // gamma5 gamma5 -> gamma5 gamma5 (do nothing)
 558                                                         // x gamma5 -> -gamma5 x
 559                                                         it->swap(*it2);
 560                                                         sign = -sign;
 561                                                         something_changed = true;
 562                                                 }
 563
 564                                         } else if (is_a<diracgammaL>(e2)) {
 565
 566                                                 if (is_a<diracgammaR>(e1)) {
 567
 568                                                         // gammaR gammaL -> 0
 569                                                         return _ex0;
 570
 571                                                 } else if (!is_a<diracgammaL>(e1) && !is_a<diracgamma5>(e1)) {
 572
 573                                                         // gammaL gammaL -> gammaL gammaL (do nothing)
 574                                                         // gamma5 gammaL -> gamma5 gammaL (do nothing)
 575                                                         // x gammaL -> gammaR x
 576                                                         it->swap(*it2);
 577                                                         *it = clifford(diracgammaR(), ex_to<clifford>(*it).get_representation_label());
 578                                                         something_changed = true;
 579                                                 }
 580
 581                                         } else if (is_a<diracgammaR>(e2)) {
 582
 583                                                 if (is_a<diracgammaL>(e1)) {
 584
 585                                                         // gammaL gammaR -> 0
 586                                                         return _ex0;
 587
 588                                                 } else if (!is_a<diracgammaR>(e1) && !is_a<diracgamma5>(e1)) {
 589
 590                                                         // gammaR gammaR -> gammaR gammaR (do nothing)
 591                                                         // gamma5 gammaR -> gamma5 gammaR (do nothing)
 592                                                         // x gammaR -> gammaL x
 593                                                         it->swap(*it2);
 594                                                         *it = clifford(diracgammaL(), ex_to<clifford>(*it).get_representation_label());
 595                                                         something_changed = true;
 596                                                 }
 597                                         }
 598                                 }
 599                                 if (it == first)
 600                                         break;
 601                                 --it;
 602                         }
 603                         if (next_to_last == first)
 604                                 break;
 605                         --next_to_last;
 606                 }
 607         }
 608
 609         // Remove equal adjacent gammas
 610         if (s.size() >= 2) {
 611                 exvector::iterator it, itend = s.end() - 1;
 612                 for (it = s.begin(); it != itend; ++it) {
 613                         ex & a = it[0];
 614                         ex & b = it[1];
 615                         if (!is_a<clifford>(a) || !is_a<clifford>(b))
 616                                 continue;
 617
 618                         const ex & ag = a.op(0);
 619                         const ex & bg = b.op(0);
 620                         bool a_is_cliffordunit = is_a<cliffordunit>(ag);
 621                         bool b_is_cliffordunit =  is_a<cliffordunit>(bg);
 622
 623                         if (a_is_cliffordunit && b_is_cliffordunit && ex_to<clifford>(a).same_metric(b)
 624                                 && (ex_to<clifford>(a).get_commutator_sign() == -1)) {
 625                                 // This is done only for Clifford algebras
 626
 627                                 const ex & ia = a.op(1);
 628                                 const ex & ib = b.op(1);
 629                                 if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
 630                                         a = ex_to<clifford>(a).get_metric(ia, ib, true);
 631                                         b = dirac_ONE(representation_label);
 632                                         something_changed = true;
 633                                 }
 634
 635                         } else if ((is_a<diracgamma5>(ag) && is_a<diracgamma5>(bg))) {
 636
 637                                 // Remove squares of gamma5
 638                                 a = dirac_ONE(representation_label);
 639                                 b = dirac_ONE(representation_label);
 640                                 something_changed = true;
 641
 642                         } else if ((is_a<diracgammaL>(ag) && is_a<diracgammaL>(bg))
 643                                 || (is_a<diracgammaR>(ag) && is_a<diracgammaR>(bg))) {
 644
 645                                 // Remove squares of gammaL/R
 646                                 b = dirac_ONE(representation_label);
 647                                 something_changed = true;
 648
 649                         } else if (is_a<diracgammaL>(ag) && is_a<diracgammaR>(bg)) {
 650
 651                                 // gammaL and gammaR are orthogonal
 652                                 return _ex0;
 653
 654                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaL>(bg)) {
 655
 656                                 // gamma5 gammaL -> -gammaL
 657                                 a = dirac_ONE(representation_label);
 658                                 sign = -sign;
 659                                 something_changed = true;
 660
 661                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaR>(bg)) {
 662
 663                                 // gamma5 gammaR -> gammaR
 664                                 a = dirac_ONE(representation_label);
 665                                 something_changed = true;
 666
 667                         } else if (!a_is_cliffordunit && !b_is_cliffordunit && ag.is_equal(bg)) {
 668
 669                                 // a\ a\ -> a^2
 670                                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).minimal_dim(ex_to<idx>(b.op(1))));
 671
 672                                 a = indexed(ag, ix) * indexed(ag, ix.toggle_variance());
 673                                 b = dirac_ONE(representation_label);
 674                                 something_changed = true;
 675                         }
 676                 }
 677         }
 678
 679         if (s.empty())
 680                 return dirac_ONE(representation_label) * sign;
 681         if (something_changed)
 682                 return reeval_ncmul(s) * sign;
 683         else
 684                 return hold_ncmul(s) * sign;
 685 }
 686
 687 ex clifford::thiscontainer(const exvector & v) const
 688 {
 689         return clifford(representation_label, metric, commutator_sign, v);
 690 }
 691
 692 ex clifford::thiscontainer(std::auto_ptr<exvector> vp) const
 693 {
 694         return clifford(representation_label, metric, commutator_sign, vp);
 695 }
 696
 697 ex diracgamma5::conjugate() const
 698 {
 699         return _ex_1 * (*this);
 700 }
 701
 702 ex diracgammaL::conjugate() const
 703 {
 704         return (new diracgammaR)->setflag(status_flags::dynallocated);
 705 }
 706
 707 ex diracgammaR::conjugate() const
 708 {
 709         return (new diracgammaL)->setflag(status_flags::dynallocated);
 710 }
 711
 712 //////////
 713 // global functions
 714 //////////
 715
 716 ex dirac_ONE(unsigned char rl)
 717 {
 718         static ex ONE = (new diracone)->setflag(status_flags::dynallocated);
 719         return clifford(ONE, rl);
 720 }
 721
 722 static unsigned get_dim_uint(const ex& e)
 723 {
 724         if (!is_a<idx>(e))
 725                 throw std::invalid_argument("get_dim_uint: argument is not an index");
 726         ex dim = ex_to<idx>(e).get_dim();
 727         if (!dim.info(info_flags::posint))
 728                 throw std::invalid_argument("get_dim_uint: dimension of index should be a positive integer");
 729         unsigned d = ex_to<numeric>(dim).to_int();
 730         return d;
 731 }
 732
 733 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl)
 734 {
 735         //static ex unit = (new cliffordunit)->setflag(status_flags::dynallocated);
 736         ex unit = (new cliffordunit)->setflag(status_flags::dynallocated);
 737
 738         if (!is_a<idx>(mu))
 739                 throw(std::invalid_argument("clifford_unit(): index of Clifford unit must be of type idx or varidx"));
 740
 741         exvector indices = metr.get_free_indices();
 742
 743         if (indices.size() == 2) {
 744                 return clifford(unit, mu, metr, rl);
 745         } else if (is_a<matrix>(metr)) {
 746                 matrix M = ex_to<matrix>(metr);
 747                 unsigned n = M.rows();
 748                 bool symmetric = true;
 749
 750                 //static idx xi((new symbol)->setflag(status_flags::dynallocated), n),
 751                 //      chi((new symbol)->setflag(status_flags::dynallocated), n);
 752                 idx xi((new symbol)->setflag(status_flags::dynallocated), n),
 753                         chi((new symbol)->setflag(status_flags::dynallocated), n);
 754                 if ((n ==  M.cols()) && (n == get_dim_uint(mu))) {
 755                         for (unsigned i = 0; i < n; i++) {
 756                                 for (unsigned j = i+1; j < n; j++) {
 757                                         if (!M(i, j).is_equal(M(j, i))) {
 758                                                 symmetric = false;
 759                                         }
 760                                 }
 761                         }
 762                         return clifford(unit, mu, indexed(metr, symmetric?symmetric2():not_symmetric(), xi, chi), rl);
 763                 } else {
 764                         throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be a square matrix with the same dimensions as index"));
 765                 }
 766         } else if (indices.size() == 0) { // a tensor or other expression without indices
 767                 //static varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim()),
 768                 //      chi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim());
 769                 varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim()),
 770                         chi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim());
 771                 return clifford(unit, mu, indexed(metr, xi, chi), rl);
 772         }  else
 773                 throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be of type tensor, matrix or an expression with two free indices"));
 774 }
 775
 776 ex dirac_gamma(const ex & mu, unsigned char rl)
 777 {
 778         static ex gamma = (new diracgamma)->setflag(status_flags::dynallocated);
 779
 780         if (!is_a<varidx>(mu))
 781                 throw(std::invalid_argument("dirac_gamma(): index of Dirac gamma must be of type varidx"));
 782
 783         static varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim()),
 784                 chi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim());
 785         return clifford(gamma, mu, indexed((new minkmetric)->setflag(status_flags::dynallocated), symmetric2(), xi, chi), rl);
 786 }
 787
 788 ex dirac_gamma5(unsigned char rl)
 789 {
 790         static ex gamma5 = (new diracgamma5)->setflag(status_flags::dynallocated);
 791         return clifford(gamma5, rl);
 792 }
 793
 794 ex dirac_gammaL(unsigned char rl)
 795 {
 796         static ex gammaL = (new diracgammaL)->setflag(status_flags::dynallocated);
 797         return clifford(gammaL, rl);
 798 }
 799
 800 ex dirac_gammaR(unsigned char rl)
 801 {
 802         static ex gammaR = (new diracgammaR)->setflag(status_flags::dynallocated);
 803         return clifford(gammaR, rl);
 804 }
 805
 806 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
 807 {
 808         // Slashed vectors are actually stored as a clifford object with the
 809         // vector as its base expression and a (dummy) index that just serves
 810         // for storing the space dimensionality
 811
 812         static varidx xi((new symbol)->setflag(status_flags::dynallocated), dim),
 813                 chi((new symbol)->setflag(status_flags::dynallocated), dim);
 814    return clifford(e, varidx(0, dim), indexed((new minkmetric)->setflag(status_flags::dynallocated), symmetric2(), xi, chi), rl);
 815 }
 816
 817 /** Extract representation label from tinfo key (as returned by
 818  *  return_type_tinfo()). */
 819 static unsigned char get_representation_label(const return_type_t& ti)
 820 {
 821         return (unsigned char)ti.rl;
 822 }
 823
 824 /** Take trace of a string of an even number of Dirac gammas given a vector
 825  *  of indices. */
 826 static ex trace_string(exvector::const_iterator ix, size_t num)
 827 {
 828         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 829         if (num == 2)
 830                 return lorentz_g(ix[0], ix[1]);
 831
 832         // 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 )
 833         else if (num == 4)
 834                 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
 835                      + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
 836                      - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
 837
 838         // Traces of 6 or more gammas are computed recursively:
 839         // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
 840         //   + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
 841         //   - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
 842         //   + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
 843         //   - ...
 844         //   + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
 845         exvector v(num - 2);
 846         int sign = 1;
 847         ex result;
 848         for (size_t i=1; i<num; i++) {
 849                 for (size_t n=1, j=0; n<num; n++) {
 850                         if (n == i)
 851                                 continue;
 852                         v[j++] = ix[n];
 853                 }
 854                 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
 855                 sign = -sign;
 856         }
 857         return result;
 858 }
 859
 860 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE)
 861 {
 862         if (is_a<clifford>(e)) {
 863
 864                 unsigned char rl = ex_to<clifford>(e).get_representation_label();
 865
 866                 // Are we taking the trace over this object's representation label?
 867                 if (rls.find(rl) == rls.end())
 868                         return e;
 869
 870                 // Yes, all elements are traceless, except for dirac_ONE and dirac_L/R
 871                 const ex & g = e.op(0);
 872                 if (is_a<diracone>(g))
 873                         return trONE;
 874                 else if (is_a<diracgammaL>(g) || is_a<diracgammaR>(g))
 875                         return trONE/2;
 876                 else
 877                         return _ex0;
 878
 879         } else if (is_exactly_a<mul>(e)) {
 880
 881                 // Trace of product: pull out non-clifford factors
 882                 ex prod = _ex1;
 883                 for (size_t i=0; i<e.nops(); i++) {
 884                         const ex &o = e.op(i);
 885                         if (is_clifford_tinfo(o.return_type_tinfo()))
 886                                 prod *= dirac_trace(o, rls, trONE);
 887                         else
 888                                 prod *= o;
 889                 }
 890                 return prod;
 891
 892         } else if (is_exactly_a<ncmul>(e)) {
 893
 894                 unsigned char rl = get_representation_label(e.return_type_tinfo());
 895
 896                 // Are we taking the trace over this string's representation label?
 897                 if (rls.find(rl) == rls.end())
 898                         return e;
 899
 900                 // Substitute gammaL/R and expand product, if necessary
 901                 ex e_expanded = e.subs(lst(
 902                         dirac_gammaL(rl) == (dirac_ONE(rl)-dirac_gamma5(rl))/2,
 903                         dirac_gammaR(rl) == (dirac_ONE(rl)+dirac_gamma5(rl))/2
 904                 ), subs_options::no_pattern).expand();
 905                 if (!is_a<ncmul>(e_expanded))
 906                         return dirac_trace(e_expanded, rls, trONE);
 907
 908                 // gamma5 gets moved to the front so this check is enough
 909                 bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
 910                 size_t num = e.nops();
 911
 912                 if (has_gamma5) {
 913
 914                         // Trace of gamma5 * odd number of gammas and trace of
 915                         // gamma5 * gamma.mu * gamma.nu are zero
 916                         if ((num & 1) == 0 || num == 3)
 917                                 return _ex0;
 918
 919                         // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
 920                         // (the epsilon is always 4-dimensional)
 921                         if (num == 5) {
 922                                 ex b1, i1, b2, i2, b3, i3, b4, i4;
 923                                 base_and_index(e.op(1), b1, i1);
 924                                 base_and_index(e.op(2), b2, i2);
 925                                 base_and_index(e.op(3), b3, i3);
 926                                 base_and_index(e.op(4), b4, i4);
 927                                 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();
 928                         }
 929
 930                         // Tr gamma5 S_2k =
 931                         //   I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
 932                         // (the epsilon is always 4-dimensional)
 933                         exvector ix(num-1), bv(num-1);
 934                         for (size_t i=1; i<num; i++)
 935                                 base_and_index(e.op(i), bv[i-1], ix[i-1]);
 936                         num--;
 937                         int *iv = new int[num];
 938                         ex result;
 939                         for (size_t i=0; i<num-3; i++) {
 940                                 ex idx1 = ix[i];
 941                                 for (size_t j=i+1; j<num-2; j++) {
 942                                         ex idx2 = ix[j];
 943                                         for (size_t k=j+1; k<num-1; k++) {
 944                                                 ex idx3 = ix[k];
 945                                                 for (size_t l=k+1; l<num; l++) {
 946                                                         ex idx4 = ix[l];
 947                                                         iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
 948                                                         exvector v;
 949                                                         v.reserve(num - 4);
 950                                                         for (size_t n=0, t=4; n<num; n++) {
 951                                                                 if (n == i || n == j || n == k || n == l)
 952                                                                         continue;
 953                                                                 iv[t++] = n;
 954                                                                 v.push_back(ix[n]);
 955                                                         }
 956                                                         int sign = permutation_sign(iv, iv + num);
 957                                                         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))
 958                                                                 * trace_string(v.begin(), num - 4);
 959                                                 }
 960                                         }
 961                                 }
 962                         }
 963                         delete[] iv;
 964                         return trONE * I * result * mul(bv);
 965
 966                 } else { // no gamma5
 967
 968                         // Trace of odd number of gammas is zero
 969                         if ((num & 1) == 1)
 970                                 return _ex0;
 971
 972                         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 973                         if (num == 2) {
 974                                 ex b1, i1, b2, i2;
 975                                 base_and_index(e.op(0), b1, i1);
 976                                 base_and_index(e.op(1), b2, i2);
 977                                 return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
 978                         }
 979
 980                         exvector iv(num), bv(num);
 981                         for (size_t i=0; i<num; i++)
 982                                 base_and_index(e.op(i), bv[i], iv[i]);
 983
 984                         return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
 985                 }
 986
 987         } else if (e.nops() > 0) {
 988
 989                 // Trace maps to all other container classes (this includes sums)
 990                 pointer_to_map_function_2args<const std::set<unsigned char> &, const ex &> fcn(dirac_trace, rls, trONE);
 991                 return e.map(fcn);
 992
 993         } else
 994                 return _ex0;
 995 }
 996
 997 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE)
 998 {
 999         // Convert list to set
1000         std::set<unsigned char> rls;
1001         for (lst::const_iterator i = rll.begin(); i != rll.end(); ++i) {
1002                 if (i->info(info_flags::nonnegint))
1003                         rls.insert(ex_to<numeric>(*i).to_int());
1004         }
1005
1006         return dirac_trace(e, rls, trONE);
1007 }
1008
1009 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
1010 {
1011         // Convert label to set
1012         std::set<unsigned char> rls;
1013         rls.insert(rl);
1014
1015         return dirac_trace(e, rls, trONE);
1016 }
1017
1018
1019 ex canonicalize_clifford(const ex & e_)
1020 {
1021         pointer_to_map_function fcn(canonicalize_clifford);
1022
1023         if (is_a<matrix>(e_)    // || is_a<pseries>(e) || is_a<integral>(e)
1024                 || e_.info(info_flags::list)) {
1025                 return e_.map(fcn);
1026         } else {
1027                 ex e=simplify_indexed(e_);
1028                 // Scan for any ncmul objects
1029                 exmap srl;
1030                 ex aux = e.to_rational(srl);
1031                 for (exmap::iterator i = srl.begin(); i != srl.end(); ++i) {
1032
1033                         ex lhs = i->first;
1034                         ex rhs = i->second;
1035
1036                         if (is_exactly_a<ncmul>(rhs)
1037                                         && rhs.return_type() == return_types::noncommutative
1038                                         && is_clifford_tinfo(rhs.return_type_tinfo())) {
1039
1040                                 // Expand product, if necessary
1041                                 ex rhs_expanded = rhs.expand();
1042                                 if (!is_a<ncmul>(rhs_expanded)) {
1043                                         i->second = canonicalize_clifford(rhs_expanded);
1044                                         continue;
1045
1046                                 } else if (!is_a<clifford>(rhs.op(0)))
1047                                         continue;
1048
1049                                 exvector v;
1050                                 v.reserve(rhs.nops());
1051                                 for (size_t j=0; j<rhs.nops(); j++)
1052                                         v.push_back(rhs.op(j));
1053
1054                                 // Stupid recursive bubble sort because we only want to swap adjacent gammas
1055                                 exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
1056                                 if (is_a<diracgamma5>(it->op(0)) || is_a<diracgammaL>(it->op(0)) || is_a<diracgammaR>(it->op(0)))
1057                                         ++it;
1058
1059                                 while (it != next_to_last) {
1060                                         if (it[0].compare(it[1]) > 0) {
1061
1062                                                 ex save0 = it[0], save1 = it[1];
1063                                                 ex b1, i1, b2, i2;
1064                                                 base_and_index(it[0], b1, i1);
1065                                                 base_and_index(it[1], b2, i2);
1066                                                 // for Clifford algebras (commutator_sign == -1) metric should be symmetrised
1067                                                 it[0] = (ex_to<clifford>(save0).get_metric(i1, i2, ex_to<clifford>(save0).get_commutator_sign() == -1) * b1 * b2).simplify_indexed();
1068                                                 it[1] = v.size() ? _ex2 * dirac_ONE(ex_to<clifford>(save0).get_representation_label()) : _ex2;
1069                                                 ex sum = ncmul(v);
1070                                                 it[0] = save1;
1071                                                 it[1] = save0;
1072                                                 sum += ex_to<clifford>(save0).get_commutator_sign() * ncmul(v, true);
1073                                                 i->second = canonicalize_clifford(sum);
1074                                                 goto next_sym;
1075                                         }
1076                                         ++it;
1077                                 }
1078 next_sym:       ;
1079                         }
1080                 }
1081                 return aux.subs(srl, subs_options::no_pattern).simplify_indexed();
1082         }
1083 }
1084
1085 ex clifford_prime(const ex & e)
1086 {
1087         pointer_to_map_function fcn(clifford_prime);
1088         if (is_a<clifford>(e) && is_a<cliffordunit>(e.op(0))) {
1089                 return -e;
1090         } else if (is_a<add>(e) || is_a<ncmul>(e) || is_a<mul>(e) //|| is_a<pseries>(e) || is_a<integral>(e)
1091                            || is_a<matrix>(e) || e.info(info_flags::list)) {
1092                 return e.map(fcn);
1093         } else if (is_a<power>(e)) {
1094                 return pow(clifford_prime(e.op(0)), e.op(1));
1095         } else
1096                 return e;
1097 }
1098
1099 ex remove_dirac_ONE(const ex & e, unsigned char rl, unsigned options)
1100 {
1101         pointer_to_map_function_2args<unsigned char, unsigned> fcn(remove_dirac_ONE, rl, options | 1);
1102         bool need_reevaluation = false;
1103         ex e1 = e;
1104         if (! (options & 1) )  { // is not a child
1105                 if (options & 2)
1106                         e1 = expand_dummy_sum(e, true);
1107                 e1 = canonicalize_clifford(e1);
1108         }
1109
1110         if (is_a<clifford>(e1) && ex_to<clifford>(e1).get_representation_label() >= rl) {
1111                 if (is_a<diracone>(e1.op(0)))
1112                         return 1;
1113                 else
1114                         throw(std::invalid_argument("remove_dirac_ONE(): expression is a non-scalar Clifford number!"));
1115         } else if (is_a<add>(e1) || is_a<ncmul>(e1) || is_a<mul>(e1)
1116                            || is_a<matrix>(e1) || e1.info(info_flags::list)) {
1117                 if (options & 3) // is a child or was already expanded
1118                         return e1.map(fcn);
1119                 else
1120                         try {
1121                                 return e1.map(fcn);
1122                         } catch (std::exception &p) {
1123                                 need_reevaluation = true;
1124                         }
1125         } else if (is_a<power>(e1)) {
1126                 if (options & 3) // is a child or was already expanded
1127                         return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
1128                 else
1129                         try {
1130                                 return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
1131                         } catch (std::exception &p) {
1132                                 need_reevaluation = true;
1133                         }
1134         }
1135         if (need_reevaluation)
1136                 return remove_dirac_ONE(e, rl, options | 2);
1137         return e1;
1138 }
1139
1140 char clifford_max_label(const ex & e, bool ignore_ONE)
1141 {
1142         if (is_a<clifford>(e))
1143                 if (ignore_ONE && is_a<diracone>(e.op(0)))
1144                         return -1;
1145                 else
1146                         return ex_to<clifford>(e).get_representation_label();
1147         else {
1148                 char rl = -1;
1149                 for (size_t i=0; i < e.nops(); i++)
1150                         rl = (rl > clifford_max_label(e.op(i), ignore_ONE)) ? rl : clifford_max_label(e.op(i), ignore_ONE);
1151                 return rl;
1152         }
1153 }
1154
1155 ex clifford_norm(const ex & e)
1156 {
1157         return sqrt(remove_dirac_ONE(e * clifford_bar(e)));
1158 }
1159
1160 ex clifford_inverse(const ex & e)
1161 {
1162         ex norm = clifford_norm(e);
1163         if (!norm.is_zero())
1164                 return clifford_bar(e) / pow(norm, 2);
1165         else
1166                 throw(std::invalid_argument("clifford_inverse(): cannot find inverse of Clifford number with zero norm!"));
1167 }
1168
1169 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl)
1170 {
1171         if (!ex_to<idx>(mu).is_dim_numeric())
1172                 throw(std::invalid_argument("lst_to_clifford(): Index should have a numeric dimension"));
1173         ex e = clifford_unit(mu, metr, rl);
1174         return lst_to_clifford(v, e);
1175 }
1176
1177 ex lst_to_clifford(const ex & v, const ex & e) {
1178         unsigned min, max;
1179
1180         if (is_a<clifford>(e)) {
1181                 ex mu = e.op(1);
1182                 ex mu_toggle
1183                         = is_a<varidx>(mu) ? ex_to<varidx>(mu).toggle_variance() : mu;
1184                 unsigned dim = get_dim_uint(mu);
1185
1186                 if (is_a<matrix>(v)) {
1187                         if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows()) {
1188                                 min = ex_to<matrix>(v).rows();
1189                                 max = ex_to<matrix>(v).cols();
1190                         } else {
1191                                 min = ex_to<matrix>(v).cols();
1192                                 max = ex_to<matrix>(v).rows();
1193                         }
1194                         if (min == 1) {
1195                                 if (dim == max)
1196                                         return indexed(v, mu_toggle) * e;
1197                                 else if (max - dim == 1) {
1198                                         if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows())
1199                                                 return v.op(0) * dirac_ONE(ex_to<clifford>(e).get_representation_label()) + indexed(sub_matrix(ex_to<matrix>(v), 0, 1, 1, dim), mu_toggle) * e;
1200                                         else
1201                                                 return v.op(0) * dirac_ONE(ex_to<clifford>(e).get_representation_label()) + indexed(sub_matrix(ex_to<matrix>(v), 1, dim, 0, 1), mu_toggle) * e;
1202                                 } else
1203                                         throw(std::invalid_argument("lst_to_clifford(): dimensions of vector and clifford unit mismatch"));
1204                         } else
1205                                 throw(std::invalid_argument("lst_to_clifford(): first argument should be a vector (nx1 or 1xn matrix)"));
1206                 } else if (v.info(info_flags::list)) {
1207                         if (dim == ex_to<lst>(v).nops())
1208                                 return indexed(matrix(dim, 1, ex_to<lst>(v)), mu_toggle) * e;
1209                         else if (ex_to<lst>(v).nops() - dim == 1)
1210                                 return v.op(0) * dirac_ONE(ex_to<clifford>(e).get_representation_label()) + indexed(sub_matrix(matrix(dim+1, 1, ex_to<lst>(v)), 1, dim, 0, 1), mu_toggle) * e;
1211                         else
1212                                 throw(std::invalid_argument("lst_to_clifford(): list length and dimension of clifford unit mismatch"));
1213                 } else
1214                         throw(std::invalid_argument("lst_to_clifford(): cannot construct from anything but list or vector"));
1215         } else
1216                 throw(std::invalid_argument("lst_to_clifford(): the second argument should be a Clifford unit"));
1217 }
1218
1219 /** Auxiliary structure to define a function for striping one Clifford unit
1220  * from vectors. Used in  clifford_to_lst(). */
1221 static ex get_clifford_comp(const ex & e, const ex & c)
1222 {
1223         pointer_to_map_function_1arg<const ex &> fcn(get_clifford_comp, c);
1224         int ival = ex_to<numeric>(ex_to<idx>(c.op(1)).get_value()).to_int();
1225
1226         if (is_a<add>(e) || e.info(info_flags::list) // || is_a<pseries>(e) || is_a<integral>(e)
1227                 || is_a<matrix>(e))
1228                 return e.map(fcn);
1229         else if (is_a<ncmul>(e) || is_a<mul>(e)) {
1230                 // find a Clifford unit with the same metric, delete it and substitute its index
1231                 size_t ind = e.nops() + 1;
1232                 for (size_t j = 0; j < e.nops(); j++)
1233                         if (is_a<clifford>(e.op(j)) && ex_to<clifford>(c).same_metric(e.op(j)))
1234                                 if (ind > e.nops())
1235                                         ind = j;
1236                                 else
1237                                         throw(std::invalid_argument("get_clifford_comp(): expression is a Clifford multi-vector"));
1238                 if (ind < e.nops()) {
1239                         ex S = 1;
1240                         bool same_value_index, found_dummy;
1241                         same_value_index = ( ex_to<idx>(e.op(ind).op(1)).is_numeric()
1242                                                                  &&  (ival == ex_to<numeric>(ex_to<idx>(e.op(ind).op(1)).get_value()).to_int()) );
1243                         found_dummy = same_value_index;
1244                         for(size_t j=0; j < e.nops(); j++)
1245                                 if (j != ind)
1246                                         if (same_value_index)
1247                                                 S = S * e.op(j);
1248                                         else {
1249                                                 exvector ind_vec = ex_to<indexed>(e.op(j)).get_dummy_indices(ex_to<indexed>(e.op(ind)));
1250                                                 if (ind_vec.size() > 0) {
1251                                                         found_dummy = true;
1252                                                         exvector::const_iterator it = ind_vec.begin(), itend = ind_vec.end();
1253                                                         while (it != itend) {
1254                                                                 ex curridx = *it;
1255                                                                 ex curridx_toggle = is_a<varidx>(curridx)
1256                                                                         ? ex_to<varidx>(curridx).toggle_variance()
1257                                                                         : curridx;
1258                                                                 S = S * e.op(j).subs(lst(curridx == ival,
1259                                                                         curridx_toggle == ival), subs_options::no_pattern);
1260                                                                 ++it;
1261                                                         }
1262                                                 } else
1263                                                         S = S * e.op(j);
1264                                         }
1265                         return (found_dummy ? S : 0);
1266                 } else
1267                         throw(std::invalid_argument("get_clifford_comp(): expression is not a Clifford vector to the given units"));
1268         } else if (e.is_zero())
1269                 return e;
1270         else if (is_a<clifford>(e) && ex_to<clifford>(e).same_metric(c))
1271                 if ( ex_to<idx>(e.op(1)).is_numeric() &&
1272                          (ival != ex_to<numeric>(ex_to<idx>(e.op(1)).get_value()).to_int()) )
1273                         return 0;
1274                 else
1275                         return 1;
1276         else
1277                 throw(std::invalid_argument("get_clifford_comp(): expression is not usable as a Clifford vector"));
1278 }
1279
1280
1281 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic)
1282 {
1283         GINAC_ASSERT(is_a<clifford>(c));
1284         ex mu = c.op(1);
1285         if (! ex_to<idx>(mu).is_dim_numeric())
1286                 throw(std::invalid_argument("clifford_to_lst(): index should have a numeric dimension"));
1287         unsigned int D = ex_to<numeric>(ex_to<idx>(mu).get_dim()).to_int();
1288
1289         if (algebraic) // check if algebraic method is applicable
1290                 for (unsigned int i = 0; i < D; i++)
1291                         if (pow(c.subs(mu == i, subs_options::no_pattern), 2).is_zero()
1292                                 or (not is_a<numeric>(pow(c.subs(mu == i, subs_options::no_pattern), 2))))
1293                                 algebraic = false;
1294         lst V;
1295         ex v0 = remove_dirac_ONE(canonicalize_clifford(e+clifford_prime(e)).normal())/2;
1296         if (not v0.is_zero())
1297                 V.append(v0);
1298         ex e1 = canonicalize_clifford(e - v0 * dirac_ONE(ex_to<clifford>(c).get_representation_label()));
1299         if (algebraic) {
1300                 for (unsigned int i = 0; i < D; i++)
1301                         V.append(remove_dirac_ONE(
1302                                                 simplify_indexed(canonicalize_clifford(e1 * c.subs(mu == i, subs_options::no_pattern) +  c.subs(mu == i, subs_options::no_pattern) * e1))
1303                                                 / (2*pow(c.subs(mu == i, subs_options::no_pattern), 2))));
1304         } else {
1305                 try {
1306                         for (unsigned int i = 0; i < D; i++)
1307                                 V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
1308                 } catch  (std::exception &p) {
1309                         /* Try to expand dummy summations to simplify the expression*/
1310                         e1 = canonicalize_clifford(expand_dummy_sum(e, true));
1311                         V.remove_all();
1312                         v0 = remove_dirac_ONE(canonicalize_clifford(e1+clifford_prime(e1)).normal())/2;
1313                         if (not v0.is_zero()) {
1314                                 V.append(v0);
1315                                 e1 = canonicalize_clifford(e1 - v0 * dirac_ONE(ex_to<clifford>(c).get_representation_label()));
1316                         }
1317                         for (unsigned int i = 0; i < D; i++)
1318                                 V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
1319                 }
1320         }
1321         return V;
1322 }
1323
1324
1325 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c, const ex & d, const ex & v, const ex & G, unsigned char rl)
1326 {
1327         ex x, D, cu;
1328
1329         if (! is_a<matrix>(v) && ! v.info(info_flags::list))
1330                 throw(std::invalid_argument("clifford_moebius_map(): parameter v should be either vector or list"));
1331
1332         if (is_a<clifford>(G)) {
1333                 cu = G;
1334         } else {
1335                 if (is_a<indexed>(G)) {
1336                         D = ex_to<idx>(G.op(1)).get_dim();
1337                         varidx mu((new symbol)->setflag(status_flags::dynallocated), D);
1338                         cu = clifford_unit(mu, G, rl);
1339                 } else if (is_a<matrix>(G)) {
1340                         D = ex_to<matrix>(G).rows();
1341                         idx mu((new symbol)->setflag(status_flags::dynallocated), D);
1342                         cu = clifford_unit(mu, G, rl);
1343                 } else throw(std::invalid_argument("clifford_moebius_map(): metric should be an indexed object, matrix, or a Clifford unit"));
1344
1345         }
1346
1347         x = lst_to_clifford(v, cu);
1348         ex e = clifford_to_lst(simplify_indexed(canonicalize_clifford((a * x + b) * clifford_inverse(c * x + d))), cu, false);
1349         return (is_a<matrix>(v) ? matrix(ex_to<matrix>(v).rows(), ex_to<matrix>(v).cols(), ex_to<lst>(e)) : e);
1350 }
1351
1352 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl)
1353 {
1354         if (is_a<matrix>(M) && (ex_to<matrix>(M).rows() == 2) && (ex_to<matrix>(M).cols() == 2))
1355                 return clifford_moebius_map(M.op(0), M.op(1), M.op(2), M.op(3), v, G, rl);
1356         else
1357                 throw(std::invalid_argument("clifford_moebius_map(): parameter M should be a 2x2 matrix"));
1358 }
1359
1360 } // namespace GiNaC