ginac/clifford.cpp

   1 /** @file clifford.cpp
   2  *
   3  *  Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2006 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 const tinfo_static_t clifford::return_type_tinfo_static[256] = {{}};
  50
  51 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracone, tensor,
  52   print_func<print_dflt>(&diracone::do_print).
  53   print_func<print_latex>(&diracone::do_print_latex))
  54
  55 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(cliffordunit, tensor,
  56   print_func<print_dflt>(&cliffordunit::do_print).
  57   print_func<print_latex>(&cliffordunit::do_print_latex))
  58
  59 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma, cliffordunit,
  60   print_func<print_dflt>(&diracgamma::do_print).
  61   print_func<print_latex>(&diracgamma::do_print_latex))
  62
  63 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma5, tensor,
  64   print_func<print_dflt>(&diracgamma5::do_print).
  65   print_func<print_latex>(&diracgamma5::do_print_latex))
  66
  67 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaL, tensor,
  68   print_func<print_context>(&diracgammaL::do_print).
  69   print_func<print_latex>(&diracgammaL::do_print_latex))
  70
  71 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaR, tensor,
  72   print_func<print_context>(&diracgammaR::do_print).
  73   print_func<print_latex>(&diracgammaR::do_print_latex))
  74
  75 //////////
  76 // default constructors
  77 //////////
  78
  79 clifford::clifford() : representation_label(0), metric(0), commutator_sign(-1)
  80 {
  81         tinfo_key = &clifford::tinfo_static;
  82 }
  83
  84 DEFAULT_CTOR(diracone)
  85 DEFAULT_CTOR(cliffordunit)
  86 DEFAULT_CTOR(diracgamma)
  87 DEFAULT_CTOR(diracgamma5)
  88 DEFAULT_CTOR(diracgammaL)
  89 DEFAULT_CTOR(diracgammaR)
  90
  91 //////////
  92 // other constructors
  93 //////////
  94
  95 /** Construct object without any indices. This constructor is for internal
  96  *  use only. Use the dirac_ONE() function instead.
  97  *  @see dirac_ONE */
  98 clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl), metric(0), commutator_sign(-1)
  99 {
 100         tinfo_key = &clifford::tinfo_static;
 101 }
 102
 103 /** Construct object with one Lorentz index. This constructor is for internal
 104  *  use only. Use the clifford_unit() or dirac_gamma() functions instead.
 105  *  @see clifford_unit
 106  *  @see dirac_gamma */
 107 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)
 108 {
 109         GINAC_ASSERT(is_a<varidx>(mu));
 110         tinfo_key = &clifford::tinfo_static;
 111 }
 112
 113 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)
 114 {
 115         tinfo_key = &clifford::tinfo_static;
 116 }
 117
 118 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)
 119 {
 120         tinfo_key = &clifford::tinfo_static;
 121 }
 122
 123 //////////
 124 // archiving
 125 //////////
 126
 127 clifford::clifford(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
 128 {
 129         unsigned rl;
 130         n.find_unsigned("label", rl);
 131         representation_label = rl;
 132         n.find_ex("metric", metric, sym_lst);
 133         n.find_unsigned("commutator_sign+1", rl);
 134         commutator_sign = rl - 1;
 135 }
 136
 137 void clifford::archive(archive_node & n) const
 138 {
 139         inherited::archive(n);
 140         n.add_unsigned("label", representation_label);
 141         n.add_ex("metric", metric);
 142         n.add_unsigned("commutator_sign+1", commutator_sign+1);
 143 }
 144
 145 DEFAULT_UNARCHIVE(clifford)
 146 DEFAULT_ARCHIVING(diracone)
 147 DEFAULT_ARCHIVING(cliffordunit)
 148 DEFAULT_ARCHIVING(diracgamma)
 149 DEFAULT_ARCHIVING(diracgamma5)
 150 DEFAULT_ARCHIVING(diracgammaL)
 151 DEFAULT_ARCHIVING(diracgammaR)
 152
 153
 154 ex clifford::get_metric(const ex & i, const ex & j, bool symmetrised) const
 155 {
 156         if (is_a<indexed>(metric)) {
 157                 if (symmetrised && !(ex_to<symmetry>(ex_to<indexed>(metric).get_symmetry()).has_symmetry())) {
 158                         if (is_a<matrix>(metric.op(0))) {
 159                                 return indexed((ex_to<matrix>(metric.op(0)).add(ex_to<matrix>(metric.op(0)).transpose())).mul(numeric(1, 2)),
 160                                                symmetric2(), i, j);
 161                         } else {
 162                                 return simplify_indexed(indexed(metric.op(0)*_ex1_2, i, j) + indexed(metric.op(0)*_ex1_2, j, i));
 163                         }
 164                 } else {
 165                         return metric.subs(lst(metric.op(1) == i, metric.op(2) == j), subs_options::no_pattern);
 166                 }
 167         } else {
 168                 exvector indices = metric.get_free_indices();
 169                 if (symmetrised)
 170                         return _ex1_2*simplify_indexed(metric.subs(lst(indices[0] == i, indices[1] == j), subs_options::no_pattern)
 171                                                                         + metric.subs(lst(indices[0] == j, indices[1] == i), subs_options::no_pattern));
 172                 else
 173                         return metric.subs(lst(indices[0] == i, indices[1] == j), subs_options::no_pattern);
 174         }
 175 }
 176
 177 bool clifford::same_metric(const ex & other) const
 178 {
 179         ex metr;
 180         if (is_a<clifford>(other))
 181                 metr = ex_to<clifford>(other).get_metric();
 182         else
 183                 metr = other;
 184
 185         if (is_a<indexed>(metr))
 186                 return metr.op(0).is_equal(get_metric().op(0));
 187         else {
 188                 exvector indices = metr.get_free_indices();
 189                 return  (indices.size() == 2)
 190                         && simplify_indexed(get_metric(indices[0], indices[1])-metr).is_zero();
 191         }
 192 }
 193
 194 //////////
 195 // functions overriding virtual functions from base classes
 196 //////////
 197
 198 ex clifford::op(size_t i) const
 199 {
 200         GINAC_ASSERT(i<nops());
 201         if (nops()-i == 1)
 202                 return representation_label;
 203         else
 204                 return inherited::op(i);
 205 }
 206
 207 ex & clifford::let_op(size_t i)
 208 {
 209         GINAC_ASSERT(i<nops());
 210
 211         static ex rl = numeric(representation_label);
 212         ensure_if_modifiable();
 213         if (nops()-i == 1)
 214                 return rl;
 215         else
 216                 return inherited::let_op(i);
 217 }
 218
 219 ex clifford::subs(const exmap & m, unsigned options) const
 220 {
 221         ex subsed = inherited::subs(m, options);
 222         if(is_a<clifford>(subsed)) {
 223                 ex prevmetric = ex_to<clifford>(subsed).metric;
 224                 ex newmetric = prevmetric.subs(m, options);
 225                 if(!are_ex_trivially_equal(prevmetric, newmetric)) {
 226                         clifford c = ex_to<clifford>(subsed);
 227                         c.metric = newmetric;
 228                         subsed = c;
 229                 }
 230         }
 231         return subsed;
 232 }
 233
 234 int clifford::compare_same_type(const basic & other) const
 235 {
 236         GINAC_ASSERT(is_a<clifford>(other));
 237         const clifford &o = static_cast<const clifford &>(other);
 238
 239         if (representation_label != o.representation_label) {
 240                 // different representation label
 241                 return representation_label < o.representation_label ? -1 : 1;
 242         }
 243
 244         return inherited::compare_same_type(other);
 245 }
 246
 247 bool clifford::match_same_type(const basic & other) const
 248 {
 249         GINAC_ASSERT(is_a<clifford>(other));
 250         const clifford &o = static_cast<const clifford &>(other);
 251
 252         return ((representation_label == o.representation_label) && (commutator_sign == o.get_commutator_sign()) && same_metric(o));
 253 }
 254
 255 static bool is_dirac_slash(const ex & seq0)
 256 {
 257         return !is_a<diracgamma5>(seq0) && !is_a<diracgammaL>(seq0) &&
 258                !is_a<diracgammaR>(seq0) && !is_a<cliffordunit>(seq0) &&
 259                !is_a<diracone>(seq0);
 260 }
 261
 262 void clifford::do_print_dflt(const print_dflt & c, unsigned level) const
 263 {
 264         // dirac_slash() object is printed differently
 265         if (is_dirac_slash(seq[0])) {
 266                 seq[0].print(c, precedence());
 267                 c.s << "\\";
 268         } else
 269                 this->print_dispatch<inherited>(c, level);
 270 }
 271
 272 void clifford::do_print_latex(const print_latex & c, unsigned level) const
 273 {
 274         // dirac_slash() object is printed differently
 275         if (is_dirac_slash(seq[0])) {
 276                 c.s << "{";
 277                 seq[0].print(c, precedence());
 278                 c.s << "\\hspace{-1.0ex}/}";
 279         } else {
 280                 c.s << "\\clifford[" << int(representation_label) << "]";
 281                 this->print_dispatch<inherited>(c, level);
 282         }
 283 }
 284
 285 DEFAULT_COMPARE(diracone)
 286 DEFAULT_COMPARE(cliffordunit)
 287 DEFAULT_COMPARE(diracgamma)
 288 DEFAULT_COMPARE(diracgamma5)
 289 DEFAULT_COMPARE(diracgammaL)
 290 DEFAULT_COMPARE(diracgammaR)
 291
 292 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbf{1}")
 293 DEFAULT_PRINT_LATEX(cliffordunit, "e", "e")
 294 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
 295 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
 296 DEFAULT_PRINT_LATEX(diracgammaL, "gammaL", "{\\gamma_L}")
 297 DEFAULT_PRINT_LATEX(diracgammaR, "gammaR", "{\\gamma_R}")
 298
 299 /** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
 300 static void base_and_index(const ex & c, ex & b, ex & i)
 301 {
 302         GINAC_ASSERT(is_a<clifford>(c));
 303         GINAC_ASSERT(c.nops() == 2+1);
 304
 305         if (is_a<cliffordunit>(c.op(0))) { // proper dirac gamma object or clifford unit
 306                 i = c.op(1);
 307                 b = _ex1;
 308         } else if (is_a<diracgamma5>(c.op(0)) || is_a<diracgammaL>(c.op(0)) || is_a<diracgammaR>(c.op(0))) { // gamma5/L/R
 309                 i = _ex0;
 310                 b = _ex1;
 311         } else { // slash object, generate new dummy index
 312                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(c.op(1)).get_dim());
 313                 b = indexed(c.op(0), ix.toggle_variance());
 314                 i = ix;
 315         }
 316 }
 317
 318 /** Predicate for finding non-clifford objects. */
 319 struct is_not_a_clifford : public std::unary_function<ex, bool> {
 320         bool operator()(const ex & e)
 321         {
 322                 return !is_a<clifford>(e);
 323         }
 324 };
 325
 326 /** Contraction of a gamma matrix with something else. */
 327 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 328 {
 329         GINAC_ASSERT(is_a<clifford>(*self));
 330         GINAC_ASSERT(is_a<indexed>(*other));
 331         GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
 332         unsigned char rl = ex_to<clifford>(*self).get_representation_label();
 333
 334         ex dim = ex_to<idx>(self->op(1)).get_dim();
 335         if (other->nops() > 1)
 336                 dim = minimal_dim(dim, ex_to<idx>(other->op(1)).get_dim());
 337
 338         if (is_a<clifford>(*other)) {
 339
 340                 // Contraction only makes sense if the represenation labels are equal
 341                 if (ex_to<clifford>(*other).get_representation_label() != rl)
 342                         return false;
 343
 344                 size_t num = other - self;
 345
 346                 // gamma~mu gamma.mu = dim ONE
 347                 if (num == 1) {
 348                         *self = dim;
 349                         *other = dirac_ONE(rl);
 350                         return true;
 351
 352                 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
 353                 } else if (num == 2
 354                         && is_a<clifford>(self[1])) {
 355                         *self = 2 - dim;
 356                         *other = _ex1;
 357                         return true;
 358
 359                 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
 360                 } else if (num == 3
 361                         && is_a<clifford>(self[1])
 362                         && is_a<clifford>(self[2])) {
 363                         ex b1, i1, b2, i2;
 364                         base_and_index(self[1], b1, i1);
 365                         base_and_index(self[2], b2, i2);
 366                         *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
 367                         self[1] = _ex1;
 368                         self[2] = _ex1;
 369                         *other = _ex1;
 370                         return true;
 371
 372                 // 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
 373                 } else if (num == 4
 374                         && is_a<clifford>(self[1])
 375                         && is_a<clifford>(self[2])
 376                         && is_a<clifford>(self[3])) {
 377                         *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3];
 378                         self[1] = _ex1;
 379                         self[2] = _ex1;
 380                         self[3] = _ex1;
 381                         *other = _ex1;
 382                         return true;
 383
 384                 // gamma~mu Sodd gamma.mu = -2 Sodd_R
 385                 // (Chisholm identity in 4 dimensions)
 386                 } else if (!((other - self) & 1) && dim.is_equal(4)) {
 387                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
 388                                 return false;
 389
 390                         *self = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(other), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
 391                         std::fill(self + 1, other, _ex1);
 392                         *other = _ex_2;
 393                         return true;
 394
 395                 // gamma~mu Sodd gamma~alpha gamma.mu = 2 gamma~alpha Sodd + 2 Sodd_R gamma~alpha
 396                 // (commutate contracted indices towards each other, then use
 397                 // Chisholm identity in 4 dimensions)
 398                 } else if (((other - self) & 1) && dim.is_equal(4)) {
 399                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
 400                                 return false;
 401
 402                         exvector::iterator next_to_last = other - 1;
 403                         ex S = ncmul(exvector(self + 1, next_to_last), true);
 404                         ex SR = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(next_to_last), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
 405
 406                         *self = (*next_to_last) * S + SR * (*next_to_last);
 407                         std::fill(self + 1, other, _ex1);
 408                         *other = _ex2;
 409                         return true;
 410
 411                 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
 412                 // (commutate contracted indices towards each other, simplify_indexed()
 413                 // will re-expand and re-run the simplification)
 414                 } else {
 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
 421                         *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
 422                         std::fill(self + 1, other + 1, _ex1);
 423                         return true;
 424                 }
 425
 426         } else if (is_a<symbol>(other->op(0)) && other->nops() == 2) {
 427
 428                 // x.mu gamma~mu -> x-slash
 429                 *self = dirac_slash(other->op(0), dim, rl);
 430                 *other = _ex1;
 431                 return true;
 432         }
 433
 434         return false;
 435 }
 436
 437 /** Contraction of a Clifford unit with something else. */
 438 bool cliffordunit::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 439 {
 440         GINAC_ASSERT(is_a<clifford>(*self));
 441         GINAC_ASSERT(is_a<indexed>(*other));
 442         GINAC_ASSERT(is_a<cliffordunit>(self->op(0)));
 443         clifford unit = ex_to<clifford>(*self);
 444         unsigned char rl = unit.get_representation_label();
 445
 446         if (is_a<clifford>(*other)) {
 447                 // Contraction only makes sense if the represenation labels are equal
 448                 // and the metrics are the same
 449                 if ((ex_to<clifford>(*other).get_representation_label() != rl)
 450                     && unit.same_metric(*other))
 451                         return false;
 452
 453                 exvector::iterator before_other = other - 1;
 454                 ex mu = self->op(1);
 455                 ex mu_toggle = other->op(1);
 456                 ex alpha = before_other->op(1);
 457
 458                 // e~mu e.mu = Tr ONE
 459                 if (other - self == 1) {
 460                         *self = unit.get_metric(mu, mu_toggle, true);
 461                         *other = dirac_ONE(rl);
 462                         return true;
 463
 464                 } else if (other - self == 2) {
 465                         if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
 466                                 // e~mu e~alpha e.mu = 2*e~mu B(alpha, mu.toggle_variance())-Tr(B) e~alpha
 467                                 *self = 2 * (*self) * unit.get_metric(alpha, mu_toggle, true) - unit.get_metric(mu, mu_toggle, true) * (*before_other);
 468                                 *before_other = _ex1;
 469                                 *other = _ex1;
 470                                 return true;
 471
 472                         } else {
 473                                 // e~mu S e.mu = Tr S ONE
 474                                 *self = unit.get_metric(mu, mu_toggle, true);
 475                                 *other = dirac_ONE(rl);
 476                                 return true;
 477                         }
 478                 } else {
 479                 // e~mu S e~alpha e.mu = 2 e~mu S B(alpha, mu.toggle_variance()) - e~mu S e.mu e~alpha
 480                 // (commutate contracted indices towards each other, simplify_indexed()
 481                 // will re-expand and re-run the simplification)
 482                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other) {
 483                                 return false;
 484                         }
 485
 486                         ex S = ncmul(exvector(self + 1, before_other), true);
 487
 488                         if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
 489                                 *self = 2 * (*self) * S * unit.get_metric(alpha, mu_toggle, true) - (*self) * S * (*other) * (*before_other);
 490                         } else {
 491                                 // simply commutes
 492                                 *self = (*self) * S * (*other) * (*before_other);
 493                         }
 494
 495                         std::fill(self + 1, other + 1, _ex1);
 496                         return true;
 497                 }
 498         }
 499         return false;
 500 }
 501
 502 /** Perform automatic simplification on noncommutative product of clifford
 503  *  objects. This removes superfluous ONEs, permutes gamma5/L/R's to the front
 504  *  and removes squares of gamma objects. */
 505 ex clifford::eval_ncmul(const exvector & v) const
 506 {
 507         exvector s;
 508         s.reserve(v.size());
 509
 510         // Remove superfluous ONEs
 511         exvector::const_iterator cit = v.begin(), citend = v.end();
 512         while (cit != citend) {
 513                 if (!is_a<clifford>(*cit) || !is_a<diracone>(cit->op(0)))
 514                         s.push_back(*cit);
 515                 cit++;
 516         }
 517
 518         bool something_changed = false;
 519         int sign = 1;
 520
 521         // Anticommutate gamma5/L/R's to the front
 522         if (s.size() >= 2) {
 523                 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
 524                 while (true) {
 525                         exvector::iterator it = next_to_last;
 526                         while (true) {
 527                                 exvector::iterator it2 = it + 1;
 528                                 if (is_a<clifford>(*it) && is_a<clifford>(*it2)) {
 529                                         ex e1 = it->op(0), e2 = it2->op(0);
 530
 531                                         if (is_a<diracgamma5>(e2)) {
 532
 533                                                 if (is_a<diracgammaL>(e1) || is_a<diracgammaR>(e1)) {
 534
 535                                                         // gammaL/R gamma5 -> gamma5 gammaL/R
 536                                                         it->swap(*it2);
 537                                                         something_changed = true;
 538
 539                                                 } else if (!is_a<diracgamma5>(e1)) {
 540
 541                                                         // gamma5 gamma5 -> gamma5 gamma5 (do nothing)
 542                                                         // x gamma5 -> -gamma5 x
 543                                                         it->swap(*it2);
 544                                                         sign = -sign;
 545                                                         something_changed = true;
 546                                                 }
 547
 548                                         } else if (is_a<diracgammaL>(e2)) {
 549
 550                                                 if (is_a<diracgammaR>(e1)) {
 551
 552                                                         // gammaR gammaL -> 0
 553                                                         return _ex0;
 554
 555                                                 } else if (!is_a<diracgammaL>(e1) && !is_a<diracgamma5>(e1)) {
 556
 557                                                         // gammaL gammaL -> gammaL gammaL (do nothing)
 558                                                         // gamma5 gammaL -> gamma5 gammaL (do nothing)
 559                                                         // x gammaL -> gammaR x
 560                                                         it->swap(*it2);
 561                                                         *it = clifford(diracgammaR(), ex_to<clifford>(*it).get_representation_label());
 562                                                         something_changed = true;
 563                                                 }
 564
 565                                         } else if (is_a<diracgammaR>(e2)) {
 566
 567                                                 if (is_a<diracgammaL>(e1)) {
 568
 569                                                         // gammaL gammaR -> 0
 570                                                         return _ex0;
 571
 572                                                 } else if (!is_a<diracgammaR>(e1) && !is_a<diracgamma5>(e1)) {
 573
 574                                                         // gammaR gammaR -> gammaR gammaR (do nothing)
 575                                                         // gamma5 gammaR -> gamma5 gammaR (do nothing)
 576                                                         // x gammaR -> gammaL x
 577                                                         it->swap(*it2);
 578                                                         *it = clifford(diracgammaL(), ex_to<clifford>(*it).get_representation_label());
 579                                                         something_changed = true;
 580                                                 }
 581                                         }
 582                                 }
 583                                 if (it == first)
 584                                         break;
 585                                 --it;
 586                         }
 587                         if (next_to_last == first)
 588                                 break;
 589                         --next_to_last;
 590                 }
 591         }
 592
 593         // Remove equal adjacent gammas
 594         if (s.size() >= 2) {
 595                 exvector::iterator it, itend = s.end() - 1;
 596                 for (it = s.begin(); it != itend; ++it) {
 597                         ex & a = it[0];
 598                         ex & b = it[1];
 599                         if (!is_a<clifford>(a) || !is_a<clifford>(b))
 600                                 continue;
 601
 602                         const ex & ag = a.op(0);
 603                         const ex & bg = b.op(0);
 604                         bool a_is_cliffordunit = is_a<cliffordunit>(ag);
 605                         bool b_is_cliffordunit =  is_a<cliffordunit>(bg);
 606
 607                         if (a_is_cliffordunit && b_is_cliffordunit && ex_to<clifford>(a).same_metric(b)
 608                                 && (ex_to<clifford>(a).get_commutator_sign() == -1)) {
 609                                 // This is done only for Clifford algebras
 610
 611                                 const ex & ia = a.op(1);
 612                                 const ex & ib = b.op(1);
 613                                 if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
 614                                         a = ex_to<clifford>(a).get_metric(ia, ib, true);
 615                                         b = dirac_ONE(representation_label);
 616                                         something_changed = true;
 617                                 }
 618
 619                         } else if ((is_a<diracgamma5>(ag) && is_a<diracgamma5>(bg))) {
 620
 621                                 // Remove squares of gamma5
 622                                 a = dirac_ONE(representation_label);
 623                                 b = dirac_ONE(representation_label);
 624                                 something_changed = true;
 625
 626                         } else if ((is_a<diracgammaL>(ag) && is_a<diracgammaL>(bg))
 627                                 || (is_a<diracgammaR>(ag) && is_a<diracgammaR>(bg))) {
 628
 629                                 // Remove squares of gammaL/R
 630                                 b = dirac_ONE(representation_label);
 631                                 something_changed = true;
 632
 633                         } else if (is_a<diracgammaL>(ag) && is_a<diracgammaR>(bg)) {
 634
 635                                 // gammaL and gammaR are orthogonal
 636                                 return _ex0;
 637
 638                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaL>(bg)) {
 639
 640                                 // gamma5 gammaL -> -gammaL
 641                                 a = dirac_ONE(representation_label);
 642                                 sign = -sign;
 643                                 something_changed = true;
 644
 645                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaR>(bg)) {
 646
 647                                 // gamma5 gammaR -> gammaR
 648                                 a = dirac_ONE(representation_label);
 649                                 something_changed = true;
 650
 651                         } else if (!a_is_cliffordunit && !b_is_cliffordunit && ag.is_equal(bg)) {
 652
 653                                 // a\ a\ -> a^2
 654                                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).minimal_dim(ex_to<idx>(b.op(1))));
 655
 656                                 a = indexed(ag, ix) * indexed(ag, ix.toggle_variance());
 657                                 b = dirac_ONE(representation_label);
 658                                 something_changed = true;
 659                         }
 660                 }
 661         }
 662
 663         if (s.empty())
 664                 return dirac_ONE(representation_label) * sign;
 665         if (something_changed)
 666                 return reeval_ncmul(s) * sign;
 667         else
 668                 return hold_ncmul(s) * sign;
 669 }
 670
 671 ex clifford::thiscontainer(const exvector & v) const
 672 {
 673         return clifford(representation_label, metric, commutator_sign, v);
 674 }
 675
 676 ex clifford::thiscontainer(std::auto_ptr<exvector> vp) const
 677 {
 678         return clifford(representation_label, metric, commutator_sign, vp);
 679 }
 680
 681 ex diracgamma5::conjugate() const
 682 {
 683         return _ex_1 * (*this);
 684 }
 685
 686 ex diracgammaL::conjugate() const
 687 {
 688         return (new diracgammaR)->setflag(status_flags::dynallocated);
 689 }
 690
 691 ex diracgammaR::conjugate() const
 692 {
 693         return (new diracgammaL)->setflag(status_flags::dynallocated);
 694 }
 695
 696 //////////
 697 // global functions
 698 //////////
 699
 700 ex dirac_ONE(unsigned char rl)
 701 {
 702         static ex ONE = (new diracone)->setflag(status_flags::dynallocated);
 703         return clifford(ONE, rl);
 704 }
 705
 706 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl)
 707 {
 708         static ex unit = (new cliffordunit)->setflag(status_flags::dynallocated);
 709
 710         if (!is_a<idx>(mu))
 711                 throw(std::invalid_argument("clifford_unit(): index of Clifford unit must be of type idx or varidx"));
 712
 713         exvector indices = metr.get_free_indices();
 714
 715         if (indices.size() == 2) {
 716                 return clifford(unit, mu, metr, rl);
 717         } else if (is_a<matrix>(metr)) {
 718                 matrix M = ex_to<matrix>(metr);
 719                 unsigned n = M.rows();
 720                 bool symmetric = true;
 721
 722                 static idx xi((new symbol)->setflag(status_flags::dynallocated), n),
 723                         chi((new symbol)->setflag(status_flags::dynallocated), n);
 724                 if ((n ==  M.cols()) && (n == ex_to<idx>(mu).get_dim())) {
 725                         for (unsigned i = 0; i < n; i++) {
 726                                 for (unsigned j = i+1; j < n; j++) {
 727                                         if (M(i, j) != M(j, i)) {
 728                                                 symmetric = false;
 729                                         }
 730                                 }
 731                         }
 732                         return clifford(unit, mu, indexed(metr, symmetric?symmetric2():not_symmetric(), xi, chi), rl);
 733                 } else {
 734                         throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be a square matrix with the same dimensions as index"));
 735                 }
 736         } else if (indices.size() == 0) { // a tensor or other expression without indices
 737                 static varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim()),
 738                         chi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim());
 739                 return clifford(unit, mu, indexed(metr, xi, chi), rl);
 740         }  else
 741                 throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be of type tensor, matrix or an expression with two free indices"));
 742 }
 743
 744 ex dirac_gamma(const ex & mu, unsigned char rl)
 745 {
 746         static ex gamma = (new diracgamma)->setflag(status_flags::dynallocated);
 747
 748         if (!is_a<varidx>(mu))
 749                 throw(std::invalid_argument("dirac_gamma(): index of Dirac gamma must be of type varidx"));
 750
 751         static varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim()),
 752                 chi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim());
 753         return clifford(gamma, mu, indexed((new minkmetric)->setflag(status_flags::dynallocated), symmetric2(), xi, chi), rl);
 754 }
 755
 756 ex dirac_gamma5(unsigned char rl)
 757 {
 758         static ex gamma5 = (new diracgamma5)->setflag(status_flags::dynallocated);
 759         return clifford(gamma5, rl);
 760 }
 761
 762 ex dirac_gammaL(unsigned char rl)
 763 {
 764         static ex gammaL = (new diracgammaL)->setflag(status_flags::dynallocated);
 765         return clifford(gammaL, rl);
 766 }
 767
 768 ex dirac_gammaR(unsigned char rl)
 769 {
 770         static ex gammaR = (new diracgammaR)->setflag(status_flags::dynallocated);
 771         return clifford(gammaR, rl);
 772 }
 773
 774 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
 775 {
 776         // Slashed vectors are actually stored as a clifford object with the
 777         // vector as its base expression and a (dummy) index that just serves
 778         // for storing the space dimensionality
 779
 780         static varidx xi((new symbol)->setflag(status_flags::dynallocated), dim),
 781                 chi((new symbol)->setflag(status_flags::dynallocated), dim);
 782    return clifford(e, varidx(0, dim), indexed((new minkmetric)->setflag(status_flags::dynallocated), symmetric2(), xi, chi), rl);
 783 }
 784
 785 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 786  *  is that of a clifford object (with an arbitrary representation label). */
 787 bool is_clifford_tinfo(tinfo_t ti)
 788 {
 789         p_int start_loc=(p_int)&clifford::return_type_tinfo_static;
 790         return (p_int)ti>=start_loc && (p_int)ti<start_loc+256;
 791 }
 792
 793 /** Extract representation label from tinfo key (as returned by
 794  *  return_type_tinfo()). */
 795 static unsigned char get_representation_label(tinfo_t ti)
 796 {
 797         return (unsigned char)((p_int)ti-(p_int)&clifford::return_type_tinfo_static);
 798 }
 799
 800 /** Take trace of a string of an even number of Dirac gammas given a vector
 801  *  of indices. */
 802 static ex trace_string(exvector::const_iterator ix, size_t num)
 803 {
 804         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 805         if (num == 2)
 806                 return lorentz_g(ix[0], ix[1]);
 807
 808         // 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 )
 809         else if (num == 4)
 810                 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
 811                      + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
 812                      - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
 813
 814         // Traces of 6 or more gammas are computed recursively:
 815         // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
 816         //   + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
 817         //   - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
 818         //   + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
 819         //   - ...
 820         //   + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
 821         exvector v(num - 2);
 822         int sign = 1;
 823         ex result;
 824         for (size_t i=1; i<num; i++) {
 825                 for (size_t n=1, j=0; n<num; n++) {
 826                         if (n == i)
 827                                 continue;
 828                         v[j++] = ix[n];
 829                 }
 830                 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
 831                 sign = -sign;
 832         }
 833         return result;
 834 }
 835
 836 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE)
 837 {
 838         if (is_a<clifford>(e)) {
 839
 840                 unsigned char rl = ex_to<clifford>(e).get_representation_label();
 841
 842                 // Are we taking the trace over this object's representation label?
 843                 if (rls.find(rl) == rls.end())
 844                         return e;
 845
 846                 // Yes, all elements are traceless, except for dirac_ONE and dirac_L/R
 847                 const ex & g = e.op(0);
 848                 if (is_a<diracone>(g))
 849                         return trONE;
 850                 else if (is_a<diracgammaL>(g) || is_a<diracgammaR>(g))
 851                         return trONE/2;
 852                 else
 853                         return _ex0;
 854
 855         } else if (is_exactly_a<mul>(e)) {
 856
 857                 // Trace of product: pull out non-clifford factors
 858                 ex prod = _ex1;
 859                 for (size_t i=0; i<e.nops(); i++) {
 860                         const ex &o = e.op(i);
 861                         if (is_clifford_tinfo(o.return_type_tinfo()))
 862                                 prod *= dirac_trace(o, rls, trONE);
 863                         else
 864                                 prod *= o;
 865                 }
 866                 return prod;
 867
 868         } else if (is_exactly_a<ncmul>(e)) {
 869
 870                 unsigned char rl = get_representation_label(e.return_type_tinfo());
 871
 872                 // Are we taking the trace over this string's representation label?
 873                 if (rls.find(rl) == rls.end())
 874                         return e;
 875
 876                 // Substitute gammaL/R and expand product, if necessary
 877                 ex e_expanded = e.subs(lst(
 878                         dirac_gammaL(rl) == (dirac_ONE(rl)-dirac_gamma5(rl))/2,
 879                         dirac_gammaR(rl) == (dirac_ONE(rl)+dirac_gamma5(rl))/2
 880                 ), subs_options::no_pattern).expand();
 881                 if (!is_a<ncmul>(e_expanded))
 882                         return dirac_trace(e_expanded, rls, trONE);
 883
 884                 // gamma5 gets moved to the front so this check is enough
 885                 bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
 886                 size_t num = e.nops();
 887
 888                 if (has_gamma5) {
 889
 890                         // Trace of gamma5 * odd number of gammas and trace of
 891                         // gamma5 * gamma.mu * gamma.nu are zero
 892                         if ((num & 1) == 0 || num == 3)
 893                                 return _ex0;
 894
 895                         // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
 896                         // (the epsilon is always 4-dimensional)
 897                         if (num == 5) {
 898                                 ex b1, i1, b2, i2, b3, i3, b4, i4;
 899                                 base_and_index(e.op(1), b1, i1);
 900                                 base_and_index(e.op(2), b2, i2);
 901                                 base_and_index(e.op(3), b3, i3);
 902                                 base_and_index(e.op(4), b4, i4);
 903                                 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();
 904                         }
 905
 906                         // Tr gamma5 S_2k =
 907                         //   I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
 908                         // (the epsilon is always 4-dimensional)
 909                         exvector ix(num-1), bv(num-1);
 910                         for (size_t i=1; i<num; i++)
 911                                 base_and_index(e.op(i), bv[i-1], ix[i-1]);
 912                         num--;
 913                         int *iv = new int[num];
 914                         ex result;
 915                         for (size_t i=0; i<num-3; i++) {
 916                                 ex idx1 = ix[i];
 917                                 for (size_t j=i+1; j<num-2; j++) {
 918                                         ex idx2 = ix[j];
 919                                         for (size_t k=j+1; k<num-1; k++) {
 920                                                 ex idx3 = ix[k];
 921                                                 for (size_t l=k+1; l<num; l++) {
 922                                                         ex idx4 = ix[l];
 923                                                         iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
 924                                                         exvector v;
 925                                                         v.reserve(num - 4);
 926                                                         for (size_t n=0, t=4; n<num; n++) {
 927                                                                 if (n == i || n == j || n == k || n == l)
 928                                                                         continue;
 929                                                                 iv[t++] = n;
 930                                                                 v.push_back(ix[n]);
 931                                                         }
 932                                                         int sign = permutation_sign(iv, iv + num);
 933                                                         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))
 934                                                                 * trace_string(v.begin(), num - 4);
 935                                                 }
 936                                         }
 937                                 }
 938                         }
 939                         delete[] iv;
 940                         return trONE * I * result * mul(bv);
 941
 942                 } else { // no gamma5
 943
 944                         // Trace of odd number of gammas is zero
 945                         if ((num & 1) == 1)
 946                                 return _ex0;
 947
 948                         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 949                         if (num == 2) {
 950                                 ex b1, i1, b2, i2;
 951                                 base_and_index(e.op(0), b1, i1);
 952                                 base_and_index(e.op(1), b2, i2);
 953                                 return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
 954                         }
 955
 956                         exvector iv(num), bv(num);
 957                         for (size_t i=0; i<num; i++)
 958                                 base_and_index(e.op(i), bv[i], iv[i]);
 959
 960                         return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
 961                 }
 962
 963         } else if (e.nops() > 0) {
 964
 965                 // Trace maps to all other container classes (this includes sums)
 966                 pointer_to_map_function_2args<const std::set<unsigned char> &, const ex &> fcn(dirac_trace, rls, trONE);
 967                 return e.map(fcn);
 968
 969         } else
 970                 return _ex0;
 971 }
 972
 973 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE)
 974 {
 975         // Convert list to set
 976         std::set<unsigned char> rls;
 977         for (lst::const_iterator i = rll.begin(); i != rll.end(); ++i) {
 978                 if (i->info(info_flags::nonnegint))
 979                         rls.insert(ex_to<numeric>(*i).to_int());
 980         }
 981
 982         return dirac_trace(e, rls, trONE);
 983 }
 984
 985 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
 986 {
 987         // Convert label to set
 988         std::set<unsigned char> rls;
 989         rls.insert(rl);
 990
 991         return dirac_trace(e, rls, trONE);
 992 }
 993
 994
 995 ex canonicalize_clifford(const ex & e_)
 996 {
 997         pointer_to_map_function fcn(canonicalize_clifford);
 998
 999         if (is_a<matrix>(e_)    // || is_a<pseries>(e) || is_a<integral>(e)
1000                 || e_.info(info_flags::list)) {
1001                 return e_.map(fcn);
1002         } else {
1003                 ex e=simplify_indexed(e_);
1004                 // Scan for any ncmul objects
1005                 exmap srl;
1006                 ex aux = e.to_rational(srl);
1007                 for (exmap::iterator i = srl.begin(); i != srl.end(); ++i) {
1008
1009                         ex lhs = i->first;
1010                         ex rhs = i->second;
1011
1012                         if (is_exactly_a<ncmul>(rhs)
1013                                         && rhs.return_type() == return_types::noncommutative
1014                                         && is_clifford_tinfo(rhs.return_type_tinfo())) {
1015
1016                                 // Expand product, if necessary
1017                                 ex rhs_expanded = rhs.expand();
1018                                 if (!is_a<ncmul>(rhs_expanded)) {
1019                                         i->second = canonicalize_clifford(rhs_expanded);
1020                                         continue;
1021
1022                                 } else if (!is_a<clifford>(rhs.op(0)))
1023                                         continue;
1024
1025                                 exvector v;
1026                                 v.reserve(rhs.nops());
1027                                 for (size_t j=0; j<rhs.nops(); j++)
1028                                         v.push_back(rhs.op(j));
1029
1030                                 // Stupid recursive bubble sort because we only want to swap adjacent gammas
1031                                 exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
1032                                 if (is_a<diracgamma5>(it->op(0)) || is_a<diracgammaL>(it->op(0)) || is_a<diracgammaR>(it->op(0)))
1033                                         ++it;
1034
1035                                 while (it != next_to_last) {
1036                                         if (it[0].compare(it[1]) > 0) {
1037
1038                                                 ex save0 = it[0], save1 = it[1];
1039                                                 ex b1, i1, b2, i2;
1040                                                 base_and_index(it[0], b1, i1);
1041                                                 base_and_index(it[1], b2, i2);
1042                                                 // for Clifford algebras (commutator_sign == -1) metric should be symmetrised
1043                                                 it[0] = (ex_to<clifford>(save0).get_metric(i1, i2, ex_to<clifford>(save0).get_commutator_sign() == -1) * b1 * b2).simplify_indexed();
1044                                                 it[1] = v.size() ? _ex2 * dirac_ONE(ex_to<clifford>(save0).get_representation_label()) : _ex2;
1045                                                 ex sum = ncmul(v);
1046                                                 it[0] = save1;
1047                                                 it[1] = save0;
1048                                                 sum += ex_to<clifford>(save0).get_commutator_sign() * ncmul(v, true);
1049                                                 i->second = canonicalize_clifford(sum);
1050                                                 goto next_sym;
1051                                         }
1052                                         ++it;
1053                                 }
1054 next_sym:       ;
1055                         }
1056                 }
1057                 return aux.subs(srl, subs_options::no_pattern).simplify_indexed();
1058         }
1059 }
1060
1061 ex clifford_prime(const ex & e)
1062 {
1063         pointer_to_map_function fcn(clifford_prime);
1064         if (is_a<clifford>(e) && is_a<cliffordunit>(e.op(0))) {
1065                 return -e;
1066         } else if (is_a<add>(e) || is_a<ncmul>(e) || is_a<mul>(e) //|| is_a<pseries>(e) || is_a<integral>(e)
1067                            || is_a<matrix>(e) || e.info(info_flags::list)) {
1068                 return e.map(fcn);
1069         } else if (is_a<power>(e)) {
1070                 return pow(clifford_prime(e.op(0)), e.op(1));
1071         } else
1072                 return e;
1073 }
1074
1075 ex remove_dirac_ONE(const ex & e, unsigned char rl, unsigned options)
1076 {
1077         pointer_to_map_function_2args<unsigned char, unsigned> fcn(remove_dirac_ONE, rl, options | 1);
1078         bool need_reevaluation = false;
1079         ex e1 = e;
1080         if (! (options & 1) )  { // is not a child
1081                 if (options & 2)
1082                         e1 = expand_dummy_sum(e, true);
1083                 e1 = canonicalize_clifford(e1);
1084         }
1085
1086         if (is_a<clifford>(e1) && ex_to<clifford>(e1).get_representation_label() >= rl) {
1087                 if (is_a<diracone>(e1.op(0)))
1088                         return 1;
1089                 else
1090                         throw(std::invalid_argument("remove_dirac_ONE(): expression is a non-scalar Clifford number!"));
1091         } else if (is_a<add>(e1) || is_a<ncmul>(e1) || is_a<mul>(e1)
1092                            || is_a<matrix>(e1) || e1.info(info_flags::list)) {
1093                 if (options & 3) // is a child or was already expanded
1094                         return e1.map(fcn);
1095                 else
1096                         try {
1097                                 return e1.map(fcn);
1098                         } catch (std::exception &p) {
1099                                 need_reevaluation = true;
1100                         }
1101         } else if (is_a<power>(e1)) {
1102                 if (options & 3) // is a child or was already expanded
1103                         return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
1104                 else
1105                         try {
1106                                 return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
1107                         } catch (std::exception &p) {
1108                                 need_reevaluation = true;
1109                         }
1110         }
1111         if (need_reevaluation)
1112                 return remove_dirac_ONE(e, rl, options | 2);
1113         return e1;
1114 }
1115
1116 char clifford_max_label(const ex & e, bool ignore_ONE)
1117 {
1118         if (is_a<clifford>(e))
1119                 if (ignore_ONE && is_a<diracone>(e.op(0)))
1120                         return -1;
1121                 else
1122                         return ex_to<clifford>(e).get_representation_label();
1123         else {
1124                 char rl = -1;
1125                 for (size_t i=0; i < e.nops(); i++)
1126                         rl = (rl > clifford_max_label(e.op(i), ignore_ONE)) ? rl : clifford_max_label(e.op(i), ignore_ONE);
1127                 return rl;
1128         }
1129 }
1130
1131 ex clifford_norm(const ex & e)
1132 {
1133         return sqrt(remove_dirac_ONE(e * clifford_bar(e)));
1134 }
1135
1136 ex clifford_inverse(const ex & e)
1137 {
1138         ex norm = clifford_norm(e);
1139         if (!norm.is_zero())
1140                 return clifford_bar(e) / pow(norm, 2);
1141         else
1142                 throw(std::invalid_argument("clifford_inverse(): cannot find inverse of Clifford number with zero norm!"));
1143 }
1144
1145 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl)
1146 {
1147         if (!ex_to<idx>(mu).is_dim_numeric())
1148                 throw(std::invalid_argument("lst_to_clifford(): Index should have a numeric dimension"));
1149         ex e = clifford_unit(mu, metr, rl);
1150         return lst_to_clifford(v, e);
1151 }
1152
1153 ex lst_to_clifford(const ex & v, const ex & e) {
1154         unsigned min, max;
1155
1156         if (is_a<clifford>(e)) {
1157                 ex mu = e.op(1);
1158                 ex mu_toggle
1159                         = is_a<varidx>(mu) ? ex_to<varidx>(mu).toggle_variance() : mu;
1160                 unsigned dim = (ex_to<numeric>(ex_to<idx>(mu).get_dim())).to_int();
1161
1162                 if (is_a<matrix>(v)) {
1163                         if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows()) {
1164                                 min = ex_to<matrix>(v).rows();
1165                                 max = ex_to<matrix>(v).cols();
1166                         } else {
1167                                 min = ex_to<matrix>(v).cols();
1168                                 max = ex_to<matrix>(v).rows();
1169                         }
1170                         if (min == 1) {
1171                                 if (dim == max)
1172                                         return indexed(v, mu_toggle) * e;
1173                                 else
1174                                         throw(std::invalid_argument("lst_to_clifford(): dimensions of vector and clifford unit mismatch"));
1175                         } else
1176                                 throw(std::invalid_argument("lst_to_clifford(): first argument should be a vector (nx1 or 1xn matrix)"));
1177                 } else if (v.info(info_flags::list)) {
1178                         if (dim == ex_to<lst>(v).nops())
1179                                 return indexed(matrix(dim, 1, ex_to<lst>(v)), mu_toggle) * e;
1180                         else
1181                                 throw(std::invalid_argument("lst_to_clifford(): list length and dimension of clifford unit mismatch"));
1182                 } else
1183                         throw(std::invalid_argument("lst_to_clifford(): cannot construct from anything but list or vector"));
1184         } else
1185                 throw(std::invalid_argument("lst_to_clifford(): the second argument should be a Clifford unit"));
1186 }
1187
1188 /** Auxiliary structure to define a function for striping one Clifford unit
1189  * from vectors. Used in  clifford_to_lst(). */
1190 static ex get_clifford_comp(const ex & e, const ex & c)
1191 {
1192         pointer_to_map_function_1arg<const ex &> fcn(get_clifford_comp, c);
1193         int ival = ex_to<numeric>(ex_to<idx>(c.op(1)).get_value()).to_int();
1194
1195         if (is_a<add>(e) || e.info(info_flags::list) // || is_a<pseries>(e) || is_a<integral>(e)
1196                 || is_a<matrix>(e))
1197                 return e.map(fcn);
1198         else if (is_a<ncmul>(e) || is_a<mul>(e)) {
1199                 // find a Clifford unit with the same metric, delete it and substitute its index
1200                 size_t ind = e.nops() + 1;
1201                 for (size_t j = 0; j < e.nops(); j++)
1202                         if (is_a<clifford>(e.op(j)) && ex_to<clifford>(c).same_metric(e.op(j)))
1203                                 if (ind > e.nops())
1204                                         ind = j;
1205                                 else
1206                                         throw(std::invalid_argument("get_clifford_comp(): expression is a Clifford multi-vector"));
1207                 if (ind < e.nops()) {
1208                         ex S = 1;
1209                         bool same_value_index, found_dummy;
1210                         same_value_index = ( ex_to<idx>(e.op(ind).op(1)).is_numeric()
1211                                                                  &&  (ival == ex_to<numeric>(ex_to<idx>(e.op(ind).op(1)).get_value()).to_int()) );
1212                         found_dummy = same_value_index;
1213                         for(size_t j=0; j < e.nops(); j++)
1214                                 if (j != ind)
1215                                         if (same_value_index)
1216                                                 S = S * e.op(j);
1217                                         else {
1218                                                 exvector ind_vec = ex_to<indexed>(e.op(j)).get_dummy_indices(ex_to<indexed>(e.op(ind)));
1219                                                 if (ind_vec.size() > 0) {
1220                                                         found_dummy = true;
1221                                                         exvector::const_iterator it = ind_vec.begin(), itend = ind_vec.end();
1222                                                         while (it != itend) {
1223                                                                 ex curridx = *it;
1224                                                                 ex curridx_toggle = is_a<varidx>(curridx)
1225                                                                         ? ex_to<varidx>(curridx).toggle_variance()
1226                                                                         : curridx;
1227                                                                 S = S * e.op(j).subs(lst(curridx == ival,
1228                                                                         curridx_toggle == ival), subs_options::no_pattern);
1229                                                                 ++it;
1230                                                         }
1231                                                 } else
1232                                                         S = S * e.op(j);
1233                                         }
1234                         return (found_dummy ? S : 0);
1235                 } else
1236                         throw(std::invalid_argument("get_clifford_comp(): expression is not a Clifford vector to the given units"));
1237         } else if (e.is_zero())
1238                 return e;
1239         else if (is_a<clifford>(e) && ex_to<clifford>(e).same_metric(c))
1240                 if ( ex_to<idx>(e.op(1)).is_numeric() &&
1241                          (ival != ex_to<numeric>(ex_to<idx>(e.op(1)).get_value()).to_int()) )
1242                         return 0;
1243                 else
1244                         return 1;
1245         else
1246                 throw(std::invalid_argument("get_clifford_comp(): expression is not usable as a Clifford vector"));
1247 }
1248
1249
1250 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic)
1251 {
1252         GINAC_ASSERT(is_a<clifford>(c));
1253         ex mu = c.op(1);
1254         if (! ex_to<idx>(mu).is_dim_numeric())
1255                 throw(std::invalid_argument("clifford_to_lst(): index should have a numeric dimension"));
1256         unsigned int D = ex_to<numeric>(ex_to<idx>(mu).get_dim()).to_int();
1257
1258         if (algebraic) // check if algebraic method is applicable
1259                 for (unsigned int i = 0; i < D; i++)
1260                         if (pow(c.subs(mu == i, subs_options::no_pattern), 2).is_zero()
1261                                 or (not is_a<numeric>(pow(c.subs(mu == i, subs_options::no_pattern), 2))))
1262                                 algebraic = false;
1263         lst V;
1264         if (algebraic) {
1265                 for (unsigned int i = 0; i < D; i++)
1266                         V.append(remove_dirac_ONE(
1267                                                 simplify_indexed(canonicalize_clifford(e * c.subs(mu == i, subs_options::no_pattern) +  c.subs(mu == i, subs_options::no_pattern) * e))
1268                                                 / (2*pow(c.subs(mu == i, subs_options::no_pattern), 2))));
1269         } else {
1270                 ex e1 = canonicalize_clifford(e);
1271                 try {
1272                         for (unsigned int i = 0; i < D; i++)
1273                                 V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
1274                 } catch  (std::exception &p) {
1275                         /* Try to expand dummy summations to simplify the expression*/
1276                         e1 = canonicalize_clifford(expand_dummy_sum(e1, true));
1277                         for (unsigned int i = 0; i < D; i++)
1278                                 V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
1279                 }
1280         }
1281         return V;
1282 }
1283
1284
1285 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)
1286 {
1287         ex x, D, cu;
1288
1289         if (! is_a<matrix>(v) && ! v.info(info_flags::list))
1290                 throw(std::invalid_argument("clifford_moebius_map(): parameter v should be either vector or list"));
1291
1292         if (is_a<clifford>(G)) {
1293                 cu = G;
1294         } else {
1295                 if (is_a<indexed>(G)) {
1296                         D = ex_to<idx>(G.op(1)).get_dim();
1297                         varidx mu((new symbol)->setflag(status_flags::dynallocated), D);
1298                         cu = clifford_unit(mu, G, rl);
1299                 } else if (is_a<matrix>(G)) {
1300                         D = ex_to<matrix>(G).rows();
1301                         idx mu((new symbol)->setflag(status_flags::dynallocated), D);
1302                         cu = clifford_unit(mu, G, rl);
1303                 } else throw(std::invalid_argument("clifford_moebius_map(): metric should be an indexed object, matrix, or a Clifford unit"));
1304
1305         }
1306
1307         x = lst_to_clifford(v, cu);
1308         ex e = clifford_to_lst(simplify_indexed(canonicalize_clifford((a * x + b) * clifford_inverse(c * x + d))), cu, false);
1309         return (is_a<matrix>(v) ? matrix(ex_to<matrix>(v).rows(), ex_to<matrix>(v).cols(), ex_to<lst>(e)) : e);
1310 }
1311
1312 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl)
1313 {
1314         if (is_a<matrix>(M))
1315                 return clifford_moebius_map(ex_to<matrix>(M)(0,0), ex_to<matrix>(M)(0,1),
1316                                             ex_to<matrix>(M)(1,0), ex_to<matrix>(M)(1,1), v, G, rl);
1317         else
1318                 throw(std::invalid_argument("clifford_moebius_map(): parameter M should be a matrix"));
1319 }
1320
1321 } // namespace GiNaC