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 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 { // We do not print representation label if it is 0
 269                 if (representation_label == 0) {
 270                         this->print_dispatch<inherited>(c, level);
 271                 } else { // otherwise we put it before indices in square brackets; the code is borrowed from indexed.cpp
 272                         if (precedence() <= level) {
 273                                 c.s << '(';
 274                         }
 275                         seq[0].print(c, precedence());
 276                         c.s << '[' << int(representation_label) << ']';
 277                         printindices(c, level);
 278                         if (precedence() <= level) {
 279                                 c.s << ')';
 280                         }
 281                 }
 282         }
 283 }
 284
 285 void clifford::do_print_latex(const print_latex & c, unsigned level) const
 286 {
 287         // dirac_slash() object is printed differently
 288         if (is_dirac_slash(seq[0])) {
 289                 c.s << "{";
 290                 seq[0].print(c, precedence());
 291                 c.s << "\\hspace{-1.0ex}/}";
 292         } else {
 293                 c.s << "\\clifford[" << int(representation_label) << "]";
 294                 this->print_dispatch<inherited>(c, level);
 295         }
 296 }
 297
 298 DEFAULT_COMPARE(diracone)
 299 DEFAULT_COMPARE(cliffordunit)
 300 DEFAULT_COMPARE(diracgamma)
 301 DEFAULT_COMPARE(diracgamma5)
 302 DEFAULT_COMPARE(diracgammaL)
 303 DEFAULT_COMPARE(diracgammaR)
 304
 305 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbf{1}")
 306 DEFAULT_PRINT_LATEX(cliffordunit, "e", "e")
 307 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
 308 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
 309 DEFAULT_PRINT_LATEX(diracgammaL, "gammaL", "{\\gamma_L}")
 310 DEFAULT_PRINT_LATEX(diracgammaR, "gammaR", "{\\gamma_R}")
 311
 312 /** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
 313 static void base_and_index(const ex & c, ex & b, ex & i)
 314 {
 315         GINAC_ASSERT(is_a<clifford>(c));
 316         GINAC_ASSERT(c.nops() == 2+1);
 317
 318         if (is_a<cliffordunit>(c.op(0))) { // proper dirac gamma object or clifford unit
 319                 i = c.op(1);
 320                 b = _ex1;
 321         } else if (is_a<diracgamma5>(c.op(0)) || is_a<diracgammaL>(c.op(0)) || is_a<diracgammaR>(c.op(0))) { // gamma5/L/R
 322                 i = _ex0;
 323                 b = _ex1;
 324         } else { // slash object, generate new dummy index
 325                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(c.op(1)).get_dim());
 326                 b = indexed(c.op(0), ix.toggle_variance());
 327                 i = ix;
 328         }
 329 }
 330
 331 /** Predicate for finding non-clifford objects. */
 332 struct is_not_a_clifford : public std::unary_function<ex, bool> {
 333         bool operator()(const ex & e)
 334         {
 335                 return !is_a<clifford>(e);
 336         }
 337 };
 338
 339 /** Contraction of a gamma matrix with something else. */
 340 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 341 {
 342         GINAC_ASSERT(is_a<clifford>(*self));
 343         GINAC_ASSERT(is_a<indexed>(*other));
 344         GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
 345         unsigned char rl = ex_to<clifford>(*self).get_representation_label();
 346
 347         ex dim = ex_to<idx>(self->op(1)).get_dim();
 348         if (other->nops() > 1)
 349                 dim = minimal_dim(dim, ex_to<idx>(other->op(1)).get_dim());
 350
 351         if (is_a<clifford>(*other)) {
 352
 353                 // Contraction only makes sense if the represenation labels are equal
 354                 if (ex_to<clifford>(*other).get_representation_label() != rl)
 355                         return false;
 356
 357                 size_t num = other - self;
 358
 359                 // gamma~mu gamma.mu = dim ONE
 360                 if (num == 1) {
 361                         *self = dim;
 362                         *other = dirac_ONE(rl);
 363                         return true;
 364
 365                 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
 366                 } else if (num == 2
 367                         && is_a<clifford>(self[1])) {
 368                         *self = 2 - dim;
 369                         *other = _ex1;
 370                         return true;
 371
 372                 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
 373                 } else if (num == 3
 374                         && is_a<clifford>(self[1])
 375                         && is_a<clifford>(self[2])) {
 376                         ex b1, i1, b2, i2;
 377                         base_and_index(self[1], b1, i1);
 378                         base_and_index(self[2], b2, i2);
 379                         *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
 380                         self[1] = _ex1;
 381                         self[2] = _ex1;
 382                         *other = _ex1;
 383                         return true;
 384
 385                 // 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
 386                 } else if (num == 4
 387                         && is_a<clifford>(self[1])
 388                         && is_a<clifford>(self[2])
 389                         && is_a<clifford>(self[3])) {
 390                         *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3];
 391                         self[1] = _ex1;
 392                         self[2] = _ex1;
 393                         self[3] = _ex1;
 394                         *other = _ex1;
 395                         return true;
 396
 397                 // gamma~mu Sodd gamma.mu = -2 Sodd_R
 398                 // (Chisholm identity in 4 dimensions)
 399                 } else if (!((other - self) & 1) && dim.is_equal(4)) {
 400                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
 401                                 return false;
 402
 403                         *self = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(other), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
 404                         std::fill(self + 1, other, _ex1);
 405                         *other = _ex_2;
 406                         return true;
 407
 408                 // gamma~mu Sodd gamma~alpha gamma.mu = 2 gamma~alpha Sodd + 2 Sodd_R gamma~alpha
 409                 // (commutate contracted indices towards each other, then use
 410                 // Chisholm identity in 4 dimensions)
 411                 } else if (((other - self) & 1) && dim.is_equal(4)) {
 412                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
 413                                 return false;
 414
 415                         exvector::iterator next_to_last = other - 1;
 416                         ex S = ncmul(exvector(self + 1, next_to_last), true);
 417                         ex SR = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(next_to_last), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
 418
 419                         *self = (*next_to_last) * S + SR * (*next_to_last);
 420                         std::fill(self + 1, other, _ex1);
 421                         *other = _ex2;
 422                         return true;
 423
 424                 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
 425                 // (commutate contracted indices towards each other, simplify_indexed()
 426                 // will re-expand and re-run the simplification)
 427                 } else {
 428                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
 429                                 return false;
 430
 431                         exvector::iterator next_to_last = other - 1;
 432                         ex S = ncmul(exvector(self + 1, next_to_last), true);
 433
 434                         *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
 435                         std::fill(self + 1, other + 1, _ex1);
 436                         return true;
 437                 }
 438
 439         } else if (is_a<symbol>(other->op(0)) && other->nops() == 2) {
 440
 441                 // x.mu gamma~mu -> x-slash
 442                 *self = dirac_slash(other->op(0), dim, rl);
 443                 *other = _ex1;
 444                 return true;
 445         }
 446
 447         return false;
 448 }
 449
 450 /** Contraction of a Clifford unit with something else. */
 451 bool cliffordunit::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 452 {
 453         GINAC_ASSERT(is_a<clifford>(*self));
 454         GINAC_ASSERT(is_a<indexed>(*other));
 455         GINAC_ASSERT(is_a<cliffordunit>(self->op(0)));
 456         clifford unit = ex_to<clifford>(*self);
 457         unsigned char rl = unit.get_representation_label();
 458
 459         if (is_a<clifford>(*other)) {
 460                 // Contraction only makes sense if the represenation labels are equal
 461                 // and the metrics are the same
 462                 if ((ex_to<clifford>(*other).get_representation_label() != rl)
 463                     && unit.same_metric(*other))
 464                         return false;
 465
 466                 exvector::iterator before_other = other - 1;
 467                 ex mu = self->op(1);
 468                 ex mu_toggle = other->op(1);
 469                 ex alpha = before_other->op(1);
 470
 471                 // e~mu e.mu = Tr ONE
 472                 if (other - self == 1) {
 473                         *self = unit.get_metric(mu, mu_toggle, true);
 474                         *other = dirac_ONE(rl);
 475                         return true;
 476
 477                 } else if (other - self == 2) {
 478                         if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
 479                                 // e~mu e~alpha e.mu = 2*e~mu B(alpha, mu.toggle_variance())-Tr(B) e~alpha
 480                                 *self = 2 * (*self) * unit.get_metric(alpha, mu_toggle, true) - unit.get_metric(mu, mu_toggle, true) * (*before_other);
 481                                 *before_other = _ex1;
 482                                 *other = _ex1;
 483                                 return true;
 484
 485                         } else {
 486                                 // e~mu S e.mu = Tr S ONE
 487                                 *self = unit.get_metric(mu, mu_toggle, true);
 488                                 *other = dirac_ONE(rl);
 489                                 return true;
 490                         }
 491                 } else {
 492                 // e~mu S e~alpha e.mu = 2 e~mu S B(alpha, mu.toggle_variance()) - e~mu S e.mu e~alpha
 493                 // (commutate contracted indices towards each other, simplify_indexed()
 494                 // will re-expand and re-run the simplification)
 495                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other) {
 496                                 return false;
 497                         }
 498
 499                         ex S = ncmul(exvector(self + 1, before_other), true);
 500
 501                         if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
 502                                 *self = 2 * (*self) * S * unit.get_metric(alpha, mu_toggle, true) - (*self) * S * (*other) * (*before_other);
 503                         } else {
 504                                 // simply commutes
 505                                 *self = (*self) * S * (*other) * (*before_other);
 506                         }
 507
 508                         std::fill(self + 1, other + 1, _ex1);
 509                         return true;
 510                 }
 511         }
 512         return false;
 513 }
 514
 515 /** Perform automatic simplification on noncommutative product of clifford
 516  *  objects. This removes superfluous ONEs, permutes gamma5/L/R's to the front
 517  *  and removes squares of gamma objects. */
 518 ex clifford::eval_ncmul(const exvector & v) const
 519 {
 520         exvector s;
 521         s.reserve(v.size());
 522
 523         // Remove superfluous ONEs
 524         exvector::const_iterator cit = v.begin(), citend = v.end();
 525         while (cit != citend) {
 526                 if (!is_a<clifford>(*cit) || !is_a<diracone>(cit->op(0)))
 527                         s.push_back(*cit);
 528                 cit++;
 529         }
 530
 531         bool something_changed = false;
 532         int sign = 1;
 533
 534         // Anticommutate gamma5/L/R's to the front
 535         if (s.size() >= 2) {
 536                 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
 537                 while (true) {
 538                         exvector::iterator it = next_to_last;
 539                         while (true) {
 540                                 exvector::iterator it2 = it + 1;
 541                                 if (is_a<clifford>(*it) && is_a<clifford>(*it2)) {
 542                                         ex e1 = it->op(0), e2 = it2->op(0);
 543
 544                                         if (is_a<diracgamma5>(e2)) {
 545
 546                                                 if (is_a<diracgammaL>(e1) || is_a<diracgammaR>(e1)) {
 547
 548                                                         // gammaL/R gamma5 -> gamma5 gammaL/R
 549                                                         it->swap(*it2);
 550                                                         something_changed = true;
 551
 552                                                 } else if (!is_a<diracgamma5>(e1)) {
 553
 554                                                         // gamma5 gamma5 -> gamma5 gamma5 (do nothing)
 555                                                         // x gamma5 -> -gamma5 x
 556                                                         it->swap(*it2);
 557                                                         sign = -sign;
 558                                                         something_changed = true;
 559                                                 }
 560
 561                                         } else if (is_a<diracgammaL>(e2)) {
 562
 563                                                 if (is_a<diracgammaR>(e1)) {
 564
 565                                                         // gammaR gammaL -> 0
 566                                                         return _ex0;
 567
 568                                                 } else if (!is_a<diracgammaL>(e1) && !is_a<diracgamma5>(e1)) {
 569
 570                                                         // gammaL gammaL -> gammaL gammaL (do nothing)
 571                                                         // gamma5 gammaL -> gamma5 gammaL (do nothing)
 572                                                         // x gammaL -> gammaR x
 573                                                         it->swap(*it2);
 574                                                         *it = clifford(diracgammaR(), ex_to<clifford>(*it).get_representation_label());
 575                                                         something_changed = true;
 576                                                 }
 577
 578                                         } else if (is_a<diracgammaR>(e2)) {
 579
 580                                                 if (is_a<diracgammaL>(e1)) {
 581
 582                                                         // gammaL gammaR -> 0
 583                                                         return _ex0;
 584
 585                                                 } else if (!is_a<diracgammaR>(e1) && !is_a<diracgamma5>(e1)) {
 586
 587                                                         // gammaR gammaR -> gammaR gammaR (do nothing)
 588                                                         // gamma5 gammaR -> gamma5 gammaR (do nothing)
 589                                                         // x gammaR -> gammaL x
 590                                                         it->swap(*it2);
 591                                                         *it = clifford(diracgammaL(), ex_to<clifford>(*it).get_representation_label());
 592                                                         something_changed = true;
 593                                                 }
 594                                         }
 595                                 }
 596                                 if (it == first)
 597                                         break;
 598                                 --it;
 599                         }
 600                         if (next_to_last == first)
 601                                 break;
 602                         --next_to_last;
 603                 }
 604         }
 605
 606         // Remove equal adjacent gammas
 607         if (s.size() >= 2) {
 608                 exvector::iterator it, itend = s.end() - 1;
 609                 for (it = s.begin(); it != itend; ++it) {
 610                         ex & a = it[0];
 611                         ex & b = it[1];
 612                         if (!is_a<clifford>(a) || !is_a<clifford>(b))
 613                                 continue;
 614
 615                         const ex & ag = a.op(0);
 616                         const ex & bg = b.op(0);
 617                         bool a_is_cliffordunit = is_a<cliffordunit>(ag);
 618                         bool b_is_cliffordunit =  is_a<cliffordunit>(bg);
 619
 620                         if (a_is_cliffordunit && b_is_cliffordunit && ex_to<clifford>(a).same_metric(b)
 621                                 && (ex_to<clifford>(a).get_commutator_sign() == -1)) {
 622                                 // This is done only for Clifford algebras
 623
 624                                 const ex & ia = a.op(1);
 625                                 const ex & ib = b.op(1);
 626                                 if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
 627                                         a = ex_to<clifford>(a).get_metric(ia, ib, true);
 628                                         b = dirac_ONE(representation_label);
 629                                         something_changed = true;
 630                                 }
 631
 632                         } else if ((is_a<diracgamma5>(ag) && is_a<diracgamma5>(bg))) {
 633
 634                                 // Remove squares of gamma5
 635                                 a = dirac_ONE(representation_label);
 636                                 b = dirac_ONE(representation_label);
 637                                 something_changed = true;
 638
 639                         } else if ((is_a<diracgammaL>(ag) && is_a<diracgammaL>(bg))
 640                                 || (is_a<diracgammaR>(ag) && is_a<diracgammaR>(bg))) {
 641
 642                                 // Remove squares of gammaL/R
 643                                 b = dirac_ONE(representation_label);
 644                                 something_changed = true;
 645
 646                         } else if (is_a<diracgammaL>(ag) && is_a<diracgammaR>(bg)) {
 647
 648                                 // gammaL and gammaR are orthogonal
 649                                 return _ex0;
 650
 651                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaL>(bg)) {
 652
 653                                 // gamma5 gammaL -> -gammaL
 654                                 a = dirac_ONE(representation_label);
 655                                 sign = -sign;
 656                                 something_changed = true;
 657
 658                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaR>(bg)) {
 659
 660                                 // gamma5 gammaR -> gammaR
 661                                 a = dirac_ONE(representation_label);
 662                                 something_changed = true;
 663
 664                         } else if (!a_is_cliffordunit && !b_is_cliffordunit && ag.is_equal(bg)) {
 665
 666                                 // a\ a\ -> a^2
 667                                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).minimal_dim(ex_to<idx>(b.op(1))));
 668
 669                                 a = indexed(ag, ix) * indexed(ag, ix.toggle_variance());
 670                                 b = dirac_ONE(representation_label);
 671                                 something_changed = true;
 672                         }
 673                 }
 674         }
 675
 676         if (s.empty())
 677                 return dirac_ONE(representation_label) * sign;
 678         if (something_changed)
 679                 return reeval_ncmul(s) * sign;
 680         else
 681                 return hold_ncmul(s) * sign;
 682 }
 683
 684 ex clifford::thiscontainer(const exvector & v) const
 685 {
 686         return clifford(representation_label, metric, commutator_sign, v);
 687 }
 688
 689 ex clifford::thiscontainer(std::auto_ptr<exvector> vp) const
 690 {
 691         return clifford(representation_label, metric, commutator_sign, vp);
 692 }
 693
 694 ex diracgamma5::conjugate() const
 695 {
 696         return _ex_1 * (*this);
 697 }
 698
 699 ex diracgammaL::conjugate() const
 700 {
 701         return (new diracgammaR)->setflag(status_flags::dynallocated);
 702 }
 703
 704 ex diracgammaR::conjugate() const
 705 {
 706         return (new diracgammaL)->setflag(status_flags::dynallocated);
 707 }
 708
 709 //////////
 710 // global functions
 711 //////////
 712
 713 ex dirac_ONE(unsigned char rl)
 714 {
 715         static ex ONE = (new diracone)->setflag(status_flags::dynallocated);
 716         return clifford(ONE, rl);
 717 }
 718
 719 static unsigned get_dim_uint(const ex& e)
 720 {
 721         if (!is_a<idx>(e))
 722                 throw std::invalid_argument("get_dim_uint: argument is not an index");
 723         ex dim = ex_to<idx>(e).get_dim();
 724         if (!dim.info(info_flags::posint))
 725                 throw std::invalid_argument("get_dim_uint: dimension of index should be a positive integer");
 726         unsigned d = ex_to<numeric>(dim).to_int();
 727         return d;
 728 }
 729
 730 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl)
 731 {
 732         //static ex unit = (new cliffordunit)->setflag(status_flags::dynallocated);
 733         ex unit = (new cliffordunit)->setflag(status_flags::dynallocated);
 734
 735         if (!is_a<idx>(mu))
 736                 throw(std::invalid_argument("clifford_unit(): index of Clifford unit must be of type idx or varidx"));
 737
 738         exvector indices = metr.get_free_indices();
 739
 740         if (indices.size() == 2) {
 741                 return clifford(unit, mu, metr, rl);
 742         } else if (is_a<matrix>(metr)) {
 743                 matrix M = ex_to<matrix>(metr);
 744                 unsigned n = M.rows();
 745                 bool symmetric = true;
 746
 747                 //static idx xi((new symbol)->setflag(status_flags::dynallocated), n),
 748                 //      chi((new symbol)->setflag(status_flags::dynallocated), n);
 749                 idx xi((new symbol)->setflag(status_flags::dynallocated), n),
 750                         chi((new symbol)->setflag(status_flags::dynallocated), n);
 751                 if ((n ==  M.cols()) && (n == get_dim_uint(mu))) {
 752                         for (unsigned i = 0; i < n; i++) {
 753                                 for (unsigned j = i+1; j < n; j++) {
 754                                         if (!M(i, j).is_equal(M(j, i))) {
 755                                                 symmetric = false;
 756                                         }
 757                                 }
 758                         }
 759                         return clifford(unit, mu, indexed(metr, symmetric?symmetric2():not_symmetric(), xi, chi), rl);
 760                 } else {
 761                         throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be a square matrix with the same dimensions as index"));
 762                 }
 763         } else if (indices.size() == 0) { // a tensor or other expression without indices
 764                 //static varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim()),
 765                 //      chi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim());
 766                 varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim()),
 767                         chi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim());
 768                 return clifford(unit, mu, indexed(metr, xi, chi), rl);
 769         }  else
 770                 throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be of type tensor, matrix or an expression with two free indices"));
 771 }
 772
 773 ex dirac_gamma(const ex & mu, unsigned char rl)
 774 {
 775         static ex gamma = (new diracgamma)->setflag(status_flags::dynallocated);
 776
 777         if (!is_a<varidx>(mu))
 778                 throw(std::invalid_argument("dirac_gamma(): index of Dirac gamma must be of type varidx"));
 779
 780         static varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim()),
 781                 chi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim());
 782         return clifford(gamma, mu, indexed((new minkmetric)->setflag(status_flags::dynallocated), symmetric2(), xi, chi), rl);
 783 }
 784
 785 ex dirac_gamma5(unsigned char rl)
 786 {
 787         static ex gamma5 = (new diracgamma5)->setflag(status_flags::dynallocated);
 788         return clifford(gamma5, rl);
 789 }
 790
 791 ex dirac_gammaL(unsigned char rl)
 792 {
 793         static ex gammaL = (new diracgammaL)->setflag(status_flags::dynallocated);
 794         return clifford(gammaL, rl);
 795 }
 796
 797 ex dirac_gammaR(unsigned char rl)
 798 {
 799         static ex gammaR = (new diracgammaR)->setflag(status_flags::dynallocated);
 800         return clifford(gammaR, rl);
 801 }
 802
 803 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
 804 {
 805         // Slashed vectors are actually stored as a clifford object with the
 806         // vector as its base expression and a (dummy) index that just serves
 807         // for storing the space dimensionality
 808
 809         static varidx xi((new symbol)->setflag(status_flags::dynallocated), dim),
 810                 chi((new symbol)->setflag(status_flags::dynallocated), dim);
 811    return clifford(e, varidx(0, dim), indexed((new minkmetric)->setflag(status_flags::dynallocated), symmetric2(), xi, chi), rl);
 812 }
 813
 814 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 815  *  is that of a clifford object (with an arbitrary representation label). */
 816 bool is_clifford_tinfo(tinfo_t ti)
 817 {
 818         p_int start_loc=(p_int)&clifford::return_type_tinfo_static;
 819         return (p_int)ti>=start_loc && (p_int)ti<start_loc+256;
 820 }
 821
 822 /** Extract representation label from tinfo key (as returned by
 823  *  return_type_tinfo()). */
 824 static unsigned char get_representation_label(tinfo_t ti)
 825 {
 826         return (unsigned char)((p_int)ti-(p_int)&clifford::return_type_tinfo_static);
 827 }
 828
 829 /** Take trace of a string of an even number of Dirac gammas given a vector
 830  *  of indices. */
 831 static ex trace_string(exvector::const_iterator ix, size_t num)
 832 {
 833         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 834         if (num == 2)
 835                 return lorentz_g(ix[0], ix[1]);
 836
 837         // 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 )
 838         else if (num == 4)
 839                 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
 840                      + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
 841                      - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
 842
 843         // Traces of 6 or more gammas are computed recursively:
 844         // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
 845         //   + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
 846         //   - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
 847         //   + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
 848         //   - ...
 849         //   + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
 850         exvector v(num - 2);
 851         int sign = 1;
 852         ex result;
 853         for (size_t i=1; i<num; i++) {
 854                 for (size_t n=1, j=0; n<num; n++) {
 855                         if (n == i)
 856                                 continue;
 857                         v[j++] = ix[n];
 858                 }
 859                 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
 860                 sign = -sign;
 861         }
 862         return result;
 863 }
 864
 865 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE)
 866 {
 867         if (is_a<clifford>(e)) {
 868
 869                 unsigned char rl = ex_to<clifford>(e).get_representation_label();
 870
 871                 // Are we taking the trace over this object's representation label?
 872                 if (rls.find(rl) == rls.end())
 873                         return e;
 874
 875                 // Yes, all elements are traceless, except for dirac_ONE and dirac_L/R
 876                 const ex & g = e.op(0);
 877                 if (is_a<diracone>(g))
 878                         return trONE;
 879                 else if (is_a<diracgammaL>(g) || is_a<diracgammaR>(g))
 880                         return trONE/2;
 881                 else
 882                         return _ex0;
 883
 884         } else if (is_exactly_a<mul>(e)) {
 885
 886                 // Trace of product: pull out non-clifford factors
 887                 ex prod = _ex1;
 888                 for (size_t i=0; i<e.nops(); i++) {
 889                         const ex &o = e.op(i);
 890                         if (is_clifford_tinfo(o.return_type_tinfo()))
 891                                 prod *= dirac_trace(o, rls, trONE);
 892                         else
 893                                 prod *= o;
 894                 }
 895                 return prod;
 896
 897         } else if (is_exactly_a<ncmul>(e)) {
 898
 899                 unsigned char rl = get_representation_label(e.return_type_tinfo());
 900
 901                 // Are we taking the trace over this string's representation label?
 902                 if (rls.find(rl) == rls.end())
 903                         return e;
 904
 905                 // Substitute gammaL/R and expand product, if necessary
 906                 ex e_expanded = e.subs(lst(
 907                         dirac_gammaL(rl) == (dirac_ONE(rl)-dirac_gamma5(rl))/2,
 908                         dirac_gammaR(rl) == (dirac_ONE(rl)+dirac_gamma5(rl))/2
 909                 ), subs_options::no_pattern).expand();
 910                 if (!is_a<ncmul>(e_expanded))
 911                         return dirac_trace(e_expanded, rls, trONE);
 912
 913                 // gamma5 gets moved to the front so this check is enough
 914                 bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
 915                 size_t num = e.nops();
 916
 917                 if (has_gamma5) {
 918
 919                         // Trace of gamma5 * odd number of gammas and trace of
 920                         // gamma5 * gamma.mu * gamma.nu are zero
 921                         if ((num & 1) == 0 || num == 3)
 922                                 return _ex0;
 923
 924                         // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
 925                         // (the epsilon is always 4-dimensional)
 926                         if (num == 5) {
 927                                 ex b1, i1, b2, i2, b3, i3, b4, i4;
 928                                 base_and_index(e.op(1), b1, i1);
 929                                 base_and_index(e.op(2), b2, i2);
 930                                 base_and_index(e.op(3), b3, i3);
 931                                 base_and_index(e.op(4), b4, i4);
 932                                 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();
 933                         }
 934
 935                         // Tr gamma5 S_2k =
 936                         //   I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
 937                         // (the epsilon is always 4-dimensional)
 938                         exvector ix(num-1), bv(num-1);
 939                         for (size_t i=1; i<num; i++)
 940                                 base_and_index(e.op(i), bv[i-1], ix[i-1]);
 941                         num--;
 942                         int *iv = new int[num];
 943                         ex result;
 944                         for (size_t i=0; i<num-3; i++) {
 945                                 ex idx1 = ix[i];
 946                                 for (size_t j=i+1; j<num-2; j++) {
 947                                         ex idx2 = ix[j];
 948                                         for (size_t k=j+1; k<num-1; k++) {
 949                                                 ex idx3 = ix[k];
 950                                                 for (size_t l=k+1; l<num; l++) {
 951                                                         ex idx4 = ix[l];
 952                                                         iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
 953                                                         exvector v;
 954                                                         v.reserve(num - 4);
 955                                                         for (size_t n=0, t=4; n<num; n++) {
 956                                                                 if (n == i || n == j || n == k || n == l)
 957                                                                         continue;
 958                                                                 iv[t++] = n;
 959                                                                 v.push_back(ix[n]);
 960                                                         }
 961                                                         int sign = permutation_sign(iv, iv + num);
 962                                                         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))
 963                                                                 * trace_string(v.begin(), num - 4);
 964                                                 }
 965                                         }
 966                                 }
 967                         }
 968                         delete[] iv;
 969                         return trONE * I * result * mul(bv);
 970
 971                 } else { // no gamma5
 972
 973                         // Trace of odd number of gammas is zero
 974                         if ((num & 1) == 1)
 975                                 return _ex0;
 976
 977                         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 978                         if (num == 2) {
 979                                 ex b1, i1, b2, i2;
 980                                 base_and_index(e.op(0), b1, i1);
 981                                 base_and_index(e.op(1), b2, i2);
 982                                 return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
 983                         }
 984
 985                         exvector iv(num), bv(num);
 986                         for (size_t i=0; i<num; i++)
 987                                 base_and_index(e.op(i), bv[i], iv[i]);
 988
 989                         return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
 990                 }
 991
 992         } else if (e.nops() > 0) {
 993
 994                 // Trace maps to all other container classes (this includes sums)
 995                 pointer_to_map_function_2args<const std::set<unsigned char> &, const ex &> fcn(dirac_trace, rls, trONE);
 996                 return e.map(fcn);
 997
 998         } else
 999                 return _ex0;
1000 }
1001
1002 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE)
1003 {
1004         // Convert list to set
1005         std::set<unsigned char> rls;
1006         for (lst::const_iterator i = rll.begin(); i != rll.end(); ++i) {
1007                 if (i->info(info_flags::nonnegint))
1008                         rls.insert(ex_to<numeric>(*i).to_int());
1009         }
1010
1011         return dirac_trace(e, rls, trONE);
1012 }
1013
1014 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
1015 {
1016         // Convert label to set
1017         std::set<unsigned char> rls;
1018         rls.insert(rl);
1019
1020         return dirac_trace(e, rls, trONE);
1021 }
1022
1023
1024 ex canonicalize_clifford(const ex & e_)
1025 {
1026         pointer_to_map_function fcn(canonicalize_clifford);
1027
1028         if (is_a<matrix>(e_)    // || is_a<pseries>(e) || is_a<integral>(e)
1029                 || e_.info(info_flags::list)) {
1030                 return e_.map(fcn);
1031         } else {
1032                 ex e=simplify_indexed(e_);
1033                 // Scan for any ncmul objects
1034                 exmap srl;
1035                 ex aux = e.to_rational(srl);
1036                 for (exmap::iterator i = srl.begin(); i != srl.end(); ++i) {
1037
1038                         ex lhs = i->first;
1039                         ex rhs = i->second;
1040
1041                         if (is_exactly_a<ncmul>(rhs)
1042                                         && rhs.return_type() == return_types::noncommutative
1043                                         && is_clifford_tinfo(rhs.return_type_tinfo())) {
1044
1045                                 // Expand product, if necessary
1046                                 ex rhs_expanded = rhs.expand();
1047                                 if (!is_a<ncmul>(rhs_expanded)) {
1048                                         i->second = canonicalize_clifford(rhs_expanded);
1049                                         continue;
1050
1051                                 } else if (!is_a<clifford>(rhs.op(0)))
1052                                         continue;
1053
1054                                 exvector v;
1055                                 v.reserve(rhs.nops());
1056                                 for (size_t j=0; j<rhs.nops(); j++)
1057                                         v.push_back(rhs.op(j));
1058
1059                                 // Stupid recursive bubble sort because we only want to swap adjacent gammas
1060                                 exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
1061                                 if (is_a<diracgamma5>(it->op(0)) || is_a<diracgammaL>(it->op(0)) || is_a<diracgammaR>(it->op(0)))
1062                                         ++it;
1063
1064                                 while (it != next_to_last) {
1065                                         if (it[0].compare(it[1]) > 0) {
1066
1067                                                 ex save0 = it[0], save1 = it[1];
1068                                                 ex b1, i1, b2, i2;
1069                                                 base_and_index(it[0], b1, i1);
1070                                                 base_and_index(it[1], b2, i2);
1071                                                 // for Clifford algebras (commutator_sign == -1) metric should be symmetrised
1072                                                 it[0] = (ex_to<clifford>(save0).get_metric(i1, i2, ex_to<clifford>(save0).get_commutator_sign() == -1) * b1 * b2).simplify_indexed();
1073                                                 it[1] = v.size() ? _ex2 * dirac_ONE(ex_to<clifford>(save0).get_representation_label()) : _ex2;
1074                                                 ex sum = ncmul(v);
1075                                                 it[0] = save1;
1076                                                 it[1] = save0;
1077                                                 sum += ex_to<clifford>(save0).get_commutator_sign() * ncmul(v, true);
1078                                                 i->second = canonicalize_clifford(sum);
1079                                                 goto next_sym;
1080                                         }
1081                                         ++it;
1082                                 }
1083 next_sym:       ;
1084                         }
1085                 }
1086                 return aux.subs(srl, subs_options::no_pattern).simplify_indexed();
1087         }
1088 }
1089
1090 ex clifford_prime(const ex & e)
1091 {
1092         pointer_to_map_function fcn(clifford_prime);
1093         if (is_a<clifford>(e) && is_a<cliffordunit>(e.op(0))) {
1094                 return -e;
1095         } else if (is_a<add>(e) || is_a<ncmul>(e) || is_a<mul>(e) //|| is_a<pseries>(e) || is_a<integral>(e)
1096                            || is_a<matrix>(e) || e.info(info_flags::list)) {
1097                 return e.map(fcn);
1098         } else if (is_a<power>(e)) {
1099                 return pow(clifford_prime(e.op(0)), e.op(1));
1100         } else
1101                 return e;
1102 }
1103
1104 ex remove_dirac_ONE(const ex & e, unsigned char rl, unsigned options)
1105 {
1106         pointer_to_map_function_2args<unsigned char, unsigned> fcn(remove_dirac_ONE, rl, options | 1);
1107         bool need_reevaluation = false;
1108         ex e1 = e;
1109         if (! (options & 1) )  { // is not a child
1110                 if (options & 2)
1111                         e1 = expand_dummy_sum(e, true);
1112                 e1 = canonicalize_clifford(e1);
1113         }
1114
1115         if (is_a<clifford>(e1) && ex_to<clifford>(e1).get_representation_label() >= rl) {
1116                 if (is_a<diracone>(e1.op(0)))
1117                         return 1;
1118                 else
1119                         throw(std::invalid_argument("remove_dirac_ONE(): expression is a non-scalar Clifford number!"));
1120         } else if (is_a<add>(e1) || is_a<ncmul>(e1) || is_a<mul>(e1)
1121                            || is_a<matrix>(e1) || e1.info(info_flags::list)) {
1122                 if (options & 3) // is a child or was already expanded
1123                         return e1.map(fcn);
1124                 else
1125                         try {
1126                                 return e1.map(fcn);
1127                         } catch (std::exception &p) {
1128                                 need_reevaluation = true;
1129                         }
1130         } else if (is_a<power>(e1)) {
1131                 if (options & 3) // is a child or was already expanded
1132                         return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
1133                 else
1134                         try {
1135                                 return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
1136                         } catch (std::exception &p) {
1137                                 need_reevaluation = true;
1138                         }
1139         }
1140         if (need_reevaluation)
1141                 return remove_dirac_ONE(e, rl, options | 2);
1142         return e1;
1143 }
1144
1145 char clifford_max_label(const ex & e, bool ignore_ONE)
1146 {
1147         if (is_a<clifford>(e))
1148                 if (ignore_ONE && is_a<diracone>(e.op(0)))
1149                         return -1;
1150                 else
1151                         return ex_to<clifford>(e).get_representation_label();
1152         else {
1153                 char rl = -1;
1154                 for (size_t i=0; i < e.nops(); i++)
1155                         rl = (rl > clifford_max_label(e.op(i), ignore_ONE)) ? rl : clifford_max_label(e.op(i), ignore_ONE);
1156                 return rl;
1157         }
1158 }
1159
1160 ex clifford_norm(const ex & e)
1161 {
1162         return sqrt(remove_dirac_ONE(e * clifford_bar(e)));
1163 }
1164
1165 ex clifford_inverse(const ex & e)
1166 {
1167         ex norm = clifford_norm(e);
1168         if (!norm.is_zero())
1169                 return clifford_bar(e) / pow(norm, 2);
1170         else
1171                 throw(std::invalid_argument("clifford_inverse(): cannot find inverse of Clifford number with zero norm!"));
1172 }
1173
1174 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl)
1175 {
1176         if (!ex_to<idx>(mu).is_dim_numeric())
1177                 throw(std::invalid_argument("lst_to_clifford(): Index should have a numeric dimension"));
1178         ex e = clifford_unit(mu, metr, rl);
1179         return lst_to_clifford(v, e);
1180 }
1181
1182 ex lst_to_clifford(const ex & v, const ex & e) {
1183         unsigned min, max;
1184
1185         if (is_a<clifford>(e)) {
1186                 ex mu = e.op(1);
1187                 ex mu_toggle
1188                         = is_a<varidx>(mu) ? ex_to<varidx>(mu).toggle_variance() : mu;
1189                 unsigned dim = get_dim_uint(mu);
1190
1191                 if (is_a<matrix>(v)) {
1192                         if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows()) {
1193                                 min = ex_to<matrix>(v).rows();
1194                                 max = ex_to<matrix>(v).cols();
1195                         } else {
1196                                 min = ex_to<matrix>(v).cols();
1197                                 max = ex_to<matrix>(v).rows();
1198                         }
1199                         if (min == 1) {
1200                                 if (dim == max)
1201                                         return indexed(v, mu_toggle) * e;
1202                                 else if (max - dim == 1) {
1203                                         if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows())
1204                                                 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;
1205                                         else
1206                                                 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;
1207                                 } else
1208                                         throw(std::invalid_argument("lst_to_clifford(): dimensions of vector and clifford unit mismatch"));
1209                         } else
1210                                 throw(std::invalid_argument("lst_to_clifford(): first argument should be a vector (nx1 or 1xn matrix)"));
1211                 } else if (v.info(info_flags::list)) {
1212                         if (dim == ex_to<lst>(v).nops())
1213                                 return indexed(matrix(dim, 1, ex_to<lst>(v)), mu_toggle) * e;
1214                         else if (ex_to<lst>(v).nops() - dim == 1)
1215                                 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;
1216                         else
1217                                 throw(std::invalid_argument("lst_to_clifford(): list length and dimension of clifford unit mismatch"));
1218                 } else
1219                         throw(std::invalid_argument("lst_to_clifford(): cannot construct from anything but list or vector"));
1220         } else
1221                 throw(std::invalid_argument("lst_to_clifford(): the second argument should be a Clifford unit"));
1222 }
1223
1224 /** Auxiliary structure to define a function for striping one Clifford unit
1225  * from vectors. Used in  clifford_to_lst(). */
1226 static ex get_clifford_comp(const ex & e, const ex & c)
1227 {
1228         pointer_to_map_function_1arg<const ex &> fcn(get_clifford_comp, c);
1229         int ival = ex_to<numeric>(ex_to<idx>(c.op(1)).get_value()).to_int();
1230
1231         if (is_a<add>(e) || e.info(info_flags::list) // || is_a<pseries>(e) || is_a<integral>(e)
1232                 || is_a<matrix>(e))
1233                 return e.map(fcn);
1234         else if (is_a<ncmul>(e) || is_a<mul>(e)) {
1235                 // find a Clifford unit with the same metric, delete it and substitute its index
1236                 size_t ind = e.nops() + 1;
1237                 for (size_t j = 0; j < e.nops(); j++)
1238                         if (is_a<clifford>(e.op(j)) && ex_to<clifford>(c).same_metric(e.op(j)))
1239                                 if (ind > e.nops())
1240                                         ind = j;
1241                                 else
1242                                         throw(std::invalid_argument("get_clifford_comp(): expression is a Clifford multi-vector"));
1243                 if (ind < e.nops()) {
1244                         ex S = 1;
1245                         bool same_value_index, found_dummy;
1246                         same_value_index = ( ex_to<idx>(e.op(ind).op(1)).is_numeric()
1247                                                                  &&  (ival == ex_to<numeric>(ex_to<idx>(e.op(ind).op(1)).get_value()).to_int()) );
1248                         found_dummy = same_value_index;
1249                         for(size_t j=0; j < e.nops(); j++)
1250                                 if (j != ind)
1251                                         if (same_value_index)
1252                                                 S = S * e.op(j);
1253                                         else {
1254                                                 exvector ind_vec = ex_to<indexed>(e.op(j)).get_dummy_indices(ex_to<indexed>(e.op(ind)));
1255                                                 if (ind_vec.size() > 0) {
1256                                                         found_dummy = true;
1257                                                         exvector::const_iterator it = ind_vec.begin(), itend = ind_vec.end();
1258                                                         while (it != itend) {
1259                                                                 ex curridx = *it;
1260                                                                 ex curridx_toggle = is_a<varidx>(curridx)
1261                                                                         ? ex_to<varidx>(curridx).toggle_variance()
1262                                                                         : curridx;
1263                                                                 S = S * e.op(j).subs(lst(curridx == ival,
1264                                                                         curridx_toggle == ival), subs_options::no_pattern);
1265                                                                 ++it;
1266                                                         }
1267                                                 } else
1268                                                         S = S * e.op(j);
1269                                         }
1270                         return (found_dummy ? S : 0);
1271                 } else
1272                         throw(std::invalid_argument("get_clifford_comp(): expression is not a Clifford vector to the given units"));
1273         } else if (e.is_zero())
1274                 return e;
1275         else if (is_a<clifford>(e) && ex_to<clifford>(e).same_metric(c))
1276                 if ( ex_to<idx>(e.op(1)).is_numeric() &&
1277                          (ival != ex_to<numeric>(ex_to<idx>(e.op(1)).get_value()).to_int()) )
1278                         return 0;
1279                 else
1280                         return 1;
1281         else
1282                 throw(std::invalid_argument("get_clifford_comp(): expression is not usable as a Clifford vector"));
1283 }
1284
1285
1286 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic)
1287 {
1288         GINAC_ASSERT(is_a<clifford>(c));
1289         ex mu = c.op(1);
1290         if (! ex_to<idx>(mu).is_dim_numeric())
1291                 throw(std::invalid_argument("clifford_to_lst(): index should have a numeric dimension"));
1292         unsigned int D = ex_to<numeric>(ex_to<idx>(mu).get_dim()).to_int();
1293
1294         if (algebraic) // check if algebraic method is applicable
1295                 for (unsigned int i = 0; i < D; i++)
1296                         if (pow(c.subs(mu == i, subs_options::no_pattern), 2).is_zero()
1297                                 or (not is_a<numeric>(pow(c.subs(mu == i, subs_options::no_pattern), 2))))
1298                                 algebraic = false;
1299         lst V;
1300         ex v0 = remove_dirac_ONE(canonicalize_clifford(e+clifford_prime(e)).normal())/2;
1301         if (not v0.is_zero())
1302                 V.append(v0);
1303         ex e1 = canonicalize_clifford(e - v0 * dirac_ONE(ex_to<clifford>(c).get_representation_label()));
1304         if (algebraic) {
1305                 for (unsigned int i = 0; i < D; i++)
1306                         V.append(remove_dirac_ONE(
1307                                                 simplify_indexed(canonicalize_clifford(e1 * c.subs(mu == i, subs_options::no_pattern) +  c.subs(mu == i, subs_options::no_pattern) * e1))
1308                                                 / (2*pow(c.subs(mu == i, subs_options::no_pattern), 2))));
1309         } else {
1310                 try {
1311                         for (unsigned int i = 0; i < D; i++)
1312                                 V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
1313                 } catch  (std::exception &p) {
1314                         /* Try to expand dummy summations to simplify the expression*/
1315                         e1 = canonicalize_clifford(expand_dummy_sum(e, true));
1316                         V.remove_all();
1317                         v0 = remove_dirac_ONE(canonicalize_clifford(e1+clifford_prime(e1)).normal())/2;
1318                         if (not v0.is_zero()) {
1319                                 V.append(v0);
1320                                 e1 = canonicalize_clifford(e1 - v0 * dirac_ONE(ex_to<clifford>(c).get_representation_label()));
1321                         }
1322                         for (unsigned int i = 0; i < D; i++)
1323                                 V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
1324                 }
1325         }
1326         return V;
1327 }
1328
1329
1330 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)
1331 {
1332         ex x, D, cu;
1333
1334         if (! is_a<matrix>(v) && ! v.info(info_flags::list))
1335                 throw(std::invalid_argument("clifford_moebius_map(): parameter v should be either vector or list"));
1336
1337         if (is_a<clifford>(G)) {
1338                 cu = G;
1339         } else {
1340                 if (is_a<indexed>(G)) {
1341                         D = ex_to<idx>(G.op(1)).get_dim();
1342                         varidx mu((new symbol)->setflag(status_flags::dynallocated), D);
1343                         cu = clifford_unit(mu, G, rl);
1344                 } else if (is_a<matrix>(G)) {
1345                         D = ex_to<matrix>(G).rows();
1346                         idx mu((new symbol)->setflag(status_flags::dynallocated), D);
1347                         cu = clifford_unit(mu, G, rl);
1348                 } else throw(std::invalid_argument("clifford_moebius_map(): metric should be an indexed object, matrix, or a Clifford unit"));
1349
1350         }
1351
1352         x = lst_to_clifford(v, cu);
1353         ex e = clifford_to_lst(simplify_indexed(canonicalize_clifford((a * x + b) * clifford_inverse(c * x + d))), cu, false);
1354         return (is_a<matrix>(v) ? matrix(ex_to<matrix>(v).rows(), ex_to<matrix>(v).cols(), ex_to<lst>(e)) : e);
1355 }
1356
1357 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl)
1358 {
1359         if (is_a<matrix>(M) && (ex_to<matrix>(M).rows() == 2) && (ex_to<matrix>(M).cols() == 2))
1360                 return clifford_moebius_map(M.op(0), M.op(1), M.op(2), M.op(3), v, G, rl);
1361         else
1362                 throw(std::invalid_argument("clifford_moebius_map(): parameter M should be a 2x2 matrix"));
1363 }
1364
1365 } // namespace GiNaC