ginac/clifford.cpp

   1 /** @file clifford.cpp
   2  *
   3  *  Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2005 Johannes Gutenberg University Mainz, Germany
   7  *
   8  *  This program is free software; you can redistribute it and/or modify
   9  *  it under the terms of the GNU General Public License as published by
  10  *  the Free Software Foundation; either version 2 of the License, or
  11  *  (at your option) any later version.
  12  *
  13  *  This program is distributed in the hope that it will be useful,
  14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  16  *  GNU General Public License for more details.
  17  *
  18  *  You should have received a copy of the GNU General Public License
  19  *  along with this program; if not, write to the Free Software
  20  *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  21  */
  22
  23 #include <stdexcept>
  24
  25 #include "clifford.h"
  26
  27 #include "ex.h"
  28 #include "idx.h"
  29 #include "ncmul.h"
  30 #include "symbol.h"
  31 #include "numeric.h" // for I
  32 #include "symmetry.h"
  33 #include "lst.h"
  34 #include "relational.h"
  35 #include "operators.h"
  36 #include "add.h"
  37 #include "mul.h"
  38 #include "power.h"
  39 #include "matrix.h"
  40 #include "archive.h"
  41 #include "utils.h"
  42
  43 namespace GiNaC {
  44
  45 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(clifford, indexed,
  46   print_func<print_dflt>(&clifford::do_print_dflt).
  47   print_func<print_latex>(&clifford::do_print_latex))
  48
  49 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracone, tensor,
  50   print_func<print_dflt>(&diracone::do_print).
  51   print_func<print_latex>(&diracone::do_print_latex))
  52
  53 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(cliffordunit, tensor,
  54   print_func<print_dflt>(&cliffordunit::do_print).
  55   print_func<print_latex>(&cliffordunit::do_print_latex))
  56
  57 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma, cliffordunit,
  58   print_func<print_dflt>(&diracgamma::do_print).
  59   print_func<print_latex>(&diracgamma::do_print_latex))
  60
  61 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma5, tensor,
  62   print_func<print_dflt>(&diracgamma5::do_print).
  63   print_func<print_latex>(&diracgamma5::do_print_latex))
  64
  65 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaL, tensor,
  66   print_func<print_context>(&diracgammaL::do_print).
  67   print_func<print_latex>(&diracgammaL::do_print_latex))
  68
  69 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaR, tensor,
  70   print_func<print_context>(&diracgammaR::do_print).
  71   print_func<print_latex>(&diracgammaR::do_print_latex))
  72
  73 //////////
  74 // default constructors
  75 //////////
  76
  77 clifford::clifford() : representation_label(0), metric(0), anticommuting(true), commutator_sign(-1)
  78 {
  79         tinfo_key = TINFO_clifford;
  80 }
  81
  82 DEFAULT_CTOR(diracone)
  83 DEFAULT_CTOR(cliffordunit)
  84 DEFAULT_CTOR(diracgamma)
  85 DEFAULT_CTOR(diracgamma5)
  86 DEFAULT_CTOR(diracgammaL)
  87 DEFAULT_CTOR(diracgammaR)
  88
  89 //////////
  90 // other constructors
  91 //////////
  92
  93 /** Construct object without any indices. This constructor is for internal
  94  *  use only. Use the dirac_ONE() function instead.
  95  *  @see dirac_ONE */
  96 clifford::clifford(const ex & b, unsigned char rl, bool anticommut) : inherited(b), representation_label(rl), metric(0), anticommuting(anticommut), commutator_sign(-1)
  97 {
  98         tinfo_key = TINFO_clifford;
  99 }
 100
 101 /** Construct object with one Lorentz index. This constructor is for internal
 102  *  use only. Use the clifford_unit() or dirac_gamma() functions instead.
 103  *  @see clifford_unit
 104  *  @see dirac_gamma */
 105 clifford::clifford(const ex & b, const ex & mu, const ex & metr, unsigned char rl, bool anticommut, int comm_sign) : inherited(b, mu), representation_label(rl), metric(metr), anticommuting(anticommut), commutator_sign(comm_sign)
 106 {
 107         GINAC_ASSERT(is_a<varidx>(mu));
 108         tinfo_key = TINFO_clifford;
 109 }
 110
 111 clifford::clifford(unsigned char rl, const ex & metr, bool anticommut, int comm_sign, const exvector & v, bool discardable) : inherited(not_symmetric(), v, discardable), representation_label(rl), metric(metr), anticommuting(anticommut), commutator_sign(comm_sign)
 112 {
 113         tinfo_key = TINFO_clifford;
 114 }
 115
 116 clifford::clifford(unsigned char rl, const ex & metr, bool anticommut, int comm_sign, std::auto_ptr<exvector> vp) : inherited(not_symmetric(), vp), representation_label(rl), metric(metr), anticommuting(anticommut), commutator_sign(comm_sign)
 117 {
 118         tinfo_key = TINFO_clifford;
 119 }
 120
 121 //////////
 122 // archiving
 123 //////////
 124
 125 clifford::clifford(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
 126 {
 127         unsigned rl;
 128         n.find_unsigned("label", rl);
 129         representation_label = rl;
 130         n.find_ex("metric", metric, sym_lst);
 131         n.find_bool("anticommuting", anticommuting);
 132         n.find_unsigned("commutator_sign+1", rl);
 133         commutator_sign = rl - 1;
 134 }
 135
 136 void clifford::archive(archive_node & n) const
 137 {
 138         inherited::archive(n);
 139         n.add_unsigned("label", representation_label);
 140         n.add_ex("metric", metric);
 141         n.add_bool("anticommuting", anticommuting);
 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 indexed(metric.op(0), ex_to<symmetry>(ex_to<indexed>(metric).get_symmetry()), i, j);
 166                         return metric.subs(lst(metric.op(1) == i, metric.op(2) == j), subs_options::no_pattern);
 167                 }
 168         } else {
 169                 // should not really happen since all constructors but clifford() make the metric an indexed object
 170                 return indexed(metric, i, j);
 171         }
 172 }
 173
 174 bool clifford::same_metric(const ex & other) const
 175 {
 176         if (is_a<clifford>(other)) {
 177                 return same_metric(ex_to<clifford>(other).get_metric());
 178         } else if (is_a<indexed>(other)) {
 179                 return get_metric(other.op(1), other.op(2)).is_equal(other);
 180         } else
 181                 return false;
 182 }
 183
 184 //////////
 185 // functions overriding virtual functions from base classes
 186 //////////
 187
 188 ex clifford::op(size_t i) const
 189 {
 190         GINAC_ASSERT(i<nops());
 191         if (nops()-i == 1)
 192                 return representation_label;
 193         else
 194                 return inherited::op(i);
 195 }
 196
 197 ex & clifford::let_op(size_t i)
 198 {
 199         GINAC_ASSERT(i<nops());
 200
 201         static ex rl = numeric(representation_label);
 202         ensure_if_modifiable();
 203         if (nops()-i == 1)
 204                 return rl;
 205         else
 206                 return inherited::let_op(i);
 207 }
 208
 209 ex clifford::subs(const exmap & m, unsigned options) const
 210 {
 211         ex subsed = inherited::subs(m, options);
 212         if(is_a<clifford>(subsed)) {
 213                 ex prevmetric = ex_to<clifford>(subsed).metric;
 214                 ex newmetric = prevmetric.subs(m, options);
 215                 if(!are_ex_trivially_equal(prevmetric, newmetric)) {
 216                         clifford c = ex_to<clifford>(subsed);
 217                         c.metric = newmetric;
 218                         subsed = c;
 219                 }
 220         }
 221         return subsed;
 222 }
 223
 224 int clifford::compare_same_type(const basic & other) const
 225 {
 226         GINAC_ASSERT(is_a<clifford>(other));
 227         const clifford &o = static_cast<const clifford &>(other);
 228
 229         if (representation_label != o.representation_label) {
 230                 // different representation label
 231                 return representation_label < o.representation_label ? -1 : 1;
 232         }
 233
 234         return inherited::compare_same_type(other);
 235 }
 236
 237 bool clifford::match_same_type(const basic & other) const
 238 {
 239         GINAC_ASSERT(is_a<clifford>(other));
 240         const clifford &o = static_cast<const clifford &>(other);
 241
 242         return ((representation_label == o.representation_label) && (commutator_sign == o.get_commutator_sign()) && same_metric(o));
 243 }
 244
 245 static bool is_dirac_slash(const ex & seq0)
 246 {
 247         return !is_a<diracgamma5>(seq0) && !is_a<diracgammaL>(seq0) &&
 248                !is_a<diracgammaR>(seq0) && !is_a<cliffordunit>(seq0) &&
 249                !is_a<diracone>(seq0);
 250 }
 251
 252 void clifford::do_print_dflt(const print_dflt & c, unsigned level) const
 253 {
 254         // dirac_slash() object is printed differently
 255         if (is_dirac_slash(seq[0])) {
 256                 seq[0].print(c, precedence());
 257                 c.s << "\\";
 258         } else
 259                 this->print_dispatch<inherited>(c, level);
 260 }
 261
 262 void clifford::do_print_latex(const print_latex & c, unsigned level) const
 263 {
 264         // dirac_slash() object is printed differently
 265         if (is_dirac_slash(seq[0])) {
 266                 c.s << "{";
 267                 seq[0].print(c, precedence());
 268                 c.s << "\\hspace{-1.0ex}/}";
 269         } else {
 270                 c.s << "\\clifford[" << int(representation_label) << "]";
 271                 this->print_dispatch<inherited>(c, level);
 272         }
 273 }
 274
 275 DEFAULT_COMPARE(diracone)
 276 DEFAULT_COMPARE(cliffordunit)
 277 DEFAULT_COMPARE(diracgamma)
 278 DEFAULT_COMPARE(diracgamma5)
 279 DEFAULT_COMPARE(diracgammaL)
 280 DEFAULT_COMPARE(diracgammaR)
 281
 282 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbf{1}")
 283 DEFAULT_PRINT_LATEX(cliffordunit, "e", "e")
 284 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
 285 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
 286 DEFAULT_PRINT_LATEX(diracgammaL, "gammaL", "{\\gamma_L}")
 287 DEFAULT_PRINT_LATEX(diracgammaR, "gammaR", "{\\gamma_R}")
 288
 289 /** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
 290 static void base_and_index(const ex & c, ex & b, ex & i)
 291 {
 292         GINAC_ASSERT(is_a<clifford>(c));
 293         GINAC_ASSERT(c.nops() == 2+1);
 294
 295         if (is_a<cliffordunit>(c.op(0))) { // proper dirac gamma object or clifford unit
 296                 i = c.op(1);
 297                 b = _ex1;
 298         } else if (is_a<diracgamma5>(c.op(0)) || is_a<diracgammaL>(c.op(0)) || is_a<diracgammaR>(c.op(0))) { // gamma5/L/R
 299                 i = _ex0;
 300                 b = _ex1;
 301         } else { // slash object, generate new dummy index
 302                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(c.op(1)).get_dim());
 303                 b = indexed(c.op(0), ix.toggle_variance());
 304                 i = ix;
 305         }
 306 }
 307
 308 /** Predicate for finding non-clifford objects. */
 309 struct is_not_a_clifford : public std::unary_function<ex, bool> {
 310         bool operator()(const ex & e)
 311         {
 312                 return !is_a<clifford>(e);
 313         }
 314 };
 315
 316 /** Contraction of a gamma matrix with something else. */
 317 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 318 {
 319         GINAC_ASSERT(is_a<clifford>(*self));
 320         GINAC_ASSERT(is_a<indexed>(*other));
 321         GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
 322         unsigned char rl = ex_to<clifford>(*self).get_representation_label();
 323
 324         ex dim = ex_to<idx>(self->op(1)).get_dim();
 325         if (other->nops() > 1)
 326                 dim = minimal_dim(dim, ex_to<idx>(other->op(1)).get_dim());
 327
 328         if (is_a<clifford>(*other)) {
 329
 330                 // Contraction only makes sense if the represenation labels are equal
 331                 if (ex_to<clifford>(*other).get_representation_label() != rl)
 332                         return false;
 333
 334                 size_t num = other - self;
 335
 336                 // gamma~mu gamma.mu = dim ONE
 337                 if (num == 1) {
 338                         *self = dim;
 339                         *other = dirac_ONE(rl);
 340                         return true;
 341
 342                 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
 343                 } else if (num == 2
 344                         && is_a<clifford>(self[1])) {
 345                         *self = 2 - dim;
 346                         *other = _ex1;
 347                         return true;
 348
 349                 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
 350                 } else if (num == 3
 351                         && is_a<clifford>(self[1])
 352                         && is_a<clifford>(self[2])) {
 353                         ex b1, i1, b2, i2;
 354                         base_and_index(self[1], b1, i1);
 355                         base_and_index(self[2], b2, i2);
 356                         *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
 357                         self[1] = _ex1;
 358                         self[2] = _ex1;
 359                         *other = _ex1;
 360                         return true;
 361
 362                 // 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
 363                 } else if (num == 4
 364                         && is_a<clifford>(self[1])
 365                         && is_a<clifford>(self[2])
 366                         && is_a<clifford>(self[3])) {
 367                         *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3];
 368                         self[1] = _ex1;
 369                         self[2] = _ex1;
 370                         self[3] = _ex1;
 371                         *other = _ex1;
 372                         return true;
 373
 374                 // gamma~mu Sodd gamma.mu = -2 Sodd_R
 375                 // (Chisholm identity in 4 dimensions)
 376                 } else if (!((other - self) & 1) && dim.is_equal(4)) {
 377                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
 378                                 return false;
 379
 380                         *self = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(other), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
 381                         std::fill(self + 1, other, _ex1);
 382                         *other = _ex_2;
 383                         return true;
 384
 385                 // gamma~mu Sodd gamma~alpha gamma.mu = 2 gamma~alpha Sodd + 2 Sodd_R gamma~alpha
 386                 // (commutate contracted indices towards each other, then use
 387                 // Chisholm identity in 4 dimensions)
 388                 } else if (((other - self) & 1) && dim.is_equal(4)) {
 389                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
 390                                 return false;
 391
 392                         exvector::iterator next_to_last = other - 1;
 393                         ex S = ncmul(exvector(self + 1, next_to_last), true);
 394                         ex SR = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(next_to_last), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
 395
 396                         *self = (*next_to_last) * S + SR * (*next_to_last);
 397                         std::fill(self + 1, other, _ex1);
 398                         *other = _ex2;
 399                         return true;
 400
 401                 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
 402                 // (commutate contracted indices towards each other, simplify_indexed()
 403                 // will re-expand and re-run the simplification)
 404                 } else {
 405                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
 406                                 return false;
 407
 408                         exvector::iterator next_to_last = other - 1;
 409                         ex S = ncmul(exvector(self + 1, next_to_last), true);
 410
 411                         *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
 412                         std::fill(self + 1, other + 1, _ex1);
 413                         return true;
 414                 }
 415
 416         } else if (is_a<symbol>(other->op(0)) && other->nops() == 2) {
 417
 418                 // x.mu gamma~mu -> x-slash
 419                 *self = dirac_slash(other->op(0), dim, rl);
 420                 *other = _ex1;
 421                 return true;
 422         }
 423
 424         return false;
 425 }
 426
 427 /** An utility function looking for a given metric within an exvector,
 428  *  used in cliffordunit::contract_with(). */
 429 static int find_same_metric(exvector & v, ex & c)
 430 {
 431         for (size_t i=0; i<v.size(); i++) {
 432                 if (is_a<indexed>(v[i]) && !is_a<clifford>(v[i])
 433                     && ((ex_to<varidx>(c.op(1)) == ex_to<indexed>(v[i]).get_indices()[0]
 434                     && ex_to<varidx>(c.op(1)) == ex_to<indexed>(v[i]).get_indices()[1])
 435                     || (ex_to<varidx>(c.op(1)).toggle_variance() == ex_to<indexed>(v[i]).get_indices()[0]
 436                     && ex_to<varidx>(c.op(1)).toggle_variance() == ex_to<indexed>(v[i]).get_indices()[1]))) {
 437                         return i; // the index of the found
 438                 }
 439         }
 440         return -1; //nothing found
 441 }
 442
 443 /** Contraction of a Clifford unit with something else. */
 444 bool cliffordunit::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 445 {
 446         GINAC_ASSERT(is_a<clifford>(*self));
 447         GINAC_ASSERT(is_a<indexed>(*other));
 448         GINAC_ASSERT(is_a<cliffordunit>(self->op(0)));
 449         clifford unit = ex_to<clifford>(*self);
 450         unsigned char rl = unit.get_representation_label();
 451
 452         if (is_a<clifford>(*other)) {
 453                 // Contraction only makes sense if the represenation labels are equal
 454                 // and the metrics are the same
 455                 if ((ex_to<clifford>(*other).get_representation_label() != rl)
 456                     && unit.same_metric(*other))
 457                         return false;
 458
 459                 // Find if a previous contraction produces the square of self
 460                 int prev_square = find_same_metric(v, *self);
 461                 const varidx d((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(self->op(1)).get_dim()),
 462                         in1((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(self->op(1)).get_dim()),
 463                         in2((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(self->op(1)).get_dim());
 464                 ex squared_metric;
 465                 if (prev_square > -1)
 466                         squared_metric = simplify_indexed(indexed(v[prev_square].op(0), in1, d)
 467                                                                                           * unit.get_metric(d.toggle_variance(), in2, true)).op(0);
 468
 469                 exvector::iterator before_other = other - 1;
 470                 const varidx & mu = ex_to<varidx>(self->op(1));
 471                 const varidx & mu_toggle = ex_to<varidx>(other->op(1));
 472                 const varidx & alpha = ex_to<varidx>(before_other->op(1));
 473
 474                 // e~mu e.mu = Tr ONE
 475                 if (other - self == 1) {
 476                         if (prev_square > -1) {
 477                                 *self = indexed(squared_metric, mu, mu_toggle);
 478                                 v[prev_square] = _ex1;
 479                         } else {
 480                                 *self = unit.get_metric(mu, mu_toggle, true);
 481                         }
 482                         *other = dirac_ONE(rl);
 483                         return true;
 484
 485                 } else if (other - self == 2) {
 486                         if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
 487                                 if (ex_to<clifford>(*self).is_anticommuting()) {
 488                                         // e~mu e~alpha e.mu = (2*pow(e~alpha, 2) -Tr(B)) e~alpha
 489                                         if (prev_square > -1) {
 490                                                 *self = 2 * indexed(squared_metric, alpha, alpha)
 491                                                         - indexed(squared_metric, mu, mu_toggle);
 492                                                 v[prev_square] = _ex1;
 493                                         } else {
 494                                                 *self = 2 * unit.get_metric(alpha, alpha, true) - unit.get_metric(mu, mu_toggle, true);
 495                                         }
 496                                         *other = _ex1;
 497                                         return true;
 498
 499                                 } else {
 500                                         // e~mu e~alpha e.mu = 2*e~mu B(alpha, mu.toggle_variance())-Tr(B) e~alpha
 501                                         *self = 2 * (*self) * unit.get_metric(alpha, mu_toggle, true) - unit.get_metric(mu, mu_toggle, true) * (*before_other);
 502                                         *before_other = _ex1;
 503                                         *other = _ex1;
 504                                         return true;
 505                                 }
 506                         } else {
 507                                 // e~mu S e.mu = Tr S ONE
 508                                 *self = unit.get_metric(mu, mu_toggle, true);
 509                                 *other = dirac_ONE(rl);
 510                                 return true;
 511                         }
 512                 } else {
 513                 // e~mu S e~alpha e.mu = 2 e~mu S B(alpha, mu.toggle_variance()) - e~mu S e.mu e~alpha
 514                 // (commutate contracted indices towards each other, simplify_indexed()
 515                 // will re-expand and re-run the simplification)
 516                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other) {
 517                                 return false;
 518                         }
 519
 520                         ex S = ncmul(exvector(self + 1, before_other), true);
 521
 522                         if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
 523                                 if (ex_to<clifford>(*self).is_anticommuting()) {
 524                                         if (prev_square > -1) {
 525                                                 *self = 2 * (*before_other) * S * indexed(squared_metric, alpha, alpha)
 526                                                         - (*self) * S * (*other) * (*before_other);
 527                                         } else {
 528                                                 *self = 2 * (*before_other) * S * unit.get_metric(alpha, alpha, true) - (*self) * S * (*other) * (*before_other);
 529                                         }
 530                                 } else {
 531                                         *self = 2 * (*self) * S * unit.get_metric(alpha, mu_toggle, true) - (*self) * S * (*other) * (*before_other);
 532                                 }
 533                         } else {
 534                                 // simply commutes
 535                                 *self = (*self) * S * (*other) * (*before_other);
 536                         }
 537
 538                         std::fill(self + 1, other + 1, _ex1);
 539                         return true;
 540                 }
 541         }
 542         return false;
 543 }
 544
 545 /** Perform automatic simplification on noncommutative product of clifford
 546  *  objects. This removes superfluous ONEs, permutes gamma5/L/R's to the front
 547  *  and removes squares of gamma objects. */
 548 ex clifford::eval_ncmul(const exvector & v) const
 549 {
 550         exvector s;
 551         s.reserve(v.size());
 552
 553         // Remove superfluous ONEs
 554         exvector::const_iterator cit = v.begin(), citend = v.end();
 555         while (cit != citend) {
 556                 if (!is_a<clifford>(*cit) || !is_a<diracone>(cit->op(0)))
 557                         s.push_back(*cit);
 558                 cit++;
 559         }
 560
 561         bool something_changed = false;
 562         int sign = 1;
 563
 564         // Anticommutate gamma5/L/R's to the front
 565         if (s.size() >= 2) {
 566                 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
 567                 while (true) {
 568                         exvector::iterator it = next_to_last;
 569                         while (true) {
 570                                 exvector::iterator it2 = it + 1;
 571                                 if (is_a<clifford>(*it) && is_a<clifford>(*it2)) {
 572                                         ex e1 = it->op(0), e2 = it2->op(0);
 573
 574                                         if (is_a<diracgamma5>(e2)) {
 575
 576                                                 if (is_a<diracgammaL>(e1) || is_a<diracgammaR>(e1)) {
 577
 578                                                         // gammaL/R gamma5 -> gamma5 gammaL/R
 579                                                         it->swap(*it2);
 580                                                         something_changed = true;
 581
 582                                                 } else if (!is_a<diracgamma5>(e1)) {
 583
 584                                                         // gamma5 gamma5 -> gamma5 gamma5 (do nothing)
 585                                                         // x gamma5 -> -gamma5 x
 586                                                         it->swap(*it2);
 587                                                         sign = -sign;
 588                                                         something_changed = true;
 589                                                 }
 590
 591                                         } else if (is_a<diracgammaL>(e2)) {
 592
 593                                                 if (is_a<diracgammaR>(e1)) {
 594
 595                                                         // gammaR gammaL -> 0
 596                                                         return _ex0;
 597
 598                                                 } else if (!is_a<diracgammaL>(e1) && !is_a<diracgamma5>(e1)) {
 599
 600                                                         // gammaL gammaL -> gammaL gammaL (do nothing)
 601                                                         // gamma5 gammaL -> gamma5 gammaL (do nothing)
 602                                                         // x gammaL -> gammaR x
 603                                                         it->swap(*it2);
 604                                                         *it = clifford(diracgammaR(), ex_to<clifford>(*it).get_representation_label());
 605                                                         something_changed = true;
 606                                                 }
 607
 608                                         } else if (is_a<diracgammaR>(e2)) {
 609
 610                                                 if (is_a<diracgammaL>(e1)) {
 611
 612                                                         // gammaL gammaR -> 0
 613                                                         return _ex0;
 614
 615                                                 } else if (!is_a<diracgammaR>(e1) && !is_a<diracgamma5>(e1)) {
 616
 617                                                         // gammaR gammaR -> gammaR gammaR (do nothing)
 618                                                         // gamma5 gammaR -> gamma5 gammaR (do nothing)
 619                                                         // x gammaR -> gammaL x
 620                                                         it->swap(*it2);
 621                                                         *it = clifford(diracgammaL(), ex_to<clifford>(*it).get_representation_label());
 622                                                         something_changed = true;
 623                                                 }
 624                                         }
 625                                 }
 626                                 if (it == first)
 627                                         break;
 628                                 --it;
 629                         }
 630                         if (next_to_last == first)
 631                                 break;
 632                         --next_to_last;
 633                 }
 634         }
 635
 636         // Remove equal adjacent gammas
 637         if (s.size() >= 2) {
 638                 exvector::iterator it, itend = s.end() - 1;
 639                 for (it = s.begin(); it != itend; ++it) {
 640                         ex & a = it[0];
 641                         ex & b = it[1];
 642                         if (!is_a<clifford>(a) || !is_a<clifford>(b))
 643                                 continue;
 644
 645                         const ex & ag = a.op(0);
 646                         const ex & bg = b.op(0);
 647                         bool a_is_cliffordunit = is_a<cliffordunit>(ag);
 648                         bool b_is_cliffordunit =  is_a<cliffordunit>(bg);
 649
 650                         if (a_is_cliffordunit && b_is_cliffordunit && ex_to<clifford>(a).same_metric(b)
 651                                 && (ex_to<clifford>(a).get_commutator_sign() == -1)) {
 652                                 // This is done only for Clifford algebras
 653
 654                                 const ex & ia = a.op(1);
 655                                 const ex & ib = b.op(1);
 656                                 if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
 657                                         a = ex_to<clifford>(a).get_metric(ia, ib, true);
 658                                         b = dirac_ONE(representation_label);
 659                                         something_changed = true;
 660                                 }
 661
 662                         } else if ((is_a<diracgamma5>(ag) && is_a<diracgamma5>(bg))) {
 663
 664                                 // Remove squares of gamma5
 665                                 a = dirac_ONE(representation_label);
 666                                 b = dirac_ONE(representation_label);
 667                                 something_changed = true;
 668
 669                         } else if ((is_a<diracgammaL>(ag) && is_a<diracgammaL>(bg))
 670                                 || (is_a<diracgammaR>(ag) && is_a<diracgammaR>(bg))) {
 671
 672                                 // Remove squares of gammaL/R
 673                                 b = dirac_ONE(representation_label);
 674                                 something_changed = true;
 675
 676                         } else if (is_a<diracgammaL>(ag) && is_a<diracgammaR>(bg)) {
 677
 678                                 // gammaL and gammaR are orthogonal
 679                                 return _ex0;
 680
 681                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaL>(bg)) {
 682
 683                                 // gamma5 gammaL -> -gammaL
 684                                 a = dirac_ONE(representation_label);
 685                                 sign = -sign;
 686                                 something_changed = true;
 687
 688                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaR>(bg)) {
 689
 690                                 // gamma5 gammaR -> gammaR
 691                                 a = dirac_ONE(representation_label);
 692                                 something_changed = true;
 693
 694                         } else if (!a_is_cliffordunit && !b_is_cliffordunit && ag.is_equal(bg)) {
 695
 696                                 // a\ a\ -> a^2
 697                                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).minimal_dim(ex_to<idx>(b.op(1))));
 698
 699                                 a = indexed(ag, ix) * indexed(ag, ix.toggle_variance());
 700                                 b = dirac_ONE(representation_label);
 701                                 something_changed = true;
 702                         }
 703                 }
 704         }
 705
 706         if (s.empty())
 707                 return dirac_ONE(representation_label) * sign;
 708         if (something_changed)
 709                 return reeval_ncmul(s) * sign;
 710         else
 711                 return hold_ncmul(s) * sign;
 712 }
 713
 714 ex clifford::thiscontainer(const exvector & v) const
 715 {
 716         return clifford(representation_label, metric, anticommuting, commutator_sign, v);
 717 }
 718
 719 ex clifford::thiscontainer(std::auto_ptr<exvector> vp) const
 720 {
 721         return clifford(representation_label, metric, anticommuting, commutator_sign, vp);
 722 }
 723
 724 ex diracgamma5::conjugate() const
 725 {
 726         return _ex_1 * (*this);
 727 }
 728
 729 ex diracgammaL::conjugate() const
 730 {
 731         return (new diracgammaR)->setflag(status_flags::dynallocated);
 732 }
 733
 734 ex diracgammaR::conjugate() const
 735 {
 736         return (new diracgammaL)->setflag(status_flags::dynallocated);
 737 }
 738
 739 //////////
 740 // global functions
 741 //////////
 742
 743 ex dirac_ONE(unsigned char rl)
 744 {
 745         static ex ONE = (new diracone)->setflag(status_flags::dynallocated);
 746         return clifford(ONE, rl, false);
 747 }
 748
 749 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl, bool anticommuting)
 750 {
 751         static ex unit = (new cliffordunit)->setflag(status_flags::dynallocated);
 752
 753         if (!is_a<idx>(mu))
 754                 throw(std::invalid_argument("clifford_unit(): index of Clifford unit must be of type idx or varidx"));
 755
 756         if (ex_to<idx>(mu).is_symbolic() && !is_a<varidx>(mu))
 757                 throw(std::invalid_argument("clifford_unit(): symbolic index of Clifford unit must be of type varidx (not idx)"));
 758
 759         if (is_a<indexed>(metr)) {
 760                 exvector indices = ex_to<indexed>(metr).get_indices();
 761                 if ((indices.size() == 2) && is_a<varidx>(indices[0]) && is_a<varidx>(indices[1])) {
 762                         return clifford(unit, mu, metr, rl, anticommuting);
 763                 } else {
 764                         throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be indexed exactly by two indices of same type as the given index"));
 765                 }
 766         } else if (is_a<tensor>(metr)) {
 767                 static varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim()),
 768                         chi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim());
 769                 return clifford(unit, mu, indexed(metr, xi, chi), rl, anticommuting);
 770         } else if (is_a<matrix>(metr)) {
 771                 matrix M = ex_to<matrix>(metr);
 772                 unsigned n = M.rows();
 773                 bool symmetric = true;
 774                 anticommuting = true;
 775
 776                 static varidx xi((new symbol)->setflag(status_flags::dynallocated), n),
 777                         chi((new symbol)->setflag(status_flags::dynallocated), n);
 778                 if ((n ==  M.cols()) && (n == ex_to<varidx>(mu).get_dim())) {
 779                         for (unsigned i = 0; i < n; i++) {
 780                                 for (unsigned j = i+1; j < n; j++) {
 781                                         if (M(i, j) != M(j, i)) {
 782                                                 symmetric = false;
 783                                         }
 784                                         if (M(i, j) != -M(j, i)) {
 785                                                 anticommuting = false;
 786                                         }
 787                                 }
 788                         }
 789                         return clifford(unit, mu, indexed(metr, symmetric?symmetric2():not_symmetric(), xi, chi), rl, anticommuting);
 790                 } else {
 791                         throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be a square matrix with the same dimensions as index"));
 792                 }
 793         } else {
 794                 throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be of type indexed, tensor or matrix"));
 795         }
 796 }
 797
 798 ex dirac_gamma(const ex & mu, unsigned char rl)
 799 {
 800         static ex gamma = (new diracgamma)->setflag(status_flags::dynallocated);
 801
 802         if (!is_a<varidx>(mu))
 803                 throw(std::invalid_argument("dirac_gamma(): index of Dirac gamma must be of type varidx"));
 804
 805         static varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim()),
 806                 chi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim());
 807         return clifford(gamma, mu, indexed((new minkmetric)->setflag(status_flags::dynallocated), symmetric2(), xi, chi), rl, true);
 808 }
 809
 810 ex dirac_gamma5(unsigned char rl)
 811 {
 812         static ex gamma5 = (new diracgamma5)->setflag(status_flags::dynallocated);
 813         return clifford(gamma5, rl);
 814 }
 815
 816 ex dirac_gammaL(unsigned char rl)
 817 {
 818         static ex gammaL = (new diracgammaL)->setflag(status_flags::dynallocated);
 819         return clifford(gammaL, rl);
 820 }
 821
 822 ex dirac_gammaR(unsigned char rl)
 823 {
 824         static ex gammaR = (new diracgammaR)->setflag(status_flags::dynallocated);
 825         return clifford(gammaR, rl);
 826 }
 827
 828 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
 829 {
 830         // Slashed vectors are actually stored as a clifford object with the
 831         // vector as its base expression and a (dummy) index that just serves
 832         // for storing the space dimensionality
 833
 834         static varidx xi((new symbol)->setflag(status_flags::dynallocated), dim),
 835                 chi((new symbol)->setflag(status_flags::dynallocated), dim);
 836    return clifford(e, varidx(0, dim), indexed((new minkmetric)->setflag(status_flags::dynallocated), symmetric2(), xi, chi), rl, true);
 837 }
 838
 839 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 840  *  is that of a clifford object with the specified representation label. */
 841 static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
 842 {
 843         return ti == (TINFO_clifford + rl);
 844 }
 845
 846 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 847  *  is that of a clifford object (with an arbitrary representation label). */
 848 static bool is_clifford_tinfo(unsigned ti)
 849 {
 850         return (ti & ~0xff) == TINFO_clifford;
 851 }
 852
 853 /** Extract representation label from tinfo key (as returned by
 854  *  return_type_tinfo()). */
 855 static unsigned char get_representation_label(unsigned ti)
 856 {
 857         return ti & 0xff;
 858 }
 859
 860 /** Take trace of a string of an even number of Dirac gammas given a vector
 861  *  of indices. */
 862 static ex trace_string(exvector::const_iterator ix, size_t num)
 863 {
 864         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 865         if (num == 2)
 866                 return lorentz_g(ix[0], ix[1]);
 867
 868         // 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 )
 869         else if (num == 4)
 870                 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
 871                      + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
 872                      - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
 873
 874         // Traces of 6 or more gammas are computed recursively:
 875         // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
 876         //   + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
 877         //   - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
 878         //   + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
 879         //   - ...
 880         //   + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
 881         exvector v(num - 2);
 882         int sign = 1;
 883         ex result;
 884         for (size_t i=1; i<num; i++) {
 885                 for (size_t n=1, j=0; n<num; n++) {
 886                         if (n == i)
 887                                 continue;
 888                         v[j++] = ix[n];
 889                 }
 890                 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
 891                 sign = -sign;
 892         }
 893         return result;
 894 }
 895
 896 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE)
 897 {
 898         if (is_a<clifford>(e)) {
 899
 900                 unsigned char rl = ex_to<clifford>(e).get_representation_label();
 901
 902                 // Are we taking the trace over this object's representation label?
 903                 if (rls.find(rl) == rls.end())
 904                         return e;
 905
 906                 // Yes, all elements are traceless, except for dirac_ONE and dirac_L/R
 907                 const ex & g = e.op(0);
 908                 if (is_a<diracone>(g))
 909                         return trONE;
 910                 else if (is_a<diracgammaL>(g) || is_a<diracgammaR>(g))
 911                         return trONE/2;
 912                 else
 913                         return _ex0;
 914
 915         } else if (is_exactly_a<mul>(e)) {
 916
 917                 // Trace of product: pull out non-clifford factors
 918                 ex prod = _ex1;
 919                 for (size_t i=0; i<e.nops(); i++) {
 920                         const ex &o = e.op(i);
 921                         if (is_clifford_tinfo(o.return_type_tinfo()))
 922                                 prod *= dirac_trace(o, rls, trONE);
 923                         else
 924                                 prod *= o;
 925                 }
 926                 return prod;
 927
 928         } else if (is_exactly_a<ncmul>(e)) {
 929
 930                 unsigned char rl = get_representation_label(e.return_type_tinfo());
 931
 932                 // Are we taking the trace over this string's representation label?
 933                 if (rls.find(rl) == rls.end())
 934                         return e;
 935
 936                 // Substitute gammaL/R and expand product, if necessary
 937                 ex e_expanded = e.subs(lst(
 938                         dirac_gammaL(rl) == (dirac_ONE(rl)-dirac_gamma5(rl))/2,
 939                         dirac_gammaR(rl) == (dirac_ONE(rl)+dirac_gamma5(rl))/2
 940                 ), subs_options::no_pattern).expand();
 941                 if (!is_a<ncmul>(e_expanded))
 942                         return dirac_trace(e_expanded, rls, trONE);
 943
 944                 // gamma5 gets moved to the front so this check is enough
 945                 bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
 946                 size_t num = e.nops();
 947
 948                 if (has_gamma5) {
 949
 950                         // Trace of gamma5 * odd number of gammas and trace of
 951                         // gamma5 * gamma.mu * gamma.nu are zero
 952                         if ((num & 1) == 0 || num == 3)
 953                                 return _ex0;
 954
 955                         // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
 956                         // (the epsilon is always 4-dimensional)
 957                         if (num == 5) {
 958                                 ex b1, i1, b2, i2, b3, i3, b4, i4;
 959                                 base_and_index(e.op(1), b1, i1);
 960                                 base_and_index(e.op(2), b2, i2);
 961                                 base_and_index(e.op(3), b3, i3);
 962                                 base_and_index(e.op(4), b4, i4);
 963                                 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();
 964                         }
 965
 966                         // Tr gamma5 S_2k =
 967                         //   I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
 968                         // (the epsilon is always 4-dimensional)
 969                         exvector ix(num-1), bv(num-1);
 970                         for (size_t i=1; i<num; i++)
 971                                 base_and_index(e.op(i), bv[i-1], ix[i-1]);
 972                         num--;
 973                         int *iv = new int[num];
 974                         ex result;
 975                         for (size_t i=0; i<num-3; i++) {
 976                                 ex idx1 = ix[i];
 977                                 for (size_t j=i+1; j<num-2; j++) {
 978                                         ex idx2 = ix[j];
 979                                         for (size_t k=j+1; k<num-1; k++) {
 980                                                 ex idx3 = ix[k];
 981                                                 for (size_t l=k+1; l<num; l++) {
 982                                                         ex idx4 = ix[l];
 983                                                         iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
 984                                                         exvector v;
 985                                                         v.reserve(num - 4);
 986                                                         for (size_t n=0, t=4; n<num; n++) {
 987                                                                 if (n == i || n == j || n == k || n == l)
 988                                                                         continue;
 989                                                                 iv[t++] = n;
 990                                                                 v.push_back(ix[n]);
 991                                                         }
 992                                                         int sign = permutation_sign(iv, iv + num);
 993                                                         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))
 994                                                                 * trace_string(v.begin(), num - 4);
 995                                                 }
 996                                         }
 997                                 }
 998                         }
 999                         delete[] iv;
1000                         return trONE * I * result * mul(bv);
1001
1002                 } else { // no gamma5
1003
1004                         // Trace of odd number of gammas is zero
1005                         if ((num & 1) == 1)
1006                                 return _ex0;
1007
1008                         // Tr gamma.mu gamma.nu = 4 g.mu.nu
1009                         if (num == 2) {
1010                                 ex b1, i1, b2, i2;
1011                                 base_and_index(e.op(0), b1, i1);
1012                                 base_and_index(e.op(1), b2, i2);
1013                                 return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
1014                         }
1015
1016                         exvector iv(num), bv(num);
1017                         for (size_t i=0; i<num; i++)
1018                                 base_and_index(e.op(i), bv[i], iv[i]);
1019
1020                         return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
1021                 }
1022
1023         } else if (e.nops() > 0) {
1024
1025                 // Trace maps to all other container classes (this includes sums)
1026                 pointer_to_map_function_2args<const std::set<unsigned char> &, const ex &> fcn(dirac_trace, rls, trONE);
1027                 return e.map(fcn);
1028
1029         } else
1030                 return _ex0;
1031 }
1032
1033 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE)
1034 {
1035         // Convert list to set
1036         std::set<unsigned char> rls;
1037         for (lst::const_iterator i = rll.begin(); i != rll.end(); ++i) {
1038                 if (i->info(info_flags::nonnegint))
1039                         rls.insert(ex_to<numeric>(*i).to_int());
1040         }
1041
1042         return dirac_trace(e, rls, trONE);
1043 }
1044
1045 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
1046 {
1047         // Convert label to set
1048         std::set<unsigned char> rls;
1049         rls.insert(rl);
1050
1051         return dirac_trace(e, rls, trONE);
1052 }
1053
1054
1055 ex canonicalize_clifford(const ex & e_)
1056 {
1057         pointer_to_map_function fcn(canonicalize_clifford);
1058
1059         if (is_a<matrix>(e_)    // || is_a<pseries>(e) || is_a<integral>(e)
1060                 || is_a<lst>(e_)) {
1061                 return e_.map(fcn);
1062         } else {
1063                 ex e=simplify_indexed(e_);
1064                 // Scan for any ncmul objects
1065                 exmap srl;
1066                 ex aux = e.to_rational(srl);
1067                 for (exmap::iterator i = srl.begin(); i != srl.end(); ++i) {
1068
1069                         ex lhs = i->first;
1070                         ex rhs = i->second;
1071
1072                         if (is_exactly_a<ncmul>(rhs)
1073                                         && rhs.return_type() == return_types::noncommutative
1074                                         && is_clifford_tinfo(rhs.return_type_tinfo())) {
1075
1076                                 // Expand product, if necessary
1077                                 ex rhs_expanded = rhs.expand();
1078                                 if (!is_a<ncmul>(rhs_expanded)) {
1079                                         i->second = canonicalize_clifford(rhs_expanded);
1080                                         continue;
1081
1082                                 } else if (!is_a<clifford>(rhs.op(0)))
1083                                         continue;
1084
1085                                 exvector v;
1086                                 v.reserve(rhs.nops());
1087                                 for (size_t j=0; j<rhs.nops(); j++)
1088                                         v.push_back(rhs.op(j));
1089
1090                                 // Stupid recursive bubble sort because we only want to swap adjacent gammas
1091                                 exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
1092                                 if (is_a<diracgamma5>(it->op(0)) || is_a<diracgammaL>(it->op(0)) || is_a<diracgammaR>(it->op(0)))
1093                                         ++it;
1094
1095                                 while (it != next_to_last) {
1096                                         if (it[0].compare(it[1]) > 0) {
1097
1098                                                 ex save0 = it[0], save1 = it[1];
1099                                                 ex b1, i1, b2, i2;
1100                                                 base_and_index(it[0], b1, i1);
1101                                                 base_and_index(it[1], b2, i2);
1102                                                 // for Clifford algebras (commutator_sign == -1) metric should be symmetrised
1103                                                 it[0] = (ex_to<clifford>(save0).get_metric(i1, i2, ex_to<clifford>(save0).get_commutator_sign() == -1) * b1 * b2).simplify_indexed();
1104                                                 it[1] = v.size() ? _ex2 * dirac_ONE(ex_to<clifford>(save0).get_representation_label()) : _ex2;
1105                                                 ex sum = ncmul(v);
1106                                                 it[0] = save1;
1107                                                 it[1] = save0;
1108                                                 sum += ex_to<clifford>(save0).get_commutator_sign() * ncmul(v, true);
1109                                                 i->second = canonicalize_clifford(sum);
1110                                                 goto next_sym;
1111                                         }
1112                                         ++it;
1113                                 }
1114 next_sym:       ;
1115                         }
1116                 }
1117                 return aux.subs(srl, subs_options::no_pattern).simplify_indexed();
1118         }
1119 }
1120
1121 ex clifford_prime(const ex & e)
1122 {
1123         pointer_to_map_function fcn(clifford_prime);
1124         if (is_a<clifford>(e) && is_a<cliffordunit>(e.op(0))) {
1125                 return -e;
1126         } else if (is_a<add>(e) || is_a<ncmul>(e) || is_a<mul>(e) //|| is_a<pseries>(e) || is_a<integral>(e)
1127                            || is_a<matrix>(e) || is_a<lst>(e)) {
1128                 return e.map(fcn);
1129         } else if (is_a<power>(e)) {
1130                 return pow(clifford_prime(e.op(0)), e.op(1));
1131         } else
1132                 return e;
1133 }
1134
1135 ex remove_dirac_ONE(const ex & e, unsigned char rl, unsigned options)
1136 {
1137         pointer_to_map_function_2args<unsigned char, unsigned> fcn(remove_dirac_ONE, rl, options | 1);
1138         bool need_reevaluation = false;
1139         ex e1 = e;
1140         if (! (options & 1) )  { // is not a child
1141                 if (options & 2)
1142                         e1 = expand_dummy_sum(e, true);
1143                 e1 = canonicalize_clifford(e1);
1144         }
1145
1146         if (is_a<clifford>(e1) && ex_to<clifford>(e1).get_representation_label() >= rl) {
1147                 if (is_a<diracone>(e1.op(0)))
1148                         return 1;
1149                 else
1150                         throw(std::invalid_argument("remove_dirac_ONE(): expression is a non-scalar Clifford number!"));
1151         } else if (is_a<add>(e1) || is_a<ncmul>(e1) || is_a<mul>(e1)
1152                            || is_a<matrix>(e1) || is_a<lst>(e1)) {
1153                 if (options & 3) // is a child or was already expanded
1154                         return e1.map(fcn);
1155                 else
1156                         try {
1157                                 return e1.map(fcn);
1158                         } catch (std::exception &p) {
1159                                 need_reevaluation = true;
1160                         }
1161         } else if (is_a<power>(e1)) {
1162                 if (options & 3) // is a child or was already expanded
1163                         return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
1164                 else
1165                         try {
1166                                 return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
1167                         } catch (std::exception &p) {
1168                                 need_reevaluation = true;
1169                         }
1170         }
1171         if (need_reevaluation)
1172                 return remove_dirac_ONE(e, rl, options | 2);
1173         return e1;
1174 }
1175
1176 char clifford_max_label(const ex & e, bool ignore_ONE)
1177 {
1178         if (is_a<clifford>(e))
1179                 if (ignore_ONE && is_a<diracone>(e.op(0)))
1180                         return -1;
1181                 else
1182                         return ex_to<clifford>(e).get_representation_label();
1183         else {
1184                 char rl = -1;
1185                 for (size_t i=0; i < e.nops(); i++)
1186                         rl = (rl > clifford_max_label(e.op(i), ignore_ONE)) ? rl : clifford_max_label(e.op(i), ignore_ONE);
1187                 return rl;
1188         }
1189 }
1190
1191 ex clifford_norm(const ex & e)
1192 {
1193         return sqrt(remove_dirac_ONE(e * clifford_bar(e)));
1194 }
1195
1196 ex clifford_inverse(const ex & e)
1197 {
1198         ex norm = clifford_norm(e);
1199         if (!norm.is_zero())
1200                 return clifford_bar(e) / pow(norm, 2);
1201         else
1202                 throw(std::invalid_argument("clifford_inverse(): cannot find inverse of Clifford number with zero norm!"));
1203 }
1204
1205 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl, bool anticommuting)
1206 {
1207         if (!ex_to<idx>(mu).is_dim_numeric())
1208                 throw(std::invalid_argument("lst_to_clifford(): Index should have a numeric dimension"));
1209         ex e = clifford_unit(mu, metr, rl, anticommuting);
1210         return lst_to_clifford(v, e);
1211 }
1212
1213 ex lst_to_clifford(const ex & v, const ex & e) {
1214         unsigned min, max;
1215
1216         if (is_a<clifford>(e)) {
1217                 varidx mu = ex_to<varidx>(e.op(1));
1218                 unsigned dim = (ex_to<numeric>(mu.get_dim())).to_int();
1219
1220                 if (is_a<matrix>(v)) {
1221                         if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows()) {
1222                                 min = ex_to<matrix>(v).rows();
1223                                 max = ex_to<matrix>(v).cols();
1224                         } else {
1225                                 min = ex_to<matrix>(v).cols();
1226                                 max = ex_to<matrix>(v).rows();
1227                         }
1228                         if (min == 1) {
1229                                 if (dim == max)
1230                                         return indexed(v, ex_to<varidx>(mu).toggle_variance()) * e;
1231                                 else
1232                                         throw(std::invalid_argument("lst_to_clifford(): dimensions of vector and clifford unit mismatch"));
1233                         } else
1234                                 throw(std::invalid_argument("lst_to_clifford(): first argument should be a vector vector"));
1235                 } else if (is_a<lst>(v)) {
1236                         if (dim == ex_to<lst>(v).nops())
1237                                 return indexed(matrix(dim, 1, ex_to<lst>(v)), ex_to<varidx>(mu).toggle_variance()) * e;
1238                         else
1239                                 throw(std::invalid_argument("lst_to_clifford(): list length and dimension of clifford unit mismatch"));
1240                 } else
1241                         throw(std::invalid_argument("lst_to_clifford(): cannot construct from anything but list or vector"));
1242         } else
1243                 throw(std::invalid_argument("lst_to_clifford(): the second argument should be a Clifford unit"));
1244 }
1245
1246 /** Auxiliary structure to define a function for striping one Clifford unit
1247  * from vectors. Used in  clifford_to_lst(). */
1248 static ex get_clifford_comp(const ex & e, const ex & c)
1249 {
1250         pointer_to_map_function_1arg<const ex &> fcn(get_clifford_comp, c);
1251         int ival = ex_to<numeric>(ex_to<varidx>(c.op(1)).get_value()).to_int();
1252
1253         if (is_a<add>(e) || is_a<lst>(e) // || is_a<pseries>(e) || is_a<integral>(e)
1254                 || is_a<matrix>(e))
1255                 return e.map(fcn);
1256         else if (is_a<ncmul>(e) || is_a<mul>(e)) {
1257                 // find a Clifford unit with the same metric, delete it and substitute its index
1258                 size_t ind = e.nops() + 1;
1259                 for (size_t j = 0; j < e.nops(); j++)
1260                         if (is_a<clifford>(e.op(j)) && ex_to<clifford>(c).same_metric(e.op(j)))
1261                                 if (ind > e.nops())
1262                                         ind = j;
1263                                 else
1264                                         throw(std::invalid_argument("get_clifford_comp(): expression is a Clifford multi-vector"));
1265                 if (ind < e.nops()) {
1266                         ex S = 1;
1267                         bool same_value_index, found_dummy;
1268                         same_value_index = ( ex_to<varidx>(e.op(ind).op(1)).is_numeric()
1269                                                                  &&  (ival == ex_to<numeric>(ex_to<varidx>(e.op(ind).op(1)).get_value()).to_int()) );
1270                         found_dummy = same_value_index;
1271                         for(size_t j=0; j < e.nops(); j++)
1272                                 if (j != ind)
1273                                         if (same_value_index)
1274                                                 S = S * e.op(j);
1275                                         else {
1276                                                 exvector ind_vec = ex_to<indexed>(e.op(j)).get_dummy_indices(ex_to<indexed>(e.op(ind)));
1277                                                 if (ind_vec.size() > 0) {
1278                                                         found_dummy = true;
1279                                                         exvector::const_iterator it = ind_vec.begin(), itend = ind_vec.end();
1280                                                         while (it != itend) {
1281                                                                 S = S * e.op(j).subs(lst(ex_to<varidx>(*it) == ival, ex_to<varidx>(*it).toggle_variance() == ival), subs_options::no_pattern);
1282                                                                 ++it;
1283                                                         }
1284                                                 } else
1285                                                         S = S * e.op(j);
1286                                         }
1287                         return (found_dummy ? S : 0);
1288                 } else
1289                         throw(std::invalid_argument("get_clifford_comp(): expression is not a Clifford vector to the given units"));
1290         } else if (e.is_zero())
1291                 return e;
1292         else if (is_a<clifford>(e) && ex_to<clifford>(e).same_metric(c))
1293                 if ( ex_to<varidx>(e.op(1)).is_numeric() &&
1294                          (ival != ex_to<numeric>(ex_to<varidx>(e.op(1)).get_value()).to_int()) )
1295                         return 0;
1296                 else
1297                         return 1;
1298         else
1299                 throw(std::invalid_argument("get_clifford_comp(): expression is not usable as a Clifford vector"));
1300 }
1301
1302
1303 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic)
1304 {
1305         GINAC_ASSERT(is_a<clifford>(c));
1306         varidx mu = ex_to<varidx>(c.op(1));
1307         if (! mu.is_dim_numeric())
1308                 throw(std::invalid_argument("clifford_to_lst(): index should have a numeric dimension"));
1309         unsigned int D = ex_to<numeric>(mu.get_dim()).to_int();
1310
1311         if (algebraic) // check if algebraic method is applicable
1312                 for (unsigned int i = 0; i < D; i++)
1313                         if (pow(c.subs(mu == i, subs_options::no_pattern), 2).is_zero()
1314                                 or (not is_a<numeric>(pow(c.subs(mu == i, subs_options::no_pattern), 2))))
1315                                 algebraic = false;
1316         lst V;
1317         if (algebraic) {
1318                 for (unsigned int i = 0; i < D; i++)
1319                         V.append(remove_dirac_ONE(
1320                                                 simplify_indexed(canonicalize_clifford(e * c.subs(mu == i, subs_options::no_pattern) +  c.subs(mu == i, subs_options::no_pattern) * e))
1321                                                 / (2*pow(c.subs(mu == i, subs_options::no_pattern), 2))));
1322         } else {
1323                 ex e1 = canonicalize_clifford(e);
1324                 try {
1325                         for (unsigned int i = 0; i < D; i++)
1326                                 V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
1327                 } catch  (std::exception &p) {
1328                         /* Try to expand dummy summations to simplify the expression*/
1329                         e1 = canonicalize_clifford(expand_dummy_sum(e1, true));
1330                         for (unsigned int i = 0; i < D; i++)
1331                                 V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
1332                 }
1333         }
1334         return V;
1335 }
1336
1337
1338 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c, const ex & d, const ex & v, const ex & G, unsigned char rl, bool anticommuting)
1339 {
1340         ex x, D, cu;
1341
1342         if (! is_a<matrix>(v) && ! is_a<lst>(v))
1343                 throw(std::invalid_argument("clifford_moebius_map(): parameter v should be either vector or list"));
1344
1345         if (is_a<clifford>(G)) {
1346                 cu = G;
1347         } else {
1348                 if (is_a<indexed>(G))
1349                         D = ex_to<varidx>(G.op(1)).get_dim();
1350                 else if (is_a<matrix>(G))
1351                         D = ex_to<matrix>(G).rows();
1352                 else throw(std::invalid_argument("clifford_moebius_map(): metric should be an indexed object, matrix, or a Clifford unit"));
1353
1354                 varidx mu((new symbol)->setflag(status_flags::dynallocated), D);
1355                 cu = clifford_unit(mu, G, rl, anticommuting);
1356         }
1357
1358         x = lst_to_clifford(v, cu);
1359         ex e = simplify_indexed(canonicalize_clifford((a * x + b) * clifford_inverse(c * x + d)));
1360         return clifford_to_lst(e, cu, false);
1361 }
1362
1363 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl, bool anticommuting)
1364 {
1365         if (is_a<matrix>(M))
1366                 return clifford_moebius_map(ex_to<matrix>(M)(0,0), ex_to<matrix>(M)(0,1),
1367                                             ex_to<matrix>(M)(1,0), ex_to<matrix>(M)(1,1), v, G, rl, anticommuting);
1368         else
1369                 throw(std::invalid_argument("clifford_moebius_map(): parameter M should be a matrix"));
1370 }
1371
1372 } // namespace GiNaC