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