ginac/clifford.cpp

   1 /** @file clifford.cpp
   2  *
   3  *  Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2004 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., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  21  */
  22
  23 #include "clifford.h"
  24
  25 #include "ex.h"
  26 #include "idx.h"
  27 #include "ncmul.h"
  28 #include "symbol.h"
  29 #include "numeric.h" // for I
  30 #include "symmetry.h"
  31 #include "lst.h"
  32 #include "relational.h"
  33 #include "operators.h"
  34 #include "add.h"
  35 #include "mul.h"
  36 #include "power.h"
  37 #include "matrix.h"
  38 #include "archive.h"
  39 #include "utils.h"
  40
  41 namespace GiNaC {
  42
  43 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(clifford, indexed,
  44   print_func<print_dflt>(&clifford::do_print_dflt).
  45   print_func<print_latex>(&clifford::do_print_latex))
  46
  47 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracone, tensor,
  48   print_func<print_dflt>(&diracone::do_print).
  49   print_func<print_latex>(&diracone::do_print_latex))
  50
  51 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(cliffordunit, tensor,
  52   print_func<print_dflt>(&cliffordunit::do_print).
  53   print_func<print_latex>(&cliffordunit::do_print_latex))
  54
  55 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma, cliffordunit,
  56   print_func<print_dflt>(&diracgamma::do_print).
  57   print_func<print_latex>(&diracgamma::do_print_latex))
  58
  59 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma5, tensor,
  60   print_func<print_dflt>(&diracgamma5::do_print).
  61   print_func<print_latex>(&diracgamma5::do_print_latex))
  62
  63 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaL, tensor,
  64   print_func<print_context>(&diracgammaL::do_print).
  65   print_func<print_latex>(&diracgammaL::do_print_latex))
  66
  67 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaR, tensor,
  68   print_func<print_context>(&diracgammaR::do_print).
  69   print_func<print_latex>(&diracgammaR::do_print_latex))
  70
  71 //////////
  72 // default constructors
  73 //////////
  74
  75 static ex default_metric()
  76 {
  77         static ex m = (new minkmetric)->setflag(status_flags::dynallocated);
  78         return m;
  79 }
  80
  81 clifford::clifford() : representation_label(0), metric(default_metric())
  82 {
  83         tinfo_key = TINFO_clifford;
  84 }
  85
  86 DEFAULT_CTOR(diracone)
  87 DEFAULT_CTOR(cliffordunit)
  88 DEFAULT_CTOR(diracgamma)
  89 DEFAULT_CTOR(diracgamma5)
  90 DEFAULT_CTOR(diracgammaL)
  91 DEFAULT_CTOR(diracgammaR)
  92
  93 //////////
  94 // other constructors
  95 //////////
  96
  97 /** Construct object without any indices. This constructor is for internal
  98  *  use only. Use the dirac_ONE() function instead.
  99  *  @see dirac_ONE */
 100 clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl), metric(0)
 101 {
 102         tinfo_key = TINFO_clifford;
 103 }
 104
 105 /** Construct object with one Lorentz index. This constructor is for internal
 106  *  use only. Use the clifford_unit() or dirac_gamma() functions instead.
 107  *  @see clifford_unit
 108  *  @see dirac_gamma */
 109 clifford::clifford(const ex & b, const ex & mu, const ex & metr, unsigned char rl) : inherited(b, mu), representation_label(rl), metric(metr)
 110 {
 111         GINAC_ASSERT(is_a<varidx>(mu));
 112         tinfo_key = TINFO_clifford;
 113 }
 114
 115 clifford::clifford(unsigned char rl, const ex & metr, const exvector & v, bool discardable) : inherited(not_symmetric(), v, discardable), representation_label(rl), metric(metr)
 116 {
 117         tinfo_key = TINFO_clifford;
 118 }
 119
 120 clifford::clifford(unsigned char rl, const ex & metr, std::auto_ptr<exvector> vp) : inherited(not_symmetric(), vp), representation_label(rl), metric(metr)
 121 {
 122         tinfo_key = TINFO_clifford;
 123 }
 124
 125 //////////
 126 // archiving
 127 //////////
 128
 129 clifford::clifford(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
 130 {
 131         unsigned rl;
 132         n.find_unsigned("label", rl);
 133         representation_label = rl;
 134         n.find_ex("metric", metric, sym_lst);
 135 }
 136
 137 void clifford::archive(archive_node &n) const
 138 {
 139         inherited::archive(n);
 140         n.add_unsigned("label", representation_label);
 141         n.add_ex("metric", metric);
 142 }
 143
 144 DEFAULT_UNARCHIVE(clifford)
 145 DEFAULT_ARCHIVING(diracone)
 146 DEFAULT_ARCHIVING(cliffordunit)
 147 DEFAULT_ARCHIVING(diracgamma)
 148 DEFAULT_ARCHIVING(diracgamma5)
 149 DEFAULT_ARCHIVING(diracgammaL)
 150 DEFAULT_ARCHIVING(diracgammaR)
 151
 152 //////////
 153 // functions overriding virtual functions from base classes
 154 //////////
 155
 156 ex clifford::get_metric(const ex & i, const ex & j) const
 157 {
 158         return indexed(metric, symmetric2(), i, j);
 159 }
 160
 161 bool clifford::same_metric(const ex & other) const
 162 {
 163         if (is_a<clifford>(other)) {
 164                 return get_metric().is_equal(ex_to<clifford>(other).get_metric());
 165         } else if (is_a<indexed>(other)) {
 166                 return get_metric(other.op(1), other.op(2)).is_equal(other);
 167         } else
 168                 return false;
 169 }
 170
 171 int clifford::compare_same_type(const basic & other) const
 172 {
 173         GINAC_ASSERT(is_a<clifford>(other));
 174         const clifford &o = static_cast<const clifford &>(other);
 175
 176         if (representation_label != o.representation_label) {
 177                 // different representation label
 178                 return representation_label < o.representation_label ? -1 : 1;
 179         }
 180
 181         return inherited::compare_same_type(other);
 182 }
 183
 184 bool clifford::match_same_type(const basic & other) const
 185 {
 186         GINAC_ASSERT(is_a<clifford>(other));
 187         const clifford &o = static_cast<const clifford &>(other);
 188
 189         return (representation_label == o.representation_label) && same_metric(o);
 190 }
 191
 192 static bool is_dirac_slash(const ex & seq0)
 193 {
 194         return !is_a<diracgamma5>(seq0) && !is_a<diracgammaL>(seq0) &&
 195                !is_a<diracgammaR>(seq0) && !is_a<cliffordunit>(seq0) &&
 196                !is_a<diracone>(seq0);
 197 }
 198
 199 void clifford::do_print_dflt(const print_dflt & c, unsigned level) const
 200 {
 201         // dirac_slash() object is printed differently
 202         if (is_dirac_slash(seq[0])) {
 203                 seq[0].print(c, level);
 204                 c.s << "\\";
 205         } else
 206                 this->print_dispatch<inherited>(c, level);
 207 }
 208
 209 void clifford::do_print_latex(const print_latex & c, unsigned level) const
 210 {
 211         // dirac_slash() object is printed differently
 212         if (is_dirac_slash(seq[0])) {
 213                 c.s << "{";
 214                 seq[0].print(c, level);
 215                 c.s << "\\hspace{-1.0ex}/}";
 216         } else
 217                 this->print_dispatch<inherited>(c, level);
 218 }
 219
 220 DEFAULT_COMPARE(diracone)
 221 DEFAULT_COMPARE(cliffordunit)
 222 DEFAULT_COMPARE(diracgamma)
 223 DEFAULT_COMPARE(diracgamma5)
 224 DEFAULT_COMPARE(diracgammaL)
 225 DEFAULT_COMPARE(diracgammaR)
 226
 227 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
 228 DEFAULT_PRINT_LATEX(cliffordunit, "e", "e")
 229 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
 230 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
 231 DEFAULT_PRINT_LATEX(diracgammaL, "gammaL", "{\\gamma_L}")
 232 DEFAULT_PRINT_LATEX(diracgammaR, "gammaR", "{\\gamma_R}")
 233
 234 /** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
 235 static void base_and_index(const ex & c, ex & b, ex & i)
 236 {
 237         GINAC_ASSERT(is_a<clifford>(c));
 238         GINAC_ASSERT(c.nops() == 2);
 239
 240         if (is_a<cliffordunit>(c.op(0))) { // proper dirac gamma object or clifford unit
 241                 i = c.op(1);
 242                 b = _ex1;
 243         } else if (is_a<diracgamma5>(c.op(0)) || is_a<diracgammaL>(c.op(0)) || is_a<diracgammaR>(c.op(0))) { // gamma5/L/R
 244                 i = _ex0;
 245                 b = _ex1;
 246         } else { // slash object, generate new dummy index
 247                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(c.op(1)).get_dim());
 248                 b = indexed(c.op(0), ix.toggle_variance());
 249                 i = ix;
 250         }
 251 }
 252
 253 /** Predicate for finding non-clifford objects. */
 254 struct is_not_a_clifford : public std::unary_function<ex, bool> {
 255         bool operator()(const ex & e)
 256         {
 257                 return !is_a<clifford>(e);
 258         }
 259 };
 260
 261 /** Contraction of a gamma matrix with something else. */
 262 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 263 {
 264         GINAC_ASSERT(is_a<clifford>(*self));
 265         GINAC_ASSERT(is_a<indexed>(*other));
 266         GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
 267         unsigned char rl = ex_to<clifford>(*self).get_representation_label();
 268
 269         ex dim = ex_to<idx>(self->op(1)).get_dim();
 270         if (other->nops() > 1)
 271                 dim = minimal_dim(dim, ex_to<idx>(other->op(1)).get_dim());
 272
 273         if (is_a<clifford>(*other)) {
 274
 275                 // Contraction only makes sense if the represenation labels are equal
 276                 if (ex_to<clifford>(*other).get_representation_label() != rl)
 277                         return false;
 278
 279                 size_t num = other - self;
 280
 281                 // gamma~mu gamma.mu = dim ONE
 282                 if (num == 1) {
 283                         *self = dim;
 284                         *other = dirac_ONE(rl);
 285                         return true;
 286
 287                 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
 288                 } else if (num == 2
 289                         && is_a<clifford>(self[1])) {
 290                         *self = 2 - dim;
 291                         *other = _ex1;
 292                         return true;
 293
 294                 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
 295                 } else if (num == 3
 296                         && is_a<clifford>(self[1])
 297                         && is_a<clifford>(self[2])) {
 298                         ex b1, i1, b2, i2;
 299                         base_and_index(self[1], b1, i1);
 300                         base_and_index(self[2], b2, i2);
 301                         *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
 302                         self[1] = _ex1;
 303                         self[2] = _ex1;
 304                         *other = _ex1;
 305                         return true;
 306
 307                 // 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
 308                 } else if (num == 4
 309                         && is_a<clifford>(self[1])
 310                         && is_a<clifford>(self[2])
 311                         && is_a<clifford>(self[3])) {
 312                         *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3];
 313                         self[1] = _ex1;
 314                         self[2] = _ex1;
 315                         self[3] = _ex1;
 316                         *other = _ex1;
 317                         return true;
 318
 319                 // gamma~mu Sodd gamma.mu = -2 Sodd_R
 320                 // (Chisholm identity in 4 dimensions)
 321                 } else if (!((other - self) & 1) && dim.is_equal(4)) {
 322                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
 323                                 return false;
 324
 325                         *self = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(other), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
 326                         std::fill(self + 1, other, _ex1);
 327                         *other = _ex_2;
 328                         return true;
 329
 330                 // gamma~mu Sodd gamma~alpha gamma.mu = 2 gamma~alpha Sodd + 2 Sodd_R gamma~alpha
 331                 // (commutate contracted indices towards each other, then use
 332                 // Chisholm identity in 4 dimensions)
 333                 } else if (((other - self) & 1) && dim.is_equal(4)) {
 334                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
 335                                 return false;
 336
 337                         exvector::iterator next_to_last = other - 1;
 338                         ex S = ncmul(exvector(self + 1, next_to_last), true);
 339                         ex SR = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(next_to_last), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
 340
 341                         *self = (*next_to_last) * S + SR * (*next_to_last);
 342                         std::fill(self + 1, other, _ex1);
 343                         *other = _ex2;
 344                         return true;
 345
 346                 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
 347                 // (commutate contracted indices towards each other, simplify_indexed()
 348                 // will re-expand and re-run the simplification)
 349                 } else {
 350                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
 351                                 return false;
 352
 353                         exvector::iterator next_to_last = other - 1;
 354                         ex S = ncmul(exvector(self + 1, next_to_last), true);
 355
 356                         *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
 357                         std::fill(self + 1, other + 1, _ex1);
 358                         return true;
 359                 }
 360
 361         } else if (is_a<symbol>(other->op(0)) && other->nops() == 2) {
 362
 363                 // x.mu gamma~mu -> x-slash
 364                 *self = dirac_slash(other->op(0), dim, rl);
 365                 *other = _ex1;
 366                 return true;
 367         }
 368
 369         return false;
 370 }
 371
 372 /** An utility function looking for a given metric within an exvector,
 373  *  used in cliffordunit::contract_with(). */
 374 static int find_same_metric(exvector & v, ex & c)
 375 {
 376         for (int i=0; i<v.size();i++) {
 377                 if (!is_a<clifford>(v[i]) && is_a<indexed>(v[i])
 378                     && ex_to<clifford>(c).same_metric(v[i])
 379                     && (ex_to<varidx>(c.op(1)) == ex_to<indexed>(v[i]).get_indices()[0]
 380                         || ex_to<varidx>(c.op(1)).toggle_variance() == ex_to<indexed>(v[i]).get_indices()[0])) {
 381                         return ++i; // next to found
 382                 }
 383         }
 384         return 0; //nothing found
 385 }
 386
 387 /** Contraction of a Clifford unit with something else. */
 388 bool cliffordunit::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 389 {
 390         GINAC_ASSERT(is_a<clifford>(*self));
 391         GINAC_ASSERT(is_a<indexed>(*other));
 392         GINAC_ASSERT(is_a<cliffordunit>(self->op(0)));
 393         clifford unit = ex_to<clifford>(*self);
 394         unsigned char rl = unit.get_representation_label();
 395
 396         if (is_a<clifford>(*other)) {
 397                 // Contraction only makes sense if the represenation labels are equal
 398                 // and the metrics are the same
 399                 if ((ex_to<clifford>(*other).get_representation_label() != rl)
 400                     && unit.same_metric(*other))
 401                         return false;
 402
 403                 // Find if a previous contraction produces the square of self
 404                 int prev_square = find_same_metric(v, self[0]);
 405                 varidx d((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(ex_to<idx>(self->op(1)).get_dim()));
 406                 ex squared_metric = unit.get_metric(self->op(1), d) * unit.get_metric(d.toggle_variance(), other->op(1));
 407
 408                 // e~mu e.mu = Tr ONE
 409                 if (other - self == 1) {
 410                         if (prev_square != 0) {
 411                                 *self = squared_metric;
 412                                 v[prev_square-1] = _ex1;
 413                         } else
 414                                 *self = unit.get_metric(self->op(1), other->op(1));
 415                         *other = dirac_ONE(rl);
 416                         return true;
 417
 418                 // e~mu e~alpha e.mu = (2e~alpha^2-Tr) e~alpha
 419                 } else if (other - self == 2
 420                         && is_a<clifford>(self[1])) {
 421
 422                         const ex & ia = self[1].op(1);
 423                         const ex & ib = self[1].op(1);
 424                         if (is_a<tensmetric>(unit.get_metric()))
 425                                 *self = 2 - unit.get_metric(self->op(1), other->op(1));
 426                         else if (prev_square != 0) {
 427                                 *self = 2-squared_metric;
 428                                 v[prev_square-1] = _ex1;
 429                         } else
 430                                 *self = 2*unit.get_metric(ia, ib) - unit.get_metric(self->op(1), other->op(1));
 431                         *other = _ex1;
 432                         return true;
 433
 434                 // e~mu S e~alpha e.mu = 2 e~alpha^3 S - e~mu S e.mu e~alpha
 435                 // (commutate contracted indices towards each other, simplify_indexed()
 436                 // will re-expand and re-run the simplification)
 437                 } else {
 438                         exvector::iterator it = self + 1, next_to_last = other - 1;
 439                         while (it != other) {
 440                                 if (!is_a<clifford>(*it))
 441                                         return false;
 442                                 ++it;
 443                         }
 444
 445                         it = self + 1;
 446                         ex S = _ex1;
 447                         while (it != next_to_last) {
 448                                 S *= *it;
 449                                 *it++ = _ex1;
 450                         }
 451
 452                         const ex & ia = next_to_last->op(1);
 453                         const ex & ib = next_to_last->op(1);
 454                         if (is_a<tensmetric>(unit.get_metric()))
 455                                 *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
 456                         else if (prev_square != 0) {
 457                                 *self = 2 * (*next_to_last) * S  - (*self) * S * (*other) * (*next_to_last)*unit.get_metric(self->op(1),self->op(1));
 458                                 v[prev_square-1] = _ex1;
 459                         } else
 460                                 *self = 2 * (*next_to_last) * S* unit.get_metric(ia,ib) - (*self) * S * (*other) * (*next_to_last);
 461                         *next_to_last = _ex1;
 462                         *other = _ex1;
 463                         return true;
 464                 }
 465
 466         }
 467
 468         return false;
 469 }
 470
 471 /** Perform automatic simplification on noncommutative product of clifford
 472  *  objects. This removes superfluous ONEs, permutes gamma5/L/R's to the front
 473  *  and removes squares of gamma objects. */
 474 ex clifford::eval_ncmul(const exvector & v) const
 475 {
 476         exvector s;
 477         s.reserve(v.size());
 478
 479         // Remove superfluous ONEs
 480         exvector::const_iterator cit = v.begin(), citend = v.end();
 481         while (cit != citend) {
 482                 if (!is_a<clifford>(*cit) || !is_a<diracone>(cit->op(0)))
 483                         s.push_back(*cit);
 484                 cit++;
 485         }
 486
 487         bool something_changed = false;
 488         int sign = 1;
 489
 490         // Anticommutate gamma5/L/R's to the front
 491         if (s.size() >= 2) {
 492                 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
 493                 while (true) {
 494                         exvector::iterator it = next_to_last;
 495                         while (true) {
 496                                 exvector::iterator it2 = it + 1;
 497                                 if (is_a<clifford>(*it) && is_a<clifford>(*it2)) {
 498                                         ex e1 = it->op(0), e2 = it2->op(0);
 499
 500                                         if (is_a<diracgamma5>(e2)) {
 501
 502                                                 if (is_a<diracgammaL>(e1) || is_a<diracgammaR>(e1)) {
 503
 504                                                         // gammaL/R gamma5 -> gamma5 gammaL/R
 505                                                         it->swap(*it2);
 506                                                         something_changed = true;
 507
 508                                                 } else if (!is_a<diracgamma5>(e1)) {
 509
 510                                                         // gamma5 gamma5 -> gamma5 gamma5 (do nothing)
 511                                                         // x gamma5 -> -gamma5 x
 512                                                         it->swap(*it2);
 513                                                         sign = -sign;
 514                                                         something_changed = true;
 515                                                 }
 516
 517                                         } else if (is_a<diracgammaL>(e2)) {
 518
 519                                                 if (is_a<diracgammaR>(e1)) {
 520
 521                                                         // gammaR gammaL -> 0
 522                                                         return _ex0;
 523
 524                                                 } else if (!is_a<diracgammaL>(e1) && !is_a<diracgamma5>(e1)) {
 525
 526                                                         // gammaL gammaL -> gammaL gammaL (do nothing)
 527                                                         // gamma5 gammaL -> gamma5 gammaL (do nothing)
 528                                                         // x gammaL -> gammaR x
 529                                                         it->swap(*it2);
 530                                                         *it = clifford(diracgammaR(), ex_to<clifford>(*it).get_representation_label());
 531                                                         something_changed = true;
 532                                                 }
 533
 534                                         } else if (is_a<diracgammaR>(e2)) {
 535
 536                                                 if (is_a<diracgammaL>(e1)) {
 537
 538                                                         // gammaL gammaR -> 0
 539                                                         return _ex0;
 540
 541                                                 } else if (!is_a<diracgammaR>(e1) && !is_a<diracgamma5>(e1)) {
 542
 543                                                         // gammaR gammaR -> gammaR gammaR (do nothing)
 544                                                         // gamma5 gammaR -> gamma5 gammaR (do nothing)
 545                                                         // x gammaR -> gammaL x
 546                                                         it->swap(*it2);
 547                                                         *it = clifford(diracgammaL(), ex_to<clifford>(*it).get_representation_label());
 548                                                         something_changed = true;
 549                                                 }
 550                                         }
 551                                 }
 552                                 if (it == first)
 553                                         break;
 554                                 --it;
 555                         }
 556                         if (next_to_last == first)
 557                                 break;
 558                         --next_to_last;
 559                 }
 560         }
 561
 562         // Remove equal adjacent gammas
 563         if (s.size() >= 2) {
 564                 exvector::iterator it, itend = s.end() - 1;
 565                 for (it = s.begin(); it != itend; ++it) {
 566                         ex & a = it[0];
 567                         ex & b = it[1];
 568                         if (!is_a<clifford>(a) || !is_a<clifford>(b))
 569                                 continue;
 570
 571                         const ex & ag = a.op(0);
 572                         const ex & bg = b.op(0);
 573                         bool a_is_cliffordunit = is_a<cliffordunit>(ag);
 574                         bool b_is_cliffordunit =  is_a<cliffordunit>(bg);
 575
 576                         if (a_is_cliffordunit && b_is_cliffordunit && ex_to<clifford>(a).same_metric(b)) {
 577
 578                                 const ex & ia = a.op(1);
 579                                 const ex & ib = b.op(1);
 580                                 if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
 581                                         a = ex_to<clifford>(a).get_metric(ia, ib);
 582                                         b = dirac_ONE(representation_label);
 583                                         something_changed = true;
 584                                 }
 585
 586                         } else if ((is_a<diracgamma5>(ag) && is_a<diracgamma5>(bg))) {
 587
 588                                 // Remove squares of gamma5
 589                                 a = dirac_ONE(representation_label);
 590                                 b = dirac_ONE(representation_label);
 591                                 something_changed = true;
 592
 593                         } else if ((is_a<diracgammaL>(ag) && is_a<diracgammaL>(bg))
 594                                 || (is_a<diracgammaR>(ag) && is_a<diracgammaR>(bg))) {
 595
 596                                 // Remove squares of gammaL/R
 597                                 b = dirac_ONE(representation_label);
 598                                 something_changed = true;
 599
 600                         } else if (is_a<diracgammaL>(ag) && is_a<diracgammaR>(bg)) {
 601
 602                                 // gammaL and gammaR are orthogonal
 603                                 return _ex0;
 604
 605                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaL>(bg)) {
 606
 607                                 // gamma5 gammaL -> -gammaL
 608                                 a = dirac_ONE(representation_label);
 609                                 sign = -sign;
 610                                 something_changed = true;
 611
 612                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaR>(bg)) {
 613
 614                                 // gamma5 gammaR -> gammaR
 615                                 a = dirac_ONE(representation_label);
 616                                 something_changed = true;
 617
 618                         } else if (!a_is_cliffordunit && !b_is_cliffordunit && ag.is_equal(bg)) {
 619
 620                                 // a\ a\ -> a^2
 621                                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).minimal_dim(ex_to<idx>(b.op(1))));
 622
 623                                 a = indexed(ag, ix) * indexed(ag, ix.toggle_variance());
 624                                 b = dirac_ONE(representation_label);
 625                                 something_changed = true;
 626                         }
 627                 }
 628         }
 629
 630         if (s.empty())
 631                 return clifford(diracone(), representation_label) * sign;
 632         if (something_changed)
 633                 return reeval_ncmul(s) * sign;
 634         else
 635                 return hold_ncmul(s) * sign;
 636 }
 637
 638 ex clifford::thiscontainer(const exvector & v) const
 639 {
 640         return clifford(representation_label, get_metric(), v);
 641 }
 642
 643 ex clifford::thiscontainer(std::auto_ptr<exvector> vp) const
 644 {
 645         return clifford(representation_label, get_metric(), vp);
 646 }
 647
 648 ex diracgamma5::conjugate() const
 649 {
 650         return _ex_1 * (*this);
 651 }
 652
 653 ex diracgammaL::conjugate() const
 654 {
 655         return (new diracgammaR)->setflag(status_flags::dynallocated);
 656 }
 657
 658 ex diracgammaR::conjugate() const
 659 {
 660         return (new diracgammaL)->setflag(status_flags::dynallocated);
 661 }
 662
 663 //////////
 664 // global functions
 665 //////////
 666
 667 ex dirac_ONE(unsigned char rl)
 668 {
 669         static ex ONE = (new diracone)->setflag(status_flags::dynallocated);
 670         return clifford(ONE, rl);
 671 }
 672
 673 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl)
 674 {
 675         static ex unit = (new cliffordunit)->setflag(status_flags::dynallocated);
 676
 677         if (!is_a<varidx>(mu))
 678                 throw(std::invalid_argument("index of Clifford unit must be of type varidx"));
 679
 680         return clifford(unit, mu, metr, rl);
 681 }
 682
 683 ex dirac_gamma(const ex & mu, unsigned char rl)
 684 {
 685         static ex gamma = (new diracgamma)->setflag(status_flags::dynallocated);
 686
 687         if (!is_a<varidx>(mu))
 688                 throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
 689
 690         return clifford(gamma, mu, default_metric(), rl);
 691 }
 692
 693 ex dirac_gamma5(unsigned char rl)
 694 {
 695         static ex gamma5 = (new diracgamma5)->setflag(status_flags::dynallocated);
 696         return clifford(gamma5, rl);
 697 }
 698
 699 ex dirac_gammaL(unsigned char rl)
 700 {
 701         static ex gammaL = (new diracgammaL)->setflag(status_flags::dynallocated);
 702         return clifford(gammaL, rl);
 703 }
 704
 705 ex dirac_gammaR(unsigned char rl)
 706 {
 707         static ex gammaR = (new diracgammaR)->setflag(status_flags::dynallocated);
 708         return clifford(gammaR, rl);
 709 }
 710
 711 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
 712 {
 713         // Slashed vectors are actually stored as a clifford object with the
 714         // vector as its base expression and a (dummy) index that just serves
 715         // for storing the space dimensionality
 716         return clifford(e, varidx(0, dim), default_metric(), rl);
 717 }
 718
 719 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 720  *  is that of a clifford object with the specified representation label. */
 721 static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
 722 {
 723         return ti == (TINFO_clifford + rl);
 724 }
 725
 726 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 727  *  is that of a clifford object (with an arbitrary representation label). */
 728 static bool is_clifford_tinfo(unsigned ti)
 729 {
 730         return (ti & ~0xff) == TINFO_clifford;
 731 }
 732
 733 /** Extract representation label from tinfo key (as returned by
 734  *  return_type_tinfo()). */
 735 static unsigned char get_representation_label(unsigned ti)
 736 {
 737         return ti & 0xff;
 738 }
 739
 740 /** Take trace of a string of an even number of Dirac gammas given a vector
 741  *  of indices. */
 742 static ex trace_string(exvector::const_iterator ix, size_t num)
 743 {
 744         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 745         if (num == 2)
 746                 return lorentz_g(ix[0], ix[1]);
 747
 748         // 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 )
 749         else if (num == 4)
 750                 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
 751                      + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
 752                      - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
 753
 754         // Traces of 6 or more gammas are computed recursively:
 755         // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
 756         //   + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
 757         //   - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
 758         //   + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
 759         //   - ...
 760         //   + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
 761         exvector v(num - 2);
 762         int sign = 1;
 763         ex result;
 764         for (size_t i=1; i<num; i++) {
 765                 for (size_t n=1, j=0; n<num; n++) {
 766                         if (n == i)
 767                                 continue;
 768                         v[j++] = ix[n];
 769                 }
 770                 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
 771                 sign = -sign;
 772         }
 773         return result;
 774 }
 775
 776 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE)
 777 {
 778         if (is_a<clifford>(e)) {
 779
 780                 unsigned char rl = ex_to<clifford>(e).get_representation_label();
 781
 782                 // Are we taking the trace over this object's representation label?
 783                 if (rls.find(rl) == rls.end())
 784                         return e;
 785
 786                 // Yes, all elements are traceless, except for dirac_ONE and dirac_L/R
 787                 const ex & g = e.op(0);
 788                 if (is_a<diracone>(g))
 789                         return trONE;
 790                 else if (is_a<diracgammaL>(g) || is_a<diracgammaR>(g))
 791                         return trONE/2;
 792                 else
 793                         return _ex0;
 794
 795         } else if (is_exactly_a<mul>(e)) {
 796
 797                 // Trace of product: pull out non-clifford factors
 798                 ex prod = _ex1;
 799                 for (size_t i=0; i<e.nops(); i++) {
 800                         const ex &o = e.op(i);
 801                         if (is_clifford_tinfo(o.return_type_tinfo()))
 802                                 prod *= dirac_trace(o, rls, trONE);
 803                         else
 804                                 prod *= o;
 805                 }
 806                 return prod;
 807
 808         } else if (is_exactly_a<ncmul>(e)) {
 809
 810                 unsigned char rl = get_representation_label(e.return_type_tinfo());
 811
 812                 // Are we taking the trace over this string's representation label?
 813                 if (rls.find(rl) == rls.end())
 814                         return e;
 815
 816                 // Substitute gammaL/R and expand product, if necessary
 817                 ex e_expanded = e.subs(lst(
 818                         dirac_gammaL(rl) == (dirac_ONE(rl)-dirac_gamma5(rl))/2,
 819                         dirac_gammaR(rl) == (dirac_ONE(rl)+dirac_gamma5(rl))/2
 820                 ), subs_options::no_pattern).expand();
 821                 if (!is_a<ncmul>(e_expanded))
 822                         return dirac_trace(e_expanded, rls, trONE);
 823
 824                 // gamma5 gets moved to the front so this check is enough
 825                 bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
 826                 size_t num = e.nops();
 827
 828                 if (has_gamma5) {
 829
 830                         // Trace of gamma5 * odd number of gammas and trace of
 831                         // gamma5 * gamma.mu * gamma.nu are zero
 832                         if ((num & 1) == 0 || num == 3)
 833                                 return _ex0;
 834
 835                         // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
 836                         // (the epsilon is always 4-dimensional)
 837                         if (num == 5) {
 838                                 ex b1, i1, b2, i2, b3, i3, b4, i4;
 839                                 base_and_index(e.op(1), b1, i1);
 840                                 base_and_index(e.op(2), b2, i2);
 841                                 base_and_index(e.op(3), b3, i3);
 842                                 base_and_index(e.op(4), b4, i4);
 843                                 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();
 844                         }
 845
 846                         // Tr gamma5 S_2k =
 847                         //   I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
 848                         // (the epsilon is always 4-dimensional)
 849                         exvector ix(num-1), bv(num-1);
 850                         for (size_t i=1; i<num; i++)
 851                                 base_and_index(e.op(i), bv[i-1], ix[i-1]);
 852                         num--;
 853                         int *iv = new int[num];
 854                         ex result;
 855                         for (size_t i=0; i<num-3; i++) {
 856                                 ex idx1 = ix[i];
 857                                 for (size_t j=i+1; j<num-2; j++) {
 858                                         ex idx2 = ix[j];
 859                                         for (size_t k=j+1; k<num-1; k++) {
 860                                                 ex idx3 = ix[k];
 861                                                 for (size_t l=k+1; l<num; l++) {
 862                                                         ex idx4 = ix[l];
 863                                                         iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
 864                                                         exvector v;
 865                                                         v.reserve(num - 4);
 866                                                         for (size_t n=0, t=4; n<num; n++) {
 867                                                                 if (n == i || n == j || n == k || n == l)
 868                                                                         continue;
 869                                                                 iv[t++] = n;
 870                                                                 v.push_back(ix[n]);
 871                                                         }
 872                                                         int sign = permutation_sign(iv, iv + num);
 873                                                         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))
 874                                                                 * trace_string(v.begin(), num - 4);
 875                                                 }
 876                                         }
 877                                 }
 878                         }
 879                         delete[] iv;
 880                         return trONE * I * result * mul(bv);
 881
 882                 } else { // no gamma5
 883
 884                         // Trace of odd number of gammas is zero
 885                         if ((num & 1) == 1)
 886                                 return _ex0;
 887
 888                         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 889                         if (num == 2) {
 890                                 ex b1, i1, b2, i2;
 891                                 base_and_index(e.op(0), b1, i1);
 892                                 base_and_index(e.op(1), b2, i2);
 893                                 return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
 894                         }
 895
 896                         exvector iv(num), bv(num);
 897                         for (size_t i=0; i<num; i++)
 898                                 base_and_index(e.op(i), bv[i], iv[i]);
 899
 900                         return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
 901                 }
 902
 903         } else if (e.nops() > 0) {
 904
 905                 // Trace maps to all other container classes (this includes sums)
 906                 pointer_to_map_function_2args<const std::set<unsigned char> &, const ex &> fcn(dirac_trace, rls, trONE);
 907                 return e.map(fcn);
 908
 909         } else
 910                 return _ex0;
 911 }
 912
 913 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE)
 914 {
 915         // Convert list to set
 916         std::set<unsigned char> rls;
 917         for (lst::const_iterator i = rll.begin(); i != rll.end(); ++i) {
 918                 if (i->info(info_flags::nonnegint))
 919                         rls.insert(ex_to<numeric>(*i).to_int());
 920         }
 921
 922         return dirac_trace(e, rls, trONE);
 923 }
 924
 925 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
 926 {
 927         // Convert label to set
 928         std::set<unsigned char> rls;
 929         rls.insert(rl);
 930
 931         return dirac_trace(e, rls, trONE);
 932 }
 933
 934
 935 ex canonicalize_clifford(const ex & e)
 936 {
 937         // Scan for any ncmul objects
 938         exmap srl;
 939         ex aux = e.to_rational(srl);
 940         for (exmap::iterator i = srl.begin(); i != srl.end(); ++i) {
 941
 942                 ex lhs = i->first;
 943                 ex rhs = i->second;
 944
 945                 if (is_exactly_a<ncmul>(rhs)
 946                  && rhs.return_type() == return_types::noncommutative
 947                  && is_clifford_tinfo(rhs.return_type_tinfo())) {
 948
 949                         // Expand product, if necessary
 950                         ex rhs_expanded = rhs.expand();
 951                         if (!is_a<ncmul>(rhs_expanded)) {
 952                                 i->second = canonicalize_clifford(rhs_expanded);
 953                                 continue;
 954
 955                         } else if (!is_a<clifford>(rhs.op(0)))
 956                                 continue;
 957
 958                         exvector v;
 959                         v.reserve(rhs.nops());
 960                         for (size_t j=0; j<rhs.nops(); j++)
 961                                 v.push_back(rhs.op(j));
 962
 963                         // Stupid recursive bubble sort because we only want to swap adjacent gammas
 964                         exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
 965                         if (is_a<diracgamma5>(it->op(0)) || is_a<diracgammaL>(it->op(0)) || is_a<diracgammaR>(it->op(0)))
 966                                 ++it;
 967                         while (it != next_to_last) {
 968                                 if (it[0].compare(it[1]) > 0) {
 969                                         ex save0 = it[0], save1 = it[1];
 970                                         ex b1, i1, b2, i2;
 971                                         base_and_index(it[0], b1, i1);
 972                                         base_and_index(it[1], b2, i2);
 973                                         it[0] = (ex_to<clifford>(save0).get_metric(i1, i2) * b1 * b2).simplify_indexed();
 974                                         it[1] = v.size() == 2 ? _ex2 * dirac_ONE(ex_to<clifford>(it[1]).get_representation_label()) : _ex2;
 975                                         ex sum = ncmul(v);
 976                                         it[0] = save1;
 977                                         it[1] = save0;
 978                                         sum -= ncmul(v, true);
 979                                         i->second = canonicalize_clifford(sum);
 980                                         goto next_sym;
 981                                 }
 982                                 ++it;
 983                         }
 984 next_sym:       ;
 985                 }
 986         }
 987         return aux.subs(srl, subs_options::no_pattern).simplify_indexed();
 988 }
 989
 990 ex clifford_prime(const ex &e)
 991 {
 992         pointer_to_map_function fcn(clifford_prime);
 993         if (is_a<clifford>(e) && is_a<cliffordunit>(e.op(0))) {
 994                 return -e;
 995         } else if (is_a<add>(e)) {
 996                 return e.map(fcn);
 997         } else if (is_a<ncmul>(e)) {
 998                 return e.map(fcn);
 999         } else if (is_a<power>(e)) {
1000                 return pow(clifford_prime(e.op(0)), e.op(1));
1001         } else
1002                 return e;
1003 }
1004
1005 ex delete_ONE(const ex &e)
1006 {
1007         pointer_to_map_function fcn(delete_ONE);
1008         if (is_a<clifford>(e) && is_a<diracone>(e.op(0))) {
1009                 return 1;
1010         } else if (is_a<add>(e)) {
1011                 return e.map(fcn);
1012         } else if (is_a<ncmul>(e)) {
1013                 return e.map(fcn);
1014         } else if (is_a<mul>(e)) {
1015                 return e.map(fcn);
1016         } else if (is_a<power>(e)) {
1017                 return pow(delete_ONE(e.op(0)), e.op(1));
1018         } else
1019                 return e;
1020 }
1021
1022 ex clifford_norm(const ex &e)
1023 {
1024         return sqrt(delete_ONE((e * clifford_bar(e)).simplify_indexed()));
1025 }
1026
1027 ex clifford_inverse(const ex &e)
1028 {
1029         ex norm = clifford_norm(e);
1030         if (!norm.is_zero())
1031                 return clifford_bar(e) / pow(norm, 2);
1032 }
1033
1034 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl)
1035 {
1036         unsigned min, max;
1037         if (!ex_to<idx>(mu).is_dim_numeric())
1038                 throw(std::invalid_argument("Index should have a numeric dimension"));
1039         unsigned dim = (ex_to<numeric>(ex_to<idx>(mu).get_dim())).to_int();
1040         ex c = clifford_unit(mu, metr, rl);
1041
1042         if (is_a<matrix>(v)) {
1043                 if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows()) {
1044                         min = ex_to<matrix>(v).rows();
1045                         max = ex_to<matrix>(v).cols();
1046                 } else {
1047                         min = ex_to<matrix>(v).cols();
1048                         max = ex_to<matrix>(v).rows();
1049                 }
1050                 if (min == 1) {
1051                         if (dim == max)
1052                                 if (is_a<varidx>(mu)) // need to swap variance
1053                                         return indexed(v, ex_to<varidx>(mu).toggle_variance()) * c;
1054                                 else
1055                                         return indexed(v, mu) * c;
1056                         else
1057                                 throw(std::invalid_argument("Dimensions of vector and clifford unit mismatch"));
1058                 } else
1059                         throw(std::invalid_argument("First argument should be a vector vector"));
1060         } else if (is_a<lst>(v)) {
1061                 if (dim == ex_to<lst>(v).nops())
1062                         return indexed(matrix(dim, 1, ex_to<lst>(v)), ex_to<varidx>(mu).toggle_variance()) * c;
1063                 else
1064                         throw(std::invalid_argument("List length and dimension of clifford unit mismatch"));
1065         } else
1066                 throw(std::invalid_argument("Cannot construct from anything but list or vector"));
1067 }
1068
1069 } // namespace GiNaC