ginac/clifford.cpp

   1 /** @file clifford.cpp
   2  *
   3  *  Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2003 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 <iostream>
  24 #include <stdexcept>
  25
  26 #include "clifford.h"
  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 "mul.h"
  36 #include "print.h"
  37 #include "archive.h"
  38 #include "utils.h"
  39
  40 namespace GiNaC {
  41
  42 GINAC_IMPLEMENT_REGISTERED_CLASS(clifford, indexed)
  43 GINAC_IMPLEMENT_REGISTERED_CLASS(diracone, tensor)
  44 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma, tensor)
  45 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma5, tensor)
  46 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgammaL, tensor)
  47 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgammaR, tensor)
  48
  49 //////////
  50 // default ctor, dtor, copy ctor, assignment operator and helpers
  51 //////////
  52
  53 clifford::clifford() : representation_label(0)
  54 {
  55         tinfo_key = TINFO_clifford;
  56 }
  57
  58 void clifford::copy(const clifford & other)
  59 {
  60         inherited::copy(other);
  61         representation_label = other.representation_label;
  62 }
  63
  64 DEFAULT_DESTROY(clifford)
  65 DEFAULT_CTORS(diracone)
  66 DEFAULT_CTORS(diracgamma)
  67 DEFAULT_CTORS(diracgamma5)
  68 DEFAULT_CTORS(diracgammaL)
  69 DEFAULT_CTORS(diracgammaR)
  70
  71 //////////
  72 // other constructors
  73 //////////
  74
  75 /** Construct object without any indices. This constructor is for internal
  76  *  use only. Use the dirac_ONE() function instead.
  77  *  @see dirac_ONE */
  78 clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
  79 {
  80         tinfo_key = TINFO_clifford;
  81 }
  82
  83 /** Construct object with one Lorentz index. This constructor is for internal
  84  *  use only. Use the dirac_gamma() function instead.
  85  *  @see dirac_gamma */
  86 clifford::clifford(const ex & b, const ex & mu, unsigned char rl) : inherited(b, mu), representation_label(rl)
  87 {
  88         GINAC_ASSERT(is_a<varidx>(mu));
  89         tinfo_key = TINFO_clifford;
  90 }
  91
  92 clifford::clifford(unsigned char rl, const exvector & v, bool discardable) : inherited(sy_none(), v, discardable), representation_label(rl)
  93 {
  94         tinfo_key = TINFO_clifford;
  95 }
  96
  97 clifford::clifford(unsigned char rl, exvector * vp) : inherited(sy_none(), vp), representation_label(rl)
  98 {
  99         tinfo_key = TINFO_clifford;
 100 }
 101
 102 //////////
 103 // archiving
 104 //////////
 105
 106 clifford::clifford(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 107 {
 108         unsigned rl;
 109         n.find_unsigned("label", rl);
 110         representation_label = rl;
 111 }
 112
 113 void clifford::archive(archive_node &n) const
 114 {
 115         inherited::archive(n);
 116         n.add_unsigned("label", representation_label);
 117 }
 118
 119 DEFAULT_UNARCHIVE(clifford)
 120 DEFAULT_ARCHIVING(diracone)
 121 DEFAULT_ARCHIVING(diracgamma)
 122 DEFAULT_ARCHIVING(diracgamma5)
 123 DEFAULT_ARCHIVING(diracgammaL)
 124 DEFAULT_ARCHIVING(diracgammaR)
 125
 126 //////////
 127 // functions overriding virtual functions from base classes
 128 //////////
 129
 130 int clifford::compare_same_type(const basic & other) const
 131 {
 132         GINAC_ASSERT(is_a<clifford>(other));
 133         const clifford &o = static_cast<const clifford &>(other);
 134
 135         if (representation_label != o.representation_label) {
 136                 // different representation label
 137                 return representation_label < o.representation_label ? -1 : 1;
 138         }
 139
 140         return inherited::compare_same_type(other);
 141 }
 142
 143 bool clifford::match_same_type(const basic & other) const
 144 {
 145         GINAC_ASSERT(is_a<clifford>(other));
 146         const clifford &o = static_cast<const clifford &>(other);
 147
 148         return representation_label == o.representation_label;
 149 }
 150
 151 void clifford::print(const print_context & c, unsigned level) const
 152 {
 153         if (!is_a<diracgamma5>(seq[0]) && !is_a<diracgammaL>(seq[0]) &&
 154             !is_a<diracgammaR>(seq[0]) && !is_a<diracgamma>(seq[0]) &&
 155             !is_a<diracone>(seq[0])) {
 156
 157                 // dirac_slash() object is printed differently
 158                 if (is_a<print_tree>(c))
 159                         inherited::print(c, level);
 160                 else if (is_a<print_latex>(c)) {
 161                         c.s << "{";
 162                         seq[0].print(c, level);
 163                         c.s << "\\hspace{-1.0ex}/}";
 164                 } else {
 165                         seq[0].print(c, level);
 166                         c.s << "\\";
 167                 }
 168
 169         } else
 170                 inherited::print(c, level);
 171 }
 172
 173 DEFAULT_COMPARE(diracone)
 174 DEFAULT_COMPARE(diracgamma)
 175 DEFAULT_COMPARE(diracgamma5)
 176 DEFAULT_COMPARE(diracgammaL)
 177 DEFAULT_COMPARE(diracgammaR)
 178
 179 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
 180 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
 181 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
 182 DEFAULT_PRINT_LATEX(diracgammaL, "gammaL", "{\\gamma_L}")
 183 DEFAULT_PRINT_LATEX(diracgammaR, "gammaR", "{\\gamma_R}")
 184
 185 /** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
 186 static void base_and_index(const ex & c, ex & b, ex & i)
 187 {
 188         GINAC_ASSERT(is_a<clifford>(c));
 189         GINAC_ASSERT(c.nops() == 2);
 190
 191         if (is_a<diracgamma>(c.op(0))) { // proper dirac gamma object
 192                 i = c.op(1);
 193                 b = _ex1;
 194         } else if (is_a<diracgamma5>(c.op(0)) || is_a<diracgammaL>(c.op(0)) || is_a<diracgammaR>(c.op(0))) { // gamma5/L/R
 195                 i = _ex0;
 196                 b = _ex1;
 197         } else { // slash object, generate new dummy index
 198                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(c.op(1)).get_dim());
 199                 b = indexed(c.op(0), ix.toggle_variance());
 200                 i = ix;
 201         }
 202 }
 203
 204 /** Contraction of a gamma matrix with something else. */
 205 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 206 {
 207         GINAC_ASSERT(is_a<clifford>(*self));
 208         GINAC_ASSERT(is_a<indexed>(*other));
 209         GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
 210         unsigned char rl = ex_to<clifford>(*self).get_representation_label();
 211
 212         ex dim = ex_to<idx>(self->op(1)).get_dim();
 213         if (other->nops() > 1)
 214                 dim = minimal_dim(dim, ex_to<idx>(self->op(1)).get_dim());
 215
 216         if (is_a<clifford>(*other)) {
 217
 218                 // Contraction only makes sense if the represenation labels are equal
 219                 if (ex_to<clifford>(*other).get_representation_label() != rl)
 220                         return false;
 221
 222                 // gamma~mu gamma.mu = dim ONE
 223                 if (other - self == 1) {
 224                         *self = dim;
 225                         *other = dirac_ONE(rl);
 226                         return true;
 227
 228                 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
 229                 } else if (other - self == 2
 230                         && is_a<clifford>(self[1])) {
 231                         *self = 2 - dim;
 232                         *other = _ex1;
 233                         return true;
 234
 235                 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
 236                 } else if (other - self == 3
 237                         && is_a<clifford>(self[1])
 238                         && is_a<clifford>(self[2])) {
 239                         ex b1, i1, b2, i2;
 240                         base_and_index(self[1], b1, i1);
 241                         base_and_index(self[2], b2, i2);
 242                         *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
 243                         self[1] = _ex1;
 244                         self[2] = _ex1;
 245                         *other = _ex1;
 246                         return true;
 247
 248                 // 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
 249                 } else if (other - self == 4
 250                         && is_a<clifford>(self[1])
 251                         && is_a<clifford>(self[2])
 252                         && is_a<clifford>(self[3])) {
 253                         *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3];
 254                         self[1] = _ex1;
 255                         self[2] = _ex1;
 256                         self[3] = _ex1;
 257                         *other = _ex1;
 258                         return true;
 259
 260                 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
 261                 // (commutate contracted indices towards each other, simplify_indexed()
 262                 // will re-expand and re-run the simplification)
 263                 } else {
 264                         exvector::iterator it = self + 1, next_to_last = other - 1;
 265                         while (it != other) {
 266                                 if (!is_a<clifford>(*it))
 267                                         return false;
 268                                 ++it;
 269                         }
 270
 271                         it = self + 1;
 272                         ex S = _ex1;
 273                         while (it != next_to_last) {
 274                                 S *= *it;
 275                                 *it++ = _ex1;
 276                         }
 277
 278                         *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
 279                         *next_to_last = _ex1;
 280                         *other = _ex1;
 281                         return true;
 282                 }
 283
 284         } else if (is_a<symbol>(other->op(0)) && other->nops() == 2) {
 285
 286                 // x.mu gamma~mu -> x-slash
 287                 *self = dirac_slash(other->op(0), dim, rl);
 288                 *other = _ex1;
 289                 return true;
 290         }
 291
 292         return false;
 293 }
 294
 295 /** Perform automatic simplification on noncommutative product of clifford
 296  *  objects. This removes superfluous ONEs, permutes gamma5/L/R's to the front
 297  *  and removes squares of gamma objects. */
 298 ex clifford::simplify_ncmul(const exvector & v) const
 299 {
 300         exvector s;
 301         s.reserve(v.size());
 302
 303         // Remove superfluous ONEs
 304         exvector::const_iterator cit = v.begin(), citend = v.end();
 305         while (cit != citend) {
 306                 if (!is_a<clifford>(*cit) || !is_a<diracone>(cit->op(0)))
 307                         s.push_back(*cit);
 308                 cit++;
 309         }
 310
 311         bool something_changed = false;
 312         int sign = 1;
 313
 314         // Anticommute gamma5/L/R's to the front
 315         if (s.size() >= 2) {
 316                 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
 317                 while (true) {
 318                         exvector::iterator it = next_to_last;
 319                         while (true) {
 320                                 exvector::iterator it2 = it + 1;
 321                                 if (is_a<clifford>(*it) && is_a<clifford>(*it2)) {
 322                                         ex e1 = it->op(0), e2 = it2->op(0);
 323
 324                                         if (is_a<diracgamma5>(e2)) {
 325
 326                                                 if (is_a<diracgammaL>(e1) || is_a<diracgammaR>(e1)) {
 327
 328                                                         // gammaL/R gamma5 -> gamma5 gammaL/R
 329                                                         it->swap(*it2);
 330                                                         something_changed = true;
 331
 332                                                 } else if (!is_a<diracgamma5>(e1)) {
 333
 334                                                         // gamma5 gamma5 -> gamma5 gamma5 (do nothing)
 335                                                         // x gamma5 -> -gamma5 x
 336                                                         it->swap(*it2);
 337                                                         sign = -sign;
 338                                                         something_changed = true;
 339                                                 }
 340
 341                                         } else if (is_a<diracgammaL>(e2)) {
 342
 343                                                 if (is_a<diracgammaR>(e1)) {
 344
 345                                                         // gammaR gammaL -> 0
 346                                                         return _ex0;
 347
 348                                                 } else if (!is_a<diracgammaL>(e1) && !is_a<diracgamma5>(e1)) {
 349
 350                                                         // gammaL gammaL -> gammaL gammaL (do nothing)
 351                                                         // gamma5 gammaL -> gamma5 gammaL (do nothing)
 352                                                         // x gammaL -> gammaR x
 353                                                         it->swap(*it2);
 354                                                         *it = clifford(diracgammaR(), ex_to<clifford>(*it).get_representation_label());
 355                                                         something_changed = true;
 356                                                 }
 357
 358                                         } else if (is_a<diracgammaR>(e2)) {
 359
 360                                                 if (is_a<diracgammaL>(e1)) {
 361
 362                                                         // gammaL gammaR -> 0
 363                                                         return _ex0;
 364
 365                                                 } else if (!is_a<diracgammaR>(e1) && !is_a<diracgamma5>(e1)) {
 366
 367                                                         // gammaR gammaR -> gammaR gammaR (do nothing)
 368                                                         // gamma5 gammaR -> gamma5 gammaR (do nothing)
 369                                                         // x gammaR -> gammaL x
 370                                                         it->swap(*it2);
 371                                                         *it = clifford(diracgammaL(), ex_to<clifford>(*it).get_representation_label());
 372                                                         something_changed = true;
 373                                                 }
 374                                         }
 375                                 }
 376                                 if (it == first)
 377                                         break;
 378                                 --it;
 379                         }
 380                         if (next_to_last == first)
 381                                 break;
 382                         --next_to_last;
 383                 }
 384         }
 385
 386         // Remove equal adjacent gammas
 387         if (s.size() >= 2) {
 388                 exvector::iterator it, itend = s.end() - 1;
 389                 for (it = s.begin(); it != itend; ++it) {
 390                         ex & a = it[0];
 391                         ex & b = it[1];
 392                         if (!is_a<clifford>(a) || !is_a<clifford>(b))
 393                                 continue;
 394
 395                         const ex & ag = a.op(0);
 396                         const ex & bg = b.op(0);
 397                         bool a_is_diracgamma = is_a<diracgamma>(ag);
 398                         bool b_is_diracgamma = is_a<diracgamma>(bg);
 399
 400                         if (a_is_diracgamma && b_is_diracgamma) {
 401
 402                                 const ex & ia = a.op(1);
 403                                 const ex & ib = b.op(1);
 404                                 if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
 405                                         a = lorentz_g(ia, ib);
 406                                         b = dirac_ONE(representation_label);
 407                                         something_changed = true;
 408                                 }
 409
 410                         } else if ((is_a<diracgamma5>(ag) && is_a<diracgamma5>(bg))) {
 411
 412                                 // Remove squares of gamma5
 413                                 a = dirac_ONE(representation_label);
 414                                 b = dirac_ONE(representation_label);
 415                                 something_changed = true;
 416
 417                         } else if ((is_a<diracgammaL>(ag) && is_a<diracgammaL>(bg))
 418                                 || (is_a<diracgammaR>(ag) && is_a<diracgammaR>(bg))) {
 419
 420                                 // Remove squares of gammaL/R
 421                                 b = dirac_ONE(representation_label);
 422                                 something_changed = true;
 423
 424                         } else if (is_a<diracgammaL>(ag) && is_a<diracgammaR>(bg)) {
 425
 426                                 // gammaL and gammaR are orthogonal
 427                                 return _ex0;
 428
 429                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaL>(bg)) {
 430
 431                                 // gamma5 gammaL -> -gammaL
 432                                 a = dirac_ONE(representation_label);
 433                                 sign = -sign;
 434                                 something_changed = true;
 435
 436                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaR>(bg)) {
 437
 438                                 // gamma5 gammaR -> gammaR
 439                                 a = dirac_ONE(representation_label);
 440                                 something_changed = true;
 441
 442                         } else if (!a_is_diracgamma && !b_is_diracgamma && ag.is_equal(bg)) {
 443
 444                                 // a\ a\ -> a^2
 445                                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).minimal_dim(ex_to<idx>(b.op(1))));
 446                                 a = indexed(ag, ix) * indexed(ag, ix.toggle_variance());
 447                                 b = dirac_ONE(representation_label);
 448                                 something_changed = true;
 449                         }
 450                 }
 451         }
 452
 453         if (s.empty())
 454                 return clifford(diracone(), representation_label) * sign;
 455         if (something_changed)
 456                 return nonsimplified_ncmul(s) * sign;
 457         else
 458                 return simplified_ncmul(s) * sign;
 459 }
 460
 461 ex clifford::thisexprseq(const exvector & v) const
 462 {
 463         return clifford(representation_label, v);
 464 }
 465
 466 ex clifford::thisexprseq(exvector * vp) const
 467 {
 468         return clifford(representation_label, vp);
 469 }
 470
 471 //////////
 472 // global functions
 473 //////////
 474
 475 ex dirac_ONE(unsigned char rl)
 476 {
 477         return clifford(diracone(), rl);
 478 }
 479
 480 ex dirac_gamma(const ex & mu, unsigned char rl)
 481 {
 482         if (!is_a<varidx>(mu))
 483                 throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
 484
 485         return clifford(diracgamma(), mu, rl);
 486 }
 487
 488 ex dirac_gamma5(unsigned char rl)
 489 {
 490         return clifford(diracgamma5(), rl);
 491 }
 492
 493 ex dirac_gammaL(unsigned char rl)
 494 {
 495         return clifford(diracgammaL(), rl);
 496 }
 497
 498 ex dirac_gammaR(unsigned char rl)
 499 {
 500         return clifford(diracgammaR(), rl);
 501 }
 502
 503 ex dirac_gamma6(unsigned char rl)
 504 {
 505         return clifford(diracone(), rl) + clifford(diracgamma5(), rl);
 506 }
 507
 508 ex dirac_gamma7(unsigned char rl)
 509 {
 510         return clifford(diracone(), rl) - clifford(diracgamma5(), rl);
 511 }
 512
 513 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
 514 {
 515         // Slashed vectors are actually stored as a clifford object with the
 516         // vector as its base expression and a (dummy) index that just serves
 517         // for storing the space dimensionality
 518         return clifford(e, varidx(0, dim), rl);
 519 }
 520
 521 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 522  *  is that of a clifford object with the specified representation label. */
 523 static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
 524 {
 525         return ti == (TINFO_clifford + rl);
 526 }
 527
 528 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 529  *  is that of a clifford object (with an arbitrary representation label). */
 530 static bool is_clifford_tinfo(unsigned ti)
 531 {
 532         return (ti & ~0xff) == TINFO_clifford;
 533 }
 534
 535 /** Take trace of a string of an even number of Dirac gammas given a vector
 536  *  of indices. */
 537 static ex trace_string(exvector::const_iterator ix, unsigned num)
 538 {
 539         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 540         if (num == 2)
 541                 return lorentz_g(ix[0], ix[1]);
 542
 543         // 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
 544         else if (num == 4)
 545                 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
 546                      + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
 547                      - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
 548
 549         // Traces of 6 or more gammas are computed recursively:
 550         // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
 551         //   + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
 552         //   - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
 553         //   + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
 554         //   - ...
 555         //   + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
 556         exvector v(num - 2);
 557         int sign = 1;
 558         ex result;
 559         for (unsigned i=1; i<num; i++) {
 560                 for (unsigned n=1, j=0; n<num; n++) {
 561                         if (n == i)
 562                                 continue;
 563                         v[j++] = ix[n];
 564                 }
 565                 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
 566                 sign = -sign;
 567         }
 568         return result;
 569 }
 570
 571 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
 572 {
 573         if (is_a<clifford>(e)) {
 574
 575                 if (!ex_to<clifford>(e).get_representation_label() == rl)
 576                         return _ex0;
 577                 const ex & g = e.op(0);
 578                 if (is_a<diracone>(g))
 579                         return trONE;
 580                 else if (is_a<diracgammaL>(g) || is_a<diracgammaR>(g))
 581                         return trONE/2;
 582                 else
 583                         return _ex0;
 584
 585         } else if (is_ex_exactly_of_type(e, mul)) {
 586
 587                 // Trace of product: pull out non-clifford factors
 588                 ex prod = _ex1;
 589                 for (unsigned i=0; i<e.nops(); i++) {
 590                         const ex &o = e.op(i);
 591                         if (is_clifford_tinfo(o.return_type_tinfo(), rl))
 592                                 prod *= dirac_trace(o, rl, trONE);
 593                         else
 594                                 prod *= o;
 595                 }
 596                 return prod;
 597
 598         } else if (is_ex_exactly_of_type(e, ncmul)) {
 599
 600                 if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
 601                         return _ex0;
 602
 603                 // Substitute gammaL/R and expand product, if necessary
 604                 ex e_expanded = e.subs(lst(
 605                         dirac_gammaL(rl) == (dirac_ONE(rl)-dirac_gamma5(rl))/2,
 606                         dirac_gammaR(rl) == (dirac_ONE(rl)+dirac_gamma5(rl))/2
 607                 )).expand();
 608                 if (!is_a<ncmul>(e_expanded))
 609                         return dirac_trace(e_expanded, rl, trONE);
 610
 611                 // gamma5 gets moved to the front so this check is enough
 612                 bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
 613                 unsigned num = e.nops();
 614
 615                 if (has_gamma5) {
 616
 617                         // Trace of gamma5 * odd number of gammas and trace of
 618                         // gamma5 * gamma.mu * gamma.nu are zero
 619                         if ((num & 1) == 0 || num == 3)
 620                                 return _ex0;
 621
 622                         // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
 623                         // (the epsilon is always 4-dimensional)
 624                         if (num == 5) {
 625                                 ex b1, i1, b2, i2, b3, i3, b4, i4;
 626                                 base_and_index(e.op(1), b1, i1);
 627                                 base_and_index(e.op(2), b2, i2);
 628                                 base_and_index(e.op(3), b3, i3);
 629                                 base_and_index(e.op(4), b4, i4);
 630                                 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();
 631                         }
 632
 633                         // Tr gamma5 S_2k =
 634                         //   I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
 635                         // (the epsilon is always 4-dimensional)
 636                         exvector ix(num-1), bv(num-1);
 637                         for (unsigned i=1; i<num; i++)
 638                                 base_and_index(e.op(i), bv[i-1], ix[i-1]);
 639                         num--;
 640                         int *iv = new int[num];
 641                         ex result;
 642                         for (unsigned i=0; i<num-3; i++) {
 643                                 ex idx1 = ix[i];
 644                                 for (unsigned j=i+1; j<num-2; j++) {
 645                                         ex idx2 = ix[j];
 646                                         for (unsigned k=j+1; k<num-1; k++) {
 647                                                 ex idx3 = ix[k];
 648                                                 for (unsigned l=k+1; l<num; l++) {
 649                                                         ex idx4 = ix[l];
 650                                                         iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
 651                                                         exvector v;
 652                                                         v.reserve(num - 4);
 653                                                         for (unsigned n=0, t=4; n<num; n++) {
 654                                                                 if (n == i || n == j || n == k || n == l)
 655                                                                         continue;
 656                                                                 iv[t++] = n;
 657                                                                 v.push_back(ix[n]);
 658                                                         }
 659                                                         int sign = permutation_sign(iv, iv + num);
 660                                                         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))
 661                                                                 * trace_string(v.begin(), num - 4);
 662                                                 }
 663                                         }
 664                                 }
 665                         }
 666                         delete[] iv;
 667                         return trONE * I * result * mul(bv);
 668
 669                 } else { // no gamma5
 670
 671                         // Trace of odd number of gammas is zero
 672                         if ((num & 1) == 1)
 673                                 return _ex0;
 674
 675                         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 676                         if (num == 2) {
 677                                 ex b1, i1, b2, i2;
 678                                 base_and_index(e.op(0), b1, i1);
 679                                 base_and_index(e.op(1), b2, i2);
 680                                 return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
 681                         }
 682
 683                         exvector iv(num), bv(num);
 684                         for (unsigned i=0; i<num; i++)
 685                                 base_and_index(e.op(i), bv[i], iv[i]);
 686
 687                         return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
 688                 }
 689
 690         } else if (e.nops() > 0) {
 691
 692                 // Trace maps to all other container classes (this includes sums)
 693                 pointer_to_map_function_2args<unsigned char, const ex &> fcn(dirac_trace, rl, trONE);
 694                 return e.map(fcn);
 695
 696         } else
 697                 return _ex0;
 698 }
 699
 700 ex canonicalize_clifford(const ex & e)
 701 {
 702         // Scan for any ncmul objects
 703         lst srl;
 704         ex aux = e.to_rational(srl);
 705         for (unsigned i=0; i<srl.nops(); i++) {
 706
 707                 ex lhs = srl.op(i).lhs();
 708                 ex rhs = srl.op(i).rhs();
 709
 710                 if (is_ex_exactly_of_type(rhs, ncmul)
 711                  && rhs.return_type() == return_types::noncommutative
 712                  && is_clifford_tinfo(rhs.return_type_tinfo())) {
 713
 714                         // Expand product, if necessary
 715                         ex rhs_expanded = rhs.expand();
 716                         if (!is_a<ncmul>(rhs_expanded)) {
 717                                 srl.let_op(i) = (lhs == canonicalize_clifford(rhs_expanded));
 718                                 continue;
 719
 720                         } else if (!is_a<clifford>(rhs.op(0)))
 721                                 continue;
 722
 723                         exvector v;
 724                         v.reserve(rhs.nops());
 725                         for (unsigned j=0; j<rhs.nops(); j++)
 726                                 v.push_back(rhs.op(j));
 727
 728                         // Stupid recursive bubble sort because we only want to swap adjacent gammas
 729                         exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
 730                         if (is_a<diracgamma5>(it->op(0)) || is_a<diracgammaL>(it->op(0)) || is_a<diracgammaR>(it->op(0)))
 731                                 ++it;
 732                         while (it != next_to_last) {
 733                                 if (it[0].compare(it[1]) > 0) {
 734                                         ex save0 = it[0], save1 = it[1];
 735                                         ex b1, i1, b2, i2;
 736                                         base_and_index(it[0], b1, i1);
 737                                         base_and_index(it[1], b2, i2);
 738                                         it[0] = (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
 739                                         it[1] = _ex2;
 740                                         ex sum = ncmul(v);
 741                                         it[0] = save1;
 742                                         it[1] = save0;
 743                                         sum -= ncmul(v, true);
 744                                         srl.let_op(i) = (lhs == canonicalize_clifford(sum));
 745                                         goto next_sym;
 746                                 }
 747                                 ++it;
 748                         }
 749 next_sym:       ;
 750                 }
 751         }
 752         return aux.subs(srl).simplify_indexed();
 753 }
 754
 755 } // namespace GiNaC