ginac/clifford.cpp

   1 /** @file clifford.cpp
   2  *
   3  *  Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2002 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         ex dim = ex_to<idx>(self->op(1)).get_dim();
 212
 213         if (is_a<clifford>(*other)) {
 214
 215                 // Contraction only makes sense if the represenation labels are equal
 216                 if (ex_to<clifford>(*other).get_representation_label() != rl)
 217                         return false;
 218
 219                 // gamma~mu gamma.mu = dim ONE
 220                 if (other - self == 1) {
 221                         *self = dim;
 222                         *other = dirac_ONE(rl);
 223                         return true;
 224
 225                 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
 226                 } else if (other - self == 2
 227                         && is_a<clifford>(self[1])) {
 228                         *self = 2 - dim;
 229                         *other = _ex1;
 230                         return true;
 231
 232                 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
 233                 } else if (other - self == 3
 234                         && is_a<clifford>(self[1])
 235                         && is_a<clifford>(self[2])) {
 236                         ex b1, i1, b2, i2;
 237                         base_and_index(self[1], b1, i1);
 238                         base_and_index(self[2], b2, i2);
 239                         *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
 240                         self[1] = _ex1;
 241                         self[2] = _ex1;
 242                         *other = _ex1;
 243                         return true;
 244
 245                 // 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
 246                 } else if (other - self == 4
 247                         && is_a<clifford>(self[1])
 248                         && is_a<clifford>(self[2])
 249                         && is_a<clifford>(self[3])) {
 250                         *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3];
 251                         self[1] = _ex1;
 252                         self[2] = _ex1;
 253                         self[3] = _ex1;
 254                         *other = _ex1;
 255                         return true;
 256
 257                 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
 258                 // (commutate contracted indices towards each other, simplify_indexed()
 259                 // will re-expand and re-run the simplification)
 260                 } else {
 261                         exvector::iterator it = self + 1, next_to_last = other - 1;
 262                         while (it != other) {
 263                                 if (!is_a<clifford>(*it))
 264                                         return false;
 265                                 ++it;
 266                         }
 267
 268                         it = self + 1;
 269                         ex S = _ex1;
 270                         while (it != next_to_last) {
 271                                 S *= *it;
 272                                 *it++ = _ex1;
 273                         }
 274
 275                         *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
 276                         *next_to_last = _ex1;
 277                         *other = _ex1;
 278                         return true;
 279                 }
 280
 281         } else if (is_a<symbol>(other->op(0)) && other->nops() == 2) {
 282
 283                 // x.mu gamma~mu -> x-slash
 284                 *self = dirac_slash(other->op(0), dim, rl);
 285                 *other = _ex1;
 286                 return true;
 287         }
 288
 289         return false;
 290 }
 291
 292 /** Perform automatic simplification on noncommutative product of clifford
 293  *  objects. This removes superfluous ONEs, permutes gamma5/L/R's to the front
 294  *  and removes squares of gamma objects. */
 295 ex clifford::simplify_ncmul(const exvector & v) const
 296 {
 297         exvector s;
 298         s.reserve(v.size());
 299
 300         // Remove superfluous ONEs
 301         exvector::const_iterator cit = v.begin(), citend = v.end();
 302         while (cit != citend) {
 303                 if (!is_a<clifford>(*cit) || !is_a<diracone>(cit->op(0)))
 304                         s.push_back(*cit);
 305                 cit++;
 306         }
 307
 308         bool something_changed = false;
 309         int sign = 1;
 310
 311         // Anticommute gamma5/L/R's to the front
 312         if (s.size() >= 2) {
 313                 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
 314                 while (true) {
 315                         exvector::iterator it = next_to_last;
 316                         while (true) {
 317                                 exvector::iterator it2 = it + 1;
 318                                 if (is_a<clifford>(*it) && is_a<clifford>(*it2)) {
 319                                         ex e1 = it->op(0), e2 = it2->op(0);
 320
 321                                         if (is_a<diracgamma5>(e2)) {
 322
 323                                                 if (is_a<diracgammaL>(e1) || is_a<diracgammaR>(e1)) {
 324
 325                                                         // gammaL/R gamma5 -> gamma5 gammaL/R
 326                                                         it->swap(*it2);
 327                                                         something_changed = true;
 328
 329                                                 } else if (!is_a<diracgamma5>(e1)) {
 330
 331                                                         // gamma5 gamma5 -> gamma5 gamma5 (do nothing)
 332                                                         // x gamma5 -> -gamma5 x
 333                                                         it->swap(*it2);
 334                                                         sign = -sign;
 335                                                         something_changed = true;
 336                                                 }
 337
 338                                         } else if (is_a<diracgammaL>(e2)) {
 339
 340                                                 if (is_a<diracgammaR>(e1)) {
 341
 342                                                         // gammaR gammaL -> 0
 343                                                         return _ex0;
 344
 345                                                 } else if (!is_a<diracgammaL>(e1) && !is_a<diracgamma5>(e1)) {
 346
 347                                                         // gammaL gammaL -> gammaL gammaL (do nothing)
 348                                                         // gamma5 gammaL -> gamma5 gammaL (do nothing)
 349                                                         // x gammaL -> gammaR x
 350                                                         it->swap(*it2);
 351                                                         *it = clifford(diracgammaR(), ex_to<clifford>(*it).get_representation_label());
 352                                                         something_changed = true;
 353                                                 }
 354
 355                                         } else if (is_a<diracgammaR>(e2)) {
 356
 357                                                 if (is_a<diracgammaL>(e1)) {
 358
 359                                                         // gammaL gammaR -> 0
 360                                                         return _ex0;
 361
 362                                                 } else if (!is_a<diracgammaR>(e1) && !is_a<diracgamma5>(e1)) {
 363
 364                                                         // gammaR gammaR -> gammaR gammaR (do nothing)
 365                                                         // gamma5 gammaR -> gamma5 gammaR (do nothing)
 366                                                         // x gammaR -> gammaL x
 367                                                         it->swap(*it2);
 368                                                         *it = clifford(diracgammaL(), ex_to<clifford>(*it).get_representation_label());
 369                                                         something_changed = true;
 370                                                 }
 371                                         }
 372                                 }
 373                                 if (it == first)
 374                                         break;
 375                                 --it;
 376                         }
 377                         if (next_to_last == first)
 378                                 break;
 379                         --next_to_last;
 380                 }
 381         }
 382
 383         // Remove equal adjacent gammas
 384         if (s.size() >= 2) {
 385                 exvector::iterator it, itend = s.end() - 1;
 386                 for (it = s.begin(); it != itend; ++it) {
 387                         ex & a = it[0];
 388                         ex & b = it[1];
 389                         if (!is_a<clifford>(a) || !is_a<clifford>(b))
 390                                 continue;
 391
 392                         const ex & ag = a.op(0);
 393                         const ex & bg = b.op(0);
 394                         bool a_is_diracgamma = is_a<diracgamma>(ag);
 395                         bool b_is_diracgamma = is_a<diracgamma>(bg);
 396
 397                         if (a_is_diracgamma && b_is_diracgamma) {
 398
 399                                 const ex & ia = a.op(1);
 400                                 const ex & ib = b.op(1);
 401                                 if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
 402                                         a = lorentz_g(ia, ib);
 403                                         b = dirac_ONE(representation_label);
 404                                         something_changed = true;
 405                                 }
 406
 407                         } else if ((is_a<diracgamma5>(ag) && is_a<diracgamma5>(bg))) {
 408
 409                                 // Remove squares of gamma5
 410                                 a = dirac_ONE(representation_label);
 411                                 b = dirac_ONE(representation_label);
 412                                 something_changed = true;
 413
 414                         } else if ((is_a<diracgammaL>(ag) && is_a<diracgammaL>(bg))
 415                                 || (is_a<diracgammaR>(ag) && is_a<diracgammaR>(bg))) {
 416
 417                                 // Remove squares of gammaL/R
 418                                 b = dirac_ONE(representation_label);
 419                                 something_changed = true;
 420
 421                         } else if (is_a<diracgammaL>(ag) && is_a<diracgammaR>(bg)) {
 422
 423                                 // gammaL and gammaR are orthogonal
 424                                 return _ex0;
 425
 426                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaL>(bg)) {
 427
 428                                 // gamma5 gammaL -> -gammaL
 429                                 a = dirac_ONE(representation_label);
 430                                 sign = -sign;
 431                                 something_changed = true;
 432
 433                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaR>(bg)) {
 434
 435                                 // gamma5 gammaR -> gammaR
 436                                 a = dirac_ONE(representation_label);
 437                                 something_changed = true;
 438
 439                         } else if (!a_is_diracgamma && !b_is_diracgamma && ag.is_equal(bg)) {
 440
 441                                 // a\ a\ -> a^2
 442                                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).get_dim());
 443                                 a = indexed(ag, ix) * indexed(ag, ix.toggle_variance());
 444                                 b = dirac_ONE(representation_label);
 445                                 something_changed = true;
 446                         }
 447                 }
 448         }
 449
 450         if (s.empty())
 451                 return clifford(diracone(), representation_label) * sign;
 452         if (something_changed)
 453                 return nonsimplified_ncmul(s) * sign;
 454         else
 455                 return simplified_ncmul(s) * sign;
 456 }
 457
 458 ex clifford::thisexprseq(const exvector & v) const
 459 {
 460         return clifford(representation_label, v);
 461 }
 462
 463 ex clifford::thisexprseq(exvector * vp) const
 464 {
 465         return clifford(representation_label, vp);
 466 }
 467
 468 //////////
 469 // global functions
 470 //////////
 471
 472 ex dirac_ONE(unsigned char rl)
 473 {
 474         return clifford(diracone(), rl);
 475 }
 476
 477 ex dirac_gamma(const ex & mu, unsigned char rl)
 478 {
 479         if (!is_a<varidx>(mu))
 480                 throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
 481
 482         return clifford(diracgamma(), mu, rl);
 483 }
 484
 485 ex dirac_gamma5(unsigned char rl)
 486 {
 487         return clifford(diracgamma5(), rl);
 488 }
 489
 490 ex dirac_gammaL(unsigned char rl)
 491 {
 492         return clifford(diracgammaL(), rl);
 493 }
 494
 495 ex dirac_gammaR(unsigned char rl)
 496 {
 497         return clifford(diracgammaR(), rl);
 498 }
 499
 500 ex dirac_gamma6(unsigned char rl)
 501 {
 502         return clifford(diracone(), rl) + clifford(diracgamma5(), rl);
 503 }
 504
 505 ex dirac_gamma7(unsigned char rl)
 506 {
 507         return clifford(diracone(), rl) - clifford(diracgamma5(), rl);
 508 }
 509
 510 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
 511 {
 512         // Slashed vectors are actually stored as a clifford object with the
 513         // vector as its base expression and a (dummy) index that just serves
 514         // for storing the space dimensionality
 515         return clifford(e, varidx(0, dim), rl);
 516 }
 517
 518 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 519  *  is that of a clifford object with the specified representation label. */
 520 static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
 521 {
 522         return ti == (TINFO_clifford + rl);
 523 }
 524
 525 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 526  *  is that of a clifford object (with an arbitrary representation label). */
 527 static bool is_clifford_tinfo(unsigned ti)
 528 {
 529         return (ti & ~0xff) == TINFO_clifford;
 530 }
 531
 532 /** Take trace of a string of an even number of Dirac gammas given a vector
 533  *  of indices. */
 534 static ex trace_string(exvector::const_iterator ix, unsigned num)
 535 {
 536         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 537         if (num == 2)
 538                 return lorentz_g(ix[0], ix[1]);
 539
 540         // 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
 541         else if (num == 4)
 542                 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
 543                      + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
 544                      - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
 545
 546         // Traces of 6 or more gammas are computed recursively:
 547         // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
 548         //   + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
 549         //   - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
 550         //   + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
 551         //   - ...
 552         //   + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
 553         exvector v(num - 2);
 554         int sign = 1;
 555         ex result;
 556         for (unsigned i=1; i<num; i++) {
 557                 for (unsigned n=1, j=0; n<num; n++) {
 558                         if (n == i)
 559                                 continue;
 560                         v[j++] = ix[n];
 561                 }
 562                 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
 563                 sign = -sign;
 564         }
 565         return result;
 566 }
 567
 568 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
 569 {
 570         if (is_a<clifford>(e)) {
 571
 572                 if (!ex_to<clifford>(e).get_representation_label() == rl)
 573                         return _ex0;
 574                 const ex & g = e.op(0);
 575                 if (is_a<diracone>(g))
 576                         return trONE;
 577                 else if (is_a<diracgammaL>(g) || is_a<diracgammaR>(g))
 578                         return trONE/2;
 579                 else
 580                         return _ex0;
 581
 582         } else if (is_ex_exactly_of_type(e, mul)) {
 583
 584                 // Trace of product: pull out non-clifford factors
 585                 ex prod = _ex1;
 586                 for (unsigned i=0; i<e.nops(); i++) {
 587                         const ex &o = e.op(i);
 588                         if (is_clifford_tinfo(o.return_type_tinfo(), rl))
 589                                 prod *= dirac_trace(o, rl, trONE);
 590                         else
 591                                 prod *= o;
 592                 }
 593                 return prod;
 594
 595         } else if (is_ex_exactly_of_type(e, ncmul)) {
 596
 597                 if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
 598                         return _ex0;
 599
 600                 // Substitute gammaL/R and expand product, if necessary
 601                 ex e_expanded = e.subs(lst(
 602                         dirac_gammaL(rl) == (dirac_ONE(rl)-dirac_gamma5(rl))/2,
 603                         dirac_gammaR(rl) == (dirac_ONE(rl)+dirac_gamma5(rl))/2
 604                 )).expand();
 605                 if (!is_a<ncmul>(e_expanded))
 606                         return dirac_trace(e_expanded, rl, trONE);
 607
 608                 // gamma5 gets moved to the front so this check is enough
 609                 bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
 610                 unsigned num = e.nops();
 611
 612                 if (has_gamma5) {
 613
 614                         // Trace of gamma5 * odd number of gammas and trace of
 615                         // gamma5 * gamma.mu * gamma.nu are zero
 616                         if ((num & 1) == 0 || num == 3)
 617                                 return _ex0;
 618
 619                         // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
 620                         // (the epsilon is always 4-dimensional)
 621                         if (num == 5) {
 622                                 ex b1, i1, b2, i2, b3, i3, b4, i4;
 623                                 base_and_index(e.op(1), b1, i1);
 624                                 base_and_index(e.op(2), b2, i2);
 625                                 base_and_index(e.op(3), b3, i3);
 626                                 base_and_index(e.op(4), b4, i4);
 627                                 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();
 628                         }
 629
 630                         // Tr gamma5 S_2k =
 631                         //   I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
 632                         // (the epsilon is always 4-dimensional)
 633                         exvector ix(num-1), bv(num-1);
 634                         for (unsigned i=1; i<num; i++)
 635                                 base_and_index(e.op(i), bv[i-1], ix[i-1]);
 636                         num--;
 637                         int *iv = new int[num];
 638                         ex result;
 639                         for (unsigned i=0; i<num-3; i++) {
 640                                 ex idx1 = ix[i];
 641                                 for (unsigned j=i+1; j<num-2; j++) {
 642                                         ex idx2 = ix[j];
 643                                         for (unsigned k=j+1; k<num-1; k++) {
 644                                                 ex idx3 = ix[k];
 645                                                 for (unsigned l=k+1; l<num; l++) {
 646                                                         ex idx4 = ix[l];
 647                                                         iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
 648                                                         exvector v;
 649                                                         v.reserve(num - 4);
 650                                                         for (unsigned n=0, t=4; n<num; n++) {
 651                                                                 if (n == i || n == j || n == k || n == l)
 652                                                                         continue;
 653                                                                 iv[t++] = n;
 654                                                                 v.push_back(ix[n]);
 655                                                         }
 656                                                         int sign = permutation_sign(iv, iv + num);
 657                                                         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))
 658                                                                 * trace_string(v.begin(), num - 4);
 659                                                 }
 660                                         }
 661                                 }
 662                         }
 663                         delete[] iv;
 664                         return trONE * I * result * mul(bv);
 665
 666                 } else { // no gamma5
 667
 668                         // Trace of odd number of gammas is zero
 669                         if ((num & 1) == 1)
 670                                 return _ex0;
 671
 672                         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 673                         if (num == 2) {
 674                                 ex b1, i1, b2, i2;
 675                                 base_and_index(e.op(0), b1, i1);
 676                                 base_and_index(e.op(1), b2, i2);
 677                                 return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
 678                         }
 679
 680                         exvector iv(num), bv(num);
 681                         for (unsigned i=0; i<num; i++)
 682                                 base_and_index(e.op(i), bv[i], iv[i]);
 683
 684                         return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
 685                 }
 686
 687         } else if (e.nops() > 0) {
 688
 689                 // Trace maps to all other container classes (this includes sums)
 690                 pointer_to_map_function_2args<unsigned char, const ex &> fcn(dirac_trace, rl, trONE);
 691                 return e.map(fcn);
 692
 693         } else
 694                 return _ex0;
 695 }
 696
 697 ex canonicalize_clifford(const ex & e)
 698 {
 699         // Scan for any ncmul objects
 700         lst srl;
 701         ex aux = e.to_rational(srl);
 702         for (unsigned i=0; i<srl.nops(); i++) {
 703
 704                 ex lhs = srl.op(i).lhs();
 705                 ex rhs = srl.op(i).rhs();
 706
 707                 if (is_ex_exactly_of_type(rhs, ncmul)
 708                  && rhs.return_type() == return_types::noncommutative
 709                  && is_clifford_tinfo(rhs.return_type_tinfo())) {
 710
 711                         // Expand product, if necessary
 712                         ex rhs_expanded = rhs.expand();
 713                         if (!is_a<ncmul>(rhs_expanded)) {
 714                                 srl.let_op(i) = (lhs == canonicalize_clifford(rhs_expanded));
 715                                 continue;
 716
 717                         } else if (!is_a<clifford>(rhs.op(0)))
 718                                 continue;
 719
 720                         exvector v;
 721                         v.reserve(rhs.nops());
 722                         for (unsigned j=0; j<rhs.nops(); j++)
 723                                 v.push_back(rhs.op(j));
 724
 725                         // Stupid recursive bubble sort because we only want to swap adjacent gammas
 726                         exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
 727                         if (is_a<diracgamma5>(it->op(0)) || is_a<diracgammaL>(it->op(0)) || is_a<diracgammaR>(it->op(0)))
 728                                 ++it;
 729                         while (it != next_to_last) {
 730                                 if (it[0].compare(it[1]) > 0) {
 731                                         ex save0 = it[0], save1 = it[1];
 732                                         ex b1, i1, b2, i2;
 733                                         base_and_index(it[0], b1, i1);
 734                                         base_and_index(it[1], b2, i2);
 735                                         it[0] = (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
 736                                         it[1] = _ex2;
 737                                         ex sum = ncmul(v);
 738                                         it[0] = save1;
 739                                         it[1] = save0;
 740                                         sum -= ncmul(v, true);
 741                                         srl.let_op(i) = (lhs == canonicalize_clifford(sum));
 742                                         goto next_sym;
 743                                 }
 744                                 ++it;
 745                         }
 746 next_sym:       ;
 747                 }
 748         }
 749         return aux.subs(srl).simplify_indexed();
 750 }
 751
 752 } // namespace GiNaC