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 "operators.h"
  36 #include "mul.h"
  37 #include "archive.h"
  38 #include "utils.h"
  39
  40 namespace GiNaC {
  41
  42 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(clifford, indexed,
  43   print_func<print_dflt>(&clifford::do_print_dflt).
  44   print_func<print_latex>(&clifford::do_print_latex))
  45
  46 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracone, tensor,
  47   print_func<print_dflt>(&diracone::do_print).
  48   print_func<print_latex>(&diracone::do_print_latex))
  49
  50 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma, tensor,
  51   print_func<print_dflt>(&diracgamma::do_print).
  52   print_func<print_latex>(&diracgamma::do_print_latex))
  53
  54 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma5, tensor,
  55   print_func<print_dflt>(&diracgamma5::do_print).
  56   print_func<print_latex>(&diracgamma5::do_print_latex))
  57
  58 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaL, tensor,
  59   print_func<print_context>(&diracgammaL::do_print).
  60   print_func<print_latex>(&diracgammaL::do_print_latex))
  61
  62 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaR, tensor,
  63   print_func<print_context>(&diracgammaR::do_print).
  64   print_func<print_latex>(&diracgammaR::do_print_latex))
  65
  66 //////////
  67 // default constructors
  68 //////////
  69
  70 clifford::clifford() : representation_label(0)
  71 {
  72         tinfo_key = TINFO_clifford;
  73 }
  74
  75 DEFAULT_CTOR(diracone)
  76 DEFAULT_CTOR(diracgamma)
  77 DEFAULT_CTOR(diracgamma5)
  78 DEFAULT_CTOR(diracgammaL)
  79 DEFAULT_CTOR(diracgammaR)
  80
  81 //////////
  82 // other constructors
  83 //////////
  84
  85 /** Construct object without any indices. This constructor is for internal
  86  *  use only. Use the dirac_ONE() function instead.
  87  *  @see dirac_ONE */
  88 clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
  89 {
  90         tinfo_key = TINFO_clifford;
  91 }
  92
  93 /** Construct object with one Lorentz index. This constructor is for internal
  94  *  use only. Use the dirac_gamma() function instead.
  95  *  @see dirac_gamma */
  96 clifford::clifford(const ex & b, const ex & mu, unsigned char rl) : inherited(b, mu), representation_label(rl)
  97 {
  98         GINAC_ASSERT(is_a<varidx>(mu));
  99         tinfo_key = TINFO_clifford;
 100 }
 101
 102 clifford::clifford(unsigned char rl, const exvector & v, bool discardable) : inherited(sy_none(), v, discardable), representation_label(rl)
 103 {
 104         tinfo_key = TINFO_clifford;
 105 }
 106
 107 clifford::clifford(unsigned char rl, exvector * vp) : inherited(sy_none(), vp), representation_label(rl)
 108 {
 109         tinfo_key = TINFO_clifford;
 110 }
 111
 112 //////////
 113 // archiving
 114 //////////
 115
 116 clifford::clifford(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
 117 {
 118         unsigned rl;
 119         n.find_unsigned("label", rl);
 120         representation_label = rl;
 121 }
 122
 123 void clifford::archive(archive_node &n) const
 124 {
 125         inherited::archive(n);
 126         n.add_unsigned("label", representation_label);
 127 }
 128
 129 DEFAULT_UNARCHIVE(clifford)
 130 DEFAULT_ARCHIVING(diracone)
 131 DEFAULT_ARCHIVING(diracgamma)
 132 DEFAULT_ARCHIVING(diracgamma5)
 133 DEFAULT_ARCHIVING(diracgammaL)
 134 DEFAULT_ARCHIVING(diracgammaR)
 135
 136 //////////
 137 // functions overriding virtual functions from base classes
 138 //////////
 139
 140 int clifford::compare_same_type(const basic & other) const
 141 {
 142         GINAC_ASSERT(is_a<clifford>(other));
 143         const clifford &o = static_cast<const clifford &>(other);
 144
 145         if (representation_label != o.representation_label) {
 146                 // different representation label
 147                 return representation_label < o.representation_label ? -1 : 1;
 148         }
 149
 150         return inherited::compare_same_type(other);
 151 }
 152
 153 bool clifford::match_same_type(const basic & other) const
 154 {
 155         GINAC_ASSERT(is_a<clifford>(other));
 156         const clifford &o = static_cast<const clifford &>(other);
 157
 158         return representation_label == o.representation_label;
 159 }
 160
 161 static bool is_dirac_slash(const ex & seq0)
 162 {
 163         return !is_a<diracgamma5>(seq0) && !is_a<diracgammaL>(seq0) &&
 164                !is_a<diracgammaR>(seq0) && !is_a<diracgamma>(seq0) &&
 165                !is_a<diracone>(seq0);
 166 }
 167
 168 void clifford::do_print_dflt(const print_dflt & c, unsigned level) const
 169 {
 170         // dirac_slash() object is printed differently
 171         if (is_dirac_slash(seq[0])) {
 172                 seq[0].print(c, level);
 173                 c.s << "\\";
 174         } else
 175                 this->print_dispatch<inherited>(c, level);
 176 }
 177
 178 void clifford::do_print_latex(const print_latex & c, unsigned level) const
 179 {
 180         // dirac_slash() object is printed differently
 181         if (is_dirac_slash(seq[0])) {
 182                 c.s << "{";
 183                 seq[0].print(c, level);
 184                 c.s << "\\hspace{-1.0ex}/}";
 185         } else
 186                 this->print_dispatch<inherited>(c, level);
 187 }
 188
 189 DEFAULT_COMPARE(diracone)
 190 DEFAULT_COMPARE(diracgamma)
 191 DEFAULT_COMPARE(diracgamma5)
 192 DEFAULT_COMPARE(diracgammaL)
 193 DEFAULT_COMPARE(diracgammaR)
 194
 195 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
 196 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
 197 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
 198 DEFAULT_PRINT_LATEX(diracgammaL, "gammaL", "{\\gamma_L}")
 199 DEFAULT_PRINT_LATEX(diracgammaR, "gammaR", "{\\gamma_R}")
 200
 201 /** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
 202 static void base_and_index(const ex & c, ex & b, ex & i)
 203 {
 204         GINAC_ASSERT(is_a<clifford>(c));
 205         GINAC_ASSERT(c.nops() == 2);
 206
 207         if (is_a<diracgamma>(c.op(0))) { // proper dirac gamma object
 208                 i = c.op(1);
 209                 b = _ex1;
 210         } else if (is_a<diracgamma5>(c.op(0)) || is_a<diracgammaL>(c.op(0)) || is_a<diracgammaR>(c.op(0))) { // gamma5/L/R
 211                 i = _ex0;
 212                 b = _ex1;
 213         } else { // slash object, generate new dummy index
 214                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(c.op(1)).get_dim());
 215                 b = indexed(c.op(0), ix.toggle_variance());
 216                 i = ix;
 217         }
 218 }
 219
 220 /** Contraction of a gamma matrix with something else. */
 221 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 222 {
 223         GINAC_ASSERT(is_a<clifford>(*self));
 224         GINAC_ASSERT(is_a<indexed>(*other));
 225         GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
 226         unsigned char rl = ex_to<clifford>(*self).get_representation_label();
 227
 228         ex dim = ex_to<idx>(self->op(1)).get_dim();
 229         if (other->nops() > 1)
 230                 dim = minimal_dim(dim, ex_to<idx>(other->op(1)).get_dim());
 231
 232         if (is_a<clifford>(*other)) {
 233
 234                 // Contraction only makes sense if the represenation labels are equal
 235                 if (ex_to<clifford>(*other).get_representation_label() != rl)
 236                         return false;
 237
 238                 // gamma~mu gamma.mu = dim ONE
 239                 if (other - self == 1) {
 240                         *self = dim;
 241                         *other = dirac_ONE(rl);
 242                         return true;
 243
 244                 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
 245                 } else if (other - self == 2
 246                         && is_a<clifford>(self[1])) {
 247                         *self = 2 - dim;
 248                         *other = _ex1;
 249                         return true;
 250
 251                 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
 252                 } else if (other - self == 3
 253                         && is_a<clifford>(self[1])
 254                         && is_a<clifford>(self[2])) {
 255                         ex b1, i1, b2, i2;
 256                         base_and_index(self[1], b1, i1);
 257                         base_and_index(self[2], b2, i2);
 258                         *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
 259                         self[1] = _ex1;
 260                         self[2] = _ex1;
 261                         *other = _ex1;
 262                         return true;
 263
 264                 // 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
 265                 } else if (other - self == 4
 266                         && is_a<clifford>(self[1])
 267                         && is_a<clifford>(self[2])
 268                         && is_a<clifford>(self[3])) {
 269                         *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3];
 270                         self[1] = _ex1;
 271                         self[2] = _ex1;
 272                         self[3] = _ex1;
 273                         *other = _ex1;
 274                         return true;
 275
 276                 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
 277                 // (commutate contracted indices towards each other, simplify_indexed()
 278                 // will re-expand and re-run the simplification)
 279                 } else {
 280                         exvector::iterator it = self + 1, next_to_last = other - 1;
 281                         while (it != other) {
 282                                 if (!is_a<clifford>(*it))
 283                                         return false;
 284                                 ++it;
 285                         }
 286
 287                         it = self + 1;
 288                         ex S = _ex1;
 289                         while (it != next_to_last) {
 290                                 S *= *it;
 291                                 *it++ = _ex1;
 292                         }
 293
 294                         *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
 295                         *next_to_last = _ex1;
 296                         *other = _ex1;
 297                         return true;
 298                 }
 299
 300         } else if (is_a<symbol>(other->op(0)) && other->nops() == 2) {
 301
 302                 // x.mu gamma~mu -> x-slash
 303                 *self = dirac_slash(other->op(0), dim, rl);
 304                 *other = _ex1;
 305                 return true;
 306         }
 307
 308         return false;
 309 }
 310
 311 /** Perform automatic simplification on noncommutative product of clifford
 312  *  objects. This removes superfluous ONEs, permutes gamma5/L/R's to the front
 313  *  and removes squares of gamma objects. */
 314 ex clifford::eval_ncmul(const exvector & v) const
 315 {
 316         exvector s;
 317         s.reserve(v.size());
 318
 319         // Remove superfluous ONEs
 320         exvector::const_iterator cit = v.begin(), citend = v.end();
 321         while (cit != citend) {
 322                 if (!is_a<clifford>(*cit) || !is_a<diracone>(cit->op(0)))
 323                         s.push_back(*cit);
 324                 cit++;
 325         }
 326
 327         bool something_changed = false;
 328         int sign = 1;
 329
 330         // Anticommute gamma5/L/R's to the front
 331         if (s.size() >= 2) {
 332                 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
 333                 while (true) {
 334                         exvector::iterator it = next_to_last;
 335                         while (true) {
 336                                 exvector::iterator it2 = it + 1;
 337                                 if (is_a<clifford>(*it) && is_a<clifford>(*it2)) {
 338                                         ex e1 = it->op(0), e2 = it2->op(0);
 339
 340                                         if (is_a<diracgamma5>(e2)) {
 341
 342                                                 if (is_a<diracgammaL>(e1) || is_a<diracgammaR>(e1)) {
 343
 344                                                         // gammaL/R gamma5 -> gamma5 gammaL/R
 345                                                         it->swap(*it2);
 346                                                         something_changed = true;
 347
 348                                                 } else if (!is_a<diracgamma5>(e1)) {
 349
 350                                                         // gamma5 gamma5 -> gamma5 gamma5 (do nothing)
 351                                                         // x gamma5 -> -gamma5 x
 352                                                         it->swap(*it2);
 353                                                         sign = -sign;
 354                                                         something_changed = true;
 355                                                 }
 356
 357                                         } else if (is_a<diracgammaL>(e2)) {
 358
 359                                                 if (is_a<diracgammaR>(e1)) {
 360
 361                                                         // gammaR gammaL -> 0
 362                                                         return _ex0;
 363
 364                                                 } else if (!is_a<diracgammaL>(e1) && !is_a<diracgamma5>(e1)) {
 365
 366                                                         // gammaL gammaL -> gammaL gammaL (do nothing)
 367                                                         // gamma5 gammaL -> gamma5 gammaL (do nothing)
 368                                                         // x gammaL -> gammaR x
 369                                                         it->swap(*it2);
 370                                                         *it = clifford(diracgammaR(), ex_to<clifford>(*it).get_representation_label());
 371                                                         something_changed = true;
 372                                                 }
 373
 374                                         } else if (is_a<diracgammaR>(e2)) {
 375
 376                                                 if (is_a<diracgammaL>(e1)) {
 377
 378                                                         // gammaL gammaR -> 0
 379                                                         return _ex0;
 380
 381                                                 } else if (!is_a<diracgammaR>(e1) && !is_a<diracgamma5>(e1)) {
 382
 383                                                         // gammaR gammaR -> gammaR gammaR (do nothing)
 384                                                         // gamma5 gammaR -> gamma5 gammaR (do nothing)
 385                                                         // x gammaR -> gammaL x
 386                                                         it->swap(*it2);
 387                                                         *it = clifford(diracgammaL(), ex_to<clifford>(*it).get_representation_label());
 388                                                         something_changed = true;
 389                                                 }
 390                                         }
 391                                 }
 392                                 if (it == first)
 393                                         break;
 394                                 --it;
 395                         }
 396                         if (next_to_last == first)
 397                                 break;
 398                         --next_to_last;
 399                 }
 400         }
 401
 402         // Remove equal adjacent gammas
 403         if (s.size() >= 2) {
 404                 exvector::iterator it, itend = s.end() - 1;
 405                 for (it = s.begin(); it != itend; ++it) {
 406                         ex & a = it[0];
 407                         ex & b = it[1];
 408                         if (!is_a<clifford>(a) || !is_a<clifford>(b))
 409                                 continue;
 410
 411                         const ex & ag = a.op(0);
 412                         const ex & bg = b.op(0);
 413                         bool a_is_diracgamma = is_a<diracgamma>(ag);
 414                         bool b_is_diracgamma = is_a<diracgamma>(bg);
 415
 416                         if (a_is_diracgamma && b_is_diracgamma) {
 417
 418                                 const ex & ia = a.op(1);
 419                                 const ex & ib = b.op(1);
 420                                 if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
 421                                         a = lorentz_g(ia, ib);
 422                                         b = dirac_ONE(representation_label);
 423                                         something_changed = true;
 424                                 }
 425
 426                         } else if ((is_a<diracgamma5>(ag) && is_a<diracgamma5>(bg))) {
 427
 428                                 // Remove squares of gamma5
 429                                 a = dirac_ONE(representation_label);
 430                                 b = dirac_ONE(representation_label);
 431                                 something_changed = true;
 432
 433                         } else if ((is_a<diracgammaL>(ag) && is_a<diracgammaL>(bg))
 434                                 || (is_a<diracgammaR>(ag) && is_a<diracgammaR>(bg))) {
 435
 436                                 // Remove squares of gammaL/R
 437                                 b = dirac_ONE(representation_label);
 438                                 something_changed = true;
 439
 440                         } else if (is_a<diracgammaL>(ag) && is_a<diracgammaR>(bg)) {
 441
 442                                 // gammaL and gammaR are orthogonal
 443                                 return _ex0;
 444
 445                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaL>(bg)) {
 446
 447                                 // gamma5 gammaL -> -gammaL
 448                                 a = dirac_ONE(representation_label);
 449                                 sign = -sign;
 450                                 something_changed = true;
 451
 452                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaR>(bg)) {
 453
 454                                 // gamma5 gammaR -> gammaR
 455                                 a = dirac_ONE(representation_label);
 456                                 something_changed = true;
 457
 458                         } else if (!a_is_diracgamma && !b_is_diracgamma && ag.is_equal(bg)) {
 459
 460                                 // a\ a\ -> a^2
 461                                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).minimal_dim(ex_to<idx>(b.op(1))));
 462                                 a = indexed(ag, ix) * indexed(ag, ix.toggle_variance());
 463                                 b = dirac_ONE(representation_label);
 464                                 something_changed = true;
 465                         }
 466                 }
 467         }
 468
 469         if (s.empty())
 470                 return clifford(diracone(), representation_label) * sign;
 471         if (something_changed)
 472                 return reeval_ncmul(s) * sign;
 473         else
 474                 return hold_ncmul(s) * sign;
 475 }
 476
 477 ex clifford::thiscontainer(const exvector & v) const
 478 {
 479         return clifford(representation_label, v);
 480 }
 481
 482 ex clifford::thiscontainer(exvector * vp) const
 483 {
 484         return clifford(representation_label, vp);
 485 }
 486
 487 //////////
 488 // global functions
 489 //////////
 490
 491 ex dirac_ONE(unsigned char rl)
 492 {
 493         return clifford(diracone(), rl);
 494 }
 495
 496 ex dirac_gamma(const ex & mu, unsigned char rl)
 497 {
 498         if (!is_a<varidx>(mu))
 499                 throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
 500
 501         return clifford(diracgamma(), mu, rl);
 502 }
 503
 504 ex dirac_gamma5(unsigned char rl)
 505 {
 506         return clifford(diracgamma5(), rl);
 507 }
 508
 509 ex dirac_gammaL(unsigned char rl)
 510 {
 511         return clifford(diracgammaL(), rl);
 512 }
 513
 514 ex dirac_gammaR(unsigned char rl)
 515 {
 516         return clifford(diracgammaR(), rl);
 517 }
 518
 519 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
 520 {
 521         // Slashed vectors are actually stored as a clifford object with the
 522         // vector as its base expression and a (dummy) index that just serves
 523         // for storing the space dimensionality
 524         return clifford(e, varidx(0, dim), rl);
 525 }
 526
 527 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 528  *  is that of a clifford object with the specified representation label. */
 529 static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
 530 {
 531         return ti == (TINFO_clifford + rl);
 532 }
 533
 534 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 535  *  is that of a clifford object (with an arbitrary representation label). */
 536 static bool is_clifford_tinfo(unsigned ti)
 537 {
 538         return (ti & ~0xff) == TINFO_clifford;
 539 }
 540
 541 /** Take trace of a string of an even number of Dirac gammas given a vector
 542  *  of indices. */
 543 static ex trace_string(exvector::const_iterator ix, size_t num)
 544 {
 545         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 546         if (num == 2)
 547                 return lorentz_g(ix[0], ix[1]);
 548
 549         // 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 )
 550         else if (num == 4)
 551                 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
 552                      + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
 553                      - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
 554
 555         // Traces of 6 or more gammas are computed recursively:
 556         // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
 557         //   + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
 558         //   - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
 559         //   + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
 560         //   - ...
 561         //   + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
 562         exvector v(num - 2);
 563         int sign = 1;
 564         ex result;
 565         for (size_t i=1; i<num; i++) {
 566                 for (size_t n=1, j=0; n<num; n++) {
 567                         if (n == i)
 568                                 continue;
 569                         v[j++] = ix[n];
 570                 }
 571                 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
 572                 sign = -sign;
 573         }
 574         return result;
 575 }
 576
 577 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
 578 {
 579         if (is_a<clifford>(e)) {
 580
 581                 if (!ex_to<clifford>(e).get_representation_label() == rl)
 582                         return _ex0;
 583                 const ex & g = e.op(0);
 584                 if (is_a<diracone>(g))
 585                         return trONE;
 586                 else if (is_a<diracgammaL>(g) || is_a<diracgammaR>(g))
 587                         return trONE/2;
 588                 else
 589                         return _ex0;
 590
 591         } else if (is_exactly_a<mul>(e)) {
 592
 593                 // Trace of product: pull out non-clifford factors
 594                 ex prod = _ex1;
 595                 for (size_t i=0; i<e.nops(); i++) {
 596                         const ex &o = e.op(i);
 597                         if (is_clifford_tinfo(o.return_type_tinfo(), rl))
 598                                 prod *= dirac_trace(o, rl, trONE);
 599                         else
 600                                 prod *= o;
 601                 }
 602                 return prod;
 603
 604         } else if (is_exactly_a<ncmul>(e)) {
 605
 606                 if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
 607                         return _ex0;
 608
 609                 // Substitute gammaL/R and expand product, if necessary
 610                 ex e_expanded = e.subs(lst(
 611                         dirac_gammaL(rl) == (dirac_ONE(rl)-dirac_gamma5(rl))/2,
 612                         dirac_gammaR(rl) == (dirac_ONE(rl)+dirac_gamma5(rl))/2
 613                 ), subs_options::no_pattern).expand();
 614                 if (!is_a<ncmul>(e_expanded))
 615                         return dirac_trace(e_expanded, rl, trONE);
 616
 617                 // gamma5 gets moved to the front so this check is enough
 618                 bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
 619                 size_t num = e.nops();
 620
 621                 if (has_gamma5) {
 622
 623                         // Trace of gamma5 * odd number of gammas and trace of
 624                         // gamma5 * gamma.mu * gamma.nu are zero
 625                         if ((num & 1) == 0 || num == 3)
 626                                 return _ex0;
 627
 628                         // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
 629                         // (the epsilon is always 4-dimensional)
 630                         if (num == 5) {
 631                                 ex b1, i1, b2, i2, b3, i3, b4, i4;
 632                                 base_and_index(e.op(1), b1, i1);
 633                                 base_and_index(e.op(2), b2, i2);
 634                                 base_and_index(e.op(3), b3, i3);
 635                                 base_and_index(e.op(4), b4, i4);
 636                                 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();
 637                         }
 638
 639                         // Tr gamma5 S_2k =
 640                         //   I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
 641                         // (the epsilon is always 4-dimensional)
 642                         exvector ix(num-1), bv(num-1);
 643                         for (size_t i=1; i<num; i++)
 644                                 base_and_index(e.op(i), bv[i-1], ix[i-1]);
 645                         num--;
 646                         int *iv = new int[num];
 647                         ex result;
 648                         for (size_t i=0; i<num-3; i++) {
 649                                 ex idx1 = ix[i];
 650                                 for (size_t j=i+1; j<num-2; j++) {
 651                                         ex idx2 = ix[j];
 652                                         for (size_t k=j+1; k<num-1; k++) {
 653                                                 ex idx3 = ix[k];
 654                                                 for (size_t l=k+1; l<num; l++) {
 655                                                         ex idx4 = ix[l];
 656                                                         iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
 657                                                         exvector v;
 658                                                         v.reserve(num - 4);
 659                                                         for (size_t n=0, t=4; n<num; n++) {
 660                                                                 if (n == i || n == j || n == k || n == l)
 661                                                                         continue;
 662                                                                 iv[t++] = n;
 663                                                                 v.push_back(ix[n]);
 664                                                         }
 665                                                         int sign = permutation_sign(iv, iv + num);
 666                                                         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))
 667                                                                 * trace_string(v.begin(), num - 4);
 668                                                 }
 669                                         }
 670                                 }
 671                         }
 672                         delete[] iv;
 673                         return trONE * I * result * mul(bv);
 674
 675                 } else { // no gamma5
 676
 677                         // Trace of odd number of gammas is zero
 678                         if ((num & 1) == 1)
 679                                 return _ex0;
 680
 681                         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 682                         if (num == 2) {
 683                                 ex b1, i1, b2, i2;
 684                                 base_and_index(e.op(0), b1, i1);
 685                                 base_and_index(e.op(1), b2, i2);
 686                                 return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
 687                         }
 688
 689                         exvector iv(num), bv(num);
 690                         for (size_t i=0; i<num; i++)
 691                                 base_and_index(e.op(i), bv[i], iv[i]);
 692
 693                         return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
 694                 }
 695
 696         } else if (e.nops() > 0) {
 697
 698                 // Trace maps to all other container classes (this includes sums)
 699                 pointer_to_map_function_2args<unsigned char, const ex &> fcn(dirac_trace, rl, trONE);
 700                 return e.map(fcn);
 701
 702         } else
 703                 return _ex0;
 704 }
 705
 706 ex canonicalize_clifford(const ex & e)
 707 {
 708         // Scan for any ncmul objects
 709         lst srl;
 710         ex aux = e.to_rational(srl);
 711         for (size_t i=0; i<srl.nops(); i++) {
 712
 713                 ex o = srl.op(i);
 714                 ex lhs = o.lhs();
 715                 ex rhs = o.rhs();
 716
 717                 if (is_exactly_a<ncmul>(rhs)
 718                  && rhs.return_type() == return_types::noncommutative
 719                  && is_clifford_tinfo(rhs.return_type_tinfo())) {
 720
 721                         // Expand product, if necessary
 722                         ex rhs_expanded = rhs.expand();
 723                         if (!is_a<ncmul>(rhs_expanded)) {
 724                                 srl[i] = (lhs == canonicalize_clifford(rhs_expanded));
 725                                 continue;
 726
 727                         } else if (!is_a<clifford>(rhs.op(0)))
 728                                 continue;
 729
 730                         exvector v;
 731                         v.reserve(rhs.nops());
 732                         for (size_t j=0; j<rhs.nops(); j++)
 733                                 v.push_back(rhs.op(j));
 734
 735                         // Stupid recursive bubble sort because we only want to swap adjacent gammas
 736                         exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
 737                         if (is_a<diracgamma5>(it->op(0)) || is_a<diracgammaL>(it->op(0)) || is_a<diracgammaR>(it->op(0)))
 738                                 ++it;
 739                         while (it != next_to_last) {
 740                                 if (it[0].compare(it[1]) > 0) {
 741                                         ex save0 = it[0], save1 = it[1];
 742                                         ex b1, i1, b2, i2;
 743                                         base_and_index(it[0], b1, i1);
 744                                         base_and_index(it[1], b2, i2);
 745                                         it[0] = (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
 746                                         it[1] = _ex2;
 747                                         ex sum = ncmul(v);
 748                                         it[0] = save1;
 749                                         it[1] = save0;
 750                                         sum -= ncmul(v, true);
 751                                         srl[i] = (lhs == canonicalize_clifford(sum));
 752                                         goto next_sym;
 753                                 }
 754                                 ++it;
 755                         }
 756 next_sym:       ;
 757                 }
 758         }
 759         return aux.subs(srl, subs_options::no_pattern).simplify_indexed();
 760 }
 761
 762 } // namespace GiNaC