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