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