ginac/clifford.cpp

   1 /** @file clifford.cpp
   2  *
   3  *  Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2001 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 "clifford.h"
  24 #include "ex.h"
  25 #include "idx.h"
  26 #include "ncmul.h"
  27 #include "symbol.h"
  28 #include "numeric.h" // for I
  29 #include "symmetry.h"
  30 #include "lst.h"
  31 #include "relational.h"
  32 #include "mul.h"
  33 #include "print.h"
  34 #include "archive.h"
  35 #include "utils.h"
  36
  37 #include <stdexcept>
  38
  39 namespace GiNaC {
  40
  41 GINAC_IMPLEMENT_REGISTERED_CLASS(clifford, indexed)
  42 GINAC_IMPLEMENT_REGISTERED_CLASS(diracone, tensor)
  43 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma, tensor)
  44 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma5, tensor)
  45
  46 //////////
  47 // default ctor, dtor, copy ctor, assignment operator and helpers
  48 //////////
  49
  50 clifford::clifford() : representation_label(0)
  51 {
  52         tinfo_key = TINFO_clifford;
  53 }
  54
  55 void clifford::copy(const clifford & other)
  56 {
  57         inherited::copy(other);
  58         representation_label = other.representation_label;
  59 }
  60
  61 DEFAULT_DESTROY(clifford)
  62 DEFAULT_CTORS(diracone)
  63 DEFAULT_CTORS(diracgamma)
  64 DEFAULT_CTORS(diracgamma5)
  65
  66 //////////
  67 // other constructors
  68 //////////
  69
  70 /** Construct object without any indices. This constructor is for internal
  71  *  use only. Use the dirac_ONE() function instead.
  72  *  @see dirac_ONE */
  73 clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
  74 {
  75         tinfo_key = TINFO_clifford;
  76 }
  77
  78 /** Construct object with one Lorentz index. This constructor is for internal
  79  *  use only. Use the dirac_gamma() function instead.
  80  *  @see dirac_gamma */
  81 clifford::clifford(const ex & b, const ex & mu, unsigned char rl) : inherited(b, mu), representation_label(rl)
  82 {
  83         GINAC_ASSERT(is_a<varidx>(mu));
  84         tinfo_key = TINFO_clifford;
  85 }
  86
  87 clifford::clifford(unsigned char rl, const exvector & v, bool discardable) : inherited(sy_none(), v, discardable), representation_label(rl)
  88 {
  89         tinfo_key = TINFO_clifford;
  90 }
  91
  92 clifford::clifford(unsigned char rl, exvector * vp) : inherited(sy_none(), vp), representation_label(rl)
  93 {
  94         tinfo_key = TINFO_clifford;
  95 }
  96
  97 //////////
  98 // archiving
  99 //////////
 100
 101 clifford::clifford(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 102 {
 103         unsigned rl;
 104         n.find_unsigned("label", rl);
 105         representation_label = rl;
 106 }
 107
 108 void clifford::archive(archive_node &n) const
 109 {
 110         inherited::archive(n);
 111         n.add_unsigned("label", representation_label);
 112 }
 113
 114 DEFAULT_UNARCHIVE(clifford)
 115 DEFAULT_ARCHIVING(diracone)
 116 DEFAULT_ARCHIVING(diracgamma)
 117 DEFAULT_ARCHIVING(diracgamma5)
 118
 119 //////////
 120 // functions overriding virtual functions from base classes
 121 //////////
 122
 123 int clifford::compare_same_type(const basic & other) const
 124 {
 125         GINAC_ASSERT(is_a<clifford>(other));
 126         const clifford &o = static_cast<const clifford &>(other);
 127
 128         if (representation_label != o.representation_label) {
 129                 // different representation label
 130                 return representation_label < o.representation_label ? -1 : 1;
 131         }
 132
 133         return inherited::compare_same_type(other);
 134 }
 135
 136 bool clifford::match_same_type(const basic & other) const
 137 {
 138         GINAC_ASSERT(is_a<clifford>(other));
 139         const clifford &o = static_cast<const clifford &>(other);
 140
 141         return representation_label == o.representation_label;
 142 }
 143
 144 void clifford::print(const print_context & c, unsigned level) const
 145 {
 146         if (!is_a<diracgamma5>(seq[0]) && !is_a<diracgamma>(seq[0]) && !is_a<diracone>(seq[0])) {
 147
 148                 // dirac_slash() object is printed differently
 149                 if (is_a<print_tree>(c))
 150                         inherited::print(c, level);
 151                 else if (is_a<print_latex>(c)) {
 152                         c.s << "{";
 153                         seq[0].print(c, level);
 154                         c.s << "\\hspace{-1.0ex}/}";
 155                 } else {
 156                         seq[0].print(c, level);
 157                         c.s << "\\";
 158                 }
 159
 160         } else
 161                 inherited::print(c, level);
 162 }
 163
 164 DEFAULT_COMPARE(diracone)
 165 DEFAULT_COMPARE(diracgamma)
 166 DEFAULT_COMPARE(diracgamma5)
 167
 168 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
 169 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
 170 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
 171
 172 /** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
 173 static void base_and_index(const ex & c, ex & b, ex & i)
 174 {
 175         GINAC_ASSERT(is_a<clifford>(c));
 176         GINAC_ASSERT(c.nops() == 2);
 177
 178         if (is_a<diracgamma>(c.op(0))) { // proper dirac gamma object
 179                 i = c.op(1);
 180                 b = _ex1;
 181         } else { // slash object, generate new dummy index
 182                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(c.op(1)).get_dim());
 183                 b = indexed(c.op(0), ix.toggle_variance());
 184                 i = ix;
 185         }
 186 }
 187
 188 /** Contraction of a gamma matrix with something else. */
 189 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 190 {
 191         GINAC_ASSERT(is_a<clifford>(*self));
 192         GINAC_ASSERT(is_a<indexed>(*other));
 193         GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
 194         unsigned char rl = ex_to<clifford>(*self).get_representation_label();
 195
 196         if (is_a<clifford>(*other)) {
 197
 198                 ex dim = ex_to<idx>(self->op(1)).get_dim();
 199
 200                 // gamma~mu gamma.mu = dim ONE
 201                 if (other - self == 1) {
 202                         *self = dim;
 203                         *other = dirac_ONE(rl);
 204                         return true;
 205
 206                 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
 207                 } else if (other - self == 2
 208                         && is_a<clifford>(self[1])) {
 209                         *self = 2 - dim;
 210                         *other = _ex1;
 211                         return true;
 212
 213                 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
 214                 } else if (other - self == 3
 215                         && is_a<clifford>(self[1])
 216                         && is_a<clifford>(self[2])) {
 217                         ex b1, i1, b2, i2;
 218                         base_and_index(self[1], b1, i1);
 219                         base_and_index(self[2], b2, i2);
 220                         *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
 221                         self[1] = _ex1;
 222                         self[2] = _ex1;
 223                         *other = _ex1;
 224                         return true;
 225
 226                 // 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
 227                 } else if (other - self == 4
 228                         && is_a<clifford>(self[1])
 229                         && is_a<clifford>(self[2])
 230                         && is_a<clifford>(self[3])) {
 231                         *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3];
 232                         self[1] = _ex1;
 233                         self[2] = _ex1;
 234                         self[3] = _ex1;
 235                         *other = _ex1;
 236                         return true;
 237
 238                 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
 239                 // (commutate contracted indices towards each other, simplify_indexed()
 240                 // will re-expand and re-run the simplification)
 241                 } else {
 242                         exvector::iterator it = self + 1, next_to_last = other - 1;
 243                         while (it != other) {
 244                                 if (!is_a<clifford>(*it))
 245                                         return false;
 246                                 ++it;
 247                         }
 248
 249                         it = self + 1;
 250                         ex S = _ex1;
 251                         while (it != next_to_last) {
 252                                 S *= *it;
 253                                 *it++ = _ex1;
 254                         }
 255
 256                         *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
 257                         *next_to_last = _ex1;
 258                         *other = _ex1;
 259                         return true;
 260                 }
 261         }
 262
 263         return false;
 264 }
 265
 266 /** Perform automatic simplification on noncommutative product of clifford
 267  *  objects. This removes superfluous ONEs, permutes gamma5's to the front
 268  *  and removes squares of gamma objects. */
 269 ex clifford::simplify_ncmul(const exvector & v) const
 270 {
 271         exvector s;
 272         s.reserve(v.size());
 273
 274         // Remove superfluous ONEs
 275         exvector::const_iterator cit = v.begin(), citend = v.end();
 276         while (cit != citend) {
 277                 if (!is_a<clifford>(*cit) || !is_a<diracone>(cit->op(0)))
 278                         s.push_back(*cit);
 279                 cit++;
 280         }
 281
 282         bool something_changed = false;
 283         int sign = 1;
 284
 285         // Anticommute gamma5's to the front
 286         if (s.size() >= 2) {
 287                 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
 288                 while (true) {
 289                         exvector::iterator it = next_to_last;
 290                         while (true) {
 291                                 exvector::iterator it2 = it + 1;
 292                                 if (is_a<clifford>(*it) && is_a<clifford>(*it2) && !is_a<diracgamma5>(it->op(0)) && is_a<diracgamma5>(it2->op(0))) {
 293                                         it->swap(*it2);
 294                                         sign = -sign;
 295                                         something_changed = true;
 296                                 }
 297                                 if (it == first)
 298                                         break;
 299                                 --it;
 300                         }
 301                         if (next_to_last == first)
 302                                 break;
 303                         --next_to_last;
 304                 }
 305         }
 306
 307         // Remove squares of gamma5
 308         while (s.size() >= 2 && is_a<clifford>(s[0]) && is_a<clifford>(s[1]) && is_a<diracgamma5>(s[0].op(0)) && is_a<diracgamma5>(s[1].op(0))) {
 309                 s.erase(s.begin(), s.begin() + 2);
 310                 something_changed = true;
 311         }
 312
 313         // Remove equal adjacent gammas
 314         if (s.size() >= 2) {
 315                 exvector::iterator it, itend = s.end() - 1;
 316                 for (it = s.begin(); it != itend; ++it) {
 317                         ex & a = it[0];
 318                         ex & b = it[1];
 319                         if (!is_a<clifford>(a) || !is_a<clifford>(b))
 320                                 continue;
 321                         if (is_a<diracgamma>(a.op(0)) && is_a<diracgamma>(b.op(0))) {
 322                                 const ex & ia = a.op(1);
 323                                 const ex & ib = b.op(1);
 324                                 if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
 325                                         a = lorentz_g(ia, ib);
 326                                         b = dirac_ONE(representation_label);
 327                                         something_changed = true;
 328                                 }
 329                         } else if (!is_a<diracgamma>(a.op(0)) && !is_a<diracgamma>(b.op(0))) {
 330                                 const ex & ba = a.op(0);
 331                                 const ex & bb = b.op(0);
 332                                 if (ba.is_equal(bb)) { // a\ a\ -> a^2
 333                                         varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).get_dim());
 334                                         a = indexed(ba, ix) * indexed(bb, ix.toggle_variance());
 335                                         b = dirac_ONE(representation_label);
 336                                         something_changed = true;
 337                                 }
 338                         }
 339                 }
 340         }
 341
 342         if (s.empty())
 343                 return clifford(diracone(), representation_label) * sign;
 344         if (something_changed)
 345                 return nonsimplified_ncmul(s) * sign;
 346         else
 347                 return simplified_ncmul(s) * sign;
 348 }
 349
 350 ex clifford::thisexprseq(const exvector & v) const
 351 {
 352         return clifford(representation_label, v);
 353 }
 354
 355 ex clifford::thisexprseq(exvector * vp) const
 356 {
 357         return clifford(representation_label, vp);
 358 }
 359
 360 //////////
 361 // global functions
 362 //////////
 363
 364 ex dirac_ONE(unsigned char rl)
 365 {
 366         return clifford(diracone(), rl);
 367 }
 368
 369 ex dirac_gamma(const ex & mu, unsigned char rl)
 370 {
 371         if (!is_a<varidx>(mu))
 372                 throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
 373
 374         return clifford(diracgamma(), mu, rl);
 375 }
 376
 377 ex dirac_gamma5(unsigned char rl)
 378 {
 379         return clifford(diracgamma5(), rl);
 380 }
 381
 382 ex dirac_gamma6(unsigned char rl)
 383 {
 384         return clifford(diracone(), rl) + clifford(diracgamma5(), rl);
 385 }
 386
 387 ex dirac_gamma7(unsigned char rl)
 388 {
 389         return clifford(diracone(), rl) - clifford(diracgamma5(), rl);
 390 }
 391
 392 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
 393 {
 394         // Slashed vectors are actually stored as a clifford object with the
 395         // vector as its base expression and a (dummy) index that just serves
 396         // for storing the space dimensionality
 397         return clifford(e, varidx(0, dim), rl);
 398 }
 399
 400 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 401  *  is that of a clifford object with the specified representation label. */
 402 static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
 403 {
 404         return ti == (TINFO_clifford + rl);
 405 }
 406
 407 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 408  *  is that of a clifford object (with an arbitrary representation label). */
 409 static bool is_clifford_tinfo(unsigned ti)
 410 {
 411         return (ti & ~0xff) == TINFO_clifford;
 412 }
 413
 414 /** Take trace of a string of an even number of Dirac gammas given a vector
 415  *  of indices. */
 416 static ex trace_string(exvector::const_iterator ix, unsigned num)
 417 {
 418         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 419         if (num == 2)
 420                 return lorentz_g(ix[0], ix[1]);
 421
 422         // 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
 423         else if (num == 4)
 424                 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
 425                      + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
 426                      - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
 427
 428         // Traces of 6 or more gammas are computed recursively:
 429         // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
 430         //   + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
 431         //   - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
 432         //   + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
 433         //   - ...
 434         //   + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
 435         exvector v(num - 2);
 436         int sign = 1;
 437         ex result;
 438         for (unsigned i=1; i<num; i++) {
 439                 for (unsigned n=1, j=0; n<num; n++) {
 440                         if (n == i)
 441                                 continue;
 442                         v[j++] = ix[n];
 443                 }
 444                 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
 445                 sign = -sign;
 446         }
 447         return result;
 448 }
 449
 450 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
 451 {
 452         if (is_a<clifford>(e)) {
 453
 454                 if (ex_to<clifford>(e).get_representation_label() == rl
 455                  && is_a<diracone>(e.op(0)))
 456                         return trONE;
 457                 else
 458                         return _ex0;
 459
 460         } else if (is_ex_exactly_of_type(e, mul)) {
 461
 462                 // Trace of product: pull out non-clifford factors
 463                 ex prod = _ex1;
 464                 for (unsigned i=0; i<e.nops(); i++) {
 465                         const ex &o = e.op(i);
 466                         if (is_clifford_tinfo(o.return_type_tinfo(), rl))
 467                                 prod *= dirac_trace(o, rl, trONE);
 468                         else
 469                                 prod *= o;
 470                 }
 471                 return prod;
 472
 473         } else if (is_ex_exactly_of_type(e, ncmul)) {
 474
 475                 if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
 476                         return _ex0;
 477
 478                 // Expand product, if necessary
 479                 ex e_expanded = e.expand();
 480                 if (!is_a<ncmul>(e_expanded))
 481                         return dirac_trace(e_expanded, rl, trONE);
 482
 483                 // gamma5 gets moved to the front so this check is enough
 484                 bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
 485                 unsigned num = e.nops();
 486
 487                 if (has_gamma5) {
 488
 489                         // Trace of gamma5 * odd number of gammas and trace of
 490                         // gamma5 * gamma.mu * gamma.nu are zero
 491                         if ((num & 1) == 0 || num == 3)
 492                                 return _ex0;
 493
 494                         // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
 495                         if (num == 5) {
 496                                 ex b1, i1, b2, i2, b3, i3, b4, i4;
 497                                 base_and_index(e.op(1), b1, i1);
 498                                 base_and_index(e.op(2), b2, i2);
 499                                 base_and_index(e.op(3), b3, i3);
 500                                 base_and_index(e.op(4), b4, i4);
 501                                 return trONE * I * (eps0123(i1, i2, i3, i4) * b1 * b2 * b3 * b4).simplify_indexed();
 502                         }
 503
 504                         // Tr gamma5 S_2k =
 505                         //   I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
 506                         exvector ix(num-1), bv(num-1);
 507                         for (unsigned i=1; i<num; i++)
 508                                 base_and_index(e.op(i), bv[i-1], ix[i-1]);
 509                         num--;
 510                         int *iv = new int[num];
 511                         ex result;
 512                         for (unsigned i=0; i<num-3; i++) {
 513                                 ex idx1 = ix[i];
 514                                 for (unsigned j=i+1; j<num-2; j++) {
 515                                         ex idx2 = ix[j];
 516                                         for (unsigned k=j+1; k<num-1; k++) {
 517                                                 ex idx3 = ix[k];
 518                                                 for (unsigned l=k+1; l<num; l++) {
 519                                                         ex idx4 = ix[l];
 520                                                         iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
 521                                                         exvector v;
 522                                                         v.reserve(num - 4);
 523                                                         for (unsigned n=0, t=4; n<num; n++) {
 524                                                                 if (n == i || n == j || n == k || n == l)
 525                                                                         continue;
 526                                                                 iv[t++] = n;
 527                                                                 v.push_back(ix[n]);
 528                                                         }
 529                                                         int sign = permutation_sign(iv, iv + num);
 530                                                         result += sign * eps0123(idx1, idx2, idx3, idx4)
 531                                                                 * trace_string(v.begin(), num - 4);
 532                                                 }
 533                                         }
 534                                 }
 535                         }
 536                         delete[] iv;
 537                         return trONE * I * result * mul(bv);
 538
 539                 } else { // no gamma5
 540
 541                         // Trace of odd number of gammas is zero
 542                         if ((num & 1) == 1)
 543                                 return _ex0;
 544
 545                         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 546                         if (num == 2) {
 547                                 ex b1, i1, b2, i2;
 548                                 base_and_index(e.op(0), b1, i1);
 549                                 base_and_index(e.op(1), b2, i2);
 550                                 return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
 551                         }
 552
 553                         exvector iv(num), bv(num);
 554                         for (unsigned i=0; i<num; i++)
 555                                 base_and_index(e.op(i), bv[i], iv[i]);
 556
 557                         return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
 558                 }
 559
 560         } else if (e.nops() > 0) {
 561
 562                 // Trace maps to all other container classes (this includes sums)
 563                 pointer_to_map_function_2args<unsigned char, const ex &> fcn(dirac_trace, rl, trONE);
 564                 return e.map(fcn);
 565
 566         } else
 567                 return _ex0;
 568 }
 569
 570 ex canonicalize_clifford(const ex & e)
 571 {
 572         // Scan for any ncmul objects
 573         lst srl;
 574         ex aux = e.to_rational(srl);
 575         for (unsigned i=0; i<srl.nops(); i++) {
 576
 577                 ex lhs = srl.op(i).lhs();
 578                 ex rhs = srl.op(i).rhs();
 579
 580                 if (is_ex_exactly_of_type(rhs, ncmul)
 581                  && rhs.return_type() == return_types::noncommutative
 582                  && is_clifford_tinfo(rhs.return_type_tinfo())) {
 583
 584                         // Expand product, if necessary
 585                         ex rhs_expanded = rhs.expand();
 586                         if (!is_a<ncmul>(rhs_expanded)) {
 587                                 srl.let_op(i) = (lhs == canonicalize_clifford(rhs_expanded));
 588                                 continue;
 589
 590                         } else if (!is_a<clifford>(rhs.op(0)))
 591                                 continue;
 592
 593                         exvector v;
 594                         v.reserve(rhs.nops());
 595                         for (unsigned j=0; j<rhs.nops(); j++)
 596                                 v.push_back(rhs.op(j));
 597
 598                         // Stupid recursive bubble sort because we only want to swap adjacent gammas
 599                         exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
 600                         if (is_a<diracgamma5>(it->op(0)))
 601                                 ++it;
 602                         while (it != next_to_last) {
 603                                 if (it[0].compare(it[1]) > 0) {
 604                                         ex save0 = it[0], save1 = it[1];
 605                                         ex b1, i1, b2, i2;
 606                                         base_and_index(it[0], b1, i1);
 607                                         base_and_index(it[1], b2, i2);
 608                                         it[0] = (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
 609                                         it[1] = _ex2;
 610                                         ex sum = ncmul(v);
 611                                         it[0] = save1;
 612                                         it[1] = save0;
 613                                         sum -= ncmul(v, true);
 614                                         srl.let_op(i) = (lhs == canonicalize_clifford(sum));
 615                                         goto next_sym;
 616                                 }
 617                                 ++it;
 618                         }
 619 next_sym:       ;
 620                 }
 621         }
 622         return aux.subs(srl).simplify_indexed();
 623 }
 624
 625 } // namespace GiNaC