ginac/clifford.cpp

   1 /** @file clifford.cpp
   2  *
   3  *  Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2002 Johannes Gutenberg University Mainz, Germany
   7  *
   8  *  This program is free software; you can redistribute it and/or modify
   9  *  it under the terms of the GNU General Public License as published by
  10  *  the Free Software Foundation; either version 2 of the License, or
  11  *  (at your option) any later version.
  12  *
  13  *  This program is distributed in the hope that it will be useful,
  14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  16  *  GNU General Public License for more details.
  17  *
  18  *  You should have received a copy of the GNU General Public License
  19  *  along with this program; if not, write to the Free Software
  20  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  21  */
  22
  23 #include <iostream>
  24 #include <stdexcept>
  25
  26 #include "clifford.h"
  27 #include "ex.h"
  28 #include "idx.h"
  29 #include "ncmul.h"
  30 #include "symbol.h"
  31 #include "numeric.h" // for I
  32 #include "symmetry.h"
  33 #include "lst.h"
  34 #include "relational.h"
  35 #include "mul.h"
  36 #include "print.h"
  37 #include "archive.h"
  38 #include "utils.h"
  39
  40 namespace GiNaC {
  41
  42 GINAC_IMPLEMENT_REGISTERED_CLASS(clifford, indexed)
  43 GINAC_IMPLEMENT_REGISTERED_CLASS(diracone, tensor)
  44 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma, tensor)
  45 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma5, tensor)
  46
  47 //////////
  48 // default ctor, dtor, copy ctor, assignment operator and helpers
  49 //////////
  50
  51 clifford::clifford() : representation_label(0)
  52 {
  53         tinfo_key = TINFO_clifford;
  54 }
  55
  56 void clifford::copy(const clifford & other)
  57 {
  58         inherited::copy(other);
  59         representation_label = other.representation_label;
  60 }
  61
  62 DEFAULT_DESTROY(clifford)
  63 DEFAULT_CTORS(diracone)
  64 DEFAULT_CTORS(diracgamma)
  65 DEFAULT_CTORS(diracgamma5)
  66
  67 //////////
  68 // other constructors
  69 //////////
  70
  71 /** Construct object without any indices. This constructor is for internal
  72  *  use only. Use the dirac_ONE() function instead.
  73  *  @see dirac_ONE */
  74 clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
  75 {
  76         tinfo_key = TINFO_clifford;
  77 }
  78
  79 /** Construct object with one Lorentz index. This constructor is for internal
  80  *  use only. Use the dirac_gamma() function instead.
  81  *  @see dirac_gamma */
  82 clifford::clifford(const ex & b, const ex & mu, unsigned char rl) : inherited(b, mu), representation_label(rl)
  83 {
  84         GINAC_ASSERT(is_a<varidx>(mu));
  85         tinfo_key = TINFO_clifford;
  86 }
  87
  88 clifford::clifford(unsigned char rl, const exvector & v, bool discardable) : inherited(sy_none(), v, discardable), representation_label(rl)
  89 {
  90         tinfo_key = TINFO_clifford;
  91 }
  92
  93 clifford::clifford(unsigned char rl, exvector * vp) : inherited(sy_none(), vp), representation_label(rl)
  94 {
  95         tinfo_key = TINFO_clifford;
  96 }
  97
  98 //////////
  99 // archiving
 100 //////////
 101
 102 clifford::clifford(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 103 {
 104         unsigned rl;
 105         n.find_unsigned("label", rl);
 106         representation_label = rl;
 107 }
 108
 109 void clifford::archive(archive_node &n) const
 110 {
 111         inherited::archive(n);
 112         n.add_unsigned("label", representation_label);
 113 }
 114
 115 DEFAULT_UNARCHIVE(clifford)
 116 DEFAULT_ARCHIVING(diracone)
 117 DEFAULT_ARCHIVING(diracgamma)
 118 DEFAULT_ARCHIVING(diracgamma5)
 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<diracgamma>(seq[0]) && !is_a<diracone>(seq[0])) {
 148
 149                 // dirac_slash() object is printed differently
 150                 if (is_a<print_tree>(c))
 151                         inherited::print(c, level);
 152                 else if (is_a<print_latex>(c)) {
 153                         c.s << "{";
 154                         seq[0].print(c, level);
 155                         c.s << "\\hspace{-1.0ex}/}";
 156                 } else {
 157                         seq[0].print(c, level);
 158                         c.s << "\\";
 159                 }
 160
 161         } else
 162                 inherited::print(c, level);
 163 }
 164
 165 DEFAULT_COMPARE(diracone)
 166 DEFAULT_COMPARE(diracgamma)
 167 DEFAULT_COMPARE(diracgamma5)
 168
 169 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
 170 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
 171 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
 172
 173 /** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
 174 static void base_and_index(const ex & c, ex & b, ex & i)
 175 {
 176         GINAC_ASSERT(is_a<clifford>(c));
 177         GINAC_ASSERT(c.nops() == 2);
 178
 179         if (is_a<diracgamma>(c.op(0))) { // proper dirac gamma object
 180                 i = c.op(1);
 181                 b = _ex1;
 182         } else { // slash object, generate new dummy index
 183                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(c.op(1)).get_dim());
 184                 b = indexed(c.op(0), ix.toggle_variance());
 185                 i = ix;
 186         }
 187 }
 188
 189 /** Contraction of a gamma matrix with something else. */
 190 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 191 {
 192         GINAC_ASSERT(is_a<clifford>(*self));
 193         GINAC_ASSERT(is_a<indexed>(*other));
 194         GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
 195         unsigned char rl = ex_to<clifford>(*self).get_representation_label();
 196
 197         if (is_a<clifford>(*other)) {
 198
 199                 ex dim = ex_to<idx>(self->op(1)).get_dim();
 200
 201                 // gamma~mu gamma.mu = dim ONE
 202                 if (other - self == 1) {
 203                         *self = dim;
 204                         *other = dirac_ONE(rl);
 205                         return true;
 206
 207                 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
 208                 } else if (other - self == 2
 209                         && is_a<clifford>(self[1])) {
 210                         *self = 2 - dim;
 211                         *other = _ex1;
 212                         return true;
 213
 214                 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
 215                 } else if (other - self == 3
 216                         && is_a<clifford>(self[1])
 217                         && is_a<clifford>(self[2])) {
 218                         ex b1, i1, b2, i2;
 219                         base_and_index(self[1], b1, i1);
 220                         base_and_index(self[2], b2, i2);
 221                         *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
 222                         self[1] = _ex1;
 223                         self[2] = _ex1;
 224                         *other = _ex1;
 225                         return true;
 226
 227                 // 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
 228                 } else if (other - self == 4
 229                         && is_a<clifford>(self[1])
 230                         && is_a<clifford>(self[2])
 231                         && is_a<clifford>(self[3])) {
 232                         *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3];
 233                         self[1] = _ex1;
 234                         self[2] = _ex1;
 235                         self[3] = _ex1;
 236                         *other = _ex1;
 237                         return true;
 238
 239                 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
 240                 // (commutate contracted indices towards each other, simplify_indexed()
 241                 // will re-expand and re-run the simplification)
 242                 } else {
 243                         exvector::iterator it = self + 1, next_to_last = other - 1;
 244                         while (it != other) {
 245                                 if (!is_a<clifford>(*it))
 246                                         return false;
 247                                 ++it;
 248                         }
 249
 250                         it = self + 1;
 251                         ex S = _ex1;
 252                         while (it != next_to_last) {
 253                                 S *= *it;
 254                                 *it++ = _ex1;
 255                         }
 256
 257                         *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
 258                         *next_to_last = _ex1;
 259                         *other = _ex1;
 260                         return true;
 261                 }
 262         }
 263
 264         return false;
 265 }
 266
 267 /** Perform automatic simplification on noncommutative product of clifford
 268  *  objects. This removes superfluous ONEs, permutes gamma5's to the front
 269  *  and removes squares of gamma objects. */
 270 ex clifford::simplify_ncmul(const exvector & v) const
 271 {
 272         exvector s;
 273         s.reserve(v.size());
 274
 275         // Remove superfluous ONEs
 276         exvector::const_iterator cit = v.begin(), citend = v.end();
 277         while (cit != citend) {
 278                 if (!is_a<clifford>(*cit) || !is_a<diracone>(cit->op(0)))
 279                         s.push_back(*cit);
 280                 cit++;
 281         }
 282
 283         bool something_changed = false;
 284         int sign = 1;
 285
 286         // Anticommute gamma5's to the front
 287         if (s.size() >= 2) {
 288                 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
 289                 while (true) {
 290                         exvector::iterator it = next_to_last;
 291                         while (true) {
 292                                 exvector::iterator it2 = it + 1;
 293                                 if (is_a<clifford>(*it) && is_a<clifford>(*it2) && !is_a<diracgamma5>(it->op(0)) && is_a<diracgamma5>(it2->op(0))) {
 294                                         it->swap(*it2);
 295                                         sign = -sign;
 296                                         something_changed = true;
 297                                 }
 298                                 if (it == first)
 299                                         break;
 300                                 --it;
 301                         }
 302                         if (next_to_last == first)
 303                                 break;
 304                         --next_to_last;
 305                 }
 306         }
 307
 308         // Remove equal adjacent gammas
 309         if (s.size() >= 2) {
 310                 exvector::iterator it, itend = s.end() - 1;
 311                 for (it = s.begin(); it != itend; ++it) {
 312                         ex & a = it[0];
 313                         ex & b = it[1];
 314                         if (!is_a<clifford>(a) || !is_a<clifford>(b))
 315                                 continue;
 316                         bool a_is_diracgamma = is_a<diracgamma>(a.op(0));
 317                         bool b_is_diracgamma = is_a<diracgamma>(b.op(0));
 318                         if (a_is_diracgamma && b_is_diracgamma) {
 319                                 const ex & ia = a.op(1);
 320                                 const ex & ib = b.op(1);
 321                                 if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
 322                                         a = lorentz_g(ia, ib);
 323                                         b = dirac_ONE(representation_label);
 324                                         something_changed = true;
 325                                 }
 326                         } else if (is_a<diracgamma5>(a.op(0)) && is_a<diracgamma5>(b.op(0))) {
 327                                 // Remove squares of gamma5
 328                                 a = dirac_ONE(representation_label);
 329                                 b = dirac_ONE(representation_label);
 330                                 something_changed = true;
 331                         } else if (!a_is_diracgamma && !b_is_diracgamma) {
 332                                 const ex & ba = a.op(0);
 333                                 const ex & bb = b.op(0);
 334                                 if (ba.is_equal(bb)) { // a\ a\ -> a^2
 335                                         varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).get_dim());
 336                                         a = indexed(ba, ix) * indexed(bb, ix.toggle_variance());
 337                                         b = dirac_ONE(representation_label);
 338                                         something_changed = true;
 339                                 }
 340                         }
 341                 }
 342         }
 343
 344         if (s.empty())
 345                 return clifford(diracone(), representation_label) * sign;
 346         if (something_changed)
 347                 return nonsimplified_ncmul(s) * sign;
 348         else
 349                 return simplified_ncmul(s) * sign;
 350 }
 351
 352 ex clifford::thisexprseq(const exvector & v) const
 353 {
 354         return clifford(representation_label, v);
 355 }
 356
 357 ex clifford::thisexprseq(exvector * vp) const
 358 {
 359         return clifford(representation_label, vp);
 360 }
 361
 362 //////////
 363 // global functions
 364 //////////
 365
 366 ex dirac_ONE(unsigned char rl)
 367 {
 368         return clifford(diracone(), rl);
 369 }
 370
 371 ex dirac_gamma(const ex & mu, unsigned char rl)
 372 {
 373         if (!is_a<varidx>(mu))
 374                 throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
 375
 376         return clifford(diracgamma(), mu, rl);
 377 }
 378
 379 ex dirac_gamma5(unsigned char rl)
 380 {
 381         return clifford(diracgamma5(), rl);
 382 }
 383
 384 ex dirac_gamma6(unsigned char rl)
 385 {
 386         return clifford(diracone(), rl) + clifford(diracgamma5(), rl);
 387 }
 388
 389 ex dirac_gamma7(unsigned char rl)
 390 {
 391         return clifford(diracone(), rl) - clifford(diracgamma5(), rl);
 392 }
 393
 394 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
 395 {
 396         // Slashed vectors are actually stored as a clifford object with the
 397         // vector as its base expression and a (dummy) index that just serves
 398         // for storing the space dimensionality
 399         return clifford(e, varidx(0, dim), rl);
 400 }
 401
 402 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 403  *  is that of a clifford object with the specified representation label. */
 404 static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
 405 {
 406         return ti == (TINFO_clifford + rl);
 407 }
 408
 409 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 410  *  is that of a clifford object (with an arbitrary representation label). */
 411 static bool is_clifford_tinfo(unsigned ti)
 412 {
 413         return (ti & ~0xff) == TINFO_clifford;
 414 }
 415
 416 /** Take trace of a string of an even number of Dirac gammas given a vector
 417  *  of indices. */
 418 static ex trace_string(exvector::const_iterator ix, unsigned num)
 419 {
 420         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 421         if (num == 2)
 422                 return lorentz_g(ix[0], ix[1]);
 423
 424         // 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
 425         else if (num == 4)
 426                 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
 427                      + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
 428                      - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
 429
 430         // Traces of 6 or more gammas are computed recursively:
 431         // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
 432         //   + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
 433         //   - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
 434         //   + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
 435         //   - ...
 436         //   + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
 437         exvector v(num - 2);
 438         int sign = 1;
 439         ex result;
 440         for (unsigned i=1; i<num; i++) {
 441                 for (unsigned n=1, j=0; n<num; n++) {
 442                         if (n == i)
 443                                 continue;
 444                         v[j++] = ix[n];
 445                 }
 446                 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
 447                 sign = -sign;
 448         }
 449         return result;
 450 }
 451
 452 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
 453 {
 454         if (is_a<clifford>(e)) {
 455
 456                 if (ex_to<clifford>(e).get_representation_label() == rl
 457                  && is_a<diracone>(e.op(0)))
 458                         return trONE;
 459                 else
 460                         return _ex0;
 461
 462         } else if (is_ex_exactly_of_type(e, mul)) {
 463
 464                 // Trace of product: pull out non-clifford factors
 465                 ex prod = _ex1;
 466                 for (unsigned i=0; i<e.nops(); i++) {
 467                         const ex &o = e.op(i);
 468                         if (is_clifford_tinfo(o.return_type_tinfo(), rl))
 469                                 prod *= dirac_trace(o, rl, trONE);
 470                         else
 471                                 prod *= o;
 472                 }
 473                 return prod;
 474
 475         } else if (is_ex_exactly_of_type(e, ncmul)) {
 476
 477                 if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
 478                         return _ex0;
 479
 480                 // Expand product, if necessary
 481                 ex e_expanded = e.expand();
 482                 if (!is_a<ncmul>(e_expanded))
 483                         return dirac_trace(e_expanded, rl, trONE);
 484
 485                 // gamma5 gets moved to the front so this check is enough
 486                 bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
 487                 unsigned num = e.nops();
 488
 489                 if (has_gamma5) {
 490
 491                         // Trace of gamma5 * odd number of gammas and trace of
 492                         // gamma5 * gamma.mu * gamma.nu are zero
 493                         if ((num & 1) == 0 || num == 3)
 494                                 return _ex0;
 495
 496                         // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
 497                         if (num == 5) {
 498                                 ex b1, i1, b2, i2, b3, i3, b4, i4;
 499                                 base_and_index(e.op(1), b1, i1);
 500                                 base_and_index(e.op(2), b2, i2);
 501                                 base_and_index(e.op(3), b3, i3);
 502                                 base_and_index(e.op(4), b4, i4);
 503                                 return trONE * I * (eps0123(i1, i2, i3, i4) * b1 * b2 * b3 * b4).simplify_indexed();
 504                         }
 505
 506                         // Tr gamma5 S_2k =
 507                         //   I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
 508                         exvector ix(num-1), bv(num-1);
 509                         for (unsigned i=1; i<num; i++)
 510                                 base_and_index(e.op(i), bv[i-1], ix[i-1]);
 511                         num--;
 512                         int *iv = new int[num];
 513                         ex result;
 514                         for (unsigned i=0; i<num-3; i++) {
 515                                 ex idx1 = ix[i];
 516                                 for (unsigned j=i+1; j<num-2; j++) {
 517                                         ex idx2 = ix[j];
 518                                         for (unsigned k=j+1; k<num-1; k++) {
 519                                                 ex idx3 = ix[k];
 520                                                 for (unsigned l=k+1; l<num; l++) {
 521                                                         ex idx4 = ix[l];
 522                                                         iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
 523                                                         exvector v;
 524                                                         v.reserve(num - 4);
 525                                                         for (unsigned n=0, t=4; n<num; n++) {
 526                                                                 if (n == i || n == j || n == k || n == l)
 527                                                                         continue;
 528                                                                 iv[t++] = n;
 529                                                                 v.push_back(ix[n]);
 530                                                         }
 531                                                         int sign = permutation_sign(iv, iv + num);
 532                                                         result += sign * eps0123(idx1, idx2, idx3, idx4)
 533                                                                 * trace_string(v.begin(), num - 4);
 534                                                 }
 535                                         }
 536                                 }
 537                         }
 538                         delete[] iv;
 539                         return trONE * I * result * mul(bv);
 540
 541                 } else { // no gamma5
 542
 543                         // Trace of odd number of gammas is zero
 544                         if ((num & 1) == 1)
 545                                 return _ex0;
 546
 547                         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 548                         if (num == 2) {
 549                                 ex b1, i1, b2, i2;
 550                                 base_and_index(e.op(0), b1, i1);
 551                                 base_and_index(e.op(1), b2, i2);
 552                                 return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
 553                         }
 554
 555                         exvector iv(num), bv(num);
 556                         for (unsigned i=0; i<num; i++)
 557                                 base_and_index(e.op(i), bv[i], iv[i]);
 558
 559                         return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
 560                 }
 561
 562         } else if (e.nops() > 0) {
 563
 564                 // Trace maps to all other container classes (this includes sums)
 565                 pointer_to_map_function_2args<unsigned char, const ex &> fcn(dirac_trace, rl, trONE);
 566                 return e.map(fcn);
 567
 568         } else
 569                 return _ex0;
 570 }
 571
 572 ex canonicalize_clifford(const ex & e)
 573 {
 574         // Scan for any ncmul objects
 575         lst srl;
 576         ex aux = e.to_rational(srl);
 577         for (unsigned i=0; i<srl.nops(); i++) {
 578
 579                 ex lhs = srl.op(i).lhs();
 580                 ex rhs = srl.op(i).rhs();
 581
 582                 if (is_ex_exactly_of_type(rhs, ncmul)
 583                  && rhs.return_type() == return_types::noncommutative
 584                  && is_clifford_tinfo(rhs.return_type_tinfo())) {
 585
 586                         // Expand product, if necessary
 587                         ex rhs_expanded = rhs.expand();
 588                         if (!is_a<ncmul>(rhs_expanded)) {
 589                                 srl.let_op(i) = (lhs == canonicalize_clifford(rhs_expanded));
 590                                 continue;
 591
 592                         } else if (!is_a<clifford>(rhs.op(0)))
 593                                 continue;
 594
 595                         exvector v;
 596                         v.reserve(rhs.nops());
 597                         for (unsigned j=0; j<rhs.nops(); j++)
 598                                 v.push_back(rhs.op(j));
 599
 600                         // Stupid recursive bubble sort because we only want to swap adjacent gammas
 601                         exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
 602                         if (is_a<diracgamma5>(it->op(0)))
 603                                 ++it;
 604                         while (it != next_to_last) {
 605                                 if (it[0].compare(it[1]) > 0) {
 606                                         ex save0 = it[0], save1 = it[1];
 607                                         ex b1, i1, b2, i2;
 608                                         base_and_index(it[0], b1, i1);
 609                                         base_and_index(it[1], b2, i2);
 610                                         it[0] = (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
 611                                         it[1] = _ex2;
 612                                         ex sum = ncmul(v);
 613                                         it[0] = save1;
 614                                         it[1] = save0;
 615                                         sum -= ncmul(v, true);
 616                                         srl.let_op(i) = (lhs == canonicalize_clifford(sum));
 617                                         goto next_sym;
 618                                 }
 619                                 ++it;
 620                         }
 621 next_sym:       ;
 622                 }
 623         }
 624         return aux.subs(srl).simplify_indexed();
 625 }
 626
 627 } // namespace GiNaC