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 "debugmsg.h"
  36 #include "utils.h"
  37
  38 #include <stdexcept>
  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 constructor, destructor, copy constructor assignment operator and helpers
  49 //////////
  50
  51 clifford::clifford() : representation_label(0)
  52 {
  53         debugmsg("clifford default constructor", LOGLEVEL_CONSTRUCT);
  54         tinfo_key = TINFO_clifford;
  55 }
  56
  57 void clifford::copy(const clifford & other)
  58 {
  59         inherited::copy(other);
  60         representation_label = other.representation_label;
  61 }
  62
  63 DEFAULT_DESTROY(clifford)
  64 DEFAULT_CTORS(diracone)
  65 DEFAULT_CTORS(diracgamma)
  66 DEFAULT_CTORS(diracgamma5)
  67
  68 //////////
  69 // other constructors
  70 //////////
  71
  72 /** Construct object without any indices. This constructor is for internal
  73  *  use only. Use the dirac_ONE() function instead.
  74  *  @see dirac_ONE */
  75 clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
  76 {
  77         debugmsg("clifford constructor from ex", LOGLEVEL_CONSTRUCT);
  78         tinfo_key = TINFO_clifford;
  79 }
  80
  81 /** Construct object with one Lorentz index. This constructor is for internal
  82  *  use only. Use the dirac_gamma() function instead.
  83  *  @see dirac_gamma */
  84 clifford::clifford(const ex & b, const ex & mu, unsigned char rl) : inherited(b, mu), representation_label(rl)
  85 {
  86         debugmsg("clifford constructor from ex,ex", LOGLEVEL_CONSTRUCT);
  87         GINAC_ASSERT(is_a<varidx>(mu));
  88         tinfo_key = TINFO_clifford;
  89 }
  90
  91 clifford::clifford(unsigned char rl, const exvector & v, bool discardable) : inherited(sy_none(), v, discardable), representation_label(rl)
  92 {
  93         debugmsg("clifford constructor from unsigned char,exvector", LOGLEVEL_CONSTRUCT);
  94         tinfo_key = TINFO_clifford;
  95 }
  96
  97 clifford::clifford(unsigned char rl, exvector * vp) : inherited(sy_none(), vp), representation_label(rl)
  98 {
  99         debugmsg("clifford constructor from unsigned char,exvector *", LOGLEVEL_CONSTRUCT);
 100         tinfo_key = TINFO_clifford;
 101 }
 102
 103 //////////
 104 // archiving
 105 //////////
 106
 107 clifford::clifford(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 108 {
 109         debugmsg("clifford constructor from archive_node", LOGLEVEL_CONSTRUCT);
 110         unsigned rl;
 111         n.find_unsigned("label", rl);
 112         representation_label = rl;
 113 }
 114
 115 void clifford::archive(archive_node &n) const
 116 {
 117         inherited::archive(n);
 118         n.add_unsigned("label", representation_label);
 119 }
 120
 121 DEFAULT_UNARCHIVE(clifford)
 122 DEFAULT_ARCHIVING(diracone)
 123 DEFAULT_ARCHIVING(diracgamma)
 124 DEFAULT_ARCHIVING(diracgamma5)
 125
 126 //////////
 127 // functions overriding virtual functions from base classes
 128 //////////
 129
 130 int clifford::compare_same_type(const basic & other) const
 131 {
 132         GINAC_ASSERT(is_of_type(other, clifford));
 133         const clifford &o = static_cast<const clifford &>(other);
 134
 135         if (representation_label != o.representation_label) {
 136                 // different representation label
 137                 return representation_label < o.representation_label ? -1 : 1;
 138         }
 139
 140         return inherited::compare_same_type(other);
 141 }
 142
 143 bool clifford::match_same_type(const basic & other) const
 144 {
 145         GINAC_ASSERT(is_of_type(other, clifford));
 146         const clifford &o = static_cast<const clifford &>(other);
 147
 148         return representation_label == o.representation_label;
 149 }
 150
 151 void clifford::print(const print_context & c, unsigned level) const
 152 {
 153         if (!is_a<diracgamma5>(seq[0]) && !is_a<diracgamma>(seq[0]) && !is_a<diracone>(seq[0])) {
 154
 155                 // dirac_slash() object is printed differently
 156                 if (is_a<print_tree>(c))
 157                         inherited::print(c, level);
 158                 else if (is_a<print_latex>(c)) {
 159                         c.s << "{";
 160                         seq[0].print(c, level);
 161                         c.s << "\\hspace{-1.0ex}/}";
 162                 } else {
 163                         seq[0].print(c, level);
 164                         c.s << "\\";
 165                 }
 166
 167         } else
 168                 inherited::print(c, level);
 169 }
 170
 171 DEFAULT_COMPARE(diracone)
 172 DEFAULT_COMPARE(diracgamma)
 173 DEFAULT_COMPARE(diracgamma5)
 174
 175 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
 176 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
 177 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
 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 { // slash object, generate new dummy index
 189                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(c.op(1)).get_dim());
 190                 b = indexed(c.op(0), ix.toggle_variance());
 191                 i = ix;
 192         }
 193 }
 194
 195 /** Contraction of a gamma matrix with something else. */
 196 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 197 {
 198         GINAC_ASSERT(is_a<clifford>(*self));
 199         GINAC_ASSERT(is_a<indexed>(*other));
 200         GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
 201         unsigned char rl = ex_to<clifford>(*self).get_representation_label();
 202
 203         if (is_a<clifford>(*other)) {
 204
 205                 ex dim = ex_to<idx>(self->op(1)).get_dim();
 206
 207                 // gamma~mu gamma.mu = dim ONE
 208                 if (other - self == 1) {
 209                         *self = dim;
 210                         *other = dirac_ONE(rl);
 211                         return true;
 212
 213                 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
 214                 } else if (other - self == 2
 215                         && is_a<clifford>(self[1])) {
 216                         *self = 2 - dim;
 217                         *other = _ex1();
 218                         return true;
 219
 220                 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
 221                 } else if (other - self == 3
 222                         && is_a<clifford>(self[1])
 223                         && is_a<clifford>(self[2])) {
 224                         ex b1, i1, b2, i2;
 225                         base_and_index(self[1], b1, i1);
 226                         base_and_index(self[2], b2, i2);
 227                         *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
 228                         self[1] = _ex1();
 229                         self[2] = _ex1();
 230                         *other = _ex1();
 231                         return true;
 232
 233                 // 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
 234                 } else if (other - self == 4
 235                         && is_a<clifford>(self[1])
 236                         && is_a<clifford>(self[2])
 237                         && is_a<clifford>(self[3])) {
 238                         *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3];
 239                         self[1] = _ex1();
 240                         self[2] = _ex1();
 241                         self[3] = _ex1();
 242                         *other = _ex1();
 243                         return true;
 244
 245                 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
 246                 // (commutate contracted indices towards each other, simplify_indexed()
 247                 // will re-expand and re-run the simplification)
 248                 } else {
 249                         exvector::iterator it = self + 1, next_to_last = other - 1;
 250                         while (it != other) {
 251                                 if (!is_a<clifford>(*it))
 252                                         return false;
 253                                 ++it;
 254                         }
 255
 256                         it = self + 1;
 257                         ex S = _ex1();
 258                         while (it != next_to_last) {
 259                                 S *= *it;
 260                                 *it++ = _ex1();
 261                         }
 262
 263                         *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
 264                         *next_to_last = _ex1();
 265                         *other = _ex1();
 266                         return true;
 267                 }
 268         }
 269
 270         return false;
 271 }
 272
 273 /** Perform automatic simplification on noncommutative product of clifford
 274  *  objects. This removes superfluous ONEs, permutes gamma5's to the front
 275  *  and removes squares of gamma objects. */
 276 ex clifford::simplify_ncmul(const exvector & v) const
 277 {
 278         exvector s;
 279         s.reserve(v.size());
 280
 281         // Remove superfluous ONEs
 282         exvector::const_iterator cit = v.begin(), citend = v.end();
 283         while (cit != citend) {
 284                 if (!is_a<clifford>(*cit) || !is_a<diracone>(cit->op(0)))
 285                         s.push_back(*cit);
 286                 cit++;
 287         }
 288
 289         bool something_changed = false;
 290         int sign = 1;
 291
 292         // Anticommute gamma5's to the front
 293         if (s.size() >= 2) {
 294                 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
 295                 while (true) {
 296                         exvector::iterator it = next_to_last;
 297                         while (true) {
 298                                 exvector::iterator it2 = it + 1;
 299                                 if (is_a<clifford>(*it) && is_a<clifford>(*it2) && !is_a<diracgamma5>(it->op(0)) && is_a<diracgamma5>(it2->op(0))) {
 300                                         it->swap(*it2);
 301                                         sign = -sign;
 302                                         something_changed = true;
 303                                 }
 304                                 if (it == first)
 305                                         break;
 306                                 --it;
 307                         }
 308                         if (next_to_last == first)
 309                                 break;
 310                         --next_to_last;
 311                 }
 312         }
 313
 314         // Remove squares of gamma5
 315         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))) {
 316                 s.erase(s.begin(), s.begin() + 2);
 317                 something_changed = true;
 318         }
 319
 320         // Remove equal adjacent gammas
 321         if (s.size() >= 2) {
 322                 exvector::iterator it, itend = s.end() - 1;
 323                 for (it = s.begin(); it != itend; ++it) {
 324                         ex & a = it[0];
 325                         ex & b = it[1];
 326                         if (!is_a<clifford>(a) || !is_a<clifford>(b))
 327                                 continue;
 328                         if (is_a<diracgamma>(a.op(0)) && is_a<diracgamma>(b.op(0))) {
 329                                 const ex & ia = a.op(1);
 330                                 const ex & ib = b.op(1);
 331                                 if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
 332                                         a = lorentz_g(ia, ib);
 333                                         b = dirac_ONE(representation_label);
 334                                         something_changed = true;
 335                                 }
 336                         } else if (!is_a<diracgamma>(a.op(0)) && !is_a<diracgamma>(b.op(0))) {
 337                                 const ex & ba = a.op(0);
 338                                 const ex & bb = b.op(0);
 339                                 if (ba.is_equal(bb)) { // a\ a\ -> a^2
 340                                         varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).get_dim());
 341                                         a = indexed(ba, ix) * indexed(bb, ix.toggle_variance());
 342                                         b = dirac_ONE(representation_label);
 343                                         something_changed = true;
 344                                 }
 345                         }
 346                 }
 347         }
 348
 349         if (s.empty())
 350                 return clifford(diracone(), representation_label) * sign;
 351         if (something_changed)
 352                 return nonsimplified_ncmul(s) * sign;
 353         else
 354                 return simplified_ncmul(s) * sign;
 355 }
 356
 357 ex clifford::thisexprseq(const exvector & v) const
 358 {
 359         return clifford(representation_label, v);
 360 }
 361
 362 ex clifford::thisexprseq(exvector * vp) const
 363 {
 364         return clifford(representation_label, vp);
 365 }
 366
 367 //////////
 368 // global functions
 369 //////////
 370
 371 ex dirac_ONE(unsigned char rl)
 372 {
 373         return clifford(diracone(), rl);
 374 }
 375
 376 ex dirac_gamma(const ex & mu, unsigned char rl)
 377 {
 378         if (!is_a<varidx>(mu))
 379                 throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
 380
 381         return clifford(diracgamma(), mu, rl);
 382 }
 383
 384 ex dirac_gamma5(unsigned char rl)
 385 {
 386         return clifford(diracgamma5(), rl);
 387 }
 388
 389 ex dirac_gamma6(unsigned char rl)
 390 {
 391         return clifford(diracone(), rl) + clifford(diracgamma5(), rl);
 392 }
 393
 394 ex dirac_gamma7(unsigned char rl)
 395 {
 396         return clifford(diracone(), rl) - clifford(diracgamma5(), rl);
 397 }
 398
 399 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
 400 {
 401         // Slashed vectors are actually stored as a clifford object with the
 402         // vector as its base expression and a (dummy) index that just serves
 403         // for storing the space dimensionality
 404         return clifford(e, varidx(0, dim), rl);
 405 }
 406
 407 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 408  *  is that of a clifford object with the specified representation label. */
 409 static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
 410 {
 411         return ti == (TINFO_clifford + rl);
 412 }
 413
 414 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 415  *  is that of a clifford object (with an arbitrary representation label). */
 416 static bool is_clifford_tinfo(unsigned ti)
 417 {
 418         return (ti & ~0xff) == TINFO_clifford;
 419 }
 420
 421 /** Take trace of a string of an even number of Dirac gammas given a vector
 422  *  of indices. */
 423 static ex trace_string(exvector::const_iterator ix, unsigned num)
 424 {
 425         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 426         if (num == 2)
 427                 return lorentz_g(ix[0], ix[1]);
 428
 429         // 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
 430         else if (num == 4)
 431                 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
 432                      + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
 433                      - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
 434
 435         // Traces of 6 or more gammas are computed recursively:
 436         // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
 437         //   + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
 438         //   - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
 439         //   + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
 440         //   - ...
 441         //   + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
 442         exvector v(num - 2);
 443         int sign = 1;
 444         ex result;
 445         for (unsigned i=1; i<num; i++) {
 446                 for (unsigned n=1, j=0; n<num; n++) {
 447                         if (n == i)
 448                                 continue;
 449                         v[j++] = ix[n];
 450                 }
 451                 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
 452                 sign = -sign;
 453         }
 454         return result;
 455 }
 456
 457 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
 458 {
 459         if (is_a<clifford>(e)) {
 460
 461                 if (ex_to<clifford>(e).get_representation_label() == rl
 462                  && is_a<diracone>(e.op(0)))
 463                         return trONE;
 464                 else
 465                         return _ex0();
 466
 467         } else if (is_ex_exactly_of_type(e, mul)) {
 468
 469                 // Trace of product: pull out non-clifford factors
 470                 ex prod = _ex1();
 471                 for (unsigned i=0; i<e.nops(); i++) {
 472                         const ex &o = e.op(i);
 473                         if (is_clifford_tinfo(o.return_type_tinfo(), rl))
 474                                 prod *= dirac_trace(o, rl, trONE);
 475                         else
 476                                 prod *= o;
 477                 }
 478                 return prod;
 479
 480         } else if (is_ex_exactly_of_type(e, ncmul)) {
 481
 482                 if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
 483                         return _ex0();
 484
 485                 // Expand product, if necessary
 486                 ex e_expanded = e.expand();
 487                 if (!is_a<ncmul>(e_expanded))
 488                         return dirac_trace(e_expanded, rl, trONE);
 489
 490                 // gamma5 gets moved to the front so this check is enough
 491                 bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
 492                 unsigned num = e.nops();
 493
 494                 if (has_gamma5) {
 495
 496                         // Trace of gamma5 * odd number of gammas and trace of
 497                         // gamma5 * gamma.mu * gamma.nu are zero
 498                         if ((num & 1) == 0 || num == 3)
 499                                 return _ex0();
 500
 501                         // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
 502                         if (num == 5) {
 503                                 ex b1, i1, b2, i2, b3, i3, b4, i4;
 504                                 base_and_index(e.op(1), b1, i1);
 505                                 base_and_index(e.op(2), b2, i2);
 506                                 base_and_index(e.op(3), b3, i3);
 507                                 base_and_index(e.op(4), b4, i4);
 508                                 return trONE * I * (eps0123(i1, i2, i3, i4) * b1 * b2 * b3 * b4).simplify_indexed();
 509                         }
 510
 511                         // Tr gamma5 S_2k =
 512                         //   I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
 513                         exvector ix(num-1), bv(num-1);
 514                         for (unsigned i=1; i<num; i++)
 515                                 base_and_index(e.op(i), bv[i-1], ix[i-1]);
 516                         num--;
 517                         int *iv = new int[num];
 518                         ex result;
 519                         for (unsigned i=0; i<num-3; i++) {
 520                                 ex idx1 = ix[i];
 521                                 for (unsigned j=i+1; j<num-2; j++) {
 522                                         ex idx2 = ix[j];
 523                                         for (unsigned k=j+1; k<num-1; k++) {
 524                                                 ex idx3 = ix[k];
 525                                                 for (unsigned l=k+1; l<num; l++) {
 526                                                         ex idx4 = ix[l];
 527                                                         iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
 528                                                         exvector v;
 529                                                         v.reserve(num - 4);
 530                                                         for (unsigned n=0, t=4; n<num; n++) {
 531                                                                 if (n == i || n == j || n == k || n == l)
 532                                                                         continue;
 533                                                                 iv[t++] = n;
 534                                                                 v.push_back(ix[n]);
 535                                                         }
 536                                                         int sign = permutation_sign(iv, iv + num);
 537                                                         result += sign * eps0123(idx1, idx2, idx3, idx4)
 538                                                                 * trace_string(v.begin(), num - 4);
 539                                                 }
 540                                         }
 541                                 }
 542                         }
 543                         delete[] iv;
 544                         return trONE * I * result * mul(bv);
 545
 546                 } else { // no gamma5
 547
 548                         // Trace of odd number of gammas is zero
 549                         if ((num & 1) == 1)
 550                                 return _ex0();
 551
 552                         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 553                         if (num == 2) {
 554                                 ex b1, i1, b2, i2;
 555                                 base_and_index(e.op(0), b1, i1);
 556                                 base_and_index(e.op(1), b2, i2);
 557                                 return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
 558                         }
 559
 560                         exvector iv(num), bv(num);
 561                         for (unsigned i=0; i<num; i++)
 562                                 base_and_index(e.op(i), bv[i], iv[i]);
 563
 564                         return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
 565                 }
 566
 567         } else if (e.nops() > 0) {
 568
 569                 // Trace maps to all other container classes (this includes sums)
 570                 pointer_to_map_function_2args<unsigned char, const ex &> fcn(dirac_trace, rl, trONE);
 571                 return e.map(fcn);
 572
 573         } else
 574                 return _ex0();
 575 }
 576
 577 ex canonicalize_clifford(const ex & e)
 578 {
 579         // Scan for any ncmul objects
 580         lst srl;
 581         ex aux = e.to_rational(srl);
 582         for (unsigned i=0; i<srl.nops(); i++) {
 583
 584                 ex lhs = srl.op(i).lhs();
 585                 ex rhs = srl.op(i).rhs();
 586
 587                 if (is_ex_exactly_of_type(rhs, ncmul)
 588                  && rhs.return_type() == return_types::noncommutative
 589                  && is_clifford_tinfo(rhs.return_type_tinfo())) {
 590
 591                         // Expand product, if necessary
 592                         ex rhs_expanded = rhs.expand();
 593                         if (!is_a<ncmul>(rhs_expanded)) {
 594                                 srl.let_op(i) = (lhs == canonicalize_clifford(rhs_expanded));
 595                                 continue;
 596
 597                         } else if (!is_a<clifford>(rhs.op(0)))
 598                                 continue;
 599
 600                         exvector v;
 601                         v.reserve(rhs.nops());
 602                         for (unsigned j=0; j<rhs.nops(); j++)
 603                                 v.push_back(rhs.op(j));
 604
 605                         // Stupid recursive bubble sort because we only want to swap adjacent gammas
 606                         exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
 607                         if (is_a<diracgamma5>(it->op(0)))
 608                                 ++it;
 609                         while (it != next_to_last) {
 610                                 if (it[0].compare(it[1]) > 0) {
 611                                         ex save0 = it[0], save1 = it[1];
 612                                         ex b1, i1, b2, i2;
 613                                         base_and_index(it[0], b1, i1);
 614                                         base_and_index(it[1], b2, i2);
 615                                         it[0] = (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
 616                                         it[1] = _ex2();
 617                                         ex sum = ncmul(v);
 618                                         it[0] = save1;
 619                                         it[1] = save0;
 620                                         sum -= ncmul(v, true);
 621                                         srl.let_op(i) = (lhs == canonicalize_clifford(sum));
 622                                         goto next_sym;
 623                                 }
 624                                 ++it;
 625                         }
 626 next_sym:       ;
 627                 }
 628         }
 629         return aux.subs(srl).simplify_indexed();
 630 }
 631
 632 } // namespace GiNaC