ginac/clifford.cpp

   1 /** @file clifford.cpp
   2  *
   3  *  Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2004 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
  25 #include "ex.h"
  26 #include "idx.h"
  27 #include "ncmul.h"
  28 #include "symbol.h"
  29 #include "numeric.h" // for I
  30 #include "symmetry.h"
  31 #include "lst.h"
  32 #include "relational.h"
  33 #include "operators.h"
  34 #include "add.h"
  35 #include "mul.h"
  36 #include "power.h"
  37 #include "matrix.h"
  38 #include "archive.h"
  39 #include "utils.h"
  40
  41 namespace GiNaC {
  42
  43 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(clifford, indexed,
  44   print_func<print_dflt>(&clifford::do_print_dflt).
  45   print_func<print_latex>(&clifford::do_print_latex))
  46
  47 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracone, tensor,
  48   print_func<print_dflt>(&diracone::do_print).
  49   print_func<print_latex>(&diracone::do_print_latex))
  50
  51 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(cliffordunit, tensor,
  52   print_func<print_dflt>(&cliffordunit::do_print).
  53   print_func<print_latex>(&cliffordunit::do_print_latex))
  54
  55 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma, cliffordunit,
  56   print_func<print_dflt>(&diracgamma::do_print).
  57   print_func<print_latex>(&diracgamma::do_print_latex))
  58
  59 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma5, tensor,
  60   print_func<print_dflt>(&diracgamma5::do_print).
  61   print_func<print_latex>(&diracgamma5::do_print_latex))
  62
  63 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaL, tensor,
  64   print_func<print_context>(&diracgammaL::do_print).
  65   print_func<print_latex>(&diracgammaL::do_print_latex))
  66
  67 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaR, tensor,
  68   print_func<print_context>(&diracgammaR::do_print).
  69   print_func<print_latex>(&diracgammaR::do_print_latex))
  70
  71 //////////
  72 // default constructors
  73 //////////
  74
  75 static ex default_metric()
  76 {
  77         static ex m = (new minkmetric)->setflag(status_flags::dynallocated);
  78         return m;
  79 }
  80
  81 clifford::clifford() : representation_label(0), metric(default_metric())
  82 {
  83         tinfo_key = TINFO_clifford;
  84 }
  85
  86 DEFAULT_CTOR(diracone)
  87 DEFAULT_CTOR(cliffordunit)
  88 DEFAULT_CTOR(diracgamma)
  89 DEFAULT_CTOR(diracgamma5)
  90 DEFAULT_CTOR(diracgammaL)
  91 DEFAULT_CTOR(diracgammaR)
  92
  93 //////////
  94 // other constructors
  95 //////////
  96
  97 /** Construct object without any indices. This constructor is for internal
  98  *  use only. Use the dirac_ONE() function instead.
  99  *  @see dirac_ONE */
 100 clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl), metric(0)
 101 {
 102         tinfo_key = TINFO_clifford;
 103 }
 104
 105 /** Construct object with one Lorentz index. This constructor is for internal
 106  *  use only. Use the clifford_unit() or dirac_gamma() functions instead.
 107  *  @see clifford_unit
 108  *  @see dirac_gamma */
 109 clifford::clifford(const ex & b, const ex & mu, const ex & metr, unsigned char rl) : inherited(b, mu), representation_label(rl), metric(metr)
 110 {
 111         GINAC_ASSERT(is_a<varidx>(mu));
 112         tinfo_key = TINFO_clifford;
 113 }
 114
 115 clifford::clifford(unsigned char rl, const ex & metr, const exvector & v, bool discardable) : inherited(not_symmetric(), v, discardable), representation_label(rl), metric(metr)
 116 {
 117         tinfo_key = TINFO_clifford;
 118 }
 119
 120 clifford::clifford(unsigned char rl, const ex & metr, std::auto_ptr<exvector> vp) : inherited(not_symmetric(), vp), representation_label(rl), metric(metr)
 121 {
 122         tinfo_key = TINFO_clifford;
 123 }
 124
 125 //////////
 126 // archiving
 127 //////////
 128
 129 clifford::clifford(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
 130 {
 131         unsigned rl;
 132         n.find_unsigned("label", rl);
 133         representation_label = rl;
 134         n.find_ex("metric", metric, sym_lst);
 135 }
 136
 137 void clifford::archive(archive_node &n) const
 138 {
 139         inherited::archive(n);
 140         n.add_unsigned("label", representation_label);
 141         n.add_ex("metric", metric);
 142 }
 143
 144 DEFAULT_UNARCHIVE(clifford)
 145 DEFAULT_ARCHIVING(diracone)
 146 DEFAULT_ARCHIVING(cliffordunit)
 147 DEFAULT_ARCHIVING(diracgamma)
 148 DEFAULT_ARCHIVING(diracgamma5)
 149 DEFAULT_ARCHIVING(diracgammaL)
 150 DEFAULT_ARCHIVING(diracgammaR)
 151
 152 //////////
 153 // functions overriding virtual functions from base classes
 154 //////////
 155
 156 ex clifford::get_metric(const ex & i, const ex & j) const
 157 {
 158         return indexed(metric, symmetric2(), i, j);
 159 }
 160
 161 bool clifford::same_metric(const ex & other) const
 162 {
 163         if (is_a<clifford>(other)) {
 164                 return get_metric().is_equal(ex_to<clifford>(other).get_metric());
 165         } else if (is_a<indexed>(other)) {
 166                 return get_metric(other.op(1), other.op(2)).is_equal(other);
 167         } else
 168                 return false;
 169 }
 170
 171 int clifford::compare_same_type(const basic & other) const
 172 {
 173         GINAC_ASSERT(is_a<clifford>(other));
 174         const clifford &o = static_cast<const clifford &>(other);
 175
 176         if (representation_label != o.representation_label) {
 177                 // different representation label
 178                 return representation_label < o.representation_label ? -1 : 1;
 179         }
 180
 181         return inherited::compare_same_type(other);
 182 }
 183
 184 bool clifford::match_same_type(const basic & other) const
 185 {
 186         GINAC_ASSERT(is_a<clifford>(other));
 187         const clifford &o = static_cast<const clifford &>(other);
 188
 189         return (representation_label == o.representation_label) && same_metric(o);
 190 }
 191
 192 static bool is_dirac_slash(const ex & seq0)
 193 {
 194         return !is_a<diracgamma5>(seq0) && !is_a<diracgammaL>(seq0) &&
 195                !is_a<diracgammaR>(seq0) && !is_a<cliffordunit>(seq0) &&
 196                !is_a<diracone>(seq0);
 197 }
 198
 199 void clifford::do_print_dflt(const print_dflt & c, unsigned level) const
 200 {
 201         // dirac_slash() object is printed differently
 202         if (is_dirac_slash(seq[0])) {
 203                 seq[0].print(c, level);
 204                 c.s << "\\";
 205         } else
 206                 this->print_dispatch<inherited>(c, level);
 207 }
 208
 209 void clifford::do_print_latex(const print_latex & c, unsigned level) const
 210 {
 211         // dirac_slash() object is printed differently
 212         if (is_dirac_slash(seq[0])) {
 213                 c.s << "{";
 214                 seq[0].print(c, level);
 215                 c.s << "\\hspace{-1.0ex}/}";
 216         } else
 217                 this->print_dispatch<inherited>(c, level);
 218 }
 219
 220 DEFAULT_COMPARE(diracone)
 221 DEFAULT_COMPARE(cliffordunit)
 222 DEFAULT_COMPARE(diracgamma)
 223 DEFAULT_COMPARE(diracgamma5)
 224 DEFAULT_COMPARE(diracgammaL)
 225 DEFAULT_COMPARE(diracgammaR)
 226
 227 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
 228 DEFAULT_PRINT_LATEX(cliffordunit, "e", "e")
 229 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
 230 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
 231 DEFAULT_PRINT_LATEX(diracgammaL, "gammaL", "{\\gamma_L}")
 232 DEFAULT_PRINT_LATEX(diracgammaR, "gammaR", "{\\gamma_R}")
 233
 234 /** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
 235 static void base_and_index(const ex & c, ex & b, ex & i)
 236 {
 237         GINAC_ASSERT(is_a<clifford>(c));
 238         GINAC_ASSERT(c.nops() == 2);
 239
 240         if (is_a<cliffordunit>(c.op(0))) { // proper dirac gamma object or clifford unit
 241                 i = c.op(1);
 242                 b = _ex1;
 243         } else if (is_a<diracgamma5>(c.op(0)) || is_a<diracgammaL>(c.op(0)) || is_a<diracgammaR>(c.op(0))) { // gamma5/L/R
 244                 i = _ex0;
 245                 b = _ex1;
 246         } else { // slash object, generate new dummy index
 247                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(c.op(1)).get_dim());
 248                 b = indexed(c.op(0), ix.toggle_variance());
 249                 i = ix;
 250         }
 251 }
 252
 253 /** Contraction of a gamma matrix with something else. */
 254 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 255 {
 256         GINAC_ASSERT(is_a<clifford>(*self));
 257         GINAC_ASSERT(is_a<indexed>(*other));
 258         GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
 259         unsigned char rl = ex_to<clifford>(*self).get_representation_label();
 260
 261         ex dim = ex_to<idx>(self->op(1)).get_dim();
 262         if (other->nops() > 1)
 263                 dim = minimal_dim(dim, ex_to<idx>(other->op(1)).get_dim());
 264
 265         if (is_a<clifford>(*other)) {
 266
 267                 // Contraction only makes sense if the represenation labels are equal
 268                 if (ex_to<clifford>(*other).get_representation_label() != rl)
 269                         return false;
 270
 271                 // gamma~mu gamma.mu = dim ONE
 272                 if (other - self == 1) {
 273                         *self = dim;
 274                         *other = dirac_ONE(rl);
 275                         return true;
 276
 277                 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
 278                 } else if (other - self == 2
 279                         && is_a<clifford>(self[1])) {
 280                         *self = 2 - dim;
 281                         *other = _ex1;
 282                         return true;
 283
 284                 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
 285                 } else if (other - self == 3
 286                         && is_a<clifford>(self[1])
 287                         && is_a<clifford>(self[2])) {
 288                         ex b1, i1, b2, i2;
 289                         base_and_index(self[1], b1, i1);
 290                         base_and_index(self[2], b2, i2);
 291                         *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
 292                         self[1] = _ex1;
 293                         self[2] = _ex1;
 294                         *other = _ex1;
 295                         return true;
 296
 297                 // 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
 298                 } else if (other - self == 4
 299                         && is_a<clifford>(self[1])
 300                         && is_a<clifford>(self[2])
 301                         && is_a<clifford>(self[3])) {
 302                         *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3];
 303                         self[1] = _ex1;
 304                         self[2] = _ex1;
 305                         self[3] = _ex1;
 306                         *other = _ex1;
 307                         return true;
 308
 309                 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
 310                 // (commutate contracted indices towards each other, simplify_indexed()
 311                 // will re-expand and re-run the simplification)
 312                 } else {
 313                         exvector::iterator it = self + 1, next_to_last = other - 1;
 314                         while (it != other) {
 315                                 if (!is_a<clifford>(*it))
 316                                         return false;
 317                                 ++it;
 318                         }
 319
 320                         it = self + 1;
 321                         ex S = _ex1;
 322                         while (it != next_to_last) {
 323                                 S *= *it;
 324                                 *it++ = _ex1;
 325                         }
 326
 327                         *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
 328                         *next_to_last = _ex1;
 329                         *other = _ex1;
 330                         return true;
 331                 }
 332
 333         } else if (is_a<symbol>(other->op(0)) && other->nops() == 2) {
 334
 335                 // x.mu gamma~mu -> x-slash
 336                 *self = dirac_slash(other->op(0), dim, rl);
 337                 *other = _ex1;
 338                 return true;
 339         }
 340
 341         return false;
 342 }
 343
 344 /** An utility function looking for a given metric within an exvector,
 345  *  used in cliffordunit::contract_with(). */
 346 static int find_same_metric(exvector & v, ex & c)
 347 {
 348         for (int i=0; i<v.size();i++) {
 349                 if (!is_a<clifford>(v[i]) && is_a<indexed>(v[i])
 350                     && ex_to<clifford>(c).same_metric(v[i])
 351                     && (ex_to<varidx>(c.op(1)) == ex_to<indexed>(v[i]).get_indices()[0]
 352                         || ex_to<varidx>(c.op(1)).toggle_variance() == ex_to<indexed>(v[i]).get_indices()[0])) {
 353                         return ++i; // next to found
 354                 }
 355         }
 356         return 0; //nothing found
 357 }
 358
 359 /** Contraction of a Clifford unit with something else. */
 360 bool cliffordunit::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 361 {
 362         GINAC_ASSERT(is_a<clifford>(*self));
 363         GINAC_ASSERT(is_a<indexed>(*other));
 364         GINAC_ASSERT(is_a<cliffordunit>(self->op(0)));
 365         clifford unit = ex_to<clifford>(*self);
 366         unsigned char rl = unit.get_representation_label();
 367
 368         if (is_a<clifford>(*other)) {
 369                 // Contraction only makes sense if the represenation labels are equal
 370                 // and the metrics are the same
 371                 if ((ex_to<clifford>(*other).get_representation_label() != rl)
 372                     && unit.same_metric(*other))
 373                         return false;
 374
 375                 // Find if a previous contraction produces the square of self
 376                 int prev_square = find_same_metric(v, self[0]);
 377                 varidx d((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(ex_to<idx>(self->op(1)).get_dim()));
 378                 ex squared_metric = unit.get_metric(self->op(1), d) * unit.get_metric(d.toggle_variance(), other->op(1));
 379
 380                 // e~mu e.mu = Tr ONE
 381                 if (other - self == 1) {
 382                         if (prev_square != 0) {
 383                                 *self = squared_metric;
 384                                 v[prev_square-1] = _ex1;
 385                         } else
 386                                 *self = unit.get_metric(self->op(1), other->op(1));
 387                         *other = dirac_ONE(rl);
 388                         return true;
 389
 390                 // e~mu e~alpha e.mu = (2e~alpha^2-Tr) e~alpha
 391                 } else if (other - self == 2
 392                         && is_a<clifford>(self[1])) {
 393
 394                         const ex & ia = self[1].op(1);
 395                         const ex & ib = self[1].op(1);
 396                         if (is_a<tensmetric>(unit.get_metric()))
 397                                 *self = 2 - unit.get_metric(self->op(1), other->op(1));
 398                         else if (prev_square != 0) {
 399                                 *self = 2-squared_metric;
 400                                 v[prev_square-1] = _ex1;
 401                         } else
 402                                 *self = 2*unit.get_metric(ia, ib) - unit.get_metric(self->op(1), other->op(1));
 403                         *other = _ex1;
 404                         return true;
 405
 406                 // e~mu S e~alpha e.mu = 2 e~alpha^3 S - e~mu S e.mu e~alpha
 407                 // (commutate contracted indices towards each other, simplify_indexed()
 408                 // will re-expand and re-run the simplification)
 409                 } else {
 410                         exvector::iterator it = self + 1, next_to_last = other - 1;
 411                         while (it != other) {
 412                                 if (!is_a<clifford>(*it))
 413                                         return false;
 414                                 ++it;
 415                         }
 416
 417                         it = self + 1;
 418                         ex S = _ex1;
 419                         while (it != next_to_last) {
 420                                 S *= *it;
 421                                 *it++ = _ex1;
 422                         }
 423
 424                         const ex & ia = next_to_last->op(1);
 425                         const ex & ib = next_to_last->op(1);
 426                         if (is_a<tensmetric>(unit.get_metric()))
 427                                 *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
 428                         else if (prev_square != 0) {
 429                                 *self = 2 * (*next_to_last) * S  - (*self) * S * (*other) * (*next_to_last)*unit.get_metric(self->op(1),self->op(1));
 430                                 v[prev_square-1] = _ex1;
 431                         } else
 432                                 *self = 2 * (*next_to_last) * S* unit.get_metric(ia,ib) - (*self) * S * (*other) * (*next_to_last);
 433                         *next_to_last = _ex1;
 434                         *other = _ex1;
 435                         return true;
 436                 }
 437
 438         }
 439
 440         return false;
 441 }
 442
 443 /** Perform automatic simplification on noncommutative product of clifford
 444  *  objects. This removes superfluous ONEs, permutes gamma5/L/R's to the front
 445  *  and removes squares of gamma objects. */
 446 ex clifford::eval_ncmul(const exvector & v) const
 447 {
 448         exvector s;
 449         s.reserve(v.size());
 450
 451         // Remove superfluous ONEs
 452         exvector::const_iterator cit = v.begin(), citend = v.end();
 453         while (cit != citend) {
 454                 if (!is_a<clifford>(*cit) || !is_a<diracone>(cit->op(0)))
 455                         s.push_back(*cit);
 456                 cit++;
 457         }
 458
 459         bool something_changed = false;
 460         int sign = 1;
 461
 462         // Anticommutate gamma5/L/R's to the front
 463         if (s.size() >= 2) {
 464                 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
 465                 while (true) {
 466                         exvector::iterator it = next_to_last;
 467                         while (true) {
 468                                 exvector::iterator it2 = it + 1;
 469                                 if (is_a<clifford>(*it) && is_a<clifford>(*it2)) {
 470                                         ex e1 = it->op(0), e2 = it2->op(0);
 471
 472                                         if (is_a<diracgamma5>(e2)) {
 473
 474                                                 if (is_a<diracgammaL>(e1) || is_a<diracgammaR>(e1)) {
 475
 476                                                         // gammaL/R gamma5 -> gamma5 gammaL/R
 477                                                         it->swap(*it2);
 478                                                         something_changed = true;
 479
 480                                                 } else if (!is_a<diracgamma5>(e1)) {
 481
 482                                                         // gamma5 gamma5 -> gamma5 gamma5 (do nothing)
 483                                                         // x gamma5 -> -gamma5 x
 484                                                         it->swap(*it2);
 485                                                         sign = -sign;
 486                                                         something_changed = true;
 487                                                 }
 488
 489                                         } else if (is_a<diracgammaL>(e2)) {
 490
 491                                                 if (is_a<diracgammaR>(e1)) {
 492
 493                                                         // gammaR gammaL -> 0
 494                                                         return _ex0;
 495
 496                                                 } else if (!is_a<diracgammaL>(e1) && !is_a<diracgamma5>(e1)) {
 497
 498                                                         // gammaL gammaL -> gammaL gammaL (do nothing)
 499                                                         // gamma5 gammaL -> gamma5 gammaL (do nothing)
 500                                                         // x gammaL -> gammaR x
 501                                                         it->swap(*it2);
 502                                                         *it = clifford(diracgammaR(), ex_to<clifford>(*it).get_representation_label());
 503                                                         something_changed = true;
 504                                                 }
 505
 506                                         } else if (is_a<diracgammaR>(e2)) {
 507
 508                                                 if (is_a<diracgammaL>(e1)) {
 509
 510                                                         // gammaL gammaR -> 0
 511                                                         return _ex0;
 512
 513                                                 } else if (!is_a<diracgammaR>(e1) && !is_a<diracgamma5>(e1)) {
 514
 515                                                         // gammaR gammaR -> gammaR gammaR (do nothing)
 516                                                         // gamma5 gammaR -> gamma5 gammaR (do nothing)
 517                                                         // x gammaR -> gammaL x
 518                                                         it->swap(*it2);
 519                                                         *it = clifford(diracgammaL(), ex_to<clifford>(*it).get_representation_label());
 520                                                         something_changed = true;
 521                                                 }
 522                                         }
 523                                 }
 524                                 if (it == first)
 525                                         break;
 526                                 --it;
 527                         }
 528                         if (next_to_last == first)
 529                                 break;
 530                         --next_to_last;
 531                 }
 532         }
 533
 534         // Remove equal adjacent gammas
 535         if (s.size() >= 2) {
 536                 exvector::iterator it, itend = s.end() - 1;
 537                 for (it = s.begin(); it != itend; ++it) {
 538                         ex & a = it[0];
 539                         ex & b = it[1];
 540                         if (!is_a<clifford>(a) || !is_a<clifford>(b))
 541                                 continue;
 542
 543                         const ex & ag = a.op(0);
 544                         const ex & bg = b.op(0);
 545                         bool a_is_cliffordunit = is_a<cliffordunit>(ag);
 546                         bool b_is_cliffordunit =  is_a<cliffordunit>(bg);
 547
 548                         if (a_is_cliffordunit && b_is_cliffordunit && ex_to<clifford>(a).same_metric(b)) {
 549
 550                                 const ex & ia = a.op(1);
 551                                 const ex & ib = b.op(1);
 552                                 if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
 553                                         a = ex_to<clifford>(a).get_metric(ia, ib);
 554                                         b = dirac_ONE(representation_label);
 555                                         something_changed = true;
 556                                 }
 557
 558                         } else if ((is_a<diracgamma5>(ag) && is_a<diracgamma5>(bg))) {
 559
 560                                 // Remove squares of gamma5
 561                                 a = dirac_ONE(representation_label);
 562                                 b = dirac_ONE(representation_label);
 563                                 something_changed = true;
 564
 565                         } else if ((is_a<diracgammaL>(ag) && is_a<diracgammaL>(bg))
 566                                 || (is_a<diracgammaR>(ag) && is_a<diracgammaR>(bg))) {
 567
 568                                 // Remove squares of gammaL/R
 569                                 b = dirac_ONE(representation_label);
 570                                 something_changed = true;
 571
 572                         } else if (is_a<diracgammaL>(ag) && is_a<diracgammaR>(bg)) {
 573
 574                                 // gammaL and gammaR are orthogonal
 575                                 return _ex0;
 576
 577                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaL>(bg)) {
 578
 579                                 // gamma5 gammaL -> -gammaL
 580                                 a = dirac_ONE(representation_label);
 581                                 sign = -sign;
 582                                 something_changed = true;
 583
 584                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaR>(bg)) {
 585
 586                                 // gamma5 gammaR -> gammaR
 587                                 a = dirac_ONE(representation_label);
 588                                 something_changed = true;
 589
 590                         } else if (!a_is_cliffordunit && !b_is_cliffordunit && ag.is_equal(bg)) {
 591
 592                                 // a\ a\ -> a^2
 593                                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).minimal_dim(ex_to<idx>(b.op(1))));
 594
 595                                 a = indexed(ag, ix) * indexed(ag, ix.toggle_variance());
 596                                 b = dirac_ONE(representation_label);
 597                                 something_changed = true;
 598                         }
 599                 }
 600         }
 601
 602         if (s.empty())
 603                 return clifford(diracone(), representation_label) * sign;
 604         if (something_changed)
 605                 return reeval_ncmul(s) * sign;
 606         else
 607                 return hold_ncmul(s) * sign;
 608 }
 609
 610 ex clifford::thiscontainer(const exvector & v) const
 611 {
 612         return clifford(representation_label, get_metric(), v);
 613 }
 614
 615 ex clifford::thiscontainer(std::auto_ptr<exvector> vp) const
 616 {
 617         return clifford(representation_label, get_metric(), vp);
 618 }
 619
 620 ex diracgamma5::conjugate() const
 621 {
 622         return _ex_1 * (*this);
 623 }
 624
 625 ex diracgammaL::conjugate() const
 626 {
 627         return (new diracgammaR)->setflag(status_flags::dynallocated);
 628 }
 629
 630 ex diracgammaR::conjugate() const
 631 {
 632         return (new diracgammaL)->setflag(status_flags::dynallocated);
 633 }
 634
 635 //////////
 636 // global functions
 637 //////////
 638
 639 ex dirac_ONE(unsigned char rl)
 640 {
 641         static ex ONE = (new diracone)->setflag(status_flags::dynallocated);
 642         return clifford(ONE, rl);
 643 }
 644
 645 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl)
 646 {
 647         static ex unit = (new cliffordunit)->setflag(status_flags::dynallocated);
 648
 649         if (!is_a<varidx>(mu))
 650                 throw(std::invalid_argument("index of Clifford unit must be of type varidx"));
 651
 652         return clifford(unit, mu, metr, rl);
 653 }
 654
 655 ex dirac_gamma(const ex & mu, unsigned char rl)
 656 {
 657         static ex gamma = (new diracgamma)->setflag(status_flags::dynallocated);
 658
 659         if (!is_a<varidx>(mu))
 660                 throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
 661
 662         return clifford(gamma, mu, default_metric(), rl);
 663 }
 664
 665 ex dirac_gamma5(unsigned char rl)
 666 {
 667         static ex gamma5 = (new diracgamma5)->setflag(status_flags::dynallocated);
 668         return clifford(gamma5, rl);
 669 }
 670
 671 ex dirac_gammaL(unsigned char rl)
 672 {
 673         static ex gammaL = (new diracgammaL)->setflag(status_flags::dynallocated);
 674         return clifford(gammaL, rl);
 675 }
 676
 677 ex dirac_gammaR(unsigned char rl)
 678 {
 679         static ex gammaR = (new diracgammaR)->setflag(status_flags::dynallocated);
 680         return clifford(gammaR, rl);
 681 }
 682
 683 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
 684 {
 685         // Slashed vectors are actually stored as a clifford object with the
 686         // vector as its base expression and a (dummy) index that just serves
 687         // for storing the space dimensionality
 688         return clifford(e, varidx(0, dim), default_metric(), rl);
 689 }
 690
 691 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 692  *  is that of a clifford object with the specified representation label. */
 693 static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
 694 {
 695         return ti == (TINFO_clifford + rl);
 696 }
 697
 698 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 699  *  is that of a clifford object (with an arbitrary representation label). */
 700 static bool is_clifford_tinfo(unsigned ti)
 701 {
 702         return (ti & ~0xff) == TINFO_clifford;
 703 }
 704
 705 /** Take trace of a string of an even number of Dirac gammas given a vector
 706  *  of indices. */
 707 static ex trace_string(exvector::const_iterator ix, size_t num)
 708 {
 709         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 710         if (num == 2)
 711                 return lorentz_g(ix[0], ix[1]);
 712
 713         // 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 )
 714         else if (num == 4)
 715                 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
 716                      + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
 717                      - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
 718
 719         // Traces of 6 or more gammas are computed recursively:
 720         // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
 721         //   + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
 722         //   - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
 723         //   + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
 724         //   - ...
 725         //   + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
 726         exvector v(num - 2);
 727         int sign = 1;
 728         ex result;
 729         for (size_t i=1; i<num; i++) {
 730                 for (size_t n=1, j=0; n<num; n++) {
 731                         if (n == i)
 732                                 continue;
 733                         v[j++] = ix[n];
 734                 }
 735                 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
 736                 sign = -sign;
 737         }
 738         return result;
 739 }
 740
 741 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
 742 {
 743         if (is_a<clifford>(e)) {
 744
 745                 if (!ex_to<clifford>(e).get_representation_label() == rl)
 746                         return _ex0;
 747                 const ex & g = e.op(0);
 748                 if (is_a<diracone>(g))
 749                         return trONE;
 750                 else if (is_a<diracgammaL>(g) || is_a<diracgammaR>(g))
 751                         return trONE/2;
 752                 else
 753                         return _ex0;
 754
 755         } else if (is_exactly_a<mul>(e)) {
 756
 757                 // Trace of product: pull out non-clifford factors
 758                 ex prod = _ex1;
 759                 for (size_t i=0; i<e.nops(); i++) {
 760                         const ex &o = e.op(i);
 761                         if (is_clifford_tinfo(o.return_type_tinfo(), rl))
 762                                 prod *= dirac_trace(o, rl, trONE);
 763                         else
 764                                 prod *= o;
 765                 }
 766                 return prod;
 767
 768         } else if (is_exactly_a<ncmul>(e)) {
 769
 770                 if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
 771                         return _ex0;
 772
 773                 // Substitute gammaL/R and expand product, if necessary
 774                 ex e_expanded = e.subs(lst(
 775                         dirac_gammaL(rl) == (dirac_ONE(rl)-dirac_gamma5(rl))/2,
 776                         dirac_gammaR(rl) == (dirac_ONE(rl)+dirac_gamma5(rl))/2
 777                 ), subs_options::no_pattern).expand();
 778                 if (!is_a<ncmul>(e_expanded))
 779                         return dirac_trace(e_expanded, rl, trONE);
 780
 781                 // gamma5 gets moved to the front so this check is enough
 782                 bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
 783                 size_t num = e.nops();
 784
 785                 if (has_gamma5) {
 786
 787                         // Trace of gamma5 * odd number of gammas and trace of
 788                         // gamma5 * gamma.mu * gamma.nu are zero
 789                         if ((num & 1) == 0 || num == 3)
 790                                 return _ex0;
 791
 792                         // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
 793                         // (the epsilon is always 4-dimensional)
 794                         if (num == 5) {
 795                                 ex b1, i1, b2, i2, b3, i3, b4, i4;
 796                                 base_and_index(e.op(1), b1, i1);
 797                                 base_and_index(e.op(2), b2, i2);
 798                                 base_and_index(e.op(3), b3, i3);
 799                                 base_and_index(e.op(4), b4, i4);
 800                                 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();
 801                         }
 802
 803                         // Tr gamma5 S_2k =
 804                         //   I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
 805                         // (the epsilon is always 4-dimensional)
 806                         exvector ix(num-1), bv(num-1);
 807                         for (size_t i=1; i<num; i++)
 808                                 base_and_index(e.op(i), bv[i-1], ix[i-1]);
 809                         num--;
 810                         int *iv = new int[num];
 811                         ex result;
 812                         for (size_t i=0; i<num-3; i++) {
 813                                 ex idx1 = ix[i];
 814                                 for (size_t j=i+1; j<num-2; j++) {
 815                                         ex idx2 = ix[j];
 816                                         for (size_t k=j+1; k<num-1; k++) {
 817                                                 ex idx3 = ix[k];
 818                                                 for (size_t l=k+1; l<num; l++) {
 819                                                         ex idx4 = ix[l];
 820                                                         iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
 821                                                         exvector v;
 822                                                         v.reserve(num - 4);
 823                                                         for (size_t n=0, t=4; n<num; n++) {
 824                                                                 if (n == i || n == j || n == k || n == l)
 825                                                                         continue;
 826                                                                 iv[t++] = n;
 827                                                                 v.push_back(ix[n]);
 828                                                         }
 829                                                         int sign = permutation_sign(iv, iv + num);
 830                                                         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))
 831                                                                 * trace_string(v.begin(), num - 4);
 832                                                 }
 833                                         }
 834                                 }
 835                         }
 836                         delete[] iv;
 837                         return trONE * I * result * mul(bv);
 838
 839                 } else { // no gamma5
 840
 841                         // Trace of odd number of gammas is zero
 842                         if ((num & 1) == 1)
 843                                 return _ex0;
 844
 845                         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 846                         if (num == 2) {
 847                                 ex b1, i1, b2, i2;
 848                                 base_and_index(e.op(0), b1, i1);
 849                                 base_and_index(e.op(1), b2, i2);
 850                                 return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
 851                         }
 852
 853                         exvector iv(num), bv(num);
 854                         for (size_t i=0; i<num; i++)
 855                                 base_and_index(e.op(i), bv[i], iv[i]);
 856
 857                         return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
 858                 }
 859
 860         } else if (e.nops() > 0) {
 861
 862                 // Trace maps to all other container classes (this includes sums)
 863                 pointer_to_map_function_2args<unsigned char, const ex &> fcn(dirac_trace, rl, trONE);
 864                 return e.map(fcn);
 865
 866         } else
 867                 return _ex0;
 868 }
 869
 870 ex canonicalize_clifford(const ex & e)
 871 {
 872         // Scan for any ncmul objects
 873         exmap srl;
 874         ex aux = e.to_rational(srl);
 875         for (exmap::iterator i = srl.begin(); i != srl.end(); ++i) {
 876
 877                 ex lhs = i->first;
 878                 ex rhs = i->second;
 879
 880                 if (is_exactly_a<ncmul>(rhs)
 881                  && rhs.return_type() == return_types::noncommutative
 882                  && is_clifford_tinfo(rhs.return_type_tinfo())) {
 883
 884                         // Expand product, if necessary
 885                         ex rhs_expanded = rhs.expand();
 886                         if (!is_a<ncmul>(rhs_expanded)) {
 887                                 i->second = canonicalize_clifford(rhs_expanded);
 888                                 continue;
 889
 890                         } else if (!is_a<clifford>(rhs.op(0)))
 891                                 continue;
 892
 893                         exvector v;
 894                         v.reserve(rhs.nops());
 895                         for (size_t j=0; j<rhs.nops(); j++)
 896                                 v.push_back(rhs.op(j));
 897
 898                         // Stupid recursive bubble sort because we only want to swap adjacent gammas
 899                         exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
 900                         if (is_a<diracgamma5>(it->op(0)) || is_a<diracgammaL>(it->op(0)) || is_a<diracgammaR>(it->op(0)))
 901                                 ++it;
 902                         while (it != next_to_last) {
 903                                 if (it[0].compare(it[1]) > 0) {
 904                                         ex save0 = it[0], save1 = it[1];
 905                                         ex b1, i1, b2, i2;
 906                                         base_and_index(it[0], b1, i1);
 907                                         base_and_index(it[1], b2, i2);
 908                                         it[0] = (ex_to<clifford>(save0).get_metric(i1, i2) * b1 * b2).simplify_indexed();
 909                                         it[1] = v.size() == 2 ? _ex2 * dirac_ONE(ex_to<clifford>(it[1]).get_representation_label()) : _ex2;
 910                                         ex sum = ncmul(v);
 911                                         it[0] = save1;
 912                                         it[1] = save0;
 913                                         sum -= ncmul(v, true);
 914                                         i->second = canonicalize_clifford(sum);
 915                                         goto next_sym;
 916                                 }
 917                                 ++it;
 918                         }
 919 next_sym:       ;
 920                 }
 921         }
 922         return aux.subs(srl, subs_options::no_pattern).simplify_indexed();
 923 }
 924
 925 ex clifford_prime(const ex &e)
 926 {
 927         pointer_to_map_function fcn(clifford_prime);
 928         if (is_a<clifford>(e) && is_a<cliffordunit>(e.op(0))) {
 929                 return -e;
 930         } else if (is_a<add>(e)) {
 931                 return e.map(fcn);
 932         } else if (is_a<ncmul>(e)) {
 933                 return e.map(fcn);
 934         } else if (is_a<power>(e)) {
 935                 return pow(clifford_prime(e.op(0)), e.op(1));
 936         } else
 937                 return e;
 938 }
 939
 940 ex delete_ONE(const ex &e)
 941 {
 942         pointer_to_map_function fcn(delete_ONE);
 943         if (is_a<clifford>(e) && is_a<diracone>(e.op(0))) {
 944                 return 1;
 945         } else if (is_a<add>(e)) {
 946                 return e.map(fcn);
 947         } else if (is_a<ncmul>(e)) {
 948                 return e.map(fcn);
 949         } else if (is_a<mul>(e)) {
 950                 return e.map(fcn);
 951         } else if (is_a<power>(e)) {
 952                 return pow(delete_ONE(e.op(0)), e.op(1));
 953         } else
 954                 return e;
 955 }
 956
 957 ex clifford_norm(const ex &e)
 958 {
 959         return sqrt(delete_ONE((e * clifford_bar(e)).simplify_indexed()));
 960 }
 961
 962 ex clifford_inverse(const ex &e)
 963 {
 964         ex norm = clifford_norm(e);
 965         if (!norm.is_zero())
 966                 return clifford_bar(e) / pow(norm, 2);
 967 }
 968
 969 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl)
 970 {
 971         unsigned min, max;
 972         if (!ex_to<idx>(mu).is_dim_numeric())
 973                 throw(std::invalid_argument("Index should have a numeric dimension"));
 974         unsigned dim = (ex_to<numeric>(ex_to<idx>(mu).get_dim())).to_int();
 975         ex c = clifford_unit(mu, metr, rl);
 976
 977         if (is_a<matrix>(v)) {
 978                 if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows()) {
 979                         min = ex_to<matrix>(v).rows();
 980                         max = ex_to<matrix>(v).cols();
 981                 } else {
 982                         min = ex_to<matrix>(v).cols();
 983                         max = ex_to<matrix>(v).rows();
 984                 }
 985                 if (min == 1) {
 986                         if (dim == max)
 987                                 if (is_a<varidx>(mu)) // need to swap variance
 988                                         return indexed(v, ex_to<varidx>(mu).toggle_variance()) * c;
 989                                 else
 990                                         return indexed(v, mu) * c;
 991                         else
 992                                 throw(std::invalid_argument("Dimensions of vector and clifford unit mismatch"));
 993                 } else
 994                         throw(std::invalid_argument("First argument should be a vector vector"));
 995         } else if (is_a<lst>(v)) {
 996                 if (dim == ex_to<lst>(v).nops())
 997                         return indexed(matrix(dim, 1, ex_to<lst>(v)), ex_to<varidx>(mu).toggle_variance()) * c;
 998                 else
 999                         throw(std::invalid_argument("List length and dimension of clifford unit mismatch"));
1000         } else
1001                 throw(std::invalid_argument("Cannot construct from anything but list or vector"));
1002 }
1003
1004 } // namespace GiNaC