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 clifford::clifford() : representation_label(0), metric(lorentz_g(varidx((new symbol)->setflag(status_flags::dynallocated), 4), varidx((new symbol)->setflag(status_flags::dynallocated), 4)))
  76 {
  77         tinfo_key = TINFO_clifford;
  78 }
  79
  80 DEFAULT_CTOR(diracone)
  81 DEFAULT_CTOR(cliffordunit)
  82 DEFAULT_CTOR(diracgamma)
  83 DEFAULT_CTOR(diracgamma5)
  84 DEFAULT_CTOR(diracgammaL)
  85 DEFAULT_CTOR(diracgammaR)
  86
  87 //////////
  88 // other constructors
  89 //////////
  90
  91 /** Construct object without any indices. This constructor is for internal
  92  *  use only. Use the dirac_ONE() function instead.
  93  *  @see dirac_ONE */
  94 clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl), metric(0)
  95 {
  96         tinfo_key = TINFO_clifford;
  97 }
  98
  99 /** Construct object with one Lorentz index. This constructor is for internal
 100  *  use only. Use the clifford_unit() or dirac_gamma() functions instead.
 101  *  @see clifford_unit
 102  *  @see dirac_gamma */
 103 clifford::clifford(const ex & b, const ex & mu, const ex & metr, unsigned char rl) : inherited(b, mu), representation_label(rl), metric(metr)
 104 {
 105         GINAC_ASSERT(is_a<varidx>(mu));
 106         tinfo_key = TINFO_clifford;
 107 }
 108
 109 clifford::clifford(unsigned char rl, const ex & metr, const exvector & v, bool discardable) : inherited(not_symmetric(), v, discardable), representation_label(rl), metric(metr)
 110 {
 111         tinfo_key = TINFO_clifford;
 112 }
 113
 114 clifford::clifford(unsigned char rl, const ex & metr, std::auto_ptr<exvector> vp) : inherited(not_symmetric(), vp), representation_label(rl), metric(metr)
 115 {
 116         tinfo_key = TINFO_clifford;
 117 }
 118
 119 //////////
 120 // archiving
 121 //////////
 122
 123 clifford::clifford(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
 124 {
 125         unsigned rl;
 126         n.find_unsigned("label", rl);
 127         representation_label = rl;
 128         n.find_ex("metric", metric, sym_lst);
 129 }
 130
 131 void clifford::archive(archive_node &n) const
 132 {
 133         inherited::archive(n);
 134         n.add_unsigned("label", representation_label);
 135         n.add_ex("metric", metric);
 136 }
 137
 138 DEFAULT_UNARCHIVE(clifford)
 139 DEFAULT_ARCHIVING(diracone)
 140 DEFAULT_ARCHIVING(cliffordunit)
 141 DEFAULT_ARCHIVING(diracgamma)
 142 DEFAULT_ARCHIVING(diracgamma5)
 143 DEFAULT_ARCHIVING(diracgammaL)
 144 DEFAULT_ARCHIVING(diracgammaR)
 145
 146 //////////
 147 // functions overriding virtual functions from base classes
 148 //////////
 149
 150 ex clifford::get_metric(const ex & i, const ex & j) const
 151 {
 152         return metric.subs(metric.op(1) == i).subs(metric.op(2) == j);
 153 }
 154
 155 bool clifford::same_metric(const ex & other) const
 156 {
 157         if (is_a<clifford>(other)) {
 158                 ex m = get_metric();
 159                 return m.is_equal(ex_to<clifford>(other).get_metric(m.op(1), m.op(2)));
 160         } else if (is_a<indexed>(other)) {
 161                 ex m = get_metric(other.op(1), other.op(2));
 162                 return m.is_equal(other);
 163         } else
 164                 return false;
 165 }
 166
 167 int clifford::compare_same_type(const basic & other) const
 168 {
 169         GINAC_ASSERT(is_a<clifford>(other));
 170         const clifford &o = static_cast<const clifford &>(other);
 171
 172         if (representation_label != o.representation_label) {
 173                 // different representation label
 174                 return representation_label < o.representation_label ? -1 : 1;
 175         }
 176
 177         return inherited::compare_same_type(other);
 178 }
 179
 180 bool clifford::match_same_type(const basic & other) const
 181 {
 182         GINAC_ASSERT(is_a<clifford>(other));
 183         const clifford &o = static_cast<const clifford &>(other);
 184
 185         return (representation_label == o.representation_label) && same_metric(o);
 186 }
 187
 188 static bool is_dirac_slash(const ex & seq0)
 189 {
 190         return !is_a<diracgamma5>(seq0) && !is_a<diracgammaL>(seq0) &&
 191                !is_a<diracgammaR>(seq0) && !is_a<cliffordunit>(seq0) &&
 192                !is_a<diracone>(seq0);
 193 }
 194
 195 void clifford::do_print_dflt(const print_dflt & c, unsigned level) const
 196 {
 197         // dirac_slash() object is printed differently
 198         if (is_dirac_slash(seq[0])) {
 199                 seq[0].print(c, level);
 200                 c.s << "\\";
 201         } else
 202                 this->print_dispatch<inherited>(c, level);
 203 }
 204
 205 void clifford::do_print_latex(const print_latex & c, unsigned level) const
 206 {
 207         // dirac_slash() object is printed differently
 208         if (is_dirac_slash(seq[0])) {
 209                 c.s << "{";
 210                 seq[0].print(c, level);
 211                 c.s << "\\hspace{-1.0ex}/}";
 212         } else
 213                 this->print_dispatch<inherited>(c, level);
 214 }
 215
 216 DEFAULT_COMPARE(diracone)
 217 DEFAULT_COMPARE(cliffordunit)
 218 DEFAULT_COMPARE(diracgamma)
 219 DEFAULT_COMPARE(diracgamma5)
 220 DEFAULT_COMPARE(diracgammaL)
 221 DEFAULT_COMPARE(diracgammaR)
 222
 223 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
 224 DEFAULT_PRINT_LATEX(cliffordunit, "e", "e")
 225 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
 226 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
 227 DEFAULT_PRINT_LATEX(diracgammaL, "gammaL", "{\\gamma_L}")
 228 DEFAULT_PRINT_LATEX(diracgammaR, "gammaR", "{\\gamma_R}")
 229
 230 /** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
 231 static void base_and_index(const ex & c, ex & b, ex & i)
 232 {
 233         GINAC_ASSERT(is_a<clifford>(c));
 234         GINAC_ASSERT(c.nops() == 2);
 235
 236         if (is_a<cliffordunit>(c.op(0))) { // proper dirac gamma object or clifford unit
 237                 i = c.op(1);
 238                 b = _ex1;
 239         } else if (is_a<diracgamma5>(c.op(0)) || is_a<diracgammaL>(c.op(0)) || is_a<diracgammaR>(c.op(0))) { // gamma5/L/R
 240                 i = _ex0;
 241                 b = _ex1;
 242         } else { // slash object, generate new dummy index
 243                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(c.op(1)).get_dim());
 244                 b = indexed(c.op(0), ix.toggle_variance());
 245                 i = ix;
 246         }
 247 }
 248
 249 /** Contraction of a gamma matrix with something else. */
 250 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 251 {
 252         GINAC_ASSERT(is_a<clifford>(*self));
 253         GINAC_ASSERT(is_a<indexed>(*other));
 254         GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
 255         unsigned char rl = ex_to<clifford>(*self).get_representation_label();
 256
 257         ex dim = ex_to<idx>(self->op(1)).get_dim();
 258         if (other->nops() > 1)
 259                 dim = minimal_dim(dim, ex_to<idx>(other->op(1)).get_dim());
 260
 261         if (is_a<clifford>(*other)) {
 262
 263                 // Contraction only makes sense if the represenation labels are equal
 264                 if (ex_to<clifford>(*other).get_representation_label() != rl)
 265                         return false;
 266
 267                 // gamma~mu gamma.mu = dim ONE
 268                 if (other - self == 1) {
 269                         *self = dim;
 270                         *other = dirac_ONE(rl);
 271                         return true;
 272
 273                 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
 274                 } else if (other - self == 2
 275                         && is_a<clifford>(self[1])) {
 276                         *self = 2 - dim;
 277                         *other = _ex1;
 278                         return true;
 279
 280                 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
 281                 } else if (other - self == 3
 282                         && is_a<clifford>(self[1])
 283                         && is_a<clifford>(self[2])) {
 284                         ex b1, i1, b2, i2;
 285                         base_and_index(self[1], b1, i1);
 286                         base_and_index(self[2], b2, i2);
 287                         *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
 288                         self[1] = _ex1;
 289                         self[2] = _ex1;
 290                         *other = _ex1;
 291                         return true;
 292
 293                 // 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
 294                 } else if (other - self == 4
 295                         && is_a<clifford>(self[1])
 296                         && is_a<clifford>(self[2])
 297                         && is_a<clifford>(self[3])) {
 298                         *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3];
 299                         self[1] = _ex1;
 300                         self[2] = _ex1;
 301                         self[3] = _ex1;
 302                         *other = _ex1;
 303                         return true;
 304
 305                 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
 306                 // (commutate contracted indices towards each other, simplify_indexed()
 307                 // will re-expand and re-run the simplification)
 308                 } else {
 309                         exvector::iterator it = self + 1, next_to_last = other - 1;
 310                         while (it != other) {
 311                                 if (!is_a<clifford>(*it))
 312                                         return false;
 313                                 ++it;
 314                         }
 315
 316                         it = self + 1;
 317                         ex S = _ex1;
 318                         while (it != next_to_last) {
 319                                 S *= *it;
 320                                 *it++ = _ex1;
 321                         }
 322
 323                         *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
 324                         *next_to_last = _ex1;
 325                         *other = _ex1;
 326                         return true;
 327                 }
 328
 329         } else if (is_a<symbol>(other->op(0)) && other->nops() == 2) {
 330
 331                 // x.mu gamma~mu -> x-slash
 332                 *self = dirac_slash(other->op(0), dim, rl);
 333                 *other = _ex1;
 334                 return true;
 335         }
 336
 337         return false;
 338 }
 339
 340 /** An utility function looking for a given metric within an exvector,
 341  *  used in cliffordunit::contract_with(). */
 342 static int find_same_metric(exvector & v, ex & c)
 343 {
 344         for (int i=0; i<v.size();i++) {
 345                 if (!is_a<clifford>(v[i]) && is_a<indexed>(v[i])
 346                     && ex_to<clifford>(c).same_metric(v[i])
 347                     && (ex_to<varidx>(c.op(1)) == ex_to<indexed>(v[i]).get_indices()[0]
 348                         || ex_to<varidx>(c.op(1)).toggle_variance() == ex_to<indexed>(v[i]).get_indices()[0])) {
 349                         return ++i; // next to found
 350                 }
 351         }
 352         return 0; //nothing found
 353 }
 354
 355 /** Contraction of a Clifford unit with something else. */
 356 bool cliffordunit::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 357 {
 358         GINAC_ASSERT(is_a<clifford>(*self));
 359         GINAC_ASSERT(is_a<indexed>(*other));
 360         GINAC_ASSERT(is_a<cliffordunit>(self->op(0)));
 361         clifford unit = ex_to<clifford>(*self);
 362         unsigned char rl = unit.get_representation_label();
 363
 364         if (is_a<clifford>(*other)) {
 365                 // Contraction only makes sense if the represenation labels are equal
 366                 // and the metrics are the same
 367                 if ((ex_to<clifford>(*other).get_representation_label() != rl)
 368                     && unit.same_metric(*other))
 369                         return false;
 370
 371                 // Find if a previous contraction produces the square of self
 372                 int prev_square = find_same_metric(v, self[0]);
 373                 varidx d((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(ex_to<idx>(self->op(1)).get_dim()));
 374                 ex squared_metric = unit.get_metric(self->op(1), d)*unit.get_metric(d.toggle_variance(), other->op(1));
 375
 376                 // e~mu e.mu = Tr ONE
 377                 if (other - self == 1) {
 378                         if (prev_square != 0) {
 379                                 *self = squared_metric;
 380                                 v[prev_square-1] = _ex1;
 381                         } else
 382                                 *self = unit.get_metric(self->op(1), other->op(1));
 383                         *other = dirac_ONE(rl);
 384                         return true;
 385
 386                 // e~mu e~alpha e.mu = (2e~alpha^2-Tr) e~alpha
 387                 } else if (other - self == 2
 388                         && is_a<clifford>(self[1])) {
 389
 390                         const ex & ia = self[1].op(1);
 391                         const ex & ib = self[1].op(1);
 392                         if (is_a<tensmetric>(unit.get_metric().op(0)))
 393                                 *self = 2 - unit.get_metric(self->op(1), other->op(1));
 394                         else if (prev_square != 0) {
 395                                 *self = 2-squared_metric;
 396                                 v[prev_square-1] = _ex1;
 397                         } else
 398                                 *self = 2*unit.get_metric(ia, ib) - unit.get_metric(self->op(1), other->op(1));
 399                         *other = _ex1;
 400                         return true;
 401
 402                 // e~mu S e~alpha e.mu = 2 e~alpha^3 S - e~mu S e.mu e~alpha
 403                 // (commutate contracted indices towards each other, simplify_indexed()
 404                 // will re-expand and re-run the simplification)
 405                 } else {
 406                         exvector::iterator it = self + 1, next_to_last = other - 1;
 407                         while (it != other) {
 408                                 if (!is_a<clifford>(*it))
 409                                         return false;
 410                                 ++it;
 411                         }
 412
 413                         it = self + 1;
 414                         ex S = _ex1;
 415                         while (it != next_to_last) {
 416                                 S *= *it;
 417                                 *it++ = _ex1;
 418                         }
 419
 420                         const ex & ia = next_to_last->op(1);
 421                         const ex & ib = next_to_last->op(1);
 422                         if (is_a<tensmetric>(unit.get_metric().op(0)))
 423                                 *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
 424                         else if (prev_square != 0) {
 425                                 *self = 2 * (*next_to_last) * S  - (*self) * S * (*other) * (*next_to_last)*unit.get_metric(self->op(1),self->op(1));
 426                                 v[prev_square-1] = _ex1;
 427                         } else
 428                                 *self = 2 * (*next_to_last) * S* unit.get_metric(ia,ib) - (*self) * S * (*other) * (*next_to_last);
 429                         *next_to_last = _ex1;
 430                         *other = _ex1;
 431                         return true;
 432                 }
 433
 434         }
 435
 436         return false;
 437 }
 438
 439 /** Perform automatic simplification on noncommutative product of clifford
 440  *  objects. This removes superfluous ONEs, permutes gamma5/L/R's to the front
 441  *  and removes squares of gamma objects. */
 442 ex clifford::eval_ncmul(const exvector & v) const
 443 {
 444         exvector s;
 445         s.reserve(v.size());
 446
 447         // Remove superfluous ONEs
 448         exvector::const_iterator cit = v.begin(), citend = v.end();
 449         while (cit != citend) {
 450                 if (!is_a<clifford>(*cit) || !is_a<diracone>(cit->op(0)))
 451                         s.push_back(*cit);
 452                 cit++;
 453         }
 454
 455         bool something_changed = false;
 456         int sign = 1;
 457
 458         // Anticommute gamma5/L/R's to the front
 459         if (s.size() >= 2) {
 460                 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
 461                 while (true) {
 462                         exvector::iterator it = next_to_last;
 463                         while (true) {
 464                                 exvector::iterator it2 = it + 1;
 465                                 if (is_a<clifford>(*it) && is_a<clifford>(*it2)) {
 466                                         ex e1 = it->op(0), e2 = it2->op(0);
 467
 468                                         if (is_a<diracgamma5>(e2)) {
 469
 470                                                 if (is_a<diracgammaL>(e1) || is_a<diracgammaR>(e1)) {
 471
 472                                                         // gammaL/R gamma5 -> gamma5 gammaL/R
 473                                                         it->swap(*it2);
 474                                                         something_changed = true;
 475
 476                                                 } else if (!is_a<diracgamma5>(e1)) {
 477
 478                                                         // gamma5 gamma5 -> gamma5 gamma5 (do nothing)
 479                                                         // x gamma5 -> -gamma5 x
 480                                                         it->swap(*it2);
 481                                                         sign = -sign;
 482                                                         something_changed = true;
 483                                                 }
 484
 485                                         } else if (is_a<diracgammaL>(e2)) {
 486
 487                                                 if (is_a<diracgammaR>(e1)) {
 488
 489                                                         // gammaR gammaL -> 0
 490                                                         return _ex0;
 491
 492                                                 } else if (!is_a<diracgammaL>(e1) && !is_a<diracgamma5>(e1)) {
 493
 494                                                         // gammaL gammaL -> gammaL gammaL (do nothing)
 495                                                         // gamma5 gammaL -> gamma5 gammaL (do nothing)
 496                                                         // x gammaL -> gammaR x
 497                                                         it->swap(*it2);
 498                                                         *it = clifford(diracgammaR(), ex_to<clifford>(*it).get_representation_label());
 499                                                         something_changed = true;
 500                                                 }
 501
 502                                         } else if (is_a<diracgammaR>(e2)) {
 503
 504                                                 if (is_a<diracgammaL>(e1)) {
 505
 506                                                         // gammaL gammaR -> 0
 507                                                         return _ex0;
 508
 509                                                 } else if (!is_a<diracgammaR>(e1) && !is_a<diracgamma5>(e1)) {
 510
 511                                                         // gammaR gammaR -> gammaR gammaR (do nothing)
 512                                                         // gamma5 gammaR -> gamma5 gammaR (do nothing)
 513                                                         // x gammaR -> gammaL x
 514                                                         it->swap(*it2);
 515                                                         *it = clifford(diracgammaL(), ex_to<clifford>(*it).get_representation_label());
 516                                                         something_changed = true;
 517                                                 }
 518                                         }
 519                                 }
 520                                 if (it == first)
 521                                         break;
 522                                 --it;
 523                         }
 524                         if (next_to_last == first)
 525                                 break;
 526                         --next_to_last;
 527                 }
 528         }
 529
 530         // Remove equal adjacent gammas
 531         if (s.size() >= 2) {
 532                 exvector::iterator it, itend = s.end() - 1;
 533                 for (it = s.begin(); it != itend; ++it) {
 534                         ex & a = it[0];
 535                         ex & b = it[1];
 536                         if (!is_a<clifford>(a) || !is_a<clifford>(b))
 537                                 continue;
 538
 539                         const ex & ag = a.op(0);
 540                         const ex & bg = b.op(0);
 541                         bool a_is_cliffordunit = is_a<cliffordunit>(ag);
 542                         bool b_is_cliffordunit =  is_a<cliffordunit>(bg);
 543
 544                         if (a_is_cliffordunit && b_is_cliffordunit && ex_to<clifford>(a).same_metric(b)) {
 545
 546                                 const ex & ia = a.op(1);
 547                                 const ex & ib = b.op(1);
 548                                 if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
 549                                         a = ex_to<clifford>(a).get_metric(ia,ib);
 550                                         b = dirac_ONE(representation_label);
 551                                         something_changed = true;
 552                                 }
 553
 554                         } else if ((is_a<diracgamma5>(ag) && is_a<diracgamma5>(bg))) {
 555
 556                                 // Remove squares of gamma5
 557                                 a = dirac_ONE(representation_label);
 558                                 b = dirac_ONE(representation_label);
 559                                 something_changed = true;
 560
 561                         } else if ((is_a<diracgammaL>(ag) && is_a<diracgammaL>(bg))
 562                                 || (is_a<diracgammaR>(ag) && is_a<diracgammaR>(bg))) {
 563
 564                                 // Remove squares of gammaL/R
 565                                 b = dirac_ONE(representation_label);
 566                                 something_changed = true;
 567
 568                         } else if (is_a<diracgammaL>(ag) && is_a<diracgammaR>(bg)) {
 569
 570                                 // gammaL and gammaR are orthogonal
 571                                 return _ex0;
 572
 573                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaL>(bg)) {
 574
 575                                 // gamma5 gammaL -> -gammaL
 576                                 a = dirac_ONE(representation_label);
 577                                 sign = -sign;
 578                                 something_changed = true;
 579
 580                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaR>(bg)) {
 581
 582                                 // gamma5 gammaR -> gammaR
 583                                 a = dirac_ONE(representation_label);
 584                                 something_changed = true;
 585
 586                         } else if (!a_is_cliffordunit && !b_is_cliffordunit && ag.is_equal(bg)) {
 587
 588                                 // a\ a\ -> a^2
 589                                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).minimal_dim(ex_to<idx>(b.op(1))));
 590
 591                                 a = indexed(ag, ix) * indexed(ag, ix.toggle_variance());
 592                                 b = dirac_ONE(representation_label);
 593                                 something_changed = true;
 594                         }
 595                 }
 596         }
 597
 598         if (s.empty())
 599                 return clifford(diracone(), representation_label) * sign;
 600         if (something_changed)
 601                 return reeval_ncmul(s) * sign;
 602         else
 603                 return hold_ncmul(s) * sign;
 604 }
 605
 606 ex clifford::thiscontainer(const exvector & v) const
 607 {
 608         return clifford(representation_label, get_metric(), v);
 609 }
 610
 611 ex clifford::thiscontainer(std::auto_ptr<exvector> vp) const
 612 {
 613         return clifford(representation_label, get_metric(), vp);
 614 }
 615
 616 ex diracgamma5::conjugate() const
 617 {
 618         return _ex_1 * (*this);
 619 }
 620
 621 ex diracgammaL::conjugate() const
 622 {
 623         return (new diracgammaR)->setflag(status_flags::dynallocated);
 624 }
 625
 626 ex diracgammaR::conjugate() const
 627 {
 628         return (new diracgammaL)->setflag(status_flags::dynallocated);
 629 }
 630
 631 //////////
 632 // global functions
 633 //////////
 634
 635 ex dirac_ONE(unsigned char rl)
 636 {
 637         static ex ONE = (new diracone)->setflag(status_flags::dynallocated);
 638         return clifford(ONE, rl);
 639 }
 640
 641 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl)
 642 {
 643         if (!is_a<varidx>(mu))
 644                 throw(std::invalid_argument("index of Clifford unit must be of type varidx"));
 645         if (!is_a<indexed>(metr))
 646                 throw(std::invalid_argument("metric for Clifford unit must be of type indexed"));
 647         exvector d = ex_to<indexed>(metr).get_indices();
 648         if (d.size() > 2 || ex_to<idx>(d[0]).get_dim() != ex_to<idx>(d[1]).get_dim())
 649           //|| ex_to<idx>(d[0]).get_dim() != ex_to<idx>(mu).get_dim())
 650                 throw(std::invalid_argument("metric is not square"));
 651         else
 652                 ex_to<idx>(mu).replace_dim(ex_to<idx>(d[0]).get_dim());
 653         return clifford(cliffordunit(), mu, metr, rl);
 654 }
 655
 656 ex dirac_gamma(const ex & mu, unsigned char rl)
 657 {
 658         static ex gamma = (new diracgamma)->setflag(status_flags::dynallocated);
 659
 660         if (!is_a<varidx>(mu))
 661                 throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
 662
 663         ex dim = ex_to<idx>(mu).get_dim();
 664         return clifford(gamma, mu, lorentz_g(varidx((new symbol)->setflag(status_flags::dynallocated), dim),varidx((new symbol)->setflag(status_flags::dynallocated), dim)), rl);
 665 }
 666
 667 ex dirac_gamma5(unsigned char rl)
 668 {
 669         static ex gamma5 = (new diracgamma5)->setflag(status_flags::dynallocated);
 670         return clifford(gamma5, rl);
 671 }
 672
 673 ex dirac_gammaL(unsigned char rl)
 674 {
 675         static ex gammaL = (new diracgammaL)->setflag(status_flags::dynallocated);
 676         return clifford(gammaL, rl);
 677 }
 678
 679 ex dirac_gammaR(unsigned char rl)
 680 {
 681         static ex gammaR = (new diracgammaR)->setflag(status_flags::dynallocated);
 682         return clifford(gammaR, rl);
 683 }
 684
 685 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
 686 {
 687         // Slashed vectors are actually stored as a clifford object with the
 688         // vector as its base expression and a (dummy) index that just serves
 689         // for storing the space dimensionality
 690         return clifford(e, varidx(0, dim), rl);
 691 }
 692
 693 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 694  *  is that of a clifford object with the specified representation label. */
 695 static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
 696 {
 697         return ti == (TINFO_clifford + rl);
 698 }
 699
 700 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 701  *  is that of a clifford object (with an arbitrary representation label). */
 702 static bool is_clifford_tinfo(unsigned ti)
 703 {
 704         return (ti & ~0xff) == TINFO_clifford;
 705 }
 706
 707 /** Take trace of a string of an even number of Dirac gammas given a vector
 708  *  of indices. */
 709 static ex trace_string(exvector::const_iterator ix, size_t num)
 710 {
 711         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 712         if (num == 2)
 713                 return lorentz_g(ix[0], ix[1]);
 714
 715         // 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 )
 716         else if (num == 4)
 717                 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
 718                      + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
 719                      - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
 720
 721         // Traces of 6 or more gammas are computed recursively:
 722         // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
 723         //   + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
 724         //   - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
 725         //   + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
 726         //   - ...
 727         //   + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
 728         exvector v(num - 2);
 729         int sign = 1;
 730         ex result;
 731         for (size_t i=1; i<num; i++) {
 732                 for (size_t n=1, j=0; n<num; n++) {
 733                         if (n == i)
 734                                 continue;
 735                         v[j++] = ix[n];
 736                 }
 737                 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
 738                 sign = -sign;
 739         }
 740         return result;
 741 }
 742
 743 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
 744 {
 745         if (is_a<clifford>(e)) {
 746
 747                 if (!ex_to<clifford>(e).get_representation_label() == rl)
 748                         return _ex0;
 749                 const ex & g = e.op(0);
 750                 if (is_a<diracone>(g))
 751                         return trONE;
 752                 else if (is_a<diracgammaL>(g) || is_a<diracgammaR>(g))
 753                         return trONE/2;
 754                 else
 755                         return _ex0;
 756
 757         } else if (is_exactly_a<mul>(e)) {
 758
 759                 // Trace of product: pull out non-clifford factors
 760                 ex prod = _ex1;
 761                 for (size_t i=0; i<e.nops(); i++) {
 762                         const ex &o = e.op(i);
 763                         if (is_clifford_tinfo(o.return_type_tinfo(), rl))
 764                                 prod *= dirac_trace(o, rl, trONE);
 765                         else
 766                                 prod *= o;
 767                 }
 768                 return prod;
 769
 770         } else if (is_exactly_a<ncmul>(e)) {
 771
 772                 if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
 773                         return _ex0;
 774
 775                 // Substitute gammaL/R and expand product, if necessary
 776                 ex e_expanded = e.subs(lst(
 777                         dirac_gammaL(rl) == (dirac_ONE(rl)-dirac_gamma5(rl))/2,
 778                         dirac_gammaR(rl) == (dirac_ONE(rl)+dirac_gamma5(rl))/2
 779                 ), subs_options::no_pattern).expand();
 780                 if (!is_a<ncmul>(e_expanded))
 781                         return dirac_trace(e_expanded, rl, trONE);
 782
 783                 // gamma5 gets moved to the front so this check is enough
 784                 bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
 785                 size_t num = e.nops();
 786
 787                 if (has_gamma5) {
 788
 789                         // Trace of gamma5 * odd number of gammas and trace of
 790                         // gamma5 * gamma.mu * gamma.nu are zero
 791                         if ((num & 1) == 0 || num == 3)
 792                                 return _ex0;
 793
 794                         // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
 795                         // (the epsilon is always 4-dimensional)
 796                         if (num == 5) {
 797                                 ex b1, i1, b2, i2, b3, i3, b4, i4;
 798                                 base_and_index(e.op(1), b1, i1);
 799                                 base_and_index(e.op(2), b2, i2);
 800                                 base_and_index(e.op(3), b3, i3);
 801                                 base_and_index(e.op(4), b4, i4);
 802                                 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();
 803                         }
 804
 805                         // Tr gamma5 S_2k =
 806                         //   I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
 807                         // (the epsilon is always 4-dimensional)
 808                         exvector ix(num-1), bv(num-1);
 809                         for (size_t i=1; i<num; i++)
 810                                 base_and_index(e.op(i), bv[i-1], ix[i-1]);
 811                         num--;
 812                         int *iv = new int[num];
 813                         ex result;
 814                         for (size_t i=0; i<num-3; i++) {
 815                                 ex idx1 = ix[i];
 816                                 for (size_t j=i+1; j<num-2; j++) {
 817                                         ex idx2 = ix[j];
 818                                         for (size_t k=j+1; k<num-1; k++) {
 819                                                 ex idx3 = ix[k];
 820                                                 for (size_t l=k+1; l<num; l++) {
 821                                                         ex idx4 = ix[l];
 822                                                         iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
 823                                                         exvector v;
 824                                                         v.reserve(num - 4);
 825                                                         for (size_t n=0, t=4; n<num; n++) {
 826                                                                 if (n == i || n == j || n == k || n == l)
 827                                                                         continue;
 828                                                                 iv[t++] = n;
 829                                                                 v.push_back(ix[n]);
 830                                                         }
 831                                                         int sign = permutation_sign(iv, iv + num);
 832                                                         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))
 833                                                                 * trace_string(v.begin(), num - 4);
 834                                                 }
 835                                         }
 836                                 }
 837                         }
 838                         delete[] iv;
 839                         return trONE * I * result * mul(bv);
 840
 841                 } else { // no gamma5
 842
 843                         // Trace of odd number of gammas is zero
 844                         if ((num & 1) == 1)
 845                                 return _ex0;
 846
 847                         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 848                         if (num == 2) {
 849                                 ex b1, i1, b2, i2;
 850                                 base_and_index(e.op(0), b1, i1);
 851                                 base_and_index(e.op(1), b2, i2);
 852                                 return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
 853                         }
 854
 855                         exvector iv(num), bv(num);
 856                         for (size_t i=0; i<num; i++)
 857                                 base_and_index(e.op(i), bv[i], iv[i]);
 858
 859                         return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
 860                 }
 861
 862         } else if (e.nops() > 0) {
 863
 864                 // Trace maps to all other container classes (this includes sums)
 865                 pointer_to_map_function_2args<unsigned char, const ex &> fcn(dirac_trace, rl, trONE);
 866                 return e.map(fcn);
 867
 868         } else
 869                 return _ex0;
 870 }
 871
 872 ex canonicalize_clifford(const ex & e)
 873 {
 874         // Scan for any ncmul objects
 875         exmap srl;
 876         ex aux = e.to_rational(srl);
 877         for (exmap::iterator i = srl.begin(); i != srl.end(); ++i) {
 878
 879                 ex lhs = i->first;
 880                 ex rhs = i->second;
 881
 882                 if (is_exactly_a<ncmul>(rhs)
 883                  && rhs.return_type() == return_types::noncommutative
 884                  && is_clifford_tinfo(rhs.return_type_tinfo())) {
 885
 886                         // Expand product, if necessary
 887                         ex rhs_expanded = rhs.expand();
 888                         if (!is_a<ncmul>(rhs_expanded)) {
 889                                 i->second = canonicalize_clifford(rhs_expanded);
 890                                 continue;
 891
 892                         } else if (!is_a<clifford>(rhs.op(0)))
 893                                 continue;
 894
 895                         exvector v;
 896                         v.reserve(rhs.nops());
 897                         for (size_t j=0; j<rhs.nops(); j++)
 898                                 v.push_back(rhs.op(j));
 899
 900                         // Stupid recursive bubble sort because we only want to swap adjacent gammas
 901                         exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
 902                         if (is_a<diracgamma5>(it->op(0)) || is_a<diracgammaL>(it->op(0)) || is_a<diracgammaR>(it->op(0)))
 903                                 ++it;
 904                         while (it != next_to_last) {
 905                                 if (it[0].compare(it[1]) > 0) {
 906                                         ex save0 = it[0], save1 = it[1];
 907                                         ex b1, i1, b2, i2;
 908                                         base_and_index(it[0], b1, i1);
 909                                         base_and_index(it[1], b2, i2);
 910                                         it[0] = (ex_to<clifford>(save0).get_metric(i1, i2) * b1 * b2).simplify_indexed();
 911                                         it[1] = v.size() == 2 ? _ex2 * dirac_ONE(ex_to<clifford>(it[1]).get_representation_label()) : _ex2;
 912                                         ex sum = ncmul(v);
 913                                         it[0] = save1;
 914                                         it[1] = save0;
 915                                         sum -= ncmul(v, true);
 916                                         i->second = canonicalize_clifford(sum);
 917                                         goto next_sym;
 918                                 }
 919                                 ++it;
 920                         }
 921 next_sym:       ;
 922                 }
 923         }
 924         return aux.subs(srl, subs_options::no_pattern).simplify_indexed();
 925 }
 926
 927 ex clifford_prime(const ex &e)
 928 {
 929         pointer_to_map_function fcn(clifford_prime);
 930         if (is_a<clifford>(e) && is_a<cliffordunit>(e.op(0))) {
 931                 return -e;
 932         } else if (is_a<add>(e)) {
 933                 return e.map(fcn);
 934         } else if (is_a<ncmul>(e)) {
 935                 return e.map(fcn);
 936         } else if (is_a<power>(e)) {
 937                 return pow(clifford_prime(e.op(0)), e.op(1));
 938         } else
 939                 return e;
 940 }
 941
 942 ex delete_ONE(const ex &e)
 943 {
 944         pointer_to_map_function fcn(delete_ONE);
 945         if (is_a<clifford>(e) && is_a<diracone>(e.op(0))) {
 946                 return 1;
 947         } else if (is_a<add>(e)) {
 948                 return e.map(fcn);
 949         } else if (is_a<ncmul>(e)) {
 950                 return e.map(fcn);
 951         } else if (is_a<mul>(e)) {
 952                 return e.map(fcn);
 953         } else if (is_a<power>(e)) {
 954                 return pow(delete_ONE(e.op(0)), e.op(1));
 955         } else
 956                 return e;
 957 }
 958
 959 ex clifford_norm(const ex &e)
 960 {
 961         return sqrt(delete_ONE((e * clifford_bar(e)).simplify_indexed()));
 962 }
 963
 964 ex clifford_inverse(const ex &e)
 965 {
 966         ex norm = clifford_norm(e);
 967         if (!norm.is_zero())
 968                 return clifford_bar(e) / pow(norm, 2);
 969 }
 970
 971 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl)
 972 {
 973         unsigned min, max;
 974         if (!ex_to<idx>(mu).is_dim_numeric())
 975                 throw(std::invalid_argument("Index should have a numeric dimension"));
 976         unsigned dim = (ex_to<numeric>(ex_to<idx>(mu).get_dim())).to_int();
 977         ex c = clifford_unit(mu, metr, rl);
 978
 979         if (is_a<matrix>(v)) {
 980                 if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows()) {
 981                         min = ex_to<matrix>(v).rows();
 982                         max = ex_to<matrix>(v).cols();
 983                 } else {
 984                         min = ex_to<matrix>(v).cols();
 985                         max = ex_to<matrix>(v).rows();
 986                 }
 987                 if (min == 1) {
 988                         if (dim == max)
 989                                 if (is_a<varidx>(mu)) // need to swap variance
 990                                         return indexed(v, ex_to<varidx>(mu).toggle_variance()) * c;
 991                                 else
 992                                         return indexed(v, mu) * c;
 993                         else
 994                                 throw(std::invalid_argument("Dimensions of vector and clifford unit mismatch"));
 995                 } else
 996                         throw(std::invalid_argument("First argument should be a vector vector"));
 997         } else if (is_a<lst>(v)) {
 998                 if (dim == ex_to<lst>(v).nops())
 999                         return indexed(matrix(dim, 1, ex_to<lst>(v)), ex_to<varidx>(mu).toggle_variance()) * c;
1000                 else
1001                         throw(std::invalid_argument("List length and dimension of clifford unit mismatch"));
1002         } else
1003                 throw(std::invalid_argument("Cannot construct from anything but list or vector"));
1004 }
1005
1006 } // namespace GiNaC