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