ginac/clifford.cpp

   1 /** @file clifford.cpp
   2  *
   3  *  Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2006 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., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  21  */
  22
  23 #include <stdexcept>
  24
  25 #include "clifford.h"
  26
  27 #include "ex.h"
  28 #include "idx.h"
  29 #include "ncmul.h"
  30 #include "symbol.h"
  31 #include "numeric.h" // for I
  32 #include "symmetry.h"
  33 #include "lst.h"
  34 #include "relational.h"
  35 #include "operators.h"
  36 #include "add.h"
  37 #include "mul.h"
  38 #include "power.h"
  39 #include "matrix.h"
  40 #include "archive.h"
  41 #include "utils.h"
  42
  43 namespace GiNaC {
  44
  45 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(clifford, indexed,
  46   print_func<print_dflt>(&clifford::do_print_dflt).
  47   print_func<print_latex>(&clifford::do_print_latex))
  48
  49 const tinfo_static_t clifford::return_type_tinfo_static[256] = {{}};
  50
  51 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracone, tensor,
  52   print_func<print_dflt>(&diracone::do_print).
  53   print_func<print_latex>(&diracone::do_print_latex))
  54
  55 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(cliffordunit, tensor,
  56   print_func<print_dflt>(&cliffordunit::do_print).
  57   print_func<print_latex>(&cliffordunit::do_print_latex))
  58
  59 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma, cliffordunit,
  60   print_func<print_dflt>(&diracgamma::do_print).
  61   print_func<print_latex>(&diracgamma::do_print_latex))
  62
  63 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma5, tensor,
  64   print_func<print_dflt>(&diracgamma5::do_print).
  65   print_func<print_latex>(&diracgamma5::do_print_latex))
  66
  67 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaL, tensor,
  68   print_func<print_context>(&diracgammaL::do_print).
  69   print_func<print_latex>(&diracgammaL::do_print_latex))
  70
  71 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaR, tensor,
  72   print_func<print_context>(&diracgammaR::do_print).
  73   print_func<print_latex>(&diracgammaR::do_print_latex))
  74
  75 //////////
  76 // default constructors
  77 //////////
  78
  79 clifford::clifford() : representation_label(0), metric(0), anticommuting(true), commutator_sign(-1)
  80 {
  81         tinfo_key = &clifford::tinfo_static;
  82 }
  83
  84 DEFAULT_CTOR(diracone)
  85 DEFAULT_CTOR(cliffordunit)
  86 DEFAULT_CTOR(diracgamma)
  87 DEFAULT_CTOR(diracgamma5)
  88 DEFAULT_CTOR(diracgammaL)
  89 DEFAULT_CTOR(diracgammaR)
  90
  91 //////////
  92 // other constructors
  93 //////////
  94
  95 /** Construct object without any indices. This constructor is for internal
  96  *  use only. Use the dirac_ONE() function instead.
  97  *  @see dirac_ONE */
  98 clifford::clifford(const ex & b, unsigned char rl, bool anticommut) : inherited(b), representation_label(rl), metric(0), anticommuting(anticommut), commutator_sign(-1)
  99 {
 100         tinfo_key = &clifford::tinfo_static;
 101 }
 102
 103 /** Construct object with one Lorentz index. This constructor is for internal
 104  *  use only. Use the clifford_unit() or dirac_gamma() functions instead.
 105  *  @see clifford_unit
 106  *  @see dirac_gamma */
 107 clifford::clifford(const ex & b, const ex & mu, const ex & metr, unsigned char rl, bool anticommut, int comm_sign) : inherited(b, mu), representation_label(rl), metric(metr), anticommuting(anticommut), commutator_sign(comm_sign)
 108 {
 109         GINAC_ASSERT(is_a<varidx>(mu));
 110         tinfo_key = &clifford::tinfo_static;
 111 }
 112
 113 clifford::clifford(unsigned char rl, const ex & metr, bool anticommut, int comm_sign, const exvector & v, bool discardable) : inherited(not_symmetric(), v, discardable), representation_label(rl), metric(metr), anticommuting(anticommut), commutator_sign(comm_sign)
 114 {
 115         tinfo_key = &clifford::tinfo_static;
 116 }
 117
 118 clifford::clifford(unsigned char rl, const ex & metr, bool anticommut, int comm_sign, std::auto_ptr<exvector> vp) : inherited(not_symmetric(), vp), representation_label(rl), metric(metr), anticommuting(anticommut), commutator_sign(comm_sign)
 119 {
 120         tinfo_key = &clifford::tinfo_static;
 121 }
 122
 123 //////////
 124 // archiving
 125 //////////
 126
 127 clifford::clifford(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
 128 {
 129         unsigned rl;
 130         n.find_unsigned("label", rl);
 131         representation_label = rl;
 132         n.find_ex("metric", metric, sym_lst);
 133         n.find_bool("anticommuting", anticommuting);
 134         n.find_unsigned("commutator_sign+1", rl);
 135         commutator_sign = rl - 1;
 136 }
 137
 138 void clifford::archive(archive_node & n) const
 139 {
 140         inherited::archive(n);
 141         n.add_unsigned("label", representation_label);
 142         n.add_ex("metric", metric);
 143         n.add_bool("anticommuting", anticommuting);
 144         n.add_unsigned("commutator_sign+1", commutator_sign+1);
 145 }
 146
 147 DEFAULT_UNARCHIVE(clifford)
 148 DEFAULT_ARCHIVING(diracone)
 149 DEFAULT_ARCHIVING(cliffordunit)
 150 DEFAULT_ARCHIVING(diracgamma)
 151 DEFAULT_ARCHIVING(diracgamma5)
 152 DEFAULT_ARCHIVING(diracgammaL)
 153 DEFAULT_ARCHIVING(diracgammaR)
 154
 155
 156 ex clifford::get_metric(const ex & i, const ex & j, bool symmetrised) const
 157 {
 158         if (is_a<indexed>(metric)) {
 159                 if (symmetrised && !(ex_to<symmetry>(ex_to<indexed>(metric).get_symmetry()).has_symmetry())) {
 160                         if (is_a<matrix>(metric.op(0))) {
 161                                 return indexed((ex_to<matrix>(metric.op(0)).add(ex_to<matrix>(metric.op(0)).transpose())).mul(numeric(1,2)),
 162                                                symmetric2(), i, j);
 163                         } else {
 164                                 return simplify_indexed(indexed(metric.op(0)*_ex1_2, i, j) + indexed(metric.op(0)*_ex1_2, j, i));
 165                         }
 166                 } else {
 167                         //return indexed(metric.op(0), ex_to<symmetry>(ex_to<indexed>(metric).get_symmetry()), i, j);
 168                         return metric.subs(lst(metric.op(1) == i, metric.op(2) == j), subs_options::no_pattern);
 169                 }
 170         } else {
 171                 // should not really happen since all constructors but clifford() make the metric an indexed object
 172                 return indexed(metric, i, j);
 173         }
 174 }
 175
 176 bool clifford::same_metric(const ex & other) const
 177 {
 178         if (is_a<clifford>(other)) {
 179                 return same_metric(ex_to<clifford>(other).get_metric());
 180         } else if (is_a<indexed>(other)) {
 181                 return get_metric(other.op(1), other.op(2)).is_equal(other);
 182         } else
 183                 return false;
 184 }
 185
 186 //////////
 187 // functions overriding virtual functions from base classes
 188 //////////
 189
 190 ex clifford::op(size_t i) const
 191 {
 192         GINAC_ASSERT(i<nops());
 193         if (nops()-i == 1)
 194                 return representation_label;
 195         else
 196                 return inherited::op(i);
 197 }
 198
 199 ex & clifford::let_op(size_t i)
 200 {
 201         GINAC_ASSERT(i<nops());
 202
 203         static ex rl = numeric(representation_label);
 204         ensure_if_modifiable();
 205         if (nops()-i == 1)
 206                 return rl;
 207         else
 208                 return inherited::let_op(i);
 209 }
 210
 211 ex clifford::subs(const exmap & m, unsigned options) const
 212 {
 213         ex subsed = inherited::subs(m, options);
 214         if(is_a<clifford>(subsed)) {
 215                 ex prevmetric = ex_to<clifford>(subsed).metric;
 216                 ex newmetric = prevmetric.subs(m, options);
 217                 if(!are_ex_trivially_equal(prevmetric, newmetric)) {
 218                         clifford c = ex_to<clifford>(subsed);
 219                         c.metric = newmetric;
 220                         subsed = c;
 221                 }
 222         }
 223         return subsed;
 224 }
 225
 226 int clifford::compare_same_type(const basic & other) const
 227 {
 228         GINAC_ASSERT(is_a<clifford>(other));
 229         const clifford &o = static_cast<const clifford &>(other);
 230
 231         if (representation_label != o.representation_label) {
 232                 // different representation label
 233                 return representation_label < o.representation_label ? -1 : 1;
 234         }
 235
 236         return inherited::compare_same_type(other);
 237 }
 238
 239 bool clifford::match_same_type(const basic & other) const
 240 {
 241         GINAC_ASSERT(is_a<clifford>(other));
 242         const clifford &o = static_cast<const clifford &>(other);
 243
 244         return ((representation_label == o.representation_label) && (commutator_sign == o.get_commutator_sign()) && same_metric(o));
 245 }
 246
 247 static bool is_dirac_slash(const ex & seq0)
 248 {
 249         return !is_a<diracgamma5>(seq0) && !is_a<diracgammaL>(seq0) &&
 250                !is_a<diracgammaR>(seq0) && !is_a<cliffordunit>(seq0) &&
 251                !is_a<diracone>(seq0);
 252 }
 253
 254 void clifford::do_print_dflt(const print_dflt & c, unsigned level) const
 255 {
 256         // dirac_slash() object is printed differently
 257         if (is_dirac_slash(seq[0])) {
 258                 seq[0].print(c, precedence());
 259                 c.s << "\\";
 260         } else
 261                 this->print_dispatch<inherited>(c, level);
 262 }
 263
 264 void clifford::do_print_latex(const print_latex & c, unsigned level) const
 265 {
 266         // dirac_slash() object is printed differently
 267         if (is_dirac_slash(seq[0])) {
 268                 c.s << "{";
 269                 seq[0].print(c, precedence());
 270                 c.s << "\\hspace{-1.0ex}/}";
 271         } else {
 272                 c.s << "\\clifford[" << int(representation_label) << "]";
 273                 this->print_dispatch<inherited>(c, level);
 274         }
 275 }
 276
 277 DEFAULT_COMPARE(diracone)
 278 DEFAULT_COMPARE(cliffordunit)
 279 DEFAULT_COMPARE(diracgamma)
 280 DEFAULT_COMPARE(diracgamma5)
 281 DEFAULT_COMPARE(diracgammaL)
 282 DEFAULT_COMPARE(diracgammaR)
 283
 284 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbf{1}")
 285 DEFAULT_PRINT_LATEX(cliffordunit, "e", "e")
 286 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
 287 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
 288 DEFAULT_PRINT_LATEX(diracgammaL, "gammaL", "{\\gamma_L}")
 289 DEFAULT_PRINT_LATEX(diracgammaR, "gammaR", "{\\gamma_R}")
 290
 291 /** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
 292 static void base_and_index(const ex & c, ex & b, ex & i)
 293 {
 294         GINAC_ASSERT(is_a<clifford>(c));
 295         GINAC_ASSERT(c.nops() == 2+1);
 296
 297         if (is_a<cliffordunit>(c.op(0))) { // proper dirac gamma object or clifford unit
 298                 i = c.op(1);
 299                 b = _ex1;
 300         } else if (is_a<diracgamma5>(c.op(0)) || is_a<diracgammaL>(c.op(0)) || is_a<diracgammaR>(c.op(0))) { // gamma5/L/R
 301                 i = _ex0;
 302                 b = _ex1;
 303         } else { // slash object, generate new dummy index
 304                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(c.op(1)).get_dim());
 305                 b = indexed(c.op(0), ix.toggle_variance());
 306                 i = ix;
 307         }
 308 }
 309
 310 /** Predicate for finding non-clifford objects. */
 311 struct is_not_a_clifford : public std::unary_function<ex, bool> {
 312         bool operator()(const ex & e)
 313         {
 314                 return !is_a<clifford>(e);
 315         }
 316 };
 317
 318 /** Contraction of a gamma matrix with something else. */
 319 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 320 {
 321         GINAC_ASSERT(is_a<clifford>(*self));
 322         GINAC_ASSERT(is_a<indexed>(*other));
 323         GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
 324         unsigned char rl = ex_to<clifford>(*self).get_representation_label();
 325
 326         ex dim = ex_to<idx>(self->op(1)).get_dim();
 327         if (other->nops() > 1)
 328                 dim = minimal_dim(dim, ex_to<idx>(other->op(1)).get_dim());
 329
 330         if (is_a<clifford>(*other)) {
 331
 332                 // Contraction only makes sense if the represenation labels are equal
 333                 if (ex_to<clifford>(*other).get_representation_label() != rl)
 334                         return false;
 335
 336                 size_t num = other - self;
 337
 338                 // gamma~mu gamma.mu = dim ONE
 339                 if (num == 1) {
 340                         *self = dim;
 341                         *other = dirac_ONE(rl);
 342                         return true;
 343
 344                 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
 345                 } else if (num == 2
 346                         && is_a<clifford>(self[1])) {
 347                         *self = 2 - dim;
 348                         *other = _ex1;
 349                         return true;
 350
 351                 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
 352                 } else if (num == 3
 353                         && is_a<clifford>(self[1])
 354                         && is_a<clifford>(self[2])) {
 355                         ex b1, i1, b2, i2;
 356                         base_and_index(self[1], b1, i1);
 357                         base_and_index(self[2], b2, i2);
 358                         *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
 359                         self[1] = _ex1;
 360                         self[2] = _ex1;
 361                         *other = _ex1;
 362                         return true;
 363
 364                 // 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
 365                 } else if (num == 4
 366                         && is_a<clifford>(self[1])
 367                         && is_a<clifford>(self[2])
 368                         && is_a<clifford>(self[3])) {
 369                         *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3];
 370                         self[1] = _ex1;
 371                         self[2] = _ex1;
 372                         self[3] = _ex1;
 373                         *other = _ex1;
 374                         return true;
 375
 376                 // gamma~mu Sodd gamma.mu = -2 Sodd_R
 377                 // (Chisholm identity in 4 dimensions)
 378                 } else if (!((other - self) & 1) && dim.is_equal(4)) {
 379                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
 380                                 return false;
 381
 382                         *self = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(other), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
 383                         std::fill(self + 1, other, _ex1);
 384                         *other = _ex_2;
 385                         return true;
 386
 387                 // gamma~mu Sodd gamma~alpha gamma.mu = 2 gamma~alpha Sodd + 2 Sodd_R gamma~alpha
 388                 // (commutate contracted indices towards each other, then use
 389                 // Chisholm identity in 4 dimensions)
 390                 } else if (((other - self) & 1) && dim.is_equal(4)) {
 391                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
 392                                 return false;
 393
 394                         exvector::iterator next_to_last = other - 1;
 395                         ex S = ncmul(exvector(self + 1, next_to_last), true);
 396                         ex SR = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(next_to_last), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
 397
 398                         *self = (*next_to_last) * S + SR * (*next_to_last);
 399                         std::fill(self + 1, other, _ex1);
 400                         *other = _ex2;
 401                         return true;
 402
 403                 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
 404                 // (commutate contracted indices towards each other, simplify_indexed()
 405                 // will re-expand and re-run the simplification)
 406                 } else {
 407                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
 408                                 return false;
 409
 410                         exvector::iterator next_to_last = other - 1;
 411                         ex S = ncmul(exvector(self + 1, next_to_last), true);
 412
 413                         *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
 414                         std::fill(self + 1, other + 1, _ex1);
 415                         return true;
 416                 }
 417
 418         } else if (is_a<symbol>(other->op(0)) && other->nops() == 2) {
 419
 420                 // x.mu gamma~mu -> x-slash
 421                 *self = dirac_slash(other->op(0), dim, rl);
 422                 *other = _ex1;
 423                 return true;
 424         }
 425
 426         return false;
 427 }
 428
 429 /** An utility function looking for a given metric within an exvector,
 430  *  used in cliffordunit::contract_with(). */
 431 static int find_same_metric(exvector & v, ex & c)
 432 {
 433         for (size_t i=0; i<v.size(); i++) {
 434                 if (is_a<indexed>(v[i]) && !is_a<clifford>(v[i])
 435                     && ((ex_to<varidx>(c.op(1)) == ex_to<indexed>(v[i]).get_indices()[0]
 436                     && ex_to<varidx>(c.op(1)) == ex_to<indexed>(v[i]).get_indices()[1])
 437                     || (ex_to<varidx>(c.op(1)).toggle_variance() == ex_to<indexed>(v[i]).get_indices()[0]
 438                     && ex_to<varidx>(c.op(1)).toggle_variance() == ex_to<indexed>(v[i]).get_indices()[1]))) {
 439                         return i; // the index of the found
 440                 }
 441         }
 442         return -1; //nothing found
 443 }
 444
 445 /** Contraction of a Clifford unit with something else. */
 446 bool cliffordunit::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 447 {
 448         GINAC_ASSERT(is_a<clifford>(*self));
 449         GINAC_ASSERT(is_a<indexed>(*other));
 450         GINAC_ASSERT(is_a<cliffordunit>(self->op(0)));
 451         clifford unit = ex_to<clifford>(*self);
 452         unsigned char rl = unit.get_representation_label();
 453
 454         if (is_a<clifford>(*other)) {
 455                 // Contraction only makes sense if the represenation labels are equal
 456                 // and the metrics are the same
 457                 if ((ex_to<clifford>(*other).get_representation_label() != rl)
 458                     && unit.same_metric(*other))
 459                         return false;
 460
 461                 // Find if a previous contraction produces the square of self
 462                 int prev_square = find_same_metric(v, *self);
 463                 const varidx d((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(self->op(1)).get_dim()),
 464                         in1((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(self->op(1)).get_dim()),
 465                         in2((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(self->op(1)).get_dim());
 466                 ex squared_metric;
 467                 if (prev_square > -1)
 468                         squared_metric = simplify_indexed(indexed(v[prev_square].op(0), in1, d)
 469                                                                                           * unit.get_metric(d.toggle_variance(), in2, true)).op(0);
 470
 471                 exvector::iterator before_other = other - 1;
 472                 const varidx & mu = ex_to<varidx>(self->op(1));
 473                 const varidx & mu_toggle = ex_to<varidx>(other->op(1));
 474                 const varidx & alpha = ex_to<varidx>(before_other->op(1));
 475
 476                 // e~mu e.mu = Tr ONE
 477                 if (other - self == 1) {
 478                         if (prev_square > -1) {
 479                                 *self = indexed(squared_metric, mu, mu_toggle);
 480                                 v[prev_square] = _ex1;
 481                         } else {
 482                                 *self = unit.get_metric(mu, mu_toggle, true);
 483                         }
 484                         *other = dirac_ONE(rl);
 485                         return true;
 486
 487                 } else if (other - self == 2) {
 488                         if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
 489                                 if (ex_to<clifford>(*self).is_anticommuting()) {
 490                                         // e~mu e~alpha e.mu = (2*pow(e~alpha, 2) -Tr(B)) e~alpha
 491                                         if (prev_square > -1) {
 492                                                 *self = 2 * indexed(squared_metric, alpha, alpha)
 493                                                         - indexed(squared_metric, mu, mu_toggle);
 494                                                 v[prev_square] = _ex1;
 495                                         } else {
 496                                                 *self = 2 * unit.get_metric(alpha, alpha, true) - unit.get_metric(mu, mu_toggle, true);
 497                                         }
 498                                         *other = _ex1;
 499                                         return true;
 500
 501                                 } else {
 502                                         // e~mu e~alpha e.mu = 2*e~mu B(alpha, mu.toggle_variance())-Tr(B) e~alpha
 503                                         *self = 2 * (*self) * unit.get_metric(alpha, mu_toggle, true) - unit.get_metric(mu, mu_toggle, true) * (*before_other);
 504                                         *before_other = _ex1;
 505                                         *other = _ex1;
 506                                         return true;
 507                                 }
 508                         } else {
 509                                 // e~mu S e.mu = Tr S ONE
 510                                 *self = unit.get_metric(mu, mu_toggle, true);
 511                                 *other = dirac_ONE(rl);
 512                                 return true;
 513                         }
 514                 } else {
 515                 // e~mu S e~alpha e.mu = 2 e~mu S B(alpha, mu.toggle_variance()) - e~mu S e.mu e~alpha
 516                 // (commutate contracted indices towards each other, simplify_indexed()
 517                 // will re-expand and re-run the simplification)
 518                         if (std::find_if(self + 1, other, is_not_a_clifford()) != other) {
 519                                 return false;
 520                         }
 521
 522                         ex S = ncmul(exvector(self + 1, before_other), true);
 523
 524                         if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
 525                                 if (ex_to<clifford>(*self).is_anticommuting()) {
 526                                         if (prev_square > -1) {
 527                                                 *self = 2 * (*before_other) * S * indexed(squared_metric, alpha, alpha)
 528                                                         - (*self) * S * (*other) * (*before_other);
 529                                         } else {
 530                                                 *self = 2 * (*before_other) * S * unit.get_metric(alpha, alpha, true) - (*self) * S * (*other) * (*before_other);
 531                                         }
 532                                 } else {
 533                                         *self = 2 * (*self) * S * unit.get_metric(alpha, mu_toggle, true) - (*self) * S * (*other) * (*before_other);
 534                                 }
 535                         } else {
 536                                 // simply commutes
 537                                 *self = (*self) * S * (*other) * (*before_other);
 538                         }
 539
 540                         std::fill(self + 1, other + 1, _ex1);
 541                         return true;
 542                 }
 543         }
 544         return false;
 545 }
 546
 547 /** Perform automatic simplification on noncommutative product of clifford
 548  *  objects. This removes superfluous ONEs, permutes gamma5/L/R's to the front
 549  *  and removes squares of gamma objects. */
 550 ex clifford::eval_ncmul(const exvector & v) const
 551 {
 552         exvector s;
 553         s.reserve(v.size());
 554
 555         // Remove superfluous ONEs
 556         exvector::const_iterator cit = v.begin(), citend = v.end();
 557         while (cit != citend) {
 558                 if (!is_a<clifford>(*cit) || !is_a<diracone>(cit->op(0)))
 559                         s.push_back(*cit);
 560                 cit++;
 561         }
 562
 563         bool something_changed = false;
 564         int sign = 1;
 565
 566         // Anticommutate gamma5/L/R's to the front
 567         if (s.size() >= 2) {
 568                 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
 569                 while (true) {
 570                         exvector::iterator it = next_to_last;
 571                         while (true) {
 572                                 exvector::iterator it2 = it + 1;
 573                                 if (is_a<clifford>(*it) && is_a<clifford>(*it2)) {
 574                                         ex e1 = it->op(0), e2 = it2->op(0);
 575
 576                                         if (is_a<diracgamma5>(e2)) {
 577
 578                                                 if (is_a<diracgammaL>(e1) || is_a<diracgammaR>(e1)) {
 579
 580                                                         // gammaL/R gamma5 -> gamma5 gammaL/R
 581                                                         it->swap(*it2);
 582                                                         something_changed = true;
 583
 584                                                 } else if (!is_a<diracgamma5>(e1)) {
 585
 586                                                         // gamma5 gamma5 -> gamma5 gamma5 (do nothing)
 587                                                         // x gamma5 -> -gamma5 x
 588                                                         it->swap(*it2);
 589                                                         sign = -sign;
 590                                                         something_changed = true;
 591                                                 }
 592
 593                                         } else if (is_a<diracgammaL>(e2)) {
 594
 595                                                 if (is_a<diracgammaR>(e1)) {
 596
 597                                                         // gammaR gammaL -> 0
 598                                                         return _ex0;
 599
 600                                                 } else if (!is_a<diracgammaL>(e1) && !is_a<diracgamma5>(e1)) {
 601
 602                                                         // gammaL gammaL -> gammaL gammaL (do nothing)
 603                                                         // gamma5 gammaL -> gamma5 gammaL (do nothing)
 604                                                         // x gammaL -> gammaR x
 605                                                         it->swap(*it2);
 606                                                         *it = clifford(diracgammaR(), ex_to<clifford>(*it).get_representation_label());
 607                                                         something_changed = true;
 608                                                 }
 609
 610                                         } else if (is_a<diracgammaR>(e2)) {
 611
 612                                                 if (is_a<diracgammaL>(e1)) {
 613
 614                                                         // gammaL gammaR -> 0
 615                                                         return _ex0;
 616
 617                                                 } else if (!is_a<diracgammaR>(e1) && !is_a<diracgamma5>(e1)) {
 618
 619                                                         // gammaR gammaR -> gammaR gammaR (do nothing)
 620                                                         // gamma5 gammaR -> gamma5 gammaR (do nothing)
 621                                                         // x gammaR -> gammaL x
 622                                                         it->swap(*it2);
 623                                                         *it = clifford(diracgammaL(), ex_to<clifford>(*it).get_representation_label());
 624                                                         something_changed = true;
 625                                                 }
 626                                         }
 627                                 }
 628                                 if (it == first)
 629                                         break;
 630                                 --it;
 631                         }
 632                         if (next_to_last == first)
 633                                 break;
 634                         --next_to_last;
 635                 }
 636         }
 637
 638         // Remove equal adjacent gammas
 639         if (s.size() >= 2) {
 640                 exvector::iterator it, itend = s.end() - 1;
 641                 for (it = s.begin(); it != itend; ++it) {
 642                         ex & a = it[0];
 643                         ex & b = it[1];
 644                         if (!is_a<clifford>(a) || !is_a<clifford>(b))
 645                                 continue;
 646
 647                         const ex & ag = a.op(0);
 648                         const ex & bg = b.op(0);
 649                         bool a_is_cliffordunit = is_a<cliffordunit>(ag);
 650                         bool b_is_cliffordunit =  is_a<cliffordunit>(bg);
 651
 652                         if (a_is_cliffordunit && b_is_cliffordunit && ex_to<clifford>(a).same_metric(b)
 653                                 && (ex_to<clifford>(a).get_commutator_sign() == -1)) {
 654                                 // This is done only for Clifford algebras
 655
 656                                 const ex & ia = a.op(1);
 657                                 const ex & ib = b.op(1);
 658                                 if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
 659                                         a = ex_to<clifford>(a).get_metric(ia, ib, true);
 660                                         b = dirac_ONE(representation_label);
 661                                         something_changed = true;
 662                                 }
 663
 664                         } else if ((is_a<diracgamma5>(ag) && is_a<diracgamma5>(bg))) {
 665
 666                                 // Remove squares of gamma5
 667                                 a = dirac_ONE(representation_label);
 668                                 b = dirac_ONE(representation_label);
 669                                 something_changed = true;
 670
 671                         } else if ((is_a<diracgammaL>(ag) && is_a<diracgammaL>(bg))
 672                                 || (is_a<diracgammaR>(ag) && is_a<diracgammaR>(bg))) {
 673
 674                                 // Remove squares of gammaL/R
 675                                 b = dirac_ONE(representation_label);
 676                                 something_changed = true;
 677
 678                         } else if (is_a<diracgammaL>(ag) && is_a<diracgammaR>(bg)) {
 679
 680                                 // gammaL and gammaR are orthogonal
 681                                 return _ex0;
 682
 683                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaL>(bg)) {
 684
 685                                 // gamma5 gammaL -> -gammaL
 686                                 a = dirac_ONE(representation_label);
 687                                 sign = -sign;
 688                                 something_changed = true;
 689
 690                         } else if (is_a<diracgamma5>(ag) && is_a<diracgammaR>(bg)) {
 691
 692                                 // gamma5 gammaR -> gammaR
 693                                 a = dirac_ONE(representation_label);
 694                                 something_changed = true;
 695
 696                         } else if (!a_is_cliffordunit && !b_is_cliffordunit && ag.is_equal(bg)) {
 697
 698                                 // a\ a\ -> a^2
 699                                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).minimal_dim(ex_to<idx>(b.op(1))));
 700
 701                                 a = indexed(ag, ix) * indexed(ag, ix.toggle_variance());
 702                                 b = dirac_ONE(representation_label);
 703                                 something_changed = true;
 704                         }
 705                 }
 706         }
 707
 708         if (s.empty())
 709                 return dirac_ONE(representation_label) * sign;
 710         if (something_changed)
 711                 return reeval_ncmul(s) * sign;
 712         else
 713                 return hold_ncmul(s) * sign;
 714 }
 715
 716 ex clifford::thiscontainer(const exvector & v) const
 717 {
 718         return clifford(representation_label, metric, anticommuting, commutator_sign, v);
 719 }
 720
 721 ex clifford::thiscontainer(std::auto_ptr<exvector> vp) const
 722 {
 723         return clifford(representation_label, metric, anticommuting, commutator_sign, vp);
 724 }
 725
 726 ex diracgamma5::conjugate() const
 727 {
 728         return _ex_1 * (*this);
 729 }
 730
 731 ex diracgammaL::conjugate() const
 732 {
 733         return (new diracgammaR)->setflag(status_flags::dynallocated);
 734 }
 735
 736 ex diracgammaR::conjugate() const
 737 {
 738         return (new diracgammaL)->setflag(status_flags::dynallocated);
 739 }
 740
 741 //////////
 742 // global functions
 743 //////////
 744
 745 ex dirac_ONE(unsigned char rl)
 746 {
 747         static ex ONE = (new diracone)->setflag(status_flags::dynallocated);
 748         return clifford(ONE, rl, false);
 749 }
 750
 751 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl, bool anticommuting)
 752 {
 753         static ex unit = (new cliffordunit)->setflag(status_flags::dynallocated);
 754
 755         if (!is_a<idx>(mu))
 756                 throw(std::invalid_argument("clifford_unit(): index of Clifford unit must be of type idx or varidx"));
 757
 758         if (ex_to<idx>(mu).is_symbolic() && !is_a<varidx>(mu))
 759                 throw(std::invalid_argument("clifford_unit(): symbolic index of Clifford unit must be of type varidx (not idx)"));
 760
 761         if (is_a<indexed>(metr)) {
 762                 exvector indices = ex_to<indexed>(metr).get_indices();
 763                 if ((indices.size() == 2) && is_a<varidx>(indices[0]) && is_a<varidx>(indices[1])) {
 764                         return clifford(unit, mu, metr, rl, anticommuting);
 765                 } else {
 766                         throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be indexed exactly by two indices of same type as the given index"));
 767                 }
 768         } else if (is_a<tensor>(metr)) {
 769                 static varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim()),
 770                         chi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim());
 771                 return clifford(unit, mu, indexed(metr, xi, chi), rl, anticommuting);
 772         } else if (is_a<matrix>(metr)) {
 773                 matrix M = ex_to<matrix>(metr);
 774                 unsigned n = M.rows();
 775                 bool symmetric = true;
 776                 anticommuting = true;
 777
 778                 static varidx xi((new symbol)->setflag(status_flags::dynallocated), n),
 779                         chi((new symbol)->setflag(status_flags::dynallocated), n);
 780                 if ((n ==  M.cols()) && (n == ex_to<varidx>(mu).get_dim())) {
 781                         for (unsigned i = 0; i < n; i++) {
 782                                 for (unsigned j = i+1; j < n; j++) {
 783                                         if (M(i, j) != M(j, i)) {
 784                                                 symmetric = false;
 785                                         }
 786                                         if (M(i, j) != -M(j, i)) {
 787                                                 anticommuting = false;
 788                                         }
 789                                 }
 790                         }
 791                         return clifford(unit, mu, indexed(metr, symmetric?symmetric2():not_symmetric(), xi, chi), rl, anticommuting);
 792                 } else {
 793                         throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be a square matrix with the same dimensions as index"));
 794                 }
 795         } else {
 796                 throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be of type indexed, tensor or matrix"));
 797         }
 798 }
 799
 800 ex dirac_gamma(const ex & mu, unsigned char rl)
 801 {
 802         static ex gamma = (new diracgamma)->setflag(status_flags::dynallocated);
 803
 804         if (!is_a<varidx>(mu))
 805                 throw(std::invalid_argument("dirac_gamma(): index of Dirac gamma must be of type varidx"));
 806
 807         static varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim()),
 808                 chi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim());
 809         return clifford(gamma, mu, indexed((new minkmetric)->setflag(status_flags::dynallocated), symmetric2(), xi, chi), rl, true);
 810 }
 811
 812 ex dirac_gamma5(unsigned char rl)
 813 {
 814         static ex gamma5 = (new diracgamma5)->setflag(status_flags::dynallocated);
 815         return clifford(gamma5, rl);
 816 }
 817
 818 ex dirac_gammaL(unsigned char rl)
 819 {
 820         static ex gammaL = (new diracgammaL)->setflag(status_flags::dynallocated);
 821         return clifford(gammaL, rl);
 822 }
 823
 824 ex dirac_gammaR(unsigned char rl)
 825 {
 826         static ex gammaR = (new diracgammaR)->setflag(status_flags::dynallocated);
 827         return clifford(gammaR, rl);
 828 }
 829
 830 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
 831 {
 832         // Slashed vectors are actually stored as a clifford object with the
 833         // vector as its base expression and a (dummy) index that just serves
 834         // for storing the space dimensionality
 835
 836         static varidx xi((new symbol)->setflag(status_flags::dynallocated), dim),
 837                 chi((new symbol)->setflag(status_flags::dynallocated), dim);
 838    return clifford(e, varidx(0, dim), indexed((new minkmetric)->setflag(status_flags::dynallocated), symmetric2(), xi, chi), rl, true);
 839 }
 840
 841 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 842  *  is that of a clifford object (with an arbitrary representation label). */
 843 bool is_clifford_tinfo(tinfo_t ti)
 844 {
 845         p_int start_loc=(p_int)&clifford::return_type_tinfo_static;
 846         return (p_int)ti>=start_loc && (p_int)ti<start_loc+256;
 847 }
 848
 849 /** Extract representation label from tinfo key (as returned by
 850  *  return_type_tinfo()). */
 851 static unsigned char get_representation_label(tinfo_t ti)
 852 {
 853         return (unsigned char)((p_int)ti-(p_int)&clifford::return_type_tinfo_static);
 854 }
 855
 856 /** Take trace of a string of an even number of Dirac gammas given a vector
 857  *  of indices. */
 858 static ex trace_string(exvector::const_iterator ix, size_t num)
 859 {
 860         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 861         if (num == 2)
 862                 return lorentz_g(ix[0], ix[1]);
 863
 864         // 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 )
 865         else if (num == 4)
 866                 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
 867                      + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
 868                      - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
 869
 870         // Traces of 6 or more gammas are computed recursively:
 871         // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
 872         //   + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
 873         //   - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
 874         //   + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
 875         //   - ...
 876         //   + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
 877         exvector v(num - 2);
 878         int sign = 1;
 879         ex result;
 880         for (size_t i=1; i<num; i++) {
 881                 for (size_t n=1, j=0; n<num; n++) {
 882                         if (n == i)
 883                                 continue;
 884                         v[j++] = ix[n];
 885                 }
 886                 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
 887                 sign = -sign;
 888         }
 889         return result;
 890 }
 891
 892 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE)
 893 {
 894         if (is_a<clifford>(e)) {
 895
 896                 unsigned char rl = ex_to<clifford>(e).get_representation_label();
 897
 898                 // Are we taking the trace over this object's representation label?
 899                 if (rls.find(rl) == rls.end())
 900                         return e;
 901
 902                 // Yes, all elements are traceless, except for dirac_ONE and dirac_L/R
 903                 const ex & g = e.op(0);
 904                 if (is_a<diracone>(g))
 905                         return trONE;
 906                 else if (is_a<diracgammaL>(g) || is_a<diracgammaR>(g))
 907                         return trONE/2;
 908                 else
 909                         return _ex0;
 910
 911         } else if (is_exactly_a<mul>(e)) {
 912
 913                 // Trace of product: pull out non-clifford factors
 914                 ex prod = _ex1;
 915                 for (size_t i=0; i<e.nops(); i++) {
 916                         const ex &o = e.op(i);
 917                         if (is_clifford_tinfo(o.return_type_tinfo()))
 918                                 prod *= dirac_trace(o, rls, trONE);
 919                         else
 920                                 prod *= o;
 921                 }
 922                 return prod;
 923
 924         } else if (is_exactly_a<ncmul>(e)) {
 925
 926                 unsigned char rl = get_representation_label(e.return_type_tinfo());
 927
 928                 // Are we taking the trace over this string's representation label?
 929                 if (rls.find(rl) == rls.end())
 930                         return e;
 931
 932                 // Substitute gammaL/R and expand product, if necessary
 933                 ex e_expanded = e.subs(lst(
 934                         dirac_gammaL(rl) == (dirac_ONE(rl)-dirac_gamma5(rl))/2,
 935                         dirac_gammaR(rl) == (dirac_ONE(rl)+dirac_gamma5(rl))/2
 936                 ), subs_options::no_pattern).expand();
 937                 if (!is_a<ncmul>(e_expanded))
 938                         return dirac_trace(e_expanded, rls, trONE);
 939
 940                 // gamma5 gets moved to the front so this check is enough
 941                 bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
 942                 size_t num = e.nops();
 943
 944                 if (has_gamma5) {
 945
 946                         // Trace of gamma5 * odd number of gammas and trace of
 947                         // gamma5 * gamma.mu * gamma.nu are zero
 948                         if ((num & 1) == 0 || num == 3)
 949                                 return _ex0;
 950
 951                         // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
 952                         // (the epsilon is always 4-dimensional)
 953                         if (num == 5) {
 954                                 ex b1, i1, b2, i2, b3, i3, b4, i4;
 955                                 base_and_index(e.op(1), b1, i1);
 956                                 base_and_index(e.op(2), b2, i2);
 957                                 base_and_index(e.op(3), b3, i3);
 958                                 base_and_index(e.op(4), b4, i4);
 959                                 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();
 960                         }
 961
 962                         // Tr gamma5 S_2k =
 963                         //   I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
 964                         // (the epsilon is always 4-dimensional)
 965                         exvector ix(num-1), bv(num-1);
 966                         for (size_t i=1; i<num; i++)
 967                                 base_and_index(e.op(i), bv[i-1], ix[i-1]);
 968                         num--;
 969                         int *iv = new int[num];
 970                         ex result;
 971                         for (size_t i=0; i<num-3; i++) {
 972                                 ex idx1 = ix[i];
 973                                 for (size_t j=i+1; j<num-2; j++) {
 974                                         ex idx2 = ix[j];
 975                                         for (size_t k=j+1; k<num-1; k++) {
 976                                                 ex idx3 = ix[k];
 977                                                 for (size_t l=k+1; l<num; l++) {
 978                                                         ex idx4 = ix[l];
 979                                                         iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
 980                                                         exvector v;
 981                                                         v.reserve(num - 4);
 982                                                         for (size_t n=0, t=4; n<num; n++) {
 983                                                                 if (n == i || n == j || n == k || n == l)
 984                                                                         continue;
 985                                                                 iv[t++] = n;
 986                                                                 v.push_back(ix[n]);
 987                                                         }
 988                                                         int sign = permutation_sign(iv, iv + num);
 989                                                         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))
 990                                                                 * trace_string(v.begin(), num - 4);
 991                                                 }
 992                                         }
 993                                 }
 994                         }
 995                         delete[] iv;
 996                         return trONE * I * result * mul(bv);
 997
 998                 } else { // no gamma5
 999
1000                         // Trace of odd number of gammas is zero
1001                         if ((num & 1) == 1)
1002                                 return _ex0;
1003
1004                         // Tr gamma.mu gamma.nu = 4 g.mu.nu
1005                         if (num == 2) {
1006                                 ex b1, i1, b2, i2;
1007                                 base_and_index(e.op(0), b1, i1);
1008                                 base_and_index(e.op(1), b2, i2);
1009                                 return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
1010                         }
1011
1012                         exvector iv(num), bv(num);
1013                         for (size_t i=0; i<num; i++)
1014                                 base_and_index(e.op(i), bv[i], iv[i]);
1015
1016                         return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
1017                 }
1018
1019         } else if (e.nops() > 0) {
1020
1021                 // Trace maps to all other container classes (this includes sums)
1022                 pointer_to_map_function_2args<const std::set<unsigned char> &, const ex &> fcn(dirac_trace, rls, trONE);
1023                 return e.map(fcn);
1024
1025         } else
1026                 return _ex0;
1027 }
1028
1029 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE)
1030 {
1031         // Convert list to set
1032         std::set<unsigned char> rls;
1033         for (lst::const_iterator i = rll.begin(); i != rll.end(); ++i) {
1034                 if (i->info(info_flags::nonnegint))
1035                         rls.insert(ex_to<numeric>(*i).to_int());
1036         }
1037
1038         return dirac_trace(e, rls, trONE);
1039 }
1040
1041 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
1042 {
1043         // Convert label to set
1044         std::set<unsigned char> rls;
1045         rls.insert(rl);
1046
1047         return dirac_trace(e, rls, trONE);
1048 }
1049
1050
1051 ex canonicalize_clifford(const ex & e_)
1052 {
1053         pointer_to_map_function fcn(canonicalize_clifford);
1054
1055         if (is_a<matrix>(e_)    // || is_a<pseries>(e) || is_a<integral>(e)
1056                 || is_a<lst>(e_)) {
1057                 return e_.map(fcn);
1058         } else {
1059                 ex e=simplify_indexed(e_);
1060                 // Scan for any ncmul objects
1061                 exmap srl;
1062                 ex aux = e.to_rational(srl);
1063                 for (exmap::iterator i = srl.begin(); i != srl.end(); ++i) {
1064
1065                         ex lhs = i->first;
1066                         ex rhs = i->second;
1067
1068                         if (is_exactly_a<ncmul>(rhs)
1069                                         && rhs.return_type() == return_types::noncommutative
1070                                         && is_clifford_tinfo(rhs.return_type_tinfo())) {
1071
1072                                 // Expand product, if necessary
1073                                 ex rhs_expanded = rhs.expand();
1074                                 if (!is_a<ncmul>(rhs_expanded)) {
1075                                         i->second = canonicalize_clifford(rhs_expanded);
1076                                         continue;
1077
1078                                 } else if (!is_a<clifford>(rhs.op(0)))
1079                                         continue;
1080
1081                                 exvector v;
1082                                 v.reserve(rhs.nops());
1083                                 for (size_t j=0; j<rhs.nops(); j++)
1084                                         v.push_back(rhs.op(j));
1085
1086                                 // Stupid recursive bubble sort because we only want to swap adjacent gammas
1087                                 exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
1088                                 if (is_a<diracgamma5>(it->op(0)) || is_a<diracgammaL>(it->op(0)) || is_a<diracgammaR>(it->op(0)))
1089                                         ++it;
1090
1091                                 while (it != next_to_last) {
1092                                         if (it[0].compare(it[1]) > 0) {
1093
1094                                                 ex save0 = it[0], save1 = it[1];
1095                                                 ex b1, i1, b2, i2;
1096                                                 base_and_index(it[0], b1, i1);
1097                                                 base_and_index(it[1], b2, i2);
1098                                                 // for Clifford algebras (commutator_sign == -1) metric should be symmetrised
1099                                                 it[0] = (ex_to<clifford>(save0).get_metric(i1, i2, ex_to<clifford>(save0).get_commutator_sign() == -1) * b1 * b2).simplify_indexed();
1100                                                 it[1] = v.size() ? _ex2 * dirac_ONE(ex_to<clifford>(save0).get_representation_label()) : _ex2;
1101                                                 ex sum = ncmul(v);
1102                                                 it[0] = save1;
1103                                                 it[1] = save0;
1104                                                 sum += ex_to<clifford>(save0).get_commutator_sign() * ncmul(v, true);
1105                                                 i->second = canonicalize_clifford(sum);
1106                                                 goto next_sym;
1107                                         }
1108                                         ++it;
1109                                 }
1110 next_sym:       ;
1111                         }
1112                 }
1113                 return aux.subs(srl, subs_options::no_pattern).simplify_indexed();
1114         }
1115 }
1116
1117 ex clifford_prime(const ex & e)
1118 {
1119         pointer_to_map_function fcn(clifford_prime);
1120         if (is_a<clifford>(e) && is_a<cliffordunit>(e.op(0))) {
1121                 return -e;
1122         } else if (is_a<add>(e) || is_a<ncmul>(e) || is_a<mul>(e) //|| is_a<pseries>(e) || is_a<integral>(e)
1123                            || is_a<matrix>(e) || is_a<lst>(e)) {
1124                 return e.map(fcn);
1125         } else if (is_a<power>(e)) {
1126                 return pow(clifford_prime(e.op(0)), e.op(1));
1127         } else
1128                 return e;
1129 }
1130
1131 ex remove_dirac_ONE(const ex & e, unsigned char rl, unsigned options)
1132 {
1133         pointer_to_map_function_2args<unsigned char, unsigned> fcn(remove_dirac_ONE, rl, options | 1);
1134         bool need_reevaluation = false;
1135         ex e1 = e;
1136         if (! (options & 1) )  { // is not a child
1137                 if (options & 2)
1138                         e1 = expand_dummy_sum(e, true);
1139                 e1 = canonicalize_clifford(e1);
1140         }
1141
1142         if (is_a<clifford>(e1) && ex_to<clifford>(e1).get_representation_label() >= rl) {
1143                 if (is_a<diracone>(e1.op(0)))
1144                         return 1;
1145                 else
1146                         throw(std::invalid_argument("remove_dirac_ONE(): expression is a non-scalar Clifford number!"));
1147         } else if (is_a<add>(e1) || is_a<ncmul>(e1) || is_a<mul>(e1)
1148                            || is_a<matrix>(e1) || is_a<lst>(e1)) {
1149                 if (options & 3) // is a child or was already expanded
1150                         return e1.map(fcn);
1151                 else
1152                         try {
1153                                 return e1.map(fcn);
1154                         } catch (std::exception &p) {
1155                                 need_reevaluation = true;
1156                         }
1157         } else if (is_a<power>(e1)) {
1158                 if (options & 3) // is a child or was already expanded
1159                         return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
1160                 else
1161                         try {
1162                                 return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
1163                         } catch (std::exception &p) {
1164                                 need_reevaluation = true;
1165                         }
1166         }
1167         if (need_reevaluation)
1168                 return remove_dirac_ONE(e, rl, options | 2);
1169         return e1;
1170 }
1171
1172 char clifford_max_label(const ex & e, bool ignore_ONE)
1173 {
1174         if (is_a<clifford>(e))
1175                 if (ignore_ONE && is_a<diracone>(e.op(0)))
1176                         return -1;
1177                 else
1178                         return ex_to<clifford>(e).get_representation_label();
1179         else {
1180                 char rl = -1;
1181                 for (size_t i=0; i < e.nops(); i++)
1182                         rl = (rl > clifford_max_label(e.op(i), ignore_ONE)) ? rl : clifford_max_label(e.op(i), ignore_ONE);
1183                 return rl;
1184         }
1185 }
1186
1187 ex clifford_norm(const ex & e)
1188 {
1189         return sqrt(remove_dirac_ONE(e * clifford_bar(e)));
1190 }
1191
1192 ex clifford_inverse(const ex & e)
1193 {
1194         ex norm = clifford_norm(e);
1195         if (!norm.is_zero())
1196                 return clifford_bar(e) / pow(norm, 2);
1197         else
1198                 throw(std::invalid_argument("clifford_inverse(): cannot find inverse of Clifford number with zero norm!"));
1199 }
1200
1201 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl, bool anticommuting)
1202 {
1203         if (!ex_to<idx>(mu).is_dim_numeric())
1204                 throw(std::invalid_argument("lst_to_clifford(): Index should have a numeric dimension"));
1205         ex e = clifford_unit(mu, metr, rl, anticommuting);
1206         return lst_to_clifford(v, e);
1207 }
1208
1209 ex lst_to_clifford(const ex & v, const ex & e) {
1210         unsigned min, max;
1211
1212         if (is_a<clifford>(e)) {
1213                 varidx mu = ex_to<varidx>(e.op(1));
1214                 unsigned dim = (ex_to<numeric>(mu.get_dim())).to_int();
1215
1216                 if (is_a<matrix>(v)) {
1217                         if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows()) {
1218                                 min = ex_to<matrix>(v).rows();
1219                                 max = ex_to<matrix>(v).cols();
1220                         } else {
1221                                 min = ex_to<matrix>(v).cols();
1222                                 max = ex_to<matrix>(v).rows();
1223                         }
1224                         if (min == 1) {
1225                                 if (dim == max)
1226                                         return indexed(v, ex_to<varidx>(mu).toggle_variance()) * e;
1227                                 else
1228                                         throw(std::invalid_argument("lst_to_clifford(): dimensions of vector and clifford unit mismatch"));
1229                         } else
1230                                 throw(std::invalid_argument("lst_to_clifford(): first argument should be a vector vector"));
1231                 } else if (is_a<lst>(v)) {
1232                         if (dim == ex_to<lst>(v).nops())
1233                                 return indexed(matrix(dim, 1, ex_to<lst>(v)), ex_to<varidx>(mu).toggle_variance()) * e;
1234                         else
1235                                 throw(std::invalid_argument("lst_to_clifford(): list length and dimension of clifford unit mismatch"));
1236                 } else
1237                         throw(std::invalid_argument("lst_to_clifford(): cannot construct from anything but list or vector"));
1238         } else
1239                 throw(std::invalid_argument("lst_to_clifford(): the second argument should be a Clifford unit"));
1240 }
1241
1242 /** Auxiliary structure to define a function for striping one Clifford unit
1243  * from vectors. Used in  clifford_to_lst(). */
1244 static ex get_clifford_comp(const ex & e, const ex & c)
1245 {
1246         pointer_to_map_function_1arg<const ex &> fcn(get_clifford_comp, c);
1247         int ival = ex_to<numeric>(ex_to<varidx>(c.op(1)).get_value()).to_int();
1248
1249         if (is_a<add>(e) || is_a<lst>(e) // || is_a<pseries>(e) || is_a<integral>(e)
1250                 || is_a<matrix>(e))
1251                 return e.map(fcn);
1252         else if (is_a<ncmul>(e) || is_a<mul>(e)) {
1253                 // find a Clifford unit with the same metric, delete it and substitute its index
1254                 size_t ind = e.nops() + 1;
1255                 for (size_t j = 0; j < e.nops(); j++)
1256                         if (is_a<clifford>(e.op(j)) && ex_to<clifford>(c).same_metric(e.op(j)))
1257                                 if (ind > e.nops())
1258                                         ind = j;
1259                                 else
1260                                         throw(std::invalid_argument("get_clifford_comp(): expression is a Clifford multi-vector"));
1261                 if (ind < e.nops()) {
1262                         ex S = 1;
1263                         bool same_value_index, found_dummy;
1264                         same_value_index = ( ex_to<varidx>(e.op(ind).op(1)).is_numeric()
1265                                                                  &&  (ival == ex_to<numeric>(ex_to<varidx>(e.op(ind).op(1)).get_value()).to_int()) );
1266                         found_dummy = same_value_index;
1267                         for(size_t j=0; j < e.nops(); j++)
1268                                 if (j != ind)
1269                                         if (same_value_index)
1270                                                 S = S * e.op(j);
1271                                         else {
1272                                                 exvector ind_vec = ex_to<indexed>(e.op(j)).get_dummy_indices(ex_to<indexed>(e.op(ind)));
1273                                                 if (ind_vec.size() > 0) {
1274                                                         found_dummy = true;
1275                                                         exvector::const_iterator it = ind_vec.begin(), itend = ind_vec.end();
1276                                                         while (it != itend) {
1277                                                                 S = S * e.op(j).subs(lst(ex_to<varidx>(*it) == ival, ex_to<varidx>(*it).toggle_variance() == ival), subs_options::no_pattern);
1278                                                                 ++it;
1279                                                         }
1280                                                 } else
1281                                                         S = S * e.op(j);
1282                                         }
1283                         return (found_dummy ? S : 0);
1284                 } else
1285                         throw(std::invalid_argument("get_clifford_comp(): expression is not a Clifford vector to the given units"));
1286         } else if (e.is_zero())
1287                 return e;
1288         else if (is_a<clifford>(e) && ex_to<clifford>(e).same_metric(c))
1289                 if ( ex_to<varidx>(e.op(1)).is_numeric() &&
1290                          (ival != ex_to<numeric>(ex_to<varidx>(e.op(1)).get_value()).to_int()) )
1291                         return 0;
1292                 else
1293                         return 1;
1294         else
1295                 throw(std::invalid_argument("get_clifford_comp(): expression is not usable as a Clifford vector"));
1296 }
1297
1298
1299 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic)
1300 {
1301         GINAC_ASSERT(is_a<clifford>(c));
1302         varidx mu = ex_to<varidx>(c.op(1));
1303         if (! mu.is_dim_numeric())
1304                 throw(std::invalid_argument("clifford_to_lst(): index should have a numeric dimension"));
1305         unsigned int D = ex_to<numeric>(mu.get_dim()).to_int();
1306
1307         if (algebraic) // check if algebraic method is applicable
1308                 for (unsigned int i = 0; i < D; i++)
1309                         if (pow(c.subs(mu == i, subs_options::no_pattern), 2).is_zero()
1310                                 or (not is_a<numeric>(pow(c.subs(mu == i, subs_options::no_pattern), 2))))
1311                                 algebraic = false;
1312         lst V;
1313         if (algebraic) {
1314                 for (unsigned int i = 0; i < D; i++)
1315                         V.append(remove_dirac_ONE(
1316                                                 simplify_indexed(canonicalize_clifford(e * c.subs(mu == i, subs_options::no_pattern) +  c.subs(mu == i, subs_options::no_pattern) * e))
1317                                                 / (2*pow(c.subs(mu == i, subs_options::no_pattern), 2))));
1318         } else {
1319                 ex e1 = canonicalize_clifford(e);
1320                 try {
1321                         for (unsigned int i = 0; i < D; i++)
1322                                 V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
1323                 } catch  (std::exception &p) {
1324                         /* Try to expand dummy summations to simplify the expression*/
1325                         e1 = canonicalize_clifford(expand_dummy_sum(e1, true));
1326                         for (unsigned int i = 0; i < D; i++)
1327                                 V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
1328                 }
1329         }
1330         return V;
1331 }
1332
1333
1334 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c, const ex & d, const ex & v, const ex & G, unsigned char rl, bool anticommuting)
1335 {
1336         ex x, D, cu;
1337
1338         if (! is_a<matrix>(v) && ! is_a<lst>(v))
1339                 throw(std::invalid_argument("clifford_moebius_map(): parameter v should be either vector or list"));
1340
1341         if (is_a<clifford>(G)) {
1342                 cu = G;
1343         } else {
1344                 if (is_a<indexed>(G))
1345                         D = ex_to<varidx>(G.op(1)).get_dim();
1346                 else if (is_a<matrix>(G))
1347                         D = ex_to<matrix>(G).rows();
1348                 else throw(std::invalid_argument("clifford_moebius_map(): metric should be an indexed object, matrix, or a Clifford unit"));
1349
1350                 varidx mu((new symbol)->setflag(status_flags::dynallocated), D);
1351                 cu = clifford_unit(mu, G, rl, anticommuting);
1352         }
1353
1354         x = lst_to_clifford(v, cu);
1355         ex e = simplify_indexed(canonicalize_clifford((a * x + b) * clifford_inverse(c * x + d)));
1356         return clifford_to_lst(e, cu, false);
1357 }
1358
1359 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl, bool anticommuting)
1360 {
1361         if (is_a<matrix>(M))
1362                 return clifford_moebius_map(ex_to<matrix>(M)(0,0), ex_to<matrix>(M)(0,1),
1363                                             ex_to<matrix>(M)(1,0), ex_to<matrix>(M)(1,1), v, G, rl, anticommuting);
1364         else
1365                 throw(std::invalid_argument("clifford_moebius_map(): parameter M should be a matrix"));
1366 }
1367
1368 } // namespace GiNaC