ginac/clifford.cpp

   1 /** @file clifford.cpp
   2  *
   3  *  Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
   7  *
   8  *  This program is free software; you can redistribute it and/or modify
   9  *  it under the terms of the GNU General Public License as published by
  10  *  the Free Software Foundation; either version 2 of the License, or
  11  *  (at your option) any later version.
  12  *
  13  *  This program is distributed in the hope that it will be useful,
  14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  16  *  GNU General Public License for more details.
  17  *
  18  *  You should have received a copy of the GNU General Public License
  19  *  along with this program; if not, write to the Free Software
  20  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  21  */
  22
  23 #include "clifford.h"
  24 #include "ex.h"
  25 #include "idx.h"
  26 #include "ncmul.h"
  27 #include "print.h"
  28 #include "archive.h"
  29 #include "debugmsg.h"
  30 #include "utils.h"
  31
  32 #include <stdexcept>
  33
  34 namespace GiNaC {
  35
  36 GINAC_IMPLEMENT_REGISTERED_CLASS(clifford, indexed)
  37 GINAC_IMPLEMENT_REGISTERED_CLASS(diracone, tensor)
  38 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma, tensor)
  39 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma5, tensor)
  40
  41 //////////
  42 // default constructor, destructor, copy constructor assignment operator and helpers
  43 //////////
  44
  45 clifford::clifford() : representation_label(0)
  46 {
  47         debugmsg("clifford default constructor", LOGLEVEL_CONSTRUCT);
  48         tinfo_key = TINFO_clifford;
  49 }
  50
  51 void clifford::copy(const clifford & other)
  52 {
  53         inherited::copy(other);
  54         representation_label = other.representation_label;
  55 }
  56
  57 DEFAULT_DESTROY(clifford)
  58 DEFAULT_CTORS(diracone)
  59 DEFAULT_CTORS(diracgamma)
  60 DEFAULT_CTORS(diracgamma5)
  61
  62 //////////
  63 // other constructors
  64 //////////
  65
  66 /** Construct object without any indices. This constructor is for internal
  67  *  use only. Use the dirac_ONE() function instead.
  68  *  @see dirac_ONE */
  69 clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
  70 {
  71         debugmsg("clifford constructor from ex", LOGLEVEL_CONSTRUCT);
  72         tinfo_key = TINFO_clifford;
  73 }
  74
  75 /** Construct object with one Lorentz index. This constructor is for internal
  76  *  use only. Use the dirac_gamma() function instead.
  77  *  @see dirac_gamma */
  78 clifford::clifford(const ex & b, const ex & mu, unsigned char rl) : inherited(b, mu), representation_label(rl)
  79 {
  80         debugmsg("clifford constructor from ex,ex", LOGLEVEL_CONSTRUCT);
  81         GINAC_ASSERT(is_ex_of_type(mu, varidx));
  82         tinfo_key = TINFO_clifford;
  83 }
  84
  85 clifford::clifford(unsigned char rl, const exvector & v, bool discardable) : inherited(indexed::unknown, v, discardable), representation_label(rl)
  86 {
  87         debugmsg("clifford constructor from unsigned char,exvector", LOGLEVEL_CONSTRUCT);
  88         tinfo_key = TINFO_clifford;
  89 }
  90
  91 clifford::clifford(unsigned char rl, exvector * vp) : inherited(indexed::unknown, vp), representation_label(rl)
  92 {
  93         debugmsg("clifford constructor from unsigned char,exvector *", LOGLEVEL_CONSTRUCT);
  94         tinfo_key = TINFO_clifford;
  95 }
  96
  97 //////////
  98 // archiving
  99 //////////
 100
 101 clifford::clifford(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 102 {
 103         debugmsg("clifford constructor from archive_node", LOGLEVEL_CONSTRUCT);
 104         unsigned rl;
 105         n.find_unsigned("label", rl);
 106         representation_label = rl;
 107 }
 108
 109 void clifford::archive(archive_node &n) const
 110 {
 111         inherited::archive(n);
 112         n.add_unsigned("label", representation_label);
 113 }
 114
 115 DEFAULT_UNARCHIVE(clifford)
 116 DEFAULT_ARCHIVING(diracone)
 117 DEFAULT_ARCHIVING(diracgamma)
 118 DEFAULT_ARCHIVING(diracgamma5)
 119
 120 //////////
 121 // functions overriding virtual functions from bases classes
 122 //////////
 123
 124 int clifford::compare_same_type(const basic & other) const
 125 {
 126         GINAC_ASSERT(other.tinfo() == TINFO_clifford);
 127         const clifford &o = static_cast<const clifford &>(other);
 128
 129         if (representation_label != o.representation_label) {
 130                 // different representation label
 131                 return representation_label < o.representation_label ? -1 : 1;
 132         }
 133
 134         return inherited::compare_same_type(other);
 135 }
 136
 137 DEFAULT_COMPARE(diracone)
 138 DEFAULT_COMPARE(diracgamma)
 139 DEFAULT_COMPARE(diracgamma5)
 140
 141 DEFAULT_PRINT(diracone, "ONE")
 142 DEFAULT_PRINT(diracgamma, "gamma")
 143 DEFAULT_PRINT(diracgamma5, "gamma5")
 144
 145 /** Contraction of a gamma matrix with something else. */
 146 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 147 {
 148         GINAC_ASSERT(is_ex_of_type(*self, clifford));
 149         GINAC_ASSERT(is_ex_of_type(*other, indexed));
 150         GINAC_ASSERT(is_ex_of_type(self->op(0), diracgamma));
 151         unsigned char rl = ex_to_clifford(*self).get_representation_label();
 152
 153         if (is_ex_of_type(other->op(0), diracgamma)) {
 154
 155                 ex dim = ex_to_idx(self->op(1)).get_dim();
 156
 157                 // gamma~mu*gamma.mu = dim*ONE
 158                 if (other - self == 1) {
 159                         *self = dim;
 160                         *other = dirac_ONE(rl);
 161                         return true;
 162
 163                 // gamma~mu*gamma~alpha*gamma.mu = (2-dim)*gamma~alpha
 164                 } else if (other - self == 2
 165                         && is_ex_of_type(self[1], clifford)) {
 166                         *self = 2 - dim;
 167                         *other = _ex1();
 168                         return true;
 169
 170                 // gamma~mu*gamma~alpha*gamma~beta*gamma.mu = 4*g~alpha~beta+(dim-4)*gamam~alpha*gamma~beta
 171                 } else if (other - self == 3
 172                         && is_ex_of_type(self[1], clifford)
 173                         && is_ex_of_type(self[2], clifford)) {
 174                         *self = 4 * metric_tensor(self[1].op(1), self[2].op(1)) * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
 175                         self[1] = _ex1();
 176                         self[2] = _ex1();
 177                         *other = _ex1();
 178                         return true;
 179
 180                 // gamma~mu*gamma~alpha*gamma~beta*gamma~delta*gamma.mu = -2*gamma~delta*gamma~beta*gamma~alpha+(4-dim)*gamma~alpha*gamma~beta*gamma~delta
 181                 } else if (other - self == 4
 182                         && is_ex_of_type(self[1], clifford)
 183                         && is_ex_of_type(self[2], clifford)
 184                         && is_ex_of_type(self[3], clifford)) {
 185                         *self = -2 * self[3] * self[2] * self[1] + (4 - dim) * self[1] * self[2] * self[3];
 186                         self[1] = _ex1();
 187                         self[2] = _ex1();
 188                         self[3] = _ex1();
 189                         *other = _ex1();
 190                         return true;
 191                 }
 192         }
 193
 194         return false;
 195 }
 196
 197 /** Perform automatic simplification on noncommutative product of clifford
 198  *  objects. This removes superfluous ONEs, permutes gamma5's to the front
 199  *  and removes squares of gamma objects. */
 200 ex clifford::simplify_ncmul(const exvector & v) const
 201 {
 202         exvector s;
 203         s.reserve(v.size());
 204         unsigned rl = ex_to_clifford(v[0]).get_representation_label();
 205
 206         // Remove superfluous ONEs
 207         exvector::const_iterator cit = v.begin(), citend = v.end();
 208         while (cit != citend) {
 209                 if (!is_ex_of_type(cit->op(0), diracone))
 210                         s.push_back(*cit);
 211                 cit++;
 212         }
 213
 214         bool something_changed = false;
 215         int sign = 1;
 216
 217         // Anticommute gamma5's to the front
 218         if (s.size() >= 2) {
 219                 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
 220                 while (true) {
 221                         exvector::iterator it = next_to_last;
 222                         while (true) {
 223                                 exvector::iterator it2 = it + 1;
 224                                 if (!is_ex_of_type(it->op(0), diracgamma5) && is_ex_of_type(it2->op(0), diracgamma5)) {
 225                                         it->swap(*it2);
 226                                         sign = -sign;
 227                                         something_changed = true;
 228                                 }
 229                                 if (it == first)
 230                                         break;
 231                                 it--;
 232                         }
 233                         if (next_to_last == first)
 234                                 break;
 235                         next_to_last--;
 236                 }
 237         }
 238
 239         // Remove squares of gamma5
 240         while (s.size() >= 2 && is_ex_of_type(s[0].op(0), diracgamma5) && is_ex_of_type(s[1].op(0), diracgamma5)) {
 241                 s.erase(s.begin(), s.begin() + 2);
 242                 something_changed = true;
 243         }
 244
 245         // Remove equal adjacent gammas
 246         if (s.size() >= 2) {
 247                 exvector::iterator it = s.begin(), itend = s.end() - 1;
 248                 while (it != itend) {
 249                         ex & a = it[0];
 250                         ex & b = it[1];
 251                         if (is_ex_of_type(a.op(0), diracgamma) && is_ex_of_type(b.op(0), diracgamma)) {
 252                                 const ex & ia = a.op(1);
 253                                 const ex & ib = b.op(1);
 254                                 if (ia.is_equal(ib)) {
 255                                         a = lorentz_g(ia, ib);
 256                                         b = dirac_ONE(rl);
 257                                         something_changed = true;
 258                                 }
 259                         }
 260                         it++;
 261                 }
 262         }
 263
 264         if (s.size() == 0)
 265                 return clifford(diracone(), rl) * sign;
 266         if (something_changed)
 267                 return nonsimplified_ncmul(s) * sign;
 268         else
 269                 return simplified_ncmul(s) * sign;
 270 }
 271
 272 ex clifford::thisexprseq(const exvector & v) const
 273 {
 274         return clifford(representation_label, v);
 275 }
 276
 277 ex clifford::thisexprseq(exvector * vp) const
 278 {
 279         return clifford(representation_label, vp);
 280 }
 281
 282 //////////
 283 // global functions
 284 //////////
 285
 286 ex dirac_ONE(unsigned char rl)
 287 {
 288         return clifford(diracone(), rl);
 289 }
 290
 291 ex dirac_gamma(const ex & mu, unsigned char rl)
 292 {
 293         if (!is_ex_of_type(mu, varidx))
 294                 throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
 295
 296         return clifford(diracgamma(), mu, rl);
 297 }
 298
 299 ex dirac_gamma5(unsigned char rl)
 300 {
 301         return clifford(diracgamma5(), rl);
 302 }
 303
 304 ex dirac_trace(const ex & e, unsigned char rl = 0)
 305 {
 306         if (is_ex_of_type(e, clifford)) {
 307
 308                 if (ex_to_clifford(e).get_representation_label() == rl
 309                  && is_ex_of_type(e.op(0), diracone))
 310                         return _ex4();
 311                 else
 312                         return _ex0();
 313
 314         } else if (is_ex_exactly_of_type(e, add)) {
 315
 316                 // Trace of sum = sum of traces
 317                 ex sum = _ex0();
 318                 for (unsigned i=0; i<e.nops(); i++)
 319                         sum += dirac_trace(e.op(i), rl);
 320                 return sum;
 321
 322         } else if (is_ex_exactly_of_type(e, mul)) {
 323
 324                 // Trace of product: pull out non-clifford factors
 325                 ex prod = _ex1();
 326                 for (unsigned i=0; i<e.nops(); i++) {
 327                         const ex &o = e.op(i);
 328                         if (is_ex_of_type(o, clifford)
 329                          && ex_to_clifford(o).get_representation_label() == rl)
 330                                 prod *= dirac_trace(o, rl);
 331                         else if (is_ex_of_type(o, ncmul)
 332                          && is_ex_of_type(o.op(0), clifford)
 333                          && ex_to_clifford(o.op(0)).get_representation_label() == rl)
 334                                 prod *= dirac_trace(o, rl);
 335                         else
 336                                 prod *= o;
 337                 }
 338                 return prod;
 339
 340         } else if (is_ex_exactly_of_type(e, ncmul)) {
 341
 342                 if (!is_ex_of_type(e.op(0), clifford)
 343                  || ex_to_clifford(e.op(0)).get_representation_label() != rl)
 344                         return _ex0();
 345
 346                 // gamma5 gets moved to the front so this check is enough
 347                 bool has_gamma5 = is_ex_of_type(e.op(0).op(0), diracgamma5);
 348                 unsigned num = e.nops();
 349
 350                 if (has_gamma5) {
 351
 352                         // Trace of gamma5 * odd number of gammas and trace of
 353                         // gamma5 * gamma_mu * gamma_nu are zero
 354                         if ((num & 1) == 0 || num == 2)
 355                                 return _ex0();
 356
 357                 } else { // no gamma5
 358
 359                         // Trace of odd number of gammas is zero
 360                         if ((num & 1) == 1)
 361                                 return _ex0();
 362
 363                         // Tr gamma_mu gamma_nu = 4 g_mu_nu
 364                         if (num == 2)
 365                                 return 4 * lorentz_g(e.op(0).op(1), e.op(1).op(1));
 366                 }
 367
 368                 throw (std::logic_error("dirac_trace: don't know how to compute trace"));
 369         }
 370
 371         return _ex0();
 372 }
 373
 374 } // namespace GiNaC