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 "symbol.h"
  28 #include "print.h"
  29 #include "archive.h"
  30 #include "debugmsg.h"
  31 #include "utils.h"
  32
  33 #include <stdexcept>
  34
  35 namespace GiNaC {
  36
  37 GINAC_IMPLEMENT_REGISTERED_CLASS(clifford, indexed)
  38 GINAC_IMPLEMENT_REGISTERED_CLASS(diracone, tensor)
  39 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma, tensor)
  40 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma5, tensor)
  41
  42 //////////
  43 // default constructor, destructor, copy constructor assignment operator and helpers
  44 //////////
  45
  46 clifford::clifford() : representation_label(0)
  47 {
  48         debugmsg("clifford default constructor", LOGLEVEL_CONSTRUCT);
  49         tinfo_key = TINFO_clifford;
  50 }
  51
  52 void clifford::copy(const clifford & other)
  53 {
  54         inherited::copy(other);
  55         representation_label = other.representation_label;
  56 }
  57
  58 DEFAULT_DESTROY(clifford)
  59 DEFAULT_CTORS(diracone)
  60 DEFAULT_CTORS(diracgamma)
  61 DEFAULT_CTORS(diracgamma5)
  62
  63 //////////
  64 // other constructors
  65 //////////
  66
  67 /** Construct object without any indices. This constructor is for internal
  68  *  use only. Use the dirac_ONE() function instead.
  69  *  @see dirac_ONE */
  70 clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
  71 {
  72         debugmsg("clifford constructor from ex", LOGLEVEL_CONSTRUCT);
  73         tinfo_key = TINFO_clifford;
  74 }
  75
  76 /** Construct object with one Lorentz index. This constructor is for internal
  77  *  use only. Use the dirac_gamma() function instead.
  78  *  @see dirac_gamma */
  79 clifford::clifford(const ex & b, const ex & mu, unsigned char rl) : inherited(b, mu), representation_label(rl)
  80 {
  81         debugmsg("clifford constructor from ex,ex", LOGLEVEL_CONSTRUCT);
  82         GINAC_ASSERT(is_ex_of_type(mu, varidx));
  83         tinfo_key = TINFO_clifford;
  84 }
  85
  86 clifford::clifford(unsigned char rl, const exvector & v, bool discardable) : inherited(indexed::unknown, v, discardable), representation_label(rl)
  87 {
  88         debugmsg("clifford constructor from unsigned char,exvector", LOGLEVEL_CONSTRUCT);
  89         tinfo_key = TINFO_clifford;
  90 }
  91
  92 clifford::clifford(unsigned char rl, exvector * vp) : inherited(indexed::unknown, vp), representation_label(rl)
  93 {
  94         debugmsg("clifford constructor from unsigned char,exvector *", LOGLEVEL_CONSTRUCT);
  95         tinfo_key = TINFO_clifford;
  96 }
  97
  98 //////////
  99 // archiving
 100 //////////
 101
 102 clifford::clifford(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 103 {
 104         debugmsg("clifford constructor from archive_node", LOGLEVEL_CONSTRUCT);
 105         unsigned rl;
 106         n.find_unsigned("label", rl);
 107         representation_label = rl;
 108 }
 109
 110 void clifford::archive(archive_node &n) const
 111 {
 112         inherited::archive(n);
 113         n.add_unsigned("label", representation_label);
 114 }
 115
 116 DEFAULT_UNARCHIVE(clifford)
 117 DEFAULT_ARCHIVING(diracone)
 118 DEFAULT_ARCHIVING(diracgamma)
 119 DEFAULT_ARCHIVING(diracgamma5)
 120
 121 //////////
 122 // functions overriding virtual functions from bases classes
 123 //////////
 124
 125 int clifford::compare_same_type(const basic & other) const
 126 {
 127         GINAC_ASSERT(other.tinfo() == TINFO_clifford);
 128         const clifford &o = static_cast<const clifford &>(other);
 129
 130         if (representation_label != o.representation_label) {
 131                 // different representation label
 132                 return representation_label < o.representation_label ? -1 : 1;
 133         }
 134
 135         return inherited::compare_same_type(other);
 136 }
 137
 138 DEFAULT_COMPARE(diracone)
 139 DEFAULT_COMPARE(diracgamma)
 140 DEFAULT_COMPARE(diracgamma5)
 141
 142 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
 143 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
 144 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
 145
 146 /** Contraction of a gamma matrix with something else. */
 147 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 148 {
 149         GINAC_ASSERT(is_ex_of_type(*self, clifford));
 150         GINAC_ASSERT(is_ex_of_type(*other, indexed));
 151         GINAC_ASSERT(is_ex_of_type(self->op(0), diracgamma));
 152         unsigned char rl = ex_to_clifford(*self).get_representation_label();
 153
 154         if (is_ex_of_type(*other, clifford)) {
 155
 156                 ex dim = ex_to_idx(self->op(1)).get_dim();
 157
 158                 // gamma~mu gamma.mu = dim ONE
 159                 if (other - self == 1) {
 160                         *self = dim;
 161                         *other = dirac_ONE(rl);
 162                         return true;
 163
 164                 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
 165                 } else if (other - self == 2
 166                         && is_ex_of_type(self[1], clifford)) {
 167                         *self = 2 - dim;
 168                         *other = _ex1();
 169                         return true;
 170
 171                 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
 172                 } else if (other - self == 3
 173                         && is_ex_of_type(self[1], clifford)
 174                         && is_ex_of_type(self[2], clifford)) {
 175                         *self = 4 * lorentz_g(self[1].op(1), self[2].op(1)) * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
 176                         self[1] = _ex1();
 177                         self[2] = _ex1();
 178                         *other = _ex1();
 179                         return true;
 180
 181 #if 0
 182                 // 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
 183                 } else if (other - self == 4
 184                         && is_ex_of_type(self[1], clifford)
 185                         && is_ex_of_type(self[2], clifford)
 186                         && is_ex_of_type(self[3], clifford)) {
 187                         *self = -2 * self[3] * self[2] * self[1] + (4 - dim) * self[1] * self[2] * self[3];
 188                         self[1] = _ex1();
 189                         self[2] = _ex1();
 190                         self[3] = _ex1();
 191                         *other = _ex1();
 192                         return true;
 193 #endif
 194
 195                 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
 196                 // (commutate contracted indices towards each other, simplify_indexed()
 197                 // will re-expand and re-run the simplification)
 198                 } else {
 199                         exvector::iterator it = self + 1, next_to_last = other - 1;
 200                         while (it != other) {
 201                                 if (!is_ex_of_type(*it, clifford))
 202                                         return false;
 203                                 it++;
 204                         }
 205
 206                         it = self + 1;
 207                         ex S = _ex1();
 208                         while (it != next_to_last) {
 209                                 S *= *it;
 210                                 *it++ = _ex1();
 211                         }
 212
 213                         *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
 214                         *next_to_last = _ex1();
 215                         *other = _ex1();
 216                         return true;
 217                 }
 218         }
 219
 220         return false;
 221 }
 222
 223 /** Perform automatic simplification on noncommutative product of clifford
 224  *  objects. This removes superfluous ONEs, permutes gamma5's to the front
 225  *  and removes squares of gamma objects. */
 226 ex clifford::simplify_ncmul(const exvector & v) const
 227 {
 228         exvector s;
 229         s.reserve(v.size());
 230         unsigned rl = ex_to_clifford(v[0]).get_representation_label();
 231
 232         // Remove superfluous ONEs
 233         exvector::const_iterator cit = v.begin(), citend = v.end();
 234         while (cit != citend) {
 235                 if (!is_ex_of_type(cit->op(0), diracone))
 236                         s.push_back(*cit);
 237                 cit++;
 238         }
 239
 240         bool something_changed = false;
 241         int sign = 1;
 242
 243         // Anticommute gamma5's to the front
 244         if (s.size() >= 2) {
 245                 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
 246                 while (true) {
 247                         exvector::iterator it = next_to_last;
 248                         while (true) {
 249                                 exvector::iterator it2 = it + 1;
 250                                 if (!is_ex_of_type(it->op(0), diracgamma5) && is_ex_of_type(it2->op(0), diracgamma5)) {
 251                                         it->swap(*it2);
 252                                         sign = -sign;
 253                                         something_changed = true;
 254                                 }
 255                                 if (it == first)
 256                                         break;
 257                                 it--;
 258                         }
 259                         if (next_to_last == first)
 260                                 break;
 261                         next_to_last--;
 262                 }
 263         }
 264
 265         // Remove squares of gamma5
 266         while (s.size() >= 2 && is_ex_of_type(s[0].op(0), diracgamma5) && is_ex_of_type(s[1].op(0), diracgamma5)) {
 267                 s.erase(s.begin(), s.begin() + 2);
 268                 something_changed = true;
 269         }
 270
 271         // Remove equal adjacent gammas
 272         if (s.size() >= 2) {
 273                 exvector::iterator it = s.begin(), itend = s.end() - 1;
 274                 while (it != itend) {
 275                         ex & a = it[0];
 276                         ex & b = it[1];
 277                         if (is_ex_of_type(a.op(0), diracgamma) && is_ex_of_type(b.op(0), diracgamma)) {
 278                                 const ex & ia = a.op(1);
 279                                 const ex & ib = b.op(1);
 280                                 if (ia.is_equal(ib)) {
 281                                         a = lorentz_g(ia, ib);
 282                                         b = dirac_ONE(rl);
 283                                         something_changed = true;
 284                                 }
 285                         }
 286                         it++;
 287                 }
 288         }
 289
 290         if (s.size() == 0)
 291                 return clifford(diracone(), rl) * sign;
 292         if (something_changed)
 293                 return nonsimplified_ncmul(s) * sign;
 294         else
 295                 return simplified_ncmul(s) * sign;
 296 }
 297
 298 ex clifford::thisexprseq(const exvector & v) const
 299 {
 300         return clifford(representation_label, v);
 301 }
 302
 303 ex clifford::thisexprseq(exvector * vp) const
 304 {
 305         return clifford(representation_label, vp);
 306 }
 307
 308 //////////
 309 // global functions
 310 //////////
 311
 312 ex dirac_ONE(unsigned char rl)
 313 {
 314         return clifford(diracone(), rl);
 315 }
 316
 317 ex dirac_gamma(const ex & mu, unsigned char rl)
 318 {
 319         if (!is_ex_of_type(mu, varidx))
 320                 throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
 321
 322         return clifford(diracgamma(), mu, rl);
 323 }
 324
 325 ex dirac_gamma5(unsigned char rl)
 326 {
 327         return clifford(diracgamma5(), rl);
 328 }
 329
 330 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
 331 {
 332         varidx mu((new symbol)->setflag(status_flags::dynallocated), dim);
 333         return indexed(e, mu.toggle_variance()) * dirac_gamma(mu, rl);
 334 }
 335
 336 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 337  *  is that of a clifford object with the specified representation label. */
 338 static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
 339 {
 340         return ti == (TINFO_clifford + rl);
 341 }
 342
 343 ex dirac_trace(const ex & e, unsigned char rl)
 344 {
 345         if (is_ex_of_type(e, clifford)) {
 346
 347                 if (ex_to_clifford(e).get_representation_label() == rl
 348                  && is_ex_of_type(e.op(0), diracone))
 349                         return _ex4();
 350                 else
 351                         return _ex0();
 352
 353         } else if (is_ex_exactly_of_type(e, add)) {
 354
 355                 // Trace of sum = sum of traces
 356                 ex sum = _ex0();
 357                 for (unsigned i=0; i<e.nops(); i++)
 358                         sum += dirac_trace(e.op(i), rl);
 359                 return sum;
 360
 361         } else if (is_ex_exactly_of_type(e, mul)) {
 362
 363                 // Trace of product: pull out non-clifford factors
 364                 ex prod = _ex1();
 365                 for (unsigned i=0; i<e.nops(); i++) {
 366                         const ex &o = e.op(i);
 367                         unsigned ti = o.return_type_tinfo();
 368                         if (is_clifford_tinfo(o.return_type_tinfo(), rl))
 369                                 prod *= dirac_trace(o, rl);
 370                         else
 371                                 prod *= o;
 372                 }
 373                 return prod;
 374
 375         } else if (is_ex_exactly_of_type(e, ncmul)) {
 376
 377                 if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
 378                         return _ex0();
 379
 380                 // Expand product, if necessary
 381                 ex e_expanded = e.expand();
 382                 if (!is_ex_of_type(e_expanded, ncmul))
 383                         return dirac_trace(e_expanded, rl);
 384
 385                 // gamma5 gets moved to the front so this check is enough
 386                 bool has_gamma5 = is_ex_of_type(e.op(0).op(0), diracgamma5);
 387                 unsigned num = e.nops();
 388
 389                 if (has_gamma5) {
 390
 391                         // Trace of gamma5 * odd number of gammas and trace of
 392                         // gamma5 * gamma.mu * gamma.nu are zero
 393                         if ((num & 1) == 0 || num == 3)
 394                                 return _ex0();
 395
 396                         // Tr gamma5 S_2k =
 397                         //   epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
 398                         ex dim = ex_to_idx(e.op(1).op(1)).get_dim();
 399                         varidx mu1((new symbol)->setflag(status_flags::dynallocated), dim),
 400                                mu2((new symbol)->setflag(status_flags::dynallocated), dim),
 401                                mu3((new symbol)->setflag(status_flags::dynallocated), dim),
 402                                mu4((new symbol)->setflag(status_flags::dynallocated), dim);
 403                         exvector v;
 404                         v.reserve(num + 3);
 405                         v.push_back(dirac_gamma(mu1, rl));
 406                         v.push_back(dirac_gamma(mu2, rl));
 407                         v.push_back(dirac_gamma(mu3, rl));
 408                         v.push_back(dirac_gamma(mu4, rl));
 409                         for (int i=1; i<num; i++)
 410                                 v.push_back(e.op(i));
 411
 412                         return (eps0123(mu1.toggle_variance(), mu2.toggle_variance(), mu3.toggle_variance(), mu4.toggle_variance()) *
 413                                 dirac_trace(ncmul(v), rl)).simplify_indexed() / 24;
 414
 415                 } else { // no gamma5
 416
 417                         // Trace of odd number of gammas is zero
 418                         if ((num & 1) == 1)
 419                                 return _ex0();
 420
 421                         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 422                         if (num == 2)
 423                                 return 4 * lorentz_g(e.op(0).op(1), e.op(1).op(1));
 424
 425                         // Traces of 4 or more gammas are computed recursively:
 426                         // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
 427                         //   + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
 428                         //   - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
 429                         //   + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
 430                         //   - ...
 431                         //   + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
 432                         exvector v(num - 2);
 433                         int sign = 1;
 434                         const ex &ix1 = e.op(0).op(1);
 435                         ex result;
 436                         for (int i=1; i<num; i++) {
 437                                 for (int n=1, j=0; n<num; n++) {
 438                                         if (n == i)
 439                                                 continue;
 440                                         v[j++] = e.op(n);
 441                                 }
 442                                 result += sign * lorentz_g(ix1, e.op(i).op(1)) * dirac_trace(ncmul(v), rl);
 443                                 sign = -sign;
 444                         }
 445                         return result;
 446                 }
 447         }
 448
 449         return _ex0();
 450 }
 451
 452 } // namespace GiNaC