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(diracone, "ONE")
 143 DEFAULT_PRINT(diracgamma, "gamma")
 144 DEFAULT_PRINT(diracgamma5, "gamma5")
 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->op(0), diracgamma)) {
 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 * metric_tensor(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                 // 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
 182                 } else if (other - self == 4
 183                         && is_ex_of_type(self[1], clifford)
 184                         && is_ex_of_type(self[2], clifford)
 185                         && is_ex_of_type(self[3], clifford)) {
 186                         *self = -2 * self[3] * self[2] * self[1] + (4 - dim) * self[1] * self[2] * self[3];
 187                         self[1] = _ex1();
 188                         self[2] = _ex1();
 189                         self[3] = _ex1();
 190                         *other = _ex1();
 191                         return true;
 192                 }
 193         }
 194
 195         return false;
 196 }
 197
 198 /** Perform automatic simplification on noncommutative product of clifford
 199  *  objects. This removes superfluous ONEs, permutes gamma5's to the front
 200  *  and removes squares of gamma objects. */
 201 ex clifford::simplify_ncmul(const exvector & v) const
 202 {
 203         exvector s;
 204         s.reserve(v.size());
 205         unsigned rl = ex_to_clifford(v[0]).get_representation_label();
 206
 207         // Remove superfluous ONEs
 208         exvector::const_iterator cit = v.begin(), citend = v.end();
 209         while (cit != citend) {
 210                 if (!is_ex_of_type(cit->op(0), diracone))
 211                         s.push_back(*cit);
 212                 cit++;
 213         }
 214
 215         bool something_changed = false;
 216         int sign = 1;
 217
 218         // Anticommute gamma5's to the front
 219         if (s.size() >= 2) {
 220                 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
 221                 while (true) {
 222                         exvector::iterator it = next_to_last;
 223                         while (true) {
 224                                 exvector::iterator it2 = it + 1;
 225                                 if (!is_ex_of_type(it->op(0), diracgamma5) && is_ex_of_type(it2->op(0), diracgamma5)) {
 226                                         it->swap(*it2);
 227                                         sign = -sign;
 228                                         something_changed = true;
 229                                 }
 230                                 if (it == first)
 231                                         break;
 232                                 it--;
 233                         }
 234                         if (next_to_last == first)
 235                                 break;
 236                         next_to_last--;
 237                 }
 238         }
 239
 240         // Remove squares of gamma5
 241         while (s.size() >= 2 && is_ex_of_type(s[0].op(0), diracgamma5) && is_ex_of_type(s[1].op(0), diracgamma5)) {
 242                 s.erase(s.begin(), s.begin() + 2);
 243                 something_changed = true;
 244         }
 245
 246         // Remove equal adjacent gammas
 247         if (s.size() >= 2) {
 248                 exvector::iterator it = s.begin(), itend = s.end() - 1;
 249                 while (it != itend) {
 250                         ex & a = it[0];
 251                         ex & b = it[1];
 252                         if (is_ex_of_type(a.op(0), diracgamma) && is_ex_of_type(b.op(0), diracgamma)) {
 253                                 const ex & ia = a.op(1);
 254                                 const ex & ib = b.op(1);
 255                                 if (ia.is_equal(ib)) {
 256                                         a = lorentz_g(ia, ib);
 257                                         b = dirac_ONE(rl);
 258                                         something_changed = true;
 259                                 }
 260                         }
 261                         it++;
 262                 }
 263         }
 264
 265         if (s.size() == 0)
 266                 return clifford(diracone(), rl) * sign;
 267         if (something_changed)
 268                 return nonsimplified_ncmul(s) * sign;
 269         else
 270                 return simplified_ncmul(s) * sign;
 271 }
 272
 273 ex clifford::thisexprseq(const exvector & v) const
 274 {
 275         return clifford(representation_label, v);
 276 }
 277
 278 ex clifford::thisexprseq(exvector * vp) const
 279 {
 280         return clifford(representation_label, vp);
 281 }
 282
 283 //////////
 284 // global functions
 285 //////////
 286
 287 ex dirac_ONE(unsigned char rl)
 288 {
 289         return clifford(diracone(), rl);
 290 }
 291
 292 ex dirac_gamma(const ex & mu, unsigned char rl)
 293 {
 294         if (!is_ex_of_type(mu, varidx))
 295                 throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
 296
 297         return clifford(diracgamma(), mu, rl);
 298 }
 299
 300 ex dirac_gamma5(unsigned char rl)
 301 {
 302         return clifford(diracgamma5(), rl);
 303 }
 304
 305 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
 306 {
 307         varidx mu((new symbol)->setflag(status_flags::dynallocated), dim);
 308         return indexed(e, mu.toggle_variance()) * dirac_gamma(mu, rl);
 309 }
 310
 311 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 312  *  is that of a clifford object with the specified representation label. */
 313 static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
 314 {
 315         return ti == (TINFO_clifford + rl);
 316 }
 317
 318 ex dirac_trace(const ex & e, unsigned char rl)
 319 {
 320         if (is_ex_of_type(e, clifford)) {
 321
 322                 if (ex_to_clifford(e).get_representation_label() == rl
 323                  && is_ex_of_type(e.op(0), diracone))
 324                         return _ex4();
 325                 else
 326                         return _ex0();
 327
 328         } else if (is_ex_exactly_of_type(e, add)) {
 329
 330                 // Trace of sum = sum of traces
 331                 ex sum = _ex0();
 332                 for (unsigned i=0; i<e.nops(); i++)
 333                         sum += dirac_trace(e.op(i), rl);
 334                 return sum;
 335
 336         } else if (is_ex_exactly_of_type(e, mul)) {
 337
 338                 // Trace of product: pull out non-clifford factors
 339                 ex prod = _ex1();
 340                 for (unsigned i=0; i<e.nops(); i++) {
 341                         const ex &o = e.op(i);
 342                         unsigned ti = o.return_type_tinfo();
 343                         if (is_clifford_tinfo(o.return_type_tinfo(), rl))
 344                                 prod *= dirac_trace(o, rl);
 345                         else
 346                                 prod *= o;
 347                 }
 348                 return prod;
 349
 350         } else if (is_ex_exactly_of_type(e, ncmul)) {
 351
 352                 if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
 353                         return _ex0();
 354
 355                 // Expand product, if necessary
 356                 ex e_expanded = e.expand();
 357                 if (!is_ex_of_type(e_expanded, ncmul))
 358                         return dirac_trace(e_expanded, rl);
 359
 360                 // gamma5 gets moved to the front so this check is enough
 361                 bool has_gamma5 = is_ex_of_type(e.op(0).op(0), diracgamma5);
 362                 unsigned num = e.nops();
 363
 364                 if (has_gamma5) {
 365
 366                         // Trace of gamma5 * odd number of gammas and trace of
 367                         // gamma5 * gamma_mu * gamma_nu are zero
 368                         if ((num & 1) == 0 || num == 3)
 369                                 return _ex0();
 370
 371                         // Tr gamma5 S_2k =
 372                         //   epsilon0123_mu1_mu2_mu3_mu4 * Tr gamma_mu1 gamma_mu2 gamma_mu3 gamma_mu4 S_2k
 373                         ex dim = ex_to_idx(e.op(1).op(1)).get_dim();
 374                         varidx mu1((new symbol)->setflag(status_flags::dynallocated), dim),
 375                                mu2((new symbol)->setflag(status_flags::dynallocated), dim),
 376                                mu3((new symbol)->setflag(status_flags::dynallocated), dim),
 377                                mu4((new symbol)->setflag(status_flags::dynallocated), dim);
 378                         exvector v;
 379                         v.reserve(num + 3);
 380                         v.push_back(dirac_gamma(mu1, rl));
 381                         v.push_back(dirac_gamma(mu2, rl));
 382                         v.push_back(dirac_gamma(mu3, rl));
 383                         v.push_back(dirac_gamma(mu4, rl));
 384                         for (int i=1; i<num; i++)
 385                                 v.push_back(e.op(i));
 386
 387                         return (eps0123(mu1.toggle_variance(), mu2.toggle_variance(), mu3.toggle_variance(), mu4.toggle_variance()) *
 388                                 dirac_trace(ncmul(v), rl)).simplify_indexed() / 24;
 389
 390                 } else { // no gamma5
 391
 392                         // Trace of odd number of gammas is zero
 393                         if ((num & 1) == 1)
 394                                 return _ex0();
 395
 396                         // Tr gamma_mu gamma_nu = 4 g_mu_nu
 397                         if (num == 2)
 398                                 return 4 * lorentz_g(e.op(0).op(1), e.op(1).op(1));
 399
 400                         // Traces of 4 or more gammas are computed recursively:
 401                         // Tr gamma_mu1 gamma_mu2 ... gamma_mun =
 402                         //   + eta_mu1_mu2 * Tr gamma_mu3 ... gamma_mun
 403                         //   - eta_mu1_mu3 * Tr gamma_mu2 gamma_mu4 ... gamma_mun
 404                         //   + eta_mu1_mu4 * Tr gamma_mu3 gamma_mu3 gamma_mu5 ... gamma_mun
 405                         //   - ...
 406                         //   + eta_mu1_mun * Tr gamma_mu2 ... gamma_mu(n-1)
 407                         exvector v(num - 2);
 408                         int sign = 1;
 409                         const ex &ix1 = e.op(0).op(1);
 410                         ex result;
 411                         for (int i=1; i<num; i++) {
 412                                 for (int n=1, j=0; n<num; n++) {
 413                                         if (n == i)
 414                                                 continue;
 415                                         v[j++] = e.op(n);
 416                                 }
 417                                 result += sign * lorentz_g(ix1, e.op(i).op(1)) * dirac_trace(ncmul(v), rl);
 418                                 sign = -sign;
 419                         }
 420                         return result;
 421                 }
 422         }
 423
 424         return _ex0();
 425 }
 426
 427 } // namespace GiNaC