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 "numeric.h" // for I
  29 #include "print.h"
  30 #include "archive.h"
  31 #include "debugmsg.h"
  32 #include "utils.h"
  33
  34 #include <stdexcept>
  35
  36 namespace GiNaC {
  37
  38 GINAC_IMPLEMENT_REGISTERED_CLASS(clifford, indexed)
  39 GINAC_IMPLEMENT_REGISTERED_CLASS(diracone, tensor)
  40 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma, tensor)
  41 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma5, tensor)
  42
  43 //////////
  44 // default constructor, destructor, copy constructor assignment operator and helpers
  45 //////////
  46
  47 clifford::clifford() : representation_label(0)
  48 {
  49         debugmsg("clifford default constructor", LOGLEVEL_CONSTRUCT);
  50         tinfo_key = TINFO_clifford;
  51 }
  52
  53 void clifford::copy(const clifford & other)
  54 {
  55         inherited::copy(other);
  56         representation_label = other.representation_label;
  57 }
  58
  59 DEFAULT_DESTROY(clifford)
  60 DEFAULT_CTORS(diracone)
  61 DEFAULT_CTORS(diracgamma)
  62 DEFAULT_CTORS(diracgamma5)
  63
  64 //////////
  65 // other constructors
  66 //////////
  67
  68 /** Construct object without any indices. This constructor is for internal
  69  *  use only. Use the dirac_ONE() function instead.
  70  *  @see dirac_ONE */
  71 clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
  72 {
  73         debugmsg("clifford constructor from ex", LOGLEVEL_CONSTRUCT);
  74         tinfo_key = TINFO_clifford;
  75 }
  76
  77 /** Construct object with one Lorentz index. This constructor is for internal
  78  *  use only. Use the dirac_gamma() function instead.
  79  *  @see dirac_gamma */
  80 clifford::clifford(const ex & b, const ex & mu, unsigned char rl) : inherited(b, mu), representation_label(rl)
  81 {
  82         debugmsg("clifford constructor from ex,ex", LOGLEVEL_CONSTRUCT);
  83         GINAC_ASSERT(is_ex_of_type(mu, varidx));
  84         tinfo_key = TINFO_clifford;
  85 }
  86
  87 clifford::clifford(unsigned char rl, const exvector & v, bool discardable) : inherited(indexed::unknown, v, discardable), representation_label(rl)
  88 {
  89         debugmsg("clifford constructor from unsigned char,exvector", LOGLEVEL_CONSTRUCT);
  90         tinfo_key = TINFO_clifford;
  91 }
  92
  93 clifford::clifford(unsigned char rl, exvector * vp) : inherited(indexed::unknown, vp), representation_label(rl)
  94 {
  95         debugmsg("clifford constructor from unsigned char,exvector *", LOGLEVEL_CONSTRUCT);
  96         tinfo_key = TINFO_clifford;
  97 }
  98
  99 //////////
 100 // archiving
 101 //////////
 102
 103 clifford::clifford(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 104 {
 105         debugmsg("clifford constructor from archive_node", LOGLEVEL_CONSTRUCT);
 106         unsigned rl;
 107         n.find_unsigned("label", rl);
 108         representation_label = rl;
 109 }
 110
 111 void clifford::archive(archive_node &n) const
 112 {
 113         inherited::archive(n);
 114         n.add_unsigned("label", representation_label);
 115 }
 116
 117 DEFAULT_UNARCHIVE(clifford)
 118 DEFAULT_ARCHIVING(diracone)
 119 DEFAULT_ARCHIVING(diracgamma)
 120 DEFAULT_ARCHIVING(diracgamma5)
 121
 122 //////////
 123 // functions overriding virtual functions from bases classes
 124 //////////
 125
 126 int clifford::compare_same_type(const basic & other) const
 127 {
 128         GINAC_ASSERT(other.tinfo() == TINFO_clifford);
 129         const clifford &o = static_cast<const clifford &>(other);
 130
 131         if (representation_label != o.representation_label) {
 132                 // different representation label
 133                 return representation_label < o.representation_label ? -1 : 1;
 134         }
 135
 136         return inherited::compare_same_type(other);
 137 }
 138
 139 DEFAULT_COMPARE(diracone)
 140 DEFAULT_COMPARE(diracgamma)
 141 DEFAULT_COMPARE(diracgamma5)
 142
 143 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
 144 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
 145 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
 146
 147 /** Contraction of a gamma matrix with something else. */
 148 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 149 {
 150         GINAC_ASSERT(is_ex_of_type(*self, clifford));
 151         GINAC_ASSERT(is_ex_of_type(*other, indexed));
 152         GINAC_ASSERT(is_ex_of_type(self->op(0), diracgamma));
 153         unsigned char rl = ex_to_clifford(*self).get_representation_label();
 154
 155         if (is_ex_of_type(*other, clifford)) {
 156
 157                 ex dim = ex_to_idx(self->op(1)).get_dim();
 158
 159                 // gamma~mu gamma.mu = dim ONE
 160                 if (other - self == 1) {
 161                         *self = dim;
 162                         *other = dirac_ONE(rl);
 163                         return true;
 164
 165                 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
 166                 } else if (other - self == 2
 167                         && is_ex_of_type(self[1], clifford)) {
 168                         *self = 2 - dim;
 169                         *other = _ex1();
 170                         return true;
 171
 172                 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
 173                 } else if (other - self == 3
 174                         && is_ex_of_type(self[1], clifford)
 175                         && is_ex_of_type(self[2], clifford)) {
 176                         *self = 4 * lorentz_g(self[1].op(1), self[2].op(1)) * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
 177                         self[1] = _ex1();
 178                         self[2] = _ex1();
 179                         *other = _ex1();
 180                         return true;
 181
 182                 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
 183                 // (commutate contracted indices towards each other, simplify_indexed()
 184                 // will re-expand and re-run the simplification)
 185                 } else {
 186                         exvector::iterator it = self + 1, next_to_last = other - 1;
 187                         while (it != other) {
 188                                 if (!is_ex_of_type(*it, clifford))
 189                                         return false;
 190                                 it++;
 191                         }
 192
 193                         it = self + 1;
 194                         ex S = _ex1();
 195                         while (it != next_to_last) {
 196                                 S *= *it;
 197                                 *it++ = _ex1();
 198                         }
 199
 200                         *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
 201                         *next_to_last = _ex1();
 202                         *other = _ex1();
 203                         return true;
 204                 }
 205         }
 206
 207         return false;
 208 }
 209
 210 /** Perform automatic simplification on noncommutative product of clifford
 211  *  objects. This removes superfluous ONEs, permutes gamma5's to the front
 212  *  and removes squares of gamma objects. */
 213 ex clifford::simplify_ncmul(const exvector & v) const
 214 {
 215         exvector s;
 216         s.reserve(v.size());
 217         unsigned rl = ex_to_clifford(v[0]).get_representation_label();
 218
 219         // Remove superfluous ONEs
 220         exvector::const_iterator cit = v.begin(), citend = v.end();
 221         while (cit != citend) {
 222                 if (!is_ex_of_type(cit->op(0), diracone))
 223                         s.push_back(*cit);
 224                 cit++;
 225         }
 226
 227         bool something_changed = false;
 228         int sign = 1;
 229
 230         // Anticommute gamma5's to the front
 231         if (s.size() >= 2) {
 232                 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
 233                 while (true) {
 234                         exvector::iterator it = next_to_last;
 235                         while (true) {
 236                                 exvector::iterator it2 = it + 1;
 237                                 if (!is_ex_of_type(it->op(0), diracgamma5) && is_ex_of_type(it2->op(0), diracgamma5)) {
 238                                         it->swap(*it2);
 239                                         sign = -sign;
 240                                         something_changed = true;
 241                                 }
 242                                 if (it == first)
 243                                         break;
 244                                 it--;
 245                         }
 246                         if (next_to_last == first)
 247                                 break;
 248                         next_to_last--;
 249                 }
 250         }
 251
 252         // Remove squares of gamma5
 253         while (s.size() >= 2 && is_ex_of_type(s[0].op(0), diracgamma5) && is_ex_of_type(s[1].op(0), diracgamma5)) {
 254                 s.erase(s.begin(), s.begin() + 2);
 255                 something_changed = true;
 256         }
 257
 258         // Remove equal adjacent gammas
 259         if (s.size() >= 2) {
 260                 exvector::iterator it = s.begin(), itend = s.end() - 1;
 261                 while (it != itend) {
 262                         ex & a = it[0];
 263                         ex & b = it[1];
 264                         if (is_ex_of_type(a.op(0), diracgamma) && is_ex_of_type(b.op(0), diracgamma)) {
 265                                 const ex & ia = a.op(1);
 266                                 const ex & ib = b.op(1);
 267                                 if (ia.is_equal(ib)) {
 268                                         a = lorentz_g(ia, ib);
 269                                         b = dirac_ONE(rl);
 270                                         something_changed = true;
 271                                 }
 272                         }
 273                         it++;
 274                 }
 275         }
 276
 277         if (s.size() == 0)
 278                 return clifford(diracone(), rl) * sign;
 279         if (something_changed)
 280                 return nonsimplified_ncmul(s) * sign;
 281         else
 282                 return simplified_ncmul(s) * sign;
 283 }
 284
 285 ex clifford::thisexprseq(const exvector & v) const
 286 {
 287         return clifford(representation_label, v);
 288 }
 289
 290 ex clifford::thisexprseq(exvector * vp) const
 291 {
 292         return clifford(representation_label, vp);
 293 }
 294
 295 //////////
 296 // global functions
 297 //////////
 298
 299 ex dirac_ONE(unsigned char rl)
 300 {
 301         return clifford(diracone(), rl);
 302 }
 303
 304 ex dirac_gamma(const ex & mu, unsigned char rl)
 305 {
 306         if (!is_ex_of_type(mu, varidx))
 307                 throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
 308
 309         return clifford(diracgamma(), mu, rl);
 310 }
 311
 312 ex dirac_gamma5(unsigned char rl)
 313 {
 314         return clifford(diracgamma5(), rl);
 315 }
 316
 317 ex dirac_gamma6(unsigned char rl)
 318 {
 319         return clifford(diracone(), rl) + clifford(diracgamma5(), rl);
 320 }
 321
 322 ex dirac_gamma7(unsigned char rl)
 323 {
 324         return clifford(diracone(), rl) - clifford(diracgamma5(), rl);
 325 }
 326
 327 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
 328 {
 329         varidx mu((new symbol)->setflag(status_flags::dynallocated), dim);
 330         return indexed(e, mu.toggle_variance()) * dirac_gamma(mu, rl);
 331 }
 332
 333 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 334  *  is that of a clifford object with the specified representation label. */
 335 static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
 336 {
 337         return ti == (TINFO_clifford + rl);
 338 }
 339
 340 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
 341 {
 342         if (is_ex_of_type(e, clifford)) {
 343
 344                 if (ex_to_clifford(e).get_representation_label() == rl
 345                  && is_ex_of_type(e.op(0), diracone))
 346                         return trONE;
 347                 else
 348                         return _ex0();
 349
 350         } else if (is_ex_exactly_of_type(e, add)) {
 351
 352                 // Trace of sum = sum of traces
 353                 ex sum = _ex0();
 354                 for (unsigned i=0; i<e.nops(); i++)
 355                         sum += dirac_trace(e.op(i), rl, trONE);
 356                 return sum;
 357
 358         } else if (is_ex_exactly_of_type(e, mul)) {
 359
 360                 // Trace of product: pull out non-clifford factors
 361                 ex prod = _ex1();
 362                 for (unsigned i=0; i<e.nops(); i++) {
 363                         const ex &o = e.op(i);
 364                         unsigned ti = o.return_type_tinfo();
 365                         if (is_clifford_tinfo(o.return_type_tinfo(), rl))
 366                                 prod *= dirac_trace(o, rl, trONE);
 367                         else
 368                                 prod *= o;
 369                 }
 370                 return prod;
 371
 372         } else if (is_ex_exactly_of_type(e, ncmul)) {
 373
 374                 if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
 375                         return _ex0();
 376
 377                 // Expand product, if necessary
 378                 ex e_expanded = e.expand();
 379                 if (!is_ex_of_type(e_expanded, ncmul))
 380                         return dirac_trace(e_expanded, rl, trONE);
 381
 382                 // gamma5 gets moved to the front so this check is enough
 383                 bool has_gamma5 = is_ex_of_type(e.op(0).op(0), diracgamma5);
 384                 unsigned num = e.nops();
 385
 386                 if (has_gamma5) {
 387
 388                         // Trace of gamma5 * odd number of gammas and trace of
 389                         // gamma5 * gamma.mu * gamma.nu are zero
 390                         if ((num & 1) == 0 || num == 3)
 391                                 return _ex0();
 392
 393                         // Tr gamma5 S_2k =
 394                         //   I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
 395                         ex dim = ex_to_idx(e.op(1).op(1)).get_dim();
 396                         varidx mu1((new symbol)->setflag(status_flags::dynallocated), dim),
 397                                mu2((new symbol)->setflag(status_flags::dynallocated), dim),
 398                                mu3((new symbol)->setflag(status_flags::dynallocated), dim),
 399                                mu4((new symbol)->setflag(status_flags::dynallocated), dim);
 400                         exvector v;
 401                         v.reserve(num + 3);
 402                         v.push_back(dirac_gamma(mu1, rl));
 403                         v.push_back(dirac_gamma(mu2, rl));
 404                         v.push_back(dirac_gamma(mu3, rl));
 405                         v.push_back(dirac_gamma(mu4, rl));
 406                         for (int i=1; i<num; i++)
 407                                 v.push_back(e.op(i));
 408
 409                         return (eps0123(mu1.toggle_variance(), mu2.toggle_variance(), mu3.toggle_variance(), mu4.toggle_variance()) *
 410                                 dirac_trace(ncmul(v), rl, trONE)).simplify_indexed() * I / 24;
 411
 412                 } else { // no gamma5
 413
 414                         // Trace of odd number of gammas is zero
 415                         if ((num & 1) == 1)
 416                                 return _ex0();
 417
 418                         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 419                         if (num == 2)
 420                                 return trONE * lorentz_g(e.op(0).op(1), e.op(1).op(1));
 421
 422                         // Traces of 4 or more gammas are computed recursively:
 423                         // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
 424                         //   + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
 425                         //   - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
 426                         //   + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
 427                         //   - ...
 428                         //   + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
 429                         exvector v(num - 2);
 430                         int sign = 1;
 431                         const ex &ix1 = e.op(0).op(1);
 432                         ex result;
 433                         for (int i=1; i<num; i++) {
 434                                 for (int n=1, j=0; n<num; n++) {
 435                                         if (n == i)
 436                                                 continue;
 437                                         v[j++] = e.op(n);
 438                                 }
 439                                 result += sign * lorentz_g(ix1, e.op(i).op(1)) * dirac_trace(ncmul(v), rl, trONE);
 440                                 sign = -sign;
 441                         }
 442                         return result;
 443                 }
 444         }
 445
 446         return _ex0();
 447 }
 448
 449 } // namespace GiNaC