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 "lst.h"
  30 #include "relational.h"
  31 #include "print.h"
  32 #include "archive.h"
  33 #include "debugmsg.h"
  34 #include "utils.h"
  35
  36 #include <stdexcept>
  37
  38 namespace GiNaC {
  39
  40 GINAC_IMPLEMENT_REGISTERED_CLASS(clifford, indexed)
  41 GINAC_IMPLEMENT_REGISTERED_CLASS(diracone, tensor)
  42 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma, tensor)
  43 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma5, tensor)
  44
  45 //////////
  46 // default constructor, destructor, copy constructor assignment operator and helpers
  47 //////////
  48
  49 clifford::clifford() : representation_label(0)
  50 {
  51         debugmsg("clifford default constructor", LOGLEVEL_CONSTRUCT);
  52         tinfo_key = TINFO_clifford;
  53 }
  54
  55 void clifford::copy(const clifford & other)
  56 {
  57         inherited::copy(other);
  58         representation_label = other.representation_label;
  59 }
  60
  61 DEFAULT_DESTROY(clifford)
  62 DEFAULT_CTORS(diracone)
  63 DEFAULT_CTORS(diracgamma)
  64 DEFAULT_CTORS(diracgamma5)
  65
  66 //////////
  67 // other constructors
  68 //////////
  69
  70 /** Construct object without any indices. This constructor is for internal
  71  *  use only. Use the dirac_ONE() function instead.
  72  *  @see dirac_ONE */
  73 clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
  74 {
  75         debugmsg("clifford constructor from ex", LOGLEVEL_CONSTRUCT);
  76         tinfo_key = TINFO_clifford;
  77 }
  78
  79 /** Construct object with one Lorentz index. This constructor is for internal
  80  *  use only. Use the dirac_gamma() function instead.
  81  *  @see dirac_gamma */
  82 clifford::clifford(const ex & b, const ex & mu, unsigned char rl) : inherited(b, mu), representation_label(rl)
  83 {
  84         debugmsg("clifford constructor from ex,ex", LOGLEVEL_CONSTRUCT);
  85         GINAC_ASSERT(is_ex_of_type(mu, varidx));
  86         tinfo_key = TINFO_clifford;
  87 }
  88
  89 clifford::clifford(unsigned char rl, const exvector & v, bool discardable) : inherited(indexed::unknown, v, discardable), representation_label(rl)
  90 {
  91         debugmsg("clifford constructor from unsigned char,exvector", LOGLEVEL_CONSTRUCT);
  92         tinfo_key = TINFO_clifford;
  93 }
  94
  95 clifford::clifford(unsigned char rl, exvector * vp) : inherited(indexed::unknown, vp), representation_label(rl)
  96 {
  97         debugmsg("clifford constructor from unsigned char,exvector *", LOGLEVEL_CONSTRUCT);
  98         tinfo_key = TINFO_clifford;
  99 }
 100
 101 //////////
 102 // archiving
 103 //////////
 104
 105 clifford::clifford(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 106 {
 107         debugmsg("clifford constructor from archive_node", LOGLEVEL_CONSTRUCT);
 108         unsigned rl;
 109         n.find_unsigned("label", rl);
 110         representation_label = rl;
 111 }
 112
 113 void clifford::archive(archive_node &n) const
 114 {
 115         inherited::archive(n);
 116         n.add_unsigned("label", representation_label);
 117 }
 118
 119 DEFAULT_UNARCHIVE(clifford)
 120 DEFAULT_ARCHIVING(diracone)
 121 DEFAULT_ARCHIVING(diracgamma)
 122 DEFAULT_ARCHIVING(diracgamma5)
 123
 124 //////////
 125 // functions overriding virtual functions from bases classes
 126 //////////
 127
 128 int clifford::compare_same_type(const basic & other) const
 129 {
 130         GINAC_ASSERT(other.tinfo() == TINFO_clifford);
 131         const clifford &o = static_cast<const clifford &>(other);
 132
 133         if (representation_label != o.representation_label) {
 134                 // different representation label
 135                 return representation_label < o.representation_label ? -1 : 1;
 136         }
 137
 138         return inherited::compare_same_type(other);
 139 }
 140
 141 DEFAULT_COMPARE(diracone)
 142 DEFAULT_COMPARE(diracgamma)
 143 DEFAULT_COMPARE(diracgamma5)
 144
 145 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
 146 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
 147 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
 148
 149 /** Contraction of a gamma matrix with something else. */
 150 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 151 {
 152         GINAC_ASSERT(is_ex_of_type(*self, clifford));
 153         GINAC_ASSERT(is_ex_of_type(*other, indexed));
 154         GINAC_ASSERT(is_ex_of_type(self->op(0), diracgamma));
 155         unsigned char rl = ex_to_clifford(*self).get_representation_label();
 156
 157         if (is_ex_of_type(*other, clifford)) {
 158
 159                 ex dim = ex_to_idx(self->op(1)).get_dim();
 160
 161                 // gamma~mu gamma.mu = dim ONE
 162                 if (other - self == 1) {
 163                         *self = dim;
 164                         *other = dirac_ONE(rl);
 165                         return true;
 166
 167                 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
 168                 } else if (other - self == 2
 169                         && is_ex_of_type(self[1], clifford)) {
 170                         *self = 2 - dim;
 171                         *other = _ex1();
 172                         return true;
 173
 174                 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
 175                 } else if (other - self == 3
 176                         && is_ex_of_type(self[1], clifford)
 177                         && is_ex_of_type(self[2], clifford)) {
 178                         *self = 4 * lorentz_g(self[1].op(1), self[2].op(1)) * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
 179                         self[1] = _ex1();
 180                         self[2] = _ex1();
 181                         *other = _ex1();
 182                         return true;
 183
 184                 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
 185                 // (commutate contracted indices towards each other, simplify_indexed()
 186                 // will re-expand and re-run the simplification)
 187                 } else {
 188                         exvector::iterator it = self + 1, next_to_last = other - 1;
 189                         while (it != other) {
 190                                 if (!is_ex_of_type(*it, clifford))
 191                                         return false;
 192                                 it++;
 193                         }
 194
 195                         it = self + 1;
 196                         ex S = _ex1();
 197                         while (it != next_to_last) {
 198                                 S *= *it;
 199                                 *it++ = _ex1();
 200                         }
 201
 202                         *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
 203                         *next_to_last = _ex1();
 204                         *other = _ex1();
 205                         return true;
 206                 }
 207         }
 208
 209         return false;
 210 }
 211
 212 /** Perform automatic simplification on noncommutative product of clifford
 213  *  objects. This removes superfluous ONEs, permutes gamma5's to the front
 214  *  and removes squares of gamma objects. */
 215 ex clifford::simplify_ncmul(const exvector & v) const
 216 {
 217         exvector s;
 218         s.reserve(v.size());
 219
 220         // Remove superfluous ONEs
 221         exvector::const_iterator cit = v.begin(), citend = v.end();
 222         while (cit != citend) {
 223                 if (!is_ex_of_type(cit->op(0), diracone))
 224                         s.push_back(*cit);
 225                 cit++;
 226         }
 227
 228         bool something_changed = false;
 229         int sign = 1;
 230
 231         // Anticommute gamma5's to the front
 232         if (s.size() >= 2) {
 233                 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
 234                 while (true) {
 235                         exvector::iterator it = next_to_last;
 236                         while (true) {
 237                                 exvector::iterator it2 = it + 1;
 238                                 if (!is_ex_of_type(it->op(0), diracgamma5) && is_ex_of_type(it2->op(0), diracgamma5)) {
 239                                         it->swap(*it2);
 240                                         sign = -sign;
 241                                         something_changed = true;
 242                                 }
 243                                 if (it == first)
 244                                         break;
 245                                 it--;
 246                         }
 247                         if (next_to_last == first)
 248                                 break;
 249                         next_to_last--;
 250                 }
 251         }
 252
 253         // Remove squares of gamma5
 254         while (s.size() >= 2 && is_ex_of_type(s[0].op(0), diracgamma5) && is_ex_of_type(s[1].op(0), diracgamma5)) {
 255                 s.erase(s.begin(), s.begin() + 2);
 256                 something_changed = true;
 257         }
 258
 259         // Remove equal adjacent gammas
 260         if (s.size() >= 2) {
 261                 exvector::iterator it = s.begin(), itend = s.end() - 1;
 262                 while (it != itend) {
 263                         ex & a = it[0];
 264                         ex & b = it[1];
 265                         if (is_ex_of_type(a.op(0), diracgamma) && is_ex_of_type(b.op(0), diracgamma)) {
 266                                 const ex & ia = a.op(1);
 267                                 const ex & ib = b.op(1);
 268                                 if (ia.is_equal(ib)) {
 269                                         a = lorentz_g(ia, ib);
 270                                         b = dirac_ONE(representation_label);
 271                                         something_changed = true;
 272                                 }
 273                         }
 274                         it++;
 275                 }
 276         }
 277
 278         if (s.size() == 0)
 279                 return clifford(diracone(), representation_label) * sign;
 280         if (something_changed)
 281                 return nonsimplified_ncmul(s) * sign;
 282         else
 283                 return simplified_ncmul(s) * sign;
 284 }
 285
 286 ex clifford::thisexprseq(const exvector & v) const
 287 {
 288         return clifford(representation_label, v);
 289 }
 290
 291 ex clifford::thisexprseq(exvector * vp) const
 292 {
 293         return clifford(representation_label, vp);
 294 }
 295
 296 //////////
 297 // global functions
 298 //////////
 299
 300 ex dirac_ONE(unsigned char rl)
 301 {
 302         return clifford(diracone(), rl);
 303 }
 304
 305 ex dirac_gamma(const ex & mu, unsigned char rl)
 306 {
 307         if (!is_ex_of_type(mu, varidx))
 308                 throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
 309
 310         return clifford(diracgamma(), mu, rl);
 311 }
 312
 313 ex dirac_gamma5(unsigned char rl)
 314 {
 315         return clifford(diracgamma5(), rl);
 316 }
 317
 318 ex dirac_gamma6(unsigned char rl)
 319 {
 320         return clifford(diracone(), rl) + clifford(diracgamma5(), rl);
 321 }
 322
 323 ex dirac_gamma7(unsigned char rl)
 324 {
 325         return clifford(diracone(), rl) - clifford(diracgamma5(), rl);
 326 }
 327
 328 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
 329 {
 330         varidx mu((new symbol)->setflag(status_flags::dynallocated), dim);
 331         return indexed(e, mu.toggle_variance()) * dirac_gamma(mu, rl);
 332 }
 333
 334 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 335  *  is that of a clifford object with the specified representation label. */
 336 static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
 337 {
 338         return ti == (TINFO_clifford + rl);
 339 }
 340
 341 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 342  *  is that of a clifford object (with an arbitrary representation label). */
 343 static bool is_clifford_tinfo(unsigned ti)
 344 {
 345         return (ti & ~0xff) == TINFO_clifford;
 346 }
 347
 348 /** Take trace of a string of an even number of Dirac gammas given a vector
 349  *  of indices. */
 350 static ex trace_string(exvector::const_iterator ix, unsigned num)
 351 {
 352         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 353         if (num == 2)
 354                 return lorentz_g(ix[0], ix[1]);
 355
 356         // 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
 357         else if (num == 4)
 358                 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
 359                      + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
 360                      - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
 361
 362         // Traces of 6 or more gammas are computed recursively:
 363         // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
 364         //   + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
 365         //   - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
 366         //   + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
 367         //   - ...
 368         //   + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
 369         exvector v(num - 2);
 370         int sign = 1;
 371         ex result;
 372         for (int i=1; i<num; i++) {
 373                 for (int n=1, j=0; n<num; n++) {
 374                         if (n == i)
 375                                 continue;
 376                         v[j++] = ix[n];
 377                 }
 378                 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
 379                 sign = -sign;
 380         }
 381         return result;
 382 }
 383
 384 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
 385 {
 386         if (is_ex_of_type(e, clifford)) {
 387
 388                 if (ex_to_clifford(e).get_representation_label() == rl
 389                  && is_ex_of_type(e.op(0), diracone))
 390                         return trONE;
 391                 else
 392                         return _ex0();
 393
 394         } else if (is_ex_exactly_of_type(e, add)) {
 395
 396                 // Trace of sum = sum of traces
 397                 ex sum = _ex0();
 398                 for (unsigned i=0; i<e.nops(); i++)
 399                         sum += dirac_trace(e.op(i), rl, trONE);
 400                 return sum;
 401
 402         } else if (is_ex_exactly_of_type(e, mul)) {
 403
 404                 // Trace of product: pull out non-clifford factors
 405                 ex prod = _ex1();
 406                 for (unsigned i=0; i<e.nops(); i++) {
 407                         const ex &o = e.op(i);
 408                         unsigned ti = o.return_type_tinfo();
 409                         if (is_clifford_tinfo(o.return_type_tinfo(), rl))
 410                                 prod *= dirac_trace(o, rl, trONE);
 411                         else
 412                                 prod *= o;
 413                 }
 414                 return prod;
 415
 416         } else if (is_ex_exactly_of_type(e, ncmul)) {
 417
 418                 if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
 419                         return _ex0();
 420
 421                 // Expand product, if necessary
 422                 ex e_expanded = e.expand();
 423                 if (!is_ex_of_type(e_expanded, ncmul))
 424                         return dirac_trace(e_expanded, rl, trONE);
 425
 426                 // gamma5 gets moved to the front so this check is enough
 427                 bool has_gamma5 = is_ex_of_type(e.op(0).op(0), diracgamma5);
 428                 unsigned num = e.nops();
 429
 430                 if (has_gamma5) {
 431
 432                         // Trace of gamma5 * odd number of gammas and trace of
 433                         // gamma5 * gamma.mu * gamma.nu are zero
 434                         if ((num & 1) == 0 || num == 3)
 435                                 return _ex0();
 436
 437                         // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
 438                         if (num == 5)
 439                                 return trONE * I * eps0123(e.op(1).op(1), e.op(2).op(1), e.op(3).op(1), e.op(4).op(1));
 440
 441                         // Tr gamma5 S_2k =
 442                         //   I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
 443                         exvector ix;
 444                         ix.reserve(num - 1);
 445                         for (unsigned i=1; i<num; i++)
 446                                 ix.push_back(e.op(i).op(1));
 447                         num--;
 448                         int *iv = new int[num];
 449                         ex result;
 450                         for (int i=0; i<num-3; i++) {
 451                                 ex idx1 = ix[i];
 452                                 for (int j=i+1; j<num-2; j++) {
 453                                         ex idx2 = ix[j];
 454                                         for (int k=j+1; k<num-1; k++) {
 455                                                 ex idx3 = ix[k];
 456                                                 for (int l=k+1; l<num; l++) {
 457                                                         ex idx4 = ix[l];
 458                                                         iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
 459                                                         exvector v;
 460                                                         v.reserve(num - 4);
 461                                                         for (int n=0, t=4; n<num; n++) {
 462                                                                 if (n == i || n == j || n == k || n == l)
 463                                                                         continue;
 464                                                                 iv[t++] = n;
 465                                                                 v.push_back(ix[n]);
 466                                                         }
 467                                                         int sign = permutation_sign(iv, iv + num);
 468                                                         result += sign * eps0123(idx1, idx2, idx3, idx4)
 469                                                                 * trace_string(v.begin(), num - 4);
 470                                                 }
 471                                         }
 472                                 }
 473                         }
 474                         delete[] iv;
 475                         return trONE * I * result;
 476
 477                 } else { // no gamma5
 478
 479                         // Trace of odd number of gammas is zero
 480                         if ((num & 1) == 1)
 481                                 return _ex0();
 482
 483                         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 484                         if (num == 2)
 485                                 return trONE * lorentz_g(e.op(0).op(1), e.op(1).op(1));
 486
 487                         exvector iv;
 488                         iv.reserve(num);
 489                         for (unsigned i=0; i<num; i++)
 490                                 iv.push_back(e.op(i).op(1));
 491
 492                         return trONE * trace_string(iv.begin(), num);
 493                 }
 494         }
 495
 496         return _ex0();
 497 }
 498
 499 ex canonicalize_clifford(const ex & e)
 500 {
 501         // Scan for any ncmul objects
 502         lst srl;
 503         ex aux = e.to_rational(srl);
 504         for (unsigned i=0; i<srl.nops(); i++) {
 505
 506                 ex lhs = srl.op(i).lhs();
 507                 ex rhs = srl.op(i).rhs();
 508
 509                 if (is_ex_exactly_of_type(rhs, ncmul)
 510                  && rhs.return_type() == return_types::noncommutative
 511                  && is_clifford_tinfo(rhs.return_type_tinfo())) {
 512
 513                         // Expand product, if necessary
 514                         ex rhs_expanded = rhs.expand();
 515                         if (!is_ex_of_type(rhs_expanded, ncmul)) {
 516                                 srl.let_op(i) = (lhs == canonicalize_clifford(rhs_expanded));
 517                                 continue;
 518
 519                         } else if (!is_ex_of_type(rhs.op(0), clifford))
 520                                 continue;
 521
 522                         exvector v;
 523                         v.reserve(rhs.nops());
 524                         for (unsigned j=0; j<rhs.nops(); j++)
 525                                 v.push_back(rhs.op(j));
 526
 527                         // Stupid recursive bubble sort because we only want to swap adjacent gammas
 528                         exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
 529                         if (is_ex_of_type(it->op(0), diracgamma5))
 530                                 it++;
 531                         while (it != next_to_last) {
 532                                 if (it[0].op(1).compare(it[1].op(1)) > 0) {
 533                                         ex save0 = it[0], save1 = it[1];
 534                                         it[0] = lorentz_g(it[0].op(1), it[1].op(1));
 535                                         it[1] = _ex2();
 536                                         ex sum = ncmul(v);
 537                                         it[0] = save1;
 538                                         it[1] = save0;
 539                                         sum -= ncmul(v, true);
 540                                         srl.let_op(i) = (lhs == canonicalize_clifford(sum));
 541                                         goto next_sym;
 542                                 }
 543                                 it++;
 544                         }
 545 next_sym:       ;
 546                 }
 547         }
 548         return aux.subs(srl);
 549 }
 550
 551 } // namespace GiNaC