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 "symmetry.h"
  30 #include "lst.h"
  31 #include "relational.h"
  32 #include "print.h"
  33 #include "archive.h"
  34 #include "debugmsg.h"
  35 #include "utils.h"
  36
  37 #include <stdexcept>
  38
  39 namespace GiNaC {
  40
  41 GINAC_IMPLEMENT_REGISTERED_CLASS(clifford, indexed)
  42 GINAC_IMPLEMENT_REGISTERED_CLASS(diracone, tensor)
  43 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma, tensor)
  44 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma5, tensor)
  45
  46 //////////
  47 // default constructor, destructor, copy constructor assignment operator and helpers
  48 //////////
  49
  50 clifford::clifford() : representation_label(0)
  51 {
  52         debugmsg("clifford default constructor", LOGLEVEL_CONSTRUCT);
  53         tinfo_key = TINFO_clifford;
  54 }
  55
  56 void clifford::copy(const clifford & other)
  57 {
  58         inherited::copy(other);
  59         representation_label = other.representation_label;
  60 }
  61
  62 DEFAULT_DESTROY(clifford)
  63 DEFAULT_CTORS(diracone)
  64 DEFAULT_CTORS(diracgamma)
  65 DEFAULT_CTORS(diracgamma5)
  66
  67 //////////
  68 // other constructors
  69 //////////
  70
  71 /** Construct object without any indices. This constructor is for internal
  72  *  use only. Use the dirac_ONE() function instead.
  73  *  @see dirac_ONE */
  74 clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
  75 {
  76         debugmsg("clifford constructor from ex", LOGLEVEL_CONSTRUCT);
  77         tinfo_key = TINFO_clifford;
  78 }
  79
  80 /** Construct object with one Lorentz index. This constructor is for internal
  81  *  use only. Use the dirac_gamma() function instead.
  82  *  @see dirac_gamma */
  83 clifford::clifford(const ex & b, const ex & mu, unsigned char rl) : inherited(b, mu), representation_label(rl)
  84 {
  85         debugmsg("clifford constructor from ex,ex", LOGLEVEL_CONSTRUCT);
  86         GINAC_ASSERT(is_ex_of_type(mu, varidx));
  87         tinfo_key = TINFO_clifford;
  88 }
  89
  90 clifford::clifford(unsigned char rl, const exvector & v, bool discardable) : inherited(sy_none(), v, discardable), representation_label(rl)
  91 {
  92         debugmsg("clifford constructor from unsigned char,exvector", LOGLEVEL_CONSTRUCT);
  93         tinfo_key = TINFO_clifford;
  94 }
  95
  96 clifford::clifford(unsigned char rl, exvector * vp) : inherited(sy_none(), vp), representation_label(rl)
  97 {
  98         debugmsg("clifford constructor from unsigned char,exvector *", LOGLEVEL_CONSTRUCT);
  99         tinfo_key = TINFO_clifford;
 100 }
 101
 102 //////////
 103 // archiving
 104 //////////
 105
 106 clifford::clifford(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 107 {
 108         debugmsg("clifford constructor from archive_node", LOGLEVEL_CONSTRUCT);
 109         unsigned rl;
 110         n.find_unsigned("label", rl);
 111         representation_label = rl;
 112 }
 113
 114 void clifford::archive(archive_node &n) const
 115 {
 116         inherited::archive(n);
 117         n.add_unsigned("label", representation_label);
 118 }
 119
 120 DEFAULT_UNARCHIVE(clifford)
 121 DEFAULT_ARCHIVING(diracone)
 122 DEFAULT_ARCHIVING(diracgamma)
 123 DEFAULT_ARCHIVING(diracgamma5)
 124
 125 //////////
 126 // functions overriding virtual functions from bases classes
 127 //////////
 128
 129 int clifford::compare_same_type(const basic & other) const
 130 {
 131         GINAC_ASSERT(other.tinfo() == TINFO_clifford);
 132         const clifford &o = static_cast<const clifford &>(other);
 133
 134         if (representation_label != o.representation_label) {
 135                 // different representation label
 136                 return representation_label < o.representation_label ? -1 : 1;
 137         }
 138
 139         return inherited::compare_same_type(other);
 140 }
 141
 142 DEFAULT_COMPARE(diracone)
 143 DEFAULT_COMPARE(diracgamma)
 144 DEFAULT_COMPARE(diracgamma5)
 145
 146 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
 147 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
 148 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
 149
 150 /** Contraction of a gamma matrix with something else. */
 151 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 152 {
 153         GINAC_ASSERT(is_ex_of_type(*self, clifford));
 154         GINAC_ASSERT(is_ex_of_type(*other, indexed));
 155         GINAC_ASSERT(is_ex_of_type(self->op(0), diracgamma));
 156         unsigned char rl = ex_to_clifford(*self).get_representation_label();
 157
 158         if (is_ex_of_type(*other, clifford)) {
 159
 160                 ex dim = ex_to_idx(self->op(1)).get_dim();
 161
 162                 // gamma~mu gamma.mu = dim ONE
 163                 if (other - self == 1) {
 164                         *self = dim;
 165                         *other = dirac_ONE(rl);
 166                         return true;
 167
 168                 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
 169                 } else if (other - self == 2
 170                         && is_ex_of_type(self[1], clifford)) {
 171                         *self = 2 - dim;
 172                         *other = _ex1();
 173                         return true;
 174
 175                 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
 176                 } else if (other - self == 3
 177                         && is_ex_of_type(self[1], clifford)
 178                         && is_ex_of_type(self[2], clifford)) {
 179                         *self = 4 * lorentz_g(self[1].op(1), self[2].op(1)) * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
 180                         self[1] = _ex1();
 181                         self[2] = _ex1();
 182                         *other = _ex1();
 183                         return true;
 184
 185                 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
 186                 // (commutate contracted indices towards each other, simplify_indexed()
 187                 // will re-expand and re-run the simplification)
 188                 } else {
 189                         exvector::iterator it = self + 1, next_to_last = other - 1;
 190                         while (it != other) {
 191                                 if (!is_ex_of_type(*it, clifford))
 192                                         return false;
 193                                 it++;
 194                         }
 195
 196                         it = self + 1;
 197                         ex S = _ex1();
 198                         while (it != next_to_last) {
 199                                 S *= *it;
 200                                 *it++ = _ex1();
 201                         }
 202
 203                         *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
 204                         *next_to_last = _ex1();
 205                         *other = _ex1();
 206                         return true;
 207                 }
 208         }
 209
 210         return false;
 211 }
 212
 213 /** Perform automatic simplification on noncommutative product of clifford
 214  *  objects. This removes superfluous ONEs, permutes gamma5's to the front
 215  *  and removes squares of gamma objects. */
 216 ex clifford::simplify_ncmul(const exvector & v) const
 217 {
 218         exvector s;
 219         s.reserve(v.size());
 220
 221         // Remove superfluous ONEs
 222         exvector::const_iterator cit = v.begin(), citend = v.end();
 223         while (cit != citend) {
 224                 if (!is_ex_of_type(cit->op(0), diracone))
 225                         s.push_back(*cit);
 226                 cit++;
 227         }
 228
 229         bool something_changed = false;
 230         int sign = 1;
 231
 232         // Anticommute gamma5's to the front
 233         if (s.size() >= 2) {
 234                 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
 235                 while (true) {
 236                         exvector::iterator it = next_to_last;
 237                         while (true) {
 238                                 exvector::iterator it2 = it + 1;
 239                                 if (!is_ex_of_type(it->op(0), diracgamma5) && is_ex_of_type(it2->op(0), diracgamma5)) {
 240                                         it->swap(*it2);
 241                                         sign = -sign;
 242                                         something_changed = true;
 243                                 }
 244                                 if (it == first)
 245                                         break;
 246                                 it--;
 247                         }
 248                         if (next_to_last == first)
 249                                 break;
 250                         next_to_last--;
 251                 }
 252         }
 253
 254         // Remove squares of gamma5
 255         while (s.size() >= 2 && is_ex_of_type(s[0].op(0), diracgamma5) && is_ex_of_type(s[1].op(0), diracgamma5)) {
 256                 s.erase(s.begin(), s.begin() + 2);
 257                 something_changed = true;
 258         }
 259
 260         // Remove equal adjacent gammas
 261         if (s.size() >= 2) {
 262                 exvector::iterator it = s.begin(), itend = s.end() - 1;
 263                 while (it != itend) {
 264                         ex & a = it[0];
 265                         ex & b = it[1];
 266                         if (is_ex_of_type(a.op(0), diracgamma) && is_ex_of_type(b.op(0), diracgamma)) {
 267                                 const ex & ia = a.op(1);
 268                                 const ex & ib = b.op(1);
 269                                 if (ia.is_equal(ib)) {
 270                                         a = lorentz_g(ia, ib);
 271                                         b = dirac_ONE(representation_label);
 272                                         something_changed = true;
 273                                 }
 274                         }
 275                         it++;
 276                 }
 277         }
 278
 279         if (s.size() == 0)
 280                 return clifford(diracone(), representation_label) * sign;
 281         if (something_changed)
 282                 return nonsimplified_ncmul(s) * sign;
 283         else
 284                 return simplified_ncmul(s) * sign;
 285 }
 286
 287 ex clifford::thisexprseq(const exvector & v) const
 288 {
 289         return clifford(representation_label, v);
 290 }
 291
 292 ex clifford::thisexprseq(exvector * vp) const
 293 {
 294         return clifford(representation_label, vp);
 295 }
 296
 297 //////////
 298 // global functions
 299 //////////
 300
 301 ex dirac_ONE(unsigned char rl)
 302 {
 303         return clifford(diracone(), rl);
 304 }
 305
 306 ex dirac_gamma(const ex & mu, unsigned char rl)
 307 {
 308         if (!is_ex_of_type(mu, varidx))
 309                 throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
 310
 311         return clifford(diracgamma(), mu, rl);
 312 }
 313
 314 ex dirac_gamma5(unsigned char rl)
 315 {
 316         return clifford(diracgamma5(), rl);
 317 }
 318
 319 ex dirac_gamma6(unsigned char rl)
 320 {
 321         return clifford(diracone(), rl) + clifford(diracgamma5(), rl);
 322 }
 323
 324 ex dirac_gamma7(unsigned char rl)
 325 {
 326         return clifford(diracone(), rl) - clifford(diracgamma5(), rl);
 327 }
 328
 329 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
 330 {
 331         varidx mu((new symbol)->setflag(status_flags::dynallocated), dim);
 332         return indexed(e, mu.toggle_variance()) * dirac_gamma(mu, rl);
 333 }
 334
 335 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 336  *  is that of a clifford object with the specified representation label. */
 337 static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
 338 {
 339         return ti == (TINFO_clifford + rl);
 340 }
 341
 342 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 343  *  is that of a clifford object (with an arbitrary representation label). */
 344 static bool is_clifford_tinfo(unsigned ti)
 345 {
 346         return (ti & ~0xff) == TINFO_clifford;
 347 }
 348
 349 /** Take trace of a string of an even number of Dirac gammas given a vector
 350  *  of indices. */
 351 static ex trace_string(exvector::const_iterator ix, unsigned num)
 352 {
 353         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 354         if (num == 2)
 355                 return lorentz_g(ix[0], ix[1]);
 356
 357         // 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
 358         else if (num == 4)
 359                 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
 360                      + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
 361                      - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
 362
 363         // Traces of 6 or more gammas are computed recursively:
 364         // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
 365         //   + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
 366         //   - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
 367         //   + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
 368         //   - ...
 369         //   + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
 370         exvector v(num - 2);
 371         int sign = 1;
 372         ex result;
 373         for (int i=1; i<num; i++) {
 374                 for (int n=1, j=0; n<num; n++) {
 375                         if (n == i)
 376                                 continue;
 377                         v[j++] = ix[n];
 378                 }
 379                 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
 380                 sign = -sign;
 381         }
 382         return result;
 383 }
 384
 385 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
 386 {
 387         if (is_ex_of_type(e, clifford)) {
 388
 389                 if (ex_to_clifford(e).get_representation_label() == rl
 390                  && is_ex_of_type(e.op(0), diracone))
 391                         return trONE;
 392                 else
 393                         return _ex0();
 394
 395         } else if (is_ex_exactly_of_type(e, mul)) {
 396
 397                 // Trace of product: pull out non-clifford factors
 398                 ex prod = _ex1();
 399                 for (unsigned i=0; i<e.nops(); i++) {
 400                         const ex &o = e.op(i);
 401                         unsigned ti = o.return_type_tinfo();
 402                         if (is_clifford_tinfo(o.return_type_tinfo(), rl))
 403                                 prod *= dirac_trace(o, rl, trONE);
 404                         else
 405                                 prod *= o;
 406                 }
 407                 return prod;
 408
 409         } else if (is_ex_exactly_of_type(e, ncmul)) {
 410
 411                 if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
 412                         return _ex0();
 413
 414                 // Expand product, if necessary
 415                 ex e_expanded = e.expand();
 416                 if (!is_ex_of_type(e_expanded, ncmul))
 417                         return dirac_trace(e_expanded, rl, trONE);
 418
 419                 // gamma5 gets moved to the front so this check is enough
 420                 bool has_gamma5 = is_ex_of_type(e.op(0).op(0), diracgamma5);
 421                 unsigned num = e.nops();
 422
 423                 if (has_gamma5) {
 424
 425                         // Trace of gamma5 * odd number of gammas and trace of
 426                         // gamma5 * gamma.mu * gamma.nu are zero
 427                         if ((num & 1) == 0 || num == 3)
 428                                 return _ex0();
 429
 430                         // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
 431                         if (num == 5)
 432                                 return trONE * I * eps0123(e.op(1).op(1), e.op(2).op(1), e.op(3).op(1), e.op(4).op(1));
 433
 434                         // Tr gamma5 S_2k =
 435                         //   I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
 436                         exvector ix;
 437                         ix.reserve(num - 1);
 438                         for (unsigned i=1; i<num; i++)
 439                                 ix.push_back(e.op(i).op(1));
 440                         num--;
 441                         int *iv = new int[num];
 442                         ex result;
 443                         for (int i=0; i<num-3; i++) {
 444                                 ex idx1 = ix[i];
 445                                 for (int j=i+1; j<num-2; j++) {
 446                                         ex idx2 = ix[j];
 447                                         for (int k=j+1; k<num-1; k++) {
 448                                                 ex idx3 = ix[k];
 449                                                 for (int l=k+1; l<num; l++) {
 450                                                         ex idx4 = ix[l];
 451                                                         iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
 452                                                         exvector v;
 453                                                         v.reserve(num - 4);
 454                                                         for (int n=0, t=4; n<num; n++) {
 455                                                                 if (n == i || n == j || n == k || n == l)
 456                                                                         continue;
 457                                                                 iv[t++] = n;
 458                                                                 v.push_back(ix[n]);
 459                                                         }
 460                                                         int sign = permutation_sign(iv, iv + num);
 461                                                         result += sign * eps0123(idx1, idx2, idx3, idx4)
 462                                                                 * trace_string(v.begin(), num - 4);
 463                                                 }
 464                                         }
 465                                 }
 466                         }
 467                         delete[] iv;
 468                         return trONE * I * result;
 469
 470                 } else { // no gamma5
 471
 472                         // Trace of odd number of gammas is zero
 473                         if ((num & 1) == 1)
 474                                 return _ex0();
 475
 476                         // Tr gamma.mu gamma.nu = 4 g.mu.nu
 477                         if (num == 2)
 478                                 return trONE * lorentz_g(e.op(0).op(1), e.op(1).op(1));
 479
 480                         exvector iv;
 481                         iv.reserve(num);
 482                         for (unsigned i=0; i<num; i++)
 483                                 iv.push_back(e.op(i).op(1));
 484
 485                         return trONE * trace_string(iv.begin(), num);
 486                 }
 487
 488         } else if (e.nops() > 0) {
 489
 490                 // Trace maps to all other container classes (this includes sums)
 491                 pointer_to_map_function_2args<unsigned char, const ex &> fcn(dirac_trace, rl, trONE);
 492                 return e.map(fcn);
 493
 494         } else
 495                 return _ex0();
 496 }
 497
 498 ex canonicalize_clifford(const ex & e)
 499 {
 500         // Scan for any ncmul objects
 501         lst srl;
 502         ex aux = e.to_rational(srl);
 503         for (unsigned i=0; i<srl.nops(); i++) {
 504
 505                 ex lhs = srl.op(i).lhs();
 506                 ex rhs = srl.op(i).rhs();
 507
 508                 if (is_ex_exactly_of_type(rhs, ncmul)
 509                  && rhs.return_type() == return_types::noncommutative
 510                  && is_clifford_tinfo(rhs.return_type_tinfo())) {
 511
 512                         // Expand product, if necessary
 513                         ex rhs_expanded = rhs.expand();
 514                         if (!is_ex_of_type(rhs_expanded, ncmul)) {
 515                                 srl.let_op(i) = (lhs == canonicalize_clifford(rhs_expanded));
 516                                 continue;
 517
 518                         } else if (!is_ex_of_type(rhs.op(0), clifford))
 519                                 continue;
 520
 521                         exvector v;
 522                         v.reserve(rhs.nops());
 523                         for (unsigned j=0; j<rhs.nops(); j++)
 524                                 v.push_back(rhs.op(j));
 525
 526                         // Stupid recursive bubble sort because we only want to swap adjacent gammas
 527                         exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
 528                         if (is_ex_of_type(it->op(0), diracgamma5))
 529                                 it++;
 530                         while (it != next_to_last) {
 531                                 if (it[0].op(1).compare(it[1].op(1)) > 0) {
 532                                         ex save0 = it[0], save1 = it[1];
 533                                         it[0] = lorentz_g(it[0].op(1), it[1].op(1));
 534                                         it[1] = _ex2();
 535                                         ex sum = ncmul(v);
 536                                         it[0] = save1;
 537                                         it[1] = save0;
 538                                         sum -= ncmul(v, true);
 539                                         srl.let_op(i) = (lhs == canonicalize_clifford(sum));
 540                                         goto next_sym;
 541                                 }
 542                                 it++;
 543                         }
 544 next_sym:       ;
 545                 }
 546         }
 547         return aux.subs(srl);
 548 }
 549
 550 } // namespace GiNaC