ginac/color.cpp

   1 /** @file color.cpp
   2  *
   3  *  Implementation of GiNaC's color (SU(3) Lie algebra) objects. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2002 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 <iostream>
  24 #include <stdexcept>
  25
  26 #include "color.h"
  27 #include "idx.h"
  28 #include "ncmul.h"
  29 #include "symmetry.h"
  30 #include "operators.h"
  31 #include "numeric.h"
  32 #include "mul.h"
  33 #include "power.h" // for sqrt()
  34 #include "symbol.h"
  35 #include "print.h"
  36 #include "archive.h"
  37 #include "utils.h"
  38
  39 namespace GiNaC {
  40
  41 GINAC_IMPLEMENT_REGISTERED_CLASS(color, indexed)
  42 GINAC_IMPLEMENT_REGISTERED_CLASS(su3one, tensor)
  43 GINAC_IMPLEMENT_REGISTERED_CLASS(su3t, tensor)
  44 GINAC_IMPLEMENT_REGISTERED_CLASS(su3f, tensor)
  45 GINAC_IMPLEMENT_REGISTERED_CLASS(su3d, tensor)
  46
  47 //////////
  48 // default ctor, dtor, copy ctor, assignment operator and helpers
  49 //////////
  50
  51 color::color() : representation_label(0)
  52 {
  53         tinfo_key = TINFO_color;
  54 }
  55
  56 void color::copy(const color & other)
  57 {
  58         inherited::copy(other);
  59         representation_label = other.representation_label;
  60 }
  61
  62 DEFAULT_DESTROY(color)
  63 DEFAULT_CTORS(su3one)
  64 DEFAULT_CTORS(su3t)
  65 DEFAULT_CTORS(su3f)
  66 DEFAULT_CTORS(su3d)
  67
  68 //////////
  69 // other constructors
  70 //////////
  71
  72 /** Construct object without any color index. This constructor is for
  73  *  internal use only. Use the color_ONE() function instead.
  74  *  @see color_ONE */
  75 color::color(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
  76 {
  77         tinfo_key = TINFO_color;
  78 }
  79
  80 /** Construct object with one color index. This constructor is for internal
  81  *  use only. Use the color_T() function instead.
  82  *  @see color_T */
  83 color::color(const ex & b, const ex & i1, unsigned char rl) : inherited(b, i1), representation_label(rl)
  84 {
  85         tinfo_key = TINFO_color;
  86 }
  87
  88 color::color(unsigned char rl, const exvector & v, bool discardable) : inherited(sy_none(), v, discardable), representation_label(rl)
  89 {
  90         tinfo_key = TINFO_color;
  91 }
  92
  93 color::color(unsigned char rl, exvector * vp) : inherited(sy_none(), vp), representation_label(rl)
  94 {
  95         tinfo_key = TINFO_color;
  96 }
  97
  98 //////////
  99 // archiving
 100 //////////
 101
 102 color::color(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 103 {
 104         unsigned rl;
 105         n.find_unsigned("label", rl);
 106         representation_label = rl;
 107 }
 108
 109 void color::archive(archive_node &n) const
 110 {
 111         inherited::archive(n);
 112         n.add_unsigned("label", representation_label);
 113 }
 114
 115 DEFAULT_UNARCHIVE(color)
 116 DEFAULT_ARCHIVING(su3one)
 117 DEFAULT_ARCHIVING(su3t)
 118 DEFAULT_ARCHIVING(su3f)
 119 DEFAULT_ARCHIVING(su3d)
 120
 121 //////////
 122 // functions overriding virtual functions from base classes
 123 //////////
 124
 125 int color::compare_same_type(const basic & other) const
 126 {
 127         GINAC_ASSERT(is_a<color>(other));
 128         const color &o = static_cast<const color &>(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 bool color::match_same_type(const basic & other) const
 139 {
 140         GINAC_ASSERT(is_a<color>(other));
 141         const color &o = static_cast<const color &>(other);
 142
 143         return representation_label == o.representation_label;
 144 }
 145
 146 DEFAULT_COMPARE(su3one)
 147 DEFAULT_COMPARE(su3t)
 148 DEFAULT_COMPARE(su3f)
 149 DEFAULT_COMPARE(su3d)
 150
 151 DEFAULT_PRINT_LATEX(su3one, "ONE", "\\mathbb{1}")
 152 DEFAULT_PRINT(su3t, "T")
 153 DEFAULT_PRINT(su3f, "f")
 154 DEFAULT_PRINT(su3d, "d")
 155
 156 /** Perform automatic simplification on noncommutative product of color
 157  *  objects. This removes superfluous ONEs. */
 158 ex color::simplify_ncmul(const exvector & v) const
 159 {
 160         exvector s;
 161         s.reserve(v.size());
 162
 163         // Remove superfluous ONEs
 164         exvector::const_iterator it = v.begin(), itend = v.end();
 165         while (it != itend) {
 166                 if (!is_a<su3one>(it->op(0)))
 167                         s.push_back(*it);
 168                 it++;
 169         }
 170
 171         if (s.empty())
 172                 return color(su3one(), representation_label);
 173         else
 174                 return simplified_ncmul(s);
 175 }
 176
 177 ex color::thisexprseq(const exvector & v) const
 178 {
 179         return color(representation_label, v);
 180 }
 181
 182 ex color::thisexprseq(exvector * vp) const
 183 {
 184         return color(representation_label, vp);
 185 }
 186
 187 /** Given a vector iv3 of three indices and a vector iv2 of two indices that
 188  *  is a subset of iv3, return the (free) index that is in iv3 but not in
 189  *  iv2 and the sign introduced by permuting that index to the front.
 190  *
 191  *  @param iv3 Vector of 3 indices
 192  *  @param iv2 Vector of 2 indices, must be a subset of iv3
 193  *  @param sig Returs sign introduced by index permutation
 194  *  @return the free index (the one that is in iv3 but not in iv2) */
 195 static ex permute_free_index_to_front(const exvector & iv3, const exvector & iv2, int & sig)
 196 {
 197         GINAC_ASSERT(iv3.size() == 3);
 198         GINAC_ASSERT(iv2.size() == 2);
 199
 200         sig = 1;
 201
 202 #define TEST_PERMUTATION(A,B,C,P) \
 203         if (iv3[B].is_equal(iv2[0]) && iv3[C].is_equal(iv2[1])) { \
 204                 sig = P; \
 205                 return iv3[A]; \
 206         }
 207
 208         TEST_PERMUTATION(0,1,2,  1);
 209         TEST_PERMUTATION(0,2,1, -1);
 210         TEST_PERMUTATION(1,0,2, -1);
 211         TEST_PERMUTATION(1,2,0,  1);
 212         TEST_PERMUTATION(2,0,1,  1);
 213         TEST_PERMUTATION(2,1,0, -1);
 214
 215         throw(std::logic_error("permute_free_index_to_front(): no valid permutation found"));
 216 }
 217
 218 /** Automatic symbolic evaluation of indexed symmetric structure constant. */
 219 ex su3d::eval_indexed(const basic & i) const
 220 {
 221         GINAC_ASSERT(is_a<indexed>(i));
 222         GINAC_ASSERT(i.nops() == 4);
 223         GINAC_ASSERT(is_a<su3d>(i.op(0)));
 224
 225         // Convolutions are zero
 226         if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
 227                 return _ex0;
 228
 229         // Numeric evaluation
 230         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 231
 232                 // Sort indices
 233                 int v[3];
 234                 for (unsigned j=0; j<3; j++)
 235                         v[j] = ex_to<numeric>(ex_to<idx>(i.op(j + 1)).get_value()).to_int();
 236                 if (v[0] > v[1]) std::swap(v[0], v[1]);
 237                 if (v[0] > v[2]) std::swap(v[0], v[2]);
 238                 if (v[1] > v[2]) std::swap(v[1], v[2]);
 239
 240 #define CMPINDICES(A,B,C) ((v[0] == (A)) && (v[1] == (B)) && (v[2] == (C)))
 241
 242                 // Check for non-zero elements
 243                 if (CMPINDICES(1,4,6) || CMPINDICES(1,5,7) || CMPINDICES(2,5,6)
 244                  || CMPINDICES(3,4,4) || CMPINDICES(3,5,5))
 245                         return _ex1_2;
 246                 else if (CMPINDICES(2,4,7) || CMPINDICES(3,6,6) || CMPINDICES(3,7,7))
 247                         return _ex_1_2;
 248                 else if (CMPINDICES(1,1,8) || CMPINDICES(2,2,8) || CMPINDICES(3,3,8))
 249                         return sqrt(_ex3)*_ex1_3;
 250                 else if (CMPINDICES(8,8,8))
 251                         return sqrt(_ex3)*_ex_1_3;
 252                 else if (CMPINDICES(4,4,8) || CMPINDICES(5,5,8)
 253                       || CMPINDICES(6,6,8) || CMPINDICES(7,7,8))
 254                         return sqrt(_ex3)/_ex_6;
 255                 else
 256                         return _ex0;
 257         }
 258
 259         // No further simplifications
 260         return i.hold();
 261 }
 262
 263 /** Automatic symbolic evaluation of indexed antisymmetric structure constant. */
 264 ex su3f::eval_indexed(const basic & i) const
 265 {
 266         GINAC_ASSERT(is_a<indexed>(i));
 267         GINAC_ASSERT(i.nops() == 4);
 268         GINAC_ASSERT(is_a<su3f>(i.op(0)));
 269
 270         // Numeric evaluation
 271         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 272
 273                 // Sort indices, remember permutation sign
 274                 int v[3];
 275                 for (unsigned j=0; j<3; j++)
 276                         v[j] = ex_to<numeric>(ex_to<idx>(i.op(j + 1)).get_value()).to_int();
 277                 int sign = 1;
 278                 if (v[0] > v[1]) { std::swap(v[0], v[1]); sign = -sign; }
 279                 if (v[0] > v[2]) { std::swap(v[0], v[2]); sign = -sign; }
 280                 if (v[1] > v[2]) { std::swap(v[1], v[2]); sign = -sign; }
 281
 282                 // Check for non-zero elements
 283                 if (CMPINDICES(1,2,3))
 284                         return sign;
 285                 else if (CMPINDICES(1,4,7) || CMPINDICES(2,4,6)
 286                       || CMPINDICES(2,5,7) || CMPINDICES(3,4,5))
 287                         return _ex1_2 * sign;
 288                 else if (CMPINDICES(1,5,6) || CMPINDICES(3,6,7))
 289                         return _ex_1_2 * sign;
 290                 else if (CMPINDICES(4,5,8) || CMPINDICES(6,7,8))
 291                         return sqrt(_ex3)/2 * sign;
 292                 else
 293                         return _ex0;
 294         }
 295
 296         // No further simplifications
 297         return i.hold();
 298 }
 299
 300
 301 /** Contraction of generator with something else. */
 302 bool su3t::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 303 {
 304         GINAC_ASSERT(is_a<indexed>(*self));
 305         GINAC_ASSERT(is_a<indexed>(*other));
 306         GINAC_ASSERT(self->nops() == 2);
 307         GINAC_ASSERT(is_a<su3t>(self->op(0)));
 308         unsigned char rl = ex_to<color>(*self).get_representation_label();
 309
 310         if (is_exactly_a<su3t>(other->op(0))) {
 311
 312                 // Contraction only makes sense if the represenation labels are equal
 313                 GINAC_ASSERT(is_a<color>(*other));
 314                 if (ex_to<color>(*other).get_representation_label() != rl)
 315                         return false;
 316
 317                 // T.a T.a = 4/3 ONE
 318                 if (other - self == 1) {
 319                         *self = numeric(4, 3);
 320                         *other = color_ONE(rl);
 321                         return true;
 322
 323                 // T.a T.b T.a = -1/6 T.b
 324                 } else if (other - self == 2
 325                         && is_a<color>(self[1])) {
 326                         *self = numeric(-1, 6);
 327                         *other = _ex1;
 328                         return true;
 329
 330                 // T.a S T.a = 1/2 Tr(S) - 1/6 S
 331                 } else {
 332                         exvector::iterator it = self + 1;
 333                         while (it != other) {
 334                                 if (!is_a<color>(*it)) {
 335                                         return false;
 336                                 }
 337                                 it++;
 338                         }
 339
 340                         it = self + 1;
 341                         ex S = _ex1;
 342                         while (it != other) {
 343                                 S *= *it;
 344                                 *it++ = _ex1;
 345                         }
 346
 347                         *self = color_trace(S, rl) * color_ONE(rl) / 2 - S / 6;
 348                         *other = _ex1;
 349                         return true;
 350                 }
 351         }
 352
 353         return false;
 354 }
 355
 356 /** Contraction of an indexed symmetric structure constant with something else. */
 357 bool su3d::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 358 {
 359         GINAC_ASSERT(is_a<indexed>(*self));
 360         GINAC_ASSERT(is_a<indexed>(*other));
 361         GINAC_ASSERT(self->nops() == 4);
 362         GINAC_ASSERT(is_a<su3d>(self->op(0)));
 363
 364         if (is_exactly_a<su3d>(other->op(0))) {
 365
 366                 // Find the dummy indices of the contraction
 367                 exvector self_indices = ex_to<indexed>(*self).get_indices();
 368                 exvector other_indices = ex_to<indexed>(*other).get_indices();
 369                 exvector all_indices = self_indices;
 370                 all_indices.insert(all_indices.end(), other_indices.begin(), other_indices.end());
 371                 exvector free_indices, dummy_indices;
 372                 find_free_and_dummy(all_indices, free_indices, dummy_indices);
 373
 374                 // d.abc d.abc = 40/3
 375                 if (dummy_indices.size() == 3) {
 376                         *self = numeric(40, 3);
 377                         *other = _ex1;
 378                         return true;
 379
 380                 // d.akl d.bkl = 5/3 delta.ab
 381                 } else if (dummy_indices.size() == 2) {
 382                         exvector a;
 383                         std::back_insert_iterator<exvector> ita(a);
 384                         ita = set_difference(self_indices.begin(), self_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
 385                         ita = set_difference(other_indices.begin(), other_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
 386                         GINAC_ASSERT(a.size() == 2);
 387                         *self = numeric(5, 3) * delta_tensor(a[0], a[1]);
 388                         *other = _ex1;
 389                         return true;
 390                 }
 391
 392         } else if (is_exactly_a<su3t>(other->op(0))) {
 393
 394                 // d.abc T.b T.c = 5/6 T.a
 395                 if (other+1 != v.end()
 396                  && is_exactly_a<su3t>(other[1].op(0))
 397                  && ex_to<indexed>(*self).has_dummy_index_for(other[1].op(1))) {
 398
 399                         exvector self_indices = ex_to<indexed>(*self).get_indices();
 400                         exvector dummy_indices;
 401                         dummy_indices.push_back(other[0].op(1));
 402                         dummy_indices.push_back(other[1].op(1));
 403                         int sig;
 404                         ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
 405                         *self = numeric(5, 6);
 406                         other[0] = color_T(a, ex_to<color>(other[0]).get_representation_label());
 407                         other[1] = _ex1;
 408                         return true;
 409                 }
 410         }
 411
 412         return false;
 413 }
 414
 415 /** Contraction of an indexed antisymmetric structure constant with something else. */
 416 bool su3f::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 417 {
 418         GINAC_ASSERT(is_a<indexed>(*self));
 419         GINAC_ASSERT(is_a<indexed>(*other));
 420         GINAC_ASSERT(self->nops() == 4);
 421         GINAC_ASSERT(is_a<su3f>(self->op(0)));
 422
 423         if (is_exactly_a<su3f>(other->op(0))) { // f*d is handled by su3d class
 424
 425                 // Find the dummy indices of the contraction
 426                 exvector dummy_indices;
 427                 dummy_indices = ex_to<indexed>(*self).get_dummy_indices(ex_to<indexed>(*other));
 428
 429                 // f.abc f.abc = 24
 430                 if (dummy_indices.size() == 3) {
 431                         *self = 24;
 432                         *other = _ex1;
 433                         return true;
 434
 435                 // f.akl f.bkl = 3 delta.ab
 436                 } else if (dummy_indices.size() == 2) {
 437                         int sign1, sign2;
 438                         ex a = permute_free_index_to_front(ex_to<indexed>(*self).get_indices(), dummy_indices, sign1);
 439                         ex b = permute_free_index_to_front(ex_to<indexed>(*other).get_indices(), dummy_indices, sign2);
 440                         *self = sign1 * sign2 * 3 * delta_tensor(a, b);
 441                         *other = _ex1;
 442                         return true;
 443                 }
 444
 445         } else if (is_exactly_a<su3t>(other->op(0))) {
 446
 447                 // f.abc T.b T.c = 3/2 I T.a
 448                 if (other+1 != v.end()
 449                  && is_exactly_a<su3t>(other[1].op(0))
 450                  && ex_to<indexed>(*self).has_dummy_index_for(other[1].op(1))) {
 451
 452                         exvector self_indices = ex_to<indexed>(*self).get_indices();
 453                         exvector dummy_indices;
 454                         dummy_indices.push_back(other[0].op(1));
 455                         dummy_indices.push_back(other[1].op(1));
 456                         int sig;
 457                         ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
 458                         *self = numeric(3, 2) * sig * I;
 459                         other[0] = color_T(a, ex_to<color>(other[0]).get_representation_label());
 460                         other[1] = _ex1;
 461                         return true;
 462                 }
 463         }
 464
 465         return false;
 466 }
 467
 468 //////////
 469 // global functions
 470 //////////
 471
 472 ex color_ONE(unsigned char rl)
 473 {
 474         return color(su3one(), rl);
 475 }
 476
 477 ex color_T(const ex & a, unsigned char rl)
 478 {
 479         if (!is_a<idx>(a))
 480                 throw(std::invalid_argument("indices of color_T must be of type idx"));
 481         if (!ex_to<idx>(a).get_dim().is_equal(8))
 482                 throw(std::invalid_argument("index dimension for color_T must be 8"));
 483
 484         return color(su3t(), a, rl);
 485 }
 486
 487 ex color_f(const ex & a, const ex & b, const ex & c)
 488 {
 489         if (!is_a<idx>(a) || !is_a<idx>(b) || !is_a<idx>(c))
 490                 throw(std::invalid_argument("indices of color_f must be of type idx"));
 491         if (!ex_to<idx>(a).get_dim().is_equal(8) || !ex_to<idx>(b).get_dim().is_equal(8) || !ex_to<idx>(c).get_dim().is_equal(8))
 492                 throw(std::invalid_argument("index dimension for color_f must be 8"));
 493
 494         return indexed(su3f(), sy_anti(), a, b, c);
 495 }
 496
 497 ex color_d(const ex & a, const ex & b, const ex & c)
 498 {
 499         if (!is_a<idx>(a) || !is_a<idx>(b) || !is_a<idx>(c))
 500                 throw(std::invalid_argument("indices of color_d must be of type idx"));
 501         if (!ex_to<idx>(a).get_dim().is_equal(8) || !ex_to<idx>(b).get_dim().is_equal(8) || !ex_to<idx>(c).get_dim().is_equal(8))
 502                 throw(std::invalid_argument("index dimension for color_d must be 8"));
 503
 504         return indexed(su3d(), sy_symm(), a, b, c);
 505 }
 506
 507 ex color_h(const ex & a, const ex & b, const ex & c)
 508 {
 509         return color_d(a, b, c) + I * color_f(a, b, c);
 510 }
 511
 512 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 513  *  is that of a color object with the specified representation label. */
 514 static bool is_color_tinfo(unsigned ti, unsigned char rl)
 515 {
 516         return ti == (TINFO_color + rl);
 517 }
 518
 519 ex color_trace(const ex & e, unsigned char rl)
 520 {
 521         if (is_a<color>(e)) {
 522
 523                 if (ex_to<color>(e).get_representation_label() == rl
 524                  && is_a<su3one>(e.op(0)))
 525                         return _ex3;
 526                 else
 527                         return _ex0;
 528
 529         } else if (is_exactly_a<mul>(e)) {
 530
 531                 // Trace of product: pull out non-color factors
 532                 ex prod = _ex1;
 533                 for (unsigned i=0; i<e.nops(); i++) {
 534                         const ex &o = e.op(i);
 535                         if (is_color_tinfo(o.return_type_tinfo(), rl))
 536                                 prod *= color_trace(o, rl);
 537                         else
 538                                 prod *= o;
 539                 }
 540                 return prod;
 541
 542         } else if (is_exactly_a<ncmul>(e)) {
 543
 544                 if (!is_color_tinfo(e.return_type_tinfo(), rl))
 545                         return _ex0;
 546
 547                 // Expand product, if necessary
 548                 ex e_expanded = e.expand();
 549                 if (!is_a<ncmul>(e_expanded))
 550                         return color_trace(e_expanded, rl);
 551
 552                 unsigned num = e.nops();
 553
 554                 if (num == 2) {
 555
 556                         // Tr T_a T_b = 1/2 delta_a_b
 557                         return delta_tensor(e.op(0).op(1), e.op(1).op(1)) / 2;
 558
 559                 } else if (num == 3) {
 560
 561                         // Tr T_a T_b T_c = 1/4 h_a_b_c
 562                         return color_h(e.op(0).op(1), e.op(1).op(1), e.op(2).op(1)) / 4;
 563
 564                 } else {
 565
 566                         // Traces of 4 or more generators are computed recursively:
 567                         // Tr T_a1 .. T_an =
 568                         //     1/6 delta_a(n-1)_an Tr T_a1 .. T_a(n-2)
 569                         //   + 1/2 h_a(n-1)_an_k Tr T_a1 .. T_a(n-2) T_k
 570                         const ex &last_index = e.op(num - 1).op(1);
 571                         const ex &next_to_last_index = e.op(num - 2).op(1);
 572                         idx summation_index((new symbol)->setflag(status_flags::dynallocated), 8);
 573
 574                         exvector v1;
 575                         v1.reserve(num - 2);
 576                         for (unsigned i=0; i<num-2; i++)
 577                                 v1.push_back(e.op(i));
 578
 579                         exvector v2 = v1;
 580                         v2.push_back(color_T(summation_index, rl));
 581
 582                         return delta_tensor(next_to_last_index, last_index) * color_trace(ncmul(v1), rl) / 6
 583                                + color_h(next_to_last_index, last_index, summation_index) * color_trace(ncmul(v2), rl) / 2;
 584                 }
 585
 586         } else if (e.nops() > 0) {
 587
 588                 // Trace maps to all other container classes (this includes sums)
 589                 pointer_to_map_function_1arg<unsigned char> fcn(color_trace, rl);
 590                 return e.map(fcn);
 591
 592         } else
 593                 return _ex0;
 594 }
 595
 596 } // namespace GiNaC