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) 1999-2003 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 constructors
  49 //////////
  50
  51 color::color() : representation_label(0)
  52 {
  53         tinfo_key = TINFO_color;
  54 }
  55
  56 DEFAULT_CTOR(su3one)
  57 DEFAULT_CTOR(su3t)
  58 DEFAULT_CTOR(su3f)
  59 DEFAULT_CTOR(su3d)
  60
  61 //////////
  62 // other constructors
  63 //////////
  64
  65 /** Construct object without any color index. This constructor is for
  66  *  internal use only. Use the color_ONE() function instead.
  67  *  @see color_ONE */
  68 color::color(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
  69 {
  70         tinfo_key = TINFO_color;
  71 }
  72
  73 /** Construct object with one color index. This constructor is for internal
  74  *  use only. Use the color_T() function instead.
  75  *  @see color_T */
  76 color::color(const ex & b, const ex & i1, unsigned char rl) : inherited(b, i1), representation_label(rl)
  77 {
  78         tinfo_key = TINFO_color;
  79 }
  80
  81 color::color(unsigned char rl, const exvector & v, bool discardable) : inherited(sy_none(), v, discardable), representation_label(rl)
  82 {
  83         tinfo_key = TINFO_color;
  84 }
  85
  86 color::color(unsigned char rl, exvector * vp) : inherited(sy_none(), vp), representation_label(rl)
  87 {
  88         tinfo_key = TINFO_color;
  89 }
  90
  91 //////////
  92 // archiving
  93 //////////
  94
  95 color::color(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
  96 {
  97         unsigned rl;
  98         n.find_unsigned("label", rl);
  99         representation_label = rl;
 100 }
 101
 102 void color::archive(archive_node &n) const
 103 {
 104         inherited::archive(n);
 105         n.add_unsigned("label", representation_label);
 106 }
 107
 108 DEFAULT_UNARCHIVE(color)
 109 DEFAULT_ARCHIVING(su3one)
 110 DEFAULT_ARCHIVING(su3t)
 111 DEFAULT_ARCHIVING(su3f)
 112 DEFAULT_ARCHIVING(su3d)
 113
 114 //////////
 115 // functions overriding virtual functions from base classes
 116 //////////
 117
 118 int color::compare_same_type(const basic & other) const
 119 {
 120         GINAC_ASSERT(is_a<color>(other));
 121         const color &o = static_cast<const color &>(other);
 122
 123         if (representation_label != o.representation_label) {
 124                 // different representation label
 125                 return representation_label < o.representation_label ? -1 : 1;
 126         }
 127
 128         return inherited::compare_same_type(other);
 129 }
 130
 131 bool color::match_same_type(const basic & other) const
 132 {
 133         GINAC_ASSERT(is_a<color>(other));
 134         const color &o = static_cast<const color &>(other);
 135
 136         return representation_label == o.representation_label;
 137 }
 138
 139 DEFAULT_COMPARE(su3one)
 140 DEFAULT_COMPARE(su3t)
 141 DEFAULT_COMPARE(su3f)
 142 DEFAULT_COMPARE(su3d)
 143
 144 DEFAULT_PRINT_LATEX(su3one, "ONE", "\\mathbb{1}")
 145 DEFAULT_PRINT(su3t, "T")
 146 DEFAULT_PRINT(su3f, "f")
 147 DEFAULT_PRINT(su3d, "d")
 148
 149 /** Perform automatic simplification on noncommutative product of color
 150  *  objects. This removes superfluous ONEs. */
 151 ex color::eval_ncmul(const exvector & v) const
 152 {
 153         exvector s;
 154         s.reserve(v.size());
 155
 156         // Remove superfluous ONEs
 157         exvector::const_iterator it = v.begin(), itend = v.end();
 158         while (it != itend) {
 159                 if (!is_a<su3one>(it->op(0)))
 160                         s.push_back(*it);
 161                 it++;
 162         }
 163
 164         if (s.empty())
 165                 return color(su3one(), representation_label);
 166         else
 167                 return hold_ncmul(s);
 168 }
 169
 170 ex color::thiscontainer(const exvector & v) const
 171 {
 172         return color(representation_label, v);
 173 }
 174
 175 ex color::thiscontainer(exvector * vp) const
 176 {
 177         return color(representation_label, vp);
 178 }
 179
 180 /** Given a vector iv3 of three indices and a vector iv2 of two indices that
 181  *  is a subset of iv3, return the (free) index that is in iv3 but not in
 182  *  iv2 and the sign introduced by permuting that index to the front.
 183  *
 184  *  @param iv3 Vector of 3 indices
 185  *  @param iv2 Vector of 2 indices, must be a subset of iv3
 186  *  @param sig Returs sign introduced by index permutation
 187  *  @return the free index (the one that is in iv3 but not in iv2) */
 188 static ex permute_free_index_to_front(const exvector & iv3, const exvector & iv2, int & sig)
 189 {
 190         GINAC_ASSERT(iv3.size() == 3);
 191         GINAC_ASSERT(iv2.size() == 2);
 192
 193         sig = 1;
 194
 195 #define TEST_PERMUTATION(A,B,C,P) \
 196         if (iv3[B].is_equal(iv2[0]) && iv3[C].is_equal(iv2[1])) { \
 197                 sig = P; \
 198                 return iv3[A]; \
 199         }
 200
 201         TEST_PERMUTATION(0,1,2,  1);
 202         TEST_PERMUTATION(0,2,1, -1);
 203         TEST_PERMUTATION(1,0,2, -1);
 204         TEST_PERMUTATION(1,2,0,  1);
 205         TEST_PERMUTATION(2,0,1,  1);
 206         TEST_PERMUTATION(2,1,0, -1);
 207
 208         throw(std::logic_error("permute_free_index_to_front(): no valid permutation found"));
 209 }
 210
 211 /** Automatic symbolic evaluation of indexed symmetric structure constant. */
 212 ex su3d::eval_indexed(const basic & i) const
 213 {
 214         GINAC_ASSERT(is_a<indexed>(i));
 215         GINAC_ASSERT(i.nops() == 4);
 216         GINAC_ASSERT(is_a<su3d>(i.op(0)));
 217
 218         // Convolutions are zero
 219         if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
 220                 return _ex0;
 221
 222         // Numeric evaluation
 223         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 224
 225                 // Sort indices
 226                 int v[3];
 227                 for (unsigned j=0; j<3; j++)
 228                         v[j] = ex_to<numeric>(ex_to<idx>(i.op(j + 1)).get_value()).to_int();
 229                 if (v[0] > v[1]) std::swap(v[0], v[1]);
 230                 if (v[0] > v[2]) std::swap(v[0], v[2]);
 231                 if (v[1] > v[2]) std::swap(v[1], v[2]);
 232
 233 #define CMPINDICES(A,B,C) ((v[0] == (A)) && (v[1] == (B)) && (v[2] == (C)))
 234
 235                 // Check for non-zero elements
 236                 if (CMPINDICES(1,4,6) || CMPINDICES(1,5,7) || CMPINDICES(2,5,6)
 237                  || CMPINDICES(3,4,4) || CMPINDICES(3,5,5))
 238                         return _ex1_2;
 239                 else if (CMPINDICES(2,4,7) || CMPINDICES(3,6,6) || CMPINDICES(3,7,7))
 240                         return _ex_1_2;
 241                 else if (CMPINDICES(1,1,8) || CMPINDICES(2,2,8) || CMPINDICES(3,3,8))
 242                         return sqrt(_ex3)*_ex1_3;
 243                 else if (CMPINDICES(8,8,8))
 244                         return sqrt(_ex3)*_ex_1_3;
 245                 else if (CMPINDICES(4,4,8) || CMPINDICES(5,5,8)
 246                       || CMPINDICES(6,6,8) || CMPINDICES(7,7,8))
 247                         return sqrt(_ex3)/_ex_6;
 248                 else
 249                         return _ex0;
 250         }
 251
 252         // No further simplifications
 253         return i.hold();
 254 }
 255
 256 /** Automatic symbolic evaluation of indexed antisymmetric structure constant. */
 257 ex su3f::eval_indexed(const basic & i) const
 258 {
 259         GINAC_ASSERT(is_a<indexed>(i));
 260         GINAC_ASSERT(i.nops() == 4);
 261         GINAC_ASSERT(is_a<su3f>(i.op(0)));
 262
 263         // Numeric evaluation
 264         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 265
 266                 // Sort indices, remember permutation sign
 267                 int v[3];
 268                 for (unsigned j=0; j<3; j++)
 269                         v[j] = ex_to<numeric>(ex_to<idx>(i.op(j + 1)).get_value()).to_int();
 270                 int sign = 1;
 271                 if (v[0] > v[1]) { std::swap(v[0], v[1]); sign = -sign; }
 272                 if (v[0] > v[2]) { std::swap(v[0], v[2]); sign = -sign; }
 273                 if (v[1] > v[2]) { std::swap(v[1], v[2]); sign = -sign; }
 274
 275                 // Check for non-zero elements
 276                 if (CMPINDICES(1,2,3))
 277                         return sign;
 278                 else if (CMPINDICES(1,4,7) || CMPINDICES(2,4,6)
 279                       || CMPINDICES(2,5,7) || CMPINDICES(3,4,5))
 280                         return _ex1_2 * sign;
 281                 else if (CMPINDICES(1,5,6) || CMPINDICES(3,6,7))
 282                         return _ex_1_2 * sign;
 283                 else if (CMPINDICES(4,5,8) || CMPINDICES(6,7,8))
 284                         return sqrt(_ex3)/2 * sign;
 285                 else
 286                         return _ex0;
 287         }
 288
 289         // No further simplifications
 290         return i.hold();
 291 }
 292
 293
 294 /** Contraction of generator with something else. */
 295 bool su3t::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 296 {
 297         GINAC_ASSERT(is_a<indexed>(*self));
 298         GINAC_ASSERT(is_a<indexed>(*other));
 299         GINAC_ASSERT(self->nops() == 2);
 300         GINAC_ASSERT(is_a<su3t>(self->op(0)));
 301         unsigned char rl = ex_to<color>(*self).get_representation_label();
 302
 303         if (is_exactly_a<su3t>(other->op(0))) {
 304
 305                 // Contraction only makes sense if the represenation labels are equal
 306                 GINAC_ASSERT(is_a<color>(*other));
 307                 if (ex_to<color>(*other).get_representation_label() != rl)
 308                         return false;
 309
 310                 // T.a T.a = 4/3 ONE
 311                 if (other - self == 1) {
 312                         *self = numeric(4, 3);
 313                         *other = color_ONE(rl);
 314                         return true;
 315
 316                 // T.a T.b T.a = -1/6 T.b
 317                 } else if (other - self == 2
 318                         && is_a<color>(self[1])) {
 319                         *self = numeric(-1, 6);
 320                         *other = _ex1;
 321                         return true;
 322
 323                 // T.a S T.a = 1/2 Tr(S) - 1/6 S
 324                 } else {
 325                         exvector::iterator it = self + 1;
 326                         while (it != other) {
 327                                 if (!is_a<color>(*it)) {
 328                                         return false;
 329                                 }
 330                                 it++;
 331                         }
 332
 333                         it = self + 1;
 334                         ex S = _ex1;
 335                         while (it != other) {
 336                                 S *= *it;
 337                                 *it++ = _ex1;
 338                         }
 339
 340                         *self = color_trace(S, rl) * color_ONE(rl) / 2 - S / 6;
 341                         *other = _ex1;
 342                         return true;
 343                 }
 344         }
 345
 346         return false;
 347 }
 348
 349 /** Contraction of an indexed symmetric structure constant with something else. */
 350 bool su3d::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 351 {
 352         GINAC_ASSERT(is_a<indexed>(*self));
 353         GINAC_ASSERT(is_a<indexed>(*other));
 354         GINAC_ASSERT(self->nops() == 4);
 355         GINAC_ASSERT(is_a<su3d>(self->op(0)));
 356
 357         if (is_exactly_a<su3d>(other->op(0))) {
 358
 359                 // Find the dummy indices of the contraction
 360                 exvector self_indices = ex_to<indexed>(*self).get_indices();
 361                 exvector other_indices = ex_to<indexed>(*other).get_indices();
 362                 exvector all_indices = self_indices;
 363                 all_indices.insert(all_indices.end(), other_indices.begin(), other_indices.end());
 364                 exvector free_indices, dummy_indices;
 365                 find_free_and_dummy(all_indices, free_indices, dummy_indices);
 366
 367                 // d.abc d.abc = 40/3
 368                 if (dummy_indices.size() == 3) {
 369                         *self = numeric(40, 3);
 370                         *other = _ex1;
 371                         return true;
 372
 373                 // d.akl d.bkl = 5/3 delta.ab
 374                 } else if (dummy_indices.size() == 2) {
 375                         exvector a;
 376                         std::back_insert_iterator<exvector> ita(a);
 377                         ita = set_difference(self_indices.begin(), self_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
 378                         ita = set_difference(other_indices.begin(), other_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
 379                         GINAC_ASSERT(a.size() == 2);
 380                         *self = numeric(5, 3) * delta_tensor(a[0], a[1]);
 381                         *other = _ex1;
 382                         return true;
 383                 }
 384
 385         } else if (is_exactly_a<su3t>(other->op(0))) {
 386
 387                 // d.abc T.b T.c = 5/6 T.a
 388                 if (other+1 != v.end()
 389                  && is_exactly_a<su3t>(other[1].op(0))
 390                  && ex_to<indexed>(*self).has_dummy_index_for(other[1].op(1))) {
 391
 392                         exvector self_indices = ex_to<indexed>(*self).get_indices();
 393                         exvector dummy_indices;
 394                         dummy_indices.push_back(other[0].op(1));
 395                         dummy_indices.push_back(other[1].op(1));
 396                         int sig;
 397                         ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
 398                         *self = numeric(5, 6);
 399                         other[0] = color_T(a, ex_to<color>(other[0]).get_representation_label());
 400                         other[1] = _ex1;
 401                         return true;
 402                 }
 403         }
 404
 405         return false;
 406 }
 407
 408 /** Contraction of an indexed antisymmetric structure constant with something else. */
 409 bool su3f::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 410 {
 411         GINAC_ASSERT(is_a<indexed>(*self));
 412         GINAC_ASSERT(is_a<indexed>(*other));
 413         GINAC_ASSERT(self->nops() == 4);
 414         GINAC_ASSERT(is_a<su3f>(self->op(0)));
 415
 416         if (is_exactly_a<su3f>(other->op(0))) { // f*d is handled by su3d class
 417
 418                 // Find the dummy indices of the contraction
 419                 exvector dummy_indices;
 420                 dummy_indices = ex_to<indexed>(*self).get_dummy_indices(ex_to<indexed>(*other));
 421
 422                 // f.abc f.abc = 24
 423                 if (dummy_indices.size() == 3) {
 424                         *self = 24;
 425                         *other = _ex1;
 426                         return true;
 427
 428                 // f.akl f.bkl = 3 delta.ab
 429                 } else if (dummy_indices.size() == 2) {
 430                         int sign1, sign2;
 431                         ex a = permute_free_index_to_front(ex_to<indexed>(*self).get_indices(), dummy_indices, sign1);
 432                         ex b = permute_free_index_to_front(ex_to<indexed>(*other).get_indices(), dummy_indices, sign2);
 433                         *self = sign1 * sign2 * 3 * delta_tensor(a, b);
 434                         *other = _ex1;
 435                         return true;
 436                 }
 437
 438         } else if (is_exactly_a<su3t>(other->op(0))) {
 439
 440                 // f.abc T.b T.c = 3/2 I T.a
 441                 if (other+1 != v.end()
 442                  && is_exactly_a<su3t>(other[1].op(0))
 443                  && ex_to<indexed>(*self).has_dummy_index_for(other[1].op(1))) {
 444
 445                         exvector self_indices = ex_to<indexed>(*self).get_indices();
 446                         exvector dummy_indices;
 447                         dummy_indices.push_back(other[0].op(1));
 448                         dummy_indices.push_back(other[1].op(1));
 449                         int sig;
 450                         ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
 451                         *self = numeric(3, 2) * sig * I;
 452                         other[0] = color_T(a, ex_to<color>(other[0]).get_representation_label());
 453                         other[1] = _ex1;
 454                         return true;
 455                 }
 456         }
 457
 458         return false;
 459 }
 460
 461 //////////
 462 // global functions
 463 //////////
 464
 465 ex color_ONE(unsigned char rl)
 466 {
 467         return color(su3one(), rl);
 468 }
 469
 470 ex color_T(const ex & a, unsigned char rl)
 471 {
 472         if (!is_a<idx>(a))
 473                 throw(std::invalid_argument("indices of color_T must be of type idx"));
 474         if (!ex_to<idx>(a).get_dim().is_equal(8))
 475                 throw(std::invalid_argument("index dimension for color_T must be 8"));
 476
 477         return color(su3t(), a, rl);
 478 }
 479
 480 ex color_f(const ex & a, const ex & b, const ex & c)
 481 {
 482         if (!is_a<idx>(a) || !is_a<idx>(b) || !is_a<idx>(c))
 483                 throw(std::invalid_argument("indices of color_f must be of type idx"));
 484         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))
 485                 throw(std::invalid_argument("index dimension for color_f must be 8"));
 486
 487         return indexed(su3f(), sy_anti(), a, b, c);
 488 }
 489
 490 ex color_d(const ex & a, const ex & b, const ex & c)
 491 {
 492         if (!is_a<idx>(a) || !is_a<idx>(b) || !is_a<idx>(c))
 493                 throw(std::invalid_argument("indices of color_d must be of type idx"));
 494         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))
 495                 throw(std::invalid_argument("index dimension for color_d must be 8"));
 496
 497         return indexed(su3d(), sy_symm(), a, b, c);
 498 }
 499
 500 ex color_h(const ex & a, const ex & b, const ex & c)
 501 {
 502         return color_d(a, b, c) + I * color_f(a, b, c);
 503 }
 504
 505 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 506  *  is that of a color object with the specified representation label. */
 507 static bool is_color_tinfo(unsigned ti, unsigned char rl)
 508 {
 509         return ti == (TINFO_color + rl);
 510 }
 511
 512 ex color_trace(const ex & e, unsigned char rl)
 513 {
 514         if (is_a<color>(e)) {
 515
 516                 if (ex_to<color>(e).get_representation_label() == rl
 517                  && is_a<su3one>(e.op(0)))
 518                         return _ex3;
 519                 else
 520                         return _ex0;
 521
 522         } else if (is_exactly_a<mul>(e)) {
 523
 524                 // Trace of product: pull out non-color factors
 525                 ex prod = _ex1;
 526                 for (size_t i=0; i<e.nops(); i++) {
 527                         const ex &o = e.op(i);
 528                         if (is_color_tinfo(o.return_type_tinfo(), rl))
 529                                 prod *= color_trace(o, rl);
 530                         else
 531                                 prod *= o;
 532                 }
 533                 return prod;
 534
 535         } else if (is_exactly_a<ncmul>(e)) {
 536
 537                 if (!is_color_tinfo(e.return_type_tinfo(), rl))
 538                         return _ex0;
 539
 540                 // Expand product, if necessary
 541                 ex e_expanded = e.expand();
 542                 if (!is_a<ncmul>(e_expanded))
 543                         return color_trace(e_expanded, rl);
 544
 545                 size_t num = e.nops();
 546
 547                 if (num == 2) {
 548
 549                         // Tr T_a T_b = 1/2 delta_a_b
 550                         return delta_tensor(e.op(0).op(1), e.op(1).op(1)) / 2;
 551
 552                 } else if (num == 3) {
 553
 554                         // Tr T_a T_b T_c = 1/4 h_a_b_c
 555                         return color_h(e.op(0).op(1), e.op(1).op(1), e.op(2).op(1)) / 4;
 556
 557                 } else {
 558
 559                         // Traces of 4 or more generators are computed recursively:
 560                         // Tr T_a1 .. T_an =
 561                         //     1/6 delta_a(n-1)_an Tr T_a1 .. T_a(n-2)
 562                         //   + 1/2 h_a(n-1)_an_k Tr T_a1 .. T_a(n-2) T_k
 563                         const ex &last_index = e.op(num - 1).op(1);
 564                         const ex &next_to_last_index = e.op(num - 2).op(1);
 565                         idx summation_index((new symbol)->setflag(status_flags::dynallocated), 8);
 566
 567                         exvector v1;
 568                         v1.reserve(num - 2);
 569                         for (size_t i=0; i<num-2; i++)
 570                                 v1.push_back(e.op(i));
 571
 572                         exvector v2 = v1;
 573                         v2.push_back(color_T(summation_index, rl));
 574
 575                         return delta_tensor(next_to_last_index, last_index) * color_trace(ncmul(v1), rl) / 6
 576                                + color_h(next_to_last_index, last_index, summation_index) * color_trace(ncmul(v2), rl) / 2;
 577                 }
 578
 579         } else if (e.nops() > 0) {
 580
 581                 // Trace maps to all other container classes (this includes sums)
 582                 pointer_to_map_function_1arg<unsigned char> fcn(color_trace, rl);
 583                 return e.map(fcn);
 584
 585         } else
 586                 return _ex0;
 587 }
 588
 589 } // namespace GiNaC