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) 1999-2007 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., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  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 "archive.h"
  36 #include "utils.h"
  37
  38 namespace GiNaC {
  39
  40 GINAC_IMPLEMENT_REGISTERED_CLASS(color, indexed)
  41
  42 const tinfo_static_t color::return_type_tinfo_static[256] = {{}};
  43
  44 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(su3one, tensor,
  45   print_func<print_dflt>(&su3one::do_print).
  46   print_func<print_latex>(&su3one::do_print_latex))
  47
  48 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(su3t, tensor,
  49   print_func<print_dflt>(&su3t::do_print).
  50   print_func<print_latex>(&su3t::do_print))
  51
  52 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(su3f, tensor,
  53   print_func<print_dflt>(&su3f::do_print).
  54   print_func<print_latex>(&su3f::do_print))
  55
  56 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(su3d, tensor,
  57   print_func<print_dflt>(&su3d::do_print).
  58   print_func<print_latex>(&su3d::do_print))
  59
  60 //////////
  61 // default constructors
  62 //////////
  63
  64 color::color() : representation_label(0)
  65 {
  66         tinfo_key = &color::tinfo_static;
  67 }
  68
  69 DEFAULT_CTOR(su3one)
  70 DEFAULT_CTOR(su3t)
  71 DEFAULT_CTOR(su3f)
  72 DEFAULT_CTOR(su3d)
  73
  74 //////////
  75 // other constructors
  76 //////////
  77
  78 /** Construct object without any color index. This constructor is for
  79  *  internal use only. Use the color_ONE() function instead.
  80  *  @see color_ONE */
  81 color::color(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
  82 {
  83         tinfo_key = &color::tinfo_static;
  84 }
  85
  86 /** Construct object with one color index. This constructor is for internal
  87  *  use only. Use the color_T() function instead.
  88  *  @see color_T */
  89 color::color(const ex & b, const ex & i1, unsigned char rl) : inherited(b, i1), representation_label(rl)
  90 {
  91         tinfo_key = &color::tinfo_static;
  92 }
  93
  94 color::color(unsigned char rl, const exvector & v, bool discardable) : inherited(not_symmetric(), v, discardable), representation_label(rl)
  95 {
  96         tinfo_key = &color::tinfo_static;
  97 }
  98
  99 color::color(unsigned char rl, std::auto_ptr<exvector> vp) : inherited(not_symmetric(), vp), representation_label(rl)
 100 {
 101         tinfo_key = &color::tinfo_static;
 102 }
 103
 104 //////////
 105 // archiving
 106 //////////
 107
 108 color::color(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
 109 {
 110         unsigned rl;
 111         n.find_unsigned("label", rl);
 112         representation_label = rl;
 113 }
 114
 115 void color::archive(archive_node &n) const
 116 {
 117         inherited::archive(n);
 118         n.add_unsigned("label", representation_label);
 119 }
 120
 121 DEFAULT_UNARCHIVE(color)
 122 DEFAULT_ARCHIVING(su3one)
 123 DEFAULT_ARCHIVING(su3t)
 124 DEFAULT_ARCHIVING(su3f)
 125 DEFAULT_ARCHIVING(su3d)
 126
 127 //////////
 128 // functions overriding virtual functions from base classes
 129 //////////
 130
 131 int color::compare_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         if (representation_label != o.representation_label) {
 137                 // different representation label
 138                 return representation_label < o.representation_label ? -1 : 1;
 139         }
 140
 141         return inherited::compare_same_type(other);
 142 }
 143
 144 bool color::match_same_type(const basic & other) const
 145 {
 146         GINAC_ASSERT(is_a<color>(other));
 147         const color &o = static_cast<const color &>(other);
 148
 149         return representation_label == o.representation_label;
 150 }
 151
 152 DEFAULT_COMPARE(su3one)
 153 DEFAULT_COMPARE(su3t)
 154 DEFAULT_COMPARE(su3f)
 155 DEFAULT_COMPARE(su3d)
 156
 157 DEFAULT_PRINT_LATEX(su3one, "ONE", "\\mathbb{1}")
 158 DEFAULT_PRINT(su3t, "T")
 159 DEFAULT_PRINT(su3f, "f")
 160 DEFAULT_PRINT(su3d, "d")
 161
 162 /** Perform automatic simplification on noncommutative product of color
 163  *  objects. This removes superfluous ONEs. */
 164 ex color::eval_ncmul(const exvector & v) const
 165 {
 166         exvector s;
 167         s.reserve(v.size());
 168
 169         // Remove superfluous ONEs
 170         exvector::const_iterator it = v.begin(), itend = v.end();
 171         while (it != itend) {
 172                 if (!is_a<su3one>(it->op(0)))
 173                         s.push_back(*it);
 174                 it++;
 175         }
 176
 177         if (s.empty())
 178                 return color(su3one(), representation_label);
 179         else
 180                 return hold_ncmul(s);
 181 }
 182
 183 ex color::thiscontainer(const exvector & v) const
 184 {
 185         return color(representation_label, v);
 186 }
 187
 188 ex color::thiscontainer(std::auto_ptr<exvector> vp) const
 189 {
 190         return color(representation_label, vp);
 191 }
 192
 193 /** Given a vector iv3 of three indices and a vector iv2 of two indices that
 194  *  is a subset of iv3, return the (free) index that is in iv3 but not in
 195  *  iv2 and the sign introduced by permuting that index to the front.
 196  *
 197  *  @param iv3 Vector of 3 indices
 198  *  @param iv2 Vector of 2 indices, must be a subset of iv3
 199  *  @param sig Returs sign introduced by index permutation
 200  *  @return the free index (the one that is in iv3 but not in iv2) */
 201 static ex permute_free_index_to_front(const exvector & iv3, const exvector & iv2, int & sig)
 202 {
 203         GINAC_ASSERT(iv3.size() == 3);
 204         GINAC_ASSERT(iv2.size() == 2);
 205
 206         sig = 1;
 207
 208 #define TEST_PERMUTATION(A,B,C,P) \
 209         if (iv3[B].is_equal(iv2[0]) && iv3[C].is_equal(iv2[1])) { \
 210                 sig = P; \
 211                 return iv3[A]; \
 212         }
 213
 214         TEST_PERMUTATION(0,1,2,  1);
 215         TEST_PERMUTATION(0,2,1, -1);
 216         TEST_PERMUTATION(1,0,2, -1);
 217         TEST_PERMUTATION(1,2,0,  1);
 218         TEST_PERMUTATION(2,0,1,  1);
 219         TEST_PERMUTATION(2,1,0, -1);
 220
 221         throw(std::logic_error("permute_free_index_to_front(): no valid permutation found"));
 222 }
 223
 224 /** Automatic symbolic evaluation of indexed symmetric structure constant. */
 225 ex su3d::eval_indexed(const basic & i) const
 226 {
 227         GINAC_ASSERT(is_a<indexed>(i));
 228         GINAC_ASSERT(i.nops() == 4);
 229         GINAC_ASSERT(is_a<su3d>(i.op(0)));
 230
 231         // Convolutions are zero
 232         if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
 233                 return _ex0;
 234
 235         // Numeric evaluation
 236         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 237
 238                 // Sort indices
 239                 int v[3];
 240                 for (unsigned j=0; j<3; j++)
 241                         v[j] = ex_to<numeric>(ex_to<idx>(i.op(j + 1)).get_value()).to_int();
 242                 if (v[0] > v[1]) std::swap(v[0], v[1]);
 243                 if (v[0] > v[2]) std::swap(v[0], v[2]);
 244                 if (v[1] > v[2]) std::swap(v[1], v[2]);
 245
 246 #define CMPINDICES(A,B,C) ((v[0] == (A)) && (v[1] == (B)) && (v[2] == (C)))
 247
 248                 // Check for non-zero elements
 249                 if (CMPINDICES(1,4,6) || CMPINDICES(1,5,7) || CMPINDICES(2,5,6)
 250                  || CMPINDICES(3,4,4) || CMPINDICES(3,5,5))
 251                         return _ex1_2;
 252                 else if (CMPINDICES(2,4,7) || CMPINDICES(3,6,6) || CMPINDICES(3,7,7))
 253                         return _ex_1_2;
 254                 else if (CMPINDICES(1,1,8) || CMPINDICES(2,2,8) || CMPINDICES(3,3,8))
 255                         return sqrt(_ex3)*_ex1_3;
 256                 else if (CMPINDICES(8,8,8))
 257                         return sqrt(_ex3)*_ex_1_3;
 258                 else if (CMPINDICES(4,4,8) || CMPINDICES(5,5,8)
 259                       || CMPINDICES(6,6,8) || CMPINDICES(7,7,8))
 260                         return sqrt(_ex3)/_ex_6;
 261                 else
 262                         return _ex0;
 263         }
 264
 265         // No further simplifications
 266         return i.hold();
 267 }
 268
 269 /** Automatic symbolic evaluation of indexed antisymmetric structure constant. */
 270 ex su3f::eval_indexed(const basic & i) const
 271 {
 272         GINAC_ASSERT(is_a<indexed>(i));
 273         GINAC_ASSERT(i.nops() == 4);
 274         GINAC_ASSERT(is_a<su3f>(i.op(0)));
 275
 276         // Numeric evaluation
 277         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 278
 279                 // Sort indices, remember permutation sign
 280                 int v[3];
 281                 for (unsigned j=0; j<3; j++)
 282                         v[j] = ex_to<numeric>(ex_to<idx>(i.op(j + 1)).get_value()).to_int();
 283                 int sign = 1;
 284                 if (v[0] > v[1]) { std::swap(v[0], v[1]); sign = -sign; }
 285                 if (v[0] > v[2]) { std::swap(v[0], v[2]); sign = -sign; }
 286                 if (v[1] > v[2]) { std::swap(v[1], v[2]); sign = -sign; }
 287
 288                 // Check for non-zero elements
 289                 if (CMPINDICES(1,2,3))
 290                         return sign;
 291                 else if (CMPINDICES(1,4,7) || CMPINDICES(2,4,6)
 292                       || CMPINDICES(2,5,7) || CMPINDICES(3,4,5))
 293                         return _ex1_2 * sign;
 294                 else if (CMPINDICES(1,5,6) || CMPINDICES(3,6,7))
 295                         return _ex_1_2 * sign;
 296                 else if (CMPINDICES(4,5,8) || CMPINDICES(6,7,8))
 297                         return sqrt(_ex3)/2 * sign;
 298                 else
 299                         return _ex0;
 300         }
 301
 302         // No further simplifications
 303         return i.hold();
 304 }
 305
 306
 307 /** Contraction of generator with something else. */
 308 bool su3t::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 309 {
 310         GINAC_ASSERT(is_a<indexed>(*self));
 311         GINAC_ASSERT(is_a<indexed>(*other));
 312         GINAC_ASSERT(self->nops() == 2);
 313         GINAC_ASSERT(is_a<su3t>(self->op(0)));
 314         unsigned char rl = ex_to<color>(*self).get_representation_label();
 315
 316         if (is_exactly_a<su3t>(other->op(0))) {
 317
 318                 // Contraction only makes sense if the represenation labels are equal
 319                 GINAC_ASSERT(is_a<color>(*other));
 320                 if (ex_to<color>(*other).get_representation_label() != rl)
 321                         return false;
 322
 323                 // T.a T.a = 4/3 ONE
 324                 if (other - self == 1) {
 325                         *self = numeric(4, 3);
 326                         *other = color_ONE(rl);
 327                         return true;
 328
 329                 // T.a T.b T.a = -1/6 T.b
 330                 } else if (other - self == 2
 331                         && is_a<color>(self[1])) {
 332                         *self = numeric(-1, 6);
 333                         *other = _ex1;
 334                         return true;
 335
 336                 // T.a S T.a = 1/2 Tr(S) - 1/6 S
 337                 } else {
 338                         exvector::iterator it = self + 1;
 339                         while (it != other) {
 340                                 if (!is_a<color>(*it)) {
 341                                         return false;
 342                                 }
 343                                 it++;
 344                         }
 345
 346                         it = self + 1;
 347                         ex S = _ex1;
 348                         while (it != other) {
 349                                 S *= *it;
 350                                 *it++ = _ex1;
 351                         }
 352
 353                         *self = color_trace(S, rl) * color_ONE(rl) / 2 - S / 6;
 354                         *other = _ex1;
 355                         return true;
 356                 }
 357         }
 358
 359         return false;
 360 }
 361
 362 /** Contraction of an indexed symmetric structure constant with something else. */
 363 bool su3d::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 364 {
 365         GINAC_ASSERT(is_a<indexed>(*self));
 366         GINAC_ASSERT(is_a<indexed>(*other));
 367         GINAC_ASSERT(self->nops() == 4);
 368         GINAC_ASSERT(is_a<su3d>(self->op(0)));
 369
 370         if (is_exactly_a<su3d>(other->op(0))) {
 371
 372                 // Find the dummy indices of the contraction
 373                 exvector self_indices = ex_to<indexed>(*self).get_indices();
 374                 exvector other_indices = ex_to<indexed>(*other).get_indices();
 375                 exvector all_indices = self_indices;
 376                 all_indices.insert(all_indices.end(), other_indices.begin(), other_indices.end());
 377                 exvector free_indices, dummy_indices;
 378                 find_free_and_dummy(all_indices, free_indices, dummy_indices);
 379
 380                 // d.abc d.abc = 40/3
 381                 if (dummy_indices.size() == 3) {
 382                         *self = numeric(40, 3);
 383                         *other = _ex1;
 384                         return true;
 385
 386                 // d.akl d.bkl = 5/3 delta.ab
 387                 } else if (dummy_indices.size() == 2) {
 388                         exvector a;
 389                         std::back_insert_iterator<exvector> ita(a);
 390                         ita = set_difference(self_indices.begin(), self_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
 391                         ita = set_difference(other_indices.begin(), other_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
 392                         GINAC_ASSERT(a.size() == 2);
 393                         *self = numeric(5, 3) * delta_tensor(a[0], a[1]);
 394                         *other = _ex1;
 395                         return true;
 396                 }
 397
 398         } else if (is_exactly_a<su3t>(other->op(0))) {
 399
 400                 // d.abc T.b T.c = 5/6 T.a
 401                 if (other+1 != v.end()
 402                  && is_exactly_a<su3t>(other[1].op(0))
 403                  && ex_to<indexed>(*self).has_dummy_index_for(other[1].op(1))) {
 404
 405                         exvector self_indices = ex_to<indexed>(*self).get_indices();
 406                         exvector dummy_indices;
 407                         dummy_indices.push_back(other[0].op(1));
 408                         dummy_indices.push_back(other[1].op(1));
 409                         int sig;
 410                         ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
 411                         *self = numeric(5, 6);
 412                         other[0] = color_T(a, ex_to<color>(other[0]).get_representation_label());
 413                         other[1] = _ex1;
 414                         return true;
 415                 }
 416         }
 417
 418         return false;
 419 }
 420
 421 /** Contraction of an indexed antisymmetric structure constant with something else. */
 422 bool su3f::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 423 {
 424         GINAC_ASSERT(is_a<indexed>(*self));
 425         GINAC_ASSERT(is_a<indexed>(*other));
 426         GINAC_ASSERT(self->nops() == 4);
 427         GINAC_ASSERT(is_a<su3f>(self->op(0)));
 428
 429         if (is_exactly_a<su3f>(other->op(0))) { // f*d is handled by su3d class
 430
 431                 // Find the dummy indices of the contraction
 432                 exvector dummy_indices;
 433                 dummy_indices = ex_to<indexed>(*self).get_dummy_indices(ex_to<indexed>(*other));
 434
 435                 // f.abc f.abc = 24
 436                 if (dummy_indices.size() == 3) {
 437                         *self = 24;
 438                         *other = _ex1;
 439                         return true;
 440
 441                 // f.akl f.bkl = 3 delta.ab
 442                 } else if (dummy_indices.size() == 2) {
 443                         int sign1, sign2;
 444                         ex a = permute_free_index_to_front(ex_to<indexed>(*self).get_indices(), dummy_indices, sign1);
 445                         ex b = permute_free_index_to_front(ex_to<indexed>(*other).get_indices(), dummy_indices, sign2);
 446                         *self = sign1 * sign2 * 3 * delta_tensor(a, b);
 447                         *other = _ex1;
 448                         return true;
 449                 }
 450
 451         } else if (is_exactly_a<su3t>(other->op(0))) {
 452
 453                 // f.abc T.b T.c = 3/2 I T.a
 454                 if (other+1 != v.end()
 455                  && is_exactly_a<su3t>(other[1].op(0))
 456                  && ex_to<indexed>(*self).has_dummy_index_for(other[1].op(1))) {
 457
 458                         exvector self_indices = ex_to<indexed>(*self).get_indices();
 459                         exvector dummy_indices;
 460                         dummy_indices.push_back(other[0].op(1));
 461                         dummy_indices.push_back(other[1].op(1));
 462                         int sig;
 463                         ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
 464                         *self = numeric(3, 2) * sig * I;
 465                         other[0] = color_T(a, ex_to<color>(other[0]).get_representation_label());
 466                         other[1] = _ex1;
 467                         return true;
 468                 }
 469         }
 470
 471         return false;
 472 }
 473
 474 //////////
 475 // global functions
 476 //////////
 477
 478 ex color_ONE(unsigned char rl)
 479 {
 480         static ex ONE = (new su3one)->setflag(status_flags::dynallocated);
 481         return color(ONE, rl);
 482 }
 483
 484 ex color_T(const ex & a, unsigned char rl)
 485 {
 486         static ex t = (new su3t)->setflag(status_flags::dynallocated);
 487
 488         if (!is_a<idx>(a))
 489                 throw(std::invalid_argument("indices of color_T must be of type idx"));
 490         if (!ex_to<idx>(a).get_dim().is_equal(8))
 491                 throw(std::invalid_argument("index dimension for color_T must be 8"));
 492
 493         return color(t, a, rl);
 494 }
 495
 496 ex color_f(const ex & a, const ex & b, const ex & c)
 497 {
 498         static ex f = (new su3f)->setflag(status_flags::dynallocated);
 499
 500         if (!is_a<idx>(a) || !is_a<idx>(b) || !is_a<idx>(c))
 501                 throw(std::invalid_argument("indices of color_f must be of type idx"));
 502         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))
 503                 throw(std::invalid_argument("index dimension for color_f must be 8"));
 504
 505         return indexed(f, antisymmetric3(), a, b, c);
 506 }
 507
 508 ex color_d(const ex & a, const ex & b, const ex & c)
 509 {
 510         static ex d = (new su3d)->setflag(status_flags::dynallocated);
 511
 512         if (!is_a<idx>(a) || !is_a<idx>(b) || !is_a<idx>(c))
 513                 throw(std::invalid_argument("indices of color_d must be of type idx"));
 514         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))
 515                 throw(std::invalid_argument("index dimension for color_d must be 8"));
 516
 517         return indexed(d, symmetric3(), a, b, c);
 518 }
 519
 520 ex color_h(const ex & a, const ex & b, const ex & c)
 521 {
 522         return color_d(a, b, c) + I * color_f(a, b, c);
 523 }
 524
 525 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 526  *  is that of a color object (with an arbitrary representation label). */
 527 static bool is_color_tinfo(tinfo_t ti)
 528 {
 529         p_int start_loc=(p_int)&color::return_type_tinfo_static;
 530         return (p_int)ti>=start_loc && (p_int)ti<start_loc+256;
 531 }
 532
 533 /** Extract representation label from tinfo key (as returned by
 534  *  return_type_tinfo()). */
 535 static unsigned char get_representation_label(tinfo_t ti)
 536 {
 537         return (unsigned char)((p_int)ti-(p_int)&color::return_type_tinfo_static);
 538 }
 539
 540 ex color_trace(const ex & e, const std::set<unsigned char> & rls)
 541 {
 542         if (is_a<color>(e)) {
 543
 544                 unsigned char rl = ex_to<color>(e).get_representation_label();
 545
 546                 // Are we taking the trace over this object's representation label?
 547                 if (rls.find(rl) == rls.end())
 548                         return e;
 549
 550                 // Yes, all generators are traceless, except for color_ONE
 551                 if (is_a<su3one>(e.op(0)))
 552                         return _ex3;
 553                 else
 554                         return _ex0;
 555
 556         } else if (is_exactly_a<mul>(e)) {
 557
 558                 // Trace of product: pull out non-color factors
 559                 ex prod = _ex1;
 560                 for (size_t i=0; i<e.nops(); i++) {
 561                         const ex &o = e.op(i);
 562                         if (is_color_tinfo(o.return_type_tinfo()))
 563                                 prod *= color_trace(o, rls);
 564                         else
 565                                 prod *= o;
 566                 }
 567                 return prod;
 568
 569         } else if (is_exactly_a<ncmul>(e)) {
 570
 571                 unsigned char rl = get_representation_label(e.return_type_tinfo());
 572
 573                 // Are we taking the trace over this string's representation label?
 574                 if (rls.find(rl) == rls.end())
 575                         return e;
 576
 577                 // Yes, expand product if necessary
 578                 ex e_expanded = e.expand();
 579                 if (!is_a<ncmul>(e_expanded))
 580                         return color_trace(e_expanded, rls);
 581
 582                 size_t num = e.nops();
 583
 584                 if (num == 2) {
 585
 586                         // Tr T_a T_b = 1/2 delta_a_b
 587                         return delta_tensor(e.op(0).op(1), e.op(1).op(1)) / 2;
 588
 589                 } else if (num == 3) {
 590
 591                         // Tr T_a T_b T_c = 1/4 h_a_b_c
 592                         return color_h(e.op(0).op(1), e.op(1).op(1), e.op(2).op(1)) / 4;
 593
 594                 } else {
 595
 596                         // Traces of 4 or more generators are computed recursively:
 597                         // Tr T_a1 .. T_an =
 598                         //     1/6 delta_a(n-1)_an Tr T_a1 .. T_a(n-2)
 599                         //   + 1/2 h_a(n-1)_an_k Tr T_a1 .. T_a(n-2) T_k
 600                         const ex &last_index = e.op(num - 1).op(1);
 601                         const ex &next_to_last_index = e.op(num - 2).op(1);
 602                         idx summation_index((new symbol)->setflag(status_flags::dynallocated), 8);
 603
 604                         exvector v1;
 605                         v1.reserve(num - 2);
 606                         for (size_t i=0; i<num-2; i++)
 607                                 v1.push_back(e.op(i));
 608
 609                         exvector v2 = v1;
 610                         v2.push_back(color_T(summation_index, rl));
 611
 612                         return delta_tensor(next_to_last_index, last_index) * color_trace(ncmul(v1), rl) / 6
 613                                + color_h(next_to_last_index, last_index, summation_index) * color_trace(ncmul(v2), rl) / 2;
 614                 }
 615
 616         } else if (e.nops() > 0) {
 617
 618                 // Trace maps to all other container classes (this includes sums)
 619                 pointer_to_map_function_1arg<const std::set<unsigned char> &> fcn(color_trace, rls);
 620                 return e.map(fcn);
 621
 622         } else
 623                 return _ex0;
 624 }
 625
 626 ex color_trace(const ex & e, const lst & rll)
 627 {
 628         // Convert list to set
 629         std::set<unsigned char> rls;
 630         for (lst::const_iterator i = rll.begin(); i != rll.end(); ++i) {
 631                 if (i->info(info_flags::nonnegint))
 632                         rls.insert(ex_to<numeric>(*i).to_int());
 633         }
 634
 635         return color_trace(e, rls);
 636 }
 637
 638 ex color_trace(const ex & e, unsigned char rl)
 639 {
 640         // Convert label to set
 641         std::set<unsigned char> rls;
 642         rls.insert(rl);
 643
 644         return color_trace(e, rls);
 645 }
 646
 647 } // namespace GiNaC