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) 1999-2008 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 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(su3one, tensor,
  43   print_func<print_dflt>(&su3one::do_print).
  44   print_func<print_latex>(&su3one::do_print_latex))
  45
  46 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(su3t, tensor,
  47   print_func<print_dflt>(&su3t::do_print).
  48   print_func<print_latex>(&su3t::do_print))
  49
  50 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(su3f, tensor,
  51   print_func<print_dflt>(&su3f::do_print).
  52   print_func<print_latex>(&su3f::do_print))
  53
  54 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(su3d, tensor,
  55   print_func<print_dflt>(&su3d::do_print).
  56   print_func<print_latex>(&su3d::do_print))
  57
  58 //////////
  59 // default constructors
  60 //////////
  61
  62 color::color() : representation_label(0)
  63 {
  64         tinfo_key = &color::tinfo_static;
  65 }
  66
  67 DEFAULT_CTOR(su3one)
  68 DEFAULT_CTOR(su3t)
  69 DEFAULT_CTOR(su3f)
  70 DEFAULT_CTOR(su3d)
  71
  72 //////////
  73 // other constructors
  74 //////////
  75
  76 /** Construct object without any color index. This constructor is for
  77  *  internal use only. Use the color_ONE() function instead.
  78  *  @see color_ONE */
  79 color::color(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
  80 {
  81         tinfo_key = &color::tinfo_static;
  82 }
  83
  84 /** Construct object with one color index. This constructor is for internal
  85  *  use only. Use the color_T() function instead.
  86  *  @see color_T */
  87 color::color(const ex & b, const ex & i1, unsigned char rl) : inherited(b, i1), representation_label(rl)
  88 {
  89         tinfo_key = &color::tinfo_static;
  90 }
  91
  92 color::color(unsigned char rl, const exvector & v, bool discardable) : inherited(not_symmetric(), v, discardable), representation_label(rl)
  93 {
  94         tinfo_key = &color::tinfo_static;
  95 }
  96
  97 color::color(unsigned char rl, std::auto_ptr<exvector> vp) : inherited(not_symmetric(), vp), representation_label(rl)
  98 {
  99         tinfo_key = &color::tinfo_static;
 100 }
 101
 102 return_type_t color::return_type_tinfo() const
 103 {
 104         return make_return_type_t<color>(representation_label);
 105 }
 106
 107 //////////
 108 // archiving
 109 //////////
 110
 111 color::color(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
 112 {
 113         unsigned rl;
 114         n.find_unsigned("label", rl);
 115         representation_label = rl;
 116 }
 117
 118 void color::archive(archive_node &n) const
 119 {
 120         inherited::archive(n);
 121         n.add_unsigned("label", representation_label);
 122 }
 123
 124 DEFAULT_UNARCHIVE(color)
 125 DEFAULT_ARCHIVING(su3one)
 126 DEFAULT_ARCHIVING(su3t)
 127 DEFAULT_ARCHIVING(su3f)
 128 DEFAULT_ARCHIVING(su3d)
 129
 130 //////////
 131 // functions overriding virtual functions from base classes
 132 //////////
 133
 134 int color::compare_same_type(const basic & other) const
 135 {
 136         GINAC_ASSERT(is_a<color>(other));
 137         const color &o = static_cast<const color &>(other);
 138
 139         if (representation_label != o.representation_label) {
 140                 // different representation label
 141                 return representation_label < o.representation_label ? -1 : 1;
 142         }
 143
 144         return inherited::compare_same_type(other);
 145 }
 146
 147 bool color::match_same_type(const basic & other) const
 148 {
 149         GINAC_ASSERT(is_a<color>(other));
 150         const color &o = static_cast<const color &>(other);
 151
 152         return representation_label == o.representation_label;
 153 }
 154
 155 DEFAULT_COMPARE(su3one)
 156 DEFAULT_COMPARE(su3t)
 157 DEFAULT_COMPARE(su3f)
 158 DEFAULT_COMPARE(su3d)
 159
 160 DEFAULT_PRINT_LATEX(su3one, "ONE", "\\mathbb{1}")
 161 DEFAULT_PRINT(su3t, "T")
 162 DEFAULT_PRINT(su3f, "f")
 163 DEFAULT_PRINT(su3d, "d")
 164
 165 /** Perform automatic simplification on noncommutative product of color
 166  *  objects. This removes superfluous ONEs. */
 167 ex color::eval_ncmul(const exvector & v) const
 168 {
 169         exvector s;
 170         s.reserve(v.size());
 171
 172         // Remove superfluous ONEs
 173         exvector::const_iterator it = v.begin(), itend = v.end();
 174         while (it != itend) {
 175                 if (!is_a<su3one>(it->op(0)))
 176                         s.push_back(*it);
 177                 it++;
 178         }
 179
 180         if (s.empty())
 181                 return color(su3one(), representation_label);
 182         else
 183                 return hold_ncmul(s);
 184 }
 185
 186 ex color::thiscontainer(const exvector & v) const
 187 {
 188         return color(representation_label, v);
 189 }
 190
 191 ex color::thiscontainer(std::auto_ptr<exvector> vp) const
 192 {
 193         return color(representation_label, vp);
 194 }
 195
 196 /** Given a vector iv3 of three indices and a vector iv2 of two indices that
 197  *  is a subset of iv3, return the (free) index that is in iv3 but not in
 198  *  iv2 and the sign introduced by permuting that index to the front.
 199  *
 200  *  @param iv3 Vector of 3 indices
 201  *  @param iv2 Vector of 2 indices, must be a subset of iv3
 202  *  @param sig Returs sign introduced by index permutation
 203  *  @return the free index (the one that is in iv3 but not in iv2) */
 204 static ex permute_free_index_to_front(const exvector & iv3, const exvector & iv2, int & sig)
 205 {
 206         GINAC_ASSERT(iv3.size() == 3);
 207         GINAC_ASSERT(iv2.size() == 2);
 208
 209         sig = 1;
 210
 211 #define TEST_PERMUTATION(A,B,C,P) \
 212         if (iv3[B].is_equal(iv2[0]) && iv3[C].is_equal(iv2[1])) { \
 213                 sig = P; \
 214                 return iv3[A]; \
 215         }
 216
 217         TEST_PERMUTATION(0,1,2,  1);
 218         TEST_PERMUTATION(0,2,1, -1);
 219         TEST_PERMUTATION(1,0,2, -1);
 220         TEST_PERMUTATION(1,2,0,  1);
 221         TEST_PERMUTATION(2,0,1,  1);
 222         TEST_PERMUTATION(2,1,0, -1);
 223
 224         throw(std::logic_error("permute_free_index_to_front(): no valid permutation found"));
 225 }
 226
 227 /** Automatic symbolic evaluation of indexed symmetric structure constant. */
 228 ex su3d::eval_indexed(const basic & i) const
 229 {
 230         GINAC_ASSERT(is_a<indexed>(i));
 231         GINAC_ASSERT(i.nops() == 4);
 232         GINAC_ASSERT(is_a<su3d>(i.op(0)));
 233
 234         // Convolutions are zero
 235         if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
 236                 return _ex0;
 237
 238         // Numeric evaluation
 239         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 240
 241                 // Sort indices
 242                 int v[3];
 243                 for (unsigned j=0; j<3; j++)
 244                         v[j] = ex_to<numeric>(ex_to<idx>(i.op(j + 1)).get_value()).to_int();
 245                 if (v[0] > v[1]) std::swap(v[0], v[1]);
 246                 if (v[0] > v[2]) std::swap(v[0], v[2]);
 247                 if (v[1] > v[2]) std::swap(v[1], v[2]);
 248
 249 #define CMPINDICES(A,B,C) ((v[0] == (A)) && (v[1] == (B)) && (v[2] == (C)))
 250
 251                 // Check for non-zero elements
 252                 if (CMPINDICES(1,4,6) || CMPINDICES(1,5,7) || CMPINDICES(2,5,6)
 253                  || CMPINDICES(3,4,4) || CMPINDICES(3,5,5))
 254                         return _ex1_2;
 255                 else if (CMPINDICES(2,4,7) || CMPINDICES(3,6,6) || CMPINDICES(3,7,7))
 256                         return _ex_1_2;
 257                 else if (CMPINDICES(1,1,8) || CMPINDICES(2,2,8) || CMPINDICES(3,3,8))
 258                         return sqrt(_ex3)*_ex1_3;
 259                 else if (CMPINDICES(8,8,8))
 260                         return sqrt(_ex3)*_ex_1_3;
 261                 else if (CMPINDICES(4,4,8) || CMPINDICES(5,5,8)
 262                       || CMPINDICES(6,6,8) || CMPINDICES(7,7,8))
 263                         return sqrt(_ex3)/_ex_6;
 264                 else
 265                         return _ex0;
 266         }
 267
 268         // No further simplifications
 269         return i.hold();
 270 }
 271
 272 /** Automatic symbolic evaluation of indexed antisymmetric structure constant. */
 273 ex su3f::eval_indexed(const basic & i) const
 274 {
 275         GINAC_ASSERT(is_a<indexed>(i));
 276         GINAC_ASSERT(i.nops() == 4);
 277         GINAC_ASSERT(is_a<su3f>(i.op(0)));
 278
 279         // Numeric evaluation
 280         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 281
 282                 // Sort indices, remember permutation sign
 283                 int v[3];
 284                 for (unsigned j=0; j<3; j++)
 285                         v[j] = ex_to<numeric>(ex_to<idx>(i.op(j + 1)).get_value()).to_int();
 286                 int sign = 1;
 287                 if (v[0] > v[1]) { std::swap(v[0], v[1]); sign = -sign; }
 288                 if (v[0] > v[2]) { std::swap(v[0], v[2]); sign = -sign; }
 289                 if (v[1] > v[2]) { std::swap(v[1], v[2]); sign = -sign; }
 290
 291                 // Check for non-zero elements
 292                 if (CMPINDICES(1,2,3))
 293                         return sign;
 294                 else if (CMPINDICES(1,4,7) || CMPINDICES(2,4,6)
 295                       || CMPINDICES(2,5,7) || CMPINDICES(3,4,5))
 296                         return _ex1_2 * sign;
 297                 else if (CMPINDICES(1,5,6) || CMPINDICES(3,6,7))
 298                         return _ex_1_2 * sign;
 299                 else if (CMPINDICES(4,5,8) || CMPINDICES(6,7,8))
 300                         return sqrt(_ex3)/2 * sign;
 301                 else
 302                         return _ex0;
 303         }
 304
 305         // No further simplifications
 306         return i.hold();
 307 }
 308
 309
 310 /** Contraction of generator with something else. */
 311 bool su3t::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 312 {
 313         GINAC_ASSERT(is_a<indexed>(*self));
 314         GINAC_ASSERT(is_a<indexed>(*other));
 315         GINAC_ASSERT(self->nops() == 2);
 316         GINAC_ASSERT(is_a<su3t>(self->op(0)));
 317         unsigned char rl = ex_to<color>(*self).get_representation_label();
 318
 319         if (is_exactly_a<su3t>(other->op(0))) {
 320
 321                 // Contraction only makes sense if the represenation labels are equal
 322                 GINAC_ASSERT(is_a<color>(*other));
 323                 if (ex_to<color>(*other).get_representation_label() != rl)
 324                         return false;
 325
 326                 // T.a T.a = 4/3 ONE
 327                 if (other - self == 1) {
 328                         *self = numeric(4, 3);
 329                         *other = color_ONE(rl);
 330                         return true;
 331
 332                 // T.a T.b T.a = -1/6 T.b
 333                 } else if (other - self == 2
 334                         && is_a<color>(self[1])) {
 335                         *self = numeric(-1, 6);
 336                         *other = _ex1;
 337                         return true;
 338
 339                 // T.a S T.a = 1/2 Tr(S) - 1/6 S
 340                 } else {
 341                         exvector::iterator it = self + 1;
 342                         while (it != other) {
 343                                 if (!is_a<color>(*it)) {
 344                                         return false;
 345                                 }
 346                                 it++;
 347                         }
 348
 349                         it = self + 1;
 350                         ex S = _ex1;
 351                         while (it != other) {
 352                                 S *= *it;
 353                                 *it++ = _ex1;
 354                         }
 355
 356                         *self = color_trace(S, rl) * color_ONE(rl) / 2 - S / 6;
 357                         *other = _ex1;
 358                         return true;
 359                 }
 360         }
 361
 362         return false;
 363 }
 364
 365 /** Contraction of an indexed symmetric structure constant with something else. */
 366 bool su3d::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 367 {
 368         GINAC_ASSERT(is_a<indexed>(*self));
 369         GINAC_ASSERT(is_a<indexed>(*other));
 370         GINAC_ASSERT(self->nops() == 4);
 371         GINAC_ASSERT(is_a<su3d>(self->op(0)));
 372
 373         if (is_exactly_a<su3d>(other->op(0))) {
 374
 375                 // Find the dummy indices of the contraction
 376                 exvector self_indices = ex_to<indexed>(*self).get_indices();
 377                 exvector other_indices = ex_to<indexed>(*other).get_indices();
 378                 exvector all_indices = self_indices;
 379                 all_indices.insert(all_indices.end(), other_indices.begin(), other_indices.end());
 380                 exvector free_indices, dummy_indices;
 381                 find_free_and_dummy(all_indices, free_indices, dummy_indices);
 382
 383                 // d.abc d.abc = 40/3
 384                 if (dummy_indices.size() == 3) {
 385                         *self = numeric(40, 3);
 386                         *other = _ex1;
 387                         return true;
 388
 389                 // d.akl d.bkl = 5/3 delta.ab
 390                 } else if (dummy_indices.size() == 2) {
 391                         exvector a;
 392                         std::back_insert_iterator<exvector> ita(a);
 393                         ita = set_difference(self_indices.begin(), self_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
 394                         ita = set_difference(other_indices.begin(), other_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
 395                         GINAC_ASSERT(a.size() == 2);
 396                         *self = numeric(5, 3) * delta_tensor(a[0], a[1]);
 397                         *other = _ex1;
 398                         return true;
 399                 }
 400
 401         } else if (is_exactly_a<su3t>(other->op(0))) {
 402
 403                 // d.abc T.b T.c = 5/6 T.a
 404                 if (other+1 != v.end()
 405                  && is_exactly_a<su3t>(other[1].op(0))
 406                  && ex_to<indexed>(*self).has_dummy_index_for(other[1].op(1))) {
 407
 408                         exvector self_indices = ex_to<indexed>(*self).get_indices();
 409                         exvector dummy_indices;
 410                         dummy_indices.push_back(other[0].op(1));
 411                         dummy_indices.push_back(other[1].op(1));
 412                         int sig;
 413                         ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
 414                         *self = numeric(5, 6);
 415                         other[0] = color_T(a, ex_to<color>(other[0]).get_representation_label());
 416                         other[1] = _ex1;
 417                         return true;
 418                 }
 419         }
 420
 421         return false;
 422 }
 423
 424 /** Contraction of an indexed antisymmetric structure constant with something else. */
 425 bool su3f::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 426 {
 427         GINAC_ASSERT(is_a<indexed>(*self));
 428         GINAC_ASSERT(is_a<indexed>(*other));
 429         GINAC_ASSERT(self->nops() == 4);
 430         GINAC_ASSERT(is_a<su3f>(self->op(0)));
 431
 432         if (is_exactly_a<su3f>(other->op(0))) { // f*d is handled by su3d class
 433
 434                 // Find the dummy indices of the contraction
 435                 exvector dummy_indices;
 436                 dummy_indices = ex_to<indexed>(*self).get_dummy_indices(ex_to<indexed>(*other));
 437
 438                 // f.abc f.abc = 24
 439                 if (dummy_indices.size() == 3) {
 440                         *self = 24;
 441                         *other = _ex1;
 442                         return true;
 443
 444                 // f.akl f.bkl = 3 delta.ab
 445                 } else if (dummy_indices.size() == 2) {
 446                         int sign1, sign2;
 447                         ex a = permute_free_index_to_front(ex_to<indexed>(*self).get_indices(), dummy_indices, sign1);
 448                         ex b = permute_free_index_to_front(ex_to<indexed>(*other).get_indices(), dummy_indices, sign2);
 449                         *self = sign1 * sign2 * 3 * delta_tensor(a, b);
 450                         *other = _ex1;
 451                         return true;
 452                 }
 453
 454         } else if (is_exactly_a<su3t>(other->op(0))) {
 455
 456                 // f.abc T.b T.c = 3/2 I T.a
 457                 if (other+1 != v.end()
 458                  && is_exactly_a<su3t>(other[1].op(0))
 459                  && ex_to<indexed>(*self).has_dummy_index_for(other[1].op(1))) {
 460
 461                         exvector self_indices = ex_to<indexed>(*self).get_indices();
 462                         exvector dummy_indices;
 463                         dummy_indices.push_back(other[0].op(1));
 464                         dummy_indices.push_back(other[1].op(1));
 465                         int sig;
 466                         ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
 467                         *self = numeric(3, 2) * sig * I;
 468                         other[0] = color_T(a, ex_to<color>(other[0]).get_representation_label());
 469                         other[1] = _ex1;
 470                         return true;
 471                 }
 472         }
 473
 474         return false;
 475 }
 476
 477 //////////
 478 // global functions
 479 //////////
 480
 481 ex color_ONE(unsigned char rl)
 482 {
 483         static ex ONE = (new su3one)->setflag(status_flags::dynallocated);
 484         return color(ONE, rl);
 485 }
 486
 487 ex color_T(const ex & a, unsigned char rl)
 488 {
 489         static ex t = (new su3t)->setflag(status_flags::dynallocated);
 490
 491         if (!is_a<idx>(a))
 492                 throw(std::invalid_argument("indices of color_T must be of type idx"));
 493         if (!ex_to<idx>(a).get_dim().is_equal(8))
 494                 throw(std::invalid_argument("index dimension for color_T must be 8"));
 495
 496         return color(t, a, rl);
 497 }
 498
 499 ex color_f(const ex & a, const ex & b, const ex & c)
 500 {
 501         static ex f = (new su3f)->setflag(status_flags::dynallocated);
 502
 503         if (!is_a<idx>(a) || !is_a<idx>(b) || !is_a<idx>(c))
 504                 throw(std::invalid_argument("indices of color_f must be of type idx"));
 505         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))
 506                 throw(std::invalid_argument("index dimension for color_f must be 8"));
 507
 508         return indexed(f, antisymmetric3(), a, b, c);
 509 }
 510
 511 ex color_d(const ex & a, const ex & b, const ex & c)
 512 {
 513         static ex d = (new su3d)->setflag(status_flags::dynallocated);
 514
 515         if (!is_a<idx>(a) || !is_a<idx>(b) || !is_a<idx>(c))
 516                 throw(std::invalid_argument("indices of color_d must be of type idx"));
 517         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))
 518                 throw(std::invalid_argument("index dimension for color_d must be 8"));
 519
 520         return indexed(d, symmetric3(), a, b, c);
 521 }
 522
 523 ex color_h(const ex & a, const ex & b, const ex & c)
 524 {
 525         return color_d(a, b, c) + I * color_f(a, b, c);
 526 }
 527
 528 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 529  *  is that of a color object (with an arbitrary representation label). */
 530 static bool is_color_tinfo(const return_type_t& ti)
 531 {
 532         return *(ti.tinfo) == typeid(color);
 533 }
 534
 535 /** Extract representation label from tinfo key (as returned by
 536  *  return_type_tinfo()). */
 537 static unsigned char get_representation_label(const return_type_t& ti)
 538 {
 539         return (unsigned char)ti.rl;
 540 }
 541
 542 ex color_trace(const ex & e, const std::set<unsigned char> & rls)
 543 {
 544         if (is_a<color>(e)) {
 545
 546                 unsigned char rl = ex_to<color>(e).get_representation_label();
 547
 548                 // Are we taking the trace over this object's representation label?
 549                 if (rls.find(rl) == rls.end())
 550                         return e;
 551
 552                 // Yes, all generators are traceless, except for color_ONE
 553                 if (is_a<su3one>(e.op(0)))
 554                         return _ex3;
 555                 else
 556                         return _ex0;
 557
 558         } else if (is_exactly_a<mul>(e)) {
 559
 560                 // Trace of product: pull out non-color factors
 561                 ex prod = _ex1;
 562                 for (size_t i=0; i<e.nops(); i++) {
 563                         const ex &o = e.op(i);
 564                         if (is_color_tinfo(o.return_type_tinfo()))
 565                                 prod *= color_trace(o, rls);
 566                         else
 567                                 prod *= o;
 568                 }
 569                 return prod;
 570
 571         } else if (is_exactly_a<ncmul>(e)) {
 572
 573                 unsigned char rl = get_representation_label(e.return_type_tinfo());
 574
 575                 // Are we taking the trace over this string's representation label?
 576                 if (rls.find(rl) == rls.end())
 577                         return e;
 578
 579                 // Yes, expand product if necessary
 580                 ex e_expanded = e.expand();
 581                 if (!is_a<ncmul>(e_expanded))
 582                         return color_trace(e_expanded, rls);
 583
 584                 size_t num = e.nops();
 585
 586                 if (num == 2) {
 587
 588                         // Tr T_a T_b = 1/2 delta_a_b
 589                         return delta_tensor(e.op(0).op(1), e.op(1).op(1)) / 2;
 590
 591                 } else if (num == 3) {
 592
 593                         // Tr T_a T_b T_c = 1/4 h_a_b_c
 594                         return color_h(e.op(0).op(1), e.op(1).op(1), e.op(2).op(1)) / 4;
 595
 596                 } else {
 597
 598                         // Traces of 4 or more generators are computed recursively:
 599                         // Tr T_a1 .. T_an =
 600                         //     1/6 delta_a(n-1)_an Tr T_a1 .. T_a(n-2)
 601                         //   + 1/2 h_a(n-1)_an_k Tr T_a1 .. T_a(n-2) T_k
 602                         const ex &last_index = e.op(num - 1).op(1);
 603                         const ex &next_to_last_index = e.op(num - 2).op(1);
 604                         idx summation_index((new symbol)->setflag(status_flags::dynallocated), 8);
 605
 606                         exvector v1;
 607                         v1.reserve(num - 2);
 608                         for (size_t i=0; i<num-2; i++)
 609                                 v1.push_back(e.op(i));
 610
 611                         exvector v2 = v1;
 612                         v2.push_back(color_T(summation_index, rl));
 613
 614                         return delta_tensor(next_to_last_index, last_index) * color_trace(ncmul(v1), rl) / 6
 615                                + color_h(next_to_last_index, last_index, summation_index) * color_trace(ncmul(v2), rl) / 2;
 616                 }
 617
 618         } else if (e.nops() > 0) {
 619
 620                 // Trace maps to all other container classes (this includes sums)
 621                 pointer_to_map_function_1arg<const std::set<unsigned char> &> fcn(color_trace, rls);
 622                 return e.map(fcn);
 623
 624         } else
 625                 return _ex0;
 626 }
 627
 628 ex color_trace(const ex & e, const lst & rll)
 629 {
 630         // Convert list to set
 631         std::set<unsigned char> rls;
 632         for (lst::const_iterator i = rll.begin(); i != rll.end(); ++i) {
 633                 if (i->info(info_flags::nonnegint))
 634                         rls.insert(ex_to<numeric>(*i).to_int());
 635         }
 636
 637         return color_trace(e, rls);
 638 }
 639
 640 ex color_trace(const ex & e, unsigned char rl)
 641 {
 642         // Convert label to set
 643         std::set<unsigned char> rls;
 644         rls.insert(rl);
 645
 646         return color_trace(e, rls);
 647 }
 648
 649 } // namespace GiNaC