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) 1999-2001 Johannes Gutenberg University Mainz, Germany
   7  *
   8  *  This program is free software; you can redistribute it and/or modify
   9  *  it under the terms of the GNU General Public License as published by
  10  *  the Free Software Foundation; either version 2 of the License, or
  11  *  (at your option) any later version.
  12  *
  13  *  This program is distributed in the hope that it will be useful,
  14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  16  *  GNU General Public License for more details.
  17  *
  18  *  You should have received a copy of the GNU General Public License
  19  *  along with this program; if not, write to the Free Software
  20  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  21  */
  22
  23 #include <algorithm>
  24 #include <stdexcept>
  25
  26 #include "color.h"
  27 #include "idx.h"
  28 #include "ncmul.h"
  29 #include "symmetry.h"
  30 #include "numeric.h"
  31 #include "power.h" // for sqrt()
  32 #include "symbol.h"
  33 #include "print.h"
  34 #include "archive.h"
  35 #include "debugmsg.h"
  36 #include "utils.h"
  37
  38 namespace GiNaC {
  39
  40 GINAC_IMPLEMENT_REGISTERED_CLASS(color, indexed)
  41 GINAC_IMPLEMENT_REGISTERED_CLASS(su3one, tensor)
  42 GINAC_IMPLEMENT_REGISTERED_CLASS(su3t, tensor)
  43 GINAC_IMPLEMENT_REGISTERED_CLASS(su3f, tensor)
  44 GINAC_IMPLEMENT_REGISTERED_CLASS(su3d, tensor)
  45
  46 //////////
  47 // default constructor, destructor, copy constructor assignment operator and helpers
  48 //////////
  49
  50 color::color() : representation_label(0)
  51 {
  52         debugmsg("color default constructor", LOGLEVEL_CONSTRUCT);
  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         debugmsg("color constructor from ex,unsigned char", LOGLEVEL_CONSTRUCT);
  78         tinfo_key = TINFO_color;
  79 }
  80
  81 /** Construct object with one color index. This constructor is for internal
  82  *  use only. Use the color_T() function instead.
  83  *  @see color_T */
  84 color::color(const ex & b, const ex & i1, unsigned char rl) : inherited(b, i1), representation_label(rl)
  85 {
  86         debugmsg("color constructor from ex,ex,unsigned char", LOGLEVEL_CONSTRUCT);
  87         tinfo_key = TINFO_color;
  88 }
  89
  90 color::color(unsigned char rl, const exvector & v, bool discardable) : inherited(sy_none(), v, discardable), representation_label(rl)
  91 {
  92         debugmsg("color constructor from unsigned char,exvector", LOGLEVEL_CONSTRUCT);
  93         tinfo_key = TINFO_color;
  94 }
  95
  96 color::color(unsigned char rl, exvector * vp) : inherited(sy_none(), vp), representation_label(rl)
  97 {
  98         debugmsg("color constructor from unsigned char,exvector *", LOGLEVEL_CONSTRUCT);
  99         tinfo_key = TINFO_color;
 100 }
 101
 102 //////////
 103 // archiving
 104 //////////
 105
 106 color::color(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 107 {
 108         debugmsg("color constructor from archive_node", LOGLEVEL_CONSTRUCT);
 109         unsigned rl;
 110         n.find_unsigned("label", rl);
 111         representation_label = rl;
 112 }
 113
 114 void color::archive(archive_node &n) const
 115 {
 116         inherited::archive(n);
 117         n.add_unsigned("label", representation_label);
 118 }
 119
 120 DEFAULT_UNARCHIVE(color)
 121 DEFAULT_ARCHIVING(su3one)
 122 DEFAULT_ARCHIVING(su3t)
 123 DEFAULT_ARCHIVING(su3f)
 124 DEFAULT_ARCHIVING(su3d)
 125
 126 //////////
 127 // functions overriding virtual functions from base classes
 128 //////////
 129
 130 int color::compare_same_type(const basic & other) const
 131 {
 132         GINAC_ASSERT(is_of_type(other, color));
 133         const color &o = static_cast<const color &>(other);
 134
 135         if (representation_label != o.representation_label) {
 136                 // different representation label
 137                 return representation_label < o.representation_label ? -1 : 1;
 138         }
 139
 140         return inherited::compare_same_type(other);
 141 }
 142
 143 bool color::match_same_type(const basic & other) const
 144 {
 145         GINAC_ASSERT(is_of_type(other, color));
 146         const color &o = static_cast<const color &>(other);
 147
 148         return representation_label == o.representation_label;
 149 }
 150
 151 DEFAULT_COMPARE(su3one)
 152 DEFAULT_COMPARE(su3t)
 153 DEFAULT_COMPARE(su3f)
 154 DEFAULT_COMPARE(su3d)
 155
 156 DEFAULT_PRINT_LATEX(su3one, "ONE", "\\mathbb{1}")
 157 DEFAULT_PRINT(su3t, "T")
 158 DEFAULT_PRINT(su3f, "f")
 159 DEFAULT_PRINT(su3d, "d")
 160
 161 /** Perform automatic simplification on noncommutative product of color
 162  *  objects. This removes superfluous ONEs. */
 163 ex color::simplify_ncmul(const exvector & v) const
 164 {
 165         exvector s;
 166         s.reserve(v.size());
 167
 168         // Remove superfluous ONEs
 169         exvector::const_iterator it = v.begin(), itend = v.end();
 170         while (it != itend) {
 171                 if (!is_ex_of_type(it->op(0), su3one))
 172                         s.push_back(*it);
 173                 it++;
 174         }
 175
 176         if (s.empty())
 177                 return color(su3one(), representation_label);
 178         else
 179                 return simplified_ncmul(s);
 180 }
 181
 182 ex color::thisexprseq(const exvector & v) const
 183 {
 184         return color(representation_label, v);
 185 }
 186
 187 ex color::thisexprseq(exvector * vp) const
 188 {
 189         return color(representation_label, vp);
 190 }
 191
 192 /** Given a vector iv3 of three indices and a vector iv2 of two indices that
 193  *  is a subset of iv3, return the (free) index that is in iv3 but not in
 194  *  iv2 and the sign introduced by permuting that index to the front.
 195  *
 196  *  @param iv3 Vector of 3 indices
 197  *  @param iv2 Vector of 2 indices, must be a subset of iv3
 198  *  @param sig Returs sign introduced by index permutation
 199  *  @return the free index (the one that is in iv3 but not in iv2) */
 200 static ex permute_free_index_to_front(const exvector & iv3, const exvector & iv2, int & sig)
 201 {
 202         GINAC_ASSERT(iv3.size() == 3);
 203         GINAC_ASSERT(iv2.size() == 2);
 204
 205         sig = 1;
 206
 207 #define TEST_PERMUTATION(A,B,C,P) \
 208         if (iv3[B].is_equal(iv2[0]) && iv3[C].is_equal(iv2[1])) { \
 209                 sig = P; \
 210                 return iv3[A]; \
 211         }
 212
 213         TEST_PERMUTATION(0,1,2,  1);
 214         TEST_PERMUTATION(0,2,1, -1);
 215         TEST_PERMUTATION(1,0,2, -1);
 216         TEST_PERMUTATION(1,2,0,  1);
 217         TEST_PERMUTATION(2,0,1,  1);
 218         TEST_PERMUTATION(2,1,0, -1);
 219
 220         throw(std::logic_error("permute_free_index_to_front(): no valid permutation found"));
 221 }
 222
 223 /** Automatic symbolic evaluation of indexed symmetric structure constant. */
 224 ex su3d::eval_indexed(const basic & i) const
 225 {
 226         GINAC_ASSERT(is_of_type(i, indexed));
 227         GINAC_ASSERT(i.nops() == 4);
 228         GINAC_ASSERT(is_ex_of_type(i.op(0), su3d));
 229
 230         // Convolutions are zero
 231         if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
 232                 return _ex0();
 233
 234         // Numeric evaluation
 235         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 236
 237                 // Sort indices
 238                 int v[3];
 239                 for (unsigned j=0; j<3; j++)
 240                         v[j] = ex_to<numeric>(ex_to<idx>(i.op(j + 1)).get_value()).to_int();
 241                 if (v[0] > v[1]) std::swap(v[0], v[1]);
 242                 if (v[0] > v[2]) std::swap(v[0], v[2]);
 243                 if (v[1] > v[2]) std::swap(v[1], v[2]);
 244
 245 #define CMPINDICES(A,B,C) ((v[0] == (A)) && (v[1] == (B)) && (v[2] == (C)))
 246
 247                 // Check for non-zero elements
 248                 if (CMPINDICES(1,4,6) || CMPINDICES(1,5,7) || CMPINDICES(2,5,6)
 249                  || CMPINDICES(3,4,4) || CMPINDICES(3,5,5))
 250                         return _ex1_2();
 251                 else if (CMPINDICES(2,4,7) || CMPINDICES(3,6,6) || CMPINDICES(3,7,7))
 252                         return _ex_1_2();
 253                 else if (CMPINDICES(1,1,8) || CMPINDICES(2,2,8) || CMPINDICES(3,3,8))
 254                         return sqrt(_ex3())/3;
 255                 else if (CMPINDICES(8,8,8))
 256                         return -sqrt(_ex3())/3;
 257                 else if (CMPINDICES(4,4,8) || CMPINDICES(5,5,8)
 258                       || CMPINDICES(6,6,8) || CMPINDICES(7,7,8))
 259                         return -sqrt(_ex3())/6;
 260                 else
 261                         return _ex0();
 262         }
 263
 264         // No further simplifications
 265         return i.hold();
 266 }
 267
 268 /** Automatic symbolic evaluation of indexed antisymmetric structure constant. */
 269 ex su3f::eval_indexed(const basic & i) const
 270 {
 271         GINAC_ASSERT(is_of_type(i, indexed));
 272         GINAC_ASSERT(i.nops() == 4);
 273         GINAC_ASSERT(is_ex_of_type(i.op(0), su3f));
 274
 275         // Numeric evaluation
 276         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 277
 278                 // Sort indices, remember permutation sign
 279                 int v[3];
 280                 for (unsigned j=0; j<3; j++)
 281                         v[j] = ex_to<numeric>(ex_to<idx>(i.op(j + 1)).get_value()).to_int();
 282                 int sign = 1;
 283                 if (v[0] > v[1]) { std::swap(v[0], v[1]); sign = -sign; }
 284                 if (v[0] > v[2]) { std::swap(v[0], v[2]); sign = -sign; }
 285                 if (v[1] > v[2]) { std::swap(v[1], v[2]); sign = -sign; }
 286
 287                 // Check for non-zero elements
 288                 if (CMPINDICES(1,2,3))
 289                         return sign;
 290                 else if (CMPINDICES(1,4,7) || CMPINDICES(2,4,6)
 291                       || CMPINDICES(2,5,7) || CMPINDICES(3,4,5))
 292                         return _ex1_2() * sign;
 293                 else if (CMPINDICES(1,5,6) || CMPINDICES(3,6,7))
 294                         return _ex_1_2() * sign;
 295                 else if (CMPINDICES(4,5,8) || CMPINDICES(6,7,8))
 296                         return sqrt(_ex3())/2 * sign;
 297                 else
 298                         return _ex0();
 299         }
 300
 301         // No further simplifications
 302         return i.hold();
 303 }
 304
 305
 306 /** Contraction of generator with something else. */
 307 bool su3t::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 308 {
 309         GINAC_ASSERT(is_ex_of_type(*self, indexed));
 310         GINAC_ASSERT(is_ex_of_type(*other, indexed));
 311         GINAC_ASSERT(self->nops() == 2);
 312         GINAC_ASSERT(is_ex_of_type(self->op(0), su3t));
 313         unsigned char rl = ex_to<color>(*self).get_representation_label();
 314
 315         if (is_ex_exactly_of_type(other->op(0), su3t)) {
 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_ex_of_type(self[1], color)) {
 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_ex_of_type(*it, color)) {
 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_ex_of_type(*self, indexed));
 360         GINAC_ASSERT(is_ex_of_type(*other, indexed));
 361         GINAC_ASSERT(self->nops() == 4);
 362         GINAC_ASSERT(is_ex_of_type(self->op(0), su3d));
 363
 364         if (is_ex_exactly_of_type(other->op(0), su3d)) {
 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_ex_exactly_of_type(other->op(0), su3t)) {
 393
 394                 // d.abc T.b T.c = 5/6 T.a
 395                 if (other+1 != v.end()
 396                  && is_ex_exactly_of_type(other[1].op(0), su3t)
 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_ex_of_type(*self, indexed));
 419         GINAC_ASSERT(is_ex_of_type(*other, indexed));
 420         GINAC_ASSERT(self->nops() == 4);
 421         GINAC_ASSERT(is_ex_of_type(self->op(0), su3f));
 422
 423         if (is_ex_exactly_of_type(other->op(0), su3f)) { // 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_ex_exactly_of_type(other->op(0), su3t)) {
 446
 447                 // f.abc T.b T.c = 3/2 I T.a
 448                 if (other+1 != v.end()
 449                  && is_ex_exactly_of_type(other[1].op(0), su3t)
 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_ex_of_type(a, idx))
 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_ex_of_type(a, idx) || !is_ex_of_type(b, idx) || !is_ex_of_type(c, idx))
 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_ex_of_type(a, idx) || !is_ex_of_type(b, idx) || !is_ex_of_type(c, idx))
 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_ex_of_type(e, color)) {
 522
 523                 if (ex_to<color>(e).get_representation_label() == rl
 524                  && is_ex_of_type(e.op(0), su3one))
 525                         return _ex3();
 526                 else
 527                         return _ex0();
 528
 529         } else if (is_ex_exactly_of_type(e, mul)) {
 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_ex_exactly_of_type(e, ncmul)) {
 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_ex_of_type(e_expanded, ncmul))
 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