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 "mul.h"
  32 #include "power.h" // for sqrt()
  33 #include "symbol.h"
  34 #include "print.h"
  35 #include "archive.h"
  36 #include "debugmsg.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 constructor, destructor, copy constructor assignment operator and helpers
  49 //////////
  50
  51 color::color() : representation_label(0)
  52 {
  53         debugmsg("color default constructor", LOGLEVEL_CONSTRUCT);
  54         tinfo_key = TINFO_color;
  55 }
  56
  57 void color::copy(const color & other)
  58 {
  59         inherited::copy(other);
  60         representation_label = other.representation_label;
  61 }
  62
  63 DEFAULT_DESTROY(color)
  64 DEFAULT_CTORS(su3one)
  65 DEFAULT_CTORS(su3t)
  66 DEFAULT_CTORS(su3f)
  67 DEFAULT_CTORS(su3d)
  68
  69 //////////
  70 // other constructors
  71 //////////
  72
  73 /** Construct object without any color index. This constructor is for
  74  *  internal use only. Use the color_ONE() function instead.
  75  *  @see color_ONE */
  76 color::color(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
  77 {
  78         debugmsg("color constructor from ex,unsigned char", LOGLEVEL_CONSTRUCT);
  79         tinfo_key = TINFO_color;
  80 }
  81
  82 /** Construct object with one color index. This constructor is for internal
  83  *  use only. Use the color_T() function instead.
  84  *  @see color_T */
  85 color::color(const ex & b, const ex & i1, unsigned char rl) : inherited(b, i1), representation_label(rl)
  86 {
  87         debugmsg("color constructor from ex,ex,unsigned char", LOGLEVEL_CONSTRUCT);
  88         tinfo_key = TINFO_color;
  89 }
  90
  91 color::color(unsigned char rl, const exvector & v, bool discardable) : inherited(sy_none(), v, discardable), representation_label(rl)
  92 {
  93         debugmsg("color constructor from unsigned char,exvector", LOGLEVEL_CONSTRUCT);
  94         tinfo_key = TINFO_color;
  95 }
  96
  97 color::color(unsigned char rl, exvector * vp) : inherited(sy_none(), vp), representation_label(rl)
  98 {
  99         debugmsg("color constructor from unsigned char,exvector *", LOGLEVEL_CONSTRUCT);
 100         tinfo_key = TINFO_color;
 101 }
 102
 103 //////////
 104 // archiving
 105 //////////
 106
 107 color::color(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 108 {
 109         debugmsg("color constructor from archive_node", LOGLEVEL_CONSTRUCT);
 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::simplify_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_ex_of_type(it->op(0), su3one))
 173                         s.push_back(*it);
 174                 it++;
 175         }
 176
 177         if (s.empty())
 178                 return color(su3one(), representation_label);
 179         else
 180                 return simplified_ncmul(s);
 181 }
 182
 183 ex color::thisexprseq(const exvector & v) const
 184 {
 185         return color(representation_label, v);
 186 }
 187
 188 ex color::thisexprseq(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())/3;
 256                 else if (CMPINDICES(8,8,8))
 257                         return -sqrt(_ex3())/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())/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_ex_exactly_of_type(other->op(0), su3t)) {
 317
 318                 // T.a T.a = 4/3 ONE
 319                 if (other - self == 1) {
 320                         *self = numeric(4, 3);
 321                         *other = color_ONE(rl);
 322                         return true;
 323
 324                 // T.a T.b T.a = -1/6 T.b
 325                 } else if (other - self == 2
 326                         && is_ex_of_type(self[1], color)) {
 327                         *self = numeric(-1, 6);
 328                         *other = _ex1();
 329                         return true;
 330
 331                 // T.a S T.a = 1/2 Tr(S) - 1/6 S
 332                 } else {
 333                         exvector::iterator it = self + 1;
 334                         while (it != other) {
 335                                 if (!is_ex_of_type(*it, color)) {
 336                                         return false;
 337                                 }
 338                                 it++;
 339                         }
 340
 341                         it = self + 1;
 342                         ex S = _ex1();
 343                         while (it != other) {
 344                                 S *= *it;
 345                                 *it++ = _ex1();
 346                         }
 347
 348                         *self = color_trace(S, rl) * color_ONE(rl) / 2 - S / 6;
 349                         *other = _ex1();
 350                         return true;
 351                 }
 352         }
 353
 354         return false;
 355 }
 356
 357 /** Contraction of an indexed symmetric structure constant with something else. */
 358 bool su3d::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 359 {
 360         GINAC_ASSERT(is_a<indexed>(*self));
 361         GINAC_ASSERT(is_a<indexed>(*other));
 362         GINAC_ASSERT(self->nops() == 4);
 363         GINAC_ASSERT(is_a<su3d>(self->op(0)));
 364
 365         if (is_ex_exactly_of_type(other->op(0), su3d)) {
 366
 367                 // Find the dummy indices of the contraction
 368                 exvector self_indices = ex_to<indexed>(*self).get_indices();
 369                 exvector other_indices = ex_to<indexed>(*other).get_indices();
 370                 exvector all_indices = self_indices;
 371                 all_indices.insert(all_indices.end(), other_indices.begin(), other_indices.end());
 372                 exvector free_indices, dummy_indices;
 373                 find_free_and_dummy(all_indices, free_indices, dummy_indices);
 374
 375                 // d.abc d.abc = 40/3
 376                 if (dummy_indices.size() == 3) {
 377                         *self = numeric(40, 3);
 378                         *other = _ex1();
 379                         return true;
 380
 381                 // d.akl d.bkl = 5/3 delta.ab
 382                 } else if (dummy_indices.size() == 2) {
 383                         exvector a;
 384                         std::back_insert_iterator<exvector> ita(a);
 385                         ita = set_difference(self_indices.begin(), self_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
 386                         ita = set_difference(other_indices.begin(), other_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
 387                         GINAC_ASSERT(a.size() == 2);
 388                         *self = numeric(5, 3) * delta_tensor(a[0], a[1]);
 389                         *other = _ex1();
 390                         return true;
 391                 }
 392
 393         } else if (is_ex_exactly_of_type(other->op(0), su3t)) {
 394
 395                 // d.abc T.b T.c = 5/6 T.a
 396                 if (other+1 != v.end()
 397                  && is_ex_exactly_of_type(other[1].op(0), su3t)
 398                  && ex_to<indexed>(*self).has_dummy_index_for(other[1].op(1))) {
 399
 400                         exvector self_indices = ex_to<indexed>(*self).get_indices();
 401                         exvector dummy_indices;
 402                         dummy_indices.push_back(other[0].op(1));
 403                         dummy_indices.push_back(other[1].op(1));
 404                         int sig;
 405                         ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
 406                         *self = numeric(5, 6);
 407                         other[0] = color_T(a, ex_to<color>(other[0]).get_representation_label());
 408                         other[1] = _ex1();
 409                         return true;
 410                 }
 411         }
 412
 413         return false;
 414 }
 415
 416 /** Contraction of an indexed antisymmetric structure constant with something else. */
 417 bool su3f::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 418 {
 419         GINAC_ASSERT(is_a<indexed>(*self));
 420         GINAC_ASSERT(is_a<indexed>(*other));
 421         GINAC_ASSERT(self->nops() == 4);
 422         GINAC_ASSERT(is_a<su3f>(self->op(0)));
 423
 424         if (is_ex_exactly_of_type(other->op(0), su3f)) { // f*d is handled by su3d class
 425
 426                 // Find the dummy indices of the contraction
 427                 exvector dummy_indices;
 428                 dummy_indices = ex_to<indexed>(*self).get_dummy_indices(ex_to<indexed>(*other));
 429
 430                 // f.abc f.abc = 24
 431                 if (dummy_indices.size() == 3) {
 432                         *self = 24;
 433                         *other = _ex1();
 434                         return true;
 435
 436                 // f.akl f.bkl = 3 delta.ab
 437                 } else if (dummy_indices.size() == 2) {
 438                         int sign1, sign2;
 439                         ex a = permute_free_index_to_front(ex_to<indexed>(*self).get_indices(), dummy_indices, sign1);
 440                         ex b = permute_free_index_to_front(ex_to<indexed>(*other).get_indices(), dummy_indices, sign2);
 441                         *self = sign1 * sign2 * 3 * delta_tensor(a, b);
 442                         *other = _ex1();
 443                         return true;
 444                 }
 445
 446         } else if (is_ex_exactly_of_type(other->op(0), su3t)) {
 447
 448                 // f.abc T.b T.c = 3/2 I T.a
 449                 if (other+1 != v.end()
 450                  && is_ex_exactly_of_type(other[1].op(0), su3t)
 451                  && ex_to<indexed>(*self).has_dummy_index_for(other[1].op(1))) {
 452
 453                         exvector self_indices = ex_to<indexed>(*self).get_indices();
 454                         exvector dummy_indices;
 455                         dummy_indices.push_back(other[0].op(1));
 456                         dummy_indices.push_back(other[1].op(1));
 457                         int sig;
 458                         ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
 459                         *self = numeric(3, 2) * sig * I;
 460                         other[0] = color_T(a, ex_to<color>(other[0]).get_representation_label());
 461                         other[1] = _ex1();
 462                         return true;
 463                 }
 464         }
 465
 466         return false;
 467 }
 468
 469 //////////
 470 // global functions
 471 //////////
 472
 473 ex color_ONE(unsigned char rl)
 474 {
 475         return color(su3one(), rl);
 476 }
 477
 478 ex color_T(const ex & a, unsigned char rl)
 479 {
 480         if (!is_ex_of_type(a, idx))
 481                 throw(std::invalid_argument("indices of color_T must be of type idx"));
 482         if (!ex_to<idx>(a).get_dim().is_equal(8))
 483                 throw(std::invalid_argument("index dimension for color_T must be 8"));
 484
 485         return color(su3t(), a, rl);
 486 }
 487
 488 ex color_f(const ex & a, const ex & b, const ex & c)
 489 {
 490         if (!is_ex_of_type(a, idx) || !is_ex_of_type(b, idx) || !is_ex_of_type(c, idx))
 491                 throw(std::invalid_argument("indices of color_f must be of type idx"));
 492         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))
 493                 throw(std::invalid_argument("index dimension for color_f must be 8"));
 494
 495         return indexed(su3f(), sy_anti(), a, b, c);
 496 }
 497
 498 ex color_d(const ex & a, const ex & b, const ex & c)
 499 {
 500         if (!is_ex_of_type(a, idx) || !is_ex_of_type(b, idx) || !is_ex_of_type(c, idx))
 501                 throw(std::invalid_argument("indices of color_d 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_d must be 8"));
 504
 505         return indexed(su3d(), sy_symm(), a, b, c);
 506 }
 507
 508 ex color_h(const ex & a, const ex & b, const ex & c)
 509 {
 510         return color_d(a, b, c) + I * color_f(a, b, c);
 511 }
 512
 513 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 514  *  is that of a color object with the specified representation label. */
 515 static bool is_color_tinfo(unsigned ti, unsigned char rl)
 516 {
 517         return ti == (TINFO_color + rl);
 518 }
 519
 520 ex color_trace(const ex & e, unsigned char rl)
 521 {
 522         if (is_ex_of_type(e, color)) {
 523
 524                 if (ex_to<color>(e).get_representation_label() == rl
 525                  && is_ex_of_type(e.op(0), su3one))
 526                         return _ex3();
 527                 else
 528                         return _ex0();
 529
 530         } else if (is_ex_exactly_of_type(e, mul)) {
 531
 532                 // Trace of product: pull out non-color factors
 533                 ex prod = _ex1();
 534                 for (unsigned i=0; i<e.nops(); i++) {
 535                         const ex &o = e.op(i);
 536                         if (is_color_tinfo(o.return_type_tinfo(), rl))
 537                                 prod *= color_trace(o, rl);
 538                         else
 539                                 prod *= o;
 540                 }
 541                 return prod;
 542
 543         } else if (is_ex_exactly_of_type(e, ncmul)) {
 544
 545                 if (!is_color_tinfo(e.return_type_tinfo(), rl))
 546                         return _ex0();
 547
 548                 // Expand product, if necessary
 549                 ex e_expanded = e.expand();
 550                 if (!is_ex_of_type(e_expanded, ncmul))
 551                         return color_trace(e_expanded, rl);
 552
 553                 unsigned num = e.nops();
 554
 555                 if (num == 2) {
 556
 557                         // Tr T_a T_b = 1/2 delta_a_b
 558                         return delta_tensor(e.op(0).op(1), e.op(1).op(1)) / 2;
 559
 560                 } else if (num == 3) {
 561
 562                         // Tr T_a T_b T_c = 1/4 h_a_b_c
 563                         return color_h(e.op(0).op(1), e.op(1).op(1), e.op(2).op(1)) / 4;
 564
 565                 } else {
 566
 567                         // Traces of 4 or more generators are computed recursively:
 568                         // Tr T_a1 .. T_an =
 569                         //     1/6 delta_a(n-1)_an Tr T_a1 .. T_a(n-2)
 570                         //   + 1/2 h_a(n-1)_an_k Tr T_a1 .. T_a(n-2) T_k
 571                         const ex &last_index = e.op(num - 1).op(1);
 572                         const ex &next_to_last_index = e.op(num - 2).op(1);
 573                         idx summation_index((new symbol)->setflag(status_flags::dynallocated), 8);
 574
 575                         exvector v1;
 576                         v1.reserve(num - 2);
 577                         for (unsigned i=0; i<num-2; i++)
 578                                 v1.push_back(e.op(i));
 579
 580                         exvector v2 = v1;
 581                         v2.push_back(color_T(summation_index, rl));
 582
 583                         return delta_tensor(next_to_last_index, last_index) * color_trace(ncmul(v1), rl) / 6
 584                                + color_h(next_to_last_index, last_index, summation_index) * color_trace(ncmul(v2), rl) / 2;
 585                 }
 586
 587         } else if (e.nops() > 0) {
 588
 589                 // Trace maps to all other container classes (this includes sums)
 590                 pointer_to_map_function_1arg<unsigned char> fcn(color_trace, rl);
 591                 return e.map(fcn);
 592
 593         } else
 594                 return _ex0();
 595 }
 596
 597 } // namespace GiNaC