ginac/color.cpp

   1 /** @file color.cpp
   2  *
   3  *  Implementation of GiNaC's color objects.
   4  *  No real implementation yet, to be done.     */
   5
   6 /*
   7  *  GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
   8  *
   9  *  This program is free software; you can redistribute it and/or modify
  10  *  it under the terms of the GNU General Public License as published by
  11  *  the Free Software Foundation; either version 2 of the License, or
  12  *  (at your option) any later version.
  13  *
  14  *  This program is distributed in the hope that it will be useful,
  15  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  16  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  17  *  GNU General Public License for more details.
  18  *
  19  *  You should have received a copy of the GNU General Public License
  20  *  along with this program; if not, write to the Free Software
  21  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  22  */
  23
  24 #include <string>
  25 #include <list>
  26 #include <algorithm>
  27 #include <iostream>
  28 #include <stdexcept>
  29
  30 #include "color.h"
  31 #include "ex.h"
  32 #include "coloridx.h"
  33 #include "ncmul.h"
  34 #include "numeric.h"
  35 #include "relational.h"
  36 #include "archive.h"
  37 #include "debugmsg.h"
  38 #include "utils.h"
  39
  40 #ifndef NO_NAMESPACE_GINAC
  41 namespace GiNaC {
  42 #endif // ndef NO_NAMESPACE_GINAC
  43
  44 GINAC_IMPLEMENT_REGISTERED_CLASS(color, indexed)
  45
  46 //////////
  47 // default constructor, destructor, copy constructor assignment operator and helpers
  48 //////////
  49
  50 // public
  51
  52 color::color() : inherited(TINFO_color), type(invalid), representation_label(0)
  53 {
  54         debugmsg("color default constructor",LOGLEVEL_CONSTRUCT);
  55 }
  56
  57 color::~color()
  58 {
  59         debugmsg("color destructor",LOGLEVEL_DESTRUCT);
  60         destroy(false);
  61 }
  62
  63 color::color(const color & other)
  64 {
  65         debugmsg("color copy constructor",LOGLEVEL_CONSTRUCT);
  66         copy (other);
  67 }
  68
  69 const color & color::operator=(const color & other)
  70 {
  71         debugmsg("color operator=",LOGLEVEL_ASSIGNMENT);
  72         if (this != &other) {
  73                 destroy(true);
  74                 copy(other);
  75         }
  76         return *this;
  77 }
  78
  79 // protected
  80
  81 void color::copy(const color & other)
  82 {
  83         inherited::copy(other);
  84         type=other.type;
  85         representation_label=other.representation_label;
  86 }
  87
  88 void color::destroy(bool call_parent)
  89 {
  90         if (call_parent) {
  91                 inherited::destroy(call_parent);
  92         }
  93 }
  94
  95 //////////
  96 // other constructors
  97 //////////
  98
  99 // protected
 100
 101 /** Construct object without any color index. This constructor is for internal
 102  *  use only. Use the color_ONE() function instead.
 103  *  @see color_ONE */
 104 color::color(color_types const t, unsigned rl) : type(t), representation_label(rl)
 105 {
 106         debugmsg("color constructor from color_types,unsigned",LOGLEVEL_CONSTRUCT);
 107         GINAC_ASSERT(representation_label<MAX_REPRESENTATION_LABELS);
 108         tinfo_key=TINFO_color;
 109         GINAC_ASSERT(all_of_type_coloridx());
 110 }
 111
 112 /** Construct object with one color index. This constructor is for internal
 113  *  use only. Use the color_T() function instead.
 114  *  @see color_T */
 115 color::color(color_types const t, const ex & i1, unsigned rl)
 116   : inherited(i1), type(t), representation_label(rl)
 117 {
 118         debugmsg("color constructor from color_types,ex,unsigned",LOGLEVEL_CONSTRUCT);
 119         GINAC_ASSERT(representation_label<MAX_REPRESENTATION_LABELS);
 120         tinfo_key=TINFO_color;
 121         GINAC_ASSERT(all_of_type_coloridx());
 122 }
 123
 124 /** Construct object with two color indices. This constructor is for internal
 125  *  use only. Use the color_delta8() function instead.
 126  *  @see color_delta8 */
 127 color::color(color_types const t, const ex & i1, const ex & i2, unsigned rl)
 128   : inherited(i1,i2), type(t), representation_label(rl)
 129 {
 130         debugmsg("color constructor from color_types,ex,ex,unsigned",LOGLEVEL_CONSTRUCT);
 131         GINAC_ASSERT(representation_label<MAX_REPRESENTATION_LABELS);
 132         tinfo_key=TINFO_color;
 133         GINAC_ASSERT(all_of_type_coloridx());
 134 }
 135
 136 /** Construct object with three color indices. This constructor is for internal
 137  *  use only. Use the color_f(), color_d() and color_h() functions instead.
 138  *  @see color_f
 139  *  @see color_d
 140  *  @see color_h */
 141 color::color(color_types const t, const ex & i1, const ex & i2, const ex & i3,
 142              unsigned rl) : inherited(i1,i2,i3), type(t), representation_label(rl)
 143 {
 144         debugmsg("color constructor from color_types,ex,ex,ex,unsigned",LOGLEVEL_CONSTRUCT);
 145         GINAC_ASSERT(representation_label<MAX_REPRESENTATION_LABELS);
 146         tinfo_key=TINFO_color;
 147         GINAC_ASSERT(all_of_type_coloridx());
 148 }
 149
 150 /** Construct object with arbitrary number of color indices. This
 151  *  constructor is for internal use only. */
 152 color::color(color_types const t, const exvector & iv, unsigned rl)
 153   : inherited(iv), type(t), representation_label(rl)
 154 {
 155         debugmsg("color constructor from color_types,exvector,unsigned",LOGLEVEL_CONSTRUCT);
 156         GINAC_ASSERT(representation_label<MAX_REPRESENTATION_LABELS);
 157         tinfo_key=TINFO_color;
 158         GINAC_ASSERT(all_of_type_coloridx());
 159 }
 160
 161 color::color(color_types const t, exvector * ivp, unsigned rl)
 162   : inherited(ivp), type(t), representation_label(rl)
 163 {
 164         debugmsg("color constructor from color_types,exvector *,unsigned",LOGLEVEL_CONSTRUCT);
 165         GINAC_ASSERT(representation_label<MAX_REPRESENTATION_LABELS);
 166         tinfo_key=TINFO_color;
 167         GINAC_ASSERT(all_of_type_coloridx());
 168 }
 169
 170 //////////
 171 // archiving
 172 //////////
 173
 174 /** Construct object from archive_node. */
 175 color::color(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 176 {
 177         debugmsg("color constructor from archive_node", LOGLEVEL_CONSTRUCT);
 178         unsigned int ty;
 179         if (!(n.find_unsigned("type", ty)))
 180                 throw (std::runtime_error("unknown color type in archive"));
 181         type = (color_types)ty;
 182         if (!(n.find_unsigned("representation", representation_label)))
 183                 throw (std::runtime_error("unknown color representation label in archive"));
 184 }
 185
 186 /** Unarchive the object. */
 187 ex color::unarchive(const archive_node &n, const lst &sym_lst)
 188 {
 189         return (new color(n, sym_lst))->setflag(status_flags::dynallocated);
 190 }
 191
 192 /** Archive the object. */
 193 void color::archive(archive_node &n) const
 194 {
 195         inherited::archive(n);
 196         n.add_unsigned("type", type);
 197         n.add_unsigned("representation", representation_label);
 198 }
 199
 200 //////////
 201 // functions overriding virtual functions from bases classes
 202 //////////
 203
 204 // public
 205
 206 basic * color::duplicate() const
 207 {
 208         debugmsg("color duplicate",LOGLEVEL_DUPLICATE);
 209         return new color(*this);
 210 }
 211
 212 void color::printraw(std::ostream & os) const
 213 {
 214         debugmsg("color printraw",LOGLEVEL_PRINT);
 215         os << "color(type=" << (unsigned)type
 216            << ",representation_label=" << representation_label
 217            << ",indices=";
 218         printrawindices(os);
 219         os << ",hash=" << hashvalue << ",flags=" << flags << ")";
 220 }
 221
 222 void color::printtree(std::ostream & os, unsigned indent) const
 223 {
 224         debugmsg("color printtree",LOGLEVEL_PRINT);
 225         os << std::string(indent,' ') << "color object: "
 226            << "type=" << (unsigned)type
 227            << ",representation_label=" << representation_label << ", ";
 228         os << seq.size() << " indices" << std::endl;
 229         printtreeindices(os,indent);
 230         os << std::string(indent,' ') << "hash=" << hashvalue
 231            << " (0x" << std::hex << hashvalue << std::dec << ")"
 232            << ", flags=" << flags << std::endl;
 233 }
 234
 235 void color::print(std::ostream & os, unsigned upper_precedence) const
 236 {
 237         debugmsg("color print",LOGLEVEL_PRINT);
 238         switch (type) {
 239         case color_T:
 240                 os << "T";
 241                 if (representation_label!=0) {
 242                         os << "^(" << representation_label << ")";
 243                 }
 244                 break;
 245         case color_f:
 246                 os << "f";
 247                 break;
 248         case color_d:
 249                 os << "d";
 250                 break;
 251         case color_delta8:
 252                 os << "delta8";
 253                 break;
 254         case color_ONE:
 255                 os << "color_ONE";
 256                 break;
 257         case invalid:
 258         default:
 259                 os << "INVALID_COLOR_OBJECT";
 260                 break;
 261         }
 262         printindices(os);
 263 }
 264
 265 bool color::info(unsigned inf) const
 266 {
 267         return inherited::info(inf);
 268 }
 269
 270 #define CMPINDICES(A,B,C) ((idx1.get_value()==(A))&&(idx2.get_value()==(B))&&(idx3.get_value()==(C)))
 271
 272 ex color::eval(int level) const
 273 {
 274         // canonicalize indices
 275
 276         bool antisymmetric=false;
 277
 278         switch (type) {
 279         case color_f:
 280                 antisymmetric=true; // no break here!
 281         case color_d:
 282         case color_delta8:
 283                 {
 284                         exvector iv=seq;
 285                         int sig=canonicalize_indices(iv,antisymmetric);
 286                         if (sig!=INT_MAX) {
 287                                 // something has changed while sorting indices, more evaluations later
 288                                 if (sig==0) return _ex0();
 289                                 return ex(sig)*color(type,iv,representation_label);
 290                         }
 291                 }
 292                 break;
 293         default:
 294                 // nothing to canonicalize
 295                 break;
 296         }
 297
 298         switch (type) {
 299         case color_delta8:
 300                 {
 301                         GINAC_ASSERT(seq.size()==2);
 302                         const coloridx & idx1=ex_to_coloridx(seq[0]);
 303                         const coloridx & idx2=ex_to_coloridx(seq[1]);
 304
 305                         // check for delta8_{a,a} where a is a symbolic index, replace by 8
 306                         if ((idx1.is_symbolic())&&(idx1.is_equal_same_type(idx2))) {
 307                                 return ex(COLOR_EIGHT);
 308                         }
 309
 310                         // check for delta8_{a,b} where a and b are numeric indices, replace by 0 or 1
 311                         if ((!idx1.is_symbolic())&&(!idx2.is_symbolic())) {
 312                                 if ((idx1.get_value()!=idx2.get_value())) {
 313                                         return _ex1();
 314                                 } else {
 315                                         return _ex0();
 316                                 }
 317                         }
 318         }
 319                 break;
 320         case color_d:
 321                 // check for d_{a,a,c} (=0) when a is symbolic
 322                 {
 323                         GINAC_ASSERT(seq.size()==3);
 324                         const coloridx & idx1=ex_to_coloridx(seq[0]);
 325                         const coloridx & idx2=ex_to_coloridx(seq[1]);
 326                         const coloridx & idx3=ex_to_coloridx(seq[2]);
 327
 328                         if (idx1.is_equal_same_type(idx2) && idx1.is_symbolic()) {
 329                                 return _ex0();
 330                         } else if (idx2.is_equal_same_type(idx3) && idx2.is_symbolic()) {
 331                                 return _ex0();
 332                         }
 333
 334                         // check for three numeric indices
 335                         if (!(idx1.is_symbolic()||idx2.is_symbolic()||idx3.is_symbolic())) {
 336                                 GINAC_ASSERT(idx1.get_value()<=idx2.get_value());
 337                                 GINAC_ASSERT(idx2.get_value()<=idx3.get_value());
 338                                 if (CMPINDICES(1,4,6)||CMPINDICES(1,5,7)||CMPINDICES(2,5,6)||
 339                                         CMPINDICES(3,4,4)||CMPINDICES(3,5,5)) {
 340                                         return _ex1_2();
 341                                 } else if (CMPINDICES(2,4,7)||CMPINDICES(3,6,6)||CMPINDICES(3,7,7)) {
 342                                         return -_ex1_2();
 343                                 } else if (CMPINDICES(1,1,8)||CMPINDICES(2,2,8)||CMPINDICES(3,3,8)) {
 344                                         return 1/sqrt(numeric(3));
 345                                 } else if (CMPINDICES(8,8,8)) {
 346                                         return -1/sqrt(numeric(3));
 347                                 } else if (CMPINDICES(4,4,8)||CMPINDICES(5,5,8)||CMPINDICES(6,6,8)||CMPINDICES(7,7,8)) {
 348                                         return -1/(2*sqrt(numeric(3)));
 349                                 }
 350                                 return _ex0();
 351                         }
 352                 }
 353                 break;
 354         case color_f:
 355                 {
 356                         GINAC_ASSERT(seq.size()==3);
 357                         const coloridx & idx1=ex_to_coloridx(seq[0]);
 358                         const coloridx & idx2=ex_to_coloridx(seq[1]);
 359                         const coloridx & idx3=ex_to_coloridx(seq[2]);
 360
 361                         // check for three numeric indices
 362                         if (!(idx1.is_symbolic()||idx2.is_symbolic()||idx3.is_symbolic())) {
 363                                 GINAC_ASSERT(idx1.get_value()<=idx2.get_value());
 364                                 GINAC_ASSERT(idx2.get_value()<=idx3.get_value());
 365                                 if (CMPINDICES(1,2,3)) {
 366                                         return _ex1();
 367                                 } else if (CMPINDICES(1,4,7)||CMPINDICES(2,4,6)||
 368                                            CMPINDICES(2,5,7)||CMPINDICES(3,4,5)) {
 369                                         return _ex1_2();
 370                                 } else if (CMPINDICES(1,5,6)||CMPINDICES(3,6,7)) {
 371                                         return -_ex1_2();
 372                                 } else if (CMPINDICES(4,5,8)||CMPINDICES(6,7,8)) {
 373                                         return sqrt(numeric(3))/2;
 374                                 } else if (CMPINDICES(8,8,8)) {
 375                                         return -1/sqrt(numeric(3));
 376                                 } else if (CMPINDICES(4,4,8)||CMPINDICES(5,5,8)||CMPINDICES(6,6,8)||CMPINDICES(7,7,8)) {
 377                                         return -1/(2*sqrt(numeric(3)));
 378                                 }
 379                                 return _ex0();
 380                         }
 381                         break;
 382                 }
 383         default:
 384                 // nothing to evaluate
 385                 break;
 386         }
 387
 388         return this->hold();
 389 }
 390
 391 // protected
 392
 393 int color::compare_same_type(const basic & other) const
 394 {
 395         GINAC_ASSERT(other.tinfo() == TINFO_color);
 396         const color &o = static_cast<const color &>(other);
 397
 398         if (type != o.type) {
 399                 // different type
 400                 return type < o.type ? -1 : 1;
 401         }
 402
 403         if (representation_label != o.representation_label) {
 404                 // different representation label
 405                 return representation_label < o.representation_label ? -1 : 1;
 406         }
 407
 408         return inherited::compare_same_type(other);
 409 }
 410
 411 bool color::is_equal_same_type(const basic & other) const
 412 {
 413         GINAC_ASSERT(other.tinfo() == TINFO_color);
 414         const color &o = static_cast<const color &>(other);
 415
 416         if (type != o.type) return false;
 417         if (representation_label != o.representation_label) return false;
 418         return inherited::is_equal_same_type(other);
 419 }
 420
 421 #include <iostream>
 422
 423 ex color::simplify_ncmul(const exvector & v) const
 424 {
 425         // simplifications: contract delta8_{a,b} where possible
 426         //                  sort delta8,f,d,T(rl=0),T(rl=1),...,ONE(rl=0),ONE(rl=1),...
 427         //                  remove superfluous ONEs
 428
 429         // contract indices of delta8_{a,b} if they are different and symbolic
 430
 431         exvector v_contracted=v;
 432         unsigned replacements;
 433         bool something_changed=false;
 434
 435         exvector::iterator it=v_contracted.begin();
 436         while (it!=v_contracted.end()) {
 437                 // process only delta8 objects
 438                 if (is_ex_exactly_of_type(*it,color) && (ex_to_color(*it).type==color_delta8)) {
 439                         color & d8=ex_to_nonconst_color(*it);
 440                         GINAC_ASSERT(d8.seq.size()==2);
 441                         const coloridx & first_idx=ex_to_coloridx(d8.seq[0]);
 442                         const coloridx & second_idx=ex_to_coloridx(d8.seq[1]);
 443                         // delta8_{a,a} should have been contracted in color::eval()
 444                         GINAC_ASSERT((!first_idx.is_equal(second_idx))||(!first_idx.is_symbolic()));
 445                         ex saved_delta8=*it; // save to restore it later
 446
 447                         // try to contract first index
 448                         replacements=1;
 449                         if (first_idx.is_symbolic()) {
 450                                 replacements = subs_index_in_exvector(v_contracted,first_idx,second_idx);
 451                                 if (replacements==1) {
 452                                         // not contracted except in itself, restore delta8 object
 453                                         *it=saved_delta8;
 454                                 } else {
 455                                         // a contracted index should occur exactly twice
 456                                         GINAC_ASSERT(replacements==2);
 457                                         *it=_ex1();
 458                                         something_changed=true;
 459                                 }
 460                         }
 461
 462                         // try second index only if first was not contracted
 463                         if ((replacements==1)&&(second_idx.is_symbolic())) {
 464                                 // first index not contracted, *it is guaranteed to be the original delta8 object
 465                                 replacements = subs_index_in_exvector(v_contracted,second_idx,first_idx);
 466                                 if (replacements==1) {
 467                                         // not contracted except in itself, restore delta8 object
 468                                         *it=saved_delta8;
 469                                 } else {
 470                                         // a contracted index should occur exactly twice
 471                                         GINAC_ASSERT(replacements==2);
 472                                         *it=_ex1();
 473                                         something_changed=true;
 474                                 }
 475                         }
 476                 }
 477                 ++it;
 478         }
 479
 480         if (something_changed) {
 481                 // do more simplifications later
 482                 return nonsimplified_ncmul(v_contracted);
 483         }
 484
 485         // there were no indices to contract
 486         // sort delta8,f,d,T(rl=0),T(rl=1),...,ONE(rl=0),ONE(rl=1),...,unknown
 487         // (if there is at least one unknown object, all Ts will be unknown to not change the order)
 488
 489         exvector delta8vec;
 490         exvector fvec;
 491         exvector dvec;
 492         exvectorvector Tvecs;
 493         Tvecs.resize(MAX_REPRESENTATION_LABELS);
 494         exvectorvector ONEvecs;
 495         ONEvecs.resize(MAX_REPRESENTATION_LABELS);
 496         exvector unknownvec;
 497
 498         split_color_string_in_parts(v,delta8vec,fvec,dvec,Tvecs,ONEvecs,unknownvec);
 499
 500         // d_{a,k,l} f_{b,k,l}=0 (includes case a=b)
 501         if ((dvec.size()>=1)&&(fvec.size()>=1)) {
 502                 for (exvector::iterator it1=dvec.begin(); it1!=dvec.end(); ++it1) {
 503                         for (exvector::iterator it2=fvec.begin(); it2!=fvec.end(); ++it2) {
 504                                 GINAC_ASSERT(is_ex_exactly_of_type(*it1,color));
 505                                 GINAC_ASSERT(is_ex_exactly_of_type(*it2,color));
 506                                 const color & col1=ex_to_color(*it1);
 507                                 const color & col2=ex_to_color(*it2);
 508                                 exvector iv_intersect=idx_intersect(col1.seq,col2.seq);
 509                                 if (iv_intersect.size()>=2) return _ex0();
 510                         }
 511                 }
 512         }
 513
 514         // d_{a,k,l} d_{b,k,l}=5/3 delta8_{a,b} (includes case a=b)
 515         if (dvec.size()>=2) {
 516                 for (exvector::iterator it1=dvec.begin(); it1!=dvec.end()-1; ++it1) {
 517                         for (exvector::iterator it2=it1+1; it2!=dvec.end(); ++it2) {
 518                                 GINAC_ASSERT(is_ex_exactly_of_type(*it1,color));
 519                                 GINAC_ASSERT(is_ex_exactly_of_type(*it2,color));
 520                                 const color & col1=ex_to_color(*it1);
 521                                 const color & col2=ex_to_color(*it2);
 522                                 exvector iv_intersect=idx_intersect(col1.seq,col2.seq);
 523                                 if (iv_intersect.size()>=2) {
 524                                         if (iv_intersect.size()==3) {
 525                                                 *it1=numeric(40)/numeric(3);
 526                                                 *it2=_ex1();
 527                                         } else {
 528                                                 int dummy; // sign unimportant, since symmetric
 529                                                 ex idx1=permute_free_index_to_front(col1.seq,iv_intersect,&dummy);
 530                                                 ex idx2=permute_free_index_to_front(col2.seq,iv_intersect,&dummy);
 531                                                 *it1=numeric(5)/numeric(3)*color(color_delta8,idx1,idx2);
 532                                                 *it2=_ex1();
 533                                         }
 534                                         return nonsimplified_ncmul(recombine_color_string(
 535                                                    delta8vec,fvec,dvec,Tvecs,ONEvecs,unknownvec));
 536                                 }
 537                         }
 538                 }
 539         }
 540
 541         // f_{a,k,l} f_{b,k,l}=3 delta8_{a,b} (includes case a=b)
 542         if (fvec.size()>=2) {
 543                 for (exvector::iterator it1=fvec.begin(); it1!=fvec.end()-1; ++it1) {
 544                         for (exvector::iterator it2=it1+1; it2!=fvec.end(); ++it2) {
 545                                 GINAC_ASSERT(is_ex_exactly_of_type(*it1,color));
 546                                 GINAC_ASSERT(is_ex_exactly_of_type(*it2,color));
 547                                 const color & col1=ex_to_color(*it1);
 548                                 const color & col2=ex_to_color(*it2);
 549                                 exvector iv_intersect=idx_intersect(col1.seq,col2.seq);
 550                                 if (iv_intersect.size()>=2) {
 551                                         if (iv_intersect.size()==3) {
 552                                                 *it1=numeric(24);
 553                                                 *it2=_ex1();
 554                                         } else {
 555                                                 int sig1, sig2;
 556                                                 ex idx1=permute_free_index_to_front(col1.seq,iv_intersect,&sig1);
 557                                                 ex idx2=permute_free_index_to_front(col2.seq,iv_intersect,&sig2);
 558                                                 *it1=numeric(sig1*sig2*5)/numeric(3)*color(color_delta8,idx1,idx2);
 559                                                 *it2=_ex1();
 560                                         }
 561                                         return nonsimplified_ncmul(recombine_color_string(
 562                                                    delta8vec,fvec,dvec,Tvecs,ONEvecs,unknownvec));
 563                                 }
 564                         }
 565                 }
 566         }
 567
 568         // d_{a,b,c} T_b T_c = 5/6 T_a
 569         // f_{a,b,c} T_b T_c = 3/2 I T_a
 570         for (unsigned rl=0; rl<MAX_REPRESENTATION_LABELS; ++rl) {
 571                 if ((Tvecs[rl].size()>=2)&&((dvec.size()>=1)||(fvec.size()>=1))) {
 572                         for (exvector::iterator it1=Tvecs[rl].begin(); it1!=Tvecs[rl].end()-1; ++it1) {
 573                                 exvector iv;
 574                                 GINAC_ASSERT(is_ex_exactly_of_type(*it1,color)&&ex_to_color(*it1).type==color_T);
 575                                 GINAC_ASSERT(is_ex_exactly_of_type(*(it1+1),color)&&ex_to_color(*(it1+1)).type==color_T);
 576                                 iv.push_back(ex_to_color(*it1).seq[0]);
 577                                 iv.push_back(ex_to_color(*(it1+1)).seq[0]);
 578
 579                                 // d_{a,b,c} T_b T_c = 5/6 T_a
 580                                 for (exvector::iterator it2=dvec.begin(); it2!=dvec.end(); ++it2) {
 581                                         GINAC_ASSERT(is_ex_exactly_of_type(*it2,color)&&ex_to_color(*it2).type==color_d);
 582                                         const color & dref=ex_to_color(*it2);
 583                                         exvector iv_intersect=idx_intersect(dref.seq,iv);
 584                                         if (iv_intersect.size()==2) {
 585                                                 int dummy; // sign unimportant, since symmetric
 586                                                 ex free_idx=permute_free_index_to_front(dref.seq,iv,&dummy);
 587                                                 *it1=color(color_T,free_idx,rl);
 588                                                 *(it1+1)=color(color_ONE,rl);
 589                                                 *it2=numeric(5)/numeric(6);
 590                                                 return nonsimplified_ncmul(recombine_color_string(
 591                                                            delta8vec,fvec,dvec,Tvecs,ONEvecs,unknownvec));
 592                                         }
 593                                 }
 594
 595                                 // f_{a,b,c} T_b T_c = 3/2 I T_a
 596                                 for (exvector::iterator it2=fvec.begin(); it2!=fvec.end(); ++it2) {
 597                                         GINAC_ASSERT(is_ex_exactly_of_type(*it2,color)&&ex_to_color(*it2).type==color_f);
 598                                         const color & fref=ex_to_color(*it2);
 599                                         exvector iv_intersect=idx_intersect(fref.seq,iv);
 600                                         if (iv_intersect.size()==2) {
 601                                                 int sig;
 602                                                 ex free_idx=permute_free_index_to_front(fref.seq,iv,&sig);
 603                                                 *it1=color(color_T,free_idx,rl);
 604                                                 *(it1+1)=color(color_ONE,rl);
 605                                                 *it2=numeric(sig*3)/numeric(2)*I;
 606                                                 return nonsimplified_ncmul(recombine_color_string(
 607                                                            delta8vec,fvec,dvec,Tvecs,ONEvecs,unknownvec));
 608                                         }
 609                                 }
 610                         }
 611                 }
 612         }
 613
 614         // clear all ONEs when there is at least one corresponding color_T
 615         // in this representation, retain one ONE otherwise
 616         for (unsigned rl=0; rl<MAX_REPRESENTATION_LABELS; ++rl) {
 617                 if (Tvecs[rl].size()!=0) {
 618                         ONEvecs[rl].clear();
 619                 } else if (ONEvecs[rl].size()!=0) {
 620                         ONEvecs[rl].clear();
 621                         ONEvecs[rl].push_back(color(color_ONE,rl));
 622                 }
 623         }
 624
 625         // return a sorted vector
 626         return simplified_ncmul(recombine_color_string(delta8vec,fvec,dvec,Tvecs,
 627                                                                                                    ONEvecs,unknownvec));
 628 }
 629
 630 ex color::thisexprseq(const exvector & v) const
 631 {
 632         return color(type,v,representation_label);
 633 }
 634
 635 ex color::thisexprseq(exvector * vp) const
 636 {
 637         return color(type,vp,representation_label);
 638 }
 639
 640 /** Check whether all indices are of class coloridx or a subclass. This
 641  *  function is used internally to make sure that all constructed color
 642  *  objects really carry color indices and not some other classes. */
 643 bool color::all_of_type_coloridx(void) const
 644 {
 645         for (exvector::const_iterator cit=seq.begin(); cit!=seq.end(); ++cit) {
 646                 if (!is_ex_of_type(*cit,coloridx)) return false;
 647         }
 648         return true;
 649 }
 650
 651 //////////
 652 // friend functions
 653 //////////
 654
 655 /** Construct an object representing the unity element of su(3).
 656  *
 657  *  @param rl Representation label
 658  *  @return newly constructed object */
 659 color color_ONE(unsigned rl)
 660 {
 661         return color(color::color_ONE,rl);
 662 }
 663
 664 /** Construct an object representing the generators T_a of SU(3). The index
 665  *  must be of class coloridx.
 666  *
 667  *  @param a Index
 668  *  @param rl Representation label
 669  *  @return newly constructed object */
 670 color color_T(const ex & a, unsigned rl)
 671 {
 672         return color(color::color_T,a,rl);
 673 }
 674
 675 /** Construct an object representing the antisymmetric structure constants
 676  *  f_abc of SU(3). The indices must be of class coloridx.
 677  *
 678  *  @param a First index
 679  *  @param b Second index
 680  *  @param c Third index
 681  *  @return newly constructed object */
 682 color color_f(const ex & a, const ex & b, const ex & c)
 683 {
 684         return color(color::color_f,a,b,c);
 685 }
 686
 687 /** Construct an object representing the symmetric structure constants d_abc
 688  *  of SU(3). The indices must be of class coloridx.
 689  *
 690  *  @param a First index
 691  *  @param b Second index
 692  *  @param c Third index
 693  *  @return newly constructed object */
 694 color color_d(const ex & a, const ex & b, const ex & c)
 695 {
 696         return color(color::color_d,a,b,c);
 697 }
 698
 699 /** This returns the linear combination d_abc+I*f_abc.
 700  *
 701  *  @param a First index
 702  *  @param b Second index
 703  *  @param c Third index
 704  *  @return newly constructed object */
 705 ex color_h(const ex & a, const ex & b, const ex & c)
 706 {
 707         return color(color::color_d,a,b,c)+I*color(color::color_f,a,b,c);
 708 }
 709
 710 /** Construct an object representing the unity matrix delta8_ab in su(3).
 711  *  The indices must be of class coloridx.
 712  *
 713  *  @param a First index
 714  *  @param b Second index
 715  *  @return newly constructed object */
 716 color color_delta8(const ex & a, const ex & b)
 717 {
 718         return color(color::color_delta8,a,b);
 719 }
 720
 721 /** Given a vector of color (and possible other) objects, split it up
 722  *  according to the object type (structure constant, generator etc.) and
 723  *  representation label while preserving the order within each group. If
 724  *  there are non-color objetcs in the vector, the SU(3) generators T_a get
 725  *  sorted into the "unknown" group together with the non-color objects
 726  *  because we don't know whether these objects commute with the generators.
 727  *
 728  *  @param v Source vector of expressions
 729  *  @param delta8vec Vector of unity matrices (returned)
 730  *  @param fvec Vector of antisymmetric structure constants (returned)
 731  *  @param dvec Vector of symmetric structure constants (returned)
 732  *  @param Tvecs Vectors of generators, one for each representation label (returned)
 733  *  @param ONEvecs Vectors of unity elements, one for each representation label (returned)
 734  *  @param unknownvec Vector of all non-color objects (returned)
 735  *
 736  *  @see color::color_types
 737  *  @see recombine_color_string */
 738 void split_color_string_in_parts(const exvector & v, exvector & delta8vec,
 739                                  exvector & fvec, exvector & dvec,
 740                                  exvectorvector & Tvecs,
 741                                  exvectorvector & ONEvecs,
 742                                  exvector & unknownvec)
 743 {
 744         // if not all elements are of type color, put all Ts in unknownvec to
 745         // retain the ordering
 746         bool all_color=true;
 747         for (exvector::const_iterator cit=v.begin(); cit!=v.end(); ++cit) {
 748                 if (!is_ex_exactly_of_type(*cit,color)) {
 749                         all_color=false;
 750                         break;
 751                 }
 752         }
 753
 754         for (exvector::const_iterator cit=v.begin(); cit!=v.end(); ++cit) {
 755                 if (is_ex_exactly_of_type(*cit,color)) {
 756                         switch (ex_to_color(*cit).type) {
 757                         case color::color_delta8:
 758                                 delta8vec.push_back(*cit);
 759                                 break;
 760                         case color::color_f:
 761                                 fvec.push_back(*cit);
 762                                 break;
 763                         case color::color_d:
 764                                 dvec.push_back(*cit);
 765                                 break;
 766                         case color::color_T:
 767                                 GINAC_ASSERT(ex_to_color(*cit).representation_label<MAX_REPRESENTATION_LABELS);
 768                                 if (all_color) {
 769                                         Tvecs[ex_to_color(*cit).representation_label].push_back(*cit);
 770                                 } else {
 771                                         unknownvec.push_back(*cit);
 772                                 }
 773                                 break;
 774                         case color::color_ONE:
 775                                 GINAC_ASSERT(ex_to_color(*cit).representation_label<MAX_REPRESENTATION_LABELS);
 776                                 ONEvecs[ex_to_color(*cit).representation_label].push_back(*cit);
 777                                 break;
 778                         default:
 779                                 throw(std::logic_error("invalid type in split_color_string_in_parts()"));
 780                         }
 781                 } else {
 782                         unknownvec.push_back(*cit);
 783                 }
 784         }
 785 }
 786
 787 /** Merge vectors of color objects sorted by object type into one vector,
 788  *  retaining the order within each group. This is the inverse operation of
 789  *  split_color_string_in_parts().
 790  *
 791  *  @param delta8vec Vector of unity matrices
 792  *  @param fvec Vector of antisymmetric structure constants
 793  *  @param dvec Vector of symmetric structure constants
 794  *  @param Tvecs Vectors of generators, one for each representation label
 795  *  @param ONEvecs Vectors of unity elements, one for each representation label
 796  *  @param unknownvec Vector of all non-color objects
 797  *  @return merged vector
 798  *
 799  *  @see color::color_types
 800  *  @see split_color_string_in_parts */
 801 exvector recombine_color_string(exvector & delta8vec, exvector & fvec,
 802                                 exvector & dvec, exvectorvector & Tvecs,
 803                                 exvectorvector & ONEvecs, exvector & unknownvec)
 804 {
 805         unsigned sz=delta8vec.size()+fvec.size()+dvec.size()+unknownvec.size();
 806         for (unsigned rl=0; rl<MAX_REPRESENTATION_LABELS; ++rl) {
 807                 sz += Tvecs[rl].size();
 808                 sz += ONEvecs[rl].size();
 809         }
 810         exvector v;
 811         v.reserve(sz);
 812
 813         append_exvector_to_exvector(v,delta8vec);
 814         append_exvector_to_exvector(v,fvec);
 815         append_exvector_to_exvector(v,dvec);
 816         for (unsigned rl=0; rl<MAX_REPRESENTATION_LABELS; ++rl) {
 817                 append_exvector_to_exvector(v,Tvecs[rl]);
 818                 append_exvector_to_exvector(v,ONEvecs[rl]);
 819         }
 820         append_exvector_to_exvector(v,unknownvec);
 821         return v;
 822 }
 823
 824 ex color_trace_of_one_representation_label(const exvector & v)
 825 {
 826         if (v.size()==0) {
 827                 return numeric(COLOR_THREE);
 828         } else if (v.size()==1) {
 829                 GINAC_ASSERT(is_ex_exactly_of_type(*(v.begin()),color));
 830                 return _ex0();
 831         }
 832         exvector v1=v;
 833         ex last_element=v1.back();
 834         GINAC_ASSERT(is_ex_exactly_of_type(last_element,color));
 835         GINAC_ASSERT(ex_to_color(last_element).type==color::color_T);
 836         v1.pop_back();
 837         ex next_to_last_element=v1.back();
 838         GINAC_ASSERT(is_ex_exactly_of_type(next_to_last_element,color));
 839         GINAC_ASSERT(ex_to_color(next_to_last_element).type==color::color_T);
 840         v1.pop_back();
 841         exvector v2=v1;
 842
 843         const ex & last_index=ex_to_color(last_element).seq[0];
 844         const ex & next_to_last_index=ex_to_color(next_to_last_element).seq[0];
 845         ex summation_index=coloridx();
 846
 847         v2.push_back(color_T(summation_index)); // don't care about the representation_label
 848
 849         // FIXME: check this formula for SU(N) with N!=3
 850         return numeric(1)/numeric(2*COLOR_THREE)*color_delta8(next_to_last_index,last_index)
 851                % color_trace_of_one_representation_label(v1)
 852                +numeric(1)/numeric(2)*color_h(next_to_last_index,last_index,summation_index)
 853                % color_trace_of_one_representation_label(v2);
 854         /*
 855         ex term1=numeric(1)/numeric(2*COLOR_THREE)*color_delta8(next_to_last_index,last_index)
 856                    % color_trace_of_one_representation_label(v1);
 857         cout << "term 1 of trace of " << v.size() << " ts=" << term1 << endl;
 858         ex term2=numeric(1)/numeric(2)*color_h(next_to_last_index,last_index,summation_index)
 859                    % color_trace_of_one_representation_label(v2);
 860         cout << "term 2 of trace of " << v.size() << " ts=" << term2 << endl;
 861         return term1+term2;
 862         */
 863 }
 864
 865 /** Calculate the trace over the (hidden) indices of the su(3) Lie algebra
 866  *  elements (the SU(3) generators and the unity element) of a specified
 867  *  representation label in a string of color objects.
 868  *
 869  *  @param v Vector of color objects
 870  *  @param rl Representation label
 871  *  @return value of the trace */
 872 ex color_trace(const exvector & v, unsigned rl)
 873 {
 874         GINAC_ASSERT(rl<MAX_REPRESENTATION_LABELS);
 875
 876         exvector v_rest;
 877         v_rest.reserve(v.size()+1); // max size if trace is empty
 878
 879         exvector delta8vec;
 880         exvector fvec;
 881         exvector dvec;
 882         exvectorvector Tvecs;
 883         Tvecs.resize(MAX_REPRESENTATION_LABELS);
 884         exvectorvector ONEvecs;
 885         ONEvecs.resize(MAX_REPRESENTATION_LABELS);
 886         exvector unknownvec;
 887
 888         split_color_string_in_parts(v,delta8vec,fvec,dvec,Tvecs,ONEvecs,unknownvec);
 889
 890         if (unknownvec.size()!=0) {
 891                 throw(std::invalid_argument("color_trace(): expression must be expanded"));
 892         }
 893
 894         append_exvector_to_exvector(v_rest,delta8vec);
 895         append_exvector_to_exvector(v_rest,fvec);
 896         append_exvector_to_exvector(v_rest,dvec);
 897         for (unsigned i=0; i<MAX_REPRESENTATION_LABELS; ++i) {
 898                 if (i!=rl) {
 899                         append_exvector_to_exvector(v_rest,Tvecs[i]);
 900                         append_exvector_to_exvector(v_rest,ONEvecs[i]);
 901                 } else {
 902                         if (Tvecs[i].size()!=0) {
 903                                 v_rest.push_back(color_trace_of_one_representation_label(Tvecs[i]));
 904                         } else if (ONEvecs[i].size()!=0) {
 905                                 v_rest.push_back(numeric(COLOR_THREE));
 906                         } else {
 907                                 throw(std::logic_error("color_trace(): representation_label not in color string"));
 908                         }
 909                 }
 910         }
 911
 912         return nonsimplified_ncmul(v_rest);
 913 }
 914
 915 ex simplify_pure_color_string(const ex & e)
 916 {
 917         GINAC_ASSERT(is_ex_exactly_of_type(e,ncmul));
 918
 919         exvector delta8vec;
 920         exvector fvec;
 921         exvector dvec;
 922         exvectorvector Tvecs;
 923         Tvecs.resize(MAX_REPRESENTATION_LABELS);
 924         exvectorvector ONEvecs;
 925         ONEvecs.resize(MAX_REPRESENTATION_LABELS);
 926         exvector unknownvec;
 927
 928         split_color_string_in_parts(ex_to_ncmul(e).get_factors(),delta8vec,fvec,dvec,Tvecs,ONEvecs,unknownvec);
 929
 930         // search for T_k S T_k (=1/2 Tr(S) - 1/6 S)
 931         for (unsigned rl=0; rl<MAX_REPRESENTATION_LABELS; ++rl) {
 932                 if (Tvecs[rl].size()>=2) {
 933                         for (unsigned i=0; i<Tvecs[rl].size()-1; ++i) {
 934                                 for (unsigned j=i+1; j<Tvecs[rl].size(); ++j) {
 935                                         ex & t1=Tvecs[rl][i];
 936                                         ex & t2=Tvecs[rl][j];
 937                                         GINAC_ASSERT(is_ex_exactly_of_type(t1,color)&&
 938                                                    (ex_to_color(t1).type==color::color_T)&&
 939                                                    (ex_to_color(t1).seq.size()==1));
 940                                         GINAC_ASSERT(is_ex_exactly_of_type(t2,color)&&
 941                                                    (ex_to_color(t2).type==color::color_T)&&
 942                                                    (ex_to_color(t2).seq.size()==1));
 943                                         const coloridx & idx1=ex_to_coloridx(ex_to_color(t1).seq[0]);
 944                                         const coloridx & idx2=ex_to_coloridx(ex_to_color(t2).seq[0]);
 945
 946                                         if (idx1.is_equal(idx2) && idx1.is_symbolic()) {
 947                                                 exvector S;
 948                                                 for (unsigned k=i+1; k<j; ++k) {
 949                                                         S.push_back(Tvecs[rl][k]);
 950                                                 }
 951                                                 t1=_ex1();
 952                                                 t2=_ex1();
 953                                                 ex term1=numeric(-1)/numeric(6)*nonsimplified_ncmul(recombine_color_string(delta8vec,fvec,dvec,Tvecs,ONEvecs,unknownvec));
 954                                                 for (unsigned k=i+1; k<j; ++k) {
 955                                                         S.push_back(_ex1());
 956                                                 }
 957                                                 t1=color_trace_of_one_representation_label(S);
 958                                                 ex term2=numeric(1)/numeric(2)*nonsimplified_ncmul(recombine_color_string(delta8vec,fvec,dvec,Tvecs,ONEvecs,unknownvec));
 959                                                 return simplify_color(term1+term2);
 960                                         }
 961                                 }
 962                         }
 963                 }
 964         }
 965
 966         // FIXME: higher contractions
 967
 968         return e;
 969 }
 970
 971 /** Perform some simplifications on an expression containing color objects. */
 972 ex simplify_color(const ex & e)
 973 {
 974         // all simplification is done on expanded objects
 975         ex e_expanded=e.expand();
 976
 977         // simplification of sum=sum of simplifications
 978         if (is_ex_exactly_of_type(e_expanded,add)) {
 979                 ex sum=_ex0();
 980                 for (unsigned i=0; i<e_expanded.nops(); ++i)
 981                         sum += simplify_color(e_expanded.op(i));
 982
 983                 return sum;
 984         }
 985
 986         // simplification of commutative product=commutative product of simplifications
 987         if (is_ex_exactly_of_type(e_expanded,mul)) {
 988                 ex prod=_ex1();
 989                 for (unsigned i=0; i<e_expanded.nops(); ++i)
 990                         prod *= simplify_color(e_expanded.op(i));
 991
 992                 return prod;
 993         }
 994
 995         // simplification of noncommutative product: test if everything is color
 996         if (is_ex_exactly_of_type(e_expanded,ncmul)) {
 997                 bool all_color=true;
 998                 for (unsigned i=0; i<e_expanded.nops(); ++i) {
 999                         if (!is_ex_exactly_of_type(e_expanded.op(i),color)) {
1000                                 all_color=false;
1001                                 break;
1002                         }
1003                 }
1004                 if (all_color) {
1005                         return simplify_pure_color_string(e_expanded);
1006                 }
1007         }
1008
1009         // cannot do anything
1010         return e_expanded;
1011 }
1012
1013 ex brute_force_sum_color_indices(const ex & e)
1014 {
1015         exvector iv_all=e.get_indices();
1016         exvector iv_double;
1017
1018         // find double symbolic indices
1019         if (iv_all.size()<2) return e;
1020         for (exvector::const_iterator cit1=iv_all.begin(); cit1!=iv_all.end()-1; ++cit1) {
1021                 GINAC_ASSERT(is_ex_of_type(*cit1,coloridx));
1022                 for (exvector::const_iterator cit2=cit1+1; cit2!=iv_all.end(); ++cit2) {
1023                         GINAC_ASSERT(is_ex_of_type(*cit2,coloridx));
1024                         if (ex_to_coloridx(*cit1).is_symbolic() &&
1025                                 ex_to_coloridx(*cit1).is_equal(ex_to_coloridx(*cit2))) {
1026                                 iv_double.push_back(*cit1);
1027                                 break;
1028                         }
1029                 }
1030         }
1031
1032         std::vector<int> counter;
1033         counter.resize(iv_double.size());
1034         int l;
1035         for (l=0; unsigned(l)<iv_double.size(); ++l) {
1036                 counter[l]=1;
1037         }
1038
1039         ex sum;
1040
1041         while (1) {
1042                 ex term = e;
1043                 for (l=0; unsigned(l)<iv_double.size(); ++l) {
1044                         term=term.subs(iv_double[l]==coloridx((unsigned)(counter[l])));
1045                         //iv_double[l].print(cout);
1046                         //cout << " " << counter[l] << " ";
1047                 }
1048                 //cout << endl;
1049                 sum += term;
1050
1051                 // increment counter[]
1052                 l = iv_double.size()-1;
1053                 while ((l>=0)&&((++counter[l])>(int)COLOR_EIGHT)) {
1054                         counter[l]=1;
1055                         l--;
1056                 }
1057                 if (l<2) { std::cout << counter[0] << counter[1] << std::endl; }
1058                 if (l<0) break;
1059         }
1060
1061         return sum;
1062 }
1063
1064 #ifndef NO_NAMESPACE_GINAC
1065 } // namespace GiNaC
1066 #endif // ndef NO_NAMESPACE_GINAC