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