ginac/color.cpp

   1 /** @file color.cpp
   2  *
   3  *  Implementation of GiNaC's color (SU(3) Lie algebra) objects. */
   4
   5 /*
   6  *  GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
   7  *
   8  *  This program is free software; you can redistribute it and/or modify
   9  *  it under the terms of the GNU General Public License as published by
  10  *  the Free Software Foundation; either version 2 of the License, or
  11  *  (at your option) any later version.
  12  *
  13  *  This program is distributed in the hope that it will be useful,
  14  *  but WITHOUT ANY WARRANTY; without even the implied warranty of
  15  *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  16  *  GNU General Public License for more details.
  17  *
  18  *  You should have received a copy of the GNU General Public License
  19  *  along with this program; if not, write to the Free Software
  20  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  21  */
  22
  23 #include <algorithm>
  24 #include <stdexcept>
  25
  26 #include "color.h"
  27 #include "ex.h"
  28 #include "idx.h"
  29 #include "ncmul.h"
  30 #include "numeric.h"
  31 #include "power.h" // for sqrt()
  32 #include "symbol.h"
  33 #include "print.h"
  34 #include "archive.h"
  35 #include "debugmsg.h"
  36 #include "utils.h"
  37
  38 namespace GiNaC {
  39
  40 GINAC_IMPLEMENT_REGISTERED_CLASS(color, indexed)
  41 GINAC_IMPLEMENT_REGISTERED_CLASS(su3one, tensor)
  42 GINAC_IMPLEMENT_REGISTERED_CLASS(su3t, tensor)
  43 GINAC_IMPLEMENT_REGISTERED_CLASS(su3f, tensor)
  44 GINAC_IMPLEMENT_REGISTERED_CLASS(su3d, tensor)
  45
  46 //////////
  47 // default constructor, destructor, copy constructor assignment operator and helpers
  48 //////////
  49
  50 color::color() : representation_label(0)
  51 {
  52         debugmsg("color default constructor", LOGLEVEL_CONSTRUCT);
  53         tinfo_key = TINFO_color;
  54 }
  55
  56 void color::copy(const color & other)
  57 {
  58         inherited::copy(other);
  59         representation_label = other.representation_label;
  60 }
  61
  62 DEFAULT_DESTROY(color)
  63 DEFAULT_CTORS(su3one)
  64 DEFAULT_CTORS(su3t)
  65 DEFAULT_CTORS(su3f)
  66 DEFAULT_CTORS(su3d)
  67
  68 //////////
  69 // other constructors
  70 //////////
  71
  72 /** Construct object without any color index. This constructor is for
  73  *  internal use only. Use the color_ONE() function instead.
  74  *  @see color_ONE */
  75 color::color(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
  76 {
  77         debugmsg("color constructor from ex,unsigned char", LOGLEVEL_CONSTRUCT);
  78         tinfo_key = TINFO_color;
  79 }
  80
  81 /** Construct object with one color index. This constructor is for internal
  82  *  use only. Use the color_T() function instead.
  83  *  @see color_T */
  84 color::color(const ex & b, const ex & i1, unsigned char rl) : inherited(b, i1), representation_label(rl)
  85 {
  86         debugmsg("color constructor from ex,ex,unsigned char", LOGLEVEL_CONSTRUCT);
  87         tinfo_key = TINFO_color;
  88 }
  89
  90 color::color(unsigned char rl, const exvector & v, bool discardable) : inherited(indexed::unknown, v, discardable), representation_label(rl)
  91 {
  92         debugmsg("color constructor from unsigned char,exvector", LOGLEVEL_CONSTRUCT);
  93         tinfo_key = TINFO_color;
  94 }
  95
  96 color::color(unsigned char rl, exvector * vp) : inherited(indexed::unknown, vp), representation_label(rl)
  97 {
  98         debugmsg("color constructor from unsigned char,exvector *", LOGLEVEL_CONSTRUCT);
  99         tinfo_key = TINFO_color;
 100 }
 101
 102 //////////
 103 // archiving
 104 //////////
 105
 106 color::color(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
 107 {
 108         debugmsg("color constructor from archive_node", LOGLEVEL_CONSTRUCT);
 109         unsigned rl;
 110         n.find_unsigned("label", rl);
 111         representation_label = rl;
 112 }
 113
 114 void color::archive(archive_node &n) const
 115 {
 116         inherited::archive(n);
 117         n.add_unsigned("label", representation_label);
 118 }
 119
 120 DEFAULT_UNARCHIVE(color)
 121 DEFAULT_ARCHIVING(su3one)
 122 DEFAULT_ARCHIVING(su3t)
 123 DEFAULT_ARCHIVING(su3f)
 124 DEFAULT_ARCHIVING(su3d)
 125
 126 //////////
 127 // functions overriding virtual functions from bases classes
 128 //////////
 129
 130 int color::compare_same_type(const basic & other) const
 131 {
 132         GINAC_ASSERT(other.tinfo() == TINFO_color);
 133         const color &o = static_cast<const color &>(other);
 134
 135         if (representation_label != o.representation_label) {
 136                 // different representation label
 137                 return representation_label < o.representation_label ? -1 : 1;
 138         }
 139
 140         return inherited::compare_same_type(other);
 141 }
 142
 143 DEFAULT_COMPARE(su3one)
 144 DEFAULT_COMPARE(su3t)
 145 DEFAULT_COMPARE(su3f)
 146 DEFAULT_COMPARE(su3d)
 147
 148 DEFAULT_PRINT_LATEX(su3one, "ONE", "\\mathbb{1}")
 149 DEFAULT_PRINT(su3t, "T")
 150 DEFAULT_PRINT(su3f, "f")
 151 DEFAULT_PRINT(su3d, "d")
 152
 153 /** Perform automatic simplification on noncommutative product of color
 154  *  objects. This removes superfluous ONEs. */
 155 ex color::simplify_ncmul(const exvector & v) const
 156 {
 157         exvector s;
 158         s.reserve(v.size());
 159         unsigned rl = ex_to_color(v[0]).get_representation_label();
 160
 161         // Remove superfluous ONEs
 162         exvector::const_iterator it = v.begin(), itend = v.end();
 163         while (it != itend) {
 164                 if (!is_ex_of_type(it->op(0), su3one))
 165                         s.push_back(*it);
 166                 it++;
 167         }
 168
 169         if (s.size() == 0)
 170                 return color(su3one(), rl);
 171         else
 172                 return simplified_ncmul(s);
 173 }
 174
 175 ex color::thisexprseq(const exvector & v) const
 176 {
 177         return color(representation_label, v);
 178 }
 179
 180 ex color::thisexprseq(exvector * vp) const
 181 {
 182         return color(representation_label, vp);
 183 }
 184
 185 /** Given a vector iv3 of three indices and a vector iv2 of two indices that
 186  *  is a subset of iv3, return the (free) index that is in iv3 but not in
 187  *  iv2 and the sign introduced by permuting that index to the front.
 188  *
 189  *  @param iv3 Vector of 3 indices
 190  *  @param iv2 Vector of 2 indices, must be a subset of iv3
 191  *  @param sig Returs sign introduced by index permutation
 192  *  @return the free index (the one that is in iv3 but not in iv2) */
 193 static ex permute_free_index_to_front(const exvector & iv3, const exvector & iv2, int & sig)
 194 {
 195         GINAC_ASSERT(iv3.size() == 3);
 196         GINAC_ASSERT(iv2.size() == 2);
 197
 198         sig = 1;
 199
 200 #define TEST_PERMUTATION(A,B,C,P) \
 201         if (iv3[B].is_equal(iv2[0]) && iv3[C].is_equal(iv2[1])) { \
 202                 sig = P; \
 203                 return iv3[A]; \
 204         }
 205
 206         TEST_PERMUTATION(0,1,2,  1);
 207         TEST_PERMUTATION(0,2,1, -1);
 208         TEST_PERMUTATION(1,0,2, -1);
 209         TEST_PERMUTATION(1,2,0,  1);
 210         TEST_PERMUTATION(2,0,1,  1);
 211         TEST_PERMUTATION(2,1,0, -1);
 212
 213         throw(std::logic_error("permute_free_index_to_front(): no valid permutation found"));
 214 }
 215
 216 /** Automatic symbolic evaluation of indexed symmetric structure constant. */
 217 ex su3d::eval_indexed(const basic & i) const
 218 {
 219         GINAC_ASSERT(is_of_type(i, indexed));
 220         GINAC_ASSERT(i.nops() == 4);
 221         GINAC_ASSERT(is_ex_of_type(i.op(0), su3d));
 222
 223         // Convolutions are zero
 224         if (static_cast<const indexed &>(i).get_dummy_indices().size() != 0)
 225                 return _ex0();
 226
 227         // Numeric evaluation
 228         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 229
 230                 // Sort indices
 231                 int v[3];
 232                 for (unsigned j=0; j<3; j++)
 233                         v[j] = ex_to_numeric(ex_to_idx(i.op(j + 1)).get_value()).to_int();
 234                 if (v[0] > v[1]) std::swap(v[0], v[1]);
 235                 if (v[0] > v[2]) std::swap(v[0], v[2]);
 236                 if (v[1] > v[2]) std::swap(v[1], v[2]);
 237
 238 #define CMPINDICES(A,B,C) ((v[0] == (A)) && (v[1] == (B)) && (v[2] == (C)))
 239
 240                 // Check for non-zero elements
 241                 if (CMPINDICES(1,4,6) || CMPINDICES(1,5,7) || CMPINDICES(2,5,6)
 242                  || CMPINDICES(3,4,4) || CMPINDICES(3,5,5))
 243                         return _ex1_2();
 244                 else if (CMPINDICES(2,4,7) || CMPINDICES(3,6,6) || CMPINDICES(3,7,7))
 245                         return _ex_1_2();
 246                 else if (CMPINDICES(1,1,8) || CMPINDICES(2,2,8) || CMPINDICES(3,3,8))
 247                         return sqrt(_ex3())/3;
 248                 else if (CMPINDICES(8,8,8))
 249                         return -sqrt(_ex3())/3;
 250                 else if (CMPINDICES(4,4,8) || CMPINDICES(5,5,8)
 251                       || CMPINDICES(6,6,8) || CMPINDICES(7,7,8))
 252                         return -sqrt(_ex3())/6;
 253                 else
 254                         return _ex0();
 255         }
 256
 257         // No further simplifications
 258         return i.hold();
 259 }
 260
 261 /** Automatic symbolic evaluation of indexed antisymmetric structure constant. */
 262 ex su3f::eval_indexed(const basic & i) const
 263 {
 264         GINAC_ASSERT(is_of_type(i, indexed));
 265         GINAC_ASSERT(i.nops() == 4);
 266         GINAC_ASSERT(is_ex_of_type(i.op(0), su3f));
 267
 268         // Numeric evaluation
 269         if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
 270
 271                 // Sort indices, remember permutation sign
 272                 int v[3];
 273                 for (unsigned j=0; j<3; j++)
 274                         v[j] = ex_to_numeric(ex_to_idx(i.op(j + 1)).get_value()).to_int();
 275                 int sign = 1;
 276                 if (v[0] > v[1]) { std::swap(v[0], v[1]); sign = -sign; }
 277                 if (v[0] > v[2]) { std::swap(v[0], v[2]); sign = -sign; }
 278                 if (v[1] > v[2]) { std::swap(v[1], v[2]); sign = -sign; }
 279
 280                 // Check for non-zero elements
 281                 if (CMPINDICES(1,2,3))
 282                         return sign;
 283                 else if (CMPINDICES(1,4,7) || CMPINDICES(2,4,6)
 284                       || CMPINDICES(2,5,7) || CMPINDICES(3,4,5))
 285                         return _ex1_2() * sign;
 286                 else if (CMPINDICES(1,5,6) || CMPINDICES(3,6,7))
 287                         return _ex_1_2() * sign;
 288                 else if (CMPINDICES(4,5,8) || CMPINDICES(6,7,8))
 289                         return sqrt(_ex3())/2 * sign;
 290                 else
 291                         return _ex0();
 292         }
 293
 294         // No further simplifications
 295         return i.hold();
 296 }
 297
 298
 299 /** Contraction of generator with something else. */
 300 bool su3t::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 301 {
 302         GINAC_ASSERT(is_ex_of_type(*self, indexed));
 303         GINAC_ASSERT(is_ex_of_type(*other, indexed));
 304         GINAC_ASSERT(self->nops() == 2);
 305         GINAC_ASSERT(is_ex_of_type(self->op(0), su3t));
 306         unsigned char rl = ex_to_color(*self).get_representation_label();
 307
 308         if (is_ex_exactly_of_type(other->op(0), su3t)) {
 309
 310                 // T.a T.a = 4/3 ONE
 311                 if (other - self == 1) {
 312                         *self = numeric(4, 3);
 313                         *other = color_ONE(rl);
 314                         return true;
 315
 316                 // T.a T.b T.a = -1/6 T.b
 317                 } else if (other - self == 2
 318                         && is_ex_of_type(self[1], color)) {
 319                         *self = numeric(-1, 6);
 320                         *other = _ex1();
 321                         return true;
 322
 323                 // T.a S T.a = 1/2 Tr(S) - 1/6 S
 324                 } else {
 325                         exvector::iterator it = self + 1;
 326                         while (it != other) {
 327                                 if (!is_ex_of_type(*it, color)) {
 328                                         return false;
 329                                 }
 330                                 it++;
 331                         }
 332
 333                         it = self + 1;
 334                         ex S = _ex1();
 335                         while (it != other) {
 336                                 S *= *it;
 337                                 *it++ = _ex1();
 338                         }
 339
 340                         *self = color_trace(S, rl) * color_ONE(rl) / 2 - S / 6;
 341                         *other = _ex1();
 342                         return true;
 343                 }
 344         }
 345
 346         return false;
 347 }
 348
 349 /** Contraction of an indexed symmetric structure constant with something else. */
 350 bool su3d::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 351 {
 352         GINAC_ASSERT(is_ex_of_type(*self, indexed));
 353         GINAC_ASSERT(is_ex_of_type(*other, indexed));
 354         GINAC_ASSERT(self->nops() == 4);
 355         GINAC_ASSERT(is_ex_of_type(self->op(0), su3d));
 356
 357         if (is_ex_exactly_of_type(other->op(0), su3d)) {
 358
 359                 // Find the dummy indices of the contraction
 360                 exvector dummy_indices;
 361                 dummy_indices = ex_to_indexed(*self).get_dummy_indices(ex_to_indexed(*other));
 362
 363                 // d.abc*d.abc=40/3
 364                 if (dummy_indices.size() == 3) {
 365                         *self = numeric(40, 3);
 366                         *other = _ex1();
 367                         return true;
 368
 369                 // d.akl*d.bkl=5/3*delta.ab
 370                 } else if (dummy_indices.size() == 2) {
 371                         exvector a = index_set_difference(ex_to_indexed(*self).get_indices(), dummy_indices);
 372                         exvector b = index_set_difference(ex_to_indexed(*other).get_indices(), dummy_indices);
 373                         GINAC_ASSERT(a.size() > 0);
 374                         GINAC_ASSERT(b.size() > 0);
 375                         *self = numeric(5, 3) * delta_tensor(a[0], b[0]);
 376                         *other = _ex1();
 377                         return true;
 378                 }
 379         }
 380
 381         return false;
 382 }
 383
 384 /** Contraction of an indexed antisymmetric structure constant with something else. */
 385 bool su3f::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 386 {
 387         GINAC_ASSERT(is_ex_of_type(*self, indexed));
 388         GINAC_ASSERT(is_ex_of_type(*other, indexed));
 389         GINAC_ASSERT(self->nops() == 4);
 390         GINAC_ASSERT(is_ex_of_type(self->op(0), su3f));
 391
 392         if (is_ex_exactly_of_type(other->op(0), su3f)) { // f*d is handled by su3d class
 393
 394                 // Find the dummy indices of the contraction
 395                 exvector dummy_indices;
 396                 dummy_indices = ex_to_indexed(*self).get_dummy_indices(ex_to_indexed(*other));
 397
 398                 // f.abc*f.abc=24
 399                 if (dummy_indices.size() == 3) {
 400                         *self = 24;
 401                         *other = _ex1();
 402                         return true;
 403
 404                 // f.akl*f.bkl=3*delta.ab
 405                 } else if (dummy_indices.size() == 2) {
 406                         int sign1, sign2;
 407                         ex a = permute_free_index_to_front(ex_to_indexed(*self).get_indices(), dummy_indices, sign1);
 408                         ex b = permute_free_index_to_front(ex_to_indexed(*other).get_indices(), dummy_indices, sign2);
 409                         *self = sign1 * sign2 * 3 * delta_tensor(a, b);
 410                         *other = _ex1();
 411                         return true;
 412                 }
 413         }
 414
 415         return false;
 416 }
 417
 418 //////////
 419 // global functions
 420 //////////
 421
 422 ex color_ONE(unsigned char rl)
 423 {
 424         return color(su3one(), rl);
 425 }
 426
 427 ex color_T(const ex & a, unsigned char rl)
 428 {
 429         if (!is_ex_of_type(a, idx))
 430                 throw(std::invalid_argument("indices of color_T must be of type idx"));
 431         if (!ex_to_idx(a).get_dim().is_equal(8))
 432                 throw(std::invalid_argument("index dimension for color_T must be 8"));
 433
 434         return color(su3t(), a, rl);
 435 }
 436
 437 ex color_f(const ex & a, const ex & b, const ex & c)
 438 {
 439         if (!is_ex_of_type(a, idx) || !is_ex_of_type(b, idx) || !is_ex_of_type(c, idx))
 440                 throw(std::invalid_argument("indices of color_f must be of type idx"));
 441         if (!ex_to_idx(a).get_dim().is_equal(8) || !ex_to_idx(b).get_dim().is_equal(8) || !ex_to_idx(c).get_dim().is_equal(8))
 442                 throw(std::invalid_argument("index dimension for color_f must be 8"));
 443
 444         return indexed(su3f(), indexed::antisymmetric, a, b, c);
 445 }
 446
 447 ex color_d(const ex & a, const ex & b, const ex & c)
 448 {
 449         if (!is_ex_of_type(a, idx) || !is_ex_of_type(b, idx) || !is_ex_of_type(c, idx))
 450                 throw(std::invalid_argument("indices of color_d must be of type idx"));
 451         if (!ex_to_idx(a).get_dim().is_equal(8) || !ex_to_idx(b).get_dim().is_equal(8) || !ex_to_idx(c).get_dim().is_equal(8))
 452                 throw(std::invalid_argument("index dimension for color_d must be 8"));
 453
 454         return indexed(su3d(), indexed::symmetric, a, b, c);
 455 }
 456
 457 ex color_h(const ex & a, const ex & b, const ex & c)
 458 {
 459         return color_d(a, b, c) + I * color_f(a, b, c);
 460 }
 461
 462 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 463  *  is that of a color object with the specified representation label. */
 464 static bool is_color_tinfo(unsigned ti, unsigned char rl)
 465 {
 466         return ti == (TINFO_color + rl);
 467 }
 468
 469 ex color_trace(const ex & e, unsigned char rl)
 470 {
 471         if (is_ex_of_type(e, color)) {
 472
 473                 if (ex_to_color(e).get_representation_label() == rl
 474                  && is_ex_of_type(e.op(0), su3one))
 475                         return _ex3();
 476                 else
 477                         return _ex0();
 478
 479         } else if (is_ex_exactly_of_type(e, add)) {
 480
 481                 // Trace of sum = sum of traces
 482                 ex sum = _ex0();
 483                 for (unsigned i=0; i<e.nops(); i++)
 484                         sum += color_trace(e.op(i), rl);
 485                 return sum;
 486
 487         } else if (is_ex_exactly_of_type(e, mul)) {
 488
 489                 // Trace of product: pull out non-color factors
 490                 ex prod = _ex1();
 491                 for (unsigned i=0; i<e.nops(); i++) {
 492                         const ex &o = e.op(i);
 493                         if (is_color_tinfo(o.return_type_tinfo(), rl))
 494                                 prod *= color_trace(o, rl);
 495                         else
 496                                 prod *= o;
 497                 }
 498                 return prod;
 499
 500         } else if (is_ex_exactly_of_type(e, ncmul)) {
 501
 502                 if (!is_color_tinfo(e.return_type_tinfo(), rl))
 503                         return _ex0();
 504
 505                 // Expand product, if necessary
 506                 ex e_expanded = e.expand();
 507                 if (!is_ex_of_type(e_expanded, ncmul))
 508                         return color_trace(e_expanded, rl);
 509
 510                 unsigned num = e.nops();
 511
 512                 if (num == 2) {
 513
 514                         // Tr T_a T_b = 1/2 delta_a_b
 515                         return delta_tensor(e.op(0).op(1), e.op(1).op(1)) / 2;
 516
 517                 } else if (num == 3) {
 518
 519                         // Tr T_a T_b T_c = 1/4 h_a_b_c
 520                         return color_h(e.op(0).op(1), e.op(1).op(1), e.op(2).op(1)) / 4;
 521
 522                 } else {
 523
 524                         // Traces of 4 or more generators are computed recursively:
 525                         // Tr T_a1 .. T_an =
 526                         //     1/6 delta_a(n-1)_an Tr T_a1 .. T_a(n-2)
 527                         //   + 1/2 h_a(n-1)_an_k Tr T_a1 .. T_a(n-2) T_k
 528                         const ex &last_index = e.op(num - 1).op(1);
 529                         const ex &next_to_last_index = e.op(num - 2).op(1);
 530                         idx summation_index((new symbol)->setflag(status_flags::dynallocated), 8);
 531
 532                         exvector v1;
 533                         v1.reserve(num - 2);
 534                         for (int i=0; i<num-2; i++)
 535                                 v1.push_back(e.op(i));
 536
 537                         exvector v2 = v1;
 538                         v2.push_back(color_T(summation_index, rl));
 539
 540                         return delta_tensor(next_to_last_index, last_index) * color_trace(ncmul(v1), rl) / 6
 541                                + color_h(next_to_last_index, last_index, summation_index) * color_trace(ncmul(v2), rl) / 2;
 542                 }
 543         }
 544
 545         return _ex0();
 546 }
 547
 548 } // namespace GiNaC