3 * Implementation of GiNaC's color (SU(3) Lie algebra) objects. */
6 * GiNaC Copyright (C) 1999-2006 Johannes Gutenberg University Mainz, Germany
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.
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.
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., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
30 #include "operators.h"
33 #include "power.h" // for sqrt()
40 GINAC_IMPLEMENT_REGISTERED_CLASS(color, indexed)
42 const tinfo_static_t color::return_type_tinfo_static[256] = {{}};
44 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(su3one, tensor,
45 print_func<print_dflt>(&su3one::do_print).
46 print_func<print_latex>(&su3one::do_print_latex))
48 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(su3t, tensor,
49 print_func<print_dflt>(&su3t::do_print).
50 print_func<print_latex>(&su3t::do_print))
52 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(su3f, tensor,
53 print_func<print_dflt>(&su3f::do_print).
54 print_func<print_latex>(&su3f::do_print))
56 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(su3d, tensor,
57 print_func<print_dflt>(&su3d::do_print).
58 print_func<print_latex>(&su3d::do_print))
61 // default constructors
64 color::color() : representation_label(0)
66 tinfo_key = &color::tinfo_static;
78 /** Construct object without any color index. This constructor is for
79 * internal use only. Use the color_ONE() function instead.
81 color::color(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
83 tinfo_key = &color::tinfo_static;
86 /** Construct object with one color index. This constructor is for internal
87 * use only. Use the color_T() function instead.
89 color::color(const ex & b, const ex & i1, unsigned char rl) : inherited(b, i1), representation_label(rl)
91 tinfo_key = &color::tinfo_static;
94 color::color(unsigned char rl, const exvector & v, bool discardable) : inherited(not_symmetric(), v, discardable), representation_label(rl)
96 tinfo_key = &color::tinfo_static;
99 color::color(unsigned char rl, std::auto_ptr<exvector> vp) : inherited(not_symmetric(), vp), representation_label(rl)
101 tinfo_key = &color::tinfo_static;
108 color::color(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
111 n.find_unsigned("label", rl);
112 representation_label = rl;
115 void color::archive(archive_node &n) const
117 inherited::archive(n);
118 n.add_unsigned("label", representation_label);
121 DEFAULT_UNARCHIVE(color)
122 DEFAULT_ARCHIVING(su3one)
123 DEFAULT_ARCHIVING(su3t)
124 DEFAULT_ARCHIVING(su3f)
125 DEFAULT_ARCHIVING(su3d)
128 // functions overriding virtual functions from base classes
131 int color::compare_same_type(const basic & other) const
133 GINAC_ASSERT(is_a<color>(other));
134 const color &o = static_cast<const color &>(other);
136 if (representation_label != o.representation_label) {
137 // different representation label
138 return representation_label < o.representation_label ? -1 : 1;
141 return inherited::compare_same_type(other);
144 bool color::match_same_type(const basic & other) const
146 GINAC_ASSERT(is_a<color>(other));
147 const color &o = static_cast<const color &>(other);
149 return representation_label == o.representation_label;
152 DEFAULT_COMPARE(su3one)
153 DEFAULT_COMPARE(su3t)
154 DEFAULT_COMPARE(su3f)
155 DEFAULT_COMPARE(su3d)
157 DEFAULT_PRINT_LATEX(su3one, "ONE", "\\mathbb{1}")
158 DEFAULT_PRINT(su3t, "T")
159 DEFAULT_PRINT(su3f, "f")
160 DEFAULT_PRINT(su3d, "d")
162 /** Perform automatic simplification on noncommutative product of color
163 * objects. This removes superfluous ONEs. */
164 ex color::eval_ncmul(const exvector & v) const
169 // Remove superfluous ONEs
170 exvector::const_iterator it = v.begin(), itend = v.end();
171 while (it != itend) {
172 if (!is_a<su3one>(it->op(0)))
178 return color(su3one(), representation_label);
180 return hold_ncmul(s);
183 ex color::thiscontainer(const exvector & v) const
185 return color(representation_label, v);
188 ex color::thiscontainer(std::auto_ptr<exvector> vp) const
190 return color(representation_label, vp);
193 /** Given a vector iv3 of three indices and a vector iv2 of two indices that
194 * is a subset of iv3, return the (free) index that is in iv3 but not in
195 * iv2 and the sign introduced by permuting that index to the front.
197 * @param iv3 Vector of 3 indices
198 * @param iv2 Vector of 2 indices, must be a subset of iv3
199 * @param sig Returs sign introduced by index permutation
200 * @return the free index (the one that is in iv3 but not in iv2) */
201 static ex permute_free_index_to_front(const exvector & iv3, const exvector & iv2, int & sig)
203 GINAC_ASSERT(iv3.size() == 3);
204 GINAC_ASSERT(iv2.size() == 2);
208 #define TEST_PERMUTATION(A,B,C,P) \
209 if (iv3[B].is_equal(iv2[0]) && iv3[C].is_equal(iv2[1])) { \
214 TEST_PERMUTATION(0,1,2, 1);
215 TEST_PERMUTATION(0,2,1, -1);
216 TEST_PERMUTATION(1,0,2, -1);
217 TEST_PERMUTATION(1,2,0, 1);
218 TEST_PERMUTATION(2,0,1, 1);
219 TEST_PERMUTATION(2,1,0, -1);
221 throw(std::logic_error("permute_free_index_to_front(): no valid permutation found"));
224 /** Automatic symbolic evaluation of indexed symmetric structure constant. */
225 ex su3d::eval_indexed(const basic & i) const
227 GINAC_ASSERT(is_a<indexed>(i));
228 GINAC_ASSERT(i.nops() == 4);
229 GINAC_ASSERT(is_a<su3d>(i.op(0)));
231 // Convolutions are zero
232 if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
235 // Numeric evaluation
236 if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
240 for (unsigned j=0; j<3; j++)
241 v[j] = ex_to<numeric>(ex_to<idx>(i.op(j + 1)).get_value()).to_int();
242 if (v[0] > v[1]) std::swap(v[0], v[1]);
243 if (v[0] > v[2]) std::swap(v[0], v[2]);
244 if (v[1] > v[2]) std::swap(v[1], v[2]);
246 #define CMPINDICES(A,B,C) ((v[0] == (A)) && (v[1] == (B)) && (v[2] == (C)))
248 // Check for non-zero elements
249 if (CMPINDICES(1,4,6) || CMPINDICES(1,5,7) || CMPINDICES(2,5,6)
250 || CMPINDICES(3,4,4) || CMPINDICES(3,5,5))
252 else if (CMPINDICES(2,4,7) || CMPINDICES(3,6,6) || CMPINDICES(3,7,7))
254 else if (CMPINDICES(1,1,8) || CMPINDICES(2,2,8) || CMPINDICES(3,3,8))
255 return sqrt(_ex3)*_ex1_3;
256 else if (CMPINDICES(8,8,8))
257 return sqrt(_ex3)*_ex_1_3;
258 else if (CMPINDICES(4,4,8) || CMPINDICES(5,5,8)
259 || CMPINDICES(6,6,8) || CMPINDICES(7,7,8))
260 return sqrt(_ex3)/_ex_6;
265 // No further simplifications
269 /** Automatic symbolic evaluation of indexed antisymmetric structure constant. */
270 ex su3f::eval_indexed(const basic & i) const
272 GINAC_ASSERT(is_a<indexed>(i));
273 GINAC_ASSERT(i.nops() == 4);
274 GINAC_ASSERT(is_a<su3f>(i.op(0)));
276 // Numeric evaluation
277 if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
279 // Sort indices, remember permutation sign
281 for (unsigned j=0; j<3; j++)
282 v[j] = ex_to<numeric>(ex_to<idx>(i.op(j + 1)).get_value()).to_int();
284 if (v[0] > v[1]) { std::swap(v[0], v[1]); sign = -sign; }
285 if (v[0] > v[2]) { std::swap(v[0], v[2]); sign = -sign; }
286 if (v[1] > v[2]) { std::swap(v[1], v[2]); sign = -sign; }
288 // Check for non-zero elements
289 if (CMPINDICES(1,2,3))
291 else if (CMPINDICES(1,4,7) || CMPINDICES(2,4,6)
292 || CMPINDICES(2,5,7) || CMPINDICES(3,4,5))
293 return _ex1_2 * sign;
294 else if (CMPINDICES(1,5,6) || CMPINDICES(3,6,7))
295 return _ex_1_2 * sign;
296 else if (CMPINDICES(4,5,8) || CMPINDICES(6,7,8))
297 return sqrt(_ex3)/2 * sign;
302 // No further simplifications
307 /** Contraction of generator with something else. */
308 bool su3t::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
310 GINAC_ASSERT(is_a<indexed>(*self));
311 GINAC_ASSERT(is_a<indexed>(*other));
312 GINAC_ASSERT(self->nops() == 2);
313 GINAC_ASSERT(is_a<su3t>(self->op(0)));
314 unsigned char rl = ex_to<color>(*self).get_representation_label();
316 if (is_exactly_a<su3t>(other->op(0))) {
318 // Contraction only makes sense if the represenation labels are equal
319 GINAC_ASSERT(is_a<color>(*other));
320 if (ex_to<color>(*other).get_representation_label() != rl)
324 if (other - self == 1) {
325 *self = numeric(4, 3);
326 *other = color_ONE(rl);
329 // T.a T.b T.a = -1/6 T.b
330 } else if (other - self == 2
331 && is_a<color>(self[1])) {
332 *self = numeric(-1, 6);
336 // T.a S T.a = 1/2 Tr(S) - 1/6 S
338 exvector::iterator it = self + 1;
339 while (it != other) {
340 if (!is_a<color>(*it)) {
348 while (it != other) {
353 *self = color_trace(S, rl) * color_ONE(rl) / 2 - S / 6;
362 /** Contraction of an indexed symmetric structure constant with something else. */
363 bool su3d::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
365 GINAC_ASSERT(is_a<indexed>(*self));
366 GINAC_ASSERT(is_a<indexed>(*other));
367 GINAC_ASSERT(self->nops() == 4);
368 GINAC_ASSERT(is_a<su3d>(self->op(0)));
370 if (is_exactly_a<su3d>(other->op(0))) {
372 // Find the dummy indices of the contraction
373 exvector self_indices = ex_to<indexed>(*self).get_indices();
374 exvector other_indices = ex_to<indexed>(*other).get_indices();
375 exvector all_indices = self_indices;
376 all_indices.insert(all_indices.end(), other_indices.begin(), other_indices.end());
377 exvector free_indices, dummy_indices;
378 find_free_and_dummy(all_indices, free_indices, dummy_indices);
380 // d.abc d.abc = 40/3
381 if (dummy_indices.size() == 3) {
382 *self = numeric(40, 3);
386 // d.akl d.bkl = 5/3 delta.ab
387 } else if (dummy_indices.size() == 2) {
389 std::back_insert_iterator<exvector> ita(a);
390 ita = set_difference(self_indices.begin(), self_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
391 ita = set_difference(other_indices.begin(), other_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
392 GINAC_ASSERT(a.size() == 2);
393 *self = numeric(5, 3) * delta_tensor(a[0], a[1]);
398 } else if (is_exactly_a<su3t>(other->op(0))) {
400 // d.abc T.b T.c = 5/6 T.a
401 if (other+1 != v.end()
402 && is_exactly_a<su3t>(other[1].op(0))
403 && ex_to<indexed>(*self).has_dummy_index_for(other[1].op(1))) {
405 exvector self_indices = ex_to<indexed>(*self).get_indices();
406 exvector dummy_indices;
407 dummy_indices.push_back(other[0].op(1));
408 dummy_indices.push_back(other[1].op(1));
410 ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
411 *self = numeric(5, 6);
412 other[0] = color_T(a, ex_to<color>(other[0]).get_representation_label());
421 /** Contraction of an indexed antisymmetric structure constant with something else. */
422 bool su3f::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
424 GINAC_ASSERT(is_a<indexed>(*self));
425 GINAC_ASSERT(is_a<indexed>(*other));
426 GINAC_ASSERT(self->nops() == 4);
427 GINAC_ASSERT(is_a<su3f>(self->op(0)));
429 if (is_exactly_a<su3f>(other->op(0))) { // f*d is handled by su3d class
431 // Find the dummy indices of the contraction
432 exvector dummy_indices;
433 dummy_indices = ex_to<indexed>(*self).get_dummy_indices(ex_to<indexed>(*other));
436 if (dummy_indices.size() == 3) {
441 // f.akl f.bkl = 3 delta.ab
442 } else if (dummy_indices.size() == 2) {
444 ex a = permute_free_index_to_front(ex_to<indexed>(*self).get_indices(), dummy_indices, sign1);
445 ex b = permute_free_index_to_front(ex_to<indexed>(*other).get_indices(), dummy_indices, sign2);
446 *self = sign1 * sign2 * 3 * delta_tensor(a, b);
451 } else if (is_exactly_a<su3t>(other->op(0))) {
453 // f.abc T.b T.c = 3/2 I T.a
454 if (other+1 != v.end()
455 && is_exactly_a<su3t>(other[1].op(0))
456 && ex_to<indexed>(*self).has_dummy_index_for(other[1].op(1))) {
458 exvector self_indices = ex_to<indexed>(*self).get_indices();
459 exvector dummy_indices;
460 dummy_indices.push_back(other[0].op(1));
461 dummy_indices.push_back(other[1].op(1));
463 ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
464 *self = numeric(3, 2) * sig * I;
465 other[0] = color_T(a, ex_to<color>(other[0]).get_representation_label());
478 ex color_ONE(unsigned char rl)
480 static ex ONE = (new su3one)->setflag(status_flags::dynallocated);
481 return color(ONE, rl);
484 ex color_T(const ex & a, unsigned char rl)
486 static ex t = (new su3t)->setflag(status_flags::dynallocated);
489 throw(std::invalid_argument("indices of color_T must be of type idx"));
490 if (!ex_to<idx>(a).get_dim().is_equal(8))
491 throw(std::invalid_argument("index dimension for color_T must be 8"));
493 return color(t, a, rl);
496 ex color_f(const ex & a, const ex & b, const ex & c)
498 static ex f = (new su3f)->setflag(status_flags::dynallocated);
500 if (!is_a<idx>(a) || !is_a<idx>(b) || !is_a<idx>(c))
501 throw(std::invalid_argument("indices of color_f must be of type idx"));
502 if (!ex_to<idx>(a).get_dim().is_equal(8) || !ex_to<idx>(b).get_dim().is_equal(8) || !ex_to<idx>(c).get_dim().is_equal(8))
503 throw(std::invalid_argument("index dimension for color_f must be 8"));
505 return indexed(f, antisymmetric3(), a, b, c);
508 ex color_d(const ex & a, const ex & b, const ex & c)
510 static ex d = (new su3d)->setflag(status_flags::dynallocated);
512 if (!is_a<idx>(a) || !is_a<idx>(b) || !is_a<idx>(c))
513 throw(std::invalid_argument("indices of color_d must be of type idx"));
514 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))
515 throw(std::invalid_argument("index dimension for color_d must be 8"));
517 return indexed(d, symmetric3(), a, b, c);
520 ex color_h(const ex & a, const ex & b, const ex & c)
522 return color_d(a, b, c) + I * color_f(a, b, c);
525 /** Check whether a given tinfo key (as returned by return_type_tinfo()
526 * is that of a color object (with an arbitrary representation label). */
527 static bool is_color_tinfo(tinfo_t ti)
529 p_int start_loc=(p_int)&color::return_type_tinfo_static;
530 return (p_int)ti>=start_loc && (p_int)ti<start_loc+256;
533 /** Extract representation label from tinfo key (as returned by
534 * return_type_tinfo()). */
535 static unsigned char get_representation_label(tinfo_t ti)
537 return (unsigned char)((p_int)ti-(p_int)&color::return_type_tinfo_static);
540 ex color_trace(const ex & e, const std::set<unsigned char> & rls)
542 if (is_a<color>(e)) {
544 unsigned char rl = ex_to<color>(e).get_representation_label();
546 // Are we taking the trace over this object's representation label?
547 if (rls.find(rl) == rls.end())
550 // Yes, all generators are traceless, except for color_ONE
551 if (is_a<su3one>(e.op(0)))
556 } else if (is_exactly_a<mul>(e)) {
558 // Trace of product: pull out non-color factors
560 for (size_t i=0; i<e.nops(); i++) {
561 const ex &o = e.op(i);
562 if (is_color_tinfo(o.return_type_tinfo()))
563 prod *= color_trace(o, rls);
569 } else if (is_exactly_a<ncmul>(e)) {
571 unsigned char rl = get_representation_label(e.return_type_tinfo());
573 // Are we taking the trace over this string's representation label?
574 if (rls.find(rl) == rls.end())
577 // Yes, expand product if necessary
578 ex e_expanded = e.expand();
579 if (!is_a<ncmul>(e_expanded))
580 return color_trace(e_expanded, rls);
582 size_t num = e.nops();
586 // Tr T_a T_b = 1/2 delta_a_b
587 return delta_tensor(e.op(0).op(1), e.op(1).op(1)) / 2;
589 } else if (num == 3) {
591 // Tr T_a T_b T_c = 1/4 h_a_b_c
592 return color_h(e.op(0).op(1), e.op(1).op(1), e.op(2).op(1)) / 4;
596 // Traces of 4 or more generators are computed recursively:
598 // 1/6 delta_a(n-1)_an Tr T_a1 .. T_a(n-2)
599 // + 1/2 h_a(n-1)_an_k Tr T_a1 .. T_a(n-2) T_k
600 const ex &last_index = e.op(num - 1).op(1);
601 const ex &next_to_last_index = e.op(num - 2).op(1);
602 idx summation_index((new symbol)->setflag(status_flags::dynallocated), 8);
606 for (size_t i=0; i<num-2; i++)
607 v1.push_back(e.op(i));
610 v2.push_back(color_T(summation_index, rl));
612 return delta_tensor(next_to_last_index, last_index) * color_trace(ncmul(v1), rl) / 6
613 + color_h(next_to_last_index, last_index, summation_index) * color_trace(ncmul(v2), rl) / 2;
616 } else if (e.nops() > 0) {
618 // Trace maps to all other container classes (this includes sums)
619 pointer_to_map_function_1arg<const std::set<unsigned char> &> fcn(color_trace, rls);
626 ex color_trace(const ex & e, const lst & rll)
628 // Convert list to set
629 std::set<unsigned char> rls;
630 for (lst::const_iterator i = rll.begin(); i != rll.end(); ++i) {
631 if (i->info(info_flags::nonnegint))
632 rls.insert(ex_to<numeric>(*i).to_int());
635 return color_trace(e, rls);
638 ex color_trace(const ex & e, unsigned char rl)
640 // Convert label to set
641 std::set<unsigned char> rls;
644 return color_trace(e, rls);