3 * Implementation of GiNaC's color (SU(3) Lie algebra) objects. */
6 * GiNaC Copyright (C) 1999-2003 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., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
30 #include "operators.h"
33 #include "power.h" // for sqrt()
41 GINAC_IMPLEMENT_REGISTERED_CLASS(color, indexed)
43 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(su3one, tensor,
44 print_func<print_dflt>(&su3one::do_print).
45 print_func<print_latex>(&su3one::do_print_latex))
47 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(su3t, tensor,
48 print_func<print_dflt>(&su3t::do_print).
49 print_func<print_latex>(&su3t::do_print))
51 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(su3f, tensor,
52 print_func<print_dflt>(&su3f::do_print).
53 print_func<print_latex>(&su3f::do_print))
55 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(su3d, tensor,
56 print_func<print_dflt>(&su3d::do_print).
57 print_func<print_latex>(&su3d::do_print))
60 // default constructors
63 color::color() : representation_label(0)
65 tinfo_key = TINFO_color;
77 /** Construct object without any color index. This constructor is for
78 * internal use only. Use the color_ONE() function instead.
80 color::color(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
82 tinfo_key = TINFO_color;
85 /** Construct object with one color index. This constructor is for internal
86 * use only. Use the color_T() function instead.
88 color::color(const ex & b, const ex & i1, unsigned char rl) : inherited(b, i1), representation_label(rl)
90 tinfo_key = TINFO_color;
93 color::color(unsigned char rl, const exvector & v, bool discardable) : inherited(sy_none(), v, discardable), representation_label(rl)
95 tinfo_key = TINFO_color;
98 color::color(unsigned char rl, exvector * vp) : inherited(sy_none(), vp), representation_label(rl)
100 tinfo_key = TINFO_color;
107 color::color(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
110 n.find_unsigned("label", rl);
111 representation_label = rl;
114 void color::archive(archive_node &n) const
116 inherited::archive(n);
117 n.add_unsigned("label", representation_label);
120 DEFAULT_UNARCHIVE(color)
121 DEFAULT_ARCHIVING(su3one)
122 DEFAULT_ARCHIVING(su3t)
123 DEFAULT_ARCHIVING(su3f)
124 DEFAULT_ARCHIVING(su3d)
127 // functions overriding virtual functions from base classes
130 int color::compare_same_type(const basic & other) const
132 GINAC_ASSERT(is_a<color>(other));
133 const color &o = static_cast<const color &>(other);
135 if (representation_label != o.representation_label) {
136 // different representation label
137 return representation_label < o.representation_label ? -1 : 1;
140 return inherited::compare_same_type(other);
143 bool color::match_same_type(const basic & other) const
145 GINAC_ASSERT(is_a<color>(other));
146 const color &o = static_cast<const color &>(other);
148 return representation_label == o.representation_label;
151 DEFAULT_COMPARE(su3one)
152 DEFAULT_COMPARE(su3t)
153 DEFAULT_COMPARE(su3f)
154 DEFAULT_COMPARE(su3d)
156 DEFAULT_PRINT_LATEX(su3one, "ONE", "\\mathbb{1}")
157 DEFAULT_PRINT(su3t, "T")
158 DEFAULT_PRINT(su3f, "f")
159 DEFAULT_PRINT(su3d, "d")
161 /** Perform automatic simplification on noncommutative product of color
162 * objects. This removes superfluous ONEs. */
163 ex color::eval_ncmul(const exvector & v) const
168 // Remove superfluous ONEs
169 exvector::const_iterator it = v.begin(), itend = v.end();
170 while (it != itend) {
171 if (!is_a<su3one>(it->op(0)))
177 return color(su3one(), representation_label);
179 return hold_ncmul(s);
182 ex color::thiscontainer(const exvector & v) const
184 return color(representation_label, v);
187 ex color::thiscontainer(exvector * vp) const
189 return color(representation_label, vp);
192 /** Given a vector iv3 of three indices and a vector iv2 of two indices that
193 * is a subset of iv3, return the (free) index that is in iv3 but not in
194 * iv2 and the sign introduced by permuting that index to the front.
196 * @param iv3 Vector of 3 indices
197 * @param iv2 Vector of 2 indices, must be a subset of iv3
198 * @param sig Returs sign introduced by index permutation
199 * @return the free index (the one that is in iv3 but not in iv2) */
200 static ex permute_free_index_to_front(const exvector & iv3, const exvector & iv2, int & sig)
202 GINAC_ASSERT(iv3.size() == 3);
203 GINAC_ASSERT(iv2.size() == 2);
207 #define TEST_PERMUTATION(A,B,C,P) \
208 if (iv3[B].is_equal(iv2[0]) && iv3[C].is_equal(iv2[1])) { \
213 TEST_PERMUTATION(0,1,2, 1);
214 TEST_PERMUTATION(0,2,1, -1);
215 TEST_PERMUTATION(1,0,2, -1);
216 TEST_PERMUTATION(1,2,0, 1);
217 TEST_PERMUTATION(2,0,1, 1);
218 TEST_PERMUTATION(2,1,0, -1);
220 throw(std::logic_error("permute_free_index_to_front(): no valid permutation found"));
223 /** Automatic symbolic evaluation of indexed symmetric structure constant. */
224 ex su3d::eval_indexed(const basic & i) const
226 GINAC_ASSERT(is_a<indexed>(i));
227 GINAC_ASSERT(i.nops() == 4);
228 GINAC_ASSERT(is_a<su3d>(i.op(0)));
230 // Convolutions are zero
231 if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
234 // Numeric evaluation
235 if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
239 for (unsigned j=0; j<3; j++)
240 v[j] = ex_to<numeric>(ex_to<idx>(i.op(j + 1)).get_value()).to_int();
241 if (v[0] > v[1]) std::swap(v[0], v[1]);
242 if (v[0] > v[2]) std::swap(v[0], v[2]);
243 if (v[1] > v[2]) std::swap(v[1], v[2]);
245 #define CMPINDICES(A,B,C) ((v[0] == (A)) && (v[1] == (B)) && (v[2] == (C)))
247 // Check for non-zero elements
248 if (CMPINDICES(1,4,6) || CMPINDICES(1,5,7) || CMPINDICES(2,5,6)
249 || CMPINDICES(3,4,4) || CMPINDICES(3,5,5))
251 else if (CMPINDICES(2,4,7) || CMPINDICES(3,6,6) || CMPINDICES(3,7,7))
253 else if (CMPINDICES(1,1,8) || CMPINDICES(2,2,8) || CMPINDICES(3,3,8))
254 return sqrt(_ex3)*_ex1_3;
255 else if (CMPINDICES(8,8,8))
256 return sqrt(_ex3)*_ex_1_3;
257 else if (CMPINDICES(4,4,8) || CMPINDICES(5,5,8)
258 || CMPINDICES(6,6,8) || CMPINDICES(7,7,8))
259 return sqrt(_ex3)/_ex_6;
264 // No further simplifications
268 /** Automatic symbolic evaluation of indexed antisymmetric structure constant. */
269 ex su3f::eval_indexed(const basic & i) const
271 GINAC_ASSERT(is_a<indexed>(i));
272 GINAC_ASSERT(i.nops() == 4);
273 GINAC_ASSERT(is_a<su3f>(i.op(0)));
275 // Numeric evaluation
276 if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
278 // Sort indices, remember permutation sign
280 for (unsigned j=0; j<3; j++)
281 v[j] = ex_to<numeric>(ex_to<idx>(i.op(j + 1)).get_value()).to_int();
283 if (v[0] > v[1]) { std::swap(v[0], v[1]); sign = -sign; }
284 if (v[0] > v[2]) { std::swap(v[0], v[2]); sign = -sign; }
285 if (v[1] > v[2]) { std::swap(v[1], v[2]); sign = -sign; }
287 // Check for non-zero elements
288 if (CMPINDICES(1,2,3))
290 else if (CMPINDICES(1,4,7) || CMPINDICES(2,4,6)
291 || CMPINDICES(2,5,7) || CMPINDICES(3,4,5))
292 return _ex1_2 * sign;
293 else if (CMPINDICES(1,5,6) || CMPINDICES(3,6,7))
294 return _ex_1_2 * sign;
295 else if (CMPINDICES(4,5,8) || CMPINDICES(6,7,8))
296 return sqrt(_ex3)/2 * sign;
301 // No further simplifications
306 /** Contraction of generator with something else. */
307 bool su3t::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
309 GINAC_ASSERT(is_a<indexed>(*self));
310 GINAC_ASSERT(is_a<indexed>(*other));
311 GINAC_ASSERT(self->nops() == 2);
312 GINAC_ASSERT(is_a<su3t>(self->op(0)));
313 unsigned char rl = ex_to<color>(*self).get_representation_label();
315 if (is_exactly_a<su3t>(other->op(0))) {
317 // Contraction only makes sense if the represenation labels are equal
318 GINAC_ASSERT(is_a<color>(*other));
319 if (ex_to<color>(*other).get_representation_label() != rl)
323 if (other - self == 1) {
324 *self = numeric(4, 3);
325 *other = color_ONE(rl);
328 // T.a T.b T.a = -1/6 T.b
329 } else if (other - self == 2
330 && is_a<color>(self[1])) {
331 *self = numeric(-1, 6);
335 // T.a S T.a = 1/2 Tr(S) - 1/6 S
337 exvector::iterator it = self + 1;
338 while (it != other) {
339 if (!is_a<color>(*it)) {
347 while (it != other) {
352 *self = color_trace(S, rl) * color_ONE(rl) / 2 - S / 6;
361 /** Contraction of an indexed symmetric structure constant with something else. */
362 bool su3d::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
364 GINAC_ASSERT(is_a<indexed>(*self));
365 GINAC_ASSERT(is_a<indexed>(*other));
366 GINAC_ASSERT(self->nops() == 4);
367 GINAC_ASSERT(is_a<su3d>(self->op(0)));
369 if (is_exactly_a<su3d>(other->op(0))) {
371 // Find the dummy indices of the contraction
372 exvector self_indices = ex_to<indexed>(*self).get_indices();
373 exvector other_indices = ex_to<indexed>(*other).get_indices();
374 exvector all_indices = self_indices;
375 all_indices.insert(all_indices.end(), other_indices.begin(), other_indices.end());
376 exvector free_indices, dummy_indices;
377 find_free_and_dummy(all_indices, free_indices, dummy_indices);
379 // d.abc d.abc = 40/3
380 if (dummy_indices.size() == 3) {
381 *self = numeric(40, 3);
385 // d.akl d.bkl = 5/3 delta.ab
386 } else if (dummy_indices.size() == 2) {
388 std::back_insert_iterator<exvector> ita(a);
389 ita = set_difference(self_indices.begin(), self_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
390 ita = set_difference(other_indices.begin(), other_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
391 GINAC_ASSERT(a.size() == 2);
392 *self = numeric(5, 3) * delta_tensor(a[0], a[1]);
397 } else if (is_exactly_a<su3t>(other->op(0))) {
399 // d.abc T.b T.c = 5/6 T.a
400 if (other+1 != v.end()
401 && is_exactly_a<su3t>(other[1].op(0))
402 && ex_to<indexed>(*self).has_dummy_index_for(other[1].op(1))) {
404 exvector self_indices = ex_to<indexed>(*self).get_indices();
405 exvector dummy_indices;
406 dummy_indices.push_back(other[0].op(1));
407 dummy_indices.push_back(other[1].op(1));
409 ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
410 *self = numeric(5, 6);
411 other[0] = color_T(a, ex_to<color>(other[0]).get_representation_label());
420 /** Contraction of an indexed antisymmetric structure constant with something else. */
421 bool su3f::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
423 GINAC_ASSERT(is_a<indexed>(*self));
424 GINAC_ASSERT(is_a<indexed>(*other));
425 GINAC_ASSERT(self->nops() == 4);
426 GINAC_ASSERT(is_a<su3f>(self->op(0)));
428 if (is_exactly_a<su3f>(other->op(0))) { // f*d is handled by su3d class
430 // Find the dummy indices of the contraction
431 exvector dummy_indices;
432 dummy_indices = ex_to<indexed>(*self).get_dummy_indices(ex_to<indexed>(*other));
435 if (dummy_indices.size() == 3) {
440 // f.akl f.bkl = 3 delta.ab
441 } else if (dummy_indices.size() == 2) {
443 ex a = permute_free_index_to_front(ex_to<indexed>(*self).get_indices(), dummy_indices, sign1);
444 ex b = permute_free_index_to_front(ex_to<indexed>(*other).get_indices(), dummy_indices, sign2);
445 *self = sign1 * sign2 * 3 * delta_tensor(a, b);
450 } else if (is_exactly_a<su3t>(other->op(0))) {
452 // f.abc T.b T.c = 3/2 I T.a
453 if (other+1 != v.end()
454 && is_exactly_a<su3t>(other[1].op(0))
455 && ex_to<indexed>(*self).has_dummy_index_for(other[1].op(1))) {
457 exvector self_indices = ex_to<indexed>(*self).get_indices();
458 exvector dummy_indices;
459 dummy_indices.push_back(other[0].op(1));
460 dummy_indices.push_back(other[1].op(1));
462 ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
463 *self = numeric(3, 2) * sig * I;
464 other[0] = color_T(a, ex_to<color>(other[0]).get_representation_label());
477 ex color_ONE(unsigned char rl)
479 return color(su3one(), rl);
482 ex color_T(const ex & a, unsigned char rl)
485 throw(std::invalid_argument("indices of color_T must be of type idx"));
486 if (!ex_to<idx>(a).get_dim().is_equal(8))
487 throw(std::invalid_argument("index dimension for color_T must be 8"));
489 return color(su3t(), a, rl);
492 ex color_f(const ex & a, const ex & b, const ex & c)
494 if (!is_a<idx>(a) || !is_a<idx>(b) || !is_a<idx>(c))
495 throw(std::invalid_argument("indices of color_f must be of type idx"));
496 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))
497 throw(std::invalid_argument("index dimension for color_f must be 8"));
499 return indexed(su3f(), sy_anti(), a, b, c);
502 ex color_d(const ex & a, const ex & b, const ex & c)
504 if (!is_a<idx>(a) || !is_a<idx>(b) || !is_a<idx>(c))
505 throw(std::invalid_argument("indices of color_d must be of type idx"));
506 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))
507 throw(std::invalid_argument("index dimension for color_d must be 8"));
509 return indexed(su3d(), sy_symm(), a, b, c);
512 ex color_h(const ex & a, const ex & b, const ex & c)
514 return color_d(a, b, c) + I * color_f(a, b, c);
517 /** Check whether a given tinfo key (as returned by return_type_tinfo()
518 * is that of a color object with the specified representation label. */
519 static bool is_color_tinfo(unsigned ti, unsigned char rl)
521 return ti == (TINFO_color + rl);
524 ex color_trace(const ex & e, unsigned char rl)
526 if (is_a<color>(e)) {
528 if (ex_to<color>(e).get_representation_label() == rl
529 && is_a<su3one>(e.op(0)))
534 } else if (is_exactly_a<mul>(e)) {
536 // Trace of product: pull out non-color factors
538 for (size_t i=0; i<e.nops(); i++) {
539 const ex &o = e.op(i);
540 if (is_color_tinfo(o.return_type_tinfo(), rl))
541 prod *= color_trace(o, rl);
547 } else if (is_exactly_a<ncmul>(e)) {
549 if (!is_color_tinfo(e.return_type_tinfo(), rl))
552 // Expand product, if necessary
553 ex e_expanded = e.expand();
554 if (!is_a<ncmul>(e_expanded))
555 return color_trace(e_expanded, rl);
557 size_t num = e.nops();
561 // Tr T_a T_b = 1/2 delta_a_b
562 return delta_tensor(e.op(0).op(1), e.op(1).op(1)) / 2;
564 } else if (num == 3) {
566 // Tr T_a T_b T_c = 1/4 h_a_b_c
567 return color_h(e.op(0).op(1), e.op(1).op(1), e.op(2).op(1)) / 4;
571 // Traces of 4 or more generators are computed recursively:
573 // 1/6 delta_a(n-1)_an Tr T_a1 .. T_a(n-2)
574 // + 1/2 h_a(n-1)_an_k Tr T_a1 .. T_a(n-2) T_k
575 const ex &last_index = e.op(num - 1).op(1);
576 const ex &next_to_last_index = e.op(num - 2).op(1);
577 idx summation_index((new symbol)->setflag(status_flags::dynallocated), 8);
581 for (size_t i=0; i<num-2; i++)
582 v1.push_back(e.op(i));
585 v2.push_back(color_T(summation_index, rl));
587 return delta_tensor(next_to_last_index, last_index) * color_trace(ncmul(v1), rl) / 6
588 + color_h(next_to_last_index, last_index, summation_index) * color_trace(ncmul(v2), rl) / 2;
591 } else if (e.nops() > 0) {
593 // Trace maps to all other container classes (this includes sums)
594 pointer_to_map_function_1arg<unsigned char> fcn(color_trace, rl);