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()
40 GINAC_IMPLEMENT_REGISTERED_CLASS(color, indexed)
42 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(su3one, tensor,
43 print_func<print_dflt>(&su3one::do_print).
44 print_func<print_latex>(&su3one::do_print_latex))
46 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(su3t, tensor,
47 print_func<print_dflt>(&su3t::do_print).
48 print_func<print_latex>(&su3t::do_print))
50 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(su3f, tensor,
51 print_func<print_dflt>(&su3f::do_print).
52 print_func<print_latex>(&su3f::do_print))
54 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(su3d, tensor,
55 print_func<print_dflt>(&su3d::do_print).
56 print_func<print_latex>(&su3d::do_print))
59 // default constructors
62 color::color() : representation_label(0)
64 tinfo_key = TINFO_color;
76 /** Construct object without any color index. This constructor is for
77 * internal use only. Use the color_ONE() function instead.
79 color::color(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
81 tinfo_key = TINFO_color;
84 /** Construct object with one color index. This constructor is for internal
85 * use only. Use the color_T() function instead.
87 color::color(const ex & b, const ex & i1, unsigned char rl) : inherited(b, i1), representation_label(rl)
89 tinfo_key = TINFO_color;
92 color::color(unsigned char rl, const exvector & v, bool discardable) : inherited(sy_none(), v, discardable), representation_label(rl)
94 tinfo_key = TINFO_color;
97 color::color(unsigned char rl, std::auto_ptr<exvector> vp) : inherited(sy_none(), vp), representation_label(rl)
99 tinfo_key = TINFO_color;
106 color::color(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
109 n.find_unsigned("label", rl);
110 representation_label = rl;
113 void color::archive(archive_node &n) const
115 inherited::archive(n);
116 n.add_unsigned("label", representation_label);
119 DEFAULT_UNARCHIVE(color)
120 DEFAULT_ARCHIVING(su3one)
121 DEFAULT_ARCHIVING(su3t)
122 DEFAULT_ARCHIVING(su3f)
123 DEFAULT_ARCHIVING(su3d)
126 // functions overriding virtual functions from base classes
129 int color::compare_same_type(const basic & other) const
131 GINAC_ASSERT(is_a<color>(other));
132 const color &o = static_cast<const color &>(other);
134 if (representation_label != o.representation_label) {
135 // different representation label
136 return representation_label < o.representation_label ? -1 : 1;
139 return inherited::compare_same_type(other);
142 bool color::match_same_type(const basic & other) const
144 GINAC_ASSERT(is_a<color>(other));
145 const color &o = static_cast<const color &>(other);
147 return representation_label == o.representation_label;
150 DEFAULT_COMPARE(su3one)
151 DEFAULT_COMPARE(su3t)
152 DEFAULT_COMPARE(su3f)
153 DEFAULT_COMPARE(su3d)
155 DEFAULT_PRINT_LATEX(su3one, "ONE", "\\mathbb{1}")
156 DEFAULT_PRINT(su3t, "T")
157 DEFAULT_PRINT(su3f, "f")
158 DEFAULT_PRINT(su3d, "d")
160 /** Perform automatic simplification on noncommutative product of color
161 * objects. This removes superfluous ONEs. */
162 ex color::eval_ncmul(const exvector & v) const
167 // Remove superfluous ONEs
168 exvector::const_iterator it = v.begin(), itend = v.end();
169 while (it != itend) {
170 if (!is_a<su3one>(it->op(0)))
176 return color(su3one(), representation_label);
178 return hold_ncmul(s);
181 ex color::thiscontainer(const exvector & v) const
183 return color(representation_label, v);
186 ex color::thiscontainer(std::auto_ptr<exvector> vp) const
188 return color(representation_label, vp);
191 /** Given a vector iv3 of three indices and a vector iv2 of two indices that
192 * is a subset of iv3, return the (free) index that is in iv3 but not in
193 * iv2 and the sign introduced by permuting that index to the front.
195 * @param iv3 Vector of 3 indices
196 * @param iv2 Vector of 2 indices, must be a subset of iv3
197 * @param sig Returs sign introduced by index permutation
198 * @return the free index (the one that is in iv3 but not in iv2) */
199 static ex permute_free_index_to_front(const exvector & iv3, const exvector & iv2, int & sig)
201 GINAC_ASSERT(iv3.size() == 3);
202 GINAC_ASSERT(iv2.size() == 2);
206 #define TEST_PERMUTATION(A,B,C,P) \
207 if (iv3[B].is_equal(iv2[0]) && iv3[C].is_equal(iv2[1])) { \
212 TEST_PERMUTATION(0,1,2, 1);
213 TEST_PERMUTATION(0,2,1, -1);
214 TEST_PERMUTATION(1,0,2, -1);
215 TEST_PERMUTATION(1,2,0, 1);
216 TEST_PERMUTATION(2,0,1, 1);
217 TEST_PERMUTATION(2,1,0, -1);
219 throw(std::logic_error("permute_free_index_to_front(): no valid permutation found"));
222 /** Automatic symbolic evaluation of indexed symmetric structure constant. */
223 ex su3d::eval_indexed(const basic & i) const
225 GINAC_ASSERT(is_a<indexed>(i));
226 GINAC_ASSERT(i.nops() == 4);
227 GINAC_ASSERT(is_a<su3d>(i.op(0)));
229 // Convolutions are zero
230 if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
233 // Numeric evaluation
234 if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
238 for (unsigned j=0; j<3; j++)
239 v[j] = ex_to<numeric>(ex_to<idx>(i.op(j + 1)).get_value()).to_int();
240 if (v[0] > v[1]) std::swap(v[0], v[1]);
241 if (v[0] > v[2]) std::swap(v[0], v[2]);
242 if (v[1] > v[2]) std::swap(v[1], v[2]);
244 #define CMPINDICES(A,B,C) ((v[0] == (A)) && (v[1] == (B)) && (v[2] == (C)))
246 // Check for non-zero elements
247 if (CMPINDICES(1,4,6) || CMPINDICES(1,5,7) || CMPINDICES(2,5,6)
248 || CMPINDICES(3,4,4) || CMPINDICES(3,5,5))
250 else if (CMPINDICES(2,4,7) || CMPINDICES(3,6,6) || CMPINDICES(3,7,7))
252 else if (CMPINDICES(1,1,8) || CMPINDICES(2,2,8) || CMPINDICES(3,3,8))
253 return sqrt(_ex3)*_ex1_3;
254 else if (CMPINDICES(8,8,8))
255 return sqrt(_ex3)*_ex_1_3;
256 else if (CMPINDICES(4,4,8) || CMPINDICES(5,5,8)
257 || CMPINDICES(6,6,8) || CMPINDICES(7,7,8))
258 return sqrt(_ex3)/_ex_6;
263 // No further simplifications
267 /** Automatic symbolic evaluation of indexed antisymmetric structure constant. */
268 ex su3f::eval_indexed(const basic & i) const
270 GINAC_ASSERT(is_a<indexed>(i));
271 GINAC_ASSERT(i.nops() == 4);
272 GINAC_ASSERT(is_a<su3f>(i.op(0)));
274 // Numeric evaluation
275 if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
277 // Sort indices, remember permutation sign
279 for (unsigned j=0; j<3; j++)
280 v[j] = ex_to<numeric>(ex_to<idx>(i.op(j + 1)).get_value()).to_int();
282 if (v[0] > v[1]) { std::swap(v[0], v[1]); sign = -sign; }
283 if (v[0] > v[2]) { std::swap(v[0], v[2]); sign = -sign; }
284 if (v[1] > v[2]) { std::swap(v[1], v[2]); sign = -sign; }
286 // Check for non-zero elements
287 if (CMPINDICES(1,2,3))
289 else if (CMPINDICES(1,4,7) || CMPINDICES(2,4,6)
290 || CMPINDICES(2,5,7) || CMPINDICES(3,4,5))
291 return _ex1_2 * sign;
292 else if (CMPINDICES(1,5,6) || CMPINDICES(3,6,7))
293 return _ex_1_2 * sign;
294 else if (CMPINDICES(4,5,8) || CMPINDICES(6,7,8))
295 return sqrt(_ex3)/2 * sign;
300 // No further simplifications
305 /** Contraction of generator with something else. */
306 bool su3t::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
308 GINAC_ASSERT(is_a<indexed>(*self));
309 GINAC_ASSERT(is_a<indexed>(*other));
310 GINAC_ASSERT(self->nops() == 2);
311 GINAC_ASSERT(is_a<su3t>(self->op(0)));
312 unsigned char rl = ex_to<color>(*self).get_representation_label();
314 if (is_exactly_a<su3t>(other->op(0))) {
316 // Contraction only makes sense if the represenation labels are equal
317 GINAC_ASSERT(is_a<color>(*other));
318 if (ex_to<color>(*other).get_representation_label() != rl)
322 if (other - self == 1) {
323 *self = numeric(4, 3);
324 *other = color_ONE(rl);
327 // T.a T.b T.a = -1/6 T.b
328 } else if (other - self == 2
329 && is_a<color>(self[1])) {
330 *self = numeric(-1, 6);
334 // T.a S T.a = 1/2 Tr(S) - 1/6 S
336 exvector::iterator it = self + 1;
337 while (it != other) {
338 if (!is_a<color>(*it)) {
346 while (it != other) {
351 *self = color_trace(S, rl) * color_ONE(rl) / 2 - S / 6;
360 /** Contraction of an indexed symmetric structure constant with something else. */
361 bool su3d::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
363 GINAC_ASSERT(is_a<indexed>(*self));
364 GINAC_ASSERT(is_a<indexed>(*other));
365 GINAC_ASSERT(self->nops() == 4);
366 GINAC_ASSERT(is_a<su3d>(self->op(0)));
368 if (is_exactly_a<su3d>(other->op(0))) {
370 // Find the dummy indices of the contraction
371 exvector self_indices = ex_to<indexed>(*self).get_indices();
372 exvector other_indices = ex_to<indexed>(*other).get_indices();
373 exvector all_indices = self_indices;
374 all_indices.insert(all_indices.end(), other_indices.begin(), other_indices.end());
375 exvector free_indices, dummy_indices;
376 find_free_and_dummy(all_indices, free_indices, dummy_indices);
378 // d.abc d.abc = 40/3
379 if (dummy_indices.size() == 3) {
380 *self = numeric(40, 3);
384 // d.akl d.bkl = 5/3 delta.ab
385 } else if (dummy_indices.size() == 2) {
387 std::back_insert_iterator<exvector> ita(a);
388 ita = set_difference(self_indices.begin(), self_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
389 ita = set_difference(other_indices.begin(), other_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
390 GINAC_ASSERT(a.size() == 2);
391 *self = numeric(5, 3) * delta_tensor(a[0], a[1]);
396 } else if (is_exactly_a<su3t>(other->op(0))) {
398 // d.abc T.b T.c = 5/6 T.a
399 if (other+1 != v.end()
400 && is_exactly_a<su3t>(other[1].op(0))
401 && ex_to<indexed>(*self).has_dummy_index_for(other[1].op(1))) {
403 exvector self_indices = ex_to<indexed>(*self).get_indices();
404 exvector dummy_indices;
405 dummy_indices.push_back(other[0].op(1));
406 dummy_indices.push_back(other[1].op(1));
408 ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
409 *self = numeric(5, 6);
410 other[0] = color_T(a, ex_to<color>(other[0]).get_representation_label());
419 /** Contraction of an indexed antisymmetric structure constant with something else. */
420 bool su3f::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
422 GINAC_ASSERT(is_a<indexed>(*self));
423 GINAC_ASSERT(is_a<indexed>(*other));
424 GINAC_ASSERT(self->nops() == 4);
425 GINAC_ASSERT(is_a<su3f>(self->op(0)));
427 if (is_exactly_a<su3f>(other->op(0))) { // f*d is handled by su3d class
429 // Find the dummy indices of the contraction
430 exvector dummy_indices;
431 dummy_indices = ex_to<indexed>(*self).get_dummy_indices(ex_to<indexed>(*other));
434 if (dummy_indices.size() == 3) {
439 // f.akl f.bkl = 3 delta.ab
440 } else if (dummy_indices.size() == 2) {
442 ex a = permute_free_index_to_front(ex_to<indexed>(*self).get_indices(), dummy_indices, sign1);
443 ex b = permute_free_index_to_front(ex_to<indexed>(*other).get_indices(), dummy_indices, sign2);
444 *self = sign1 * sign2 * 3 * delta_tensor(a, b);
449 } else if (is_exactly_a<su3t>(other->op(0))) {
451 // f.abc T.b T.c = 3/2 I T.a
452 if (other+1 != v.end()
453 && is_exactly_a<su3t>(other[1].op(0))
454 && ex_to<indexed>(*self).has_dummy_index_for(other[1].op(1))) {
456 exvector self_indices = ex_to<indexed>(*self).get_indices();
457 exvector dummy_indices;
458 dummy_indices.push_back(other[0].op(1));
459 dummy_indices.push_back(other[1].op(1));
461 ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
462 *self = numeric(3, 2) * sig * I;
463 other[0] = color_T(a, ex_to<color>(other[0]).get_representation_label());
476 ex color_ONE(unsigned char rl)
478 return color(su3one(), rl);
481 ex color_T(const ex & a, unsigned char rl)
484 throw(std::invalid_argument("indices of color_T must be of type idx"));
485 if (!ex_to<idx>(a).get_dim().is_equal(8))
486 throw(std::invalid_argument("index dimension for color_T must be 8"));
488 return color(su3t(), a, rl);
491 ex color_f(const ex & a, const ex & b, const ex & c)
493 if (!is_a<idx>(a) || !is_a<idx>(b) || !is_a<idx>(c))
494 throw(std::invalid_argument("indices of color_f must be of type idx"));
495 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))
496 throw(std::invalid_argument("index dimension for color_f must be 8"));
498 return indexed(su3f(), sy_anti(), a, b, c);
501 ex color_d(const ex & a, const ex & b, const ex & c)
503 if (!is_a<idx>(a) || !is_a<idx>(b) || !is_a<idx>(c))
504 throw(std::invalid_argument("indices of color_d must be of type idx"));
505 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))
506 throw(std::invalid_argument("index dimension for color_d must be 8"));
508 return indexed(su3d(), sy_symm(), a, b, c);
511 ex color_h(const ex & a, const ex & b, const ex & c)
513 return color_d(a, b, c) + I * color_f(a, b, c);
516 /** Check whether a given tinfo key (as returned by return_type_tinfo()
517 * is that of a color object with the specified representation label. */
518 static bool is_color_tinfo(unsigned ti, unsigned char rl)
520 return ti == (TINFO_color + rl);
523 ex color_trace(const ex & e, unsigned char rl)
525 if (is_a<color>(e)) {
527 if (ex_to<color>(e).get_representation_label() == rl
528 && is_a<su3one>(e.op(0)))
533 } else if (is_exactly_a<mul>(e)) {
535 // Trace of product: pull out non-color factors
537 for (size_t i=0; i<e.nops(); i++) {
538 const ex &o = e.op(i);
539 if (is_color_tinfo(o.return_type_tinfo(), rl))
540 prod *= color_trace(o, rl);
546 } else if (is_exactly_a<ncmul>(e)) {
548 if (!is_color_tinfo(e.return_type_tinfo(), rl))
551 // Expand product, if necessary
552 ex e_expanded = e.expand();
553 if (!is_a<ncmul>(e_expanded))
554 return color_trace(e_expanded, rl);
556 size_t num = e.nops();
560 // Tr T_a T_b = 1/2 delta_a_b
561 return delta_tensor(e.op(0).op(1), e.op(1).op(1)) / 2;
563 } else if (num == 3) {
565 // Tr T_a T_b T_c = 1/4 h_a_b_c
566 return color_h(e.op(0).op(1), e.op(1).op(1), e.op(2).op(1)) / 4;
570 // Traces of 4 or more generators are computed recursively:
572 // 1/6 delta_a(n-1)_an Tr T_a1 .. T_a(n-2)
573 // + 1/2 h_a(n-1)_an_k Tr T_a1 .. T_a(n-2) T_k
574 const ex &last_index = e.op(num - 1).op(1);
575 const ex &next_to_last_index = e.op(num - 2).op(1);
576 idx summation_index((new symbol)->setflag(status_flags::dynallocated), 8);
580 for (size_t i=0; i<num-2; i++)
581 v1.push_back(e.op(i));
584 v2.push_back(color_T(summation_index, rl));
586 return delta_tensor(next_to_last_index, last_index) * color_trace(ncmul(v1), rl) / 6
587 + color_h(next_to_last_index, last_index, summation_index) * color_trace(ncmul(v2), rl) / 2;
590 } else if (e.nops() > 0) {
592 // Trace maps to all other container classes (this includes sums)
593 pointer_to_map_function_1arg<unsigned char> fcn(color_trace, rl);