3 * Implementation of GiNaC's color (SU(3) Lie algebra) objects. */
6 * GiNaC Copyright (C) 1999-2020 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
27 #include "operators.h"
30 #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)
75 /** Construct object without any color index. This constructor is for
76 * internal use only. Use the color_ONE() function instead.
78 color::color(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
82 /** Construct object with one color index. This constructor is for internal
83 * use only. Use the color_T() function instead.
85 color::color(const ex & b, const ex & i1, unsigned char rl) : inherited(b, i1), representation_label(rl)
89 color::color(unsigned char rl, const exvector & v) : inherited(not_symmetric(), v), representation_label(rl)
93 color::color(unsigned char rl, exvector && v) : inherited(not_symmetric(), std::move(v)), representation_label(rl)
97 return_type_t color::return_type_tinfo() const
99 return make_return_type_t<color>(representation_label);
106 void color::read_archive(const archive_node& n, lst& sym_lst)
108 inherited::read_archive(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 GINAC_BIND_UNARCHIVER(color);
121 GINAC_BIND_UNARCHIVER(su3one);
122 GINAC_BIND_UNARCHIVER(su3t);
123 GINAC_BIND_UNARCHIVER(su3f);
124 GINAC_BIND_UNARCHIVER(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 for (auto & it : v) {
170 if (!is_a<su3one>(it.op(0)))
175 return color(su3one(), representation_label);
177 return hold_ncmul(s);
180 ex color::thiscontainer(const exvector & v) const
182 return color(representation_label, v);
185 ex color::thiscontainer(exvector && v) const
187 return color(representation_label, std::move(v));
190 /** Given a vector iv3 of three indices and a vector iv2 of two indices that
191 * is a subset of iv3, return the (free) index that is in iv3 but not in
192 * iv2 and the sign introduced by permuting that index to the front.
194 * @param iv3 Vector of 3 indices
195 * @param iv2 Vector of 2 indices, must be a subset of iv3
196 * @param sig Returns sign introduced by index permutation
197 * @return the free index (the one that is in iv3 but not in iv2) */
198 static ex permute_free_index_to_front(const exvector & iv3, const exvector & iv2, int & sig)
200 GINAC_ASSERT(iv3.size() == 3);
201 GINAC_ASSERT(iv2.size() == 2);
205 #define TEST_PERMUTATION(A,B,C,P) \
206 if (iv3[B].is_equal(iv2[0]) && iv3[C].is_equal(iv2[1])) { \
211 TEST_PERMUTATION(0,1,2, 1);
212 TEST_PERMUTATION(0,2,1, -1);
213 TEST_PERMUTATION(1,0,2, -1);
214 TEST_PERMUTATION(1,2,0, 1);
215 TEST_PERMUTATION(2,0,1, 1);
216 TEST_PERMUTATION(2,1,0, -1);
218 throw(std::logic_error("permute_free_index_to_front(): no valid permutation found"));
221 /** Automatic symbolic evaluation of indexed symmetric structure constant. */
222 ex su3d::eval_indexed(const basic & i) const
224 GINAC_ASSERT(is_a<indexed>(i));
225 GINAC_ASSERT(i.nops() == 4);
226 GINAC_ASSERT(is_a<su3d>(i.op(0)));
228 // Convolutions are zero
229 if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
232 // Numeric evaluation
233 if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
237 for (unsigned j=0; j<3; j++)
238 v[j] = ex_to<numeric>(ex_to<idx>(i.op(j + 1)).get_value()).to_int();
239 if (v[0] > v[1]) std::swap(v[0], v[1]);
240 if (v[0] > v[2]) std::swap(v[0], v[2]);
241 if (v[1] > v[2]) std::swap(v[1], v[2]);
243 #define CMPINDICES(A,B,C) ((v[0] == (A)) && (v[1] == (B)) && (v[2] == (C)))
245 // Check for non-zero elements
246 if (CMPINDICES(1,4,6) || CMPINDICES(1,5,7) || CMPINDICES(2,5,6)
247 || CMPINDICES(3,4,4) || CMPINDICES(3,5,5))
249 else if (CMPINDICES(2,4,7) || CMPINDICES(3,6,6) || CMPINDICES(3,7,7))
251 else if (CMPINDICES(1,1,8) || CMPINDICES(2,2,8) || CMPINDICES(3,3,8))
252 return sqrt(_ex3)*_ex1_3;
253 else if (CMPINDICES(8,8,8))
254 return sqrt(_ex3)*_ex_1_3;
255 else if (CMPINDICES(4,4,8) || CMPINDICES(5,5,8)
256 || CMPINDICES(6,6,8) || CMPINDICES(7,7,8))
257 return sqrt(_ex3)/_ex_6;
262 // No further simplifications
266 /** Automatic symbolic evaluation of indexed antisymmetric structure constant. */
267 ex su3f::eval_indexed(const basic & i) const
269 GINAC_ASSERT(is_a<indexed>(i));
270 GINAC_ASSERT(i.nops() == 4);
271 GINAC_ASSERT(is_a<su3f>(i.op(0)));
273 // Numeric evaluation
274 if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
276 // Sort indices, remember permutation sign
278 for (unsigned j=0; j<3; j++)
279 v[j] = ex_to<numeric>(ex_to<idx>(i.op(j + 1)).get_value()).to_int();
281 if (v[0] > v[1]) { std::swap(v[0], v[1]); sign = -sign; }
282 if (v[0] > v[2]) { std::swap(v[0], v[2]); sign = -sign; }
283 if (v[1] > v[2]) { std::swap(v[1], v[2]); sign = -sign; }
285 // Check for non-zero elements
286 if (CMPINDICES(1,2,3))
288 else if (CMPINDICES(1,4,7) || CMPINDICES(2,4,6)
289 || CMPINDICES(2,5,7) || CMPINDICES(3,4,5))
290 return _ex1_2 * sign;
291 else if (CMPINDICES(1,5,6) || CMPINDICES(3,6,7))
292 return _ex_1_2 * sign;
293 else if (CMPINDICES(4,5,8) || CMPINDICES(6,7,8))
294 return sqrt(_ex3)/2 * sign;
299 // No further simplifications
304 /** Contraction of generator with something else. */
305 bool su3t::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
307 GINAC_ASSERT(is_a<indexed>(*self));
308 GINAC_ASSERT(is_a<indexed>(*other));
309 GINAC_ASSERT(self->nops() == 2);
310 GINAC_ASSERT(is_a<su3t>(self->op(0)));
311 unsigned char rl = ex_to<color>(*self).get_representation_label();
313 if (is_exactly_a<su3t>(other->op(0))) {
315 // Contraction only makes sense if the representation labels are equal
316 GINAC_ASSERT(is_a<color>(*other));
317 if (ex_to<color>(*other).get_representation_label() != rl)
321 if (other - self == 1) {
322 *self = numeric(4, 3);
323 *other = color_ONE(rl);
326 // T.a T.b T.a = -1/6 T.b
327 } else if (other - self == 2
328 && is_a<color>(self[1])) {
329 *self = numeric(-1, 6);
333 // T.a S T.a = 1/2 Tr(S) - 1/6 S
336 while (it != other) {
337 if (!is_a<color>(*it)) {
345 while (it != other) {
350 *self = color_trace(S, rl) * color_ONE(rl) / 2 - S / 6;
359 /** Contraction of an indexed symmetric structure constant with something else. */
360 bool su3d::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
362 GINAC_ASSERT(is_a<indexed>(*self));
363 GINAC_ASSERT(is_a<indexed>(*other));
364 GINAC_ASSERT(self->nops() == 4);
365 GINAC_ASSERT(is_a<su3d>(self->op(0)));
367 if (is_exactly_a<su3d>(other->op(0))) {
369 // Find the dummy indices of the contraction
370 exvector self_indices = ex_to<indexed>(*self).get_indices();
371 exvector other_indices = ex_to<indexed>(*other).get_indices();
372 exvector all_indices = self_indices;
373 all_indices.insert(all_indices.end(), other_indices.begin(), other_indices.end());
374 exvector free_indices, dummy_indices;
375 find_free_and_dummy(all_indices, free_indices, dummy_indices);
377 // d.abc d.abc = 40/3
378 if (dummy_indices.size() == 3) {
379 *self = numeric(40, 3);
383 // d.akl d.bkl = 5/3 delta.ab
384 } else if (dummy_indices.size() == 2) {
386 std::back_insert_iterator<exvector> ita(a);
387 ita = set_difference(self_indices.begin(), self_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
388 ita = set_difference(other_indices.begin(), other_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
389 GINAC_ASSERT(a.size() == 2);
390 *self = numeric(5, 3) * delta_tensor(a[0], a[1]);
395 } else if (is_exactly_a<su3t>(other->op(0))) {
397 // d.abc T.b T.c = 5/6 T.a
398 if (other+1 != v.end()
399 && is_exactly_a<su3t>(other[1].op(0))
400 && ex_to<indexed>(*self).has_dummy_index_for(other[1].op(1))) {
402 exvector self_indices = ex_to<indexed>(*self).get_indices();
403 exvector dummy_indices = {other[0].op(1), other[1].op(1)};
405 ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
406 *self = numeric(5, 6);
407 other[0] = color_T(a, ex_to<color>(other[0]).get_representation_label());
416 /** Contraction of an indexed antisymmetric structure constant with something else. */
417 bool su3f::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
419 GINAC_ASSERT(is_a<indexed>(*self));
420 GINAC_ASSERT(is_a<indexed>(*other));
421 GINAC_ASSERT(self->nops() == 4);
422 GINAC_ASSERT(is_a<su3f>(self->op(0)));
424 if (is_exactly_a<su3f>(other->op(0))) { // f*d is handled by su3d class
426 // Find the dummy indices of the contraction
427 exvector dummy_indices;
428 dummy_indices = ex_to<indexed>(*self).get_dummy_indices(ex_to<indexed>(*other));
431 if (dummy_indices.size() == 3) {
436 // f.akl f.bkl = 3 delta.ab
437 } else if (dummy_indices.size() == 2) {
439 ex a = permute_free_index_to_front(ex_to<indexed>(*self).get_indices(), dummy_indices, sign1);
440 ex b = permute_free_index_to_front(ex_to<indexed>(*other).get_indices(), dummy_indices, sign2);
441 *self = sign1 * sign2 * 3 * delta_tensor(a, b);
446 } else if (is_exactly_a<su3t>(other->op(0))) {
448 // f.abc T.b T.c = 3/2 I T.a
449 if (other+1 != v.end()
450 && is_exactly_a<su3t>(other[1].op(0))
451 && ex_to<indexed>(*self).has_dummy_index_for(other[1].op(1))) {
453 exvector self_indices = ex_to<indexed>(*self).get_indices();
454 exvector dummy_indices = {other[0].op(1), other[1].op(1)};
456 ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
457 *self = numeric(3, 2) * sig * I;
458 other[0] = color_T(a, ex_to<color>(other[0]).get_representation_label());
471 ex color_ONE(unsigned char rl)
473 static ex ONE = dynallocate<su3one>();
474 return color(ONE, rl);
477 ex color_T(const ex & a, unsigned char rl)
479 static ex t = dynallocate<su3t>();
482 throw(std::invalid_argument("indices of color_T must be of type idx"));
483 if (!ex_to<idx>(a).get_dim().is_equal(8))
484 throw(std::invalid_argument("index dimension for color_T must be 8"));
486 return color(t, a, rl);
489 ex color_f(const ex & a, const ex & b, const ex & c)
491 static ex f = dynallocate<su3f>();
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(f, antisymmetric3(), a, b, c);
501 ex color_d(const ex & a, const ex & b, const ex & c)
503 static ex d = dynallocate<su3d>();
505 if (!is_a<idx>(a) || !is_a<idx>(b) || !is_a<idx>(c))
506 throw(std::invalid_argument("indices of color_d must be of type idx"));
507 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))
508 throw(std::invalid_argument("index dimension for color_d must be 8"));
510 return indexed(d, symmetric3(), a, b, c);
513 ex color_h(const ex & a, const ex & b, const ex & c)
515 return color_d(a, b, c) + I * color_f(a, b, c);
518 /** Check whether a given tinfo key (as returned by return_type_tinfo()
519 * is that of a color object (with an arbitrary representation label). */
520 static bool is_color_tinfo(const return_type_t& ti)
522 return *(ti.tinfo) == typeid(color);
525 /** Extract representation label from tinfo key (as returned by
526 * return_type_tinfo()). */
527 static unsigned char get_representation_label(const return_type_t& ti)
529 return (unsigned char)ti.rl;
532 ex color_trace(const ex & e, const std::set<unsigned char> & rls)
534 if (is_a<color>(e)) {
536 unsigned char rl = ex_to<color>(e).get_representation_label();
538 // Are we taking the trace over this object's representation label?
539 if (rls.find(rl) == rls.end())
542 // Yes, all generators are traceless, except for color_ONE
543 if (is_a<su3one>(e.op(0)))
548 } else if (is_exactly_a<mul>(e)) {
550 // Trace of product: pull out non-color factors
552 for (size_t i=0; i<e.nops(); i++) {
553 const ex &o = e.op(i);
554 if (is_color_tinfo(o.return_type_tinfo()))
555 prod *= color_trace(o, rls);
561 } else if (is_exactly_a<ncmul>(e)) {
563 unsigned char rl = get_representation_label(e.return_type_tinfo());
565 // Are we taking the trace over this string's representation label?
566 if (rls.find(rl) == rls.end())
569 // Yes, expand product if necessary
570 ex e_expanded = e.expand();
571 if (!is_a<ncmul>(e_expanded))
572 return color_trace(e_expanded, rls);
574 size_t num = e.nops();
578 // Tr T_a T_b = 1/2 delta_a_b
579 return delta_tensor(e.op(0).op(1), e.op(1).op(1)) / 2;
581 } else if (num == 3) {
583 // Tr T_a T_b T_c = 1/4 h_a_b_c
584 return color_h(e.op(0).op(1), e.op(1).op(1), e.op(2).op(1)) / 4;
588 // Traces of 4 or more generators are computed recursively:
590 // 1/6 delta_a(n-1)_an Tr T_a1 .. T_a(n-2)
591 // + 1/2 h_a(n-1)_an_k Tr T_a1 .. T_a(n-2) T_k
592 const ex &last_index = e.op(num - 1).op(1);
593 const ex &next_to_last_index = e.op(num - 2).op(1);
594 idx summation_index(dynallocate<symbol>(), 8);
598 for (size_t i=0; i<num-2; i++)
599 v1.push_back(e.op(i));
602 v2.push_back(color_T(summation_index, rl));
604 return delta_tensor(next_to_last_index, last_index) * color_trace(ncmul(v1), rl) / 6
605 + color_h(next_to_last_index, last_index, summation_index) * color_trace(ncmul(v2), rl) / 2;
608 } else if (e.nops() > 0) {
610 // Trace maps to all other container classes (this includes sums)
611 pointer_to_map_function_1arg<const std::set<unsigned char> &> fcn(color_trace, rls);
618 ex color_trace(const ex & e, const lst & rll)
620 // Convert list to set
621 std::set<unsigned char> rls;
622 for (auto & it : rll) {
623 if (it.info(info_flags::nonnegint))
624 rls.insert(ex_to<numeric>(it).to_int());
627 return color_trace(e, rls);
630 ex color_trace(const ex & e, unsigned char rl)
632 // Convert label to set
633 std::set<unsigned char> rls;
636 return color_trace(e, rls);