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)
42 GINAC_IMPLEMENT_REGISTERED_CLASS(su3one, tensor)
43 GINAC_IMPLEMENT_REGISTERED_CLASS(su3t, tensor)
44 GINAC_IMPLEMENT_REGISTERED_CLASS(su3f, tensor)
45 GINAC_IMPLEMENT_REGISTERED_CLASS(su3d, tensor)
48 // default constructors
51 color::color() : representation_label(0)
53 tinfo_key = TINFO_color;
65 /** Construct object without any color index. This constructor is for
66 * internal use only. Use the color_ONE() function instead.
68 color::color(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
70 tinfo_key = TINFO_color;
73 /** Construct object with one color index. This constructor is for internal
74 * use only. Use the color_T() function instead.
76 color::color(const ex & b, const ex & i1, unsigned char rl) : inherited(b, i1), representation_label(rl)
78 tinfo_key = TINFO_color;
81 color::color(unsigned char rl, const exvector & v, bool discardable) : inherited(sy_none(), v, discardable), representation_label(rl)
83 tinfo_key = TINFO_color;
86 color::color(unsigned char rl, exvector * vp) : inherited(sy_none(), vp), representation_label(rl)
88 tinfo_key = TINFO_color;
95 color::color(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
98 n.find_unsigned("label", rl);
99 representation_label = rl;
102 void color::archive(archive_node &n) const
104 inherited::archive(n);
105 n.add_unsigned("label", representation_label);
108 DEFAULT_UNARCHIVE(color)
109 DEFAULT_ARCHIVING(su3one)
110 DEFAULT_ARCHIVING(su3t)
111 DEFAULT_ARCHIVING(su3f)
112 DEFAULT_ARCHIVING(su3d)
115 // functions overriding virtual functions from base classes
118 int color::compare_same_type(const basic & other) const
120 GINAC_ASSERT(is_a<color>(other));
121 const color &o = static_cast<const color &>(other);
123 if (representation_label != o.representation_label) {
124 // different representation label
125 return representation_label < o.representation_label ? -1 : 1;
128 return inherited::compare_same_type(other);
131 bool color::match_same_type(const basic & other) const
133 GINAC_ASSERT(is_a<color>(other));
134 const color &o = static_cast<const color &>(other);
136 return representation_label == o.representation_label;
139 DEFAULT_COMPARE(su3one)
140 DEFAULT_COMPARE(su3t)
141 DEFAULT_COMPARE(su3f)
142 DEFAULT_COMPARE(su3d)
144 DEFAULT_PRINT_LATEX(su3one, "ONE", "\\mathbb{1}")
145 DEFAULT_PRINT(su3t, "T")
146 DEFAULT_PRINT(su3f, "f")
147 DEFAULT_PRINT(su3d, "d")
149 /** Perform automatic simplification on noncommutative product of color
150 * objects. This removes superfluous ONEs. */
151 ex color::eval_ncmul(const exvector & v) const
156 // Remove superfluous ONEs
157 exvector::const_iterator it = v.begin(), itend = v.end();
158 while (it != itend) {
159 if (!is_a<su3one>(it->op(0)))
165 return color(su3one(), representation_label);
167 return hold_ncmul(s);
170 ex color::thiscontainer(const exvector & v) const
172 return color(representation_label, v);
175 ex color::thiscontainer(exvector * vp) const
177 return color(representation_label, vp);
180 /** Given a vector iv3 of three indices and a vector iv2 of two indices that
181 * is a subset of iv3, return the (free) index that is in iv3 but not in
182 * iv2 and the sign introduced by permuting that index to the front.
184 * @param iv3 Vector of 3 indices
185 * @param iv2 Vector of 2 indices, must be a subset of iv3
186 * @param sig Returs sign introduced by index permutation
187 * @return the free index (the one that is in iv3 but not in iv2) */
188 static ex permute_free_index_to_front(const exvector & iv3, const exvector & iv2, int & sig)
190 GINAC_ASSERT(iv3.size() == 3);
191 GINAC_ASSERT(iv2.size() == 2);
195 #define TEST_PERMUTATION(A,B,C,P) \
196 if (iv3[B].is_equal(iv2[0]) && iv3[C].is_equal(iv2[1])) { \
201 TEST_PERMUTATION(0,1,2, 1);
202 TEST_PERMUTATION(0,2,1, -1);
203 TEST_PERMUTATION(1,0,2, -1);
204 TEST_PERMUTATION(1,2,0, 1);
205 TEST_PERMUTATION(2,0,1, 1);
206 TEST_PERMUTATION(2,1,0, -1);
208 throw(std::logic_error("permute_free_index_to_front(): no valid permutation found"));
211 /** Automatic symbolic evaluation of indexed symmetric structure constant. */
212 ex su3d::eval_indexed(const basic & i) const
214 GINAC_ASSERT(is_a<indexed>(i));
215 GINAC_ASSERT(i.nops() == 4);
216 GINAC_ASSERT(is_a<su3d>(i.op(0)));
218 // Convolutions are zero
219 if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
222 // Numeric evaluation
223 if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
227 for (unsigned j=0; j<3; j++)
228 v[j] = ex_to<numeric>(ex_to<idx>(i.op(j + 1)).get_value()).to_int();
229 if (v[0] > v[1]) std::swap(v[0], v[1]);
230 if (v[0] > v[2]) std::swap(v[0], v[2]);
231 if (v[1] > v[2]) std::swap(v[1], v[2]);
233 #define CMPINDICES(A,B,C) ((v[0] == (A)) && (v[1] == (B)) && (v[2] == (C)))
235 // Check for non-zero elements
236 if (CMPINDICES(1,4,6) || CMPINDICES(1,5,7) || CMPINDICES(2,5,6)
237 || CMPINDICES(3,4,4) || CMPINDICES(3,5,5))
239 else if (CMPINDICES(2,4,7) || CMPINDICES(3,6,6) || CMPINDICES(3,7,7))
241 else if (CMPINDICES(1,1,8) || CMPINDICES(2,2,8) || CMPINDICES(3,3,8))
242 return sqrt(_ex3)*_ex1_3;
243 else if (CMPINDICES(8,8,8))
244 return sqrt(_ex3)*_ex_1_3;
245 else if (CMPINDICES(4,4,8) || CMPINDICES(5,5,8)
246 || CMPINDICES(6,6,8) || CMPINDICES(7,7,8))
247 return sqrt(_ex3)/_ex_6;
252 // No further simplifications
256 /** Automatic symbolic evaluation of indexed antisymmetric structure constant. */
257 ex su3f::eval_indexed(const basic & i) const
259 GINAC_ASSERT(is_a<indexed>(i));
260 GINAC_ASSERT(i.nops() == 4);
261 GINAC_ASSERT(is_a<su3f>(i.op(0)));
263 // Numeric evaluation
264 if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
266 // Sort indices, remember permutation sign
268 for (unsigned j=0; j<3; j++)
269 v[j] = ex_to<numeric>(ex_to<idx>(i.op(j + 1)).get_value()).to_int();
271 if (v[0] > v[1]) { std::swap(v[0], v[1]); sign = -sign; }
272 if (v[0] > v[2]) { std::swap(v[0], v[2]); sign = -sign; }
273 if (v[1] > v[2]) { std::swap(v[1], v[2]); sign = -sign; }
275 // Check for non-zero elements
276 if (CMPINDICES(1,2,3))
278 else if (CMPINDICES(1,4,7) || CMPINDICES(2,4,6)
279 || CMPINDICES(2,5,7) || CMPINDICES(3,4,5))
280 return _ex1_2 * sign;
281 else if (CMPINDICES(1,5,6) || CMPINDICES(3,6,7))
282 return _ex_1_2 * sign;
283 else if (CMPINDICES(4,5,8) || CMPINDICES(6,7,8))
284 return sqrt(_ex3)/2 * sign;
289 // No further simplifications
294 /** Contraction of generator with something else. */
295 bool su3t::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
297 GINAC_ASSERT(is_a<indexed>(*self));
298 GINAC_ASSERT(is_a<indexed>(*other));
299 GINAC_ASSERT(self->nops() == 2);
300 GINAC_ASSERT(is_a<su3t>(self->op(0)));
301 unsigned char rl = ex_to<color>(*self).get_representation_label();
303 if (is_exactly_a<su3t>(other->op(0))) {
305 // Contraction only makes sense if the represenation labels are equal
306 GINAC_ASSERT(is_a<color>(*other));
307 if (ex_to<color>(*other).get_representation_label() != rl)
311 if (other - self == 1) {
312 *self = numeric(4, 3);
313 *other = color_ONE(rl);
316 // T.a T.b T.a = -1/6 T.b
317 } else if (other - self == 2
318 && is_a<color>(self[1])) {
319 *self = numeric(-1, 6);
323 // T.a S T.a = 1/2 Tr(S) - 1/6 S
325 exvector::iterator it = self + 1;
326 while (it != other) {
327 if (!is_a<color>(*it)) {
335 while (it != other) {
340 *self = color_trace(S, rl) * color_ONE(rl) / 2 - S / 6;
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
352 GINAC_ASSERT(is_a<indexed>(*self));
353 GINAC_ASSERT(is_a<indexed>(*other));
354 GINAC_ASSERT(self->nops() == 4);
355 GINAC_ASSERT(is_a<su3d>(self->op(0)));
357 if (is_exactly_a<su3d>(other->op(0))) {
359 // Find the dummy indices of the contraction
360 exvector self_indices = ex_to<indexed>(*self).get_indices();
361 exvector other_indices = ex_to<indexed>(*other).get_indices();
362 exvector all_indices = self_indices;
363 all_indices.insert(all_indices.end(), other_indices.begin(), other_indices.end());
364 exvector free_indices, dummy_indices;
365 find_free_and_dummy(all_indices, free_indices, dummy_indices);
367 // d.abc d.abc = 40/3
368 if (dummy_indices.size() == 3) {
369 *self = numeric(40, 3);
373 // d.akl d.bkl = 5/3 delta.ab
374 } else if (dummy_indices.size() == 2) {
376 std::back_insert_iterator<exvector> ita(a);
377 ita = set_difference(self_indices.begin(), self_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
378 ita = set_difference(other_indices.begin(), other_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
379 GINAC_ASSERT(a.size() == 2);
380 *self = numeric(5, 3) * delta_tensor(a[0], a[1]);
385 } else if (is_exactly_a<su3t>(other->op(0))) {
387 // d.abc T.b T.c = 5/6 T.a
388 if (other+1 != v.end()
389 && is_exactly_a<su3t>(other[1].op(0))
390 && ex_to<indexed>(*self).has_dummy_index_for(other[1].op(1))) {
392 exvector self_indices = ex_to<indexed>(*self).get_indices();
393 exvector dummy_indices;
394 dummy_indices.push_back(other[0].op(1));
395 dummy_indices.push_back(other[1].op(1));
397 ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
398 *self = numeric(5, 6);
399 other[0] = color_T(a, ex_to<color>(other[0]).get_representation_label());
408 /** Contraction of an indexed antisymmetric structure constant with something else. */
409 bool su3f::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
411 GINAC_ASSERT(is_a<indexed>(*self));
412 GINAC_ASSERT(is_a<indexed>(*other));
413 GINAC_ASSERT(self->nops() == 4);
414 GINAC_ASSERT(is_a<su3f>(self->op(0)));
416 if (is_exactly_a<su3f>(other->op(0))) { // f*d is handled by su3d class
418 // Find the dummy indices of the contraction
419 exvector dummy_indices;
420 dummy_indices = ex_to<indexed>(*self).get_dummy_indices(ex_to<indexed>(*other));
423 if (dummy_indices.size() == 3) {
428 // f.akl f.bkl = 3 delta.ab
429 } else if (dummy_indices.size() == 2) {
431 ex a = permute_free_index_to_front(ex_to<indexed>(*self).get_indices(), dummy_indices, sign1);
432 ex b = permute_free_index_to_front(ex_to<indexed>(*other).get_indices(), dummy_indices, sign2);
433 *self = sign1 * sign2 * 3 * delta_tensor(a, b);
438 } else if (is_exactly_a<su3t>(other->op(0))) {
440 // f.abc T.b T.c = 3/2 I T.a
441 if (other+1 != v.end()
442 && is_exactly_a<su3t>(other[1].op(0))
443 && ex_to<indexed>(*self).has_dummy_index_for(other[1].op(1))) {
445 exvector self_indices = ex_to<indexed>(*self).get_indices();
446 exvector dummy_indices;
447 dummy_indices.push_back(other[0].op(1));
448 dummy_indices.push_back(other[1].op(1));
450 ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
451 *self = numeric(3, 2) * sig * I;
452 other[0] = color_T(a, ex_to<color>(other[0]).get_representation_label());
465 ex color_ONE(unsigned char rl)
467 return color(su3one(), rl);
470 ex color_T(const ex & a, unsigned char rl)
473 throw(std::invalid_argument("indices of color_T must be of type idx"));
474 if (!ex_to<idx>(a).get_dim().is_equal(8))
475 throw(std::invalid_argument("index dimension for color_T must be 8"));
477 return color(su3t(), a, rl);
480 ex color_f(const ex & a, const ex & b, const ex & c)
482 if (!is_a<idx>(a) || !is_a<idx>(b) || !is_a<idx>(c))
483 throw(std::invalid_argument("indices of color_f must be of type idx"));
484 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))
485 throw(std::invalid_argument("index dimension for color_f must be 8"));
487 return indexed(su3f(), sy_anti(), a, b, c);
490 ex color_d(const ex & a, const ex & b, const ex & c)
492 if (!is_a<idx>(a) || !is_a<idx>(b) || !is_a<idx>(c))
493 throw(std::invalid_argument("indices of color_d must be of type idx"));
494 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))
495 throw(std::invalid_argument("index dimension for color_d must be 8"));
497 return indexed(su3d(), sy_symm(), a, b, c);
500 ex color_h(const ex & a, const ex & b, const ex & c)
502 return color_d(a, b, c) + I * color_f(a, b, c);
505 /** Check whether a given tinfo key (as returned by return_type_tinfo()
506 * is that of a color object with the specified representation label. */
507 static bool is_color_tinfo(unsigned ti, unsigned char rl)
509 return ti == (TINFO_color + rl);
512 ex color_trace(const ex & e, unsigned char rl)
514 if (is_a<color>(e)) {
516 if (ex_to<color>(e).get_representation_label() == rl
517 && is_a<su3one>(e.op(0)))
522 } else if (is_exactly_a<mul>(e)) {
524 // Trace of product: pull out non-color factors
526 for (size_t i=0; i<e.nops(); i++) {
527 const ex &o = e.op(i);
528 if (is_color_tinfo(o.return_type_tinfo(), rl))
529 prod *= color_trace(o, rl);
535 } else if (is_exactly_a<ncmul>(e)) {
537 if (!is_color_tinfo(e.return_type_tinfo(), rl))
540 // Expand product, if necessary
541 ex e_expanded = e.expand();
542 if (!is_a<ncmul>(e_expanded))
543 return color_trace(e_expanded, rl);
545 size_t num = e.nops();
549 // Tr T_a T_b = 1/2 delta_a_b
550 return delta_tensor(e.op(0).op(1), e.op(1).op(1)) / 2;
552 } else if (num == 3) {
554 // Tr T_a T_b T_c = 1/4 h_a_b_c
555 return color_h(e.op(0).op(1), e.op(1).op(1), e.op(2).op(1)) / 4;
559 // Traces of 4 or more generators are computed recursively:
561 // 1/6 delta_a(n-1)_an Tr T_a1 .. T_a(n-2)
562 // + 1/2 h_a(n-1)_an_k Tr T_a1 .. T_a(n-2) T_k
563 const ex &last_index = e.op(num - 1).op(1);
564 const ex &next_to_last_index = e.op(num - 2).op(1);
565 idx summation_index((new symbol)->setflag(status_flags::dynallocated), 8);
569 for (size_t i=0; i<num-2; i++)
570 v1.push_back(e.op(i));
573 v2.push_back(color_T(summation_index, rl));
575 return delta_tensor(next_to_last_index, last_index) * color_trace(ncmul(v1), rl) / 6
576 + color_h(next_to_last_index, last_index, summation_index) * color_trace(ncmul(v2), rl) / 2;
579 } else if (e.nops() > 0) {
581 // Trace maps to all other container classes (this includes sums)
582 pointer_to_map_function_1arg<unsigned char> fcn(color_trace, rl);