3 * Implementation of GiNaC's color (SU(3) Lie algebra) objects. */
6 * GiNaC Copyright (C) 1999-2002 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
32 #include "power.h" // for sqrt()
40 GINAC_IMPLEMENT_REGISTERED_CLASS(color, indexed)
41 GINAC_IMPLEMENT_REGISTERED_CLASS(su3one, tensor)
42 GINAC_IMPLEMENT_REGISTERED_CLASS(su3t, tensor)
43 GINAC_IMPLEMENT_REGISTERED_CLASS(su3f, tensor)
44 GINAC_IMPLEMENT_REGISTERED_CLASS(su3d, tensor)
47 // default ctor, dtor, copy ctor, assignment operator and helpers
50 color::color() : representation_label(0)
52 tinfo_key = TINFO_color;
55 void color::copy(const color & other)
57 inherited::copy(other);
58 representation_label = other.representation_label;
61 DEFAULT_DESTROY(color)
71 /** Construct object without any color index. This constructor is for
72 * internal use only. Use the color_ONE() function instead.
74 color::color(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
76 tinfo_key = TINFO_color;
79 /** Construct object with one color index. This constructor is for internal
80 * use only. Use the color_T() function instead.
82 color::color(const ex & b, const ex & i1, unsigned char rl) : inherited(b, i1), representation_label(rl)
84 tinfo_key = TINFO_color;
87 color::color(unsigned char rl, const exvector & v, bool discardable) : inherited(sy_none(), v, discardable), representation_label(rl)
89 tinfo_key = TINFO_color;
92 color::color(unsigned char rl, exvector * vp) : inherited(sy_none(), vp), representation_label(rl)
94 tinfo_key = TINFO_color;
101 color::color(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
104 n.find_unsigned("label", rl);
105 representation_label = rl;
108 void color::archive(archive_node &n) const
110 inherited::archive(n);
111 n.add_unsigned("label", representation_label);
114 DEFAULT_UNARCHIVE(color)
115 DEFAULT_ARCHIVING(su3one)
116 DEFAULT_ARCHIVING(su3t)
117 DEFAULT_ARCHIVING(su3f)
118 DEFAULT_ARCHIVING(su3d)
121 // functions overriding virtual functions from base classes
124 int color::compare_same_type(const basic & other) const
126 GINAC_ASSERT(is_a<color>(other));
127 const color &o = static_cast<const color &>(other);
129 if (representation_label != o.representation_label) {
130 // different representation label
131 return representation_label < o.representation_label ? -1 : 1;
134 return inherited::compare_same_type(other);
137 bool color::match_same_type(const basic & other) const
139 GINAC_ASSERT(is_a<color>(other));
140 const color &o = static_cast<const color &>(other);
142 return representation_label == o.representation_label;
145 DEFAULT_COMPARE(su3one)
146 DEFAULT_COMPARE(su3t)
147 DEFAULT_COMPARE(su3f)
148 DEFAULT_COMPARE(su3d)
150 DEFAULT_PRINT_LATEX(su3one, "ONE", "\\mathbb{1}")
151 DEFAULT_PRINT(su3t, "T")
152 DEFAULT_PRINT(su3f, "f")
153 DEFAULT_PRINT(su3d, "d")
155 /** Perform automatic simplification on noncommutative product of color
156 * objects. This removes superfluous ONEs. */
157 ex color::simplify_ncmul(const exvector & v) const
162 // Remove superfluous ONEs
163 exvector::const_iterator it = v.begin(), itend = v.end();
164 while (it != itend) {
165 if (!is_a<su3one>(it->op(0)))
171 return color(su3one(), representation_label);
173 return simplified_ncmul(s);
176 ex color::thisexprseq(const exvector & v) const
178 return color(representation_label, v);
181 ex color::thisexprseq(exvector * vp) const
183 return color(representation_label, vp);
186 /** Given a vector iv3 of three indices and a vector iv2 of two indices that
187 * is a subset of iv3, return the (free) index that is in iv3 but not in
188 * iv2 and the sign introduced by permuting that index to the front.
190 * @param iv3 Vector of 3 indices
191 * @param iv2 Vector of 2 indices, must be a subset of iv3
192 * @param sig Returs sign introduced by index permutation
193 * @return the free index (the one that is in iv3 but not in iv2) */
194 static ex permute_free_index_to_front(const exvector & iv3, const exvector & iv2, int & sig)
196 GINAC_ASSERT(iv3.size() == 3);
197 GINAC_ASSERT(iv2.size() == 2);
201 #define TEST_PERMUTATION(A,B,C,P) \
202 if (iv3[B].is_equal(iv2[0]) && iv3[C].is_equal(iv2[1])) { \
207 TEST_PERMUTATION(0,1,2, 1);
208 TEST_PERMUTATION(0,2,1, -1);
209 TEST_PERMUTATION(1,0,2, -1);
210 TEST_PERMUTATION(1,2,0, 1);
211 TEST_PERMUTATION(2,0,1, 1);
212 TEST_PERMUTATION(2,1,0, -1);
214 throw(std::logic_error("permute_free_index_to_front(): no valid permutation found"));
217 /** Automatic symbolic evaluation of indexed symmetric structure constant. */
218 ex su3d::eval_indexed(const basic & i) const
220 GINAC_ASSERT(is_a<indexed>(i));
221 GINAC_ASSERT(i.nops() == 4);
222 GINAC_ASSERT(is_a<su3d>(i.op(0)));
224 // Convolutions are zero
225 if (!(static_cast<const indexed &>(i).get_dummy_indices().empty()))
228 // Numeric evaluation
229 if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
233 for (unsigned j=0; j<3; j++)
234 v[j] = ex_to<numeric>(ex_to<idx>(i.op(j + 1)).get_value()).to_int();
235 if (v[0] > v[1]) std::swap(v[0], v[1]);
236 if (v[0] > v[2]) std::swap(v[0], v[2]);
237 if (v[1] > v[2]) std::swap(v[1], v[2]);
239 #define CMPINDICES(A,B,C) ((v[0] == (A)) && (v[1] == (B)) && (v[2] == (C)))
241 // Check for non-zero elements
242 if (CMPINDICES(1,4,6) || CMPINDICES(1,5,7) || CMPINDICES(2,5,6)
243 || CMPINDICES(3,4,4) || CMPINDICES(3,5,5))
245 else if (CMPINDICES(2,4,7) || CMPINDICES(3,6,6) || CMPINDICES(3,7,7))
247 else if (CMPINDICES(1,1,8) || CMPINDICES(2,2,8) || CMPINDICES(3,3,8))
248 return sqrt(_ex3)*_ex1_3;
249 else if (CMPINDICES(8,8,8))
250 return sqrt(_ex3)*_ex_1_3;
251 else if (CMPINDICES(4,4,8) || CMPINDICES(5,5,8)
252 || CMPINDICES(6,6,8) || CMPINDICES(7,7,8))
253 return sqrt(_ex3)/_ex_6;
258 // No further simplifications
262 /** Automatic symbolic evaluation of indexed antisymmetric structure constant. */
263 ex su3f::eval_indexed(const basic & i) const
265 GINAC_ASSERT(is_a<indexed>(i));
266 GINAC_ASSERT(i.nops() == 4);
267 GINAC_ASSERT(is_a<su3f>(i.op(0)));
269 // Numeric evaluation
270 if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
272 // Sort indices, remember permutation sign
274 for (unsigned j=0; j<3; j++)
275 v[j] = ex_to<numeric>(ex_to<idx>(i.op(j + 1)).get_value()).to_int();
277 if (v[0] > v[1]) { std::swap(v[0], v[1]); sign = -sign; }
278 if (v[0] > v[2]) { std::swap(v[0], v[2]); sign = -sign; }
279 if (v[1] > v[2]) { std::swap(v[1], v[2]); sign = -sign; }
281 // Check for non-zero elements
282 if (CMPINDICES(1,2,3))
284 else if (CMPINDICES(1,4,7) || CMPINDICES(2,4,6)
285 || CMPINDICES(2,5,7) || CMPINDICES(3,4,5))
286 return _ex1_2 * sign;
287 else if (CMPINDICES(1,5,6) || CMPINDICES(3,6,7))
288 return _ex_1_2 * sign;
289 else if (CMPINDICES(4,5,8) || CMPINDICES(6,7,8))
290 return sqrt(_ex3)/2 * sign;
295 // No further simplifications
300 /** Contraction of generator with something else. */
301 bool su3t::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
303 GINAC_ASSERT(is_a<indexed>(*self));
304 GINAC_ASSERT(is_a<indexed>(*other));
305 GINAC_ASSERT(self->nops() == 2);
306 GINAC_ASSERT(is_a<su3t>(self->op(0)));
307 unsigned char rl = ex_to<color>(*self).get_representation_label();
309 if (is_exactly_a<su3t>(other->op(0))) {
311 // Contraction only makes sense if the represenation labels are equal
312 GINAC_ASSERT(is_a<color>(*other));
313 if (ex_to<color>(*other).get_representation_label() != rl)
317 if (other - self == 1) {
318 *self = numeric(4, 3);
319 *other = color_ONE(rl);
322 // T.a T.b T.a = -1/6 T.b
323 } else if (other - self == 2
324 && is_a<color>(self[1])) {
325 *self = numeric(-1, 6);
329 // T.a S T.a = 1/2 Tr(S) - 1/6 S
331 exvector::iterator it = self + 1;
332 while (it != other) {
333 if (!is_a<color>(*it)) {
341 while (it != other) {
346 *self = color_trace(S, rl) * color_ONE(rl) / 2 - S / 6;
355 /** Contraction of an indexed symmetric structure constant with something else. */
356 bool su3d::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
358 GINAC_ASSERT(is_a<indexed>(*self));
359 GINAC_ASSERT(is_a<indexed>(*other));
360 GINAC_ASSERT(self->nops() == 4);
361 GINAC_ASSERT(is_a<su3d>(self->op(0)));
363 if (is_exactly_a<su3d>(other->op(0))) {
365 // Find the dummy indices of the contraction
366 exvector self_indices = ex_to<indexed>(*self).get_indices();
367 exvector other_indices = ex_to<indexed>(*other).get_indices();
368 exvector all_indices = self_indices;
369 all_indices.insert(all_indices.end(), other_indices.begin(), other_indices.end());
370 exvector free_indices, dummy_indices;
371 find_free_and_dummy(all_indices, free_indices, dummy_indices);
373 // d.abc d.abc = 40/3
374 if (dummy_indices.size() == 3) {
375 *self = numeric(40, 3);
379 // d.akl d.bkl = 5/3 delta.ab
380 } else if (dummy_indices.size() == 2) {
382 std::back_insert_iterator<exvector> ita(a);
383 ita = set_difference(self_indices.begin(), self_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
384 ita = set_difference(other_indices.begin(), other_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
385 GINAC_ASSERT(a.size() == 2);
386 *self = numeric(5, 3) * delta_tensor(a[0], a[1]);
391 } else if (is_exactly_a<su3t>(other->op(0))) {
393 // d.abc T.b T.c = 5/6 T.a
394 if (other+1 != v.end()
395 && is_exactly_a<su3t>(other[1].op(0))
396 && ex_to<indexed>(*self).has_dummy_index_for(other[1].op(1))) {
398 exvector self_indices = ex_to<indexed>(*self).get_indices();
399 exvector dummy_indices;
400 dummy_indices.push_back(other[0].op(1));
401 dummy_indices.push_back(other[1].op(1));
403 ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
404 *self = numeric(5, 6);
405 other[0] = color_T(a, ex_to<color>(other[0]).get_representation_label());
414 /** Contraction of an indexed antisymmetric structure constant with something else. */
415 bool su3f::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
417 GINAC_ASSERT(is_a<indexed>(*self));
418 GINAC_ASSERT(is_a<indexed>(*other));
419 GINAC_ASSERT(self->nops() == 4);
420 GINAC_ASSERT(is_a<su3f>(self->op(0)));
422 if (is_exactly_a<su3f>(other->op(0))) { // f*d is handled by su3d class
424 // Find the dummy indices of the contraction
425 exvector dummy_indices;
426 dummy_indices = ex_to<indexed>(*self).get_dummy_indices(ex_to<indexed>(*other));
429 if (dummy_indices.size() == 3) {
434 // f.akl f.bkl = 3 delta.ab
435 } else if (dummy_indices.size() == 2) {
437 ex a = permute_free_index_to_front(ex_to<indexed>(*self).get_indices(), dummy_indices, sign1);
438 ex b = permute_free_index_to_front(ex_to<indexed>(*other).get_indices(), dummy_indices, sign2);
439 *self = sign1 * sign2 * 3 * delta_tensor(a, b);
444 } else if (is_exactly_a<su3t>(other->op(0))) {
446 // f.abc T.b T.c = 3/2 I T.a
447 if (other+1 != v.end()
448 && is_exactly_a<su3t>(other[1].op(0))
449 && ex_to<indexed>(*self).has_dummy_index_for(other[1].op(1))) {
451 exvector self_indices = ex_to<indexed>(*self).get_indices();
452 exvector dummy_indices;
453 dummy_indices.push_back(other[0].op(1));
454 dummy_indices.push_back(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 return color(su3one(), rl);
476 ex color_T(const ex & a, unsigned char rl)
479 throw(std::invalid_argument("indices of color_T must be of type idx"));
480 if (!ex_to<idx>(a).get_dim().is_equal(8))
481 throw(std::invalid_argument("index dimension for color_T must be 8"));
483 return color(su3t(), a, rl);
486 ex color_f(const ex & a, const ex & b, const ex & c)
488 if (!is_a<idx>(a) || !is_a<idx>(b) || !is_a<idx>(c))
489 throw(std::invalid_argument("indices of color_f must be of type idx"));
490 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))
491 throw(std::invalid_argument("index dimension for color_f must be 8"));
493 return indexed(su3f(), sy_anti(), a, b, c);
496 ex color_d(const ex & a, const ex & b, const ex & c)
498 if (!is_a<idx>(a) || !is_a<idx>(b) || !is_a<idx>(c))
499 throw(std::invalid_argument("indices of color_d must be of type idx"));
500 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))
501 throw(std::invalid_argument("index dimension for color_d must be 8"));
503 return indexed(su3d(), sy_symm(), a, b, c);
506 ex color_h(const ex & a, const ex & b, const ex & c)
508 return color_d(a, b, c) + I * color_f(a, b, c);
511 /** Check whether a given tinfo key (as returned by return_type_tinfo()
512 * is that of a color object with the specified representation label. */
513 static bool is_color_tinfo(unsigned ti, unsigned char rl)
515 return ti == (TINFO_color + rl);
518 ex color_trace(const ex & e, unsigned char rl)
520 if (is_a<color>(e)) {
522 if (ex_to<color>(e).get_representation_label() == rl
523 && is_a<su3one>(e.op(0)))
528 } else if (is_exactly_a<mul>(e)) {
530 // Trace of product: pull out non-color factors
532 for (unsigned i=0; i<e.nops(); i++) {
533 const ex &o = e.op(i);
534 if (is_color_tinfo(o.return_type_tinfo(), rl))
535 prod *= color_trace(o, rl);
541 } else if (is_exactly_a<ncmul>(e)) {
543 if (!is_color_tinfo(e.return_type_tinfo(), rl))
546 // Expand product, if necessary
547 ex e_expanded = e.expand();
548 if (!is_a<ncmul>(e_expanded))
549 return color_trace(e_expanded, rl);
551 unsigned num = e.nops();
555 // Tr T_a T_b = 1/2 delta_a_b
556 return delta_tensor(e.op(0).op(1), e.op(1).op(1)) / 2;
558 } else if (num == 3) {
560 // Tr T_a T_b T_c = 1/4 h_a_b_c
561 return color_h(e.op(0).op(1), e.op(1).op(1), e.op(2).op(1)) / 4;
565 // Traces of 4 or more generators are computed recursively:
567 // 1/6 delta_a(n-1)_an Tr T_a1 .. T_a(n-2)
568 // + 1/2 h_a(n-1)_an_k Tr T_a1 .. T_a(n-2) T_k
569 const ex &last_index = e.op(num - 1).op(1);
570 const ex &next_to_last_index = e.op(num - 2).op(1);
571 idx summation_index((new symbol)->setflag(status_flags::dynallocated), 8);
575 for (unsigned i=0; i<num-2; i++)
576 v1.push_back(e.op(i));
579 v2.push_back(color_T(summation_index, rl));
581 return delta_tensor(next_to_last_index, last_index) * color_trace(ncmul(v1), rl) / 6
582 + color_h(next_to_last_index, last_index, summation_index) * color_trace(ncmul(v2), rl) / 2;
585 } else if (e.nops() > 0) {
587 // Trace maps to all other container classes (this includes sums)
588 pointer_to_map_function_1arg<unsigned char> fcn(color_trace, rl);