3 * Implementation of GiNaC's color (SU(3) Lie algebra) objects. */
6 * GiNaC Copyright (C) 1999-2001 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 "power.h" // for sqrt()
39 GINAC_IMPLEMENT_REGISTERED_CLASS(color, indexed)
40 GINAC_IMPLEMENT_REGISTERED_CLASS(su3one, tensor)
41 GINAC_IMPLEMENT_REGISTERED_CLASS(su3t, tensor)
42 GINAC_IMPLEMENT_REGISTERED_CLASS(su3f, tensor)
43 GINAC_IMPLEMENT_REGISTERED_CLASS(su3d, tensor)
46 // default constructor, destructor, copy constructor assignment operator and helpers
49 color::color() : representation_label(0)
51 debugmsg("color default constructor", LOGLEVEL_CONSTRUCT);
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 debugmsg("color constructor from ex,unsigned char", LOGLEVEL_CONSTRUCT);
77 tinfo_key = TINFO_color;
80 /** Construct object with one color index. This constructor is for internal
81 * use only. Use the color_T() function instead.
83 color::color(const ex & b, const ex & i1, unsigned char rl) : inherited(b, i1), representation_label(rl)
85 debugmsg("color constructor from ex,ex,unsigned char", LOGLEVEL_CONSTRUCT);
86 tinfo_key = TINFO_color;
89 color::color(unsigned char rl, const exvector & v, bool discardable) : inherited(indexed::unknown, v, discardable), representation_label(rl)
91 debugmsg("color constructor from unsigned char,exvector", LOGLEVEL_CONSTRUCT);
92 tinfo_key = TINFO_color;
95 color::color(unsigned char rl, exvector * vp) : inherited(indexed::unknown, vp), representation_label(rl)
97 debugmsg("color constructor from unsigned char,exvector *", LOGLEVEL_CONSTRUCT);
98 tinfo_key = TINFO_color;
105 color::color(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
107 debugmsg("color constructor from archive_node", LOGLEVEL_CONSTRUCT);
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 bases classes
129 int color::compare_same_type(const basic & other) const
131 GINAC_ASSERT(other.tinfo() == TINFO_color);
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 DEFAULT_COMPARE(su3one)
143 DEFAULT_COMPARE(su3t)
144 DEFAULT_COMPARE(su3f)
145 DEFAULT_COMPARE(su3d)
147 DEFAULT_PRINT_LATEX(su3one, "ONE", "\\mathbb{1}")
148 DEFAULT_PRINT(su3t, "T")
149 DEFAULT_PRINT(su3f, "f")
150 DEFAULT_PRINT(su3d, "d")
152 /** Perform automatic simplification on noncommutative product of color
153 * objects. This removes superfluous ONEs. */
154 ex color::simplify_ncmul(const exvector & v) const
159 // Remove superfluous ONEs
160 exvector::const_iterator it = v.begin(), itend = v.end();
161 while (it != itend) {
162 if (!is_ex_of_type(it->op(0), su3one))
168 return color(su3one(), representation_label);
170 return simplified_ncmul(s);
173 ex color::thisexprseq(const exvector & v) const
175 return color(representation_label, v);
178 ex color::thisexprseq(exvector * vp) const
180 return color(representation_label, vp);
183 /** Given a vector iv3 of three indices and a vector iv2 of two indices that
184 * is a subset of iv3, return the (free) index that is in iv3 but not in
185 * iv2 and the sign introduced by permuting that index to the front.
187 * @param iv3 Vector of 3 indices
188 * @param iv2 Vector of 2 indices, must be a subset of iv3
189 * @param sig Returs sign introduced by index permutation
190 * @return the free index (the one that is in iv3 but not in iv2) */
191 static ex permute_free_index_to_front(const exvector & iv3, const exvector & iv2, int & sig)
193 GINAC_ASSERT(iv3.size() == 3);
194 GINAC_ASSERT(iv2.size() == 2);
198 #define TEST_PERMUTATION(A,B,C,P) \
199 if (iv3[B].is_equal(iv2[0]) && iv3[C].is_equal(iv2[1])) { \
204 TEST_PERMUTATION(0,1,2, 1);
205 TEST_PERMUTATION(0,2,1, -1);
206 TEST_PERMUTATION(1,0,2, -1);
207 TEST_PERMUTATION(1,2,0, 1);
208 TEST_PERMUTATION(2,0,1, 1);
209 TEST_PERMUTATION(2,1,0, -1);
211 throw(std::logic_error("permute_free_index_to_front(): no valid permutation found"));
214 /** Automatic symbolic evaluation of indexed symmetric structure constant. */
215 ex su3d::eval_indexed(const basic & i) const
217 GINAC_ASSERT(is_of_type(i, indexed));
218 GINAC_ASSERT(i.nops() == 4);
219 GINAC_ASSERT(is_ex_of_type(i.op(0), su3d));
221 // Convolutions are zero
222 if (static_cast<const indexed &>(i).get_dummy_indices().size() != 0)
225 // Numeric evaluation
226 if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
230 for (unsigned j=0; j<3; j++)
231 v[j] = ex_to_numeric(ex_to_idx(i.op(j + 1)).get_value()).to_int();
232 if (v[0] > v[1]) std::swap(v[0], v[1]);
233 if (v[0] > v[2]) std::swap(v[0], v[2]);
234 if (v[1] > v[2]) std::swap(v[1], v[2]);
236 #define CMPINDICES(A,B,C) ((v[0] == (A)) && (v[1] == (B)) && (v[2] == (C)))
238 // Check for non-zero elements
239 if (CMPINDICES(1,4,6) || CMPINDICES(1,5,7) || CMPINDICES(2,5,6)
240 || CMPINDICES(3,4,4) || CMPINDICES(3,5,5))
242 else if (CMPINDICES(2,4,7) || CMPINDICES(3,6,6) || CMPINDICES(3,7,7))
244 else if (CMPINDICES(1,1,8) || CMPINDICES(2,2,8) || CMPINDICES(3,3,8))
245 return sqrt(_ex3())/3;
246 else if (CMPINDICES(8,8,8))
247 return -sqrt(_ex3())/3;
248 else if (CMPINDICES(4,4,8) || CMPINDICES(5,5,8)
249 || CMPINDICES(6,6,8) || CMPINDICES(7,7,8))
250 return -sqrt(_ex3())/6;
255 // No further simplifications
259 /** Automatic symbolic evaluation of indexed antisymmetric structure constant. */
260 ex su3f::eval_indexed(const basic & i) const
262 GINAC_ASSERT(is_of_type(i, indexed));
263 GINAC_ASSERT(i.nops() == 4);
264 GINAC_ASSERT(is_ex_of_type(i.op(0), su3f));
266 // Numeric evaluation
267 if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
269 // Sort indices, remember permutation sign
271 for (unsigned j=0; j<3; j++)
272 v[j] = ex_to_numeric(ex_to_idx(i.op(j + 1)).get_value()).to_int();
274 if (v[0] > v[1]) { std::swap(v[0], v[1]); sign = -sign; }
275 if (v[0] > v[2]) { std::swap(v[0], v[2]); sign = -sign; }
276 if (v[1] > v[2]) { std::swap(v[1], v[2]); sign = -sign; }
278 // Check for non-zero elements
279 if (CMPINDICES(1,2,3))
281 else if (CMPINDICES(1,4,7) || CMPINDICES(2,4,6)
282 || CMPINDICES(2,5,7) || CMPINDICES(3,4,5))
283 return _ex1_2() * sign;
284 else if (CMPINDICES(1,5,6) || CMPINDICES(3,6,7))
285 return _ex_1_2() * sign;
286 else if (CMPINDICES(4,5,8) || CMPINDICES(6,7,8))
287 return sqrt(_ex3())/2 * sign;
292 // No further simplifications
297 /** Contraction of generator with something else. */
298 bool su3t::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
300 GINAC_ASSERT(is_ex_of_type(*self, indexed));
301 GINAC_ASSERT(is_ex_of_type(*other, indexed));
302 GINAC_ASSERT(self->nops() == 2);
303 GINAC_ASSERT(is_ex_of_type(self->op(0), su3t));
304 unsigned char rl = ex_to_color(*self).get_representation_label();
306 if (is_ex_exactly_of_type(other->op(0), su3t)) {
309 if (other - self == 1) {
310 *self = numeric(4, 3);
311 *other = color_ONE(rl);
314 // T.a T.b T.a = -1/6 T.b
315 } else if (other - self == 2
316 && is_ex_of_type(self[1], color)) {
317 *self = numeric(-1, 6);
321 // T.a S T.a = 1/2 Tr(S) - 1/6 S
323 exvector::iterator it = self + 1;
324 while (it != other) {
325 if (!is_ex_of_type(*it, color)) {
333 while (it != other) {
338 *self = color_trace(S, rl) * color_ONE(rl) / 2 - S / 6;
347 /** Contraction of an indexed symmetric structure constant with something else. */
348 bool su3d::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
350 GINAC_ASSERT(is_ex_of_type(*self, indexed));
351 GINAC_ASSERT(is_ex_of_type(*other, indexed));
352 GINAC_ASSERT(self->nops() == 4);
353 GINAC_ASSERT(is_ex_of_type(self->op(0), su3d));
355 if (is_ex_exactly_of_type(other->op(0), su3d)) {
357 // Find the dummy indices of the contraction
358 exvector self_indices = ex_to_indexed(*self).get_indices();
359 exvector other_indices = ex_to_indexed(*other).get_indices();
360 exvector all_indices = self_indices;
361 all_indices.insert(all_indices.end(), other_indices.begin(), other_indices.end());
362 exvector free_indices, dummy_indices;
363 find_free_and_dummy(all_indices, free_indices, dummy_indices);
365 // d.abc d.abc = 40/3
366 if (dummy_indices.size() == 3) {
367 *self = numeric(40, 3);
371 // d.akl d.bkl = 5/3 delta.ab
372 } else if (dummy_indices.size() == 2) {
374 back_insert_iterator<exvector> ita(a);
375 ita = set_difference(self_indices.begin(), self_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
376 ita = set_difference(other_indices.begin(), other_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
377 GINAC_ASSERT(a.size() == 2);
378 *self = numeric(5, 3) * delta_tensor(a[0], a[1]);
383 } else if (is_ex_exactly_of_type(other->op(0), su3t)) {
385 // d.abc T.b T.c = 5/6 T.a
386 if (other+1 != v.end()
387 && is_ex_exactly_of_type(other[1].op(0), su3t)
388 && ex_to_indexed(*self).has_dummy_index_for(other[1].op(1))) {
390 exvector self_indices = ex_to_indexed(*self).get_indices();
391 exvector dummy_indices;
392 dummy_indices.push_back(other[0].op(1));
393 dummy_indices.push_back(other[1].op(1));
395 ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
396 *self = numeric(5, 6);
397 other[0] = color_T(a, ex_to_color(other[0]).get_representation_label());
406 /** Contraction of an indexed antisymmetric structure constant with something else. */
407 bool su3f::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
409 GINAC_ASSERT(is_ex_of_type(*self, indexed));
410 GINAC_ASSERT(is_ex_of_type(*other, indexed));
411 GINAC_ASSERT(self->nops() == 4);
412 GINAC_ASSERT(is_ex_of_type(self->op(0), su3f));
414 if (is_ex_exactly_of_type(other->op(0), su3f)) { // f*d is handled by su3d class
416 // Find the dummy indices of the contraction
417 exvector dummy_indices;
418 dummy_indices = ex_to_indexed(*self).get_dummy_indices(ex_to_indexed(*other));
421 if (dummy_indices.size() == 3) {
426 // f.akl f.bkl = 3 delta.ab
427 } else if (dummy_indices.size() == 2) {
429 ex a = permute_free_index_to_front(ex_to_indexed(*self).get_indices(), dummy_indices, sign1);
430 ex b = permute_free_index_to_front(ex_to_indexed(*other).get_indices(), dummy_indices, sign2);
431 *self = sign1 * sign2 * 3 * delta_tensor(a, b);
436 } else if (is_ex_exactly_of_type(other->op(0), su3t)) {
438 // f.abc T.b T.c = 3/2 I T.a
439 if (other+1 != v.end()
440 && is_ex_exactly_of_type(other[1].op(0), su3t)
441 && ex_to_indexed(*self).has_dummy_index_for(other[1].op(1))) {
443 exvector self_indices = ex_to_indexed(*self).get_indices();
444 exvector dummy_indices;
445 dummy_indices.push_back(other[0].op(1));
446 dummy_indices.push_back(other[1].op(1));
448 ex a = permute_free_index_to_front(self_indices, dummy_indices, sig);
449 *self = numeric(3, 2) * sig * I;
450 other[0] = color_T(a, ex_to_color(other[0]).get_representation_label());
463 ex color_ONE(unsigned char rl)
465 return color(su3one(), rl);
468 ex color_T(const ex & a, unsigned char rl)
470 if (!is_ex_of_type(a, idx))
471 throw(std::invalid_argument("indices of color_T must be of type idx"));
472 if (!ex_to_idx(a).get_dim().is_equal(8))
473 throw(std::invalid_argument("index dimension for color_T must be 8"));
475 return color(su3t(), a, rl);
478 ex color_f(const ex & a, const ex & b, const ex & c)
480 if (!is_ex_of_type(a, idx) || !is_ex_of_type(b, idx) || !is_ex_of_type(c, idx))
481 throw(std::invalid_argument("indices of color_f must be of type idx"));
482 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))
483 throw(std::invalid_argument("index dimension for color_f must be 8"));
485 return indexed(su3f(), indexed::antisymmetric, a, b, c);
488 ex color_d(const ex & a, const ex & b, const ex & c)
490 if (!is_ex_of_type(a, idx) || !is_ex_of_type(b, idx) || !is_ex_of_type(c, idx))
491 throw(std::invalid_argument("indices of color_d must be of type idx"));
492 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))
493 throw(std::invalid_argument("index dimension for color_d must be 8"));
495 return indexed(su3d(), indexed::symmetric, a, b, c);
498 ex color_h(const ex & a, const ex & b, const ex & c)
500 return color_d(a, b, c) + I * color_f(a, b, c);
503 /** Check whether a given tinfo key (as returned by return_type_tinfo()
504 * is that of a color object with the specified representation label. */
505 static bool is_color_tinfo(unsigned ti, unsigned char rl)
507 return ti == (TINFO_color + rl);
510 ex color_trace(const ex & e, unsigned char rl)
512 if (is_ex_of_type(e, color)) {
514 if (ex_to_color(e).get_representation_label() == rl
515 && is_ex_of_type(e.op(0), su3one))
520 } else if (is_ex_exactly_of_type(e, add)) {
522 // Trace of sum = sum of traces
524 for (unsigned i=0; i<e.nops(); i++)
525 sum += color_trace(e.op(i), rl);
528 } else if (is_ex_exactly_of_type(e, mul)) {
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_ex_exactly_of_type(e, ncmul)) {
543 if (!is_color_tinfo(e.return_type_tinfo(), rl))
546 // Expand product, if necessary
547 ex e_expanded = e.expand();
548 if (!is_ex_of_type(e_expanded, ncmul))
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 (int 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;