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
31 #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 constructor, destructor, copy constructor assignment operator and helpers
50 color::color() : representation_label(0)
52 debugmsg("color default constructor", LOGLEVEL_CONSTRUCT);
53 tinfo_key = TINFO_color;
56 void color::copy(const color & other)
58 inherited::copy(other);
59 representation_label = other.representation_label;
62 DEFAULT_DESTROY(color)
72 /** Construct object without any color index. This constructor is for
73 * internal use only. Use the color_ONE() function instead.
75 color::color(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
77 debugmsg("color constructor from ex,unsigned char", LOGLEVEL_CONSTRUCT);
78 tinfo_key = TINFO_color;
81 /** Construct object with one color index. This constructor is for internal
82 * use only. Use the color_T() function instead.
84 color::color(const ex & b, const ex & i1, unsigned char rl) : inherited(b, i1), representation_label(rl)
86 debugmsg("color constructor from ex,ex,unsigned char", LOGLEVEL_CONSTRUCT);
87 tinfo_key = TINFO_color;
90 color::color(unsigned char rl, const exvector & v, bool discardable) : inherited(indexed::unknown, v, discardable), representation_label(rl)
92 debugmsg("color constructor from unsigned char,exvector", LOGLEVEL_CONSTRUCT);
93 tinfo_key = TINFO_color;
96 color::color(unsigned char rl, exvector * vp) : inherited(indexed::unknown, vp), representation_label(rl)
98 debugmsg("color constructor from unsigned char,exvector *", LOGLEVEL_CONSTRUCT);
99 tinfo_key = TINFO_color;
106 color::color(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
108 debugmsg("color constructor from archive_node", LOGLEVEL_CONSTRUCT);
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 DEFAULT_UNARCHIVE(color)
121 DEFAULT_ARCHIVING(su3one)
122 DEFAULT_ARCHIVING(su3t)
123 DEFAULT_ARCHIVING(su3f)
124 DEFAULT_ARCHIVING(su3d)
127 // functions overriding virtual functions from bases classes
130 int color::compare_same_type(const basic & other) const
132 GINAC_ASSERT(other.tinfo() == TINFO_color);
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 DEFAULT_COMPARE(su3one)
144 DEFAULT_COMPARE(su3t)
145 DEFAULT_COMPARE(su3f)
146 DEFAULT_COMPARE(su3d)
148 DEFAULT_PRINT_LATEX(su3one, "ONE", "\\mathbb{1}")
149 DEFAULT_PRINT(su3t, "T")
150 DEFAULT_PRINT(su3f, "f")
151 DEFAULT_PRINT(su3d, "d")
153 /** Perform automatic simplification on noncommutative product of color
154 * objects. This removes superfluous ONEs. */
155 ex color::simplify_ncmul(const exvector & v) const
160 // Remove superfluous ONEs
161 exvector::const_iterator it = v.begin(), itend = v.end();
162 while (it != itend) {
163 if (!is_ex_of_type(it->op(0), su3one))
169 return color(su3one(), representation_label);
171 return simplified_ncmul(s);
174 ex color::thisexprseq(const exvector & v) const
176 return color(representation_label, v);
179 ex color::thisexprseq(exvector * vp) const
181 return color(representation_label, vp);
184 /** Given a vector iv3 of three indices and a vector iv2 of two indices that
185 * is a subset of iv3, return the (free) index that is in iv3 but not in
186 * iv2 and the sign introduced by permuting that index to the front.
188 * @param iv3 Vector of 3 indices
189 * @param iv2 Vector of 2 indices, must be a subset of iv3
190 * @param sig Returs sign introduced by index permutation
191 * @return the free index (the one that is in iv3 but not in iv2) */
192 static ex permute_free_index_to_front(const exvector & iv3, const exvector & iv2, int & sig)
194 GINAC_ASSERT(iv3.size() == 3);
195 GINAC_ASSERT(iv2.size() == 2);
199 #define TEST_PERMUTATION(A,B,C,P) \
200 if (iv3[B].is_equal(iv2[0]) && iv3[C].is_equal(iv2[1])) { \
205 TEST_PERMUTATION(0,1,2, 1);
206 TEST_PERMUTATION(0,2,1, -1);
207 TEST_PERMUTATION(1,0,2, -1);
208 TEST_PERMUTATION(1,2,0, 1);
209 TEST_PERMUTATION(2,0,1, 1);
210 TEST_PERMUTATION(2,1,0, -1);
212 throw(std::logic_error("permute_free_index_to_front(): no valid permutation found"));
215 /** Automatic symbolic evaluation of indexed symmetric structure constant. */
216 ex su3d::eval_indexed(const basic & i) const
218 GINAC_ASSERT(is_of_type(i, indexed));
219 GINAC_ASSERT(i.nops() == 4);
220 GINAC_ASSERT(is_ex_of_type(i.op(0), su3d));
222 // Convolutions are zero
223 if (static_cast<const indexed &>(i).get_dummy_indices().size() != 0)
226 // Numeric evaluation
227 if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
231 for (unsigned j=0; j<3; j++)
232 v[j] = ex_to_numeric(ex_to_idx(i.op(j + 1)).get_value()).to_int();
233 if (v[0] > v[1]) std::swap(v[0], v[1]);
234 if (v[0] > v[2]) std::swap(v[0], v[2]);
235 if (v[1] > v[2]) std::swap(v[1], v[2]);
237 #define CMPINDICES(A,B,C) ((v[0] == (A)) && (v[1] == (B)) && (v[2] == (C)))
239 // Check for non-zero elements
240 if (CMPINDICES(1,4,6) || CMPINDICES(1,5,7) || CMPINDICES(2,5,6)
241 || CMPINDICES(3,4,4) || CMPINDICES(3,5,5))
243 else if (CMPINDICES(2,4,7) || CMPINDICES(3,6,6) || CMPINDICES(3,7,7))
245 else if (CMPINDICES(1,1,8) || CMPINDICES(2,2,8) || CMPINDICES(3,3,8))
246 return sqrt(_ex3())/3;
247 else if (CMPINDICES(8,8,8))
248 return -sqrt(_ex3())/3;
249 else if (CMPINDICES(4,4,8) || CMPINDICES(5,5,8)
250 || CMPINDICES(6,6,8) || CMPINDICES(7,7,8))
251 return -sqrt(_ex3())/6;
256 // No further simplifications
260 /** Automatic symbolic evaluation of indexed antisymmetric structure constant. */
261 ex su3f::eval_indexed(const basic & i) const
263 GINAC_ASSERT(is_of_type(i, indexed));
264 GINAC_ASSERT(i.nops() == 4);
265 GINAC_ASSERT(is_ex_of_type(i.op(0), su3f));
267 // Numeric evaluation
268 if (static_cast<const indexed &>(i).all_index_values_are(info_flags::nonnegint)) {
270 // Sort indices, remember permutation sign
272 for (unsigned j=0; j<3; j++)
273 v[j] = ex_to_numeric(ex_to_idx(i.op(j + 1)).get_value()).to_int();
275 if (v[0] > v[1]) { std::swap(v[0], v[1]); sign = -sign; }
276 if (v[0] > v[2]) { std::swap(v[0], v[2]); sign = -sign; }
277 if (v[1] > v[2]) { std::swap(v[1], v[2]); sign = -sign; }
279 // Check for non-zero elements
280 if (CMPINDICES(1,2,3))
282 else if (CMPINDICES(1,4,7) || CMPINDICES(2,4,6)
283 || CMPINDICES(2,5,7) || CMPINDICES(3,4,5))
284 return _ex1_2() * sign;
285 else if (CMPINDICES(1,5,6) || CMPINDICES(3,6,7))
286 return _ex_1_2() * sign;
287 else if (CMPINDICES(4,5,8) || CMPINDICES(6,7,8))
288 return sqrt(_ex3())/2 * sign;
293 // No further simplifications
298 /** Contraction of generator with something else. */
299 bool su3t::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
301 GINAC_ASSERT(is_ex_of_type(*self, indexed));
302 GINAC_ASSERT(is_ex_of_type(*other, indexed));
303 GINAC_ASSERT(self->nops() == 2);
304 GINAC_ASSERT(is_ex_of_type(self->op(0), su3t));
305 unsigned char rl = ex_to_color(*self).get_representation_label();
307 if (is_ex_exactly_of_type(other->op(0), su3t)) {
310 if (other - self == 1) {
311 *self = numeric(4, 3);
312 *other = color_ONE(rl);
315 // T.a T.b T.a = -1/6 T.b
316 } else if (other - self == 2
317 && is_ex_of_type(self[1], color)) {
318 *self = numeric(-1, 6);
322 // T.a S T.a = 1/2 Tr(S) - 1/6 S
324 exvector::iterator it = self + 1;
325 while (it != other) {
326 if (!is_ex_of_type(*it, color)) {
334 while (it != other) {
339 *self = color_trace(S, rl) * color_ONE(rl) / 2 - S / 6;
348 /** Contraction of an indexed symmetric structure constant with something else. */
349 bool su3d::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
351 GINAC_ASSERT(is_ex_of_type(*self, indexed));
352 GINAC_ASSERT(is_ex_of_type(*other, indexed));
353 GINAC_ASSERT(self->nops() == 4);
354 GINAC_ASSERT(is_ex_of_type(self->op(0), su3d));
356 if (is_ex_exactly_of_type(other->op(0), su3d)) {
358 // Find the dummy indices of the contraction
359 exvector self_indices = ex_to_indexed(*self).get_indices();
360 exvector other_indices = ex_to_indexed(*other).get_indices();
361 exvector all_indices = self_indices;
362 all_indices.insert(all_indices.end(), other_indices.begin(), other_indices.end());
363 exvector free_indices, dummy_indices;
364 find_free_and_dummy(all_indices, free_indices, dummy_indices);
367 if (dummy_indices.size() == 3) {
368 *self = numeric(40, 3);
372 // d.akl*d.bkl=5/3*delta.ab
373 } else if (dummy_indices.size() == 2) {
375 back_insert_iterator<exvector> ita(a);
376 ita = set_difference(self_indices.begin(), self_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
377 ita = set_difference(other_indices.begin(), other_indices.end(), dummy_indices.begin(), dummy_indices.end(), ita, ex_is_less());
378 GINAC_ASSERT(a.size() == 2);
379 *self = numeric(5, 3) * delta_tensor(a[0], a[1]);
388 /** Contraction of an indexed antisymmetric structure constant with something else. */
389 bool su3f::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
391 GINAC_ASSERT(is_ex_of_type(*self, indexed));
392 GINAC_ASSERT(is_ex_of_type(*other, indexed));
393 GINAC_ASSERT(self->nops() == 4);
394 GINAC_ASSERT(is_ex_of_type(self->op(0), su3f));
396 if (is_ex_exactly_of_type(other->op(0), su3f)) { // f*d is handled by su3d class
398 // Find the dummy indices of the contraction
399 exvector dummy_indices;
400 dummy_indices = ex_to_indexed(*self).get_dummy_indices(ex_to_indexed(*other));
403 if (dummy_indices.size() == 3) {
408 // f.akl*f.bkl=3*delta.ab
409 } else if (dummy_indices.size() == 2) {
411 ex a = permute_free_index_to_front(ex_to_indexed(*self).get_indices(), dummy_indices, sign1);
412 ex b = permute_free_index_to_front(ex_to_indexed(*other).get_indices(), dummy_indices, sign2);
413 *self = sign1 * sign2 * 3 * delta_tensor(a, b);
426 ex color_ONE(unsigned char rl)
428 return color(su3one(), rl);
431 ex color_T(const ex & a, unsigned char rl)
433 if (!is_ex_of_type(a, idx))
434 throw(std::invalid_argument("indices of color_T must be of type idx"));
435 if (!ex_to_idx(a).get_dim().is_equal(8))
436 throw(std::invalid_argument("index dimension for color_T must be 8"));
438 return color(su3t(), a, rl);
441 ex color_f(const ex & a, const ex & b, const ex & c)
443 if (!is_ex_of_type(a, idx) || !is_ex_of_type(b, idx) || !is_ex_of_type(c, idx))
444 throw(std::invalid_argument("indices of color_f must be of type idx"));
445 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))
446 throw(std::invalid_argument("index dimension for color_f must be 8"));
448 return indexed(su3f(), indexed::antisymmetric, a, b, c);
451 ex color_d(const ex & a, const ex & b, const ex & c)
453 if (!is_ex_of_type(a, idx) || !is_ex_of_type(b, idx) || !is_ex_of_type(c, idx))
454 throw(std::invalid_argument("indices of color_d must be of type idx"));
455 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))
456 throw(std::invalid_argument("index dimension for color_d must be 8"));
458 return indexed(su3d(), indexed::symmetric, a, b, c);
461 ex color_h(const ex & a, const ex & b, const ex & c)
463 return color_d(a, b, c) + I * color_f(a, b, c);
466 /** Check whether a given tinfo key (as returned by return_type_tinfo()
467 * is that of a color object with the specified representation label. */
468 static bool is_color_tinfo(unsigned ti, unsigned char rl)
470 return ti == (TINFO_color + rl);
473 ex color_trace(const ex & e, unsigned char rl)
475 if (is_ex_of_type(e, color)) {
477 if (ex_to_color(e).get_representation_label() == rl
478 && is_ex_of_type(e.op(0), su3one))
483 } else if (is_ex_exactly_of_type(e, add)) {
485 // Trace of sum = sum of traces
487 for (unsigned i=0; i<e.nops(); i++)
488 sum += color_trace(e.op(i), rl);
491 } else if (is_ex_exactly_of_type(e, mul)) {
493 // Trace of product: pull out non-color factors
495 for (unsigned i=0; i<e.nops(); i++) {
496 const ex &o = e.op(i);
497 if (is_color_tinfo(o.return_type_tinfo(), rl))
498 prod *= color_trace(o, rl);
504 } else if (is_ex_exactly_of_type(e, ncmul)) {
506 if (!is_color_tinfo(e.return_type_tinfo(), rl))
509 // Expand product, if necessary
510 ex e_expanded = e.expand();
511 if (!is_ex_of_type(e_expanded, ncmul))
512 return color_trace(e_expanded, rl);
514 unsigned num = e.nops();
518 // Tr T_a T_b = 1/2 delta_a_b
519 return delta_tensor(e.op(0).op(1), e.op(1).op(1)) / 2;
521 } else if (num == 3) {
523 // Tr T_a T_b T_c = 1/4 h_a_b_c
524 return color_h(e.op(0).op(1), e.op(1).op(1), e.op(2).op(1)) / 4;
528 // Traces of 4 or more generators are computed recursively:
530 // 1/6 delta_a(n-1)_an Tr T_a1 .. T_a(n-2)
531 // + 1/2 h_a(n-1)_an_k Tr T_a1 .. T_a(n-2) T_k
532 const ex &last_index = e.op(num - 1).op(1);
533 const ex &next_to_last_index = e.op(num - 2).op(1);
534 idx summation_index((new symbol)->setflag(status_flags::dynallocated), 8);
538 for (int i=0; i<num-2; i++)
539 v1.push_back(e.op(i));
542 v2.push_back(color_T(summation_index, rl));
544 return delta_tensor(next_to_last_index, last_index) * color_trace(ncmul(v1), rl) / 6
545 + color_h(next_to_last_index, last_index, summation_index) * color_trace(ncmul(v2), rl) / 2;