- simplify_indexed() renames dummy indices so, e.g., "a.i*a.i+a.j*a.j" gets
[ginac.git] / ginac / clifford.cpp
1 /** @file clifford.cpp
2  *
3  *  Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
4
5 /*
6  *  GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
7  *
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.
12  *
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.
17  *
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
21  */
22
23 #include "clifford.h"
24 #include "ex.h"
25 #include "idx.h"
26 #include "ncmul.h"
27 #include "symbol.h"
28 #include "numeric.h" // for I
29 #include "print.h"
30 #include "archive.h"
31 #include "debugmsg.h"
32 #include "utils.h"
33
34 #include <stdexcept>
35
36 namespace GiNaC {
37
38 GINAC_IMPLEMENT_REGISTERED_CLASS(clifford, indexed)
39 GINAC_IMPLEMENT_REGISTERED_CLASS(diracone, tensor)
40 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma, tensor)
41 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma5, tensor)
42
43 //////////
44 // default constructor, destructor, copy constructor assignment operator and helpers
45 //////////
46
47 clifford::clifford() : representation_label(0)
48 {
49         debugmsg("clifford default constructor", LOGLEVEL_CONSTRUCT);
50         tinfo_key = TINFO_clifford;
51 }
52
53 void clifford::copy(const clifford & other)
54 {
55         inherited::copy(other);
56         representation_label = other.representation_label;
57 }
58
59 DEFAULT_DESTROY(clifford)
60 DEFAULT_CTORS(diracone)
61 DEFAULT_CTORS(diracgamma)
62 DEFAULT_CTORS(diracgamma5)
63
64 //////////
65 // other constructors
66 //////////
67
68 /** Construct object without any indices. This constructor is for internal
69  *  use only. Use the dirac_ONE() function instead.
70  *  @see dirac_ONE */
71 clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
72 {
73         debugmsg("clifford constructor from ex", LOGLEVEL_CONSTRUCT);
74         tinfo_key = TINFO_clifford;
75 }
76
77 /** Construct object with one Lorentz index. This constructor is for internal
78  *  use only. Use the dirac_gamma() function instead.
79  *  @see dirac_gamma */
80 clifford::clifford(const ex & b, const ex & mu, unsigned char rl) : inherited(b, mu), representation_label(rl)
81 {
82         debugmsg("clifford constructor from ex,ex", LOGLEVEL_CONSTRUCT);
83         GINAC_ASSERT(is_ex_of_type(mu, varidx));
84         tinfo_key = TINFO_clifford;
85 }
86
87 clifford::clifford(unsigned char rl, const exvector & v, bool discardable) : inherited(indexed::unknown, v, discardable), representation_label(rl)
88 {
89         debugmsg("clifford constructor from unsigned char,exvector", LOGLEVEL_CONSTRUCT);
90         tinfo_key = TINFO_clifford;
91 }
92
93 clifford::clifford(unsigned char rl, exvector * vp) : inherited(indexed::unknown, vp), representation_label(rl)
94 {
95         debugmsg("clifford constructor from unsigned char,exvector *", LOGLEVEL_CONSTRUCT);
96         tinfo_key = TINFO_clifford;
97 }
98
99 //////////
100 // archiving
101 //////////
102
103 clifford::clifford(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
104 {
105         debugmsg("clifford constructor from archive_node", LOGLEVEL_CONSTRUCT);
106         unsigned rl;
107         n.find_unsigned("label", rl);
108         representation_label = rl;
109 }
110
111 void clifford::archive(archive_node &n) const
112 {
113         inherited::archive(n);
114         n.add_unsigned("label", representation_label);
115 }
116
117 DEFAULT_UNARCHIVE(clifford)
118 DEFAULT_ARCHIVING(diracone)
119 DEFAULT_ARCHIVING(diracgamma)
120 DEFAULT_ARCHIVING(diracgamma5)
121
122 //////////
123 // functions overriding virtual functions from bases classes
124 //////////
125
126 int clifford::compare_same_type(const basic & other) const
127 {
128         GINAC_ASSERT(other.tinfo() == TINFO_clifford);
129         const clifford &o = static_cast<const clifford &>(other);
130
131         if (representation_label != o.representation_label) {
132                 // different representation label
133                 return representation_label < o.representation_label ? -1 : 1;
134         }
135
136         return inherited::compare_same_type(other);
137 }
138
139 DEFAULT_COMPARE(diracone)
140 DEFAULT_COMPARE(diracgamma)
141 DEFAULT_COMPARE(diracgamma5)
142
143 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
144 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
145 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
146
147 /** Contraction of a gamma matrix with something else. */
148 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
149 {
150         GINAC_ASSERT(is_ex_of_type(*self, clifford));
151         GINAC_ASSERT(is_ex_of_type(*other, indexed));
152         GINAC_ASSERT(is_ex_of_type(self->op(0), diracgamma));
153         unsigned char rl = ex_to_clifford(*self).get_representation_label();
154
155         if (is_ex_of_type(*other, clifford)) {
156
157                 ex dim = ex_to_idx(self->op(1)).get_dim();
158
159                 // gamma~mu gamma.mu = dim ONE
160                 if (other - self == 1) {
161                         *self = dim;
162                         *other = dirac_ONE(rl);
163                         return true;
164
165                 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
166                 } else if (other - self == 2
167                         && is_ex_of_type(self[1], clifford)) {
168                         *self = 2 - dim;
169                         *other = _ex1();
170                         return true;
171
172                 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
173                 } else if (other - self == 3
174                         && is_ex_of_type(self[1], clifford)
175                         && is_ex_of_type(self[2], clifford)) {
176                         *self = 4 * lorentz_g(self[1].op(1), self[2].op(1)) * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
177                         self[1] = _ex1();
178                         self[2] = _ex1();
179                         *other = _ex1();
180                         return true;
181
182                 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
183                 // (commutate contracted indices towards each other, simplify_indexed()
184                 // will re-expand and re-run the simplification)
185                 } else {
186                         exvector::iterator it = self + 1, next_to_last = other - 1;
187                         while (it != other) {
188                                 if (!is_ex_of_type(*it, clifford))
189                                         return false;
190                                 it++;
191                         }
192
193                         it = self + 1;
194                         ex S = _ex1();
195                         while (it != next_to_last) {
196                                 S *= *it;
197                                 *it++ = _ex1();
198                         }
199
200                         *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
201                         *next_to_last = _ex1();
202                         *other = _ex1();
203                         return true;
204                 }
205         }
206
207         return false;
208 }
209
210 /** Perform automatic simplification on noncommutative product of clifford
211  *  objects. This removes superfluous ONEs, permutes gamma5's to the front
212  *  and removes squares of gamma objects. */
213 ex clifford::simplify_ncmul(const exvector & v) const
214 {
215         exvector s;
216         s.reserve(v.size());
217         unsigned rl = ex_to_clifford(v[0]).get_representation_label();
218
219         // Remove superfluous ONEs
220         exvector::const_iterator cit = v.begin(), citend = v.end();
221         while (cit != citend) {
222                 if (!is_ex_of_type(cit->op(0), diracone))
223                         s.push_back(*cit);
224                 cit++;
225         }
226
227         bool something_changed = false;
228         int sign = 1;
229
230         // Anticommute gamma5's to the front
231         if (s.size() >= 2) {
232                 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
233                 while (true) {
234                         exvector::iterator it = next_to_last;
235                         while (true) {
236                                 exvector::iterator it2 = it + 1;
237                                 if (!is_ex_of_type(it->op(0), diracgamma5) && is_ex_of_type(it2->op(0), diracgamma5)) {
238                                         it->swap(*it2);
239                                         sign = -sign;
240                                         something_changed = true;
241                                 }
242                                 if (it == first)
243                                         break;
244                                 it--;
245                         }
246                         if (next_to_last == first)
247                                 break;
248                         next_to_last--;
249                 }
250         }
251
252         // Remove squares of gamma5
253         while (s.size() >= 2 && is_ex_of_type(s[0].op(0), diracgamma5) && is_ex_of_type(s[1].op(0), diracgamma5)) {
254                 s.erase(s.begin(), s.begin() + 2);
255                 something_changed = true;
256         }
257
258         // Remove equal adjacent gammas
259         if (s.size() >= 2) {
260                 exvector::iterator it = s.begin(), itend = s.end() - 1;
261                 while (it != itend) {
262                         ex & a = it[0];
263                         ex & b = it[1];
264                         if (is_ex_of_type(a.op(0), diracgamma) && is_ex_of_type(b.op(0), diracgamma)) {
265                                 const ex & ia = a.op(1);
266                                 const ex & ib = b.op(1);
267                                 if (ia.is_equal(ib)) {
268                                         a = lorentz_g(ia, ib);
269                                         b = dirac_ONE(rl);
270                                         something_changed = true;
271                                 }
272                         }
273                         it++;
274                 }
275         }
276
277         if (s.size() == 0)
278                 return clifford(diracone(), rl) * sign;
279         if (something_changed)
280                 return nonsimplified_ncmul(s) * sign;
281         else
282                 return simplified_ncmul(s) * sign;
283 }
284
285 ex clifford::thisexprseq(const exvector & v) const
286 {
287         return clifford(representation_label, v);
288 }
289
290 ex clifford::thisexprseq(exvector * vp) const
291 {
292         return clifford(representation_label, vp);
293 }
294
295 //////////
296 // global functions
297 //////////
298
299 ex dirac_ONE(unsigned char rl)
300 {
301         return clifford(diracone(), rl);
302 }
303
304 ex dirac_gamma(const ex & mu, unsigned char rl)
305 {
306         if (!is_ex_of_type(mu, varidx))
307                 throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
308
309         return clifford(diracgamma(), mu, rl);
310 }
311
312 ex dirac_gamma5(unsigned char rl)
313 {
314         return clifford(diracgamma5(), rl);
315 }
316
317 ex dirac_gamma6(unsigned char rl)
318 {
319         return clifford(diracone(), rl) + clifford(diracgamma5(), rl);
320 }
321
322 ex dirac_gamma7(unsigned char rl)
323 {
324         return clifford(diracone(), rl) - clifford(diracgamma5(), rl);
325 }
326
327 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
328 {
329         varidx mu((new symbol)->setflag(status_flags::dynallocated), dim);
330         return indexed(e, mu.toggle_variance()) * dirac_gamma(mu, rl);
331 }
332
333 /** Check whether a given tinfo key (as returned by return_type_tinfo()
334  *  is that of a clifford object with the specified representation label. */
335 static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
336 {
337         return ti == (TINFO_clifford + rl);
338 }
339
340 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
341 {
342         if (is_ex_of_type(e, clifford)) {
343
344                 if (ex_to_clifford(e).get_representation_label() == rl
345                  && is_ex_of_type(e.op(0), diracone))
346                         return trONE;
347                 else
348                         return _ex0();
349
350         } else if (is_ex_exactly_of_type(e, add)) {
351
352                 // Trace of sum = sum of traces
353                 ex sum = _ex0();
354                 for (unsigned i=0; i<e.nops(); i++)
355                         sum += dirac_trace(e.op(i), rl, trONE);
356                 return sum;
357
358         } else if (is_ex_exactly_of_type(e, mul)) {
359
360                 // Trace of product: pull out non-clifford factors
361                 ex prod = _ex1();
362                 for (unsigned i=0; i<e.nops(); i++) {
363                         const ex &o = e.op(i);
364                         unsigned ti = o.return_type_tinfo();
365                         if (is_clifford_tinfo(o.return_type_tinfo(), rl))
366                                 prod *= dirac_trace(o, rl, trONE);
367                         else
368                                 prod *= o;
369                 }
370                 return prod;
371
372         } else if (is_ex_exactly_of_type(e, ncmul)) {
373
374                 if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
375                         return _ex0();
376
377                 // Expand product, if necessary
378                 ex e_expanded = e.expand();
379                 if (!is_ex_of_type(e_expanded, ncmul))
380                         return dirac_trace(e_expanded, rl, trONE);
381
382                 // gamma5 gets moved to the front so this check is enough
383                 bool has_gamma5 = is_ex_of_type(e.op(0).op(0), diracgamma5);
384                 unsigned num = e.nops();
385
386                 if (has_gamma5) {
387
388                         // Trace of gamma5 * odd number of gammas and trace of
389                         // gamma5 * gamma.mu * gamma.nu are zero
390                         if ((num & 1) == 0 || num == 3)
391                                 return _ex0();
392
393                         // Tr gamma5 S_2k =
394                         //   I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
395                         ex dim = ex_to_idx(e.op(1).op(1)).get_dim();
396                         varidx mu1((new symbol)->setflag(status_flags::dynallocated), dim),
397                                mu2((new symbol)->setflag(status_flags::dynallocated), dim),
398                                mu3((new symbol)->setflag(status_flags::dynallocated), dim),
399                                mu4((new symbol)->setflag(status_flags::dynallocated), dim);
400                         exvector v;
401                         v.reserve(num + 3);
402                         v.push_back(dirac_gamma(mu1, rl));
403                         v.push_back(dirac_gamma(mu2, rl));
404                         v.push_back(dirac_gamma(mu3, rl));
405                         v.push_back(dirac_gamma(mu4, rl));
406                         for (int i=1; i<num; i++)
407                                 v.push_back(e.op(i));
408
409                         return (eps0123(mu1.toggle_variance(), mu2.toggle_variance(), mu3.toggle_variance(), mu4.toggle_variance()) *
410                                 dirac_trace(ncmul(v), rl, trONE)).simplify_indexed() * I / 24;
411
412                 } else { // no gamma5
413
414                         // Trace of odd number of gammas is zero
415                         if ((num & 1) == 1)
416                                 return _ex0();
417
418                         // Tr gamma.mu gamma.nu = 4 g.mu.nu
419                         if (num == 2)
420                                 return trONE * lorentz_g(e.op(0).op(1), e.op(1).op(1));
421
422                         // Traces of 4 or more gammas are computed recursively:
423                         // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
424                         //   + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
425                         //   - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
426                         //   + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
427                         //   - ...
428                         //   + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
429                         exvector v(num - 2);
430                         int sign = 1;
431                         const ex &ix1 = e.op(0).op(1);
432                         ex result;
433                         for (int i=1; i<num; i++) {
434                                 for (int n=1, j=0; n<num; n++) {
435                                         if (n == i)
436                                                 continue;
437                                         v[j++] = e.op(n);
438                                 }
439                                 result += sign * lorentz_g(ix1, e.op(i).op(1)) * dirac_trace(ncmul(v), rl, trONE);
440                                 sign = -sign;
441                         }
442                         return result;
443                 }
444         }
445
446         return _ex0();
447 }
448
449 ex canonicalize_clifford(const ex & e)
450 {
451         if (is_ex_exactly_of_type(e, add)) {
452
453                 ex sum = _ex0();
454                 for (unsigned i=0; i<e.nops(); i++)
455                         sum += canonicalize_clifford(e.op(i));
456                 return sum;
457
458         } else if (is_ex_exactly_of_type(e, mul)) {
459
460                 ex prod = _ex1();
461                 for (unsigned i=0; i<e.nops(); i++)
462                         prod *= canonicalize_clifford(e.op(i));
463                 return prod;
464
465         } else if (is_ex_exactly_of_type(e, ncmul)) {
466
467                 // Expand product, if necessary
468                 ex e_expanded = e.expand();
469                 if (!is_ex_of_type(e_expanded, ncmul))
470                         return canonicalize_clifford(e_expanded);
471
472                 if (!is_ex_of_type(e.op(0), clifford))
473                         return e;
474
475                 exvector v;
476                 v.reserve(e.nops());
477                 for (int i=0; i<e.nops(); i++)
478                         v.push_back(e.op(i));
479
480                 // Stupid bubble sort because we only want to swap adjacent gammas
481                 exvector::iterator itstart = v.begin(), itend = v.end(), next_to_last = itend - 1;
482                 if (is_ex_of_type(itstart->op(0), diracgamma5))
483                         itstart++;
484                 while (next_to_last != itstart) {
485                         exvector::iterator it = itstart;
486                         while (it != next_to_last) {
487                                 if (it[0].op(1).compare(it[1].op(1)) > 0) {
488                                         ex save0 = it[0], save1 = it[1];
489                                         it[0] = lorentz_g(it[0].op(1), it[1].op(1));
490                                         it[1] = _ex2();
491                                         ex sum = ncmul(v);
492                                         it[0] = save1;
493                                         it[1] = save0;
494                                         sum -= ncmul(v);
495                                         return canonicalize_clifford(sum);
496                                 }
497                                 it++;
498                         }
499                         next_to_last--;
500                 }
501                 return ncmul(v);
502         }
503
504         return e;
505 }
506
507 } // namespace GiNaC