- symbols can have a LaTeX name, e.g. symbol s("s", "\\sigma");
[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 "print.h"
29 #include "archive.h"
30 #include "debugmsg.h"
31 #include "utils.h"
32
33 #include <stdexcept>
34
35 namespace GiNaC {
36
37 GINAC_IMPLEMENT_REGISTERED_CLASS(clifford, indexed)
38 GINAC_IMPLEMENT_REGISTERED_CLASS(diracone, tensor)
39 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma, tensor)
40 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma5, tensor)
41
42 //////////
43 // default constructor, destructor, copy constructor assignment operator and helpers
44 //////////
45
46 clifford::clifford() : representation_label(0)
47 {
48         debugmsg("clifford default constructor", LOGLEVEL_CONSTRUCT);
49         tinfo_key = TINFO_clifford;
50 }
51
52 void clifford::copy(const clifford & other)
53 {
54         inherited::copy(other);
55         representation_label = other.representation_label;
56 }
57
58 DEFAULT_DESTROY(clifford)
59 DEFAULT_CTORS(diracone)
60 DEFAULT_CTORS(diracgamma)
61 DEFAULT_CTORS(diracgamma5)
62
63 //////////
64 // other constructors
65 //////////
66
67 /** Construct object without any indices. This constructor is for internal
68  *  use only. Use the dirac_ONE() function instead.
69  *  @see dirac_ONE */
70 clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
71 {
72         debugmsg("clifford constructor from ex", LOGLEVEL_CONSTRUCT);
73         tinfo_key = TINFO_clifford;
74 }
75
76 /** Construct object with one Lorentz index. This constructor is for internal
77  *  use only. Use the dirac_gamma() function instead.
78  *  @see dirac_gamma */
79 clifford::clifford(const ex & b, const ex & mu, unsigned char rl) : inherited(b, mu), representation_label(rl)
80 {
81         debugmsg("clifford constructor from ex,ex", LOGLEVEL_CONSTRUCT);
82         GINAC_ASSERT(is_ex_of_type(mu, varidx));
83         tinfo_key = TINFO_clifford;
84 }
85
86 clifford::clifford(unsigned char rl, const exvector & v, bool discardable) : inherited(indexed::unknown, v, discardable), representation_label(rl)
87 {
88         debugmsg("clifford constructor from unsigned char,exvector", LOGLEVEL_CONSTRUCT);
89         tinfo_key = TINFO_clifford;
90 }
91
92 clifford::clifford(unsigned char rl, exvector * vp) : inherited(indexed::unknown, vp), representation_label(rl)
93 {
94         debugmsg("clifford constructor from unsigned char,exvector *", LOGLEVEL_CONSTRUCT);
95         tinfo_key = TINFO_clifford;
96 }
97
98 //////////
99 // archiving
100 //////////
101
102 clifford::clifford(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
103 {
104         debugmsg("clifford constructor from archive_node", LOGLEVEL_CONSTRUCT);
105         unsigned rl;
106         n.find_unsigned("label", rl);
107         representation_label = rl;
108 }
109
110 void clifford::archive(archive_node &n) const
111 {
112         inherited::archive(n);
113         n.add_unsigned("label", representation_label);
114 }
115
116 DEFAULT_UNARCHIVE(clifford)
117 DEFAULT_ARCHIVING(diracone)
118 DEFAULT_ARCHIVING(diracgamma)
119 DEFAULT_ARCHIVING(diracgamma5)
120
121 //////////
122 // functions overriding virtual functions from bases classes
123 //////////
124
125 int clifford::compare_same_type(const basic & other) const
126 {
127         GINAC_ASSERT(other.tinfo() == TINFO_clifford);
128         const clifford &o = static_cast<const clifford &>(other);
129
130         if (representation_label != o.representation_label) {
131                 // different representation label
132                 return representation_label < o.representation_label ? -1 : 1;
133         }
134
135         return inherited::compare_same_type(other);
136 }
137
138 DEFAULT_COMPARE(diracone)
139 DEFAULT_COMPARE(diracgamma)
140 DEFAULT_COMPARE(diracgamma5)
141
142 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
143 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
144 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
145
146 /** Contraction of a gamma matrix with something else. */
147 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
148 {
149         GINAC_ASSERT(is_ex_of_type(*self, clifford));
150         GINAC_ASSERT(is_ex_of_type(*other, indexed));
151         GINAC_ASSERT(is_ex_of_type(self->op(0), diracgamma));
152         unsigned char rl = ex_to_clifford(*self).get_representation_label();
153
154         if (is_ex_of_type(*other, clifford)) {
155
156                 ex dim = ex_to_idx(self->op(1)).get_dim();
157
158                 // gamma~mu gamma.mu = dim ONE
159                 if (other - self == 1) {
160                         *self = dim;
161                         *other = dirac_ONE(rl);
162                         return true;
163
164                 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
165                 } else if (other - self == 2
166                         && is_ex_of_type(self[1], clifford)) {
167                         *self = 2 - dim;
168                         *other = _ex1();
169                         return true;
170
171                 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
172                 } else if (other - self == 3
173                         && is_ex_of_type(self[1], clifford)
174                         && is_ex_of_type(self[2], clifford)) {
175                         *self = 4 * lorentz_g(self[1].op(1), self[2].op(1)) * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
176                         self[1] = _ex1();
177                         self[2] = _ex1();
178                         *other = _ex1();
179                         return true;
180
181 #if 0
182                 // gamma~mu gamma~alpha gamma~beta gamma~delta gamma.mu = -2 gamma~delta gamma~beta gamma~alpha + (4-dim) gamma~alpha gamma~beta gamma~delta
183                 } else if (other - self == 4
184                         && is_ex_of_type(self[1], clifford)
185                         && is_ex_of_type(self[2], clifford)
186                         && is_ex_of_type(self[3], clifford)) {
187                         *self = -2 * self[3] * self[2] * self[1] + (4 - dim) * self[1] * self[2] * self[3];
188                         self[1] = _ex1();
189                         self[2] = _ex1();
190                         self[3] = _ex1();
191                         *other = _ex1();
192                         return true;
193 #endif
194
195                 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
196                 // (commutate contracted indices towards each other, simplify_indexed()
197                 // will re-expand and re-run the simplification)
198                 } else {
199                         exvector::iterator it = self + 1, next_to_last = other - 1;
200                         while (it != other) {
201                                 if (!is_ex_of_type(*it, clifford))
202                                         return false;
203                                 it++;
204                         }
205
206                         it = self + 1;
207                         ex S = _ex1();
208                         while (it != next_to_last) {
209                                 S *= *it;
210                                 *it++ = _ex1();
211                         }
212
213                         *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
214                         *next_to_last = _ex1();
215                         *other = _ex1();
216                         return true;
217                 }
218         }
219
220         return false;
221 }
222
223 /** Perform automatic simplification on noncommutative product of clifford
224  *  objects. This removes superfluous ONEs, permutes gamma5's to the front
225  *  and removes squares of gamma objects. */
226 ex clifford::simplify_ncmul(const exvector & v) const
227 {
228         exvector s;
229         s.reserve(v.size());
230         unsigned rl = ex_to_clifford(v[0]).get_representation_label();
231
232         // Remove superfluous ONEs
233         exvector::const_iterator cit = v.begin(), citend = v.end();
234         while (cit != citend) {
235                 if (!is_ex_of_type(cit->op(0), diracone))
236                         s.push_back(*cit);
237                 cit++;
238         }
239
240         bool something_changed = false;
241         int sign = 1;
242
243         // Anticommute gamma5's to the front
244         if (s.size() >= 2) {
245                 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
246                 while (true) {
247                         exvector::iterator it = next_to_last;
248                         while (true) {
249                                 exvector::iterator it2 = it + 1;
250                                 if (!is_ex_of_type(it->op(0), diracgamma5) && is_ex_of_type(it2->op(0), diracgamma5)) {
251                                         it->swap(*it2);
252                                         sign = -sign;
253                                         something_changed = true;
254                                 }
255                                 if (it == first)
256                                         break;
257                                 it--;
258                         }
259                         if (next_to_last == first)
260                                 break;
261                         next_to_last--;
262                 }
263         }
264
265         // Remove squares of gamma5
266         while (s.size() >= 2 && is_ex_of_type(s[0].op(0), diracgamma5) && is_ex_of_type(s[1].op(0), diracgamma5)) {
267                 s.erase(s.begin(), s.begin() + 2);
268                 something_changed = true;
269         }
270
271         // Remove equal adjacent gammas
272         if (s.size() >= 2) {
273                 exvector::iterator it = s.begin(), itend = s.end() - 1;
274                 while (it != itend) {
275                         ex & a = it[0];
276                         ex & b = it[1];
277                         if (is_ex_of_type(a.op(0), diracgamma) && is_ex_of_type(b.op(0), diracgamma)) {
278                                 const ex & ia = a.op(1);
279                                 const ex & ib = b.op(1);
280                                 if (ia.is_equal(ib)) {
281                                         a = lorentz_g(ia, ib);
282                                         b = dirac_ONE(rl);
283                                         something_changed = true;
284                                 }
285                         }
286                         it++;
287                 }
288         }
289
290         if (s.size() == 0)
291                 return clifford(diracone(), rl) * sign;
292         if (something_changed)
293                 return nonsimplified_ncmul(s) * sign;
294         else
295                 return simplified_ncmul(s) * sign;
296 }
297
298 ex clifford::thisexprseq(const exvector & v) const
299 {
300         return clifford(representation_label, v);
301 }
302
303 ex clifford::thisexprseq(exvector * vp) const
304 {
305         return clifford(representation_label, vp);
306 }
307
308 //////////
309 // global functions
310 //////////
311
312 ex dirac_ONE(unsigned char rl)
313 {
314         return clifford(diracone(), rl);
315 }
316
317 ex dirac_gamma(const ex & mu, unsigned char rl)
318 {
319         if (!is_ex_of_type(mu, varidx))
320                 throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
321
322         return clifford(diracgamma(), mu, rl);
323 }
324
325 ex dirac_gamma5(unsigned char rl)
326 {
327         return clifford(diracgamma5(), rl);
328 }
329
330 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
331 {
332         varidx mu((new symbol)->setflag(status_flags::dynallocated), dim);
333         return indexed(e, mu.toggle_variance()) * dirac_gamma(mu, rl);
334 }
335
336 /** Check whether a given tinfo key (as returned by return_type_tinfo()
337  *  is that of a clifford object with the specified representation label. */
338 static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
339 {
340         return ti == (TINFO_clifford + rl);
341 }
342
343 ex dirac_trace(const ex & e, unsigned char rl)
344 {
345         if (is_ex_of_type(e, clifford)) {
346
347                 if (ex_to_clifford(e).get_representation_label() == rl
348                  && is_ex_of_type(e.op(0), diracone))
349                         return _ex4();
350                 else
351                         return _ex0();
352
353         } else if (is_ex_exactly_of_type(e, add)) {
354
355                 // Trace of sum = sum of traces
356                 ex sum = _ex0();
357                 for (unsigned i=0; i<e.nops(); i++)
358                         sum += dirac_trace(e.op(i), rl);
359                 return sum;
360
361         } else if (is_ex_exactly_of_type(e, mul)) {
362
363                 // Trace of product: pull out non-clifford factors
364                 ex prod = _ex1();
365                 for (unsigned i=0; i<e.nops(); i++) {
366                         const ex &o = e.op(i);
367                         unsigned ti = o.return_type_tinfo();
368                         if (is_clifford_tinfo(o.return_type_tinfo(), rl))
369                                 prod *= dirac_trace(o, rl);
370                         else
371                                 prod *= o;
372                 }
373                 return prod;
374
375         } else if (is_ex_exactly_of_type(e, ncmul)) {
376
377                 if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
378                         return _ex0();
379
380                 // Expand product, if necessary
381                 ex e_expanded = e.expand();
382                 if (!is_ex_of_type(e_expanded, ncmul))
383                         return dirac_trace(e_expanded, rl);
384
385                 // gamma5 gets moved to the front so this check is enough
386                 bool has_gamma5 = is_ex_of_type(e.op(0).op(0), diracgamma5);
387                 unsigned num = e.nops();
388
389                 if (has_gamma5) {
390
391                         // Trace of gamma5 * odd number of gammas and trace of
392                         // gamma5 * gamma.mu * gamma.nu are zero
393                         if ((num & 1) == 0 || num == 3)
394                                 return _ex0();
395
396                         // Tr gamma5 S_2k =
397                         //   epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
398                         ex dim = ex_to_idx(e.op(1).op(1)).get_dim();
399                         varidx mu1((new symbol)->setflag(status_flags::dynallocated), dim),
400                                mu2((new symbol)->setflag(status_flags::dynallocated), dim),
401                                mu3((new symbol)->setflag(status_flags::dynallocated), dim),
402                                mu4((new symbol)->setflag(status_flags::dynallocated), dim);
403                         exvector v;
404                         v.reserve(num + 3);
405                         v.push_back(dirac_gamma(mu1, rl));
406                         v.push_back(dirac_gamma(mu2, rl));
407                         v.push_back(dirac_gamma(mu3, rl));
408                         v.push_back(dirac_gamma(mu4, rl));
409                         for (int i=1; i<num; i++)
410                                 v.push_back(e.op(i));
411
412                         return (eps0123(mu1.toggle_variance(), mu2.toggle_variance(), mu3.toggle_variance(), mu4.toggle_variance()) *
413                                 dirac_trace(ncmul(v), rl)).simplify_indexed() / 24;
414
415                 } else { // no gamma5
416
417                         // Trace of odd number of gammas is zero
418                         if ((num & 1) == 1)
419                                 return _ex0();
420
421                         // Tr gamma.mu gamma.nu = 4 g.mu.nu
422                         if (num == 2)
423                                 return 4 * lorentz_g(e.op(0).op(1), e.op(1).op(1));
424
425                         // Traces of 4 or more gammas are computed recursively:
426                         // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
427                         //   + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
428                         //   - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
429                         //   + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
430                         //   - ...
431                         //   + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
432                         exvector v(num - 2);
433                         int sign = 1;
434                         const ex &ix1 = e.op(0).op(1);
435                         ex result;
436                         for (int i=1; i<num; i++) {
437                                 for (int n=1, j=0; n<num; n++) {
438                                         if (n == i)
439                                                 continue;
440                                         v[j++] = e.op(n);
441                                 }
442                                 result += sign * lorentz_g(ix1, e.op(i).op(1)) * dirac_trace(ncmul(v), rl);
443                                 sign = -sign;
444                         }
445                         return result;
446                 }
447         }
448
449         return _ex0();
450 }
451
452 } // namespace GiNaC