* Headers only include <iosfwd> from now on, since that contains all the
[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 <iostream>
24 #include <stdexcept>
25
26 #include "clifford.h"
27 #include "ex.h"
28 #include "idx.h"
29 #include "ncmul.h"
30 #include "symbol.h"
31 #include "numeric.h" // for I
32 #include "symmetry.h"
33 #include "lst.h"
34 #include "relational.h"
35 #include "mul.h"
36 #include "print.h"
37 #include "archive.h"
38 #include "utils.h"
39
40 namespace GiNaC {
41
42 GINAC_IMPLEMENT_REGISTERED_CLASS(clifford, indexed)
43 GINAC_IMPLEMENT_REGISTERED_CLASS(diracone, tensor)
44 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma, tensor)
45 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma5, tensor)
46
47 //////////
48 // default ctor, dtor, copy ctor, assignment operator and helpers
49 //////////
50
51 clifford::clifford() : representation_label(0)
52 {
53         tinfo_key = TINFO_clifford;
54 }
55
56 void clifford::copy(const clifford & other)
57 {
58         inherited::copy(other);
59         representation_label = other.representation_label;
60 }
61
62 DEFAULT_DESTROY(clifford)
63 DEFAULT_CTORS(diracone)
64 DEFAULT_CTORS(diracgamma)
65 DEFAULT_CTORS(diracgamma5)
66
67 //////////
68 // other constructors
69 //////////
70
71 /** Construct object without any indices. This constructor is for internal
72  *  use only. Use the dirac_ONE() function instead.
73  *  @see dirac_ONE */
74 clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
75 {
76         tinfo_key = TINFO_clifford;
77 }
78
79 /** Construct object with one Lorentz index. This constructor is for internal
80  *  use only. Use the dirac_gamma() function instead.
81  *  @see dirac_gamma */
82 clifford::clifford(const ex & b, const ex & mu, unsigned char rl) : inherited(b, mu), representation_label(rl)
83 {
84         GINAC_ASSERT(is_a<varidx>(mu));
85         tinfo_key = TINFO_clifford;
86 }
87
88 clifford::clifford(unsigned char rl, const exvector & v, bool discardable) : inherited(sy_none(), v, discardable), representation_label(rl)
89 {
90         tinfo_key = TINFO_clifford;
91 }
92
93 clifford::clifford(unsigned char rl, exvector * vp) : inherited(sy_none(), vp), representation_label(rl)
94 {
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         unsigned rl;
105         n.find_unsigned("label", rl);
106         representation_label = rl;
107 }
108
109 void clifford::archive(archive_node &n) const
110 {
111         inherited::archive(n);
112         n.add_unsigned("label", representation_label);
113 }
114
115 DEFAULT_UNARCHIVE(clifford)
116 DEFAULT_ARCHIVING(diracone)
117 DEFAULT_ARCHIVING(diracgamma)
118 DEFAULT_ARCHIVING(diracgamma5)
119
120 //////////
121 // functions overriding virtual functions from base classes
122 //////////
123
124 int clifford::compare_same_type(const basic & other) const
125 {
126         GINAC_ASSERT(is_a<clifford>(other));
127         const clifford &o = static_cast<const clifford &>(other);
128
129         if (representation_label != o.representation_label) {
130                 // different representation label
131                 return representation_label < o.representation_label ? -1 : 1;
132         }
133
134         return inherited::compare_same_type(other);
135 }
136
137 bool clifford::match_same_type(const basic & other) const
138 {
139         GINAC_ASSERT(is_a<clifford>(other));
140         const clifford &o = static_cast<const clifford &>(other);
141
142         return representation_label == o.representation_label;
143 }
144
145 void clifford::print(const print_context & c, unsigned level) const
146 {
147         if (!is_a<diracgamma5>(seq[0]) && !is_a<diracgamma>(seq[0]) && !is_a<diracone>(seq[0])) {
148
149                 // dirac_slash() object is printed differently
150                 if (is_a<print_tree>(c))
151                         inherited::print(c, level);
152                 else if (is_a<print_latex>(c)) {
153                         c.s << "{";
154                         seq[0].print(c, level);
155                         c.s << "\\hspace{-1.0ex}/}";
156                 } else {
157                         seq[0].print(c, level);
158                         c.s << "\\";
159                 }
160
161         } else
162                 inherited::print(c, level);
163 }
164
165 DEFAULT_COMPARE(diracone)
166 DEFAULT_COMPARE(diracgamma)
167 DEFAULT_COMPARE(diracgamma5)
168
169 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
170 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
171 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
172
173 /** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
174 static void base_and_index(const ex & c, ex & b, ex & i)
175 {
176         GINAC_ASSERT(is_a<clifford>(c));
177         GINAC_ASSERT(c.nops() == 2);
178
179         if (is_a<diracgamma>(c.op(0))) { // proper dirac gamma object
180                 i = c.op(1);
181                 b = _ex1;
182         } else { // slash object, generate new dummy index
183                 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(c.op(1)).get_dim());
184                 b = indexed(c.op(0), ix.toggle_variance());
185                 i = ix;
186         }
187 }
188
189 /** Contraction of a gamma matrix with something else. */
190 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
191 {
192         GINAC_ASSERT(is_a<clifford>(*self));
193         GINAC_ASSERT(is_a<indexed>(*other));
194         GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
195         unsigned char rl = ex_to<clifford>(*self).get_representation_label();
196
197         if (is_a<clifford>(*other)) {
198
199                 ex dim = ex_to<idx>(self->op(1)).get_dim();
200
201                 // gamma~mu gamma.mu = dim ONE
202                 if (other - self == 1) {
203                         *self = dim;
204                         *other = dirac_ONE(rl);
205                         return true;
206
207                 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
208                 } else if (other - self == 2
209                         && is_a<clifford>(self[1])) {
210                         *self = 2 - dim;
211                         *other = _ex1;
212                         return true;
213
214                 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
215                 } else if (other - self == 3
216                         && is_a<clifford>(self[1])
217                         && is_a<clifford>(self[2])) {
218                         ex b1, i1, b2, i2;
219                         base_and_index(self[1], b1, i1);
220                         base_and_index(self[2], b2, i2);
221                         *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
222                         self[1] = _ex1;
223                         self[2] = _ex1;
224                         *other = _ex1;
225                         return true;
226
227                 // gamma~mu gamma~alpha gamma~beta gamma~delta gamma.mu = -2 gamma~delta gamma~beta gamma~alpha - (dim-4) gamam~alpha gamma~beta gamma~delta
228                 } else if (other - self == 4
229                         && is_a<clifford>(self[1])
230                         && is_a<clifford>(self[2])
231                         && is_a<clifford>(self[3])) {
232                         *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3];
233                         self[1] = _ex1;
234                         self[2] = _ex1;
235                         self[3] = _ex1;
236                         *other = _ex1;
237                         return true;
238
239                 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
240                 // (commutate contracted indices towards each other, simplify_indexed()
241                 // will re-expand and re-run the simplification)
242                 } else {
243                         exvector::iterator it = self + 1, next_to_last = other - 1;
244                         while (it != other) {
245                                 if (!is_a<clifford>(*it))
246                                         return false;
247                                 ++it;
248                         }
249
250                         it = self + 1;
251                         ex S = _ex1;
252                         while (it != next_to_last) {
253                                 S *= *it;
254                                 *it++ = _ex1;
255                         }
256
257                         *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
258                         *next_to_last = _ex1;
259                         *other = _ex1;
260                         return true;
261                 }
262         }
263
264         return false;
265 }
266
267 /** Perform automatic simplification on noncommutative product of clifford
268  *  objects. This removes superfluous ONEs, permutes gamma5's to the front
269  *  and removes squares of gamma objects. */
270 ex clifford::simplify_ncmul(const exvector & v) const
271 {
272         exvector s;
273         s.reserve(v.size());
274
275         // Remove superfluous ONEs
276         exvector::const_iterator cit = v.begin(), citend = v.end();
277         while (cit != citend) {
278                 if (!is_a<clifford>(*cit) || !is_a<diracone>(cit->op(0)))
279                         s.push_back(*cit);
280                 cit++;
281         }
282
283         bool something_changed = false;
284         int sign = 1;
285
286         // Anticommute gamma5's to the front
287         if (s.size() >= 2) {
288                 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
289                 while (true) {
290                         exvector::iterator it = next_to_last;
291                         while (true) {
292                                 exvector::iterator it2 = it + 1;
293                                 if (is_a<clifford>(*it) && is_a<clifford>(*it2) && !is_a<diracgamma5>(it->op(0)) && is_a<diracgamma5>(it2->op(0))) {
294                                         it->swap(*it2);
295                                         sign = -sign;
296                                         something_changed = true;
297                                 }
298                                 if (it == first)
299                                         break;
300                                 --it;
301                         }
302                         if (next_to_last == first)
303                                 break;
304                         --next_to_last;
305                 }
306         }
307
308         // Remove squares of gamma5
309         while (s.size() >= 2 && is_a<clifford>(s[0]) && is_a<clifford>(s[1]) && is_a<diracgamma5>(s[0].op(0)) && is_a<diracgamma5>(s[1].op(0))) {
310                 s.erase(s.begin(), s.begin() + 2);
311                 something_changed = true;
312         }
313
314         // Remove equal adjacent gammas
315         if (s.size() >= 2) {
316                 exvector::iterator it, itend = s.end() - 1;
317                 for (it = s.begin(); it != itend; ++it) {
318                         ex & a = it[0];
319                         ex & b = it[1];
320                         if (!is_a<clifford>(a) || !is_a<clifford>(b))
321                                 continue;
322                         if (is_a<diracgamma>(a.op(0)) && is_a<diracgamma>(b.op(0))) {
323                                 const ex & ia = a.op(1);
324                                 const ex & ib = b.op(1);
325                                 if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
326                                         a = lorentz_g(ia, ib);
327                                         b = dirac_ONE(representation_label);
328                                         something_changed = true;
329                                 }
330                         } else if (!is_a<diracgamma>(a.op(0)) && !is_a<diracgamma>(b.op(0))) {
331                                 const ex & ba = a.op(0);
332                                 const ex & bb = b.op(0);
333                                 if (ba.is_equal(bb)) { // a\ a\ -> a^2
334                                         varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).get_dim());
335                                         a = indexed(ba, ix) * indexed(bb, ix.toggle_variance());
336                                         b = dirac_ONE(representation_label);
337                                         something_changed = true;
338                                 }
339                         }
340                 }
341         }
342
343         if (s.empty())
344                 return clifford(diracone(), representation_label) * sign;
345         if (something_changed)
346                 return nonsimplified_ncmul(s) * sign;
347         else
348                 return simplified_ncmul(s) * sign;
349 }
350
351 ex clifford::thisexprseq(const exvector & v) const
352 {
353         return clifford(representation_label, v);
354 }
355
356 ex clifford::thisexprseq(exvector * vp) const
357 {
358         return clifford(representation_label, vp);
359 }
360
361 //////////
362 // global functions
363 //////////
364
365 ex dirac_ONE(unsigned char rl)
366 {
367         return clifford(diracone(), rl);
368 }
369
370 ex dirac_gamma(const ex & mu, unsigned char rl)
371 {
372         if (!is_a<varidx>(mu))
373                 throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
374
375         return clifford(diracgamma(), mu, rl);
376 }
377
378 ex dirac_gamma5(unsigned char rl)
379 {
380         return clifford(diracgamma5(), rl);
381 }
382
383 ex dirac_gamma6(unsigned char rl)
384 {
385         return clifford(diracone(), rl) + clifford(diracgamma5(), rl);
386 }
387
388 ex dirac_gamma7(unsigned char rl)
389 {
390         return clifford(diracone(), rl) - clifford(diracgamma5(), rl);
391 }
392
393 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
394 {
395         // Slashed vectors are actually stored as a clifford object with the
396         // vector as its base expression and a (dummy) index that just serves
397         // for storing the space dimensionality
398         return clifford(e, varidx(0, dim), rl);
399 }
400
401 /** Check whether a given tinfo key (as returned by return_type_tinfo()
402  *  is that of a clifford object with the specified representation label. */
403 static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
404 {
405         return ti == (TINFO_clifford + rl);
406 }
407
408 /** Check whether a given tinfo key (as returned by return_type_tinfo()
409  *  is that of a clifford object (with an arbitrary representation label). */
410 static bool is_clifford_tinfo(unsigned ti)
411 {
412         return (ti & ~0xff) == TINFO_clifford;
413 }
414
415 /** Take trace of a string of an even number of Dirac gammas given a vector
416  *  of indices. */
417 static ex trace_string(exvector::const_iterator ix, unsigned num)
418 {
419         // Tr gamma.mu gamma.nu = 4 g.mu.nu
420         if (num == 2)
421                 return lorentz_g(ix[0], ix[1]);
422
423         // Tr gamma.mu gamma.nu gamma.rho gamma.sig = 4 (g.mu.nu g.rho.sig + g.nu.rho g.mu.sig - g.mu.rho g.nu.sig
424         else if (num == 4)
425                 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
426                      + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
427                      - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
428
429         // Traces of 6 or more gammas are computed recursively:
430         // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
431         //   + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
432         //   - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
433         //   + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
434         //   - ...
435         //   + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
436         exvector v(num - 2);
437         int sign = 1;
438         ex result;
439         for (unsigned i=1; i<num; i++) {
440                 for (unsigned n=1, j=0; n<num; n++) {
441                         if (n == i)
442                                 continue;
443                         v[j++] = ix[n];
444                 }
445                 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
446                 sign = -sign;
447         }
448         return result;
449 }
450
451 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
452 {
453         if (is_a<clifford>(e)) {
454
455                 if (ex_to<clifford>(e).get_representation_label() == rl
456                  && is_a<diracone>(e.op(0)))
457                         return trONE;
458                 else
459                         return _ex0;
460
461         } else if (is_ex_exactly_of_type(e, mul)) {
462
463                 // Trace of product: pull out non-clifford factors
464                 ex prod = _ex1;
465                 for (unsigned i=0; i<e.nops(); i++) {
466                         const ex &o = e.op(i);
467                         if (is_clifford_tinfo(o.return_type_tinfo(), rl))
468                                 prod *= dirac_trace(o, rl, trONE);
469                         else
470                                 prod *= o;
471                 }
472                 return prod;
473
474         } else if (is_ex_exactly_of_type(e, ncmul)) {
475
476                 if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
477                         return _ex0;
478
479                 // Expand product, if necessary
480                 ex e_expanded = e.expand();
481                 if (!is_a<ncmul>(e_expanded))
482                         return dirac_trace(e_expanded, rl, trONE);
483
484                 // gamma5 gets moved to the front so this check is enough
485                 bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
486                 unsigned num = e.nops();
487
488                 if (has_gamma5) {
489
490                         // Trace of gamma5 * odd number of gammas and trace of
491                         // gamma5 * gamma.mu * gamma.nu are zero
492                         if ((num & 1) == 0 || num == 3)
493                                 return _ex0;
494
495                         // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
496                         if (num == 5) {
497                                 ex b1, i1, b2, i2, b3, i3, b4, i4;
498                                 base_and_index(e.op(1), b1, i1);
499                                 base_and_index(e.op(2), b2, i2);
500                                 base_and_index(e.op(3), b3, i3);
501                                 base_and_index(e.op(4), b4, i4);
502                                 return trONE * I * (eps0123(i1, i2, i3, i4) * b1 * b2 * b3 * b4).simplify_indexed();
503                         }
504
505                         // Tr gamma5 S_2k =
506                         //   I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
507                         exvector ix(num-1), bv(num-1);
508                         for (unsigned i=1; i<num; i++)
509                                 base_and_index(e.op(i), bv[i-1], ix[i-1]);
510                         num--;
511                         int *iv = new int[num];
512                         ex result;
513                         for (unsigned i=0; i<num-3; i++) {
514                                 ex idx1 = ix[i];
515                                 for (unsigned j=i+1; j<num-2; j++) {
516                                         ex idx2 = ix[j];
517                                         for (unsigned k=j+1; k<num-1; k++) {
518                                                 ex idx3 = ix[k];
519                                                 for (unsigned l=k+1; l<num; l++) {
520                                                         ex idx4 = ix[l];
521                                                         iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
522                                                         exvector v;
523                                                         v.reserve(num - 4);
524                                                         for (unsigned n=0, t=4; n<num; n++) {
525                                                                 if (n == i || n == j || n == k || n == l)
526                                                                         continue;
527                                                                 iv[t++] = n;
528                                                                 v.push_back(ix[n]);
529                                                         }
530                                                         int sign = permutation_sign(iv, iv + num);
531                                                         result += sign * eps0123(idx1, idx2, idx3, idx4)
532                                                                 * trace_string(v.begin(), num - 4);
533                                                 }
534                                         }
535                                 }
536                         }
537                         delete[] iv;
538                         return trONE * I * result * mul(bv);
539
540                 } else { // no gamma5
541
542                         // Trace of odd number of gammas is zero
543                         if ((num & 1) == 1)
544                                 return _ex0;
545
546                         // Tr gamma.mu gamma.nu = 4 g.mu.nu
547                         if (num == 2) {
548                                 ex b1, i1, b2, i2;
549                                 base_and_index(e.op(0), b1, i1);
550                                 base_and_index(e.op(1), b2, i2);
551                                 return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
552                         }
553
554                         exvector iv(num), bv(num);
555                         for (unsigned i=0; i<num; i++)
556                                 base_and_index(e.op(i), bv[i], iv[i]);
557
558                         return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
559                 }
560
561         } else if (e.nops() > 0) {
562
563                 // Trace maps to all other container classes (this includes sums)
564                 pointer_to_map_function_2args<unsigned char, const ex &> fcn(dirac_trace, rl, trONE);
565                 return e.map(fcn);
566
567         } else
568                 return _ex0;
569 }
570
571 ex canonicalize_clifford(const ex & e)
572 {
573         // Scan for any ncmul objects
574         lst srl;
575         ex aux = e.to_rational(srl);
576         for (unsigned i=0; i<srl.nops(); i++) {
577
578                 ex lhs = srl.op(i).lhs();
579                 ex rhs = srl.op(i).rhs();
580
581                 if (is_ex_exactly_of_type(rhs, ncmul)
582                  && rhs.return_type() == return_types::noncommutative
583                  && is_clifford_tinfo(rhs.return_type_tinfo())) {
584
585                         // Expand product, if necessary
586                         ex rhs_expanded = rhs.expand();
587                         if (!is_a<ncmul>(rhs_expanded)) {
588                                 srl.let_op(i) = (lhs == canonicalize_clifford(rhs_expanded));
589                                 continue;
590
591                         } else if (!is_a<clifford>(rhs.op(0)))
592                                 continue;
593
594                         exvector v;
595                         v.reserve(rhs.nops());
596                         for (unsigned j=0; j<rhs.nops(); j++)
597                                 v.push_back(rhs.op(j));
598
599                         // Stupid recursive bubble sort because we only want to swap adjacent gammas
600                         exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
601                         if (is_a<diracgamma5>(it->op(0)))
602                                 ++it;
603                         while (it != next_to_last) {
604                                 if (it[0].compare(it[1]) > 0) {
605                                         ex save0 = it[0], save1 = it[1];
606                                         ex b1, i1, b2, i2;
607                                         base_and_index(it[0], b1, i1);
608                                         base_and_index(it[1], b2, i2);
609                                         it[0] = (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
610                                         it[1] = _ex2;
611                                         ex sum = ncmul(v);
612                                         it[0] = save1;
613                                         it[1] = save0;
614                                         sum -= ncmul(v, true);
615                                         srl.let_op(i) = (lhs == canonicalize_clifford(sum));
616                                         goto next_sym;
617                                 }
618                                 ++it;
619                         }
620 next_sym:       ;
621                 }
622         }
623         return aux.subs(srl).simplify_indexed();
624 }
625
626 } // namespace GiNaC