66ddc019ea5e76b0d10bfdb96d80e8718eb4c6b2
[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) 1999-2002 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 equal adjacent gammas
309         if (s.size() >= 2) {
310                 exvector::iterator it, itend = s.end() - 1;
311                 for (it = s.begin(); it != itend; ++it) {
312                         ex & a = it[0];
313                         ex & b = it[1];
314                         if (!is_a<clifford>(a) || !is_a<clifford>(b))
315                                 continue;
316                         bool a_is_diracgamma = is_a<diracgamma>(a.op(0));
317                         bool b_is_diracgamma = is_a<diracgamma>(b.op(0));
318                         if (a_is_diracgamma && b_is_diracgamma) {
319                                 const ex & ia = a.op(1);
320                                 const ex & ib = b.op(1);
321                                 if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
322                                         a = lorentz_g(ia, ib);
323                                         b = dirac_ONE(representation_label);
324                                         something_changed = true;
325                                 }
326                         } else if (is_a<diracgamma5>(a.op(0)) && is_a<diracgamma5>(b.op(0))) {
327                                 // Remove squares of gamma5
328                                 a = dirac_ONE(representation_label);
329                                 b = dirac_ONE(representation_label);
330                                 something_changed = true;
331                         } else if (!a_is_diracgamma && !b_is_diracgamma) {
332                                 const ex & ba = a.op(0);
333                                 const ex & bb = b.op(0);
334                                 if (ba.is_equal(bb)) { // a\ a\ -> a^2
335                                         varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).get_dim());
336                                         a = indexed(ba, ix) * indexed(bb, ix.toggle_variance());
337                                         b = dirac_ONE(representation_label);
338                                         something_changed = true;
339                                 }
340                         }
341                 }
342         }
343
344         if (s.empty())
345                 return clifford(diracone(), representation_label) * sign;
346         if (something_changed)
347                 return nonsimplified_ncmul(s) * sign;
348         else
349                 return simplified_ncmul(s) * sign;
350 }
351
352 ex clifford::thisexprseq(const exvector & v) const
353 {
354         return clifford(representation_label, v);
355 }
356
357 ex clifford::thisexprseq(exvector * vp) const
358 {
359         return clifford(representation_label, vp);
360 }
361
362 //////////
363 // global functions
364 //////////
365
366 ex dirac_ONE(unsigned char rl)
367 {
368         return clifford(diracone(), rl);
369 }
370
371 ex dirac_gamma(const ex & mu, unsigned char rl)
372 {
373         if (!is_a<varidx>(mu))
374                 throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
375
376         return clifford(diracgamma(), mu, rl);
377 }
378
379 ex dirac_gamma5(unsigned char rl)
380 {
381         return clifford(diracgamma5(), rl);
382 }
383
384 ex dirac_gamma6(unsigned char rl)
385 {
386         return clifford(diracone(), rl) + clifford(diracgamma5(), rl);
387 }
388
389 ex dirac_gamma7(unsigned char rl)
390 {
391         return clifford(diracone(), rl) - clifford(diracgamma5(), rl);
392 }
393
394 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
395 {
396         // Slashed vectors are actually stored as a clifford object with the
397         // vector as its base expression and a (dummy) index that just serves
398         // for storing the space dimensionality
399         return clifford(e, varidx(0, dim), rl);
400 }
401
402 /** Check whether a given tinfo key (as returned by return_type_tinfo()
403  *  is that of a clifford object with the specified representation label. */
404 static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
405 {
406         return ti == (TINFO_clifford + rl);
407 }
408
409 /** Check whether a given tinfo key (as returned by return_type_tinfo()
410  *  is that of a clifford object (with an arbitrary representation label). */
411 static bool is_clifford_tinfo(unsigned ti)
412 {
413         return (ti & ~0xff) == TINFO_clifford;
414 }
415
416 /** Take trace of a string of an even number of Dirac gammas given a vector
417  *  of indices. */
418 static ex trace_string(exvector::const_iterator ix, unsigned num)
419 {
420         // Tr gamma.mu gamma.nu = 4 g.mu.nu
421         if (num == 2)
422                 return lorentz_g(ix[0], ix[1]);
423
424         // 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
425         else if (num == 4)
426                 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
427                      + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
428                      - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
429
430         // Traces of 6 or more gammas are computed recursively:
431         // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
432         //   + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
433         //   - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
434         //   + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
435         //   - ...
436         //   + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
437         exvector v(num - 2);
438         int sign = 1;
439         ex result;
440         for (unsigned i=1; i<num; i++) {
441                 for (unsigned n=1, j=0; n<num; n++) {
442                         if (n == i)
443                                 continue;
444                         v[j++] = ix[n];
445                 }
446                 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
447                 sign = -sign;
448         }
449         return result;
450 }
451
452 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
453 {
454         if (is_a<clifford>(e)) {
455
456                 if (ex_to<clifford>(e).get_representation_label() == rl
457                  && is_a<diracone>(e.op(0)))
458                         return trONE;
459                 else
460                         return _ex0;
461
462         } else if (is_ex_exactly_of_type(e, mul)) {
463
464                 // Trace of product: pull out non-clifford factors
465                 ex prod = _ex1;
466                 for (unsigned i=0; i<e.nops(); i++) {
467                         const ex &o = e.op(i);
468                         if (is_clifford_tinfo(o.return_type_tinfo(), rl))
469                                 prod *= dirac_trace(o, rl, trONE);
470                         else
471                                 prod *= o;
472                 }
473                 return prod;
474
475         } else if (is_ex_exactly_of_type(e, ncmul)) {
476
477                 if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
478                         return _ex0;
479
480                 // Expand product, if necessary
481                 ex e_expanded = e.expand();
482                 if (!is_a<ncmul>(e_expanded))
483                         return dirac_trace(e_expanded, rl, trONE);
484
485                 // gamma5 gets moved to the front so this check is enough
486                 bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
487                 unsigned num = e.nops();
488
489                 if (has_gamma5) {
490
491                         // Trace of gamma5 * odd number of gammas and trace of
492                         // gamma5 * gamma.mu * gamma.nu are zero
493                         if ((num & 1) == 0 || num == 3)
494                                 return _ex0;
495
496                         // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
497                         if (num == 5) {
498                                 ex b1, i1, b2, i2, b3, i3, b4, i4;
499                                 base_and_index(e.op(1), b1, i1);
500                                 base_and_index(e.op(2), b2, i2);
501                                 base_and_index(e.op(3), b3, i3);
502                                 base_and_index(e.op(4), b4, i4);
503                                 return trONE * I * (eps0123(i1, i2, i3, i4) * b1 * b2 * b3 * b4).simplify_indexed();
504                         }
505
506                         // Tr gamma5 S_2k =
507                         //   I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
508                         exvector ix(num-1), bv(num-1);
509                         for (unsigned i=1; i<num; i++)
510                                 base_and_index(e.op(i), bv[i-1], ix[i-1]);
511                         num--;
512                         int *iv = new int[num];
513                         ex result;
514                         for (unsigned i=0; i<num-3; i++) {
515                                 ex idx1 = ix[i];
516                                 for (unsigned j=i+1; j<num-2; j++) {
517                                         ex idx2 = ix[j];
518                                         for (unsigned k=j+1; k<num-1; k++) {
519                                                 ex idx3 = ix[k];
520                                                 for (unsigned l=k+1; l<num; l++) {
521                                                         ex idx4 = ix[l];
522                                                         iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
523                                                         exvector v;
524                                                         v.reserve(num - 4);
525                                                         for (unsigned n=0, t=4; n<num; n++) {
526                                                                 if (n == i || n == j || n == k || n == l)
527                                                                         continue;
528                                                                 iv[t++] = n;
529                                                                 v.push_back(ix[n]);
530                                                         }
531                                                         int sign = permutation_sign(iv, iv + num);
532                                                         result += sign * eps0123(idx1, idx2, idx3, idx4)
533                                                                 * trace_string(v.begin(), num - 4);
534                                                 }
535                                         }
536                                 }
537                         }
538                         delete[] iv;
539                         return trONE * I * result * mul(bv);
540
541                 } else { // no gamma5
542
543                         // Trace of odd number of gammas is zero
544                         if ((num & 1) == 1)
545                                 return _ex0;
546
547                         // Tr gamma.mu gamma.nu = 4 g.mu.nu
548                         if (num == 2) {
549                                 ex b1, i1, b2, i2;
550                                 base_and_index(e.op(0), b1, i1);
551                                 base_and_index(e.op(1), b2, i2);
552                                 return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
553                         }
554
555                         exvector iv(num), bv(num);
556                         for (unsigned i=0; i<num; i++)
557                                 base_and_index(e.op(i), bv[i], iv[i]);
558
559                         return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
560                 }
561
562         } else if (e.nops() > 0) {
563
564                 // Trace maps to all other container classes (this includes sums)
565                 pointer_to_map_function_2args<unsigned char, const ex &> fcn(dirac_trace, rl, trONE);
566                 return e.map(fcn);
567
568         } else
569                 return _ex0;
570 }
571
572 ex canonicalize_clifford(const ex & e)
573 {
574         // Scan for any ncmul objects
575         lst srl;
576         ex aux = e.to_rational(srl);
577         for (unsigned i=0; i<srl.nops(); i++) {
578
579                 ex lhs = srl.op(i).lhs();
580                 ex rhs = srl.op(i).rhs();
581
582                 if (is_ex_exactly_of_type(rhs, ncmul)
583                  && rhs.return_type() == return_types::noncommutative
584                  && is_clifford_tinfo(rhs.return_type_tinfo())) {
585
586                         // Expand product, if necessary
587                         ex rhs_expanded = rhs.expand();
588                         if (!is_a<ncmul>(rhs_expanded)) {
589                                 srl.let_op(i) = (lhs == canonicalize_clifford(rhs_expanded));
590                                 continue;
591
592                         } else if (!is_a<clifford>(rhs.op(0)))
593                                 continue;
594
595                         exvector v;
596                         v.reserve(rhs.nops());
597                         for (unsigned j=0; j<rhs.nops(); j++)
598                                 v.push_back(rhs.op(j));
599
600                         // Stupid recursive bubble sort because we only want to swap adjacent gammas
601                         exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
602                         if (is_a<diracgamma5>(it->op(0)))
603                                 ++it;
604                         while (it != next_to_last) {
605                                 if (it[0].compare(it[1]) > 0) {
606                                         ex save0 = it[0], save1 = it[1];
607                                         ex b1, i1, b2, i2;
608                                         base_and_index(it[0], b1, i1);
609                                         base_and_index(it[1], b2, i2);
610                                         it[0] = (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
611                                         it[1] = _ex2;
612                                         ex sum = ncmul(v);
613                                         it[0] = save1;
614                                         it[1] = save0;
615                                         sum -= ncmul(v, true);
616                                         srl.let_op(i) = (lhs == canonicalize_clifford(sum));
617                                         goto next_sym;
618                                 }
619                                 ++it;
620                         }
621 next_sym:       ;
622                 }
623         }
624         return aux.subs(srl).simplify_indexed();
625 }
626
627 } // namespace GiNaC