3 * Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
6 * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
8 * This program is free software; you can redistribute it and/or modify
9 * it under the terms of the GNU General Public License as published by
10 * the Free Software Foundation; either version 2 of the License, or
11 * (at your option) any later version.
13 * This program is distributed in the hope that it will be useful,
14 * but WITHOUT ANY WARRANTY; without even the implied warranty of
15 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
16 * GNU General Public License for more details.
18 * You should have received a copy of the GNU General Public License
19 * along with this program; if not, write to the Free Software
20 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
28 #include "numeric.h" // for I
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)
44 // default constructor, destructor, copy constructor assignment operator and helpers
47 clifford::clifford() : representation_label(0)
49 debugmsg("clifford default constructor", LOGLEVEL_CONSTRUCT);
50 tinfo_key = TINFO_clifford;
53 void clifford::copy(const clifford & other)
55 inherited::copy(other);
56 representation_label = other.representation_label;
59 DEFAULT_DESTROY(clifford)
60 DEFAULT_CTORS(diracone)
61 DEFAULT_CTORS(diracgamma)
62 DEFAULT_CTORS(diracgamma5)
68 /** Construct object without any indices. This constructor is for internal
69 * use only. Use the dirac_ONE() function instead.
71 clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
73 debugmsg("clifford constructor from ex", LOGLEVEL_CONSTRUCT);
74 tinfo_key = TINFO_clifford;
77 /** Construct object with one Lorentz index. This constructor is for internal
78 * use only. Use the dirac_gamma() function instead.
80 clifford::clifford(const ex & b, const ex & mu, unsigned char rl) : inherited(b, mu), representation_label(rl)
82 debugmsg("clifford constructor from ex,ex", LOGLEVEL_CONSTRUCT);
83 GINAC_ASSERT(is_ex_of_type(mu, varidx));
84 tinfo_key = TINFO_clifford;
87 clifford::clifford(unsigned char rl, const exvector & v, bool discardable) : inherited(indexed::unknown, v, discardable), representation_label(rl)
89 debugmsg("clifford constructor from unsigned char,exvector", LOGLEVEL_CONSTRUCT);
90 tinfo_key = TINFO_clifford;
93 clifford::clifford(unsigned char rl, exvector * vp) : inherited(indexed::unknown, vp), representation_label(rl)
95 debugmsg("clifford constructor from unsigned char,exvector *", LOGLEVEL_CONSTRUCT);
96 tinfo_key = TINFO_clifford;
103 clifford::clifford(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
105 debugmsg("clifford constructor from archive_node", LOGLEVEL_CONSTRUCT);
107 n.find_unsigned("label", rl);
108 representation_label = rl;
111 void clifford::archive(archive_node &n) const
113 inherited::archive(n);
114 n.add_unsigned("label", representation_label);
117 DEFAULT_UNARCHIVE(clifford)
118 DEFAULT_ARCHIVING(diracone)
119 DEFAULT_ARCHIVING(diracgamma)
120 DEFAULT_ARCHIVING(diracgamma5)
123 // functions overriding virtual functions from bases classes
126 int clifford::compare_same_type(const basic & other) const
128 GINAC_ASSERT(other.tinfo() == TINFO_clifford);
129 const clifford &o = static_cast<const clifford &>(other);
131 if (representation_label != o.representation_label) {
132 // different representation label
133 return representation_label < o.representation_label ? -1 : 1;
136 return inherited::compare_same_type(other);
139 DEFAULT_COMPARE(diracone)
140 DEFAULT_COMPARE(diracgamma)
141 DEFAULT_COMPARE(diracgamma5)
143 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
144 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
145 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
147 /** Contraction of a gamma matrix with something else. */
148 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
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();
155 if (is_ex_of_type(*other, clifford)) {
157 ex dim = ex_to_idx(self->op(1)).get_dim();
159 // gamma~mu gamma.mu = dim ONE
160 if (other - self == 1) {
162 *other = dirac_ONE(rl);
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)) {
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];
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)
186 exvector::iterator it = self + 1, next_to_last = other - 1;
187 while (it != other) {
188 if (!is_ex_of_type(*it, clifford))
195 while (it != next_to_last) {
200 *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
201 *next_to_last = _ex1();
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
217 unsigned rl = ex_to_clifford(v[0]).get_representation_label();
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))
227 bool something_changed = false;
230 // Anticommute gamma5's to the front
232 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
234 exvector::iterator it = next_to_last;
236 exvector::iterator it2 = it + 1;
237 if (!is_ex_of_type(it->op(0), diracgamma5) && is_ex_of_type(it2->op(0), diracgamma5)) {
240 something_changed = true;
246 if (next_to_last == first)
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;
258 // Remove equal adjacent gammas
260 exvector::iterator it = s.begin(), itend = s.end() - 1;
261 while (it != itend) {
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);
270 something_changed = true;
278 return clifford(diracone(), rl) * sign;
279 if (something_changed)
280 return nonsimplified_ncmul(s) * sign;
282 return simplified_ncmul(s) * sign;
285 ex clifford::thisexprseq(const exvector & v) const
287 return clifford(representation_label, v);
290 ex clifford::thisexprseq(exvector * vp) const
292 return clifford(representation_label, vp);
299 ex dirac_ONE(unsigned char rl)
301 return clifford(diracone(), rl);
304 ex dirac_gamma(const ex & mu, unsigned char rl)
306 if (!is_ex_of_type(mu, varidx))
307 throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
309 return clifford(diracgamma(), mu, rl);
312 ex dirac_gamma5(unsigned char rl)
314 return clifford(diracgamma5(), rl);
317 ex dirac_gamma6(unsigned char rl)
319 return clifford(diracone(), rl) + clifford(diracgamma5(), rl);
322 ex dirac_gamma7(unsigned char rl)
324 return clifford(diracone(), rl) - clifford(diracgamma5(), rl);
327 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
329 varidx mu((new symbol)->setflag(status_flags::dynallocated), dim);
330 return indexed(e, mu.toggle_variance()) * dirac_gamma(mu, rl);
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)
337 return ti == (TINFO_clifford + rl);
340 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
342 if (is_ex_of_type(e, clifford)) {
344 if (ex_to_clifford(e).get_representation_label() == rl
345 && is_ex_of_type(e.op(0), diracone))
350 } else if (is_ex_exactly_of_type(e, add)) {
352 // Trace of sum = sum of traces
354 for (unsigned i=0; i<e.nops(); i++)
355 sum += dirac_trace(e.op(i), rl, trONE);
358 } else if (is_ex_exactly_of_type(e, mul)) {
360 // Trace of product: pull out non-clifford factors
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);
372 } else if (is_ex_exactly_of_type(e, ncmul)) {
374 if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
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);
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();
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)
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);
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));
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;
412 } else { // no gamma5
414 // Trace of odd number of gammas is zero
418 // Tr gamma.mu gamma.nu = 4 g.mu.nu
420 return trONE * lorentz_g(e.op(0).op(1), e.op(1).op(1));
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
428 // + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
431 const ex &ix1 = e.op(0).op(1);
433 for (int i=1; i<num; i++) {
434 for (int n=1, j=0; n<num; n++) {
439 result += sign * lorentz_g(ix1, e.op(i).op(1)) * dirac_trace(ncmul(v), rl, trONE);
449 ex canonicalize_clifford(const ex & e)
451 if (is_ex_exactly_of_type(e, add)) {
454 for (unsigned i=0; i<e.nops(); i++)
455 sum += canonicalize_clifford(e.op(i));
458 } else if (is_ex_exactly_of_type(e, mul)) {
461 for (unsigned i=0; i<e.nops(); i++)
462 prod *= canonicalize_clifford(e.op(i));
465 } else if (is_ex_exactly_of_type(e, ncmul)) {
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);
472 if (!is_ex_of_type(e.op(0), clifford))
477 for (int i=0; i<e.nops(); i++)
478 v.push_back(e.op(i));
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))
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));
495 return canonicalize_clifford(sum);