3 * Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
6 * GiNaC Copyright (C) 1999-2008 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., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
31 #include "numeric.h" // for I
34 #include "relational.h"
35 #include "operators.h"
45 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(clifford, indexed,
46 print_func<print_dflt>(&clifford::do_print_dflt).
47 print_func<print_latex>(&clifford::do_print_latex))
49 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracone, tensor,
50 print_func<print_dflt>(&diracone::do_print).
51 print_func<print_latex>(&diracone::do_print_latex))
53 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(cliffordunit, tensor,
54 print_func<print_dflt>(&cliffordunit::do_print).
55 print_func<print_latex>(&cliffordunit::do_print_latex))
57 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma, cliffordunit,
58 print_func<print_dflt>(&diracgamma::do_print).
59 print_func<print_latex>(&diracgamma::do_print_latex))
61 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma5, tensor,
62 print_func<print_dflt>(&diracgamma5::do_print).
63 print_func<print_latex>(&diracgamma5::do_print_latex))
65 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaL, tensor,
66 print_func<print_context>(&diracgammaL::do_print).
67 print_func<print_latex>(&diracgammaL::do_print_latex))
69 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaR, tensor,
70 print_func<print_context>(&diracgammaR::do_print).
71 print_func<print_latex>(&diracgammaR::do_print_latex))
74 // default constructors
77 clifford::clifford() : representation_label(0), metric(0), commutator_sign(-1)
79 tinfo_key = &clifford::tinfo_static;
82 DEFAULT_CTOR(diracone)
83 DEFAULT_CTOR(cliffordunit)
84 DEFAULT_CTOR(diracgamma)
85 DEFAULT_CTOR(diracgamma5)
86 DEFAULT_CTOR(diracgammaL)
87 DEFAULT_CTOR(diracgammaR)
93 /** Construct object without any indices. This constructor is for internal
94 * use only. Use the dirac_ONE() function instead.
96 clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl), metric(0), commutator_sign(-1)
98 tinfo_key = &clifford::tinfo_static;
101 /** Construct object with one Lorentz index. This constructor is for internal
102 * use only. Use the clifford_unit() or dirac_gamma() functions instead.
104 * @see dirac_gamma */
105 clifford::clifford(const ex & b, const ex & mu, const ex & metr, unsigned char rl, int comm_sign) : inherited(b, mu), representation_label(rl), metric(metr), commutator_sign(comm_sign)
107 GINAC_ASSERT(is_a<varidx>(mu));
108 tinfo_key = &clifford::tinfo_static;
111 clifford::clifford(unsigned char rl, const ex & metr, int comm_sign, const exvector & v, bool discardable) : inherited(not_symmetric(), v, discardable), representation_label(rl), metric(metr), commutator_sign(comm_sign)
113 tinfo_key = &clifford::tinfo_static;
116 clifford::clifford(unsigned char rl, const ex & metr, int comm_sign, std::auto_ptr<exvector> vp) : inherited(not_symmetric(), vp), representation_label(rl), metric(metr), commutator_sign(comm_sign)
118 tinfo_key = &clifford::tinfo_static;
121 return_type_t clifford::return_type_tinfo() const
123 return make_return_type_t<clifford>(representation_label);
130 clifford::clifford(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst)
133 n.find_unsigned("label", rl);
134 representation_label = rl;
135 n.find_ex("metric", metric, sym_lst);
136 n.find_unsigned("commutator_sign+1", rl);
137 commutator_sign = rl - 1;
140 void clifford::archive(archive_node & n) const
142 inherited::archive(n);
143 n.add_unsigned("label", representation_label);
144 n.add_ex("metric", metric);
145 n.add_unsigned("commutator_sign+1", commutator_sign+1);
148 DEFAULT_UNARCHIVE(clifford)
149 DEFAULT_ARCHIVING(diracone)
150 DEFAULT_ARCHIVING(cliffordunit)
151 DEFAULT_ARCHIVING(diracgamma)
152 DEFAULT_ARCHIVING(diracgamma5)
153 DEFAULT_ARCHIVING(diracgammaL)
154 DEFAULT_ARCHIVING(diracgammaR)
157 ex clifford::get_metric(const ex & i, const ex & j, bool symmetrised) const
159 if (is_a<indexed>(metric)) {
160 if (symmetrised && !(ex_to<symmetry>(ex_to<indexed>(metric).get_symmetry()).has_symmetry())) {
161 if (is_a<matrix>(metric.op(0))) {
162 return indexed((ex_to<matrix>(metric.op(0)).add(ex_to<matrix>(metric.op(0)).transpose())).mul(numeric(1, 2)),
165 return simplify_indexed(indexed(metric.op(0)*_ex1_2, i, j) + indexed(metric.op(0)*_ex1_2, j, i));
168 return metric.subs(lst(metric.op(1) == i, metric.op(2) == j), subs_options::no_pattern);
171 exvector indices = metric.get_free_indices();
173 return _ex1_2*simplify_indexed(metric.subs(lst(indices[0] == i, indices[1] == j), subs_options::no_pattern)
174 + metric.subs(lst(indices[0] == j, indices[1] == i), subs_options::no_pattern));
176 return metric.subs(lst(indices[0] == i, indices[1] == j), subs_options::no_pattern);
180 bool clifford::same_metric(const ex & other) const
183 if (is_a<clifford>(other))
184 metr = ex_to<clifford>(other).get_metric();
188 if (is_a<indexed>(metr))
189 return metr.op(0).is_equal(get_metric().op(0));
191 exvector indices = metr.get_free_indices();
192 return (indices.size() == 2)
193 && simplify_indexed(get_metric(indices[0], indices[1])-metr).is_zero();
198 // functions overriding virtual functions from base classes
201 ex clifford::op(size_t i) const
203 GINAC_ASSERT(i<nops());
205 return representation_label;
207 return inherited::op(i);
210 ex & clifford::let_op(size_t i)
212 GINAC_ASSERT(i<nops());
214 static ex rl = numeric(representation_label);
215 ensure_if_modifiable();
219 return inherited::let_op(i);
222 ex clifford::subs(const exmap & m, unsigned options) const
224 ex subsed = inherited::subs(m, options);
225 if(is_a<clifford>(subsed)) {
226 ex prevmetric = ex_to<clifford>(subsed).metric;
227 ex newmetric = prevmetric.subs(m, options);
228 if(!are_ex_trivially_equal(prevmetric, newmetric)) {
229 clifford c = ex_to<clifford>(subsed);
230 c.metric = newmetric;
237 int clifford::compare_same_type(const basic & other) const
239 GINAC_ASSERT(is_a<clifford>(other));
240 const clifford &o = static_cast<const clifford &>(other);
242 if (representation_label != o.representation_label) {
243 // different representation label
244 return representation_label < o.representation_label ? -1 : 1;
247 return inherited::compare_same_type(other);
250 bool clifford::match_same_type(const basic & other) const
252 GINAC_ASSERT(is_a<clifford>(other));
253 const clifford &o = static_cast<const clifford &>(other);
255 return ((representation_label == o.representation_label) && (commutator_sign == o.get_commutator_sign()) && same_metric(o));
258 static bool is_dirac_slash(const ex & seq0)
260 return !is_a<diracgamma5>(seq0) && !is_a<diracgammaL>(seq0) &&
261 !is_a<diracgammaR>(seq0) && !is_a<cliffordunit>(seq0) &&
262 !is_a<diracone>(seq0);
265 void clifford::do_print_dflt(const print_dflt & c, unsigned level) const
267 // dirac_slash() object is printed differently
268 if (is_dirac_slash(seq[0])) {
269 seq[0].print(c, precedence());
271 } else { // We do not print representation label if it is 0
272 if (representation_label == 0) {
273 this->print_dispatch<inherited>(c, level);
274 } else { // otherwise we put it before indices in square brackets; the code is borrowed from indexed.cpp
275 if (precedence() <= level) {
278 seq[0].print(c, precedence());
279 c.s << '[' << int(representation_label) << ']';
280 printindices(c, level);
281 if (precedence() <= level) {
288 void clifford::do_print_latex(const print_latex & c, unsigned level) const
290 // dirac_slash() object is printed differently
291 if (is_dirac_slash(seq[0])) {
293 seq[0].print(c, precedence());
294 c.s << "\\hspace{-1.0ex}/}";
296 c.s << "\\clifford[" << int(representation_label) << "]";
297 this->print_dispatch<inherited>(c, level);
301 DEFAULT_COMPARE(diracone)
302 DEFAULT_COMPARE(cliffordunit)
303 DEFAULT_COMPARE(diracgamma)
304 DEFAULT_COMPARE(diracgamma5)
305 DEFAULT_COMPARE(diracgammaL)
306 DEFAULT_COMPARE(diracgammaR)
308 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbf{1}")
309 DEFAULT_PRINT_LATEX(cliffordunit, "e", "e")
310 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
311 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
312 DEFAULT_PRINT_LATEX(diracgammaL, "gammaL", "{\\gamma_L}")
313 DEFAULT_PRINT_LATEX(diracgammaR, "gammaR", "{\\gamma_R}")
315 /** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
316 static void base_and_index(const ex & c, ex & b, ex & i)
318 GINAC_ASSERT(is_a<clifford>(c));
319 GINAC_ASSERT(c.nops() == 2+1);
321 if (is_a<cliffordunit>(c.op(0))) { // proper dirac gamma object or clifford unit
324 } else if (is_a<diracgamma5>(c.op(0)) || is_a<diracgammaL>(c.op(0)) || is_a<diracgammaR>(c.op(0))) { // gamma5/L/R
327 } else { // slash object, generate new dummy index
328 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(c.op(1)).get_dim());
329 b = indexed(c.op(0), ix.toggle_variance());
334 /** Predicate for finding non-clifford objects. */
335 struct is_not_a_clifford : public std::unary_function<ex, bool> {
336 bool operator()(const ex & e)
338 return !is_a<clifford>(e);
342 /** Contraction of a gamma matrix with something else. */
343 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
345 GINAC_ASSERT(is_a<clifford>(*self));
346 GINAC_ASSERT(is_a<indexed>(*other));
347 GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
348 unsigned char rl = ex_to<clifford>(*self).get_representation_label();
350 ex dim = ex_to<idx>(self->op(1)).get_dim();
351 if (other->nops() > 1)
352 dim = minimal_dim(dim, ex_to<idx>(other->op(1)).get_dim());
354 if (is_a<clifford>(*other)) {
356 // Contraction only makes sense if the represenation labels are equal
357 if (ex_to<clifford>(*other).get_representation_label() != rl)
360 size_t num = other - self;
362 // gamma~mu gamma.mu = dim ONE
365 *other = dirac_ONE(rl);
368 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
370 && is_a<clifford>(self[1])) {
375 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
377 && is_a<clifford>(self[1])
378 && is_a<clifford>(self[2])) {
380 base_and_index(self[1], b1, i1);
381 base_and_index(self[2], b2, i2);
382 *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
388 // 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
390 && is_a<clifford>(self[1])
391 && is_a<clifford>(self[2])
392 && is_a<clifford>(self[3])) {
393 *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3];
400 // gamma~mu Sodd gamma.mu = -2 Sodd_R
401 // (Chisholm identity in 4 dimensions)
402 } else if (!((other - self) & 1) && dim.is_equal(4)) {
403 if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
406 *self = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(other), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
407 std::fill(self + 1, other, _ex1);
411 // gamma~mu Sodd gamma~alpha gamma.mu = 2 gamma~alpha Sodd + 2 Sodd_R gamma~alpha
412 // (commutate contracted indices towards each other, then use
413 // Chisholm identity in 4 dimensions)
414 } else if (((other - self) & 1) && dim.is_equal(4)) {
415 if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
418 exvector::iterator next_to_last = other - 1;
419 ex S = ncmul(exvector(self + 1, next_to_last), true);
420 ex SR = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(next_to_last), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
422 *self = (*next_to_last) * S + SR * (*next_to_last);
423 std::fill(self + 1, other, _ex1);
427 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
428 // (commutate contracted indices towards each other, simplify_indexed()
429 // will re-expand and re-run the simplification)
431 if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
434 exvector::iterator next_to_last = other - 1;
435 ex S = ncmul(exvector(self + 1, next_to_last), true);
437 *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
438 std::fill(self + 1, other + 1, _ex1);
442 } else if (is_a<symbol>(other->op(0)) && other->nops() == 2) {
444 // x.mu gamma~mu -> x-slash
445 *self = dirac_slash(other->op(0), dim, rl);
453 /** Contraction of a Clifford unit with something else. */
454 bool cliffordunit::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
456 GINAC_ASSERT(is_a<clifford>(*self));
457 GINAC_ASSERT(is_a<indexed>(*other));
458 GINAC_ASSERT(is_a<cliffordunit>(self->op(0)));
459 clifford unit = ex_to<clifford>(*self);
460 unsigned char rl = unit.get_representation_label();
462 if (is_a<clifford>(*other)) {
463 // Contraction only makes sense if the represenation labels are equal
464 // and the metrics are the same
465 if ((ex_to<clifford>(*other).get_representation_label() != rl)
466 && unit.same_metric(*other))
469 exvector::iterator before_other = other - 1;
471 ex mu_toggle = other->op(1);
472 ex alpha = before_other->op(1);
474 // e~mu e.mu = Tr ONE
475 if (other - self == 1) {
476 *self = unit.get_metric(mu, mu_toggle, true);
477 *other = dirac_ONE(rl);
480 } else if (other - self == 2) {
481 if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
482 // e~mu e~alpha e.mu = 2*e~mu B(alpha, mu.toggle_variance())-Tr(B) e~alpha
483 *self = 2 * (*self) * unit.get_metric(alpha, mu_toggle, true) - unit.get_metric(mu, mu_toggle, true) * (*before_other);
484 *before_other = _ex1;
489 // e~mu S e.mu = Tr S ONE
490 *self = unit.get_metric(mu, mu_toggle, true);
491 *other = dirac_ONE(rl);
495 // e~mu S e~alpha e.mu = 2 e~mu S B(alpha, mu.toggle_variance()) - e~mu S e.mu e~alpha
496 // (commutate contracted indices towards each other, simplify_indexed()
497 // will re-expand and re-run the simplification)
498 if (std::find_if(self + 1, other, is_not_a_clifford()) != other) {
502 ex S = ncmul(exvector(self + 1, before_other), true);
504 if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
505 *self = 2 * (*self) * S * unit.get_metric(alpha, mu_toggle, true) - (*self) * S * (*other) * (*before_other);
508 *self = (*self) * S * (*other) * (*before_other);
511 std::fill(self + 1, other + 1, _ex1);
518 /** Perform automatic simplification on noncommutative product of clifford
519 * objects. This removes superfluous ONEs, permutes gamma5/L/R's to the front
520 * and removes squares of gamma objects. */
521 ex clifford::eval_ncmul(const exvector & v) const
526 // Remove superfluous ONEs
527 exvector::const_iterator cit = v.begin(), citend = v.end();
528 while (cit != citend) {
529 if (!is_a<clifford>(*cit) || !is_a<diracone>(cit->op(0)))
534 bool something_changed = false;
537 // Anticommutate gamma5/L/R's to the front
539 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
541 exvector::iterator it = next_to_last;
543 exvector::iterator it2 = it + 1;
544 if (is_a<clifford>(*it) && is_a<clifford>(*it2)) {
545 ex e1 = it->op(0), e2 = it2->op(0);
547 if (is_a<diracgamma5>(e2)) {
549 if (is_a<diracgammaL>(e1) || is_a<diracgammaR>(e1)) {
551 // gammaL/R gamma5 -> gamma5 gammaL/R
553 something_changed = true;
555 } else if (!is_a<diracgamma5>(e1)) {
557 // gamma5 gamma5 -> gamma5 gamma5 (do nothing)
558 // x gamma5 -> -gamma5 x
561 something_changed = true;
564 } else if (is_a<diracgammaL>(e2)) {
566 if (is_a<diracgammaR>(e1)) {
568 // gammaR gammaL -> 0
571 } else if (!is_a<diracgammaL>(e1) && !is_a<diracgamma5>(e1)) {
573 // gammaL gammaL -> gammaL gammaL (do nothing)
574 // gamma5 gammaL -> gamma5 gammaL (do nothing)
575 // x gammaL -> gammaR x
577 *it = clifford(diracgammaR(), ex_to<clifford>(*it).get_representation_label());
578 something_changed = true;
581 } else if (is_a<diracgammaR>(e2)) {
583 if (is_a<diracgammaL>(e1)) {
585 // gammaL gammaR -> 0
588 } else if (!is_a<diracgammaR>(e1) && !is_a<diracgamma5>(e1)) {
590 // gammaR gammaR -> gammaR gammaR (do nothing)
591 // gamma5 gammaR -> gamma5 gammaR (do nothing)
592 // x gammaR -> gammaL x
594 *it = clifford(diracgammaL(), ex_to<clifford>(*it).get_representation_label());
595 something_changed = true;
603 if (next_to_last == first)
609 // Remove equal adjacent gammas
611 exvector::iterator it, itend = s.end() - 1;
612 for (it = s.begin(); it != itend; ++it) {
615 if (!is_a<clifford>(a) || !is_a<clifford>(b))
618 const ex & ag = a.op(0);
619 const ex & bg = b.op(0);
620 bool a_is_cliffordunit = is_a<cliffordunit>(ag);
621 bool b_is_cliffordunit = is_a<cliffordunit>(bg);
623 if (a_is_cliffordunit && b_is_cliffordunit && ex_to<clifford>(a).same_metric(b)
624 && (ex_to<clifford>(a).get_commutator_sign() == -1)) {
625 // This is done only for Clifford algebras
627 const ex & ia = a.op(1);
628 const ex & ib = b.op(1);
629 if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
630 a = ex_to<clifford>(a).get_metric(ia, ib, true);
631 b = dirac_ONE(representation_label);
632 something_changed = true;
635 } else if ((is_a<diracgamma5>(ag) && is_a<diracgamma5>(bg))) {
637 // Remove squares of gamma5
638 a = dirac_ONE(representation_label);
639 b = dirac_ONE(representation_label);
640 something_changed = true;
642 } else if ((is_a<diracgammaL>(ag) && is_a<diracgammaL>(bg))
643 || (is_a<diracgammaR>(ag) && is_a<diracgammaR>(bg))) {
645 // Remove squares of gammaL/R
646 b = dirac_ONE(representation_label);
647 something_changed = true;
649 } else if (is_a<diracgammaL>(ag) && is_a<diracgammaR>(bg)) {
651 // gammaL and gammaR are orthogonal
654 } else if (is_a<diracgamma5>(ag) && is_a<diracgammaL>(bg)) {
656 // gamma5 gammaL -> -gammaL
657 a = dirac_ONE(representation_label);
659 something_changed = true;
661 } else if (is_a<diracgamma5>(ag) && is_a<diracgammaR>(bg)) {
663 // gamma5 gammaR -> gammaR
664 a = dirac_ONE(representation_label);
665 something_changed = true;
667 } else if (!a_is_cliffordunit && !b_is_cliffordunit && ag.is_equal(bg)) {
670 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).minimal_dim(ex_to<idx>(b.op(1))));
672 a = indexed(ag, ix) * indexed(ag, ix.toggle_variance());
673 b = dirac_ONE(representation_label);
674 something_changed = true;
680 return dirac_ONE(representation_label) * sign;
681 if (something_changed)
682 return reeval_ncmul(s) * sign;
684 return hold_ncmul(s) * sign;
687 ex clifford::thiscontainer(const exvector & v) const
689 return clifford(representation_label, metric, commutator_sign, v);
692 ex clifford::thiscontainer(std::auto_ptr<exvector> vp) const
694 return clifford(representation_label, metric, commutator_sign, vp);
697 ex diracgamma5::conjugate() const
699 return _ex_1 * (*this);
702 ex diracgammaL::conjugate() const
704 return (new diracgammaR)->setflag(status_flags::dynallocated);
707 ex diracgammaR::conjugate() const
709 return (new diracgammaL)->setflag(status_flags::dynallocated);
716 ex dirac_ONE(unsigned char rl)
718 static ex ONE = (new diracone)->setflag(status_flags::dynallocated);
719 return clifford(ONE, rl);
722 static unsigned get_dim_uint(const ex& e)
725 throw std::invalid_argument("get_dim_uint: argument is not an index");
726 ex dim = ex_to<idx>(e).get_dim();
727 if (!dim.info(info_flags::posint))
728 throw std::invalid_argument("get_dim_uint: dimension of index should be a positive integer");
729 unsigned d = ex_to<numeric>(dim).to_int();
733 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl)
735 //static ex unit = (new cliffordunit)->setflag(status_flags::dynallocated);
736 ex unit = (new cliffordunit)->setflag(status_flags::dynallocated);
739 throw(std::invalid_argument("clifford_unit(): index of Clifford unit must be of type idx or varidx"));
741 exvector indices = metr.get_free_indices();
743 if (indices.size() == 2) {
744 return clifford(unit, mu, metr, rl);
745 } else if (is_a<matrix>(metr)) {
746 matrix M = ex_to<matrix>(metr);
747 unsigned n = M.rows();
748 bool symmetric = true;
750 //static idx xi((new symbol)->setflag(status_flags::dynallocated), n),
751 // chi((new symbol)->setflag(status_flags::dynallocated), n);
752 idx xi((new symbol)->setflag(status_flags::dynallocated), n),
753 chi((new symbol)->setflag(status_flags::dynallocated), n);
754 if ((n == M.cols()) && (n == get_dim_uint(mu))) {
755 for (unsigned i = 0; i < n; i++) {
756 for (unsigned j = i+1; j < n; j++) {
757 if (!M(i, j).is_equal(M(j, i))) {
762 return clifford(unit, mu, indexed(metr, symmetric?symmetric2():not_symmetric(), xi, chi), rl);
764 throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be a square matrix with the same dimensions as index"));
766 } else if (indices.size() == 0) { // a tensor or other expression without indices
767 //static varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim()),
768 // chi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim());
769 varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim()),
770 chi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim());
771 return clifford(unit, mu, indexed(metr, xi, chi), rl);
773 throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be of type tensor, matrix or an expression with two free indices"));
776 ex dirac_gamma(const ex & mu, unsigned char rl)
778 static ex gamma = (new diracgamma)->setflag(status_flags::dynallocated);
780 if (!is_a<varidx>(mu))
781 throw(std::invalid_argument("dirac_gamma(): index of Dirac gamma must be of type varidx"));
783 static varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim()),
784 chi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim());
785 return clifford(gamma, mu, indexed((new minkmetric)->setflag(status_flags::dynallocated), symmetric2(), xi, chi), rl);
788 ex dirac_gamma5(unsigned char rl)
790 static ex gamma5 = (new diracgamma5)->setflag(status_flags::dynallocated);
791 return clifford(gamma5, rl);
794 ex dirac_gammaL(unsigned char rl)
796 static ex gammaL = (new diracgammaL)->setflag(status_flags::dynallocated);
797 return clifford(gammaL, rl);
800 ex dirac_gammaR(unsigned char rl)
802 static ex gammaR = (new diracgammaR)->setflag(status_flags::dynallocated);
803 return clifford(gammaR, rl);
806 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
808 // Slashed vectors are actually stored as a clifford object with the
809 // vector as its base expression and a (dummy) index that just serves
810 // for storing the space dimensionality
812 static varidx xi((new symbol)->setflag(status_flags::dynallocated), dim),
813 chi((new symbol)->setflag(status_flags::dynallocated), dim);
814 return clifford(e, varidx(0, dim), indexed((new minkmetric)->setflag(status_flags::dynallocated), symmetric2(), xi, chi), rl);
817 /** Extract representation label from tinfo key (as returned by
818 * return_type_tinfo()). */
819 static unsigned char get_representation_label(const return_type_t& ti)
821 return (unsigned char)ti.rl;
824 /** Take trace of a string of an even number of Dirac gammas given a vector
826 static ex trace_string(exvector::const_iterator ix, size_t num)
828 // Tr gamma.mu gamma.nu = 4 g.mu.nu
830 return lorentz_g(ix[0], ix[1]);
832 // 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 )
834 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
835 + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
836 - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
838 // Traces of 6 or more gammas are computed recursively:
839 // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
840 // + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
841 // - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
842 // + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
844 // + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
848 for (size_t i=1; i<num; i++) {
849 for (size_t n=1, j=0; n<num; n++) {
854 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
860 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE)
862 if (is_a<clifford>(e)) {
864 unsigned char rl = ex_to<clifford>(e).get_representation_label();
866 // Are we taking the trace over this object's representation label?
867 if (rls.find(rl) == rls.end())
870 // Yes, all elements are traceless, except for dirac_ONE and dirac_L/R
871 const ex & g = e.op(0);
872 if (is_a<diracone>(g))
874 else if (is_a<diracgammaL>(g) || is_a<diracgammaR>(g))
879 } else if (is_exactly_a<mul>(e)) {
881 // Trace of product: pull out non-clifford factors
883 for (size_t i=0; i<e.nops(); i++) {
884 const ex &o = e.op(i);
885 if (is_clifford_tinfo(o.return_type_tinfo()))
886 prod *= dirac_trace(o, rls, trONE);
892 } else if (is_exactly_a<ncmul>(e)) {
894 unsigned char rl = get_representation_label(e.return_type_tinfo());
896 // Are we taking the trace over this string's representation label?
897 if (rls.find(rl) == rls.end())
900 // Substitute gammaL/R and expand product, if necessary
901 ex e_expanded = e.subs(lst(
902 dirac_gammaL(rl) == (dirac_ONE(rl)-dirac_gamma5(rl))/2,
903 dirac_gammaR(rl) == (dirac_ONE(rl)+dirac_gamma5(rl))/2
904 ), subs_options::no_pattern).expand();
905 if (!is_a<ncmul>(e_expanded))
906 return dirac_trace(e_expanded, rls, trONE);
908 // gamma5 gets moved to the front so this check is enough
909 bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
910 size_t num = e.nops();
914 // Trace of gamma5 * odd number of gammas and trace of
915 // gamma5 * gamma.mu * gamma.nu are zero
916 if ((num & 1) == 0 || num == 3)
919 // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
920 // (the epsilon is always 4-dimensional)
922 ex b1, i1, b2, i2, b3, i3, b4, i4;
923 base_and_index(e.op(1), b1, i1);
924 base_and_index(e.op(2), b2, i2);
925 base_and_index(e.op(3), b3, i3);
926 base_and_index(e.op(4), b4, i4);
927 return trONE * I * (lorentz_eps(ex_to<idx>(i1).replace_dim(_ex4), ex_to<idx>(i2).replace_dim(_ex4), ex_to<idx>(i3).replace_dim(_ex4), ex_to<idx>(i4).replace_dim(_ex4)) * b1 * b2 * b3 * b4).simplify_indexed();
931 // I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
932 // (the epsilon is always 4-dimensional)
933 exvector ix(num-1), bv(num-1);
934 for (size_t i=1; i<num; i++)
935 base_and_index(e.op(i), bv[i-1], ix[i-1]);
937 int *iv = new int[num];
939 for (size_t i=0; i<num-3; i++) {
941 for (size_t j=i+1; j<num-2; j++) {
943 for (size_t k=j+1; k<num-1; k++) {
945 for (size_t l=k+1; l<num; l++) {
947 iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
950 for (size_t n=0, t=4; n<num; n++) {
951 if (n == i || n == j || n == k || n == l)
956 int sign = permutation_sign(iv, iv + num);
957 result += sign * lorentz_eps(ex_to<idx>(idx1).replace_dim(_ex4), ex_to<idx>(idx2).replace_dim(_ex4), ex_to<idx>(idx3).replace_dim(_ex4), ex_to<idx>(idx4).replace_dim(_ex4))
958 * trace_string(v.begin(), num - 4);
964 return trONE * I * result * mul(bv);
966 } else { // no gamma5
968 // Trace of odd number of gammas is zero
972 // Tr gamma.mu gamma.nu = 4 g.mu.nu
975 base_and_index(e.op(0), b1, i1);
976 base_and_index(e.op(1), b2, i2);
977 return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
980 exvector iv(num), bv(num);
981 for (size_t i=0; i<num; i++)
982 base_and_index(e.op(i), bv[i], iv[i]);
984 return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
987 } else if (e.nops() > 0) {
989 // Trace maps to all other container classes (this includes sums)
990 pointer_to_map_function_2args<const std::set<unsigned char> &, const ex &> fcn(dirac_trace, rls, trONE);
997 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE)
999 // Convert list to set
1000 std::set<unsigned char> rls;
1001 for (lst::const_iterator i = rll.begin(); i != rll.end(); ++i) {
1002 if (i->info(info_flags::nonnegint))
1003 rls.insert(ex_to<numeric>(*i).to_int());
1006 return dirac_trace(e, rls, trONE);
1009 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
1011 // Convert label to set
1012 std::set<unsigned char> rls;
1015 return dirac_trace(e, rls, trONE);
1019 ex canonicalize_clifford(const ex & e_)
1021 pointer_to_map_function fcn(canonicalize_clifford);
1023 if (is_a<matrix>(e_) // || is_a<pseries>(e) || is_a<integral>(e)
1024 || e_.info(info_flags::list)) {
1027 ex e=simplify_indexed(e_);
1028 // Scan for any ncmul objects
1030 ex aux = e.to_rational(srl);
1031 for (exmap::iterator i = srl.begin(); i != srl.end(); ++i) {
1036 if (is_exactly_a<ncmul>(rhs)
1037 && rhs.return_type() == return_types::noncommutative
1038 && is_clifford_tinfo(rhs.return_type_tinfo())) {
1040 // Expand product, if necessary
1041 ex rhs_expanded = rhs.expand();
1042 if (!is_a<ncmul>(rhs_expanded)) {
1043 i->second = canonicalize_clifford(rhs_expanded);
1046 } else if (!is_a<clifford>(rhs.op(0)))
1050 v.reserve(rhs.nops());
1051 for (size_t j=0; j<rhs.nops(); j++)
1052 v.push_back(rhs.op(j));
1054 // Stupid recursive bubble sort because we only want to swap adjacent gammas
1055 exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
1056 if (is_a<diracgamma5>(it->op(0)) || is_a<diracgammaL>(it->op(0)) || is_a<diracgammaR>(it->op(0)))
1059 while (it != next_to_last) {
1060 if (it[0].compare(it[1]) > 0) {
1062 ex save0 = it[0], save1 = it[1];
1064 base_and_index(it[0], b1, i1);
1065 base_and_index(it[1], b2, i2);
1066 // for Clifford algebras (commutator_sign == -1) metric should be symmetrised
1067 it[0] = (ex_to<clifford>(save0).get_metric(i1, i2, ex_to<clifford>(save0).get_commutator_sign() == -1) * b1 * b2).simplify_indexed();
1068 it[1] = v.size() ? _ex2 * dirac_ONE(ex_to<clifford>(save0).get_representation_label()) : _ex2;
1072 sum += ex_to<clifford>(save0).get_commutator_sign() * ncmul(v, true);
1073 i->second = canonicalize_clifford(sum);
1081 return aux.subs(srl, subs_options::no_pattern).simplify_indexed();
1085 ex clifford_prime(const ex & e)
1087 pointer_to_map_function fcn(clifford_prime);
1088 if (is_a<clifford>(e) && is_a<cliffordunit>(e.op(0))) {
1090 } else if (is_a<add>(e) || is_a<ncmul>(e) || is_a<mul>(e) //|| is_a<pseries>(e) || is_a<integral>(e)
1091 || is_a<matrix>(e) || e.info(info_flags::list)) {
1093 } else if (is_a<power>(e)) {
1094 return pow(clifford_prime(e.op(0)), e.op(1));
1099 ex remove_dirac_ONE(const ex & e, unsigned char rl, unsigned options)
1101 pointer_to_map_function_2args<unsigned char, unsigned> fcn(remove_dirac_ONE, rl, options | 1);
1102 bool need_reevaluation = false;
1104 if (! (options & 1) ) { // is not a child
1106 e1 = expand_dummy_sum(e, true);
1107 e1 = canonicalize_clifford(e1);
1110 if (is_a<clifford>(e1) && ex_to<clifford>(e1).get_representation_label() >= rl) {
1111 if (is_a<diracone>(e1.op(0)))
1114 throw(std::invalid_argument("remove_dirac_ONE(): expression is a non-scalar Clifford number!"));
1115 } else if (is_a<add>(e1) || is_a<ncmul>(e1) || is_a<mul>(e1)
1116 || is_a<matrix>(e1) || e1.info(info_flags::list)) {
1117 if (options & 3) // is a child or was already expanded
1122 } catch (std::exception &p) {
1123 need_reevaluation = true;
1125 } else if (is_a<power>(e1)) {
1126 if (options & 3) // is a child or was already expanded
1127 return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
1130 return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
1131 } catch (std::exception &p) {
1132 need_reevaluation = true;
1135 if (need_reevaluation)
1136 return remove_dirac_ONE(e, rl, options | 2);
1140 char clifford_max_label(const ex & e, bool ignore_ONE)
1142 if (is_a<clifford>(e))
1143 if (ignore_ONE && is_a<diracone>(e.op(0)))
1146 return ex_to<clifford>(e).get_representation_label();
1149 for (size_t i=0; i < e.nops(); i++)
1150 rl = (rl > clifford_max_label(e.op(i), ignore_ONE)) ? rl : clifford_max_label(e.op(i), ignore_ONE);
1155 ex clifford_norm(const ex & e)
1157 return sqrt(remove_dirac_ONE(e * clifford_bar(e)));
1160 ex clifford_inverse(const ex & e)
1162 ex norm = clifford_norm(e);
1163 if (!norm.is_zero())
1164 return clifford_bar(e) / pow(norm, 2);
1166 throw(std::invalid_argument("clifford_inverse(): cannot find inverse of Clifford number with zero norm!"));
1169 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl)
1171 if (!ex_to<idx>(mu).is_dim_numeric())
1172 throw(std::invalid_argument("lst_to_clifford(): Index should have a numeric dimension"));
1173 ex e = clifford_unit(mu, metr, rl);
1174 return lst_to_clifford(v, e);
1177 ex lst_to_clifford(const ex & v, const ex & e) {
1180 if (is_a<clifford>(e)) {
1183 = is_a<varidx>(mu) ? ex_to<varidx>(mu).toggle_variance() : mu;
1184 unsigned dim = get_dim_uint(mu);
1186 if (is_a<matrix>(v)) {
1187 if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows()) {
1188 min = ex_to<matrix>(v).rows();
1189 max = ex_to<matrix>(v).cols();
1191 min = ex_to<matrix>(v).cols();
1192 max = ex_to<matrix>(v).rows();
1196 return indexed(v, mu_toggle) * e;
1197 else if (max - dim == 1) {
1198 if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows())
1199 return v.op(0) * dirac_ONE(ex_to<clifford>(e).get_representation_label()) + indexed(sub_matrix(ex_to<matrix>(v), 0, 1, 1, dim), mu_toggle) * e;
1201 return v.op(0) * dirac_ONE(ex_to<clifford>(e).get_representation_label()) + indexed(sub_matrix(ex_to<matrix>(v), 1, dim, 0, 1), mu_toggle) * e;
1203 throw(std::invalid_argument("lst_to_clifford(): dimensions of vector and clifford unit mismatch"));
1205 throw(std::invalid_argument("lst_to_clifford(): first argument should be a vector (nx1 or 1xn matrix)"));
1206 } else if (v.info(info_flags::list)) {
1207 if (dim == ex_to<lst>(v).nops())
1208 return indexed(matrix(dim, 1, ex_to<lst>(v)), mu_toggle) * e;
1209 else if (ex_to<lst>(v).nops() - dim == 1)
1210 return v.op(0) * dirac_ONE(ex_to<clifford>(e).get_representation_label()) + indexed(sub_matrix(matrix(dim+1, 1, ex_to<lst>(v)), 1, dim, 0, 1), mu_toggle) * e;
1212 throw(std::invalid_argument("lst_to_clifford(): list length and dimension of clifford unit mismatch"));
1214 throw(std::invalid_argument("lst_to_clifford(): cannot construct from anything but list or vector"));
1216 throw(std::invalid_argument("lst_to_clifford(): the second argument should be a Clifford unit"));
1219 /** Auxiliary structure to define a function for striping one Clifford unit
1220 * from vectors. Used in clifford_to_lst(). */
1221 static ex get_clifford_comp(const ex & e, const ex & c)
1223 pointer_to_map_function_1arg<const ex &> fcn(get_clifford_comp, c);
1224 int ival = ex_to<numeric>(ex_to<idx>(c.op(1)).get_value()).to_int();
1226 if (is_a<add>(e) || e.info(info_flags::list) // || is_a<pseries>(e) || is_a<integral>(e)
1229 else if (is_a<ncmul>(e) || is_a<mul>(e)) {
1230 // find a Clifford unit with the same metric, delete it and substitute its index
1231 size_t ind = e.nops() + 1;
1232 for (size_t j = 0; j < e.nops(); j++)
1233 if (is_a<clifford>(e.op(j)) && ex_to<clifford>(c).same_metric(e.op(j)))
1237 throw(std::invalid_argument("get_clifford_comp(): expression is a Clifford multi-vector"));
1238 if (ind < e.nops()) {
1240 bool same_value_index, found_dummy;
1241 same_value_index = ( ex_to<idx>(e.op(ind).op(1)).is_numeric()
1242 && (ival == ex_to<numeric>(ex_to<idx>(e.op(ind).op(1)).get_value()).to_int()) );
1243 found_dummy = same_value_index;
1244 for(size_t j=0; j < e.nops(); j++)
1246 if (same_value_index)
1249 exvector ind_vec = ex_to<indexed>(e.op(j)).get_dummy_indices(ex_to<indexed>(e.op(ind)));
1250 if (ind_vec.size() > 0) {
1252 exvector::const_iterator it = ind_vec.begin(), itend = ind_vec.end();
1253 while (it != itend) {
1255 ex curridx_toggle = is_a<varidx>(curridx)
1256 ? ex_to<varidx>(curridx).toggle_variance()
1258 S = S * e.op(j).subs(lst(curridx == ival,
1259 curridx_toggle == ival), subs_options::no_pattern);
1265 return (found_dummy ? S : 0);
1267 throw(std::invalid_argument("get_clifford_comp(): expression is not a Clifford vector to the given units"));
1268 } else if (e.is_zero())
1270 else if (is_a<clifford>(e) && ex_to<clifford>(e).same_metric(c))
1271 if ( ex_to<idx>(e.op(1)).is_numeric() &&
1272 (ival != ex_to<numeric>(ex_to<idx>(e.op(1)).get_value()).to_int()) )
1277 throw(std::invalid_argument("get_clifford_comp(): expression is not usable as a Clifford vector"));
1281 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic)
1283 GINAC_ASSERT(is_a<clifford>(c));
1285 if (! ex_to<idx>(mu).is_dim_numeric())
1286 throw(std::invalid_argument("clifford_to_lst(): index should have a numeric dimension"));
1287 unsigned int D = ex_to<numeric>(ex_to<idx>(mu).get_dim()).to_int();
1289 if (algebraic) // check if algebraic method is applicable
1290 for (unsigned int i = 0; i < D; i++)
1291 if (pow(c.subs(mu == i, subs_options::no_pattern), 2).is_zero()
1292 or (not is_a<numeric>(pow(c.subs(mu == i, subs_options::no_pattern), 2))))
1295 ex v0 = remove_dirac_ONE(canonicalize_clifford(e+clifford_prime(e)).normal())/2;
1296 if (not v0.is_zero())
1298 ex e1 = canonicalize_clifford(e - v0 * dirac_ONE(ex_to<clifford>(c).get_representation_label()));
1300 for (unsigned int i = 0; i < D; i++)
1301 V.append(remove_dirac_ONE(
1302 simplify_indexed(canonicalize_clifford(e1 * c.subs(mu == i, subs_options::no_pattern) + c.subs(mu == i, subs_options::no_pattern) * e1))
1303 / (2*pow(c.subs(mu == i, subs_options::no_pattern), 2))));
1306 for (unsigned int i = 0; i < D; i++)
1307 V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
1308 } catch (std::exception &p) {
1309 /* Try to expand dummy summations to simplify the expression*/
1310 e1 = canonicalize_clifford(expand_dummy_sum(e, true));
1312 v0 = remove_dirac_ONE(canonicalize_clifford(e1+clifford_prime(e1)).normal())/2;
1313 if (not v0.is_zero()) {
1315 e1 = canonicalize_clifford(e1 - v0 * dirac_ONE(ex_to<clifford>(c).get_representation_label()));
1317 for (unsigned int i = 0; i < D; i++)
1318 V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
1325 ex clifford_moebius_map(const ex & a, const ex & b, const ex & c, const ex & d, const ex & v, const ex & G, unsigned char rl)
1329 if (! is_a<matrix>(v) && ! v.info(info_flags::list))
1330 throw(std::invalid_argument("clifford_moebius_map(): parameter v should be either vector or list"));
1332 if (is_a<clifford>(G)) {
1335 if (is_a<indexed>(G)) {
1336 D = ex_to<idx>(G.op(1)).get_dim();
1337 varidx mu((new symbol)->setflag(status_flags::dynallocated), D);
1338 cu = clifford_unit(mu, G, rl);
1339 } else if (is_a<matrix>(G)) {
1340 D = ex_to<matrix>(G).rows();
1341 idx mu((new symbol)->setflag(status_flags::dynallocated), D);
1342 cu = clifford_unit(mu, G, rl);
1343 } else throw(std::invalid_argument("clifford_moebius_map(): metric should be an indexed object, matrix, or a Clifford unit"));
1347 x = lst_to_clifford(v, cu);
1348 ex e = clifford_to_lst(simplify_indexed(canonicalize_clifford((a * x + b) * clifford_inverse(c * x + d))), cu, false);
1349 return (is_a<matrix>(v) ? matrix(ex_to<matrix>(v).rows(), ex_to<matrix>(v).cols(), ex_to<lst>(e)) : e);
1352 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl)
1354 if (is_a<matrix>(M) && (ex_to<matrix>(M).rows() == 2) && (ex_to<matrix>(M).cols() == 2))
1355 return clifford_moebius_map(M.op(0), M.op(1), M.op(2), M.op(3), v, G, rl);
1357 throw(std::invalid_argument("clifford_moebius_map(): parameter M should be a 2x2 matrix"));
1360 } // namespace GiNaC