3 * Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
6 * GiNaC Copyright (C) 1999-2010 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
29 #include "numeric.h" // for I
32 #include "relational.h"
33 #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)
81 DEFAULT_CTOR(diracone)
82 DEFAULT_CTOR(cliffordunit)
83 DEFAULT_CTOR(diracgamma)
84 DEFAULT_CTOR(diracgamma5)
85 DEFAULT_CTOR(diracgammaL)
86 DEFAULT_CTOR(diracgammaR)
92 /** Construct object without any indices. This constructor is for internal
93 * use only. Use the dirac_ONE() function instead.
95 clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl), metric(0), commutator_sign(-1)
99 /** Construct object with one Lorentz index. This constructor is for internal
100 * use only. Use the clifford_unit() or dirac_gamma() functions instead.
102 * @see dirac_gamma */
103 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)
105 GINAC_ASSERT(is_a<varidx>(mu));
108 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)
112 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)
116 return_type_t clifford::return_type_tinfo() const
118 return make_return_type_t<clifford>(representation_label);
125 void clifford::read_archive(const archive_node& n, lst& sym_lst)
127 inherited::read_archive(n, sym_lst);
129 n.find_unsigned("label", rl);
130 representation_label = rl;
131 n.find_ex("metric", metric, sym_lst);
132 n.find_unsigned("commutator_sign+1", rl);
133 commutator_sign = rl - 1;
136 void clifford::archive(archive_node & n) const
138 inherited::archive(n);
139 n.add_unsigned("label", representation_label);
140 n.add_ex("metric", metric);
141 n.add_unsigned("commutator_sign+1", commutator_sign+1);
144 GINAC_BIND_UNARCHIVER(clifford);
145 GINAC_BIND_UNARCHIVER(diracone);
146 GINAC_BIND_UNARCHIVER(diracgamma);
147 GINAC_BIND_UNARCHIVER(diracgamma5);
148 GINAC_BIND_UNARCHIVER(diracgammaL);
149 GINAC_BIND_UNARCHIVER(diracgammaR);
152 ex clifford::get_metric(const ex & i, const ex & j, bool symmetrised) const
154 if (is_a<indexed>(metric)) {
155 if (symmetrised && !(ex_to<symmetry>(ex_to<indexed>(metric).get_symmetry()).has_symmetry())) {
156 if (is_a<matrix>(metric.op(0))) {
157 return indexed((ex_to<matrix>(metric.op(0)).add(ex_to<matrix>(metric.op(0)).transpose())).mul(numeric(1, 2)),
160 return simplify_indexed(indexed(metric.op(0)*_ex1_2, i, j) + indexed(metric.op(0)*_ex1_2, j, i));
163 return metric.subs(lst(metric.op(1) == i, metric.op(2) == j), subs_options::no_pattern);
166 exvector indices = metric.get_free_indices();
168 return _ex1_2*simplify_indexed(metric.subs(lst(indices[0] == i, indices[1] == j), subs_options::no_pattern)
169 + metric.subs(lst(indices[0] == j, indices[1] == i), subs_options::no_pattern));
171 return metric.subs(lst(indices[0] == i, indices[1] == j), subs_options::no_pattern);
175 bool clifford::same_metric(const ex & other) const
178 if (is_a<clifford>(other))
179 metr = ex_to<clifford>(other).get_metric();
183 if (is_a<indexed>(metr))
184 return metr.op(0).is_equal(get_metric().op(0));
186 exvector indices = metr.get_free_indices();
187 return (indices.size() == 2)
188 && simplify_indexed(get_metric(indices[0], indices[1])-metr).is_zero();
193 // functions overriding virtual functions from base classes
196 ex clifford::op(size_t i) const
198 GINAC_ASSERT(i<nops());
200 return representation_label;
202 return inherited::op(i);
205 ex & clifford::let_op(size_t i)
207 GINAC_ASSERT(i<nops());
209 static ex rl = numeric(representation_label);
210 ensure_if_modifiable();
214 return inherited::let_op(i);
217 ex clifford::subs(const exmap & m, unsigned options) const
219 ex subsed = inherited::subs(m, options);
220 if(is_a<clifford>(subsed)) {
221 ex prevmetric = ex_to<clifford>(subsed).metric;
222 ex newmetric = prevmetric.subs(m, options);
223 if(!are_ex_trivially_equal(prevmetric, newmetric)) {
224 clifford c = ex_to<clifford>(subsed);
225 c.metric = newmetric;
232 int clifford::compare_same_type(const basic & other) const
234 GINAC_ASSERT(is_a<clifford>(other));
235 const clifford &o = static_cast<const clifford &>(other);
237 if (representation_label != o.representation_label) {
238 // different representation label
239 return representation_label < o.representation_label ? -1 : 1;
242 return inherited::compare_same_type(other);
245 bool clifford::match_same_type(const basic & other) const
247 GINAC_ASSERT(is_a<clifford>(other));
248 const clifford &o = static_cast<const clifford &>(other);
250 return ((representation_label == o.representation_label) && (commutator_sign == o.get_commutator_sign()) && same_metric(o));
253 static bool is_dirac_slash(const ex & seq0)
255 return !is_a<diracgamma5>(seq0) && !is_a<diracgammaL>(seq0) &&
256 !is_a<diracgammaR>(seq0) && !is_a<cliffordunit>(seq0) &&
257 !is_a<diracone>(seq0);
260 void clifford::do_print_dflt(const print_dflt & c, unsigned level) const
262 // dirac_slash() object is printed differently
263 if (is_dirac_slash(seq[0])) {
264 seq[0].print(c, precedence());
266 } else { // We do not print representation label if it is 0
267 if (representation_label == 0) {
268 this->print_dispatch<inherited>(c, level);
269 } else { // otherwise we put it before indices in square brackets; the code is borrowed from indexed.cpp
270 if (precedence() <= level) {
273 seq[0].print(c, precedence());
274 c.s << '[' << int(representation_label) << ']';
275 printindices(c, level);
276 if (precedence() <= level) {
283 void clifford::do_print_latex(const print_latex & c, unsigned level) const
285 // dirac_slash() object is printed differently
286 if (is_dirac_slash(seq[0])) {
288 seq[0].print(c, precedence());
289 c.s << "\\hspace{-1.0ex}/}";
291 c.s << "\\clifford[" << int(representation_label) << "]";
292 this->print_dispatch<inherited>(c, level);
296 DEFAULT_COMPARE(diracone)
297 DEFAULT_COMPARE(cliffordunit)
298 DEFAULT_COMPARE(diracgamma)
299 DEFAULT_COMPARE(diracgamma5)
300 DEFAULT_COMPARE(diracgammaL)
301 DEFAULT_COMPARE(diracgammaR)
303 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbf{1}")
304 DEFAULT_PRINT_LATEX(cliffordunit, "e", "e")
305 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
306 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
307 DEFAULT_PRINT_LATEX(diracgammaL, "gammaL", "{\\gamma_L}")
308 DEFAULT_PRINT_LATEX(diracgammaR, "gammaR", "{\\gamma_R}")
310 /** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
311 static void base_and_index(const ex & c, ex & b, ex & i)
313 GINAC_ASSERT(is_a<clifford>(c));
314 GINAC_ASSERT(c.nops() == 2+1);
316 if (is_a<cliffordunit>(c.op(0))) { // proper dirac gamma object or clifford unit
319 } else if (is_a<diracgamma5>(c.op(0)) || is_a<diracgammaL>(c.op(0)) || is_a<diracgammaR>(c.op(0))) { // gamma5/L/R
322 } else { // slash object, generate new dummy index
323 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(c.op(1)).get_dim());
324 b = indexed(c.op(0), ix.toggle_variance());
329 /** Predicate for finding non-clifford objects. */
330 struct is_not_a_clifford : public std::unary_function<ex, bool> {
331 bool operator()(const ex & e)
333 return !is_a<clifford>(e);
337 /** Contraction of a gamma matrix with something else. */
338 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
340 GINAC_ASSERT(is_a<clifford>(*self));
341 GINAC_ASSERT(is_a<indexed>(*other));
342 GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
343 unsigned char rl = ex_to<clifford>(*self).get_representation_label();
345 ex dim = ex_to<idx>(self->op(1)).get_dim();
346 if (other->nops() > 1)
347 dim = minimal_dim(dim, ex_to<idx>(other->op(1)).get_dim());
349 if (is_a<clifford>(*other)) {
351 // Contraction only makes sense if the represenation labels are equal
352 if (ex_to<clifford>(*other).get_representation_label() != rl)
355 size_t num = other - self;
357 // gamma~mu gamma.mu = dim ONE
360 *other = dirac_ONE(rl);
363 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
365 && is_a<clifford>(self[1])) {
370 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
372 && is_a<clifford>(self[1])
373 && is_a<clifford>(self[2])) {
375 base_and_index(self[1], b1, i1);
376 base_and_index(self[2], b2, i2);
377 *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
383 // 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
385 && is_a<clifford>(self[1])
386 && is_a<clifford>(self[2])
387 && is_a<clifford>(self[3])) {
388 *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3];
395 // gamma~mu Sodd gamma.mu = -2 Sodd_R
396 // (Chisholm identity in 4 dimensions)
397 } else if (!((other - self) & 1) && dim.is_equal(4)) {
398 if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
401 *self = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(other), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
402 std::fill(self + 1, other, _ex1);
406 // gamma~mu Sodd gamma~alpha gamma.mu = 2 gamma~alpha Sodd + 2 Sodd_R gamma~alpha
407 // (commutate contracted indices towards each other, then use
408 // Chisholm identity in 4 dimensions)
409 } else if (((other - self) & 1) && dim.is_equal(4)) {
410 if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
413 exvector::iterator next_to_last = other - 1;
414 ex S = ncmul(exvector(self + 1, next_to_last), true);
415 ex SR = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(next_to_last), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
417 *self = (*next_to_last) * S + SR * (*next_to_last);
418 std::fill(self + 1, other, _ex1);
422 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
423 // (commutate contracted indices towards each other, simplify_indexed()
424 // will re-expand and re-run the simplification)
426 if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
429 exvector::iterator next_to_last = other - 1;
430 ex S = ncmul(exvector(self + 1, next_to_last), true);
432 *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
433 std::fill(self + 1, other + 1, _ex1);
437 } else if (is_a<symbol>(other->op(0)) && other->nops() == 2) {
439 // x.mu gamma~mu -> x-slash
440 *self = dirac_slash(other->op(0), dim, rl);
448 /** Contraction of a Clifford unit with something else. */
449 bool cliffordunit::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
451 GINAC_ASSERT(is_a<clifford>(*self));
452 GINAC_ASSERT(is_a<indexed>(*other));
453 GINAC_ASSERT(is_a<cliffordunit>(self->op(0)));
454 clifford unit = ex_to<clifford>(*self);
455 unsigned char rl = unit.get_representation_label();
457 if (is_a<clifford>(*other)) {
458 // Contraction only makes sense if the represenation labels are equal
459 // and the metrics are the same
460 if ((ex_to<clifford>(*other).get_representation_label() != rl)
461 && unit.same_metric(*other))
464 exvector::iterator before_other = other - 1;
466 ex mu_toggle = other->op(1);
467 ex alpha = before_other->op(1);
469 // e~mu e.mu = Tr ONE
470 if (other - self == 1) {
471 *self = unit.get_metric(mu, mu_toggle, true);
472 *other = dirac_ONE(rl);
475 } else if (other - self == 2) {
476 if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
477 // e~mu e~alpha e.mu = 2*e~mu B(alpha, mu.toggle_variance())-Tr(B) e~alpha
478 *self = 2 * (*self) * unit.get_metric(alpha, mu_toggle, true) - unit.get_metric(mu, mu_toggle, true) * (*before_other);
479 *before_other = _ex1;
484 // e~mu S e.mu = Tr S ONE
485 *self = unit.get_metric(mu, mu_toggle, true);
486 *other = dirac_ONE(rl);
490 // e~mu S e~alpha e.mu = 2 e~mu S B(alpha, mu.toggle_variance()) - e~mu S e.mu e~alpha
491 // (commutate contracted indices towards each other, simplify_indexed()
492 // will re-expand and re-run the simplification)
493 if (std::find_if(self + 1, other, is_not_a_clifford()) != other) {
497 ex S = ncmul(exvector(self + 1, before_other), true);
499 if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
500 *self = 2 * (*self) * S * unit.get_metric(alpha, mu_toggle, true) - (*self) * S * (*other) * (*before_other);
503 *self = (*self) * S * (*other) * (*before_other);
506 std::fill(self + 1, other + 1, _ex1);
513 /** Perform automatic simplification on noncommutative product of clifford
514 * objects. This removes superfluous ONEs, permutes gamma5/L/R's to the front
515 * and removes squares of gamma objects. */
516 ex clifford::eval_ncmul(const exvector & v) const
521 // Remove superfluous ONEs
522 exvector::const_iterator cit = v.begin(), citend = v.end();
523 while (cit != citend) {
524 if (!is_a<clifford>(*cit) || !is_a<diracone>(cit->op(0)))
529 bool something_changed = false;
532 // Anticommutate gamma5/L/R's to the front
534 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
536 exvector::iterator it = next_to_last;
538 exvector::iterator it2 = it + 1;
539 if (is_a<clifford>(*it) && is_a<clifford>(*it2)) {
540 ex e1 = it->op(0), e2 = it2->op(0);
542 if (is_a<diracgamma5>(e2)) {
544 if (is_a<diracgammaL>(e1) || is_a<diracgammaR>(e1)) {
546 // gammaL/R gamma5 -> gamma5 gammaL/R
548 something_changed = true;
550 } else if (!is_a<diracgamma5>(e1)) {
552 // gamma5 gamma5 -> gamma5 gamma5 (do nothing)
553 // x gamma5 -> -gamma5 x
556 something_changed = true;
559 } else if (is_a<diracgammaL>(e2)) {
561 if (is_a<diracgammaR>(e1)) {
563 // gammaR gammaL -> 0
566 } else if (!is_a<diracgammaL>(e1) && !is_a<diracgamma5>(e1)) {
568 // gammaL gammaL -> gammaL gammaL (do nothing)
569 // gamma5 gammaL -> gamma5 gammaL (do nothing)
570 // x gammaL -> gammaR x
572 *it = clifford(diracgammaR(), ex_to<clifford>(*it).get_representation_label());
573 something_changed = true;
576 } else if (is_a<diracgammaR>(e2)) {
578 if (is_a<diracgammaL>(e1)) {
580 // gammaL gammaR -> 0
583 } else if (!is_a<diracgammaR>(e1) && !is_a<diracgamma5>(e1)) {
585 // gammaR gammaR -> gammaR gammaR (do nothing)
586 // gamma5 gammaR -> gamma5 gammaR (do nothing)
587 // x gammaR -> gammaL x
589 *it = clifford(diracgammaL(), ex_to<clifford>(*it).get_representation_label());
590 something_changed = true;
598 if (next_to_last == first)
604 // Remove equal adjacent gammas
606 exvector::iterator it, itend = s.end() - 1;
607 for (it = s.begin(); it != itend; ++it) {
610 if (!is_a<clifford>(a) || !is_a<clifford>(b))
613 const ex & ag = a.op(0);
614 const ex & bg = b.op(0);
615 bool a_is_cliffordunit = is_a<cliffordunit>(ag);
616 bool b_is_cliffordunit = is_a<cliffordunit>(bg);
618 if (a_is_cliffordunit && b_is_cliffordunit && ex_to<clifford>(a).same_metric(b)
619 && (ex_to<clifford>(a).get_commutator_sign() == -1)) {
620 // This is done only for Clifford algebras
622 const ex & ia = a.op(1);
623 const ex & ib = b.op(1);
624 if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
625 a = ex_to<clifford>(a).get_metric(ia, ib, true);
626 b = dirac_ONE(representation_label);
627 something_changed = true;
630 } else if ((is_a<diracgamma5>(ag) && is_a<diracgamma5>(bg))) {
632 // Remove squares of gamma5
633 a = dirac_ONE(representation_label);
634 b = dirac_ONE(representation_label);
635 something_changed = true;
637 } else if ((is_a<diracgammaL>(ag) && is_a<diracgammaL>(bg))
638 || (is_a<diracgammaR>(ag) && is_a<diracgammaR>(bg))) {
640 // Remove squares of gammaL/R
641 b = dirac_ONE(representation_label);
642 something_changed = true;
644 } else if (is_a<diracgammaL>(ag) && is_a<diracgammaR>(bg)) {
646 // gammaL and gammaR are orthogonal
649 } else if (is_a<diracgamma5>(ag) && is_a<diracgammaL>(bg)) {
651 // gamma5 gammaL -> -gammaL
652 a = dirac_ONE(representation_label);
654 something_changed = true;
656 } else if (is_a<diracgamma5>(ag) && is_a<diracgammaR>(bg)) {
658 // gamma5 gammaR -> gammaR
659 a = dirac_ONE(representation_label);
660 something_changed = true;
662 } else if (!a_is_cliffordunit && !b_is_cliffordunit && ag.is_equal(bg)) {
665 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).minimal_dim(ex_to<idx>(b.op(1))));
667 a = indexed(ag, ix) * indexed(ag, ix.toggle_variance());
668 b = dirac_ONE(representation_label);
669 something_changed = true;
675 return dirac_ONE(representation_label) * sign;
676 if (something_changed)
677 return reeval_ncmul(s) * sign;
679 return hold_ncmul(s) * sign;
682 ex clifford::thiscontainer(const exvector & v) const
684 return clifford(representation_label, metric, commutator_sign, v);
687 ex clifford::thiscontainer(std::auto_ptr<exvector> vp) const
689 return clifford(representation_label, metric, commutator_sign, vp);
692 ex diracgamma5::conjugate() const
694 return _ex_1 * (*this);
697 ex diracgammaL::conjugate() const
699 return (new diracgammaR)->setflag(status_flags::dynallocated);
702 ex diracgammaR::conjugate() const
704 return (new diracgammaL)->setflag(status_flags::dynallocated);
711 ex dirac_ONE(unsigned char rl)
713 static ex ONE = (new diracone)->setflag(status_flags::dynallocated);
714 return clifford(ONE, rl);
717 static unsigned get_dim_uint(const ex& e)
720 throw std::invalid_argument("get_dim_uint: argument is not an index");
721 ex dim = ex_to<idx>(e).get_dim();
722 if (!dim.info(info_flags::posint))
723 throw std::invalid_argument("get_dim_uint: dimension of index should be a positive integer");
724 unsigned d = ex_to<numeric>(dim).to_int();
728 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl)
730 //static ex unit = (new cliffordunit)->setflag(status_flags::dynallocated);
731 ex unit = (new cliffordunit)->setflag(status_flags::dynallocated);
734 throw(std::invalid_argument("clifford_unit(): index of Clifford unit must be of type idx or varidx"));
736 exvector indices = metr.get_free_indices();
738 if (indices.size() == 2) {
739 return clifford(unit, mu, metr, rl);
740 } else if (is_a<matrix>(metr)) {
741 matrix M = ex_to<matrix>(metr);
742 unsigned n = M.rows();
743 bool symmetric = true;
745 //static idx xi((new symbol)->setflag(status_flags::dynallocated), n),
746 // chi((new symbol)->setflag(status_flags::dynallocated), n);
747 idx xi((new symbol)->setflag(status_flags::dynallocated), n),
748 chi((new symbol)->setflag(status_flags::dynallocated), n);
749 if ((n == M.cols()) && (n == get_dim_uint(mu))) {
750 for (unsigned i = 0; i < n; i++) {
751 for (unsigned j = i+1; j < n; j++) {
752 if (!M(i, j).is_equal(M(j, i))) {
757 return clifford(unit, mu, indexed(metr, symmetric?symmetric2():not_symmetric(), xi, chi), rl);
759 throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be a square matrix with the same dimensions as index"));
761 } else if (indices.size() == 0) { // a tensor or other expression without indices
762 //static varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim()),
763 // chi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim());
764 varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim()),
765 chi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim());
766 return clifford(unit, mu, indexed(metr, xi, chi), rl);
768 throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be of type tensor, matrix or an expression with two free indices"));
771 ex dirac_gamma(const ex & mu, unsigned char rl)
773 static ex gamma = (new diracgamma)->setflag(status_flags::dynallocated);
775 if (!is_a<varidx>(mu))
776 throw(std::invalid_argument("dirac_gamma(): index of Dirac gamma must be of type varidx"));
778 static varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim()),
779 chi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim());
780 return clifford(gamma, mu, indexed((new minkmetric)->setflag(status_flags::dynallocated), symmetric2(), xi, chi), rl);
783 ex dirac_gamma5(unsigned char rl)
785 static ex gamma5 = (new diracgamma5)->setflag(status_flags::dynallocated);
786 return clifford(gamma5, rl);
789 ex dirac_gammaL(unsigned char rl)
791 static ex gammaL = (new diracgammaL)->setflag(status_flags::dynallocated);
792 return clifford(gammaL, rl);
795 ex dirac_gammaR(unsigned char rl)
797 static ex gammaR = (new diracgammaR)->setflag(status_flags::dynallocated);
798 return clifford(gammaR, rl);
801 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
803 // Slashed vectors are actually stored as a clifford object with the
804 // vector as its base expression and a (dummy) index that just serves
805 // for storing the space dimensionality
807 static varidx xi((new symbol)->setflag(status_flags::dynallocated), dim),
808 chi((new symbol)->setflag(status_flags::dynallocated), dim);
809 return clifford(e, varidx(0, dim), indexed((new minkmetric)->setflag(status_flags::dynallocated), symmetric2(), xi, chi), rl);
812 /** Extract representation label from tinfo key (as returned by
813 * return_type_tinfo()). */
814 static unsigned char get_representation_label(const return_type_t& ti)
816 return (unsigned char)ti.rl;
819 /** Take trace of a string of an even number of Dirac gammas given a vector
821 static ex trace_string(exvector::const_iterator ix, size_t num)
823 // Tr gamma.mu gamma.nu = 4 g.mu.nu
825 return lorentz_g(ix[0], ix[1]);
827 // 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 )
829 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
830 + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
831 - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
833 // Traces of 6 or more gammas are computed recursively:
834 // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
835 // + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
836 // - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
837 // + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
839 // + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
843 for (size_t i=1; i<num; i++) {
844 for (size_t n=1, j=0; n<num; n++) {
849 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
855 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE)
857 if (is_a<clifford>(e)) {
859 unsigned char rl = ex_to<clifford>(e).get_representation_label();
861 // Are we taking the trace over this object's representation label?
862 if (rls.find(rl) == rls.end())
865 // Yes, all elements are traceless, except for dirac_ONE and dirac_L/R
866 const ex & g = e.op(0);
867 if (is_a<diracone>(g))
869 else if (is_a<diracgammaL>(g) || is_a<diracgammaR>(g))
874 } else if (is_exactly_a<mul>(e)) {
876 // Trace of product: pull out non-clifford factors
878 for (size_t i=0; i<e.nops(); i++) {
879 const ex &o = e.op(i);
880 if (is_clifford_tinfo(o.return_type_tinfo()))
881 prod *= dirac_trace(o, rls, trONE);
887 } else if (is_exactly_a<ncmul>(e)) {
889 unsigned char rl = get_representation_label(e.return_type_tinfo());
891 // Are we taking the trace over this string's representation label?
892 if (rls.find(rl) == rls.end())
895 // Substitute gammaL/R and expand product, if necessary
896 ex e_expanded = e.subs(lst(
897 dirac_gammaL(rl) == (dirac_ONE(rl)-dirac_gamma5(rl))/2,
898 dirac_gammaR(rl) == (dirac_ONE(rl)+dirac_gamma5(rl))/2
899 ), subs_options::no_pattern).expand();
900 if (!is_a<ncmul>(e_expanded))
901 return dirac_trace(e_expanded, rls, trONE);
903 // gamma5 gets moved to the front so this check is enough
904 bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
905 size_t num = e.nops();
909 // Trace of gamma5 * odd number of gammas and trace of
910 // gamma5 * gamma.mu * gamma.nu are zero
911 if ((num & 1) == 0 || num == 3)
914 // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
915 // (the epsilon is always 4-dimensional)
917 ex b1, i1, b2, i2, b3, i3, b4, i4;
918 base_and_index(e.op(1), b1, i1);
919 base_and_index(e.op(2), b2, i2);
920 base_and_index(e.op(3), b3, i3);
921 base_and_index(e.op(4), b4, i4);
922 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();
926 // I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
927 // (the epsilon is always 4-dimensional)
928 exvector ix(num-1), bv(num-1);
929 for (size_t i=1; i<num; i++)
930 base_and_index(e.op(i), bv[i-1], ix[i-1]);
932 int *iv = new int[num];
934 for (size_t i=0; i<num-3; i++) {
936 for (size_t j=i+1; j<num-2; j++) {
938 for (size_t k=j+1; k<num-1; k++) {
940 for (size_t l=k+1; l<num; l++) {
942 iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
945 for (size_t n=0, t=4; n<num; n++) {
946 if (n == i || n == j || n == k || n == l)
951 int sign = permutation_sign(iv, iv + num);
952 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))
953 * trace_string(v.begin(), num - 4);
959 return trONE * I * result * mul(bv);
961 } else { // no gamma5
963 // Trace of odd number of gammas is zero
967 // Tr gamma.mu gamma.nu = 4 g.mu.nu
970 base_and_index(e.op(0), b1, i1);
971 base_and_index(e.op(1), b2, i2);
972 return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
975 exvector iv(num), bv(num);
976 for (size_t i=0; i<num; i++)
977 base_and_index(e.op(i), bv[i], iv[i]);
979 return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
982 } else if (e.nops() > 0) {
984 // Trace maps to all other container classes (this includes sums)
985 pointer_to_map_function_2args<const std::set<unsigned char> &, const ex &> fcn(dirac_trace, rls, trONE);
992 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE)
994 // Convert list to set
995 std::set<unsigned char> rls;
996 for (lst::const_iterator i = rll.begin(); i != rll.end(); ++i) {
997 if (i->info(info_flags::nonnegint))
998 rls.insert(ex_to<numeric>(*i).to_int());
1001 return dirac_trace(e, rls, trONE);
1004 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
1006 // Convert label to set
1007 std::set<unsigned char> rls;
1010 return dirac_trace(e, rls, trONE);
1014 ex canonicalize_clifford(const ex & e_)
1016 pointer_to_map_function fcn(canonicalize_clifford);
1018 if (is_a<matrix>(e_) // || is_a<pseries>(e) || is_a<integral>(e)
1019 || e_.info(info_flags::list)) {
1022 ex e=simplify_indexed(e_);
1023 // Scan for any ncmul objects
1025 ex aux = e.to_rational(srl);
1026 for (exmap::iterator i = srl.begin(); i != srl.end(); ++i) {
1031 if (is_exactly_a<ncmul>(rhs)
1032 && rhs.return_type() == return_types::noncommutative
1033 && is_clifford_tinfo(rhs.return_type_tinfo())) {
1035 // Expand product, if necessary
1036 ex rhs_expanded = rhs.expand();
1037 if (!is_a<ncmul>(rhs_expanded)) {
1038 i->second = canonicalize_clifford(rhs_expanded);
1041 } else if (!is_a<clifford>(rhs.op(0)))
1045 v.reserve(rhs.nops());
1046 for (size_t j=0; j<rhs.nops(); j++)
1047 v.push_back(rhs.op(j));
1049 // Stupid recursive bubble sort because we only want to swap adjacent gammas
1050 exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
1051 if (is_a<diracgamma5>(it->op(0)) || is_a<diracgammaL>(it->op(0)) || is_a<diracgammaR>(it->op(0)))
1054 while (it != next_to_last) {
1055 if (it[0].compare(it[1]) > 0) {
1057 ex save0 = it[0], save1 = it[1];
1059 base_and_index(it[0], b1, i1);
1060 base_and_index(it[1], b2, i2);
1061 // for Clifford algebras (commutator_sign == -1) metric should be symmetrised
1062 it[0] = (ex_to<clifford>(save0).get_metric(i1, i2, ex_to<clifford>(save0).get_commutator_sign() == -1) * b1 * b2).simplify_indexed();
1063 it[1] = v.size() ? _ex2 * dirac_ONE(ex_to<clifford>(save0).get_representation_label()) : _ex2;
1067 sum += ex_to<clifford>(save0).get_commutator_sign() * ncmul(v, true);
1068 i->second = canonicalize_clifford(sum);
1076 return aux.subs(srl, subs_options::no_pattern).simplify_indexed();
1080 ex clifford_prime(const ex & e)
1082 pointer_to_map_function fcn(clifford_prime);
1083 if (is_a<clifford>(e) && is_a<cliffordunit>(e.op(0))) {
1085 } else if (is_a<add>(e) || is_a<ncmul>(e) || is_a<mul>(e) //|| is_a<pseries>(e) || is_a<integral>(e)
1086 || is_a<matrix>(e) || e.info(info_flags::list)) {
1088 } else if (is_a<power>(e)) {
1089 return pow(clifford_prime(e.op(0)), e.op(1));
1094 ex remove_dirac_ONE(const ex & e, unsigned char rl, unsigned options)
1096 pointer_to_map_function_2args<unsigned char, unsigned> fcn(remove_dirac_ONE, rl, options | 1);
1097 bool need_reevaluation = false;
1099 if (! (options & 1) ) { // is not a child
1101 e1 = expand_dummy_sum(e, true);
1102 e1 = canonicalize_clifford(e1);
1105 if (is_a<clifford>(e1) && ex_to<clifford>(e1).get_representation_label() >= rl) {
1106 if (is_a<diracone>(e1.op(0)))
1109 throw(std::invalid_argument("remove_dirac_ONE(): expression is a non-scalar Clifford number!"));
1110 } else if (is_a<add>(e1) || is_a<ncmul>(e1) || is_a<mul>(e1)
1111 || is_a<matrix>(e1) || e1.info(info_flags::list)) {
1112 if (options & 3) // is a child or was already expanded
1117 } catch (std::exception &p) {
1118 need_reevaluation = true;
1120 } else if (is_a<power>(e1)) {
1121 if (options & 3) // is a child or was already expanded
1122 return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
1125 return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
1126 } catch (std::exception &p) {
1127 need_reevaluation = true;
1130 if (need_reevaluation)
1131 return remove_dirac_ONE(e, rl, options | 2);
1135 char clifford_max_label(const ex & e, bool ignore_ONE)
1137 if (is_a<clifford>(e))
1138 if (ignore_ONE && is_a<diracone>(e.op(0)))
1141 return ex_to<clifford>(e).get_representation_label();
1144 for (size_t i=0; i < e.nops(); i++)
1145 rl = (rl > clifford_max_label(e.op(i), ignore_ONE)) ? rl : clifford_max_label(e.op(i), ignore_ONE);
1150 ex clifford_norm(const ex & e)
1152 return sqrt(remove_dirac_ONE(e * clifford_bar(e)));
1155 ex clifford_inverse(const ex & e)
1157 ex norm = clifford_norm(e);
1158 if (!norm.is_zero())
1159 return clifford_bar(e) / pow(norm, 2);
1161 throw(std::invalid_argument("clifford_inverse(): cannot find inverse of Clifford number with zero norm!"));
1164 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl)
1166 if (!ex_to<idx>(mu).is_dim_numeric())
1167 throw(std::invalid_argument("lst_to_clifford(): Index should have a numeric dimension"));
1168 ex e = clifford_unit(mu, metr, rl);
1169 return lst_to_clifford(v, e);
1172 ex lst_to_clifford(const ex & v, const ex & e) {
1175 if (is_a<clifford>(e)) {
1178 = is_a<varidx>(mu) ? ex_to<varidx>(mu).toggle_variance() : mu;
1179 unsigned dim = get_dim_uint(mu);
1181 if (is_a<matrix>(v)) {
1182 if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows()) {
1183 min = ex_to<matrix>(v).rows();
1184 max = ex_to<matrix>(v).cols();
1186 min = ex_to<matrix>(v).cols();
1187 max = ex_to<matrix>(v).rows();
1191 return indexed(v, mu_toggle) * e;
1192 else if (max - dim == 1) {
1193 if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows())
1194 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;
1196 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;
1198 throw(std::invalid_argument("lst_to_clifford(): dimensions of vector and clifford unit mismatch"));
1200 throw(std::invalid_argument("lst_to_clifford(): first argument should be a vector (nx1 or 1xn matrix)"));
1201 } else if (v.info(info_flags::list)) {
1202 if (dim == ex_to<lst>(v).nops())
1203 return indexed(matrix(dim, 1, ex_to<lst>(v)), mu_toggle) * e;
1204 else if (ex_to<lst>(v).nops() - dim == 1)
1205 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;
1207 throw(std::invalid_argument("lst_to_clifford(): list length and dimension of clifford unit mismatch"));
1209 throw(std::invalid_argument("lst_to_clifford(): cannot construct from anything but list or vector"));
1211 throw(std::invalid_argument("lst_to_clifford(): the second argument should be a Clifford unit"));
1214 /** Auxiliary structure to define a function for striping one Clifford unit
1215 * from vectors. Used in clifford_to_lst(). */
1216 static ex get_clifford_comp(const ex & e, const ex & c)
1218 pointer_to_map_function_1arg<const ex &> fcn(get_clifford_comp, c);
1219 int ival = ex_to<numeric>(ex_to<idx>(c.op(1)).get_value()).to_int();
1221 if (is_a<add>(e) || e.info(info_flags::list) // || is_a<pseries>(e) || is_a<integral>(e)
1224 else if (is_a<ncmul>(e) || is_a<mul>(e)) {
1225 // find a Clifford unit with the same metric, delete it and substitute its index
1226 size_t ind = e.nops() + 1;
1227 for (size_t j = 0; j < e.nops(); j++) {
1228 if (is_a<clifford>(e.op(j)) && ex_to<clifford>(c).same_metric(e.op(j))) {
1229 if (ind > e.nops()) {
1233 throw(std::invalid_argument("get_clifford_comp(): expression is a Clifford multi-vector"));
1237 if (ind < e.nops()) {
1239 bool same_value_index, found_dummy;
1240 same_value_index = ( ex_to<idx>(e.op(ind).op(1)).is_numeric()
1241 && (ival == ex_to<numeric>(ex_to<idx>(e.op(ind).op(1)).get_value()).to_int()) );
1242 found_dummy = same_value_index;
1243 for (size_t j=0; j < e.nops(); j++) {
1245 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);
1267 return (found_dummy ? S : 0);
1269 throw(std::invalid_argument("get_clifford_comp(): expression is not a Clifford vector to the given units"));
1270 } else if (e.is_zero())
1272 else if (is_a<clifford>(e) && ex_to<clifford>(e).same_metric(c))
1273 if ( ex_to<idx>(e.op(1)).is_numeric() &&
1274 (ival != ex_to<numeric>(ex_to<idx>(e.op(1)).get_value()).to_int()) )
1279 throw(std::invalid_argument("get_clifford_comp(): expression is not usable as a Clifford vector"));
1283 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic)
1285 GINAC_ASSERT(is_a<clifford>(c));
1287 if (! ex_to<idx>(mu).is_dim_numeric())
1288 throw(std::invalid_argument("clifford_to_lst(): index should have a numeric dimension"));
1289 unsigned int D = ex_to<numeric>(ex_to<idx>(mu).get_dim()).to_int();
1291 if (algebraic) // check if algebraic method is applicable
1292 for (unsigned int i = 0; i < D; i++)
1293 if (pow(c.subs(mu == i, subs_options::no_pattern), 2).is_zero()
1294 || (! is_a<numeric>(pow(c.subs(mu == i, subs_options::no_pattern), 2))))
1297 ex v0 = remove_dirac_ONE(canonicalize_clifford(e+clifford_prime(e)).normal())/2;
1300 ex e1 = canonicalize_clifford(e - v0 * dirac_ONE(ex_to<clifford>(c).get_representation_label()));
1302 for (unsigned int i = 0; i < D; i++)
1303 V.append(remove_dirac_ONE(
1304 simplify_indexed(canonicalize_clifford(e1 * c.subs(mu == i, subs_options::no_pattern) + c.subs(mu == i, subs_options::no_pattern) * e1))
1305 / (2*pow(c.subs(mu == i, subs_options::no_pattern), 2))));
1308 for (unsigned int i = 0; i < D; i++)
1309 V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
1310 } catch (std::exception &p) {
1311 /* Try to expand dummy summations to simplify the expression*/
1312 e1 = canonicalize_clifford(expand_dummy_sum(e, true));
1314 v0 = remove_dirac_ONE(canonicalize_clifford(e1+clifford_prime(e1)).normal())/2;
1315 if (! v0.is_zero()) {
1317 e1 = canonicalize_clifford(e1 - v0 * dirac_ONE(ex_to<clifford>(c).get_representation_label()));
1319 for (unsigned int i = 0; i < D; i++)
1320 V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
1327 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)
1331 if (! is_a<matrix>(v) && ! v.info(info_flags::list))
1332 throw(std::invalid_argument("clifford_moebius_map(): parameter v should be either vector or list"));
1334 if (is_a<clifford>(G)) {
1337 if (is_a<indexed>(G)) {
1338 D = ex_to<idx>(G.op(1)).get_dim();
1339 varidx mu((new symbol)->setflag(status_flags::dynallocated), D);
1340 cu = clifford_unit(mu, G, rl);
1341 } else if (is_a<matrix>(G)) {
1342 D = ex_to<matrix>(G).rows();
1343 idx mu((new symbol)->setflag(status_flags::dynallocated), D);
1344 cu = clifford_unit(mu, G, rl);
1345 } else throw(std::invalid_argument("clifford_moebius_map(): metric should be an indexed object, matrix, or a Clifford unit"));
1349 x = lst_to_clifford(v, cu);
1350 ex e = clifford_to_lst(simplify_indexed(canonicalize_clifford((a * x + b) * clifford_inverse(c * x + d))), cu, false);
1351 return (is_a<matrix>(v) ? matrix(ex_to<matrix>(v).rows(), ex_to<matrix>(v).cols(), ex_to<lst>(e)) : e);
1354 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl)
1356 if (is_a<matrix>(M) && (ex_to<matrix>(M).rows() == 2) && (ex_to<matrix>(M).cols() == 2))
1357 return clifford_moebius_map(M.op(0), M.op(1), M.op(2), M.op(3), v, G, rl);
1359 throw(std::invalid_argument("clifford_moebius_map(): parameter M should be a 2x2 matrix"));
1362 } // namespace GiNaC