3 * Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
6 * GiNaC Copyright (C) 1999-2015 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<idx>(mu));
108 clifford::clifford(unsigned char rl, const ex & metr, int comm_sign, const exvector & v) : inherited(not_symmetric(), v), representation_label(rl), metric(metr), commutator_sign(comm_sign)
112 clifford::clifford(unsigned char rl, const ex & metr, int comm_sign, exvector && v) : inherited(not_symmetric(), std::move(v)), 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(cliffordunit);
146 GINAC_BIND_UNARCHIVER(diracone);
147 GINAC_BIND_UNARCHIVER(diracgamma);
148 GINAC_BIND_UNARCHIVER(diracgamma5);
149 GINAC_BIND_UNARCHIVER(diracgammaL);
150 GINAC_BIND_UNARCHIVER(diracgammaR);
153 ex clifford::get_metric(const ex & i, const ex & j, bool symmetrised) const
155 if (is_a<indexed>(metric)) {
156 if (symmetrised && !(ex_to<symmetry>(ex_to<indexed>(metric).get_symmetry()).has_symmetry())) {
157 if (is_a<matrix>(metric.op(0))) {
158 return indexed((ex_to<matrix>(metric.op(0)).add(ex_to<matrix>(metric.op(0)).transpose())).mul(numeric(1, 2)),
161 return simplify_indexed(indexed(metric.op(0)*_ex1_2, i, j) + indexed(metric.op(0)*_ex1_2, j, i));
164 return metric.subs(lst{metric.op(1) == i, metric.op(2) == j}, subs_options::no_pattern);
167 exvector indices = metric.get_free_indices();
169 return _ex1_2*simplify_indexed(metric.subs(lst{indices[0] == i, indices[1] == j}, subs_options::no_pattern)
170 + metric.subs(lst{indices[0] == j, indices[1] == i}, subs_options::no_pattern));
172 return metric.subs(lst{indices[0] == i, indices[1] == j}, subs_options::no_pattern);
176 bool clifford::same_metric(const ex & other) const
179 if (is_a<clifford>(other))
180 metr = ex_to<clifford>(other).get_metric();
184 if (is_a<indexed>(metr))
185 return metr.op(0).is_equal(get_metric().op(0));
187 exvector indices = metr.get_free_indices();
188 return (indices.size() == 2)
189 && simplify_indexed(get_metric(indices[0], indices[1])-metr).is_zero();
194 // functions overriding virtual functions from base classes
197 ex clifford::op(size_t i) const
199 GINAC_ASSERT(i<nops());
201 return representation_label;
203 return inherited::op(i);
206 ex & clifford::let_op(size_t i)
208 GINAC_ASSERT(i<nops());
210 static ex rl = numeric(representation_label);
211 ensure_if_modifiable();
215 return inherited::let_op(i);
218 ex clifford::subs(const exmap & m, unsigned options) const
220 ex subsed = inherited::subs(m, options);
221 if(is_a<clifford>(subsed)) {
222 ex prevmetric = ex_to<clifford>(subsed).metric;
223 ex newmetric = prevmetric.subs(m, options);
224 if(!are_ex_trivially_equal(prevmetric, newmetric)) {
225 clifford c = ex_to<clifford>(subsed);
226 c.metric = newmetric;
233 int clifford::compare_same_type(const basic & other) const
235 GINAC_ASSERT(is_a<clifford>(other));
236 const clifford &o = static_cast<const clifford &>(other);
238 if (representation_label != o.representation_label) {
239 // different representation label
240 return representation_label < o.representation_label ? -1 : 1;
243 return inherited::compare_same_type(other);
246 bool clifford::match_same_type(const basic & other) const
248 GINAC_ASSERT(is_a<clifford>(other));
249 const clifford &o = static_cast<const clifford &>(other);
251 return ((representation_label == o.representation_label) && (commutator_sign == o.get_commutator_sign()) && same_metric(o));
254 static bool is_dirac_slash(const ex & seq0)
256 return !is_a<diracgamma5>(seq0) && !is_a<diracgammaL>(seq0) &&
257 !is_a<diracgammaR>(seq0) && !is_a<cliffordunit>(seq0) &&
258 !is_a<diracone>(seq0);
261 void clifford::do_print_dflt(const print_dflt & c, unsigned level) const
263 // dirac_slash() object is printed differently
264 if (is_dirac_slash(seq[0])) {
265 seq[0].print(c, precedence());
267 } else { // We do not print representation label if it is 0
268 if (representation_label == 0) {
269 this->print_dispatch<inherited>(c, level);
270 } else { // otherwise we put it before indices in square brackets; the code is borrowed from indexed.cpp
271 if (precedence() <= level) {
274 seq[0].print(c, precedence());
275 c.s << '[' << int(representation_label) << ']';
276 printindices(c, level);
277 if (precedence() <= level) {
284 void clifford::do_print_latex(const print_latex & c, unsigned level) const
286 // dirac_slash() object is printed differently
287 if (is_dirac_slash(seq[0])) {
289 seq[0].print(c, precedence());
290 c.s << "\\hspace{-1.0ex}/}";
292 c.s << "\\clifford[" << int(representation_label) << "]";
293 this->print_dispatch<inherited>(c, level);
297 DEFAULT_COMPARE(diracone)
298 DEFAULT_COMPARE(cliffordunit)
299 DEFAULT_COMPARE(diracgamma)
300 DEFAULT_COMPARE(diracgamma5)
301 DEFAULT_COMPARE(diracgammaL)
302 DEFAULT_COMPARE(diracgammaR)
304 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbf{1}")
305 DEFAULT_PRINT_LATEX(cliffordunit, "e", "e")
306 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
307 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
308 DEFAULT_PRINT_LATEX(diracgammaL, "gammaL", "{\\gamma_L}")
309 DEFAULT_PRINT_LATEX(diracgammaR, "gammaR", "{\\gamma_R}")
311 /** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
312 static void base_and_index(const ex & c, ex & b, ex & i)
314 GINAC_ASSERT(is_a<clifford>(c));
315 GINAC_ASSERT(c.nops() == 2+1);
317 if (is_a<cliffordunit>(c.op(0))) { // proper dirac gamma object or clifford unit
320 } else if (is_a<diracgamma5>(c.op(0)) || is_a<diracgammaL>(c.op(0)) || is_a<diracgammaR>(c.op(0))) { // gamma5/L/R
323 } else { // slash object, generate new dummy index
324 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(c.op(1)).get_dim());
325 b = indexed(c.op(0), ix.toggle_variance());
330 /** Predicate for finding non-clifford objects. */
331 struct is_not_a_clifford : public std::unary_function<ex, bool> {
332 bool operator()(const ex & e)
334 return !is_a<clifford>(e);
338 /** Contraction of a gamma matrix with something else. */
339 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
341 GINAC_ASSERT(is_a<clifford>(*self));
342 GINAC_ASSERT(is_a<indexed>(*other));
343 GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
344 unsigned char rl = ex_to<clifford>(*self).get_representation_label();
346 ex dim = ex_to<idx>(self->op(1)).get_dim();
347 if (other->nops() > 1)
348 dim = minimal_dim(dim, ex_to<idx>(other->op(1)).get_dim());
350 if (is_a<clifford>(*other)) {
352 // Contraction only makes sense if the representation labels are equal
353 if (ex_to<clifford>(*other).get_representation_label() != rl)
356 size_t num = other - self;
358 // gamma~mu gamma.mu = dim ONE
361 *other = dirac_ONE(rl);
364 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
366 && is_a<clifford>(self[1])) {
371 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
373 && is_a<clifford>(self[1])
374 && is_a<clifford>(self[2])) {
376 base_and_index(self[1], b1, i1);
377 base_and_index(self[2], b2, i2);
378 *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
384 // 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
386 && is_a<clifford>(self[1])
387 && is_a<clifford>(self[2])
388 && is_a<clifford>(self[3])) {
389 *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3];
396 // gamma~mu Sodd gamma.mu = -2 Sodd_R
397 // (Chisholm identity in 4 dimensions)
398 } else if (!((other - self) & 1) && dim.is_equal(4)) {
399 if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
402 *self = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(other), std::reverse_iterator<exvector::const_iterator>(self + 1)));
403 std::fill(self + 1, other, _ex1);
407 // gamma~mu Sodd gamma~alpha gamma.mu = 2 gamma~alpha Sodd + 2 Sodd_R gamma~alpha
408 // (commutate contracted indices towards each other, then use
409 // Chisholm identity in 4 dimensions)
410 } else if (((other - self) & 1) && dim.is_equal(4)) {
411 if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
414 exvector::iterator next_to_last = other - 1;
415 ex S = ncmul(exvector(self + 1, next_to_last));
416 ex SR = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(next_to_last), std::reverse_iterator<exvector::const_iterator>(self + 1)));
418 *self = (*next_to_last) * S + SR * (*next_to_last);
419 std::fill(self + 1, other, _ex1);
423 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
424 // (commutate contracted indices towards each other, simplify_indexed()
425 // will re-expand and re-run the simplification)
427 if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
430 exvector::iterator next_to_last = other - 1;
431 ex S = ncmul(exvector(self + 1, next_to_last));
433 *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
434 std::fill(self + 1, other + 1, _ex1);
438 } else if (is_a<symbol>(other->op(0)) && other->nops() == 2) {
440 // x.mu gamma~mu -> x-slash
441 *self = dirac_slash(other->op(0), dim, rl);
449 /** Contraction of a Clifford unit with something else. */
450 bool cliffordunit::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
452 GINAC_ASSERT(is_a<clifford>(*self));
453 GINAC_ASSERT(is_a<indexed>(*other));
454 GINAC_ASSERT(is_a<cliffordunit>(self->op(0)));
455 clifford unit = ex_to<clifford>(*self);
456 unsigned char rl = unit.get_representation_label();
458 if (is_a<clifford>(*other)) {
459 // Contraction only makes sense if the representation labels are equal
460 // and the metrics are the same
461 if ((ex_to<clifford>(*other).get_representation_label() != rl)
462 && unit.same_metric(*other))
465 exvector::iterator before_other = other - 1;
467 ex mu_toggle = other->op(1);
468 ex alpha = before_other->op(1);
470 // e~mu e.mu = Tr ONE
471 if (other - self == 1) {
472 *self = unit.get_metric(mu, mu_toggle, true);
473 *other = dirac_ONE(rl);
476 } else if (other - self == 2) {
477 if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
478 // e~mu e~alpha e.mu = 2*e~mu B(alpha, mu.toggle_variance())-Tr(B) e~alpha
479 *self = 2 * (*self) * unit.get_metric(alpha, mu_toggle, true) - unit.get_metric(mu, mu_toggle, true) * (*before_other);
480 *before_other = _ex1;
485 // e~mu S e.mu = Tr S ONE
486 *self = unit.get_metric(mu, mu_toggle, true);
487 *other = dirac_ONE(rl);
491 // e~mu S e~alpha e.mu = 2 e~mu S B(alpha, mu.toggle_variance()) - e~mu S e.mu e~alpha
492 // (commutate contracted indices towards each other, simplify_indexed()
493 // will re-expand and re-run the simplification)
494 if (std::find_if(self + 1, other, is_not_a_clifford()) != other) {
498 ex S = ncmul(exvector(self + 1, before_other));
500 if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
501 *self = 2 * (*self) * S * unit.get_metric(alpha, mu_toggle, true) - (*self) * S * (*other) * (*before_other);
504 *self = (*self) * S * (*other) * (*before_other);
507 std::fill(self + 1, other + 1, _ex1);
514 /** Perform automatic simplification on noncommutative product of clifford
515 * objects. This removes superfluous ONEs, permutes gamma5/L/R's to the front
516 * and removes squares of gamma objects. */
517 ex clifford::eval_ncmul(const exvector & v) const
522 // Remove superfluous ONEs
523 for (auto & it : v) {
524 if (!is_a<clifford>(it) || !is_a<diracone>(it.op(0)))
528 bool something_changed = false;
531 // Anticommutate gamma5/L/R's to the front
533 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
535 exvector::iterator it = next_to_last;
537 exvector::iterator it2 = it + 1;
538 if (is_a<clifford>(*it) && is_a<clifford>(*it2)) {
539 ex e1 = it->op(0), e2 = it2->op(0);
541 if (is_a<diracgamma5>(e2)) {
543 if (is_a<diracgammaL>(e1) || is_a<diracgammaR>(e1)) {
545 // gammaL/R gamma5 -> gamma5 gammaL/R
547 something_changed = true;
549 } else if (!is_a<diracgamma5>(e1)) {
551 // gamma5 gamma5 -> gamma5 gamma5 (do nothing)
552 // x gamma5 -> -gamma5 x
555 something_changed = true;
558 } else if (is_a<diracgammaL>(e2)) {
560 if (is_a<diracgammaR>(e1)) {
562 // gammaR gammaL -> 0
565 } else if (!is_a<diracgammaL>(e1) && !is_a<diracgamma5>(e1)) {
567 // gammaL gammaL -> gammaL gammaL (do nothing)
568 // gamma5 gammaL -> gamma5 gammaL (do nothing)
569 // x gammaL -> gammaR x
571 *it = clifford(diracgammaR(), ex_to<clifford>(*it).get_representation_label());
572 something_changed = true;
575 } else if (is_a<diracgammaR>(e2)) {
577 if (is_a<diracgammaL>(e1)) {
579 // gammaL gammaR -> 0
582 } else if (!is_a<diracgammaR>(e1) && !is_a<diracgamma5>(e1)) {
584 // gammaR gammaR -> gammaR gammaR (do nothing)
585 // gamma5 gammaR -> gamma5 gammaR (do nothing)
586 // x gammaR -> gammaL x
588 *it = clifford(diracgammaL(), ex_to<clifford>(*it).get_representation_label());
589 something_changed = true;
597 if (next_to_last == first)
603 // Remove equal adjacent gammas
605 exvector::iterator it, itend = s.end() - 1;
606 for (it = s.begin(); it != itend; ++it) {
609 if (!is_a<clifford>(a) || !is_a<clifford>(b))
612 const ex & ag = a.op(0);
613 const ex & bg = b.op(0);
614 bool a_is_cliffordunit = is_a<cliffordunit>(ag);
615 bool b_is_cliffordunit = is_a<cliffordunit>(bg);
617 if (a_is_cliffordunit && b_is_cliffordunit && ex_to<clifford>(a).same_metric(b)
618 && (ex_to<clifford>(a).get_commutator_sign() == -1)) {
619 // This is done only for Clifford algebras
621 const ex & ia = a.op(1);
622 const ex & ib = b.op(1);
623 if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
624 a = ex_to<clifford>(a).get_metric(ia, ib, true);
625 b = dirac_ONE(representation_label);
626 something_changed = true;
629 } else if ((is_a<diracgamma5>(ag) && is_a<diracgamma5>(bg))) {
631 // Remove squares of gamma5
632 a = dirac_ONE(representation_label);
633 b = dirac_ONE(representation_label);
634 something_changed = true;
636 } else if ((is_a<diracgammaL>(ag) && is_a<diracgammaL>(bg))
637 || (is_a<diracgammaR>(ag) && is_a<diracgammaR>(bg))) {
639 // Remove squares of gammaL/R
640 b = dirac_ONE(representation_label);
641 something_changed = true;
643 } else if (is_a<diracgammaL>(ag) && is_a<diracgammaR>(bg)) {
645 // gammaL and gammaR are orthogonal
648 } else if (is_a<diracgamma5>(ag) && is_a<diracgammaL>(bg)) {
650 // gamma5 gammaL -> -gammaL
651 a = dirac_ONE(representation_label);
653 something_changed = true;
655 } else if (is_a<diracgamma5>(ag) && is_a<diracgammaR>(bg)) {
657 // gamma5 gammaR -> gammaR
658 a = dirac_ONE(representation_label);
659 something_changed = true;
661 } else if (!a_is_cliffordunit && !b_is_cliffordunit && ag.is_equal(bg)) {
664 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).minimal_dim(ex_to<idx>(b.op(1))));
666 a = indexed(ag, ix) * indexed(ag, ix.toggle_variance());
667 b = dirac_ONE(representation_label);
668 something_changed = true;
674 return dirac_ONE(representation_label) * sign;
675 if (something_changed)
676 return reeval_ncmul(s) * sign;
678 return hold_ncmul(s) * sign;
681 ex clifford::thiscontainer(const exvector & v) const
683 return clifford(representation_label, metric, commutator_sign, v);
686 ex clifford::thiscontainer(exvector && v) const
688 return clifford(representation_label, metric, commutator_sign, std::move(v));
691 ex diracgamma5::conjugate() const
693 return _ex_1 * (*this);
696 ex diracgammaL::conjugate() const
698 return (new diracgammaR)->setflag(status_flags::dynallocated);
701 ex diracgammaR::conjugate() const
703 return (new diracgammaL)->setflag(status_flags::dynallocated);
710 ex dirac_ONE(unsigned char rl)
712 static ex ONE = (new diracone)->setflag(status_flags::dynallocated);
713 return clifford(ONE, rl);
716 static unsigned get_dim_uint(const ex& e)
719 throw std::invalid_argument("get_dim_uint: argument is not an index");
720 ex dim = ex_to<idx>(e).get_dim();
721 if (!dim.info(info_flags::posint))
722 throw std::invalid_argument("get_dim_uint: dimension of index should be a positive integer");
723 unsigned d = ex_to<numeric>(dim).to_int();
727 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl)
729 //static ex unit = (new cliffordunit)->setflag(status_flags::dynallocated);
730 ex unit = (new cliffordunit)->setflag(status_flags::dynallocated);
733 throw(std::invalid_argument("clifford_unit(): index of Clifford unit must be of type idx or varidx"));
735 exvector indices = metr.get_free_indices();
737 if (indices.size() == 2) {
738 return clifford(unit, mu, metr, rl);
739 } else if (is_a<matrix>(metr)) {
740 matrix M = ex_to<matrix>(metr);
741 unsigned n = M.rows();
742 bool symmetric = true;
744 //static idx xi((new symbol)->setflag(status_flags::dynallocated), n),
745 // chi((new symbol)->setflag(status_flags::dynallocated), n);
746 idx xi((new symbol)->setflag(status_flags::dynallocated), n),
747 chi((new symbol)->setflag(status_flags::dynallocated), n);
748 if ((n == M.cols()) && (n == get_dim_uint(mu))) {
749 for (unsigned i = 0; i < n; i++) {
750 for (unsigned j = i+1; j < n; j++) {
751 if (!M(i, j).is_equal(M(j, i))) {
756 return clifford(unit, mu, indexed(metr, symmetric?symmetric2():not_symmetric(), xi, chi), rl);
758 throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be a square matrix with the same dimensions as index"));
760 } else if (indices.size() == 0) { // a tensor or other expression without indices
761 //static varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim()),
762 // chi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim());
763 varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim()),
764 chi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim());
765 return clifford(unit, mu, indexed(metr, xi, chi), rl);
767 throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be of type tensor, matrix or an expression with two free indices"));
770 ex dirac_gamma(const ex & mu, unsigned char rl)
772 static ex gamma = (new diracgamma)->setflag(status_flags::dynallocated);
774 if (!is_a<varidx>(mu))
775 throw(std::invalid_argument("dirac_gamma(): index of Dirac gamma must be of type varidx"));
777 static varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim()),
778 chi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim());
779 return clifford(gamma, mu, indexed((new minkmetric)->setflag(status_flags::dynallocated), symmetric2(), xi, chi), rl);
782 ex dirac_gamma5(unsigned char rl)
784 static ex gamma5 = (new diracgamma5)->setflag(status_flags::dynallocated);
785 return clifford(gamma5, rl);
788 ex dirac_gammaL(unsigned char rl)
790 static ex gammaL = (new diracgammaL)->setflag(status_flags::dynallocated);
791 return clifford(gammaL, rl);
794 ex dirac_gammaR(unsigned char rl)
796 static ex gammaR = (new diracgammaR)->setflag(status_flags::dynallocated);
797 return clifford(gammaR, rl);
800 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
802 // Slashed vectors are actually stored as a clifford object with the
803 // vector as its base expression and a (dummy) index that just serves
804 // for storing the space dimensionality
806 static varidx xi((new symbol)->setflag(status_flags::dynallocated), dim),
807 chi((new symbol)->setflag(status_flags::dynallocated), dim);
808 return clifford(e, varidx(0, dim), indexed((new minkmetric)->setflag(status_flags::dynallocated), symmetric2(), xi, chi), rl);
811 /** Extract representation label from tinfo key (as returned by
812 * return_type_tinfo()). */
813 static unsigned char get_representation_label(const return_type_t& ti)
815 return (unsigned char)ti.rl;
818 /** Take trace of a string of an even number of Dirac gammas given a vector
820 static ex trace_string(exvector::const_iterator ix, size_t num)
822 // Tr gamma.mu gamma.nu = 4 g.mu.nu
824 return lorentz_g(ix[0], ix[1]);
826 // 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 )
828 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
829 + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
830 - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
832 // Traces of 6 or more gammas are computed recursively:
833 // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
834 // + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
835 // - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
836 // + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
838 // + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
842 for (size_t i=1; i<num; i++) {
843 for (size_t n=1, j=0; n<num; n++) {
848 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
854 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE)
856 if (is_a<clifford>(e)) {
858 unsigned char rl = ex_to<clifford>(e).get_representation_label();
860 // Are we taking the trace over this object's representation label?
861 if (rls.find(rl) == rls.end())
864 // Yes, all elements are traceless, except for dirac_ONE and dirac_L/R
865 const ex & g = e.op(0);
866 if (is_a<diracone>(g))
868 else if (is_a<diracgammaL>(g) || is_a<diracgammaR>(g))
873 } else if (is_exactly_a<mul>(e)) {
875 // Trace of product: pull out non-clifford factors
877 for (size_t i=0; i<e.nops(); i++) {
878 const ex &o = e.op(i);
879 if (is_clifford_tinfo(o.return_type_tinfo()))
880 prod *= dirac_trace(o, rls, trONE);
886 } else if (is_exactly_a<ncmul>(e)) {
888 unsigned char rl = get_representation_label(e.return_type_tinfo());
890 // Are we taking the trace over this string's representation label?
891 if (rls.find(rl) == rls.end())
894 // Substitute gammaL/R and expand product, if necessary
895 ex e_expanded = e.subs(lst{
896 dirac_gammaL(rl) == (dirac_ONE(rl)-dirac_gamma5(rl))/2,
897 dirac_gammaR(rl) == (dirac_ONE(rl)+dirac_gamma5(rl))/2
898 }, subs_options::no_pattern).expand();
899 if (!is_a<ncmul>(e_expanded))
900 return dirac_trace(e_expanded, rls, trONE);
902 // gamma5 gets moved to the front so this check is enough
903 bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
904 size_t num = e.nops();
908 // Trace of gamma5 * odd number of gammas and trace of
909 // gamma5 * gamma.mu * gamma.nu are zero
910 if ((num & 1) == 0 || num == 3)
913 // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
914 // (the epsilon is always 4-dimensional)
916 ex b1, i1, b2, i2, b3, i3, b4, i4;
917 base_and_index(e.op(1), b1, i1);
918 base_and_index(e.op(2), b2, i2);
919 base_and_index(e.op(3), b3, i3);
920 base_and_index(e.op(4), b4, i4);
921 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();
925 // I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
926 // (the epsilon is always 4-dimensional)
927 exvector ix(num-1), bv(num-1);
928 for (size_t i=1; i<num; i++)
929 base_and_index(e.op(i), bv[i-1], ix[i-1]);
931 int *iv = new int[num];
933 for (size_t i=0; i<num-3; i++) {
935 for (size_t j=i+1; j<num-2; j++) {
937 for (size_t k=j+1; k<num-1; k++) {
939 for (size_t l=k+1; l<num; l++) {
941 iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
944 for (size_t n=0, t=4; n<num; n++) {
945 if (n == i || n == j || n == k || n == l)
950 int sign = permutation_sign(iv, iv + num);
951 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))
952 * trace_string(v.begin(), num - 4);
958 return trONE * I * result * mul(bv);
960 } else { // no gamma5
962 // Trace of odd number of gammas is zero
966 // Tr gamma.mu gamma.nu = 4 g.mu.nu
969 base_and_index(e.op(0), b1, i1);
970 base_and_index(e.op(1), b2, i2);
971 return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
974 exvector iv(num), bv(num);
975 for (size_t i=0; i<num; i++)
976 base_and_index(e.op(i), bv[i], iv[i]);
978 return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
981 } else if (e.nops() > 0) {
983 // Trace maps to all other container classes (this includes sums)
984 pointer_to_map_function_2args<const std::set<unsigned char> &, const ex &> fcn(dirac_trace, rls, trONE);
991 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE)
993 // Convert list to set
994 std::set<unsigned char> rls;
995 for (lst::const_iterator i = rll.begin(); i != rll.end(); ++i) {
996 if (i->info(info_flags::nonnegint))
997 rls.insert(ex_to<numeric>(*i).to_int());
1000 return dirac_trace(e, rls, trONE);
1003 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
1005 // Convert label to set
1006 std::set<unsigned char> rls;
1009 return dirac_trace(e, rls, trONE);
1013 ex canonicalize_clifford(const ex & e_)
1015 pointer_to_map_function fcn(canonicalize_clifford);
1017 if (is_a<matrix>(e_) // || is_a<pseries>(e) || is_a<integral>(e)
1018 || e_.info(info_flags::list)) {
1021 ex e=simplify_indexed(e_);
1022 // Scan for any ncmul objects
1024 ex aux = e.to_rational(srl);
1025 for (exmap::iterator i = srl.begin(); i != srl.end(); ++i) {
1030 if (is_exactly_a<ncmul>(rhs)
1031 && rhs.return_type() == return_types::noncommutative
1032 && is_clifford_tinfo(rhs.return_type_tinfo())) {
1034 // Expand product, if necessary
1035 ex rhs_expanded = rhs.expand();
1036 if (!is_a<ncmul>(rhs_expanded)) {
1037 i->second = canonicalize_clifford(rhs_expanded);
1040 } else if (!is_a<clifford>(rhs.op(0)))
1044 v.reserve(rhs.nops());
1045 for (size_t j=0; j<rhs.nops(); j++)
1046 v.push_back(rhs.op(j));
1048 // Stupid recursive bubble sort because we only want to swap adjacent gammas
1049 exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
1050 if (is_a<diracgamma5>(it->op(0)) || is_a<diracgammaL>(it->op(0)) || is_a<diracgammaR>(it->op(0)))
1053 while (it != next_to_last) {
1054 if (it[0].compare(it[1]) > 0) {
1056 ex save0 = it[0], save1 = it[1];
1058 base_and_index(it[0], b1, i1);
1059 base_and_index(it[1], b2, i2);
1060 // for Clifford algebras (commutator_sign == -1) metric should be symmetrised
1061 it[0] = (ex_to<clifford>(save0).get_metric(i1, i2, ex_to<clifford>(save0).get_commutator_sign() == -1) * b1 * b2).simplify_indexed();
1062 it[1] = v.size() ? _ex2 * dirac_ONE(ex_to<clifford>(save0).get_representation_label()) : _ex2;
1066 sum += ex_to<clifford>(save0).get_commutator_sign() * ncmul(std::move(v));
1067 i->second = canonicalize_clifford(sum);
1075 return aux.subs(srl, subs_options::no_pattern).simplify_indexed();
1079 ex clifford_prime(const ex & e)
1081 pointer_to_map_function fcn(clifford_prime);
1082 if (is_a<clifford>(e) && is_a<cliffordunit>(e.op(0))) {
1084 } else if (is_a<add>(e) || is_a<ncmul>(e) || is_a<mul>(e) //|| is_a<pseries>(e) || is_a<integral>(e)
1085 || is_a<matrix>(e) || e.info(info_flags::list)) {
1087 } else if (is_a<power>(e)) {
1088 return pow(clifford_prime(e.op(0)), e.op(1));
1093 ex remove_dirac_ONE(const ex & e, unsigned char rl, unsigned options)
1095 pointer_to_map_function_2args<unsigned char, unsigned> fcn(remove_dirac_ONE, rl, options | 1);
1096 bool need_reevaluation = false;
1098 if (! (options & 1) ) { // is not a child
1100 e1 = expand_dummy_sum(e, true);
1101 e1 = canonicalize_clifford(e1);
1104 if (is_a<clifford>(e1) && ex_to<clifford>(e1).get_representation_label() >= rl) {
1105 if (is_a<diracone>(e1.op(0)))
1108 throw(std::invalid_argument("remove_dirac_ONE(): expression is a non-scalar Clifford number!"));
1109 } else if (is_a<add>(e1) || is_a<ncmul>(e1) || is_a<mul>(e1)
1110 || is_a<matrix>(e1) || e1.info(info_flags::list)) {
1111 if (options & 3) // is a child or was already expanded
1116 } catch (std::exception &p) {
1117 need_reevaluation = true;
1119 } else if (is_a<power>(e1)) {
1120 if (options & 3) // is a child or was already expanded
1121 return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
1124 return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
1125 } catch (std::exception &p) {
1126 need_reevaluation = true;
1129 if (need_reevaluation)
1130 return remove_dirac_ONE(e, rl, options | 2);
1134 int clifford_max_label(const ex & e, bool ignore_ONE)
1136 if (is_a<clifford>(e))
1137 if (ignore_ONE && is_a<diracone>(e.op(0)))
1140 return ex_to<clifford>(e).get_representation_label();
1143 for (size_t i=0; i < e.nops(); i++)
1144 rl = (rl > clifford_max_label(e.op(i), ignore_ONE)) ? rl : clifford_max_label(e.op(i), ignore_ONE);
1149 ex clifford_norm(const ex & e)
1151 return sqrt(remove_dirac_ONE(e * clifford_bar(e)));
1154 ex clifford_inverse(const ex & e)
1156 ex norm = clifford_norm(e);
1157 if (!norm.is_zero())
1158 return clifford_bar(e) / pow(norm, 2);
1160 throw(std::invalid_argument("clifford_inverse(): cannot find inverse of Clifford number with zero norm!"));
1163 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl)
1165 if (!ex_to<idx>(mu).is_dim_numeric())
1166 throw(std::invalid_argument("lst_to_clifford(): Index should have a numeric dimension"));
1167 ex e = clifford_unit(mu, metr, rl);
1168 return lst_to_clifford(v, e);
1171 ex lst_to_clifford(const ex & v, const ex & e) {
1174 if (is_a<clifford>(e)) {
1177 = is_a<varidx>(mu) ? ex_to<varidx>(mu).toggle_variance() : mu;
1178 unsigned dim = get_dim_uint(mu);
1180 if (is_a<matrix>(v)) {
1181 if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows()) {
1182 min = ex_to<matrix>(v).rows();
1183 max = ex_to<matrix>(v).cols();
1185 min = ex_to<matrix>(v).cols();
1186 max = ex_to<matrix>(v).rows();
1190 return indexed(v, mu_toggle) * e;
1191 else if (max - dim == 1) {
1192 if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows())
1193 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;
1195 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;
1197 throw(std::invalid_argument("lst_to_clifford(): dimensions of vector and clifford unit mismatch"));
1199 throw(std::invalid_argument("lst_to_clifford(): first argument should be a vector (nx1 or 1xn matrix)"));
1200 } else if (v.info(info_flags::list)) {
1201 if (dim == ex_to<lst>(v).nops())
1202 return indexed(matrix(dim, 1, ex_to<lst>(v)), mu_toggle) * e;
1203 else if (ex_to<lst>(v).nops() - dim == 1)
1204 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;
1206 throw(std::invalid_argument("lst_to_clifford(): list length and dimension of clifford unit mismatch"));
1208 throw(std::invalid_argument("lst_to_clifford(): cannot construct from anything but list or vector"));
1210 throw(std::invalid_argument("lst_to_clifford(): the second argument should be a Clifford unit"));
1213 /** Auxiliary structure to define a function for striping one Clifford unit
1214 * from vectors. Used in clifford_to_lst(). */
1215 static ex get_clifford_comp(const ex & e, const ex & c)
1217 pointer_to_map_function_1arg<const ex &> fcn(get_clifford_comp, c);
1218 int ival = ex_to<numeric>(ex_to<idx>(c.op(1)).get_value()).to_int();
1220 if (is_a<add>(e) || e.info(info_flags::list) // || is_a<pseries>(e) || is_a<integral>(e)
1223 else if (is_a<ncmul>(e) || is_a<mul>(e)) {
1224 // find a Clifford unit with the same metric, delete it and substitute its index
1225 size_t ind = e.nops() + 1;
1226 for (size_t j = 0; j < e.nops(); j++) {
1227 if (is_a<clifford>(e.op(j)) && ex_to<clifford>(c).same_metric(e.op(j))) {
1228 if (ind > e.nops()) {
1231 throw(std::invalid_argument("get_clifford_comp(): expression is a Clifford multi-vector"));
1235 if (ind < e.nops()) {
1237 bool same_value_index, found_dummy;
1238 same_value_index = ( ex_to<idx>(e.op(ind).op(1)).is_numeric()
1239 && (ival == ex_to<numeric>(ex_to<idx>(e.op(ind).op(1)).get_value()).to_int()) );
1240 found_dummy = same_value_index;
1241 // Run through the expression collecting all non-clifford factors
1242 for (size_t j=0; j < e.nops(); j++) {
1244 if (same_value_index) {
1248 if (is_a<indexed>(e.op(j)))
1249 ind_vec = ex_to<indexed>(e.op(j)).get_dummy_indices(ex_to<indexed>(e.op(ind)));
1251 if (ind_vec.size() > 0) {
1253 for (auto & it : ind_vec) {
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, curridx_toggle == ival},
1259 subs_options::no_pattern);
1266 return (found_dummy ? S : 0);
1268 throw(std::invalid_argument("get_clifford_comp(): expression is not a Clifford vector to the given units"));
1269 } else if (e.is_zero())
1271 else if (is_a<clifford>(e) && ex_to<clifford>(e).same_metric(c))
1272 if ( ex_to<idx>(e.op(1)).is_numeric() &&
1273 (ival != ex_to<numeric>(ex_to<idx>(e.op(1)).get_value()).to_int()) )
1278 throw(std::invalid_argument("get_clifford_comp(): expression is not usable as a Clifford vector"));
1282 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic)
1284 GINAC_ASSERT(is_a<clifford>(c));
1286 if (! ex_to<idx>(mu).is_dim_numeric())
1287 throw(std::invalid_argument("clifford_to_lst(): index should have a numeric dimension"));
1288 unsigned int D = ex_to<numeric>(ex_to<idx>(mu).get_dim()).to_int();
1290 if (algebraic) // check if algebraic method is applicable
1291 for (unsigned int i = 0; i < D; i++)
1292 if (pow(c.subs(mu == i, subs_options::no_pattern), 2).is_zero()
1293 || (! is_a<numeric>(pow(c.subs(mu == i, subs_options::no_pattern), 2))))
1296 ex v0 = remove_dirac_ONE(canonicalize_clifford(e+clifford_prime(e)).normal())/2;
1299 ex e1 = canonicalize_clifford(e - v0 * dirac_ONE(ex_to<clifford>(c).get_representation_label()));
1301 for (unsigned int i = 0; i < D; i++)
1302 V.append(remove_dirac_ONE(
1303 simplify_indexed(canonicalize_clifford(e1 * c.subs(mu == i, subs_options::no_pattern) + c.subs(mu == i, subs_options::no_pattern) * e1))
1304 / (2*pow(c.subs(mu == i, subs_options::no_pattern), 2))));
1307 for (unsigned int i = 0; i < D; i++)
1308 V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
1309 } catch (std::exception &p) {
1310 /* Try to expand dummy summations to simplify the expression*/
1311 e1 = canonicalize_clifford(expand_dummy_sum(e, true));
1313 v0 = remove_dirac_ONE(canonicalize_clifford(e1+clifford_prime(e1)).normal())/2;
1314 if (! v0.is_zero()) {
1316 e1 = canonicalize_clifford(e1 - v0 * dirac_ONE(ex_to<clifford>(c).get_representation_label()));
1318 for (unsigned int i = 0; i < D; i++)
1319 V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
1326 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)
1330 if (! is_a<matrix>(v) && ! v.info(info_flags::list))
1331 throw(std::invalid_argument("clifford_moebius_map(): parameter v should be either vector or list"));
1333 if (is_a<clifford>(G)) {
1336 if (is_a<indexed>(G)) {
1337 D = ex_to<idx>(G.op(1)).get_dim();
1338 varidx mu((new symbol)->setflag(status_flags::dynallocated), D);
1339 cu = clifford_unit(mu, G, rl);
1340 } else if (is_a<matrix>(G)) {
1341 D = ex_to<matrix>(G).rows();
1342 idx mu((new symbol)->setflag(status_flags::dynallocated), D);
1343 cu = clifford_unit(mu, G, rl);
1344 } else throw(std::invalid_argument("clifford_moebius_map(): metric should be an indexed object, matrix, or a Clifford unit"));
1348 x = lst_to_clifford(v, cu);
1349 ex e = clifford_to_lst(simplify_indexed(canonicalize_clifford((a * x + b) * clifford_inverse(c * x + d))), cu, false);
1350 return (is_a<matrix>(v) ? matrix(ex_to<matrix>(v).rows(), ex_to<matrix>(v).cols(), ex_to<lst>(e)) : e);
1353 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl)
1355 if (is_a<matrix>(M) && (ex_to<matrix>(M).rows() == 2) && (ex_to<matrix>(M).cols() == 2))
1356 return clifford_moebius_map(M.op(0), M.op(1), M.op(2), M.op(3), v, G, rl);
1358 throw(std::invalid_argument("clifford_moebius_map(): parameter M should be a 2x2 matrix"));
1361 } // namespace GiNaC