3 * Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
6 * GiNaC Copyright (C) 1999-2019 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).
48 print_func<print_tree>(&clifford::do_print_tree))
50 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracone, tensor,
51 print_func<print_dflt>(&diracone::do_print).
52 print_func<print_latex>(&diracone::do_print_latex))
54 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(cliffordunit, tensor,
55 print_func<print_dflt>(&cliffordunit::do_print).
56 print_func<print_latex>(&cliffordunit::do_print_latex))
58 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma, cliffordunit,
59 print_func<print_dflt>(&diracgamma::do_print).
60 print_func<print_latex>(&diracgamma::do_print_latex))
62 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma5, tensor,
63 print_func<print_dflt>(&diracgamma5::do_print).
64 print_func<print_latex>(&diracgamma5::do_print_latex))
66 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaL, tensor,
67 print_func<print_context>(&diracgammaL::do_print).
68 print_func<print_latex>(&diracgammaL::do_print_latex))
70 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaR, tensor,
71 print_func<print_context>(&diracgammaR::do_print).
72 print_func<print_latex>(&diracgammaR::do_print_latex))
75 // default constructors
78 clifford::clifford() : representation_label(0), metric(0), commutator_sign(-1)
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)
100 /** Construct object with one Lorentz index. This constructor is for internal
101 * use only. Use the clifford_unit() or dirac_gamma() functions instead.
103 * @see dirac_gamma */
104 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)
106 GINAC_ASSERT(is_a<idx>(mu));
109 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)
113 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)
117 return_type_t clifford::return_type_tinfo() const
119 return make_return_type_t<clifford>(representation_label);
126 void clifford::read_archive(const archive_node& n, lst& sym_lst)
128 inherited::read_archive(n, sym_lst);
130 n.find_unsigned("label", rl);
131 representation_label = rl;
132 n.find_ex("metric", metric, sym_lst);
133 n.find_unsigned("commutator_sign+1", rl);
134 commutator_sign = rl - 1;
137 void clifford::archive(archive_node & n) const
139 inherited::archive(n);
140 n.add_unsigned("label", representation_label);
141 n.add_ex("metric", metric);
142 n.add_unsigned("commutator_sign+1", commutator_sign+1);
145 GINAC_BIND_UNARCHIVER(clifford);
146 GINAC_BIND_UNARCHIVER(cliffordunit);
147 GINAC_BIND_UNARCHIVER(diracone);
148 GINAC_BIND_UNARCHIVER(diracgamma);
149 GINAC_BIND_UNARCHIVER(diracgamma5);
150 GINAC_BIND_UNARCHIVER(diracgammaL);
151 GINAC_BIND_UNARCHIVER(diracgammaR);
154 ex clifford::get_metric(const ex & i, const ex & j, bool symmetrised) const
156 if (is_a<indexed>(metric)) {
157 if (symmetrised && !(ex_to<symmetry>(ex_to<indexed>(metric).get_symmetry()).has_symmetry())) {
158 if (is_a<matrix>(metric.op(0))) {
159 return indexed((ex_to<matrix>(metric.op(0)).add(ex_to<matrix>(metric.op(0)).transpose())).mul(numeric(1, 2)),
162 return simplify_indexed(indexed(metric.op(0)*_ex1_2, i, j) + indexed(metric.op(0)*_ex1_2, j, i));
165 return metric.subs(lst{metric.op(1) == i, metric.op(2) == j}, subs_options::no_pattern);
168 exvector indices = metric.get_free_indices();
170 return _ex1_2*simplify_indexed(metric.subs(lst{indices[0] == i, indices[1] == j}, subs_options::no_pattern)
171 + metric.subs(lst{indices[0] == j, indices[1] == i}, subs_options::no_pattern));
173 return metric.subs(lst{indices[0] == i, indices[1] == j}, subs_options::no_pattern);
177 bool clifford::same_metric(const ex & other) const
180 if (is_a<clifford>(other))
181 metr = ex_to<clifford>(other).get_metric();
185 if (is_a<indexed>(metr))
186 return metr.op(0).is_equal(get_metric().op(0));
188 exvector indices = metr.get_free_indices();
189 return (indices.size() == 2)
190 && simplify_indexed(get_metric(indices[0], indices[1])-metr).is_zero();
195 // functions overriding virtual functions from base classes
198 ex clifford::op(size_t i) const
200 GINAC_ASSERT(i<nops());
202 return representation_label;
204 return inherited::op(i);
207 ex & clifford::let_op(size_t i)
209 GINAC_ASSERT(i<nops());
211 static ex rl = numeric(representation_label);
212 ensure_if_modifiable();
216 return inherited::let_op(i);
219 ex clifford::subs(const exmap & m, unsigned options) const
221 ex subsed = inherited::subs(m, options);
222 if(is_a<clifford>(subsed)) {
223 ex prevmetric = ex_to<clifford>(subsed).metric;
224 ex newmetric = prevmetric.subs(m, options);
225 if(!are_ex_trivially_equal(prevmetric, newmetric)) {
226 clifford c = ex_to<clifford>(subsed);
227 c.metric = newmetric;
234 int clifford::compare_same_type(const basic & other) const
236 GINAC_ASSERT(is_a<clifford>(other));
237 const clifford &o = static_cast<const clifford &>(other);
239 if (representation_label != o.representation_label) {
240 // different representation label
241 return representation_label < o.representation_label ? -1 : 1;
244 return inherited::compare_same_type(other);
247 bool clifford::match_same_type(const basic & other) const
249 GINAC_ASSERT(is_a<clifford>(other));
250 const clifford &o = static_cast<const clifford &>(other);
252 return ((representation_label == o.representation_label) && (commutator_sign == o.get_commutator_sign()) && same_metric(o));
255 static bool is_dirac_slash(const ex & seq0)
257 return !is_a<diracgamma5>(seq0) && !is_a<diracgammaL>(seq0) &&
258 !is_a<diracgammaR>(seq0) && !is_a<cliffordunit>(seq0) &&
259 !is_a<diracone>(seq0);
262 void clifford::do_print_dflt(const print_dflt & c, unsigned level) const
264 // dirac_slash() object is printed differently
265 if (is_dirac_slash(seq[0])) {
266 seq[0].print(c, precedence());
268 } else { // We do not print representation label if it is 0
269 if (representation_label == 0) {
270 this->print_dispatch<inherited>(c, level);
271 } else { // otherwise we put it before indices in square brackets; the code is borrowed from indexed.cpp
272 if (precedence() <= level) {
275 seq[0].print(c, precedence());
276 c.s << '[' << int(representation_label) << ']';
277 printindices(c, level);
278 if (precedence() <= level) {
285 void clifford::do_print_latex(const print_latex & c, unsigned level) const
287 // dirac_slash() object is printed differently
288 if (is_dirac_slash(seq[0])) {
290 seq[0].print(c, precedence());
291 c.s << "\\hspace{-1.0ex}/}";
293 c.s << "\\clifford[" << int(representation_label) << "]";
294 this->print_dispatch<inherited>(c, level);
298 void clifford::do_print_tree(const print_tree & c, unsigned level) const
300 c.s << std::string(level, ' ') << class_name() << " @" << this
301 << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
302 << ", " << seq.size()-1 << " indices"
303 << ", symmetry=" << symtree << std::endl;
304 metric.print(c, level + c.delta_indent);
305 seq[0].print(c, level + c.delta_indent);
306 printindices(c, level + c.delta_indent);
309 DEFAULT_COMPARE(diracone)
310 DEFAULT_COMPARE(cliffordunit)
311 DEFAULT_COMPARE(diracgamma)
312 DEFAULT_COMPARE(diracgamma5)
313 DEFAULT_COMPARE(diracgammaL)
314 DEFAULT_COMPARE(diracgammaR)
316 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbf{1}")
317 DEFAULT_PRINT_LATEX(cliffordunit, "e", "e")
318 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
319 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
320 DEFAULT_PRINT_LATEX(diracgammaL, "gammaL", "{\\gamma_L}")
321 DEFAULT_PRINT_LATEX(diracgammaR, "gammaR", "{\\gamma_R}")
323 /** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
324 static void base_and_index(const ex & c, ex & b, ex & i)
326 GINAC_ASSERT(is_a<clifford>(c));
327 GINAC_ASSERT(c.nops() == 2+1);
329 if (is_a<cliffordunit>(c.op(0))) { // proper dirac gamma object or clifford unit
332 } else if (is_a<diracgamma5>(c.op(0)) || is_a<diracgammaL>(c.op(0)) || is_a<diracgammaR>(c.op(0))) { // gamma5/L/R
335 } else { // slash object, generate new dummy index
336 varidx ix(dynallocate<symbol>(), ex_to<idx>(c.op(1)).get_dim());
337 b = indexed(c.op(0), ix.toggle_variance());
342 /** Predicate for finding non-clifford objects. */
343 struct is_not_a_clifford {
344 bool operator()(const ex & e)
346 return !is_a<clifford>(e);
350 /** Contraction of a gamma matrix with something else. */
351 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
353 GINAC_ASSERT(is_a<clifford>(*self));
354 GINAC_ASSERT(is_a<indexed>(*other));
355 GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
356 unsigned char rl = ex_to<clifford>(*self).get_representation_label();
358 ex dim = ex_to<idx>(self->op(1)).get_dim();
359 if (other->nops() > 1)
360 dim = minimal_dim(dim, ex_to<idx>(other->op(1)).get_dim());
362 if (is_a<clifford>(*other)) {
364 // Contraction only makes sense if the representation labels are equal
365 if (ex_to<clifford>(*other).get_representation_label() != rl)
368 size_t num = other - self;
370 // gamma~mu gamma.mu = dim ONE
373 *other = dirac_ONE(rl);
376 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
378 && is_a<clifford>(self[1])) {
383 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
385 && is_a<clifford>(self[1])
386 && is_a<clifford>(self[2])) {
388 base_and_index(self[1], b1, i1);
389 base_and_index(self[2], b2, i2);
390 *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
396 // 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
398 && is_a<clifford>(self[1])
399 && is_a<clifford>(self[2])
400 && is_a<clifford>(self[3])) {
401 *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3];
408 // gamma~mu Sodd gamma.mu = -2 Sodd_R
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 *self = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(other), std::reverse_iterator<exvector::const_iterator>(self + 1)));
415 std::fill(self + 1, other, _ex1);
419 // gamma~mu Sodd gamma~alpha gamma.mu = 2 gamma~alpha Sodd + 2 Sodd_R gamma~alpha
420 // (commutate contracted indices towards each other, then use
421 // Chisholm identity in 4 dimensions)
422 } else if (((other - self) & 1) && dim.is_equal(4)) {
423 if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
426 auto next_to_last = other - 1;
427 ex S = ncmul(exvector(self + 1, next_to_last));
428 ex SR = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(next_to_last), std::reverse_iterator<exvector::const_iterator>(self + 1)));
430 *self = (*next_to_last) * S + SR * (*next_to_last);
431 std::fill(self + 1, other, _ex1);
435 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
436 // (commutate contracted indices towards each other, simplify_indexed()
437 // will re-expand and re-run the simplification)
439 if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
442 auto next_to_last = other - 1;
443 ex S = ncmul(exvector(self + 1, next_to_last));
445 *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
446 std::fill(self + 1, other + 1, _ex1);
450 } else if (is_a<symbol>(other->op(0)) && other->nops() == 2) {
452 // x.mu gamma~mu -> x-slash
453 *self = dirac_slash(other->op(0), dim, rl);
461 /** Contraction of a Clifford unit with something else. */
462 bool cliffordunit::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
464 GINAC_ASSERT(is_a<clifford>(*self));
465 GINAC_ASSERT(is_a<indexed>(*other));
466 GINAC_ASSERT(is_a<cliffordunit>(self->op(0)));
467 clifford unit = ex_to<clifford>(*self);
468 unsigned char rl = unit.get_representation_label();
470 if (is_a<clifford>(*other)) {
471 // Contraction only makes sense if the representation labels are equal
472 // and the metrics are the same
473 if ((ex_to<clifford>(*other).get_representation_label() != rl)
474 && unit.same_metric(*other))
477 auto before_other = other - 1;
479 ex mu_toggle = other->op(1);
480 ex alpha = before_other->op(1);
482 // e~mu e.mu = Tr ONE
483 if (other - self == 1) {
484 *self = unit.get_metric(mu, mu_toggle, true);
485 *other = dirac_ONE(rl);
488 } else if (other - self == 2) {
489 if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
490 // e~mu e~alpha e.mu = 2*e~mu B(alpha, mu.toggle_variance())-Tr(B) e~alpha
491 *self = 2 * (*self) * unit.get_metric(alpha, mu_toggle, true) - unit.get_metric(mu, mu_toggle, true) * (*before_other);
492 *before_other = _ex1;
497 // e~mu S e.mu = Tr S ONE
498 *self = unit.get_metric(mu, mu_toggle, true);
499 *other = dirac_ONE(rl);
503 // e~mu S e~alpha e.mu = 2 e~mu S B(alpha, mu.toggle_variance()) - e~mu S e.mu e~alpha
504 // (commutate contracted indices towards each other, simplify_indexed()
505 // will re-expand and re-run the simplification)
506 if (std::find_if(self + 1, other, is_not_a_clifford()) != other) {
510 ex S = ncmul(exvector(self + 1, before_other));
512 if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
513 *self = 2 * (*self) * S * unit.get_metric(alpha, mu_toggle, true) - (*self) * S * (*other) * (*before_other);
516 *self = (*self) * S * (*other) * (*before_other);
519 std::fill(self + 1, other + 1, _ex1);
526 /** Perform automatic simplification on noncommutative product of clifford
527 * objects. This removes superfluous ONEs, permutes gamma5/L/R's to the front
528 * and removes squares of gamma objects. */
529 ex clifford::eval_ncmul(const exvector & v) const
534 // Remove superfluous ONEs
535 for (auto & it : v) {
536 if (!is_a<clifford>(it) || !is_a<diracone>(it.op(0)))
540 bool something_changed = false;
543 // Anticommutate gamma5/L/R's to the front
545 auto first = s.begin(), next_to_last = s.end() - 2;
547 auto it = next_to_last;
550 if (is_a<clifford>(*it) && is_a<clifford>(*it2)) {
551 ex e1 = it->op(0), e2 = it2->op(0);
553 if (is_a<diracgamma5>(e2)) {
555 if (is_a<diracgammaL>(e1) || is_a<diracgammaR>(e1)) {
557 // gammaL/R gamma5 -> gamma5 gammaL/R
559 something_changed = true;
561 } else if (!is_a<diracgamma5>(e1)) {
563 // gamma5 gamma5 -> gamma5 gamma5 (do nothing)
564 // x gamma5 -> -gamma5 x
567 something_changed = true;
570 } else if (is_a<diracgammaL>(e2)) {
572 if (is_a<diracgammaR>(e1)) {
574 // gammaR gammaL -> 0
577 } else if (!is_a<diracgammaL>(e1) && !is_a<diracgamma5>(e1)) {
579 // gammaL gammaL -> gammaL gammaL (do nothing)
580 // gamma5 gammaL -> gamma5 gammaL (do nothing)
581 // x gammaL -> gammaR x
583 *it = clifford(diracgammaR(), ex_to<clifford>(*it).get_representation_label());
584 something_changed = true;
587 } else if (is_a<diracgammaR>(e2)) {
589 if (is_a<diracgammaL>(e1)) {
591 // gammaL gammaR -> 0
594 } else if (!is_a<diracgammaR>(e1) && !is_a<diracgamma5>(e1)) {
596 // gammaR gammaR -> gammaR gammaR (do nothing)
597 // gamma5 gammaR -> gamma5 gammaR (do nothing)
598 // x gammaR -> gammaL x
600 *it = clifford(diracgammaL(), ex_to<clifford>(*it).get_representation_label());
601 something_changed = true;
609 if (next_to_last == first)
615 // Remove equal adjacent gammas
617 exvector::iterator it, itend = s.end() - 1;
618 for (it = s.begin(); it != itend; ++it) {
621 if (!is_a<clifford>(a) || !is_a<clifford>(b))
624 const ex & ag = a.op(0);
625 const ex & bg = b.op(0);
626 bool a_is_cliffordunit = is_a<cliffordunit>(ag);
627 bool b_is_cliffordunit = is_a<cliffordunit>(bg);
629 if (a_is_cliffordunit && b_is_cliffordunit && ex_to<clifford>(a).same_metric(b)
630 && (ex_to<clifford>(a).get_commutator_sign() == -1)) {
631 // This is done only for Clifford algebras
633 const ex & ia = a.op(1);
634 const ex & ib = b.op(1);
635 if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
636 a = ex_to<clifford>(a).get_metric(ia, ib, true);
637 b = dirac_ONE(representation_label);
638 something_changed = true;
641 } else if ((is_a<diracgamma5>(ag) && is_a<diracgamma5>(bg))) {
643 // Remove squares of gamma5
644 a = dirac_ONE(representation_label);
645 b = dirac_ONE(representation_label);
646 something_changed = true;
648 } else if ((is_a<diracgammaL>(ag) && is_a<diracgammaL>(bg))
649 || (is_a<diracgammaR>(ag) && is_a<diracgammaR>(bg))) {
651 // Remove squares of gammaL/R
652 b = dirac_ONE(representation_label);
653 something_changed = true;
655 } else if (is_a<diracgammaL>(ag) && is_a<diracgammaR>(bg)) {
657 // gammaL and gammaR are orthogonal
660 } else if (is_a<diracgamma5>(ag) && is_a<diracgammaL>(bg)) {
662 // gamma5 gammaL -> -gammaL
663 a = dirac_ONE(representation_label);
665 something_changed = true;
667 } else if (is_a<diracgamma5>(ag) && is_a<diracgammaR>(bg)) {
669 // gamma5 gammaR -> gammaR
670 a = dirac_ONE(representation_label);
671 something_changed = true;
673 } else if (!a_is_cliffordunit && !b_is_cliffordunit && ag.is_equal(bg)) {
676 varidx ix(dynallocate<symbol>(), ex_to<idx>(a.op(1)).minimal_dim(ex_to<idx>(b.op(1))));
678 a = indexed(ag, ix) * indexed(ag, ix.toggle_variance());
679 b = dirac_ONE(representation_label);
680 something_changed = true;
686 return dirac_ONE(representation_label) * sign;
687 if (something_changed)
688 return reeval_ncmul(s) * sign;
690 return hold_ncmul(s) * sign;
693 ex clifford::thiscontainer(const exvector & v) const
695 return clifford(representation_label, metric, commutator_sign, v);
698 ex clifford::thiscontainer(exvector && v) const
700 return clifford(representation_label, metric, commutator_sign, std::move(v));
703 ex diracgamma5::conjugate() const
705 return _ex_1 * (*this);
708 ex diracgammaL::conjugate() const
710 return dynallocate<diracgammaR>();
713 ex diracgammaR::conjugate() const
715 return dynallocate<diracgammaL>();
722 ex dirac_ONE(unsigned char rl)
724 static ex ONE = dynallocate<diracone>();
725 return clifford(ONE, rl);
728 static unsigned get_dim_uint(const ex& e)
731 throw std::invalid_argument("get_dim_uint: argument is not an index");
732 ex dim = ex_to<idx>(e).get_dim();
733 if (!dim.info(info_flags::posint))
734 throw std::invalid_argument("get_dim_uint: dimension of index should be a positive integer");
735 unsigned d = ex_to<numeric>(dim).to_int();
739 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl)
741 ex unit = dynallocate<cliffordunit>();
744 throw(std::invalid_argument("clifford_unit(): index of Clifford unit must be of type idx or varidx"));
746 exvector indices = metr.get_free_indices();
748 if (indices.size() == 2) {
749 return clifford(unit, mu, metr, rl);
750 } else if (is_a<matrix>(metr)) {
751 matrix M = ex_to<matrix>(metr);
752 unsigned n = M.rows();
753 bool symmetric = true;
755 //static idx xi(dynallocate<symbol>(), n),
756 // chi(dynallocate<symbol>(), n);
757 idx xi(dynallocate<symbol>(), n),
758 chi(dynallocate<symbol>(), n);
759 if ((n == M.cols()) && (n == get_dim_uint(mu))) {
760 for (unsigned i = 0; i < n; i++) {
761 for (unsigned j = i+1; j < n; j++) {
762 if (!M(i, j).is_equal(M(j, i))) {
767 return clifford(unit, mu, indexed(metr, symmetric?symmetric2():not_symmetric(), xi, chi), rl);
769 throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be a square matrix with the same dimensions as index"));
771 } else if (indices.size() == 0) { // a tensor or other expression without indices
772 //static varidx xi(dynallocate<symbol>(), ex_to<idx>(mu).get_dim()),
773 // chi(dynallocate<symbol>(), ex_to<idx>(mu).get_dim());
774 varidx xi(dynallocate<symbol>(), ex_to<idx>(mu).get_dim()),
775 chi(dynallocate<symbol>(), ex_to<idx>(mu).get_dim());
776 return clifford(unit, mu, indexed(metr, xi, chi), rl);
778 throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be of type tensor, matrix or an expression with two free indices"));
781 ex dirac_gamma(const ex & mu, unsigned char rl)
783 static ex gamma = dynallocate<diracgamma>();
785 if (!is_a<varidx>(mu))
786 throw(std::invalid_argument("dirac_gamma(): index of Dirac gamma must be of type varidx"));
788 static varidx xi(dynallocate<symbol>(), ex_to<varidx>(mu).get_dim()),
789 chi(dynallocate<symbol>(), ex_to<varidx>(mu).get_dim());
790 return clifford(gamma, mu, indexed(dynallocate<minkmetric>(), symmetric2(), xi, chi), rl);
793 ex dirac_gamma5(unsigned char rl)
795 static ex gamma5 = dynallocate<diracgamma5>();
796 return clifford(gamma5, rl);
799 ex dirac_gammaL(unsigned char rl)
801 static ex gammaL = dynallocate<diracgammaL>();
802 return clifford(gammaL, rl);
805 ex dirac_gammaR(unsigned char rl)
807 static ex gammaR = dynallocate<diracgammaR>();
808 return clifford(gammaR, rl);
811 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
813 // Slashed vectors are actually stored as a clifford object with the
814 // vector as its base expression and a (dummy) index that just serves
815 // for storing the space dimensionality
817 static varidx xi(dynallocate<symbol>(), dim),
818 chi(dynallocate<symbol>(), dim);
819 return clifford(e, varidx(0, dim), indexed(dynallocate<minkmetric>(), symmetric2(), xi, chi), rl);
822 /** Extract representation label from tinfo key (as returned by
823 * return_type_tinfo()). */
824 static unsigned char get_representation_label(const return_type_t& ti)
826 return (unsigned char)ti.rl;
829 /** Take trace of a string of an even number of Dirac gammas given a vector
831 static ex trace_string(exvector::const_iterator ix, size_t num)
833 // Tr gamma.mu gamma.nu = 4 g.mu.nu
835 return lorentz_g(ix[0], ix[1]);
837 // 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 )
839 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
840 + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
841 - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
843 // Traces of 6 or more gammas are computed recursively:
844 // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
845 // + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
846 // - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
847 // + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
849 // + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
853 for (size_t i=1; i<num; i++) {
854 for (size_t n=1, j=0; n<num; n++) {
859 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
865 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE)
867 if (is_a<clifford>(e)) {
869 unsigned char rl = ex_to<clifford>(e).get_representation_label();
871 // Are we taking the trace over this object's representation label?
872 if (rls.find(rl) == rls.end())
875 // Yes, all elements are traceless, except for dirac_ONE and dirac_L/R
876 const ex & g = e.op(0);
877 if (is_a<diracone>(g))
879 else if (is_a<diracgammaL>(g) || is_a<diracgammaR>(g))
884 } else if (is_exactly_a<mul>(e)) {
886 // Trace of product: pull out non-clifford factors
888 for (size_t i=0; i<e.nops(); i++) {
889 const ex &o = e.op(i);
890 if (is_clifford_tinfo(o.return_type_tinfo()))
891 prod *= dirac_trace(o, rls, trONE);
897 } else if (is_exactly_a<ncmul>(e)) {
899 unsigned char rl = get_representation_label(e.return_type_tinfo());
901 // Are we taking the trace over this string's representation label?
902 if (rls.find(rl) == rls.end())
905 // Substitute gammaL/R and expand product, if necessary
906 ex e_expanded = e.subs(lst{
907 dirac_gammaL(rl) == (dirac_ONE(rl)-dirac_gamma5(rl))/2,
908 dirac_gammaR(rl) == (dirac_ONE(rl)+dirac_gamma5(rl))/2
909 }, subs_options::no_pattern).expand();
910 if (!is_a<ncmul>(e_expanded))
911 return dirac_trace(e_expanded, rls, trONE);
913 // gamma5 gets moved to the front so this check is enough
914 bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
915 size_t num = e.nops();
919 // Trace of gamma5 * odd number of gammas and trace of
920 // gamma5 * gamma.mu * gamma.nu are zero
921 if ((num & 1) == 0 || num == 3)
924 // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
925 // (the epsilon is always 4-dimensional)
927 ex b1, i1, b2, i2, b3, i3, b4, i4;
928 base_and_index(e.op(1), b1, i1);
929 base_and_index(e.op(2), b2, i2);
930 base_and_index(e.op(3), b3, i3);
931 base_and_index(e.op(4), b4, i4);
932 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();
936 // I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
937 // (the epsilon is always 4-dimensional)
938 exvector ix(num-1), bv(num-1);
939 for (size_t i=1; i<num; i++)
940 base_and_index(e.op(i), bv[i-1], ix[i-1]);
942 int *iv = new int[num];
944 for (size_t i=0; i<num-3; i++) {
946 for (size_t j=i+1; j<num-2; j++) {
948 for (size_t k=j+1; k<num-1; k++) {
950 for (size_t l=k+1; l<num; l++) {
952 iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
955 for (size_t n=0, t=4; n<num; n++) {
956 if (n == i || n == j || n == k || n == l)
961 int sign = permutation_sign(iv, iv + num);
962 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))
963 * trace_string(v.begin(), num - 4);
969 return trONE * I * result * mul(bv);
971 } else { // no gamma5
973 // Trace of odd number of gammas is zero
977 // Tr gamma.mu gamma.nu = 4 g.mu.nu
980 base_and_index(e.op(0), b1, i1);
981 base_and_index(e.op(1), b2, i2);
982 return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
985 exvector iv(num), bv(num);
986 for (size_t i=0; i<num; i++)
987 base_and_index(e.op(i), bv[i], iv[i]);
989 return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
992 } else if (e.nops() > 0) {
994 // Trace maps to all other container classes (this includes sums)
995 pointer_to_map_function_2args<const std::set<unsigned char> &, const ex &> fcn(dirac_trace, rls, trONE);
1002 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE)
1004 // Convert list to set
1005 std::set<unsigned char> rls;
1006 for (const auto & i : rll) {
1007 if (i.info(info_flags::nonnegint))
1008 rls.insert(ex_to<numeric>(i).to_int());
1011 return dirac_trace(e, rls, trONE);
1014 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
1016 // Convert label to set
1017 std::set<unsigned char> rls;
1020 return dirac_trace(e, rls, trONE);
1024 ex canonicalize_clifford(const ex & e_)
1026 pointer_to_map_function fcn(canonicalize_clifford);
1028 if (is_a<matrix>(e_) // || is_a<pseries>(e) || is_a<integral>(e)
1029 || e_.info(info_flags::list)) {
1032 ex e=simplify_indexed(e_);
1033 // Scan for any ncmul objects
1035 ex aux = e.to_rational(srl);
1036 for (auto & i : srl) {
1041 if (is_exactly_a<ncmul>(rhs)
1042 && rhs.return_type() == return_types::noncommutative
1043 && is_clifford_tinfo(rhs.return_type_tinfo())) {
1045 // Expand product, if necessary
1046 ex rhs_expanded = rhs.expand();
1047 if (!is_a<ncmul>(rhs_expanded)) {
1048 i.second = canonicalize_clifford(rhs_expanded);
1051 } else if (!is_a<clifford>(rhs.op(0)))
1055 v.reserve(rhs.nops());
1056 for (size_t j=0; j<rhs.nops(); j++)
1057 v.push_back(rhs.op(j));
1059 // Stupid recursive bubble sort because we only want to swap adjacent gammas
1060 auto it = v.begin(), next_to_last = v.end() - 1;
1061 if (is_a<diracgamma5>(it->op(0)) || is_a<diracgammaL>(it->op(0)) || is_a<diracgammaR>(it->op(0)))
1064 while (it != next_to_last) {
1065 if (it[0].compare(it[1]) > 0) {
1067 ex save0 = it[0], save1 = it[1];
1069 base_and_index(it[0], b1, i1);
1070 base_and_index(it[1], b2, i2);
1071 // for Clifford algebras (commutator_sign == -1) metric should be symmetrised
1072 it[0] = (ex_to<clifford>(save0).get_metric(i1, i2, ex_to<clifford>(save0).get_commutator_sign() == -1) * b1 * b2).simplify_indexed();
1073 it[1] = v.size() ? _ex2 * dirac_ONE(ex_to<clifford>(save0).get_representation_label()) : _ex2;
1077 sum += ex_to<clifford>(save0).get_commutator_sign() * ncmul(std::move(v));
1078 i.second = canonicalize_clifford(sum);
1086 return aux.subs(srl, subs_options::no_pattern).simplify_indexed();
1090 ex clifford_star_bar(const ex & e, bool do_bar, unsigned options)
1092 pointer_to_map_function_2args<bool, unsigned> fcn(clifford_star_bar, do_bar, options | 1);
1094 // is a child, no need to expand
1095 ex e1= (options & 1 ? e : e.expand());
1097 if (is_a<ncmul>(e1) ) { // reversing order of clifford units
1099 ev.reserve(e1.nops());
1100 cv.reserve(e1.nops());
1101 // separate clifford and non-clifford entries
1102 for (int i= 0; i < e1.nops(); ++i) {
1103 if (is_a<clifford>(e1.op(i)) && is_a<cliffordunit>(e1.op(i).op(0)))
1104 cv.push_back(e1.op(i));
1106 ev.push_back(e1.op(i));
1108 for (auto i=cv.rbegin(); i!=cv.rend(); ++i) { // reverse order of Clifford units
1109 ev.push_back(i->conjugate());
1111 // For clifford_bar an odd number of clifford units reverts the sign
1112 if (do_bar && (cv.size() % 2 == 1))
1113 return -dynallocate<ncmul>(std::move(ev));
1115 return dynallocate<ncmul>(std::move(ev));
1116 } else if (is_a<clifford>(e1) && is_a<cliffordunit>(e1.op(0))) {
1121 } else if (is_a<power>(e1)) {
1122 // apply the procedure to the base of a power
1123 return pow(clifford_star_bar(e1.op(0), do_bar, 0), e1.op(1));
1124 } else if (is_a<add>(e1) || is_a<mul>(e1) || e.info(info_flags::list)) {
1125 // recurse into subexpressions
1127 } else // nothing meaningful can be done
1131 ex clifford_prime(const ex & e)
1133 pointer_to_map_function fcn(clifford_prime);
1134 if (is_a<clifford>(e) && is_a<cliffordunit>(e.op(0))) {
1136 } else if (is_a<add>(e) || is_a<ncmul>(e) || is_a<mul>(e) //|| is_a<pseries>(e) || is_a<integral>(e)
1137 || is_a<matrix>(e) || e.info(info_flags::list)) {
1139 } else if (is_a<power>(e)) {
1140 return pow(clifford_prime(e.op(0)), e.op(1));
1145 ex remove_dirac_ONE(const ex & e, unsigned char rl, unsigned options)
1147 pointer_to_map_function_2args<unsigned char, unsigned> fcn(remove_dirac_ONE, rl, options | 1);
1148 bool need_reevaluation = false;
1150 if (! (options & 1) ) { // is not a child
1152 e1 = expand_dummy_sum(e, true);
1153 e1 = canonicalize_clifford(e1);
1156 if (is_a<clifford>(e1) && ex_to<clifford>(e1).get_representation_label() >= rl) {
1157 if (is_a<diracone>(e1.op(0)))
1160 throw(std::invalid_argument("remove_dirac_ONE(): expression is a non-scalar Clifford number!"));
1161 } else if (is_a<add>(e1) || is_a<ncmul>(e1) || is_a<mul>(e1)
1162 || is_a<matrix>(e1) || e1.info(info_flags::list)) {
1163 if (options & 3) // is a child or was already expanded
1168 } catch (std::exception &p) {
1169 need_reevaluation = true;
1171 } else if (is_a<power>(e1)) {
1172 if (options & 3) // is a child or was already expanded
1173 return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
1176 return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
1177 } catch (std::exception &p) {
1178 need_reevaluation = true;
1181 if (need_reevaluation)
1182 return remove_dirac_ONE(e, rl, options | 2);
1186 int clifford_max_label(const ex & e, bool ignore_ONE)
1188 if (is_a<clifford>(e))
1189 if (ignore_ONE && is_a<diracone>(e.op(0)))
1192 return ex_to<clifford>(e).get_representation_label();
1195 for (size_t i=0; i < e.nops(); i++)
1196 rl = (rl > clifford_max_label(e.op(i), ignore_ONE)) ? rl : clifford_max_label(e.op(i), ignore_ONE);
1201 ex clifford_norm(const ex & e)
1203 return sqrt(remove_dirac_ONE(e * clifford_bar(e)));
1206 ex clifford_inverse(const ex & e)
1208 ex norm = clifford_norm(e);
1209 if (!norm.is_zero())
1210 return clifford_bar(e) / pow(norm, 2);
1212 throw(std::invalid_argument("clifford_inverse(): cannot find inverse of Clifford number with zero norm!"));
1215 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl)
1217 if (!ex_to<idx>(mu).is_dim_numeric())
1218 throw(std::invalid_argument("lst_to_clifford(): Index should have a numeric dimension"));
1219 ex e = clifford_unit(mu, metr, rl);
1220 return lst_to_clifford(v, e);
1223 ex lst_to_clifford(const ex & v, const ex & e) {
1226 if (is_a<clifford>(e)) {
1229 = is_a<varidx>(mu) ? ex_to<varidx>(mu).toggle_variance() : mu;
1230 unsigned dim = get_dim_uint(mu);
1232 if (is_a<matrix>(v)) {
1233 if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows()) {
1234 min = ex_to<matrix>(v).rows();
1235 max = ex_to<matrix>(v).cols();
1237 min = ex_to<matrix>(v).cols();
1238 max = ex_to<matrix>(v).rows();
1242 return indexed(v, mu_toggle) * e;
1243 else if (max - dim == 1) {
1244 if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows())
1245 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;
1247 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;
1249 throw(std::invalid_argument("lst_to_clifford(): dimensions of vector and clifford unit mismatch"));
1251 throw(std::invalid_argument("lst_to_clifford(): first argument should be a vector (nx1 or 1xn matrix)"));
1252 } else if (v.info(info_flags::list)) {
1253 if (dim == ex_to<lst>(v).nops())
1254 return indexed(matrix(dim, 1, ex_to<lst>(v)), mu_toggle) * e;
1255 else if (ex_to<lst>(v).nops() - dim == 1)
1256 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;
1258 throw(std::invalid_argument("lst_to_clifford(): list length and dimension of clifford unit mismatch"));
1260 throw(std::invalid_argument("lst_to_clifford(): cannot construct from anything but list or vector"));
1262 throw(std::invalid_argument("lst_to_clifford(): the second argument should be a Clifford unit"));
1265 /** Auxiliary structure to define a function for striping one Clifford unit
1266 * from vectors. Used in clifford_to_lst(). */
1267 static ex get_clifford_comp(const ex & e, const ex & c, bool root=true)
1269 // make expansion on the top-level call only
1270 ex e1=(root? e.expand() : e);
1272 pointer_to_map_function_2args<const ex &, bool> fcn(get_clifford_comp, c, false);
1273 int ival = ex_to<numeric>(ex_to<idx>(c.op(1)).get_value()).to_int();
1274 int rl=ex_to<clifford>(c).get_representation_label();
1276 if ( (is_a<add>(e1) || e1.info(info_flags::list) || is_a<matrix>(e1))) {
1278 } else if (is_a<ncmul>(e1) || is_a<mul>(e1)) {
1279 // searches are done within products only
1280 exvector ev, all_dummy=get_all_dummy_indices(e1);
1281 bool found=false, same_value_found=false;
1283 ev.reserve(e1.nops());
1284 for (int i=0; i < e1.nops();++i) {
1285 // look for a Clifford unit with the same metric and representation label,
1286 // if found remember its index
1287 if (is_a<clifford>(e1.op(i)) && ex_to<clifford>(e1.op(i)).get_representation_label() == rl
1288 && is_a<cliffordunit>(e1.op(i).op(0)) && ex_to<clifford>(e1.op(i)).same_metric(c)) { // same Clifford unit
1290 throw(std::invalid_argument("get_clifford_comp(): expression is a Clifford multi-vector"));
1292 if (ex_to<idx>(e1.op(i).op(1)).is_numeric() &&
1293 (ival == ex_to<numeric>(ex_to<idx>(e1.op(i).op(1)).get_value()).to_int())) {
1294 same_value_found = true; // desired index value is found
1295 } else if ((std::find(all_dummy.begin(), all_dummy.end(), e1.op(i).op(1)) != all_dummy.end())
1296 || (is_a<varidx>(e1.op(i).op(1))
1297 && std::find(all_dummy.begin(), all_dummy.end(),
1298 ex_to<varidx>(e1.op(i).op(1)).toggle_variance()) != all_dummy.end())) {
1299 dummy_ind=(e1.op(i).op(1)); // suitable dummy index found
1301 ev.push_back(e.op(i)); // another index value
1303 ev.push_back(e1.op(i));
1306 if (! found) // no Clifford units found at all
1307 throw(std::invalid_argument("get_clifford_comp(): expression is not a Clifford vector to the given units"));
1309 ex res=dynallocate<ncmul>(std::move(ev));
1310 if (same_value_found) {
1312 } else if (! dummy_ind.is_zero()) { // a dummy index was found
1313 if (is_a<varidx>(dummy_ind))
1314 dummy_ind = ex_to<varidx>(dummy_ind).toggle_variance();
1315 return res.subs(dummy_ind==ival, subs_options::no_pattern);
1316 } else // found a Clifford unit with another index
1318 } else if (e1.is_zero()) {
1320 } else if (is_a<clifford>(e1) && is_a<cliffordunit>(e1.op(0)) && ex_to<clifford>(e1).same_metric(c)) {
1321 if (ex_to<idx>(e1.op(1)).is_numeric() &&
1322 (ival == ex_to<numeric>(ex_to<idx>(e1.op(1)).get_value()).to_int()) )
1327 throw(std::invalid_argument("get_clifford_comp(): expression is not usable as a Clifford vector"));
1330 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic)
1332 GINAC_ASSERT(is_a<clifford>(c));
1334 if (! ex_to<idx>(mu).is_dim_numeric())
1335 throw(std::invalid_argument("clifford_to_lst(): index should have a numeric dimension"));
1336 unsigned int D = ex_to<numeric>(ex_to<idx>(mu).get_dim()).to_int();
1338 if (algebraic) // check if algebraic method is applicable
1339 for (unsigned int i = 0; i < D; i++)
1340 if (pow(c.subs(mu == i, subs_options::no_pattern), 2).is_zero()
1341 || (! is_a<numeric>(pow(c.subs(mu == i, subs_options::no_pattern), 2))))
1344 ex v0 = remove_dirac_ONE(canonicalize_clifford(e+clifford_prime(e)))/2;
1347 ex e1 = canonicalize_clifford(e - v0 * dirac_ONE(ex_to<clifford>(c).get_representation_label()));
1349 for (unsigned int i = 0; i < D; i++)
1350 V.append(remove_dirac_ONE(
1351 simplify_indexed(canonicalize_clifford(e1 * c.subs(mu == i, subs_options::no_pattern) + c.subs(mu == i, subs_options::no_pattern) * e1))
1352 / (2*pow(c.subs(mu == i, subs_options::no_pattern), 2))));
1355 for (unsigned int i = 0; i < D; i++)
1356 V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
1357 } catch (std::exception &p) {
1358 /* Try to expand dummy summations to simplify the expression*/
1359 e1 = canonicalize_clifford(expand_dummy_sum(e, true));
1361 v0 = remove_dirac_ONE(canonicalize_clifford(e1+clifford_prime(e1)))/2;
1362 if (! v0.is_zero()) {
1364 e1 = canonicalize_clifford(e1 - v0 * dirac_ONE(ex_to<clifford>(c).get_representation_label()));
1366 for (unsigned int i = 0; i < D; i++)
1367 V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
1374 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)
1378 if (! is_a<matrix>(v) && ! v.info(info_flags::list))
1379 throw(std::invalid_argument("clifford_moebius_map(): parameter v should be either vector or list"));
1381 if (is_a<clifford>(G)) {
1384 if (is_a<indexed>(G)) {
1385 D = ex_to<idx>(G.op(1)).get_dim();
1386 varidx mu(dynallocate<symbol>(), D);
1387 cu = clifford_unit(mu, G, rl);
1388 } else if (is_a<matrix>(G)) {
1389 D = ex_to<matrix>(G).rows();
1390 idx mu(dynallocate<symbol>(), D);
1391 cu = clifford_unit(mu, G, rl);
1392 } else throw(std::invalid_argument("clifford_moebius_map(): metric should be an indexed object, matrix, or a Clifford unit"));
1396 x = lst_to_clifford(v, cu);
1397 ex e = clifford_to_lst(simplify_indexed(canonicalize_clifford((a * x + b) * clifford_inverse(c * x + d))), cu, false);
1398 return (is_a<matrix>(v) ? matrix(ex_to<matrix>(v).rows(), ex_to<matrix>(v).cols(), ex_to<lst>(e)) : e);
1401 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl)
1403 if (is_a<matrix>(M) && (ex_to<matrix>(M).rows() == 2) && (ex_to<matrix>(M).cols() == 2))
1404 return clifford_moebius_map(M.op(0), M.op(1), M.op(2), M.op(3), v, G, rl);
1406 throw(std::invalid_argument("clifford_moebius_map(): parameter M should be a 2x2 matrix"));
1409 } // namespace GiNaC