3 * Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
6 * GiNaC Copyright (C) 1999-2007 Johannes Gutenberg University Mainz, Germany
8 * This program is free software; you can redistribute it and/or modify
9 * it under the terms of the GNU General Public License as published by
10 * the Free Software Foundation; either version 2 of the License, or
11 * (at your option) any later version.
13 * This program is distributed in the hope that it will be useful,
14 * but WITHOUT ANY WARRANTY; without even the implied warranty of
15 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
16 * GNU General Public License for more details.
18 * You should have received a copy of the GNU General Public License
19 * along with this program; if not, write to the Free Software
20 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
31 #include "numeric.h" // for I
34 #include "relational.h"
35 #include "operators.h"
45 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(clifford, indexed,
46 print_func<print_dflt>(&clifford::do_print_dflt).
47 print_func<print_latex>(&clifford::do_print_latex))
49 const tinfo_static_t clifford::return_type_tinfo_static[256] = {{}};
51 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracone, tensor,
52 print_func<print_dflt>(&diracone::do_print).
53 print_func<print_latex>(&diracone::do_print_latex))
55 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(cliffordunit, tensor,
56 print_func<print_dflt>(&cliffordunit::do_print).
57 print_func<print_latex>(&cliffordunit::do_print_latex))
59 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma, cliffordunit,
60 print_func<print_dflt>(&diracgamma::do_print).
61 print_func<print_latex>(&diracgamma::do_print_latex))
63 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma5, tensor,
64 print_func<print_dflt>(&diracgamma5::do_print).
65 print_func<print_latex>(&diracgamma5::do_print_latex))
67 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaL, tensor,
68 print_func<print_context>(&diracgammaL::do_print).
69 print_func<print_latex>(&diracgammaL::do_print_latex))
71 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaR, tensor,
72 print_func<print_context>(&diracgammaR::do_print).
73 print_func<print_latex>(&diracgammaR::do_print_latex))
76 // default constructors
79 clifford::clifford() : representation_label(0), metric(0), commutator_sign(-1)
81 tinfo_key = &clifford::tinfo_static;
84 DEFAULT_CTOR(diracone)
85 DEFAULT_CTOR(cliffordunit)
86 DEFAULT_CTOR(diracgamma)
87 DEFAULT_CTOR(diracgamma5)
88 DEFAULT_CTOR(diracgammaL)
89 DEFAULT_CTOR(diracgammaR)
95 /** Construct object without any indices. This constructor is for internal
96 * use only. Use the dirac_ONE() function instead.
98 clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl), metric(0), commutator_sign(-1)
100 tinfo_key = &clifford::tinfo_static;
103 /** Construct object with one Lorentz index. This constructor is for internal
104 * use only. Use the clifford_unit() or dirac_gamma() functions instead.
106 * @see dirac_gamma */
107 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)
109 GINAC_ASSERT(is_a<varidx>(mu));
110 tinfo_key = &clifford::tinfo_static;
113 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)
115 tinfo_key = &clifford::tinfo_static;
118 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)
120 tinfo_key = &clifford::tinfo_static;
127 clifford::clifford(const archive_node & n, lst & sym_lst) : inherited(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 DEFAULT_UNARCHIVE(clifford)
146 DEFAULT_ARCHIVING(diracone)
147 DEFAULT_ARCHIVING(cliffordunit)
148 DEFAULT_ARCHIVING(diracgamma)
149 DEFAULT_ARCHIVING(diracgamma5)
150 DEFAULT_ARCHIVING(diracgammaL)
151 DEFAULT_ARCHIVING(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 DEFAULT_COMPARE(diracone)
299 DEFAULT_COMPARE(cliffordunit)
300 DEFAULT_COMPARE(diracgamma)
301 DEFAULT_COMPARE(diracgamma5)
302 DEFAULT_COMPARE(diracgammaL)
303 DEFAULT_COMPARE(diracgammaR)
305 DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbf{1}")
306 DEFAULT_PRINT_LATEX(cliffordunit, "e", "e")
307 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
308 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
309 DEFAULT_PRINT_LATEX(diracgammaL, "gammaL", "{\\gamma_L}")
310 DEFAULT_PRINT_LATEX(diracgammaR, "gammaR", "{\\gamma_R}")
312 /** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
313 static void base_and_index(const ex & c, ex & b, ex & i)
315 GINAC_ASSERT(is_a<clifford>(c));
316 GINAC_ASSERT(c.nops() == 2+1);
318 if (is_a<cliffordunit>(c.op(0))) { // proper dirac gamma object or clifford unit
321 } else if (is_a<diracgamma5>(c.op(0)) || is_a<diracgammaL>(c.op(0)) || is_a<diracgammaR>(c.op(0))) { // gamma5/L/R
324 } else { // slash object, generate new dummy index
325 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(c.op(1)).get_dim());
326 b = indexed(c.op(0), ix.toggle_variance());
331 /** Predicate for finding non-clifford objects. */
332 struct is_not_a_clifford : public std::unary_function<ex, bool> {
333 bool operator()(const ex & e)
335 return !is_a<clifford>(e);
339 /** Contraction of a gamma matrix with something else. */
340 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
342 GINAC_ASSERT(is_a<clifford>(*self));
343 GINAC_ASSERT(is_a<indexed>(*other));
344 GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
345 unsigned char rl = ex_to<clifford>(*self).get_representation_label();
347 ex dim = ex_to<idx>(self->op(1)).get_dim();
348 if (other->nops() > 1)
349 dim = minimal_dim(dim, ex_to<idx>(other->op(1)).get_dim());
351 if (is_a<clifford>(*other)) {
353 // Contraction only makes sense if the represenation labels are equal
354 if (ex_to<clifford>(*other).get_representation_label() != rl)
357 size_t num = other - self;
359 // gamma~mu gamma.mu = dim ONE
362 *other = dirac_ONE(rl);
365 // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
367 && is_a<clifford>(self[1])) {
372 // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
374 && is_a<clifford>(self[1])
375 && is_a<clifford>(self[2])) {
377 base_and_index(self[1], b1, i1);
378 base_and_index(self[2], b2, i2);
379 *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
385 // 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
387 && is_a<clifford>(self[1])
388 && is_a<clifford>(self[2])
389 && is_a<clifford>(self[3])) {
390 *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3];
397 // gamma~mu Sodd gamma.mu = -2 Sodd_R
398 // (Chisholm identity in 4 dimensions)
399 } else if (!((other - self) & 1) && dim.is_equal(4)) {
400 if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
403 *self = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(other), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
404 std::fill(self + 1, other, _ex1);
408 // gamma~mu Sodd gamma~alpha gamma.mu = 2 gamma~alpha Sodd + 2 Sodd_R gamma~alpha
409 // (commutate contracted indices towards each other, then use
410 // Chisholm identity in 4 dimensions)
411 } else if (((other - self) & 1) && dim.is_equal(4)) {
412 if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
415 exvector::iterator next_to_last = other - 1;
416 ex S = ncmul(exvector(self + 1, next_to_last), true);
417 ex SR = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(next_to_last), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
419 *self = (*next_to_last) * S + SR * (*next_to_last);
420 std::fill(self + 1, other, _ex1);
424 // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
425 // (commutate contracted indices towards each other, simplify_indexed()
426 // will re-expand and re-run the simplification)
428 if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
431 exvector::iterator next_to_last = other - 1;
432 ex S = ncmul(exvector(self + 1, next_to_last), true);
434 *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
435 std::fill(self + 1, other + 1, _ex1);
439 } else if (is_a<symbol>(other->op(0)) && other->nops() == 2) {
441 // x.mu gamma~mu -> x-slash
442 *self = dirac_slash(other->op(0), dim, rl);
450 /** Contraction of a Clifford unit with something else. */
451 bool cliffordunit::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
453 GINAC_ASSERT(is_a<clifford>(*self));
454 GINAC_ASSERT(is_a<indexed>(*other));
455 GINAC_ASSERT(is_a<cliffordunit>(self->op(0)));
456 clifford unit = ex_to<clifford>(*self);
457 unsigned char rl = unit.get_representation_label();
459 if (is_a<clifford>(*other)) {
460 // Contraction only makes sense if the represenation labels are equal
461 // and the metrics are the same
462 if ((ex_to<clifford>(*other).get_representation_label() != rl)
463 && unit.same_metric(*other))
466 exvector::iterator before_other = other - 1;
468 ex mu_toggle = other->op(1);
469 ex alpha = before_other->op(1);
471 // e~mu e.mu = Tr ONE
472 if (other - self == 1) {
473 *self = unit.get_metric(mu, mu_toggle, true);
474 *other = dirac_ONE(rl);
477 } else if (other - self == 2) {
478 if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
479 // e~mu e~alpha e.mu = 2*e~mu B(alpha, mu.toggle_variance())-Tr(B) e~alpha
480 *self = 2 * (*self) * unit.get_metric(alpha, mu_toggle, true) - unit.get_metric(mu, mu_toggle, true) * (*before_other);
481 *before_other = _ex1;
486 // e~mu S e.mu = Tr S ONE
487 *self = unit.get_metric(mu, mu_toggle, true);
488 *other = dirac_ONE(rl);
492 // e~mu S e~alpha e.mu = 2 e~mu S B(alpha, mu.toggle_variance()) - e~mu S e.mu e~alpha
493 // (commutate contracted indices towards each other, simplify_indexed()
494 // will re-expand and re-run the simplification)
495 if (std::find_if(self + 1, other, is_not_a_clifford()) != other) {
499 ex S = ncmul(exvector(self + 1, before_other), true);
501 if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
502 *self = 2 * (*self) * S * unit.get_metric(alpha, mu_toggle, true) - (*self) * S * (*other) * (*before_other);
505 *self = (*self) * S * (*other) * (*before_other);
508 std::fill(self + 1, other + 1, _ex1);
515 /** Perform automatic simplification on noncommutative product of clifford
516 * objects. This removes superfluous ONEs, permutes gamma5/L/R's to the front
517 * and removes squares of gamma objects. */
518 ex clifford::eval_ncmul(const exvector & v) const
523 // Remove superfluous ONEs
524 exvector::const_iterator cit = v.begin(), citend = v.end();
525 while (cit != citend) {
526 if (!is_a<clifford>(*cit) || !is_a<diracone>(cit->op(0)))
531 bool something_changed = false;
534 // Anticommutate gamma5/L/R's to the front
536 exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
538 exvector::iterator it = next_to_last;
540 exvector::iterator it2 = it + 1;
541 if (is_a<clifford>(*it) && is_a<clifford>(*it2)) {
542 ex e1 = it->op(0), e2 = it2->op(0);
544 if (is_a<diracgamma5>(e2)) {
546 if (is_a<diracgammaL>(e1) || is_a<diracgammaR>(e1)) {
548 // gammaL/R gamma5 -> gamma5 gammaL/R
550 something_changed = true;
552 } else if (!is_a<diracgamma5>(e1)) {
554 // gamma5 gamma5 -> gamma5 gamma5 (do nothing)
555 // x gamma5 -> -gamma5 x
558 something_changed = true;
561 } else if (is_a<diracgammaL>(e2)) {
563 if (is_a<diracgammaR>(e1)) {
565 // gammaR gammaL -> 0
568 } else if (!is_a<diracgammaL>(e1) && !is_a<diracgamma5>(e1)) {
570 // gammaL gammaL -> gammaL gammaL (do nothing)
571 // gamma5 gammaL -> gamma5 gammaL (do nothing)
572 // x gammaL -> gammaR x
574 *it = clifford(diracgammaR(), ex_to<clifford>(*it).get_representation_label());
575 something_changed = true;
578 } else if (is_a<diracgammaR>(e2)) {
580 if (is_a<diracgammaL>(e1)) {
582 // gammaL gammaR -> 0
585 } else if (!is_a<diracgammaR>(e1) && !is_a<diracgamma5>(e1)) {
587 // gammaR gammaR -> gammaR gammaR (do nothing)
588 // gamma5 gammaR -> gamma5 gammaR (do nothing)
589 // x gammaR -> gammaL x
591 *it = clifford(diracgammaL(), ex_to<clifford>(*it).get_representation_label());
592 something_changed = true;
600 if (next_to_last == first)
606 // Remove equal adjacent gammas
608 exvector::iterator it, itend = s.end() - 1;
609 for (it = s.begin(); it != itend; ++it) {
612 if (!is_a<clifford>(a) || !is_a<clifford>(b))
615 const ex & ag = a.op(0);
616 const ex & bg = b.op(0);
617 bool a_is_cliffordunit = is_a<cliffordunit>(ag);
618 bool b_is_cliffordunit = is_a<cliffordunit>(bg);
620 if (a_is_cliffordunit && b_is_cliffordunit && ex_to<clifford>(a).same_metric(b)
621 && (ex_to<clifford>(a).get_commutator_sign() == -1)) {
622 // This is done only for Clifford algebras
624 const ex & ia = a.op(1);
625 const ex & ib = b.op(1);
626 if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
627 a = ex_to<clifford>(a).get_metric(ia, ib, true);
628 b = dirac_ONE(representation_label);
629 something_changed = true;
632 } else if ((is_a<diracgamma5>(ag) && is_a<diracgamma5>(bg))) {
634 // Remove squares of gamma5
635 a = dirac_ONE(representation_label);
636 b = dirac_ONE(representation_label);
637 something_changed = true;
639 } else if ((is_a<diracgammaL>(ag) && is_a<diracgammaL>(bg))
640 || (is_a<diracgammaR>(ag) && is_a<diracgammaR>(bg))) {
642 // Remove squares of gammaL/R
643 b = dirac_ONE(representation_label);
644 something_changed = true;
646 } else if (is_a<diracgammaL>(ag) && is_a<diracgammaR>(bg)) {
648 // gammaL and gammaR are orthogonal
651 } else if (is_a<diracgamma5>(ag) && is_a<diracgammaL>(bg)) {
653 // gamma5 gammaL -> -gammaL
654 a = dirac_ONE(representation_label);
656 something_changed = true;
658 } else if (is_a<diracgamma5>(ag) && is_a<diracgammaR>(bg)) {
660 // gamma5 gammaR -> gammaR
661 a = dirac_ONE(representation_label);
662 something_changed = true;
664 } else if (!a_is_cliffordunit && !b_is_cliffordunit && ag.is_equal(bg)) {
667 varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).minimal_dim(ex_to<idx>(b.op(1))));
669 a = indexed(ag, ix) * indexed(ag, ix.toggle_variance());
670 b = dirac_ONE(representation_label);
671 something_changed = true;
677 return dirac_ONE(representation_label) * sign;
678 if (something_changed)
679 return reeval_ncmul(s) * sign;
681 return hold_ncmul(s) * sign;
684 ex clifford::thiscontainer(const exvector & v) const
686 return clifford(representation_label, metric, commutator_sign, v);
689 ex clifford::thiscontainer(std::auto_ptr<exvector> vp) const
691 return clifford(representation_label, metric, commutator_sign, vp);
694 ex diracgamma5::conjugate() const
696 return _ex_1 * (*this);
699 ex diracgammaL::conjugate() const
701 return (new diracgammaR)->setflag(status_flags::dynallocated);
704 ex diracgammaR::conjugate() const
706 return (new diracgammaL)->setflag(status_flags::dynallocated);
713 ex dirac_ONE(unsigned char rl)
715 static ex ONE = (new diracone)->setflag(status_flags::dynallocated);
716 return clifford(ONE, rl);
719 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl)
721 static ex unit = (new cliffordunit)->setflag(status_flags::dynallocated);
724 throw(std::invalid_argument("clifford_unit(): index of Clifford unit must be of type idx or varidx"));
726 exvector indices = metr.get_free_indices();
728 if (indices.size() == 2) {
729 return clifford(unit, mu, metr, rl);
730 } else if (is_a<matrix>(metr)) {
731 matrix M = ex_to<matrix>(metr);
732 unsigned n = M.rows();
733 bool symmetric = true;
735 static idx xi((new symbol)->setflag(status_flags::dynallocated), n),
736 chi((new symbol)->setflag(status_flags::dynallocated), n);
737 if ((n == M.cols()) && (n == ex_to<idx>(mu).get_dim())) {
738 for (unsigned i = 0; i < n; i++) {
739 for (unsigned j = i+1; j < n; j++) {
740 if (M(i, j) != M(j, i)) {
745 return clifford(unit, mu, indexed(metr, symmetric?symmetric2():not_symmetric(), xi, chi), rl);
747 throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be a square matrix with the same dimensions as index"));
749 } else if (indices.size() == 0) { // a tensor or other expression without indices
750 static varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim()),
751 chi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim());
752 return clifford(unit, mu, indexed(metr, xi, chi), rl);
754 throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be of type tensor, matrix or an expression with two free indices"));
757 ex dirac_gamma(const ex & mu, unsigned char rl)
759 static ex gamma = (new diracgamma)->setflag(status_flags::dynallocated);
761 if (!is_a<varidx>(mu))
762 throw(std::invalid_argument("dirac_gamma(): index of Dirac gamma must be of type varidx"));
764 static varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim()),
765 chi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim());
766 return clifford(gamma, mu, indexed((new minkmetric)->setflag(status_flags::dynallocated), symmetric2(), xi, chi), rl);
769 ex dirac_gamma5(unsigned char rl)
771 static ex gamma5 = (new diracgamma5)->setflag(status_flags::dynallocated);
772 return clifford(gamma5, rl);
775 ex dirac_gammaL(unsigned char rl)
777 static ex gammaL = (new diracgammaL)->setflag(status_flags::dynallocated);
778 return clifford(gammaL, rl);
781 ex dirac_gammaR(unsigned char rl)
783 static ex gammaR = (new diracgammaR)->setflag(status_flags::dynallocated);
784 return clifford(gammaR, rl);
787 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
789 // Slashed vectors are actually stored as a clifford object with the
790 // vector as its base expression and a (dummy) index that just serves
791 // for storing the space dimensionality
793 static varidx xi((new symbol)->setflag(status_flags::dynallocated), dim),
794 chi((new symbol)->setflag(status_flags::dynallocated), dim);
795 return clifford(e, varidx(0, dim), indexed((new minkmetric)->setflag(status_flags::dynallocated), symmetric2(), xi, chi), rl);
798 /** Check whether a given tinfo key (as returned by return_type_tinfo()
799 * is that of a clifford object (with an arbitrary representation label). */
800 bool is_clifford_tinfo(tinfo_t ti)
802 p_int start_loc=(p_int)&clifford::return_type_tinfo_static;
803 return (p_int)ti>=start_loc && (p_int)ti<start_loc+256;
806 /** Extract representation label from tinfo key (as returned by
807 * return_type_tinfo()). */
808 static unsigned char get_representation_label(tinfo_t ti)
810 return (unsigned char)((p_int)ti-(p_int)&clifford::return_type_tinfo_static);
813 /** Take trace of a string of an even number of Dirac gammas given a vector
815 static ex trace_string(exvector::const_iterator ix, size_t num)
817 // Tr gamma.mu gamma.nu = 4 g.mu.nu
819 return lorentz_g(ix[0], ix[1]);
821 // 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 )
823 return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
824 + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
825 - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
827 // Traces of 6 or more gammas are computed recursively:
828 // Tr gamma.mu1 gamma.mu2 ... gamma.mun =
829 // + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
830 // - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
831 // + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
833 // + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
837 for (size_t i=1; i<num; i++) {
838 for (size_t n=1, j=0; n<num; n++) {
843 result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
849 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE)
851 if (is_a<clifford>(e)) {
853 unsigned char rl = ex_to<clifford>(e).get_representation_label();
855 // Are we taking the trace over this object's representation label?
856 if (rls.find(rl) == rls.end())
859 // Yes, all elements are traceless, except for dirac_ONE and dirac_L/R
860 const ex & g = e.op(0);
861 if (is_a<diracone>(g))
863 else if (is_a<diracgammaL>(g) || is_a<diracgammaR>(g))
868 } else if (is_exactly_a<mul>(e)) {
870 // Trace of product: pull out non-clifford factors
872 for (size_t i=0; i<e.nops(); i++) {
873 const ex &o = e.op(i);
874 if (is_clifford_tinfo(o.return_type_tinfo()))
875 prod *= dirac_trace(o, rls, trONE);
881 } else if (is_exactly_a<ncmul>(e)) {
883 unsigned char rl = get_representation_label(e.return_type_tinfo());
885 // Are we taking the trace over this string's representation label?
886 if (rls.find(rl) == rls.end())
889 // Substitute gammaL/R and expand product, if necessary
890 ex e_expanded = e.subs(lst(
891 dirac_gammaL(rl) == (dirac_ONE(rl)-dirac_gamma5(rl))/2,
892 dirac_gammaR(rl) == (dirac_ONE(rl)+dirac_gamma5(rl))/2
893 ), subs_options::no_pattern).expand();
894 if (!is_a<ncmul>(e_expanded))
895 return dirac_trace(e_expanded, rls, trONE);
897 // gamma5 gets moved to the front so this check is enough
898 bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
899 size_t num = e.nops();
903 // Trace of gamma5 * odd number of gammas and trace of
904 // gamma5 * gamma.mu * gamma.nu are zero
905 if ((num & 1) == 0 || num == 3)
908 // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma)
909 // (the epsilon is always 4-dimensional)
911 ex b1, i1, b2, i2, b3, i3, b4, i4;
912 base_and_index(e.op(1), b1, i1);
913 base_and_index(e.op(2), b2, i2);
914 base_and_index(e.op(3), b3, i3);
915 base_and_index(e.op(4), b4, i4);
916 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();
920 // I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
921 // (the epsilon is always 4-dimensional)
922 exvector ix(num-1), bv(num-1);
923 for (size_t i=1; i<num; i++)
924 base_and_index(e.op(i), bv[i-1], ix[i-1]);
926 int *iv = new int[num];
928 for (size_t i=0; i<num-3; i++) {
930 for (size_t j=i+1; j<num-2; j++) {
932 for (size_t k=j+1; k<num-1; k++) {
934 for (size_t l=k+1; l<num; l++) {
936 iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
939 for (size_t n=0, t=4; n<num; n++) {
940 if (n == i || n == j || n == k || n == l)
945 int sign = permutation_sign(iv, iv + num);
946 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))
947 * trace_string(v.begin(), num - 4);
953 return trONE * I * result * mul(bv);
955 } else { // no gamma5
957 // Trace of odd number of gammas is zero
961 // Tr gamma.mu gamma.nu = 4 g.mu.nu
964 base_and_index(e.op(0), b1, i1);
965 base_and_index(e.op(1), b2, i2);
966 return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
969 exvector iv(num), bv(num);
970 for (size_t i=0; i<num; i++)
971 base_and_index(e.op(i), bv[i], iv[i]);
973 return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
976 } else if (e.nops() > 0) {
978 // Trace maps to all other container classes (this includes sums)
979 pointer_to_map_function_2args<const std::set<unsigned char> &, const ex &> fcn(dirac_trace, rls, trONE);
986 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE)
988 // Convert list to set
989 std::set<unsigned char> rls;
990 for (lst::const_iterator i = rll.begin(); i != rll.end(); ++i) {
991 if (i->info(info_flags::nonnegint))
992 rls.insert(ex_to<numeric>(*i).to_int());
995 return dirac_trace(e, rls, trONE);
998 ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
1000 // Convert label to set
1001 std::set<unsigned char> rls;
1004 return dirac_trace(e, rls, trONE);
1008 ex canonicalize_clifford(const ex & e_)
1010 pointer_to_map_function fcn(canonicalize_clifford);
1012 if (is_a<matrix>(e_) // || is_a<pseries>(e) || is_a<integral>(e)
1013 || e_.info(info_flags::list)) {
1016 ex e=simplify_indexed(e_);
1017 // Scan for any ncmul objects
1019 ex aux = e.to_rational(srl);
1020 for (exmap::iterator i = srl.begin(); i != srl.end(); ++i) {
1025 if (is_exactly_a<ncmul>(rhs)
1026 && rhs.return_type() == return_types::noncommutative
1027 && is_clifford_tinfo(rhs.return_type_tinfo())) {
1029 // Expand product, if necessary
1030 ex rhs_expanded = rhs.expand();
1031 if (!is_a<ncmul>(rhs_expanded)) {
1032 i->second = canonicalize_clifford(rhs_expanded);
1035 } else if (!is_a<clifford>(rhs.op(0)))
1039 v.reserve(rhs.nops());
1040 for (size_t j=0; j<rhs.nops(); j++)
1041 v.push_back(rhs.op(j));
1043 // Stupid recursive bubble sort because we only want to swap adjacent gammas
1044 exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
1045 if (is_a<diracgamma5>(it->op(0)) || is_a<diracgammaL>(it->op(0)) || is_a<diracgammaR>(it->op(0)))
1048 while (it != next_to_last) {
1049 if (it[0].compare(it[1]) > 0) {
1051 ex save0 = it[0], save1 = it[1];
1053 base_and_index(it[0], b1, i1);
1054 base_and_index(it[1], b2, i2);
1055 // for Clifford algebras (commutator_sign == -1) metric should be symmetrised
1056 it[0] = (ex_to<clifford>(save0).get_metric(i1, i2, ex_to<clifford>(save0).get_commutator_sign() == -1) * b1 * b2).simplify_indexed();
1057 it[1] = v.size() ? _ex2 * dirac_ONE(ex_to<clifford>(save0).get_representation_label()) : _ex2;
1061 sum += ex_to<clifford>(save0).get_commutator_sign() * ncmul(v, true);
1062 i->second = canonicalize_clifford(sum);
1070 return aux.subs(srl, subs_options::no_pattern).simplify_indexed();
1074 ex clifford_prime(const ex & e)
1076 pointer_to_map_function fcn(clifford_prime);
1077 if (is_a<clifford>(e) && is_a<cliffordunit>(e.op(0))) {
1079 } else if (is_a<add>(e) || is_a<ncmul>(e) || is_a<mul>(e) //|| is_a<pseries>(e) || is_a<integral>(e)
1080 || is_a<matrix>(e) || e.info(info_flags::list)) {
1082 } else if (is_a<power>(e)) {
1083 return pow(clifford_prime(e.op(0)), e.op(1));
1088 ex remove_dirac_ONE(const ex & e, unsigned char rl, unsigned options)
1090 pointer_to_map_function_2args<unsigned char, unsigned> fcn(remove_dirac_ONE, rl, options | 1);
1091 bool need_reevaluation = false;
1093 if (! (options & 1) ) { // is not a child
1095 e1 = expand_dummy_sum(e, true);
1096 e1 = canonicalize_clifford(e1);
1099 if (is_a<clifford>(e1) && ex_to<clifford>(e1).get_representation_label() >= rl) {
1100 if (is_a<diracone>(e1.op(0)))
1103 throw(std::invalid_argument("remove_dirac_ONE(): expression is a non-scalar Clifford number!"));
1104 } else if (is_a<add>(e1) || is_a<ncmul>(e1) || is_a<mul>(e1)
1105 || is_a<matrix>(e1) || e1.info(info_flags::list)) {
1106 if (options & 3) // is a child or was already expanded
1111 } catch (std::exception &p) {
1112 need_reevaluation = true;
1114 } else if (is_a<power>(e1)) {
1115 if (options & 3) // is a child or was already expanded
1116 return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
1119 return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
1120 } catch (std::exception &p) {
1121 need_reevaluation = true;
1124 if (need_reevaluation)
1125 return remove_dirac_ONE(e, rl, options | 2);
1129 char clifford_max_label(const ex & e, bool ignore_ONE)
1131 if (is_a<clifford>(e))
1132 if (ignore_ONE && is_a<diracone>(e.op(0)))
1135 return ex_to<clifford>(e).get_representation_label();
1138 for (size_t i=0; i < e.nops(); i++)
1139 rl = (rl > clifford_max_label(e.op(i), ignore_ONE)) ? rl : clifford_max_label(e.op(i), ignore_ONE);
1144 ex clifford_norm(const ex & e)
1146 return sqrt(remove_dirac_ONE(e * clifford_bar(e)));
1149 ex clifford_inverse(const ex & e)
1151 ex norm = clifford_norm(e);
1152 if (!norm.is_zero())
1153 return clifford_bar(e) / pow(norm, 2);
1155 throw(std::invalid_argument("clifford_inverse(): cannot find inverse of Clifford number with zero norm!"));
1158 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl)
1160 if (!ex_to<idx>(mu).is_dim_numeric())
1161 throw(std::invalid_argument("lst_to_clifford(): Index should have a numeric dimension"));
1162 ex e = clifford_unit(mu, metr, rl);
1163 return lst_to_clifford(v, e);
1166 ex lst_to_clifford(const ex & v, const ex & e) {
1169 if (is_a<clifford>(e)) {
1172 = is_a<varidx>(mu) ? ex_to<varidx>(mu).toggle_variance() : mu;
1173 unsigned dim = (ex_to<numeric>(ex_to<idx>(mu).get_dim())).to_int();
1175 if (is_a<matrix>(v)) {
1176 if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows()) {
1177 min = ex_to<matrix>(v).rows();
1178 max = ex_to<matrix>(v).cols();
1180 min = ex_to<matrix>(v).cols();
1181 max = ex_to<matrix>(v).rows();
1185 return indexed(v, mu_toggle) * e;
1186 else if (max - dim == 1) {
1187 if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows())
1188 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;
1190 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;
1192 throw(std::invalid_argument("lst_to_clifford(): dimensions of vector and clifford unit mismatch"));
1194 throw(std::invalid_argument("lst_to_clifford(): first argument should be a vector (nx1 or 1xn matrix)"));
1195 } else if (v.info(info_flags::list)) {
1196 if (dim == ex_to<lst>(v).nops())
1197 return indexed(matrix(dim, 1, ex_to<lst>(v)), mu_toggle) * e;
1198 else if (ex_to<lst>(v).nops() - dim == 1)
1199 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;
1201 throw(std::invalid_argument("lst_to_clifford(): list length and dimension of clifford unit mismatch"));
1203 throw(std::invalid_argument("lst_to_clifford(): cannot construct from anything but list or vector"));
1205 throw(std::invalid_argument("lst_to_clifford(): the second argument should be a Clifford unit"));
1208 /** Auxiliary structure to define a function for striping one Clifford unit
1209 * from vectors. Used in clifford_to_lst(). */
1210 static ex get_clifford_comp(const ex & e, const ex & c)
1212 pointer_to_map_function_1arg<const ex &> fcn(get_clifford_comp, c);
1213 int ival = ex_to<numeric>(ex_to<idx>(c.op(1)).get_value()).to_int();
1215 if (is_a<add>(e) || e.info(info_flags::list) // || is_a<pseries>(e) || is_a<integral>(e)
1218 else if (is_a<ncmul>(e) || is_a<mul>(e)) {
1219 // find a Clifford unit with the same metric, delete it and substitute its index
1220 size_t ind = e.nops() + 1;
1221 for (size_t j = 0; j < e.nops(); j++)
1222 if (is_a<clifford>(e.op(j)) && ex_to<clifford>(c).same_metric(e.op(j)))
1226 throw(std::invalid_argument("get_clifford_comp(): expression is a Clifford multi-vector"));
1227 if (ind < e.nops()) {
1229 bool same_value_index, found_dummy;
1230 same_value_index = ( ex_to<idx>(e.op(ind).op(1)).is_numeric()
1231 && (ival == ex_to<numeric>(ex_to<idx>(e.op(ind).op(1)).get_value()).to_int()) );
1232 found_dummy = same_value_index;
1233 for(size_t j=0; j < e.nops(); j++)
1235 if (same_value_index)
1238 exvector ind_vec = ex_to<indexed>(e.op(j)).get_dummy_indices(ex_to<indexed>(e.op(ind)));
1239 if (ind_vec.size() > 0) {
1241 exvector::const_iterator it = ind_vec.begin(), itend = ind_vec.end();
1242 while (it != itend) {
1244 ex curridx_toggle = is_a<varidx>(curridx)
1245 ? ex_to<varidx>(curridx).toggle_variance()
1247 S = S * e.op(j).subs(lst(curridx == ival,
1248 curridx_toggle == ival), subs_options::no_pattern);
1254 return (found_dummy ? S : 0);
1256 throw(std::invalid_argument("get_clifford_comp(): expression is not a Clifford vector to the given units"));
1257 } else if (e.is_zero())
1259 else if (is_a<clifford>(e) && ex_to<clifford>(e).same_metric(c))
1260 if ( ex_to<idx>(e.op(1)).is_numeric() &&
1261 (ival != ex_to<numeric>(ex_to<idx>(e.op(1)).get_value()).to_int()) )
1266 throw(std::invalid_argument("get_clifford_comp(): expression is not usable as a Clifford vector"));
1270 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic)
1272 GINAC_ASSERT(is_a<clifford>(c));
1274 if (! ex_to<idx>(mu).is_dim_numeric())
1275 throw(std::invalid_argument("clifford_to_lst(): index should have a numeric dimension"));
1276 unsigned int D = ex_to<numeric>(ex_to<idx>(mu).get_dim()).to_int();
1278 if (algebraic) // check if algebraic method is applicable
1279 for (unsigned int i = 0; i < D; i++)
1280 if (pow(c.subs(mu == i, subs_options::no_pattern), 2).is_zero()
1281 or (not is_a<numeric>(pow(c.subs(mu == i, subs_options::no_pattern), 2))))
1284 ex v0 = remove_dirac_ONE(canonicalize_clifford(e+clifford_prime(e)).normal())/2;
1285 if (not v0.is_zero())
1287 ex e1 = canonicalize_clifford(e - v0 * dirac_ONE(ex_to<clifford>(c).get_representation_label()));
1289 for (unsigned int i = 0; i < D; i++)
1290 V.append(remove_dirac_ONE(
1291 simplify_indexed(canonicalize_clifford(e1 * c.subs(mu == i, subs_options::no_pattern) + c.subs(mu == i, subs_options::no_pattern) * e1))
1292 / (2*pow(c.subs(mu == i, subs_options::no_pattern), 2))));
1295 for (unsigned int i = 0; i < D; i++)
1296 V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
1297 } catch (std::exception &p) {
1298 /* Try to expand dummy summations to simplify the expression*/
1299 e1 = canonicalize_clifford(expand_dummy_sum(e, true));
1301 v0 = remove_dirac_ONE(canonicalize_clifford(e1+clifford_prime(e1)).normal())/2;
1302 if (not v0.is_zero()) {
1304 e1 = canonicalize_clifford(e1 - v0 * dirac_ONE(ex_to<clifford>(c).get_representation_label()));
1306 for (unsigned int i = 0; i < D; i++)
1307 V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
1314 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)
1318 if (! is_a<matrix>(v) && ! v.info(info_flags::list))
1319 throw(std::invalid_argument("clifford_moebius_map(): parameter v should be either vector or list"));
1321 if (is_a<clifford>(G)) {
1324 if (is_a<indexed>(G)) {
1325 D = ex_to<idx>(G.op(1)).get_dim();
1326 varidx mu((new symbol)->setflag(status_flags::dynallocated), D);
1327 cu = clifford_unit(mu, G, rl);
1328 } else if (is_a<matrix>(G)) {
1329 D = ex_to<matrix>(G).rows();
1330 idx mu((new symbol)->setflag(status_flags::dynallocated), D);
1331 cu = clifford_unit(mu, G, rl);
1332 } else throw(std::invalid_argument("clifford_moebius_map(): metric should be an indexed object, matrix, or a Clifford unit"));
1336 x = lst_to_clifford(v, cu);
1337 ex e = clifford_to_lst(simplify_indexed(canonicalize_clifford((a * x + b) * clifford_inverse(c * x + d))), cu, false);
1338 return (is_a<matrix>(v) ? matrix(ex_to<matrix>(v).rows(), ex_to<matrix>(v).cols(), ex_to<lst>(e)) : e);
1341 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl)
1343 if (is_a<matrix>(M) && (ex_to<matrix>(M).rows() == 2) && (ex_to<matrix>(M).cols() == 2))
1344 return clifford_moebius_map(M.op(0), M.op(1), M.op(2), M.op(3), v, G, rl);
1346 throw(std::invalid_argument("clifford_moebius_map(): parameter M should be a 2x2 matrix"));
1349 } // namespace GiNaC