X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fclifford.cpp;h=67aa2c97343fe4d643fd36e4b86edd4284036c33;hp=37e1d8df89d43930bc0ca1f7e09e57cf49fa8f82;hb=0160f9ab1da453641e30539abcf0eaa4162582eb;hpb=e58227e1112f989f3b5417e497a61d53fc2971fa diff --git a/ginac/clifford.cpp b/ginac/clifford.cpp index 37e1d8df..67aa2c97 100644 --- a/ginac/clifford.cpp +++ b/ginac/clifford.cpp @@ -1,10 +1,9 @@ /** @file clifford.cpp * - * Implementation of GiNaC's clifford objects. - * No real implementation yet, to be done. */ + * Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */ /* - * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany + * GiNaC Copyright (C) 1999-2005 Johannes Gutenberg University Mainz, Germany * * This program is free software; you can redistribute it and/or modify * it under the terms of the GNU General Public License as published by @@ -18,206 +17,1338 @@ * * You should have received a copy of the GNU General Public License * along with this program; if not, write to the Free Software - * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA + * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA */ -#include +#include #include "clifford.h" + #include "ex.h" +#include "idx.h" #include "ncmul.h" +#include "symbol.h" +#include "numeric.h" // for I +#include "symmetry.h" +#include "lst.h" +#include "relational.h" +#include "operators.h" +#include "add.h" +#include "mul.h" +#include "power.h" +#include "matrix.h" #include "archive.h" #include "utils.h" -#include "debugmsg.h" -#ifndef NO_NAMESPACE_GINAC namespace GiNaC { -#endif // ndef NO_NAMESPACE_GINAC -GINAC_IMPLEMENT_REGISTERED_CLASS(clifford, lortensor) +GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(clifford, indexed, + print_func(&clifford::do_print_dflt). + print_func(&clifford::do_print_latex)) + +GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracone, tensor, + print_func(&diracone::do_print). + print_func(&diracone::do_print_latex)) + +GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(cliffordunit, tensor, + print_func(&cliffordunit::do_print). + print_func(&cliffordunit::do_print_latex)) + +GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma, cliffordunit, + print_func(&diracgamma::do_print). + print_func(&diracgamma::do_print_latex)) + +GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma5, tensor, + print_func(&diracgamma5::do_print). + print_func(&diracgamma5::do_print_latex)) + +GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaL, tensor, + print_func(&diracgammaL::do_print). + print_func(&diracgammaL::do_print_latex)) + +GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaR, tensor, + print_func(&diracgammaR::do_print). + print_func(&diracgammaR::do_print_latex)) ////////// -// default constructor, destructor, copy constructor assignment operator and helpers +// default constructors ////////// -// public +clifford::clifford() : representation_label(0), metric(0), anticommuting(true), commutator_sign(-1) +{ + tinfo_key = TINFO_clifford; +} + +DEFAULT_CTOR(diracone) +DEFAULT_CTOR(cliffordunit) +DEFAULT_CTOR(diracgamma) +DEFAULT_CTOR(diracgamma5) +DEFAULT_CTOR(diracgammaL) +DEFAULT_CTOR(diracgammaR) + +////////// +// other constructors +////////// -clifford::clifford() +/** Construct object without any indices. This constructor is for internal + * use only. Use the dirac_ONE() function instead. + * @see dirac_ONE */ +clifford::clifford(const ex & b, unsigned char rl, bool anticommut) : inherited(b), representation_label(rl), metric(0), anticommuting(anticommut), commutator_sign(-1) { - debugmsg("clifford default constructor",LOGLEVEL_CONSTRUCT); tinfo_key = TINFO_clifford; } -clifford::~clifford() +/** Construct object with one Lorentz index. This constructor is for internal + * use only. Use the clifford_unit() or dirac_gamma() functions instead. + * @see clifford_unit + * @see dirac_gamma */ +clifford::clifford(const ex & b, const ex & mu, const ex & metr, unsigned char rl, bool anticommut, int comm_sign) : inherited(b, mu), representation_label(rl), metric(metr), anticommuting(anticommut), commutator_sign(comm_sign) { - debugmsg("clifford destructor",LOGLEVEL_DESTRUCT); - destroy(false); + GINAC_ASSERT(is_a(mu)); + tinfo_key = TINFO_clifford; } -clifford::clifford(const clifford & other) +clifford::clifford(unsigned char rl, const ex & metr, bool anticommut, int comm_sign, const exvector & v, bool discardable) : inherited(not_symmetric(), v, discardable), representation_label(rl), metric(metr), anticommuting(anticommut), commutator_sign(comm_sign) { - debugmsg("clifford copy constructor",LOGLEVEL_CONSTRUCT); - copy(other); + tinfo_key = TINFO_clifford; } -const clifford & clifford::operator=(const clifford & other) +clifford::clifford(unsigned char rl, const ex & metr, bool anticommut, int comm_sign, std::auto_ptr vp) : inherited(not_symmetric(), vp), representation_label(rl), metric(metr), anticommuting(anticommut), commutator_sign(comm_sign) { - debugmsg("clifford operator=",LOGLEVEL_ASSIGNMENT); - if (this != &other) { - destroy(true); - copy(other); - } - return *this; + tinfo_key = TINFO_clifford; } -// protected +////////// +// archiving +////////// -void clifford::copy(const clifford & other) +clifford::clifford(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst) { - inherited::copy(other); + unsigned rl; + n.find_unsigned("label", rl); + representation_label = rl; + n.find_ex("metric", metric, sym_lst); + n.find_bool("anticommuting", anticommuting); + n.find_unsigned("commutator_sign+1", rl); + commutator_sign = rl - 1; } -void clifford::destroy(bool call_parent) +void clifford::archive(archive_node & n) const { - if (call_parent) inherited::destroy(call_parent); + inherited::archive(n); + n.add_unsigned("label", representation_label); + n.add_ex("metric", metric); + n.add_bool("anticommuting", anticommuting); + n.add_unsigned("commutator_sign+1", commutator_sign+1); +} + +DEFAULT_UNARCHIVE(clifford) +DEFAULT_ARCHIVING(diracone) +DEFAULT_ARCHIVING(cliffordunit) +DEFAULT_ARCHIVING(diracgamma) +DEFAULT_ARCHIVING(diracgamma5) +DEFAULT_ARCHIVING(diracgammaL) +DEFAULT_ARCHIVING(diracgammaR) + + +ex clifford::get_metric(const ex & i, const ex & j, bool symmetrised) const +{ + if (is_a(metric)) { + if (symmetrised && !(ex_to(ex_to(metric).get_symmetry()).has_symmetry())) { + if (is_a(metric.op(0))) { + return indexed((ex_to(metric.op(0)).add(ex_to(metric.op(0)).transpose())).mul(numeric(1,2)), + symmetric2(), i, j); + } else { + return simplify_indexed(indexed(metric.op(0)*_ex1_2, i, j) + indexed(metric.op(0)*_ex1_2, j, i)); + } + } else { + //return indexed(metric.op(0), ex_to(ex_to(metric).get_symmetry()), i, j); + return metric.subs(lst(metric.op(1) == i, metric.op(2) == j), subs_options::no_pattern); + } + } else { + // should not really happen since all constructors but clifford() make the metric an indexed object + return indexed(metric, i, j); + } +} + +bool clifford::same_metric(const ex & other) const +{ + if (is_a(other)) { + return same_metric(ex_to(other).get_metric()); + } else if (is_a(other)) { + return get_metric(other.op(1), other.op(2)).is_equal(other); + } else + return false; } ////////// -// other constructors +// functions overriding virtual functions from base classes ////////// -// public +ex clifford::op(size_t i) const +{ + GINAC_ASSERT(i(other)); + const clifford &o = static_cast(other); + + if (representation_label != o.representation_label) { + // different representation label + return representation_label < o.representation_label ? -1 : 1; + } + + return inherited::compare_same_type(other); } -clifford::clifford(const std::string & n, exvector *ivp) : inherited(lortensor_symbolic, n, *ivp) +bool clifford::match_same_type(const basic & other) const { - debugmsg("clifford constructor from string,exvector", LOGLEVEL_CONSTRUCT); - GINAC_ASSERT(all_of_type_lorentzidx()); - tinfo_key=TINFO_clifford; + GINAC_ASSERT(is_a(other)); + const clifford &o = static_cast(other); + + return ((representation_label == o.representation_label) && (commutator_sign == o.get_commutator_sign()) && same_metric(o)); } -////////// -// archiving -////////// +static bool is_dirac_slash(const ex & seq0) +{ + return !is_a(seq0) && !is_a(seq0) && + !is_a(seq0) && !is_a(seq0) && + !is_a(seq0); +} -/** Construct object from archive_node. */ -clifford::clifford(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst) +void clifford::do_print_dflt(const print_dflt & c, unsigned level) const { - debugmsg("clifford constructor from archive_node", LOGLEVEL_CONSTRUCT); + // dirac_slash() object is printed differently + if (is_dirac_slash(seq[0])) { + seq[0].print(c, precedence()); + c.s << "\\"; + } else + this->print_dispatch(c, level); } -/** Unarchive the object. */ -ex clifford::unarchive(const archive_node &n, const lst &sym_lst) +void clifford::do_print_latex(const print_latex & c, unsigned level) const { - return (new clifford(n, sym_lst))->setflag(status_flags::dynallocated); + // dirac_slash() object is printed differently + if (is_dirac_slash(seq[0])) { + c.s << "{"; + seq[0].print(c, precedence()); + c.s << "\\hspace{-1.0ex}/}"; + } else { + c.s << "\\clifford[" << int(representation_label) << "]"; + this->print_dispatch(c, level); + } } -/** Archive the object. */ -void clifford::archive(archive_node &n) const +DEFAULT_COMPARE(diracone) +DEFAULT_COMPARE(cliffordunit) +DEFAULT_COMPARE(diracgamma) +DEFAULT_COMPARE(diracgamma5) +DEFAULT_COMPARE(diracgammaL) +DEFAULT_COMPARE(diracgammaR) + +DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbf{1}") +DEFAULT_PRINT_LATEX(cliffordunit, "e", "e") +DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma") +DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}") +DEFAULT_PRINT_LATEX(diracgammaL, "gammaL", "{\\gamma_L}") +DEFAULT_PRINT_LATEX(diracgammaR, "gammaR", "{\\gamma_R}") + +/** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */ +static void base_and_index(const ex & c, ex & b, ex & i) { - inherited::archive(n); + GINAC_ASSERT(is_a(c)); + GINAC_ASSERT(c.nops() == 2+1); + + if (is_a(c.op(0))) { // proper dirac gamma object or clifford unit + i = c.op(1); + b = _ex1; + } else if (is_a(c.op(0)) || is_a(c.op(0)) || is_a(c.op(0))) { // gamma5/L/R + i = _ex0; + b = _ex1; + } else { // slash object, generate new dummy index + varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to(c.op(1)).get_dim()); + b = indexed(c.op(0), ix.toggle_variance()); + i = ix; + } +} + +/** Predicate for finding non-clifford objects. */ +struct is_not_a_clifford : public std::unary_function { + bool operator()(const ex & e) + { + return !is_a(e); + } +}; + +/** Contraction of a gamma matrix with something else. */ +bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const +{ + GINAC_ASSERT(is_a(*self)); + GINAC_ASSERT(is_a(*other)); + GINAC_ASSERT(is_a(self->op(0))); + unsigned char rl = ex_to(*self).get_representation_label(); + + ex dim = ex_to(self->op(1)).get_dim(); + if (other->nops() > 1) + dim = minimal_dim(dim, ex_to(other->op(1)).get_dim()); + + if (is_a(*other)) { + + // Contraction only makes sense if the represenation labels are equal + if (ex_to(*other).get_representation_label() != rl) + return false; + + size_t num = other - self; + + // gamma~mu gamma.mu = dim ONE + if (num == 1) { + *self = dim; + *other = dirac_ONE(rl); + return true; + + // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha + } else if (num == 2 + && is_a(self[1])) { + *self = 2 - dim; + *other = _ex1; + return true; + + // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta + } else if (num == 3 + && is_a(self[1]) + && is_a(self[2])) { + ex b1, i1, b2, i2; + base_and_index(self[1], b1, i1); + base_and_index(self[2], b2, i2); + *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2]; + self[1] = _ex1; + self[2] = _ex1; + *other = _ex1; + return true; + + // 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 + } else if (num == 4 + && is_a(self[1]) + && is_a(self[2]) + && is_a(self[3])) { + *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3]; + self[1] = _ex1; + self[2] = _ex1; + self[3] = _ex1; + *other = _ex1; + return true; + + // gamma~mu Sodd gamma.mu = -2 Sodd_R + // (Chisholm identity in 4 dimensions) + } else if (!((other - self) & 1) && dim.is_equal(4)) { + if (std::find_if(self + 1, other, is_not_a_clifford()) != other) + return false; + + *self = ncmul(exvector(std::reverse_iterator(other), std::reverse_iterator(self + 1)), true); + std::fill(self + 1, other, _ex1); + *other = _ex_2; + return true; + + // gamma~mu Sodd gamma~alpha gamma.mu = 2 gamma~alpha Sodd + 2 Sodd_R gamma~alpha + // (commutate contracted indices towards each other, then use + // Chisholm identity in 4 dimensions) + } else if (((other - self) & 1) && dim.is_equal(4)) { + if (std::find_if(self + 1, other, is_not_a_clifford()) != other) + return false; + + exvector::iterator next_to_last = other - 1; + ex S = ncmul(exvector(self + 1, next_to_last), true); + ex SR = ncmul(exvector(std::reverse_iterator(next_to_last), std::reverse_iterator(self + 1)), true); + + *self = (*next_to_last) * S + SR * (*next_to_last); + std::fill(self + 1, other, _ex1); + *other = _ex2; + return true; + + // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha + // (commutate contracted indices towards each other, simplify_indexed() + // will re-expand and re-run the simplification) + } else { + if (std::find_if(self + 1, other, is_not_a_clifford()) != other) + return false; + + exvector::iterator next_to_last = other - 1; + ex S = ncmul(exvector(self + 1, next_to_last), true); + + *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last); + std::fill(self + 1, other + 1, _ex1); + return true; + } + + } else if (is_a(other->op(0)) && other->nops() == 2) { + + // x.mu gamma~mu -> x-slash + *self = dirac_slash(other->op(0), dim, rl); + *other = _ex1; + return true; + } + + return false; +} + +/** An utility function looking for a given metric within an exvector, + * used in cliffordunit::contract_with(). */ +static int find_same_metric(exvector & v, ex & c) +{ + for (size_t i=0; i(v[i]) && !is_a(v[i]) + && ((ex_to(c.op(1)) == ex_to(v[i]).get_indices()[0] + && ex_to(c.op(1)) == ex_to(v[i]).get_indices()[1]) + || (ex_to(c.op(1)).toggle_variance() == ex_to(v[i]).get_indices()[0] + && ex_to(c.op(1)).toggle_variance() == ex_to(v[i]).get_indices()[1]))) { + return i; // the index of the found + } + } + return -1; //nothing found +} + +/** Contraction of a Clifford unit with something else. */ +bool cliffordunit::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const +{ + GINAC_ASSERT(is_a(*self)); + GINAC_ASSERT(is_a(*other)); + GINAC_ASSERT(is_a(self->op(0))); + clifford unit = ex_to(*self); + unsigned char rl = unit.get_representation_label(); + + if (is_a(*other)) { + // Contraction only makes sense if the represenation labels are equal + // and the metrics are the same + if ((ex_to(*other).get_representation_label() != rl) + && unit.same_metric(*other)) + return false; + + // Find if a previous contraction produces the square of self + int prev_square = find_same_metric(v, *self); + const varidx d((new symbol)->setflag(status_flags::dynallocated), ex_to(self->op(1)).get_dim()), + in1((new symbol)->setflag(status_flags::dynallocated), ex_to(self->op(1)).get_dim()), + in2((new symbol)->setflag(status_flags::dynallocated), ex_to(self->op(1)).get_dim()); + ex squared_metric; + if (prev_square > -1) + squared_metric = simplify_indexed(indexed(v[prev_square].op(0), in1, d) + * unit.get_metric(d.toggle_variance(), in2, true)).op(0); + + exvector::iterator before_other = other - 1; + const varidx & mu = ex_to(self->op(1)); + const varidx & mu_toggle = ex_to(other->op(1)); + const varidx & alpha = ex_to(before_other->op(1)); + + // e~mu e.mu = Tr ONE + if (other - self == 1) { + if (prev_square > -1) { + *self = indexed(squared_metric, mu, mu_toggle); + v[prev_square] = _ex1; + } else { + *self = unit.get_metric(mu, mu_toggle, true); + } + *other = dirac_ONE(rl); + return true; + + } else if (other - self == 2) { + if (is_a(*before_other) && ex_to(*before_other).get_representation_label() == rl) { + if (ex_to(*self).is_anticommuting()) { + // e~mu e~alpha e.mu = (2*pow(e~alpha, 2) -Tr(B)) e~alpha + if (prev_square > -1) { + *self = 2 * indexed(squared_metric, alpha, alpha) + - indexed(squared_metric, mu, mu_toggle); + v[prev_square] = _ex1; + } else { + *self = 2 * unit.get_metric(alpha, alpha, true) - unit.get_metric(mu, mu_toggle, true); + } + *other = _ex1; + return true; + + } else { + // e~mu e~alpha e.mu = 2*e~mu B(alpha, mu.toggle_variance())-Tr(B) e~alpha + *self = 2 * (*self) * unit.get_metric(alpha, mu_toggle, true) - unit.get_metric(mu, mu_toggle, true) * (*before_other); + *before_other = _ex1; + *other = _ex1; + return true; + } + } else { + // e~mu S e.mu = Tr S ONE + *self = unit.get_metric(mu, mu_toggle, true); + *other = dirac_ONE(rl); + return true; + } + } else { + // e~mu S e~alpha e.mu = 2 e~mu S B(alpha, mu.toggle_variance()) - e~mu S e.mu e~alpha + // (commutate contracted indices towards each other, simplify_indexed() + // will re-expand and re-run the simplification) + if (std::find_if(self + 1, other, is_not_a_clifford()) != other) { + return false; + } + + ex S = ncmul(exvector(self + 1, before_other), true); + + if (is_a(*before_other) && ex_to(*before_other).get_representation_label() == rl) { + if (ex_to(*self).is_anticommuting()) { + if (prev_square > -1) { + *self = 2 * (*before_other) * S * indexed(squared_metric, alpha, alpha) + - (*self) * S * (*other) * (*before_other); + } else { + *self = 2 * (*before_other) * S * unit.get_metric(alpha, alpha, true) - (*self) * S * (*other) * (*before_other); + } + } else { + *self = 2 * (*self) * S * unit.get_metric(alpha, mu_toggle, true) - (*self) * S * (*other) * (*before_other); + } + } else { + // simply commutes + *self = (*self) * S * (*other) * (*before_other); + } + + std::fill(self + 1, other + 1, _ex1); + return true; + } + } + return false; +} + +/** Perform automatic simplification on noncommutative product of clifford + * objects. This removes superfluous ONEs, permutes gamma5/L/R's to the front + * and removes squares of gamma objects. */ +ex clifford::eval_ncmul(const exvector & v) const +{ + exvector s; + s.reserve(v.size()); + + // Remove superfluous ONEs + exvector::const_iterator cit = v.begin(), citend = v.end(); + while (cit != citend) { + if (!is_a(*cit) || !is_a(cit->op(0))) + s.push_back(*cit); + cit++; + } + + bool something_changed = false; + int sign = 1; + + // Anticommutate gamma5/L/R's to the front + if (s.size() >= 2) { + exvector::iterator first = s.begin(), next_to_last = s.end() - 2; + while (true) { + exvector::iterator it = next_to_last; + while (true) { + exvector::iterator it2 = it + 1; + if (is_a(*it) && is_a(*it2)) { + ex e1 = it->op(0), e2 = it2->op(0); + + if (is_a(e2)) { + + if (is_a(e1) || is_a(e1)) { + + // gammaL/R gamma5 -> gamma5 gammaL/R + it->swap(*it2); + something_changed = true; + + } else if (!is_a(e1)) { + + // gamma5 gamma5 -> gamma5 gamma5 (do nothing) + // x gamma5 -> -gamma5 x + it->swap(*it2); + sign = -sign; + something_changed = true; + } + + } else if (is_a(e2)) { + + if (is_a(e1)) { + + // gammaR gammaL -> 0 + return _ex0; + + } else if (!is_a(e1) && !is_a(e1)) { + + // gammaL gammaL -> gammaL gammaL (do nothing) + // gamma5 gammaL -> gamma5 gammaL (do nothing) + // x gammaL -> gammaR x + it->swap(*it2); + *it = clifford(diracgammaR(), ex_to(*it).get_representation_label()); + something_changed = true; + } + + } else if (is_a(e2)) { + + if (is_a(e1)) { + + // gammaL gammaR -> 0 + return _ex0; + + } else if (!is_a(e1) && !is_a(e1)) { + + // gammaR gammaR -> gammaR gammaR (do nothing) + // gamma5 gammaR -> gamma5 gammaR (do nothing) + // x gammaR -> gammaL x + it->swap(*it2); + *it = clifford(diracgammaL(), ex_to(*it).get_representation_label()); + something_changed = true; + } + } + } + if (it == first) + break; + --it; + } + if (next_to_last == first) + break; + --next_to_last; + } + } + + // Remove equal adjacent gammas + if (s.size() >= 2) { + exvector::iterator it, itend = s.end() - 1; + for (it = s.begin(); it != itend; ++it) { + ex & a = it[0]; + ex & b = it[1]; + if (!is_a(a) || !is_a(b)) + continue; + + const ex & ag = a.op(0); + const ex & bg = b.op(0); + bool a_is_cliffordunit = is_a(ag); + bool b_is_cliffordunit = is_a(bg); + + if (a_is_cliffordunit && b_is_cliffordunit && ex_to(a).same_metric(b) + && (ex_to(a).get_commutator_sign() == -1)) { + // This is done only for Clifford algebras + + const ex & ia = a.op(1); + const ex & ib = b.op(1); + if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha + a = ex_to(a).get_metric(ia, ib, true); + b = dirac_ONE(representation_label); + something_changed = true; + } + + } else if ((is_a(ag) && is_a(bg))) { + + // Remove squares of gamma5 + a = dirac_ONE(representation_label); + b = dirac_ONE(representation_label); + something_changed = true; + + } else if ((is_a(ag) && is_a(bg)) + || (is_a(ag) && is_a(bg))) { + + // Remove squares of gammaL/R + b = dirac_ONE(representation_label); + something_changed = true; + + } else if (is_a(ag) && is_a(bg)) { + + // gammaL and gammaR are orthogonal + return _ex0; + + } else if (is_a(ag) && is_a(bg)) { + + // gamma5 gammaL -> -gammaL + a = dirac_ONE(representation_label); + sign = -sign; + something_changed = true; + + } else if (is_a(ag) && is_a(bg)) { + + // gamma5 gammaR -> gammaR + a = dirac_ONE(representation_label); + something_changed = true; + + } else if (!a_is_cliffordunit && !b_is_cliffordunit && ag.is_equal(bg)) { + + // a\ a\ -> a^2 + varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to(a.op(1)).minimal_dim(ex_to(b.op(1)))); + + a = indexed(ag, ix) * indexed(ag, ix.toggle_variance()); + b = dirac_ONE(representation_label); + something_changed = true; + } + } + } + + if (s.empty()) + return dirac_ONE(representation_label) * sign; + if (something_changed) + return reeval_ncmul(s) * sign; + else + return hold_ncmul(s) * sign; +} + +ex clifford::thiscontainer(const exvector & v) const +{ + return clifford(representation_label, metric, anticommuting, commutator_sign, v); +} + +ex clifford::thiscontainer(std::auto_ptr vp) const +{ + return clifford(representation_label, metric, anticommuting, commutator_sign, vp); +} + +ex diracgamma5::conjugate() const +{ + return _ex_1 * (*this); +} + +ex diracgammaL::conjugate() const +{ + return (new diracgammaR)->setflag(status_flags::dynallocated); +} + +ex diracgammaR::conjugate() const +{ + return (new diracgammaL)->setflag(status_flags::dynallocated); } ////////// -// functions overriding virtual functions from bases classes +// global functions ////////// -// public +ex dirac_ONE(unsigned char rl) +{ + static ex ONE = (new diracone)->setflag(status_flags::dynallocated); + return clifford(ONE, rl, false); +} -basic * clifford::duplicate() const +ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl, bool anticommuting) { - debugmsg("clifford duplicate",LOGLEVEL_DUPLICATE); - return new clifford(*this); + static ex unit = (new cliffordunit)->setflag(status_flags::dynallocated); + + if (!is_a(mu)) + throw(std::invalid_argument("clifford_unit(): index of Clifford unit must be of type idx or varidx")); + + if (ex_to(mu).is_symbolic() && !is_a(mu)) + throw(std::invalid_argument("clifford_unit(): symbolic index of Clifford unit must be of type varidx (not idx)")); + + if (is_a(metr)) { + exvector indices = ex_to(metr).get_indices(); + if ((indices.size() == 2) && is_a(indices[0]) && is_a(indices[1])) { + return clifford(unit, mu, metr, rl, anticommuting); + } else { + throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be indexed exactly by two indices of same type as the given index")); + } + } else if (is_a(metr)) { + static varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to(mu).get_dim()), + chi((new symbol)->setflag(status_flags::dynallocated), ex_to(mu).get_dim()); + return clifford(unit, mu, indexed(metr, xi, chi), rl, anticommuting); + } else if (is_a(metr)) { + matrix M = ex_to(metr); + unsigned n = M.rows(); + bool symmetric = true; + anticommuting = true; + + static varidx xi((new symbol)->setflag(status_flags::dynallocated), n), + chi((new symbol)->setflag(status_flags::dynallocated), n); + if ((n == M.cols()) && (n == ex_to(mu).get_dim())) { + for (unsigned i = 0; i < n; i++) { + for (unsigned j = i+1; j < n; j++) { + if (M(i, j) != M(j, i)) { + symmetric = false; + } + if (M(i, j) != -M(j, i)) { + anticommuting = false; + } + } + } + return clifford(unit, mu, indexed(metr, symmetric?symmetric2():not_symmetric(), xi, chi), rl, anticommuting); + } else { + throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be a square matrix with the same dimensions as index")); + } + } else { + throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be of type indexed, tensor or matrix")); + } } -void clifford::printraw(std::ostream & os) const +ex dirac_gamma(const ex & mu, unsigned char rl) { - debugmsg("clifford printraw",LOGLEVEL_PRINT); - os << "clifford(" << "indices="; - printrawindices(os); - os << ",hash=" << hashvalue << ",flags=" << flags << ")"; + static ex gamma = (new diracgamma)->setflag(status_flags::dynallocated); + + if (!is_a(mu)) + throw(std::invalid_argument("dirac_gamma(): index of Dirac gamma must be of type varidx")); + + static varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to(mu).get_dim()), + chi((new symbol)->setflag(status_flags::dynallocated), ex_to(mu).get_dim()); + return clifford(gamma, mu, indexed((new minkmetric)->setflag(status_flags::dynallocated), symmetric2(), xi, chi), rl, true); +} + +ex dirac_gamma5(unsigned char rl) +{ + static ex gamma5 = (new diracgamma5)->setflag(status_flags::dynallocated); + return clifford(gamma5, rl); } -void clifford::printtree(std::ostream & os, unsigned indent) const +ex dirac_gammaL(unsigned char rl) { - debugmsg("clifford printtree",LOGLEVEL_PRINT); - os << std::string(indent,' ') << " (clifford): " - << seq.size() << "indices="; - printtreeindices(os, indent); - os << ", hash=" << hashvalue - << " (0x" << std::hex << hashvalue << std::dec << ")" - << ", flags=" << flags << std::endl; + static ex gammaL = (new diracgammaL)->setflag(status_flags::dynallocated); + return clifford(gammaL, rl); } -void clifford::print(std::ostream & os, unsigned upper_precedence) const +ex dirac_gammaR(unsigned char rl) { - debugmsg("clifford print",LOGLEVEL_PRINT); - os << name; - printindices(os); + static ex gammaR = (new diracgammaR)->setflag(status_flags::dynallocated); + return clifford(gammaR, rl); } -bool clifford::info(unsigned inf) const +ex dirac_slash(const ex & e, const ex & dim, unsigned char rl) { - return inherited::info(inf); + // Slashed vectors are actually stored as a clifford object with the + // vector as its base expression and a (dummy) index that just serves + // for storing the space dimensionality + return clifford(e, varidx(0, dim), 0, rl); } -// protected +/** Check whether a given tinfo key (as returned by return_type_tinfo() + * is that of a clifford object with the specified representation label. */ +static bool is_clifford_tinfo(unsigned ti, unsigned char rl) +{ + return ti == (TINFO_clifford + rl); +} -int clifford::compare_same_type(const basic & other) const +/** Check whether a given tinfo key (as returned by return_type_tinfo() + * is that of a clifford object (with an arbitrary representation label). */ +static bool is_clifford_tinfo(unsigned ti) +{ + return (ti & ~0xff) == TINFO_clifford; +} + +/** Extract representation label from tinfo key (as returned by + * return_type_tinfo()). */ +static unsigned char get_representation_label(unsigned ti) +{ + return ti & 0xff; +} + +/** Take trace of a string of an even number of Dirac gammas given a vector + * of indices. */ +static ex trace_string(exvector::const_iterator ix, size_t num) +{ + // Tr gamma.mu gamma.nu = 4 g.mu.nu + if (num == 2) + return lorentz_g(ix[0], ix[1]); + + // 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 ) + else if (num == 4) + return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3]) + + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3]) + - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]); + + // Traces of 6 or more gammas are computed recursively: + // Tr gamma.mu1 gamma.mu2 ... gamma.mun = + // + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun + // - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun + // + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun + // - ... + // + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1) + exvector v(num - 2); + int sign = 1; + ex result; + for (size_t i=1; i & rls, const ex & trONE) +{ + if (is_a(e)) { + + unsigned char rl = ex_to(e).get_representation_label(); + + // Are we taking the trace over this object's representation label? + if (rls.find(rl) == rls.end()) + return e; + + // Yes, all elements are traceless, except for dirac_ONE and dirac_L/R + const ex & g = e.op(0); + if (is_a(g)) + return trONE; + else if (is_a(g) || is_a(g)) + return trONE/2; + else + return _ex0; + + } else if (is_exactly_a(e)) { + + // Trace of product: pull out non-clifford factors + ex prod = _ex1; + for (size_t i=0; i(e)) { + + unsigned char rl = get_representation_label(e.return_type_tinfo()); + + // Are we taking the trace over this string's representation label? + if (rls.find(rl) == rls.end()) + return e; + + // Substitute gammaL/R and expand product, if necessary + ex e_expanded = e.subs(lst( + dirac_gammaL(rl) == (dirac_ONE(rl)-dirac_gamma5(rl))/2, + dirac_gammaR(rl) == (dirac_ONE(rl)+dirac_gamma5(rl))/2 + ), subs_options::no_pattern).expand(); + if (!is_a(e_expanded)) + return dirac_trace(e_expanded, rls, trONE); + + // gamma5 gets moved to the front so this check is enough + bool has_gamma5 = is_a(e.op(0).op(0)); + size_t num = e.nops(); + + if (has_gamma5) { + + // Trace of gamma5 * odd number of gammas and trace of + // gamma5 * gamma.mu * gamma.nu are zero + if ((num & 1) == 0 || num == 3) + return _ex0; + + // Tr gamma5 gamma.mu gamma.nu gamma.rho gamma.sigma = 4I * epsilon(mu, nu, rho, sigma) + // (the epsilon is always 4-dimensional) + if (num == 5) { + ex b1, i1, b2, i2, b3, i3, b4, i4; + base_and_index(e.op(1), b1, i1); + base_and_index(e.op(2), b2, i2); + base_and_index(e.op(3), b3, i3); + base_and_index(e.op(4), b4, i4); + return trONE * I * (lorentz_eps(ex_to(i1).replace_dim(_ex4), ex_to(i2).replace_dim(_ex4), ex_to(i3).replace_dim(_ex4), ex_to(i4).replace_dim(_ex4)) * b1 * b2 * b3 * b4).simplify_indexed(); + } + + // Tr gamma5 S_2k = + // I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k + // (the epsilon is always 4-dimensional) + exvector ix(num-1), bv(num-1); + for (size_t i=1; i(idx1).replace_dim(_ex4), ex_to(idx2).replace_dim(_ex4), ex_to(idx3).replace_dim(_ex4), ex_to(idx4).replace_dim(_ex4)) + * trace_string(v.begin(), num - 4); + } + } + } + } + delete[] iv; + return trONE * I * result * mul(bv); + + } else { // no gamma5 + + // Trace of odd number of gammas is zero + if ((num & 1) == 1) + return _ex0; + + // Tr gamma.mu gamma.nu = 4 g.mu.nu + if (num == 2) { + ex b1, i1, b2, i2; + base_and_index(e.op(0), b1, i1); + base_and_index(e.op(1), b2, i2); + return trONE * (lorentz_g(i1, i2) * b1 * b2).simplify_indexed(); + } + + exvector iv(num), bv(num); + for (size_t i=0; i 0) { + + // Trace maps to all other container classes (this includes sums) + pointer_to_map_function_2args &, const ex &> fcn(dirac_trace, rls, trONE); + return e.map(fcn); + + } else + return _ex0; +} + +ex dirac_trace(const ex & e, const lst & rll, const ex & trONE) { - GINAC_ASSERT(is_of_type(other,clifford)); - // only compare indices - return exprseq::compare_same_type(other); + // Convert list to set + std::set rls; + for (lst::const_iterator i = rll.begin(); i != rll.end(); ++i) { + if (i->info(info_flags::nonnegint)) + rls.insert(ex_to(*i).to_int()); + } + + return dirac_trace(e, rls, trONE); } -bool clifford::is_equal_same_type(const basic & other) const +ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE) { - GINAC_ASSERT(is_of_type(other,clifford)); - // only compare indices - return exprseq::is_equal_same_type(other); + // Convert label to set + std::set rls; + rls.insert(rl); + + return dirac_trace(e, rls, trONE); } -ex clifford::thisexprseq(const exvector & v) const + +ex canonicalize_clifford(const ex & e_) { - return clifford(name, v); + pointer_to_map_function fcn(canonicalize_clifford); + + if (is_a(e_) // || is_a(e) || is_a(e) + || is_a(e_)) { + return e_.map(fcn); + } else { + ex e=simplify_indexed(e_); + // Scan for any ncmul objects + exmap srl; + ex aux = e.to_rational(srl); + for (exmap::iterator i = srl.begin(); i != srl.end(); ++i) { + + ex lhs = i->first; + ex rhs = i->second; + + if (is_exactly_a(rhs) + && rhs.return_type() == return_types::noncommutative + && is_clifford_tinfo(rhs.return_type_tinfo())) { + + // Expand product, if necessary + ex rhs_expanded = rhs.expand(); + if (!is_a(rhs_expanded)) { + i->second = canonicalize_clifford(rhs_expanded); + continue; + + } else if (!is_a(rhs.op(0))) + continue; + + exvector v; + v.reserve(rhs.nops()); + for (size_t j=0; j(it->op(0)) || is_a(it->op(0)) || is_a(it->op(0))) + ++it; + + while (it != next_to_last) { + if (it[0].compare(it[1]) > 0) { + + ex save0 = it[0], save1 = it[1]; + ex b1, i1, b2, i2; + base_and_index(it[0], b1, i1); + base_and_index(it[1], b2, i2); + // for Clifford algebras (commutator_sign == -1) metric should be symmetrised + it[0] = (ex_to(save0).get_metric(i1, i2, ex_to(save0).get_commutator_sign() == -1) * b1 * b2).simplify_indexed(); + it[1] = v.size() ? _ex2 * dirac_ONE(ex_to(save0).get_representation_label()) : _ex2; + ex sum = ncmul(v); + it[0] = save1; + it[1] = save0; + sum += ex_to(save0).get_commutator_sign() * ncmul(v, true); + i->second = canonicalize_clifford(sum); + goto next_sym; + } + ++it; + } +next_sym: ; + } + } + return aux.subs(srl, subs_options::no_pattern).simplify_indexed(); + } } -ex clifford::thisexprseq(exvector *vp) const +ex clifford_prime(const ex & e) { - return clifford(name, vp); + pointer_to_map_function fcn(clifford_prime); + if (is_a(e) && is_a(e.op(0))) { + return -e; + } else if (is_a(e) || is_a(e) || is_a(e) //|| is_a(e) || is_a(e) + || is_a(e) || is_a(e)) { + return e.map(fcn); + } else if (is_a(e)) { + return pow(clifford_prime(e.op(0)), e.op(1)); + } else + return e; } -ex clifford::simplify_ncmul(const exvector & v) const +ex remove_dirac_ONE(const ex & e, unsigned char rl, unsigned options) { - return simplified_ncmul(v); + pointer_to_map_function_2args fcn(remove_dirac_ONE, rl, options | 1); + bool need_reevaluation = false; + ex e1 = e; + if (! (options & 1) ) { // is not a child + if (options & 2) + e1 = expand_dummy_sum(e, true); + e1 = canonicalize_clifford(e1); + } + + if (is_a(e1) && ex_to(e1).get_representation_label() >= rl) { + if (is_a(e1.op(0))) + return 1; + else + throw(std::invalid_argument("remove_dirac_ONE(): expression is a non-scalar Clifford number!")); + } else if (is_a(e1) || is_a(e1) || is_a(e1) + || is_a(e1) || is_a(e1)) { + if (options & 3) // is a child or was already expanded + return e1.map(fcn); + else + try { + return e1.map(fcn); + } catch (std::exception &p) { + need_reevaluation = true; + } + } else if (is_a(e1)) { + if (options & 3) // is a child or was already expanded + return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1)); + else + try { + return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1)); + } catch (std::exception &p) { + need_reevaluation = true; + } + } + if (need_reevaluation) + return remove_dirac_ONE(e, rl, options | 2); + return e1; } -////////// -// friend functions -////////// +char clifford_max_label(const ex & e, bool ignore_ONE) +{ + if (is_a(e)) + if (ignore_ONE && is_a(e.op(0))) + return -1; + else + return ex_to(e).get_representation_label(); + else { + char rl = -1; + for (size_t i=0; i < e.nops(); i++) + rl = (rl > clifford_max_label(e.op(i), ignore_ONE)) ? rl : clifford_max_label(e.op(i), ignore_ONE); + return rl; + } +} -/** Construct an object representing a Dirac gamma matrix. The index must - * be of class lorentzidx. - * - * @param mu Index - * @return newly constructed object */ -clifford clifford_gamma(const ex & mu) +ex clifford_norm(const ex & e) +{ + return sqrt(remove_dirac_ONE(e * clifford_bar(e))); +} + +ex clifford_inverse(const ex & e) +{ + ex norm = clifford_norm(e); + if (!norm.is_zero()) + return clifford_bar(e) / pow(norm, 2); + else + throw(std::invalid_argument("clifford_inverse(): cannot find inverse of Clifford number with zero norm!")); +} + +ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl, bool anticommuting) +{ + if (!ex_to(mu).is_dim_numeric()) + throw(std::invalid_argument("lst_to_clifford(): Index should have a numeric dimension")); + ex e = clifford_unit(mu, metr, rl, anticommuting); + return lst_to_clifford(v, e); +} + +ex lst_to_clifford(const ex & v, const ex & e) { + unsigned min, max; + + if (is_a(e)) { + varidx mu = ex_to(e.op(1)); + unsigned dim = (ex_to(mu.get_dim())).to_int(); + + if (is_a(v)) { + if (ex_to(v).cols() > ex_to(v).rows()) { + min = ex_to(v).rows(); + max = ex_to(v).cols(); + } else { + min = ex_to(v).cols(); + max = ex_to(v).rows(); + } + if (min == 1) { + if (dim == max) + return indexed(v, ex_to(mu).toggle_variance()) * e; + else + throw(std::invalid_argument("lst_to_clifford(): dimensions of vector and clifford unit mismatch")); + } else + throw(std::invalid_argument("lst_to_clifford(): first argument should be a vector vector")); + } else if (is_a(v)) { + if (dim == ex_to(v).nops()) + return indexed(matrix(dim, 1, ex_to(v)), ex_to(mu).toggle_variance()) * e; + else + throw(std::invalid_argument("lst_to_clifford(): list length and dimension of clifford unit mismatch")); + } else + throw(std::invalid_argument("lst_to_clifford(): cannot construct from anything but list or vector")); + } else + throw(std::invalid_argument("lst_to_clifford(): the second argument should be a Clifford unit")); +} + +/** Auxiliary structure to define a function for striping one Clifford unit + * from vectors. Used in clifford_to_lst(). */ +static ex get_clifford_comp(const ex & e, const ex & c) +{ + pointer_to_map_function_1arg fcn(get_clifford_comp, c); + int ival = ex_to(ex_to(c.op(1)).get_value()).to_int(); + + if (is_a(e) || is_a(e) // || is_a(e) || is_a(e) + || is_a(e)) + return e.map(fcn); + else if (is_a(e) || is_a(e)) { + // find a Clifford unit with the same metric, delete it and substitute its index + size_t ind = e.nops() + 1; + for (size_t j = 0; j < e.nops(); j++) + if (is_a(e.op(j)) && ex_to(c).same_metric(e.op(j))) + if (ind > e.nops()) + ind = j; + else + throw(std::invalid_argument("get_clifford_comp(): expression is a Clifford multi-vector")); + if (ind < e.nops()) { + ex S = 1; + bool same_value_index, found_dummy; + same_value_index = ( ex_to(e.op(ind).op(1)).is_numeric() + && (ival == ex_to(ex_to(e.op(ind).op(1)).get_value()).to_int()) ); + found_dummy = same_value_index; + for(size_t j=0; j < e.nops(); j++) + if (j != ind) + if (same_value_index) + S = S * e.op(j); + else { + exvector ind_vec = ex_to(e.op(j)).get_dummy_indices(ex_to(e.op(ind))); + if (ind_vec.size() > 0) { + found_dummy = true; + exvector::const_iterator it = ind_vec.begin(), itend = ind_vec.end(); + while (it != itend) { + S = S * e.op(j).subs(lst(ex_to(*it) == ival, ex_to(*it).toggle_variance() == ival), subs_options::no_pattern); + ++it; + } + } else + S = S * e.op(j); + } + return (found_dummy ? S : 0); + } else + throw(std::invalid_argument("get_clifford_comp(): expression is not a Clifford vector to the given units")); + } else if (e.is_zero()) + return e; + else if (is_a(e) && ex_to(e).same_metric(c)) + if ( ex_to(e.op(1)).is_numeric() && + (ival != ex_to(ex_to(e.op(1)).get_value()).to_int()) ) + return 0; + else + return 1; + else + throw(std::invalid_argument("get_clifford_comp(): expression is not usable as a Clifford vector")); +} + + +lst clifford_to_lst(const ex & e, const ex & c, bool algebraic) +{ + GINAC_ASSERT(is_a(c)); + varidx mu = ex_to(c.op(1)); + if (! mu.is_dim_numeric()) + throw(std::invalid_argument("clifford_to_lst(): index should have a numeric dimension")); + unsigned int D = ex_to(mu.get_dim()).to_int(); + + if (algebraic) // check if algebraic method is applicable + for (unsigned int i = 0; i < D; i++) + if (pow(c.subs(mu == i, subs_options::no_pattern), 2).is_zero() + or (not is_a(pow(c.subs(mu == i, subs_options::no_pattern), 2)))) + algebraic = false; + lst V; + if (algebraic) { + for (unsigned int i = 0; i < D; i++) + V.append(remove_dirac_ONE( + simplify_indexed(canonicalize_clifford(e * c.subs(mu == i, subs_options::no_pattern) + c.subs(mu == i, subs_options::no_pattern) * e)) + / (2*pow(c.subs(mu == i, subs_options::no_pattern), 2)))); + } else { + ex e1 = canonicalize_clifford(e); + try { + for (unsigned int i = 0; i < D; i++) + V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern))); + } catch (std::exception &p) { + /* Try to expand dummy summations to simplify the expression*/ + e1 = canonicalize_clifford(expand_dummy_sum(e1, true)); + for (unsigned int i = 0; i < D; i++) + V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern))); + } + } + return V; +} + + +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, bool anticommuting) +{ + ex x, D, cu; + + if (! is_a(v) && ! is_a(v)) + throw(std::invalid_argument("clifford_moebius_map(): parameter v should be either vector or list")); + + if (is_a(G)) { + cu = G; + } else { + if (is_a(G)) + D = ex_to(G.op(1)).get_dim(); + else if (is_a(G)) + D = ex_to(G).rows(); + else throw(std::invalid_argument("clifford_moebius_map(): metric should be an indexed object, matrix, or a Clifford unit")); + + varidx mu((new symbol)->setflag(status_flags::dynallocated), D); + cu = clifford_unit(mu, G, rl, anticommuting); + } + + x = lst_to_clifford(v, cu); + ex e = simplify_indexed(canonicalize_clifford((a * x + b) * clifford_inverse(c * x + d))); + return clifford_to_lst(e, cu, false); +} + +ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl, bool anticommuting) { - return clifford("gamma", mu); + if (is_a(M)) + return clifford_moebius_map(ex_to(M)(0,0), ex_to(M)(0,1), + ex_to(M)(1,0), ex_to(M)(1,1), v, G, rl, anticommuting); + else + throw(std::invalid_argument("clifford_moebius_map(): parameter M should be a matrix")); } -#ifndef NO_NAMESPACE_GINAC } // namespace GiNaC -#endif // ndef NO_NAMESPACE_GINAC