clifford.cpp

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/*
 *  GiNaC Copyright (C) 1999-2023 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
 *  the Free Software Foundation; either version 2 of the License, or
 *  (at your option) any later version.
 *
 *  This program is distributed in the hope that it will be useful,
 *  but WITHOUT ANY WARRANTY; without even the implied warranty of
 *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *  GNU General Public License for more details.
 *
 *  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., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
 */
 
#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 <stdexcept>
 
namespace GiNaC {
 
GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(clifford, indexed,
  print_func<print_dflt>(&clifford::do_print_dflt).
  print_func<print_latex>(&clifford::do_print_latex).
  print_func<print_tree>(&clifford::do_print_tree))
 
GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracone, tensor,
  print_func<print_dflt>(&diracone::do_print).
  print_func<print_latex>(&diracone::do_print_latex))
 
GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(cliffordunit, tensor,
  print_func<print_dflt>(&cliffordunit::do_print).
  print_func<print_latex>(&cliffordunit::do_print_latex))
 
GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma, cliffordunit,
  print_func<print_dflt>(&diracgamma::do_print).
  print_func<print_latex>(&diracgamma::do_print_latex))
 
GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma5, tensor,
  print_func<print_dflt>(&diracgamma5::do_print).
  print_func<print_latex>(&diracgamma5::do_print_latex))
 
GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaL, tensor,
  print_func<print_context>(&diracgammaL::do_print).
  print_func<print_latex>(&diracgammaL::do_print_latex))
 
GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaR, tensor,
  print_func<print_context>(&diracgammaR::do_print).
  print_func<print_latex>(&diracgammaR::do_print_latex))
 
// default constructors
 
clifford::clifford() : representation_label(0), metric(0), commutator_sign(-1)
{
}
 
DEFAULT_CTOR(diracone)
DEFAULT_CTOR(cliffordunit)
DEFAULT_CTOR(diracgamma)
DEFAULT_CTOR(diracgamma5)
DEFAULT_CTOR(diracgammaL)
DEFAULT_CTOR(diracgammaR)
 
 
// other constructors
 
clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl), metric(0), commutator_sign(-1)
{
}
 
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)
{
    GINAC_ASSERT(is_a<idx>(mu));
}
 
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)
{
}
 
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)
{
}
 
return_type_t clifford::return_type_tinfo() const
{
    return make_return_type_t<clifford>(representation_label);
}
 
// archiving
 
void clifford::read_archive(const archive_node& n, lst& sym_lst)
{
    inherited::read_archive(n, sym_lst);
    unsigned rl;
    n.find_unsigned("label", rl);
    representation_label = rl;
    n.find_ex("metric", metric, sym_lst);
    n.find_unsigned("commutator_sign+1", rl);
    commutator_sign = rl - 1;
}
 
void clifford::archive(archive_node & n) const
{
    inherited::archive(n);
    n.add_unsigned("label", representation_label);
    n.add_ex("metric", metric);
    n.add_unsigned("commutator_sign+1", commutator_sign+1);
}
 
GINAC_BIND_UNARCHIVER(clifford);
GINAC_BIND_UNARCHIVER(cliffordunit);
GINAC_BIND_UNARCHIVER(diracone);
GINAC_BIND_UNARCHIVER(diracgamma);
GINAC_BIND_UNARCHIVER(diracgamma5);
GINAC_BIND_UNARCHIVER(diracgammaL);
GINAC_BIND_UNARCHIVER(diracgammaR);
 
 
ex clifford::get_metric(const ex & i, const ex & j, bool symmetrised) const
{
    if (is_a<indexed>(metric)) {
        if (symmetrised && !(ex_to<symmetry>(ex_to<indexed>(metric).get_symmetry()).has_symmetry())) {
            if (is_a<matrix>(metric.op(0))) {
                return indexed((ex_to<matrix>(metric.op(0)).add(ex_to<matrix>(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 metric.subs(lst{metric.op(1) == i, metric.op(2) == j}, subs_options::no_pattern);
        }
    } else {
        exvector indices = metric.get_free_indices();
        if (symmetrised)
            return _ex1_2*simplify_indexed(metric.subs(lst{indices[0] == i, indices[1] == j}, subs_options::no_pattern)
                                         + metric.subs(lst{indices[0] == j, indices[1] == i}, subs_options::no_pattern));
        else
            return metric.subs(lst{indices[0] == i, indices[1] == j}, subs_options::no_pattern);
    }
}
 
bool clifford::same_metric(const ex & other) const
{
    ex metr;
    if (is_a<clifford>(other)) 
        metr = ex_to<clifford>(other).get_metric();
    else 
        metr = other;
 
    if (is_a<indexed>(metr))
        return metr.op(0).is_equal(get_metric().op(0));
    else {
        exvector indices = metr.get_free_indices();
        return  (indices.size() == 2) 
            && simplify_indexed(get_metric(indices[0], indices[1])-metr).is_zero();
    }
}
 
// functions overriding virtual functions from base classes
 
ex clifford::op(size_t i) const
{
    GINAC_ASSERT(i<nops());
    if (nops()-i == 1)
        return representation_label;
    else 
        return inherited::op(i);
}
 
ex & clifford::let_op(size_t i)
{
        GINAC_ASSERT(i<nops());
 
    static ex rl = numeric(representation_label);
    ensure_if_modifiable();
    if (nops()-i == 1)
        return rl;
    else 
        return inherited::let_op(i);
}
 
ex clifford::subs(const exmap & m, unsigned options) const
{
    ex subsed = inherited::subs(m, options);
    if(is_a<clifford>(subsed)) {
        ex prevmetric = ex_to<clifford>(subsed).metric;
        ex newmetric = prevmetric.subs(m, options);
        if(!are_ex_trivially_equal(prevmetric, newmetric)) {
            clifford c = ex_to<clifford>(subsed);
            c.metric = newmetric;
            subsed = c;
        }
    }
    return subsed;
}
 
int clifford::compare_same_type(const basic & other) const
{
    GINAC_ASSERT(is_a<clifford>(other));
    const clifford &o = static_cast<const clifford &>(other);
 
    if (representation_label != o.representation_label) {
        // different representation label
        return representation_label < o.representation_label ? -1 : 1;
    }
 
    return inherited::compare_same_type(other);
}
 
bool clifford::match_same_type(const basic & other) const
{
    GINAC_ASSERT(is_a<clifford>(other));
    const clifford &o = static_cast<const clifford &>(other);
 
    return ((representation_label == o.representation_label) && (commutator_sign == o.get_commutator_sign()) && same_metric(o));
}
 
static bool is_dirac_slash(const ex & seq0)
{
    return !is_a<diracgamma5>(seq0) && !is_a<diracgammaL>(seq0) &&
           !is_a<diracgammaR>(seq0) && !is_a<cliffordunit>(seq0) &&
           !is_a<diracone>(seq0);
}
 
void clifford::do_print_dflt(const print_dflt & c, unsigned level) const
{
    // dirac_slash() object is printed differently
    if (is_dirac_slash(seq[0])) {
        seq[0].print(c, precedence());
        c.s << "\\";
    } else { // We do not print representation label if it is 0
        if (representation_label == 0) {
            this->print_dispatch<inherited>(c, level);
        } else { // otherwise we put it before indices in square brackets; the code is borrowed from indexed.cpp 
            if (precedence() <= level) {
                c.s << '(';
            }
            seq[0].print(c, precedence());
            c.s << '[' << int(representation_label) << ']';
            printindices(c, level);
            if (precedence() <= level) {
                c.s << ')';
            }
        }
    }
}
 
void clifford::do_print_latex(const print_latex & c, unsigned level) const
{
    // 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<inherited>(c, level);
    }
}
 
void clifford::do_print_tree(const print_tree & c, unsigned level) const
{
    c.s << std::string(level, ' ') << class_name() << " @" << this
        << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
        << ", " << seq.size()-1 << " indices"
        << ", symmetry=" << symtree << std::endl;
    metric.print(c, level + c.delta_indent);
    seq[0].print(c, level + c.delta_indent);
    printindices(c, level + c.delta_indent);
}
 
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}")
 
static void base_and_index(const ex & c, ex & b, ex & i)
{
    GINAC_ASSERT(is_a<clifford>(c));
    GINAC_ASSERT(c.nops() == 2+1);
 
    if (is_a<cliffordunit>(c.op(0))) { // proper dirac gamma object or clifford unit
        i = c.op(1);
        b = _ex1;
    } else if (is_a<diracgamma5>(c.op(0)) || is_a<diracgammaL>(c.op(0)) || is_a<diracgammaR>(c.op(0))) { // gamma5/L/R
        i = _ex0;
        b = _ex1;
    } else { // slash object, generate new dummy index
        varidx ix(dynallocate<symbol>(), ex_to<idx>(c.op(1)).get_dim());
        b = indexed(c.op(0), ix.toggle_variance());
        i = ix;
    }
}
 
struct is_not_a_clifford {
    bool operator()(const ex & e)
    {
        return !is_a<clifford>(e);
    }
};
 
bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
{
    GINAC_ASSERT(is_a<clifford>(*self));
    GINAC_ASSERT(is_a<indexed>(*other));
    GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
    unsigned char rl = ex_to<clifford>(*self).get_representation_label();
 
    ex dim = ex_to<idx>(self->op(1)).get_dim();
    if (other->nops() > 1)
        dim = minimal_dim(dim, ex_to<idx>(other->op(1)).get_dim());
 
    if (is_a<clifford>(*other)) {
 
        // Contraction only makes sense if the representation labels are equal
        if (ex_to<clifford>(*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<clifford>(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<clifford>(self[1])
                && is_a<clifford>(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<clifford>(self[1])
                && is_a<clifford>(self[2])
                && is_a<clifford>(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<exvector::const_iterator>(other), std::reverse_iterator<exvector::const_iterator>(self + 1)));
            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;
 
            auto next_to_last = other - 1;
            ex S = ncmul(exvector(self + 1, next_to_last));
            ex SR = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(next_to_last), std::reverse_iterator<exvector::const_iterator>(self + 1)));
 
            *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;
 
            auto next_to_last = other - 1;
            ex S = ncmul(exvector(self + 1, next_to_last));
 
            *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<symbol>(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;
}
 
bool cliffordunit::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
{
    GINAC_ASSERT(is_a<clifford>(*self));
    GINAC_ASSERT(is_a<indexed>(*other));
    GINAC_ASSERT(is_a<cliffordunit>(self->op(0)));
    clifford unit = ex_to<clifford>(*self);
    unsigned char rl = unit.get_representation_label();
 
    if (is_a<clifford>(*other)) {
        // Contraction only makes sense if the representation labels are equal
        // and the metrics are the same
        if ((ex_to<clifford>(*other).get_representation_label() != rl) 
            && unit.same_metric(*other))
            return false;
 
        auto before_other = other - 1;
        ex mu = self->op(1);
        ex mu_toggle = other->op(1);
        ex alpha = before_other->op(1);
 
        // e~mu e.mu = Tr ONE
        if (other - self == 1) {
            *self = unit.get_metric(mu, mu_toggle, true);
            *other = dirac_ONE(rl);
            return true;
 
        } else if (other - self == 2) {
            if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
                // 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));
 
            if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
                *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;
}
 
ex clifford::eval_ncmul(const exvector & v) const
{
    exvector s;
    s.reserve(v.size());
 
    // Remove superfluous ONEs
    for (auto & it : v) {
        if (!is_a<clifford>(it) || !is_a<diracone>(it.op(0)))
            s.push_back(it);
    }
 
    bool something_changed = false;
    int sign = 1;
 
    // Anticommutate gamma5/L/R's to the front
    if (s.size() >= 2) {
        auto first = s.begin(), next_to_last = s.end() - 2;
        while (true) {
            auto it = next_to_last;
            while (true) {
                auto it2 = it + 1;
                if (is_a<clifford>(*it) && is_a<clifford>(*it2)) {
                    ex e1 = it->op(0), e2 = it2->op(0);
 
                    if (is_a<diracgamma5>(e2)) {
 
                        if (is_a<diracgammaL>(e1) || is_a<diracgammaR>(e1)) {
 
                            // gammaL/R gamma5 -> gamma5 gammaL/R
                            it->swap(*it2);
                            something_changed = true;
 
                        } else if (!is_a<diracgamma5>(e1)) {
 
                            // gamma5 gamma5 -> gamma5 gamma5 (do nothing)
                            // x gamma5 -> -gamma5 x
                            it->swap(*it2);
                            sign = -sign;
                            something_changed = true;
                        }
 
                    } else if (is_a<diracgammaL>(e2)) {
 
                        if (is_a<diracgammaR>(e1)) {
 
                            // gammaR gammaL -> 0
                            return _ex0;
 
                        } else if (!is_a<diracgammaL>(e1) && !is_a<diracgamma5>(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<clifford>(*it).get_representation_label());
                            something_changed = true;
                        }
 
                    } else if (is_a<diracgammaR>(e2)) {
 
                        if (is_a<diracgammaL>(e1)) {
 
                            // gammaL gammaR -> 0
                            return _ex0;
 
                        } else if (!is_a<diracgammaR>(e1) && !is_a<diracgamma5>(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<clifford>(*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<clifford>(a) || !is_a<clifford>(b))
                continue;
 
            const ex & ag = a.op(0);
            const ex & bg = b.op(0);
            bool a_is_cliffordunit = is_a<cliffordunit>(ag);
            bool b_is_cliffordunit =  is_a<cliffordunit>(bg);
 
            if (a_is_cliffordunit && b_is_cliffordunit && ex_to<clifford>(a).same_metric(b)
                && (ex_to<clifford>(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<clifford>(a).get_metric(ia, ib, true);
                    b = dirac_ONE(representation_label);
                    something_changed = true;
                }
 
            } else if ((is_a<diracgamma5>(ag) && is_a<diracgamma5>(bg))) {
 
                // Remove squares of gamma5
                a = dirac_ONE(representation_label);
                b = dirac_ONE(representation_label);
                something_changed = true;
 
            } else if ((is_a<diracgammaL>(ag) && is_a<diracgammaL>(bg))
                    || (is_a<diracgammaR>(ag) && is_a<diracgammaR>(bg))) {
 
                // Remove squares of gammaL/R
                b = dirac_ONE(representation_label);
                something_changed = true;
 
            } else if (is_a<diracgammaL>(ag) && is_a<diracgammaR>(bg)) {
 
                // gammaL and gammaR are orthogonal
                return _ex0;
 
            } else if (is_a<diracgamma5>(ag) && is_a<diracgammaL>(bg)) {
 
                // gamma5 gammaL -> -gammaL
                a = dirac_ONE(representation_label);
                sign = -sign;
                something_changed = true;
 
            } else if (is_a<diracgamma5>(ag) && is_a<diracgammaR>(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(dynallocate<symbol>(), ex_to<idx>(a.op(1)).minimal_dim(ex_to<idx>(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, commutator_sign, v);
}
 
ex clifford::thiscontainer(exvector && v) const
{
    return clifford(representation_label, metric, commutator_sign, std::move(v));
}
 
ex diracgamma5::conjugate() const
{   
    return _ex_1 * (*this);
}
 
ex diracgammaL::conjugate() const
{
    return dynallocate<diracgammaR>();
}
 
ex diracgammaR::conjugate() const
{
    return dynallocate<diracgammaL>();
}
 
// global functions
 
ex dirac_ONE(unsigned char rl)
{
    static ex ONE = dynallocate<diracone>();
    return clifford(ONE, rl);
}
 
static unsigned get_dim_uint(const ex& e)
{
    if (!is_a<idx>(e))
        throw std::invalid_argument("get_dim_uint: argument is not an index");
    ex dim = ex_to<idx>(e).get_dim();
    if (!dim.info(info_flags::posint))
        throw std::invalid_argument("get_dim_uint: dimension of index should be a positive integer");
    unsigned d = ex_to<numeric>(dim).to_int();
    return d;
}
 
ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl)
{
    ex unit = dynallocate<cliffordunit>();
 
    if (!is_a<idx>(mu))
        throw(std::invalid_argument("clifford_unit(): index of Clifford unit must be of type idx or varidx"));
 
    exvector indices = metr.get_free_indices();
 
    if (indices.size() == 2) {
        return clifford(unit, mu, metr, rl);
    } else if (is_a<matrix>(metr)) {
        matrix M = ex_to<matrix>(metr);
        unsigned n = M.rows();
        bool symmetric = true;
 
        //static idx xi(dynallocate<symbol>(), n),
        //           chi(dynallocate<symbol>(), n);
        idx xi(dynallocate<symbol>(), n),
            chi(dynallocate<symbol>(), n);
        if ((n ==  M.cols()) && (n == get_dim_uint(mu))) {
            for (unsigned i = 0; i < n; i++) {
                for (unsigned j = i+1; j < n; j++) {
                    if (!M(i, j).is_equal(M(j, i))) {
                        symmetric = false;
                    }
                }
            }
            return clifford(unit, mu, indexed(metr, symmetric?symmetric2():not_symmetric(), xi, chi), rl);
        } else {
            throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be a square matrix with the same dimensions as index"));
        }
    } else if (indices.size() == 0) { // a tensor or other expression without indices
        //static varidx xi(dynallocate<symbol>(), ex_to<idx>(mu).get_dim()),
        //              chi(dynallocate<symbol>(), ex_to<idx>(mu).get_dim());
        varidx xi(dynallocate<symbol>(), ex_to<idx>(mu).get_dim()),
               chi(dynallocate<symbol>(), ex_to<idx>(mu).get_dim());
        return clifford(unit, mu, indexed(metr, xi, chi), rl);
    }  else 
        throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be of type tensor, matrix or an expression with two free indices"));
}
 
ex dirac_gamma(const ex & mu, unsigned char rl)
{
    static ex gamma = dynallocate<diracgamma>();
 
    if (!is_a<varidx>(mu))
        throw(std::invalid_argument("dirac_gamma(): index of Dirac gamma must be of type varidx"));
 
    static varidx xi(dynallocate<symbol>(), ex_to<varidx>(mu).get_dim()),
                  chi(dynallocate<symbol>(), ex_to<varidx>(mu).get_dim());
    return clifford(gamma, mu, indexed(dynallocate<minkmetric>(), symmetric2(), xi, chi), rl);
}
 
ex dirac_gamma5(unsigned char rl)
{
    static ex gamma5 = dynallocate<diracgamma5>();
    return clifford(gamma5, rl);
}
 
ex dirac_gammaL(unsigned char rl)
{
    static ex gammaL = dynallocate<diracgammaL>();
    return clifford(gammaL, rl);
}
 
ex dirac_gammaR(unsigned char rl)
{
    static ex gammaR = dynallocate<diracgammaR>();
    return clifford(gammaR, rl);
}
 
ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
{
    // 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
 
    static varidx xi(dynallocate<symbol>(), dim),
                  chi(dynallocate<symbol>(), dim);
    return clifford(e, varidx(0, dim), indexed(dynallocate<minkmetric>(), symmetric2(), xi, chi), rl);
}
 
static unsigned char get_representation_label(const return_type_t& ti)
{
    return (unsigned char)ti.rl;
}
 
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<num; i++) {
        for (size_t n=1, j=0; n<num; n++) {
            if (n == i)
                continue;
            v[j++] = ix[n];
        }
        result += sign * lorentz_g(ix[0], ix[i]) * trace_string(v.begin(), num-2);
        sign = -sign;
    }
    return result;
}
 
ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE)
{
    if (is_a<clifford>(e)) {
 
        unsigned char rl = ex_to<clifford>(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<diracone>(g))
            return trONE;
        else if (is_a<diracgammaL>(g) || is_a<diracgammaR>(g))
            return trONE/2;
        else
            return _ex0;
 
    } else if (is_exactly_a<mul>(e)) {
 
        // Trace of product: pull out non-clifford factors
        ex prod = _ex1;
        for (size_t i=0; i<e.nops(); i++) {
            const ex &o = e.op(i);
            if (is_clifford_tinfo(o.return_type_tinfo()))
                prod *= dirac_trace(o, rls, trONE);
            else
                prod *= o;
        }
        return prod;
 
    } else if (is_exactly_a<ncmul>(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<ncmul>(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<diracgamma5>(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<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();
            }
 
            // 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<num; i++)
                base_and_index(e.op(i), bv[i-1], ix[i-1]);
            num--;
            int *iv = new int[num];
            ex result;
            for (size_t i=0; i<num-3; i++) {
                ex idx1 = ix[i];
                for (size_t j=i+1; j<num-2; j++) {
                    ex idx2 = ix[j];
                    for (size_t k=j+1; k<num-1; k++) {
                        ex idx3 = ix[k];
                        for (size_t l=k+1; l<num; l++) {
                            ex idx4 = ix[l];
                            iv[0] = i; iv[1] = j; iv[2] = k; iv[3] = l;
                            exvector v;
                            v.reserve(num - 4);
                            for (size_t n=0, t=4; n<num; n++) {
                                if (n == i || n == j || n == k || n == l)
                                    continue;
                                iv[t++] = n;
                                v.push_back(ix[n]);
                            }
                            int sign = permutation_sign(iv, iv + num);
                            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))
                                    * 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<num; i++)
                base_and_index(e.op(i), bv[i], iv[i]);
 
            return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
        }
 
    } else if (e.nops() > 0) {
 
        // Trace maps to all other container classes (this includes sums)
        pointer_to_map_function_2args<const std::set<unsigned char> &, 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)
{
    // Convert list to set
    std::set<unsigned char> rls;
    for (const auto & i : rll) {
        if (i.info(info_flags::nonnegint))
            rls.insert(ex_to<numeric>(i).to_int());
    }
 
    return dirac_trace(e, rls, trONE);
}
 
ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
{
    // Convert label to set
    std::set<unsigned char> rls;
    rls.insert(rl);
 
    return dirac_trace(e, rls, trONE);
}
 
 
ex canonicalize_clifford(const ex & e_)
{
    pointer_to_map_function fcn(canonicalize_clifford);
 
    if (is_a<matrix>(e_)    // || is_a<pseries>(e) || is_a<integral>(e)
        || e_.info(info_flags::list)) {
        return e_.map(fcn);
    } else {
        ex e=simplify_indexed(e_);
        // Scan for any ncmul objects
        exmap srl;
        ex aux = e.to_rational(srl);
        for (auto & i : srl) {
 
            ex lhs = i.first;
            ex rhs = i.second;
 
            if (is_exactly_a<ncmul>(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<ncmul>(rhs_expanded)) {
                    i.second = canonicalize_clifford(rhs_expanded);
                    continue;
 
                } else if (!is_a<clifford>(rhs.op(0)))
                    continue;
 
                exvector v;
                v.reserve(rhs.nops());
                for (size_t j=0; j<rhs.nops(); j++)
                    v.push_back(rhs.op(j));
 
                // Stupid recursive bubble sort because we only want to swap adjacent gammas
                auto it = v.begin(), next_to_last = v.end() - 1;
                if (is_a<diracgamma5>(it->op(0)) || is_a<diracgammaL>(it->op(0)) || is_a<diracgammaR>(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<clifford>(save0).get_metric(i1, i2, ex_to<clifford>(save0).get_commutator_sign() == -1) * b1 * b2).simplify_indexed();
                        it[1] = v.size() ? _ex2 * dirac_ONE(ex_to<clifford>(save0).get_representation_label()) : _ex2;
                        ex sum = ncmul(v);
                        it[0] = save1;
                        it[1] = save0;
                        sum += ex_to<clifford>(save0).get_commutator_sign() * ncmul(std::move(v));
                        i.second = canonicalize_clifford(sum);
                        goto next_sym;
                    }
                    ++it;
                }
next_sym:   ;
            }
        }
        return aux.subs(srl, subs_options::no_pattern).simplify_indexed();
    }
}
 
ex clifford_star_bar(const ex & e, bool do_bar, unsigned options)
{
    pointer_to_map_function_2args<bool, unsigned> fcn(clifford_star_bar, do_bar, options | 1);
 
    // is a child, no need to expand
    ex e1= (options & 1 ? e : e.expand());
 
    if (is_a<ncmul>(e1) ) { // reversing order of clifford units
        exvector ev, cv;
        ev.reserve(e1.nops());
        cv.reserve(e1.nops());
        // separate clifford and non-clifford entries
        for (size_t i= 0; i < e1.nops(); ++i) {
            if (is_a<clifford>(e1.op(i)) && is_a<cliffordunit>(e1.op(i).op(0)))
                cv.push_back(e1.op(i));
            else
                ev.push_back(e1.op(i));
        }
        for (auto i=cv.rbegin(); i!=cv.rend(); ++i) { // reverse order of Clifford units
            ev.push_back(i->conjugate());
        }
        // For clifford_bar an odd number of clifford units reverts the sign
        if (do_bar && (cv.size() % 2 == 1))
            return -dynallocate<ncmul>(std::move(ev));
        else
            return dynallocate<ncmul>(std::move(ev));
    } else if (is_a<clifford>(e1) && is_a<cliffordunit>(e1.op(0))) {
        if (do_bar)
            return -e;
        else
            return e;
    } else if (is_a<power>(e1)) {
        // apply the procedure to the base of a power
        return pow(clifford_star_bar(e1.op(0), do_bar, 0), e1.op(1));
    } else if (is_a<add>(e1) || is_a<mul>(e1) || e.info(info_flags::list)) {
        // recurse into subexpressions
        return e1.map(fcn);
    } else  // nothing meaningful can be done
        return e;
}
 
ex clifford_prime(const ex & e)
{
    pointer_to_map_function fcn(clifford_prime);
    if (is_a<clifford>(e) && is_a<cliffordunit>(e.op(0))) {
        return -e;
    } else if (is_a<add>(e) || is_a<ncmul>(e) || is_a<mul>(e) //|| is_a<pseries>(e) || is_a<integral>(e)
               || is_a<matrix>(e) || e.info(info_flags::list)) {
        return e.map(fcn);
    } else if (is_a<power>(e)) {
        return pow(clifford_prime(e.op(0)), e.op(1));
    } else
        return e;
}
 
ex remove_dirac_ONE(const ex & e, unsigned char rl, unsigned options)
{
    pointer_to_map_function_2args<unsigned char, unsigned> 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<clifford>(e1) && ex_to<clifford>(e1).get_representation_label() >= rl) {
        if (is_a<diracone>(e1.op(0)))
            return 1;
        else 
            throw(std::invalid_argument("remove_dirac_ONE(): expression is a non-scalar Clifford number!"));
    } else if (is_a<add>(e1) || is_a<ncmul>(e1) || is_a<mul>(e1)  
               || is_a<matrix>(e1) || e1.info(info_flags::list)) {
        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<power>(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;
}
 
int clifford_max_label(const ex & e, bool ignore_ONE)
{
    if (is_a<clifford>(e))
        if (ignore_ONE && is_a<diracone>(e.op(0)))
            return -1;
        else
            return ex_to<clifford>(e).get_representation_label();
    else {
        int 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;
    }
}
 
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)
{
    if (!ex_to<idx>(mu).is_dim_numeric())
        throw(std::invalid_argument("lst_to_clifford(): Index should have a numeric dimension"));
    ex e = clifford_unit(mu, metr, rl);
    return lst_to_clifford(v, e);
}
 
ex lst_to_clifford(const ex & v, const ex & e) {
    unsigned min, max;
 
    if (is_a<clifford>(e)) {
        ex mu = e.op(1);
        ex mu_toggle
            = is_a<varidx>(mu) ? ex_to<varidx>(mu).toggle_variance() : mu;
        unsigned dim = get_dim_uint(mu);
 
        if (is_a<matrix>(v)) {
            if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows()) {
                min = ex_to<matrix>(v).rows();
                max = ex_to<matrix>(v).cols();
            } else {
                min = ex_to<matrix>(v).cols();
                max = ex_to<matrix>(v).rows();
            }
            if (min == 1) {
                if (dim == max)
                    return indexed(v, mu_toggle) * e;
                else if (max - dim == 1) {
                    if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows())
                        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;
                    else 
                        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;
                } 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 (nx1 or 1xn matrix)"));
        } else if (v.info(info_flags::list)) {
            if (dim == ex_to<lst>(v).nops())
                return indexed(matrix(dim, 1, ex_to<lst>(v)), mu_toggle) * e;
            else if (ex_to<lst>(v).nops() - dim == 1)
                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;
            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"));
}
 
static ex get_clifford_comp(const ex & e, const ex & c, bool root=true)
{
    // make expansion on the top-level call only
    ex e1=(root? e.expand() : e);
 
    pointer_to_map_function_2args<const ex &, bool> fcn(get_clifford_comp, c, false);
    int ival = ex_to<numeric>(ex_to<idx>(c.op(1)).get_value()).to_int();
    int rl=ex_to<clifford>(c).get_representation_label();
 
    if ( (is_a<add>(e1) || e1.info(info_flags::list) || is_a<matrix>(e1))) {
        return e1.map(fcn);
    } else if (is_a<ncmul>(e1) || is_a<mul>(e1)) {
        // searches are done within products only
        exvector ev, all_dummy=get_all_dummy_indices(e1);
        bool found=false, same_value_found=false;
        ex dummy_ind=0;
        ev.reserve(e1.nops());
        for (size_t i=0; i < e1.nops(); ++i) {
            // look for a Clifford unit with the same metric and representation label,
            // if found remember its index
            if (is_a<clifford>(e1.op(i)) && ex_to<clifford>(e1.op(i)).get_representation_label() == rl
                && is_a<cliffordunit>(e1.op(i).op(0)) && ex_to<clifford>(e1.op(i)).same_metric(c)) { // same Clifford unit
                if (found)
                    throw(std::invalid_argument("get_clifford_comp(): expression is a Clifford multi-vector"));
                found=true;
                if (ex_to<idx>(e1.op(i).op(1)).is_numeric() &&
                    (ival == ex_to<numeric>(ex_to<idx>(e1.op(i).op(1)).get_value()).to_int())) {
                    same_value_found = true; // desired index value is found
                } else if ((std::find(all_dummy.begin(), all_dummy.end(), e1.op(i).op(1)) != all_dummy.end())
                           || (is_a<varidx>(e1.op(i).op(1))
                               && std::find(all_dummy.begin(), all_dummy.end(),
                                            ex_to<varidx>(e1.op(i).op(1)).toggle_variance()) != all_dummy.end())) {
                    dummy_ind=(e1.op(i).op(1)); // suitable dummy index found
                } else
                    ev.push_back(e.op(i)); // another index value
            } else
                ev.push_back(e1.op(i));
        }
 
        if (! found) // no Clifford units found at all
            throw(std::invalid_argument("get_clifford_comp(): expression is not a Clifford vector to the given units"));
 
        ex res=dynallocate<ncmul>(std::move(ev));
        if (same_value_found) {
            return  res;
        } else if (! dummy_ind.is_zero()) { // a dummy index was found
            if (is_a<varidx>(dummy_ind))
                dummy_ind = ex_to<varidx>(dummy_ind).toggle_variance();
            return res.subs(dummy_ind==ival, subs_options::no_pattern);
        } else // found a Clifford unit with another index
            return 0;
    } else if (e1.is_zero()) {
        return 0;
    } else if (is_a<clifford>(e1) && is_a<cliffordunit>(e1.op(0)) && ex_to<clifford>(e1).same_metric(c)) {
        if (ex_to<idx>(e1.op(1)).is_numeric() &&
            (ival == ex_to<numeric>(ex_to<idx>(e1.op(1)).get_value()).to_int()) )
            return 1;
        else
            return 0;
    } 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<clifford>(c));
    ex mu = c.op(1);
    if (! ex_to<idx>(mu).is_dim_numeric())
        throw(std::invalid_argument("clifford_to_lst(): index should have a numeric dimension"));
    unsigned int D = ex_to<numeric>(ex_to<idx>(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() 
                || (! is_a<numeric>(pow(c.subs(mu == i, subs_options::no_pattern), 2))))
                algebraic = false;
    lst V; 
    ex v0 = remove_dirac_ONE(canonicalize_clifford(e+clifford_prime(e)))/2;
    if (! v0.is_zero())
        V.append(v0);
    ex e1 = canonicalize_clifford(e - v0 * dirac_ONE(ex_to<clifford>(c).get_representation_label())); 
    if (algebraic) {
        for (unsigned int i = 0; i < D; i++) 
            V.append(remove_dirac_ONE(
                        simplify_indexed(canonicalize_clifford(e1 * c.subs(mu == i, subs_options::no_pattern) +  c.subs(mu == i, subs_options::no_pattern) * e1))
                        / (2*pow(c.subs(mu == i, subs_options::no_pattern), 2))));
    } else {
        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(e, true));
            V.remove_all();
            v0 = remove_dirac_ONE(canonicalize_clifford(e1+clifford_prime(e1)))/2;
            if (! v0.is_zero()) {
                V.append(v0);
                e1 = canonicalize_clifford(e1 - v0 * dirac_ONE(ex_to<clifford>(c).get_representation_label())); 
            }
            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)
{
    ex x, D, cu;
    
    if (! is_a<matrix>(v) && ! v.info(info_flags::list))
        throw(std::invalid_argument("clifford_moebius_map(): parameter v should be either vector or list"));
    
    if (is_a<clifford>(G)) {
        cu = G;
    } else {
        if (is_a<indexed>(G)) {
            D = ex_to<idx>(G.op(1)).get_dim();
            varidx mu(dynallocate<symbol>(), D);
            cu = clifford_unit(mu, G, rl);
        } else if (is_a<matrix>(G)) {
            D = ex_to<matrix>(G).rows(); 
            idx mu(dynallocate<symbol>(), D);
            cu = clifford_unit(mu, G, rl);
        } else throw(std::invalid_argument("clifford_moebius_map(): metric should be an indexed object, matrix, or a Clifford unit"));
        
    }
    
    x = lst_to_clifford(v, cu); 
    ex e = clifford_to_lst(simplify_indexed(canonicalize_clifford((a * x + b) * clifford_inverse(c * x + d))), cu, false);
    return (is_a<matrix>(v) ? matrix(ex_to<matrix>(v).rows(), ex_to<matrix>(v).cols(), ex_to<lst>(e)) : e);
}
 
ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl)
{
    if (is_a<matrix>(M) && (ex_to<matrix>(M).rows() == 2) && (ex_to<matrix>(M).cols() == 2)) 
        return clifford_moebius_map(M.op(0), M.op(1), M.op(2), M.op(3), v, G, rl);
    else
        throw(std::invalid_argument("clifford_moebius_map(): parameter M should be a 2x2 matrix"));
}
 
} // namespace GiNaC