ginac-1.8.2/ginac/clifford.cpp

/** @file clifford.cpp
 *
 *  Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */

/*
 *  GiNaC Copyright (C) 1999-2022 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
//////////

/** 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) : inherited(b), representation_label(rl), metric(0), commutator_sign(-1)
{
}

/** 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, 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}")

/** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
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;
	}
}

/** Predicate for finding non-clifford objects. */
struct is_not_a_clifford {
	bool operator()(const ex & e)
	{
		return !is_a<clifford>(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<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;
}

/** 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<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;
}

/** 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
	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);
}

/** Extract representation label from tinfo key (as returned by
 *  return_type_tinfo()). */
static unsigned char get_representation_label(const return_type_t& ti)
{
	return (unsigned char)ti.rl;
}

/** 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<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 (int 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"));
}

/** 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, 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 (int 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