X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fclifford.cpp;h=b337cd70c6c848490cba467236eac6c8bc86f487;hp=8a8bfe09294e942e88023d0770784f09d5d9f289;hb=1431a02f0f4a6acacce20282a1431bdf0097b482;hpb=43e0a8f5ca5e1c48cef5daaf014acdbca4e44568

diff --git a/ginac/clifford.cpp b/ginac/clifford.cpp
index 8a8bfe09..b337cd70 100644
--- a/ginac/clifford.cpp
+++ b/ginac/clifford.cpp
@@ -3,7 +3,7 @@
  *  Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
 
 /*
- *  GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
+ *  GiNaC Copyright (C) 1999-2005 Johannes Gutenberg University Mainz, Germany
  *
  *  This program is free software; you can redistribute it and/or modify
  *  it under the terms of the GNU General Public License as published by
@@ -21,44 +21,74 @@
  */
 
 #include "clifford.h"
+
 #include "ex.h"
 #include "idx.h"
 #include "ncmul.h"
 #include "symbol.h"
-#include "print.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 "debugmsg.h"
 #include "utils.h"
 
-#include <stdexcept>
-
 namespace GiNaC {
 
-GINAC_IMPLEMENT_REGISTERED_CLASS(clifford, indexed)
-GINAC_IMPLEMENT_REGISTERED_CLASS(diracone, tensor)
-GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma, tensor)
-GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma5, tensor)
+GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(clifford, indexed,
+  print_func<print_dflt>(&clifford::do_print_dflt).
+  print_func<print_latex>(&clifford::do_print_latex))
+
+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 constructor, destructor, copy constructor assignment operator and helpers
+// default constructors
 //////////
 
-clifford::clifford() : representation_label(0)
+static ex default_metric()
 {
-	debugmsg("clifford default constructor", LOGLEVEL_CONSTRUCT);
-	tinfo_key = TINFO_clifford;
+	static ex m = (new minkmetric)->setflag(status_flags::dynallocated);
+	return m;
 }
 
-void clifford::copy(const clifford & other)
+clifford::clifford() : representation_label(0), metric(default_metric())
 {
-	inherited::copy(other);
-	representation_label = other.representation_label;
+	tinfo_key = TINFO_clifford;
 }
 
-DEFAULT_DESTROY(clifford)
-DEFAULT_CTORS(diracone)
-DEFAULT_CTORS(diracgamma)
-DEFAULT_CTORS(diracgamma5)
+DEFAULT_CTOR(diracone)
+DEFAULT_CTOR(cliffordunit)
+DEFAULT_CTOR(diracgamma)
+DEFAULT_CTOR(diracgamma5)
+DEFAULT_CTOR(diracgammaL)
+DEFAULT_CTOR(diracgammaR)
 
 //////////
 // other constructors
@@ -67,31 +97,28 @@ DEFAULT_CTORS(diracgamma5)
 /** 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)
+clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl), metric(0)
 {
-	debugmsg("clifford constructor from ex", LOGLEVEL_CONSTRUCT);
 	tinfo_key = TINFO_clifford;
 }
 
 /** Construct object with one Lorentz index. This constructor is for internal
- *  use only. Use the dirac_gamma() function instead.
+ *  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, unsigned char rl) : inherited(b, mu), representation_label(rl)
+clifford::clifford(const ex & b, const ex & mu, const ex & metr, unsigned char rl) : inherited(b, mu), representation_label(rl), metric(metr)
 {
-	debugmsg("clifford constructor from ex,ex", LOGLEVEL_CONSTRUCT);
-	GINAC_ASSERT(is_ex_of_type(mu, varidx));
+	GINAC_ASSERT(is_a<varidx>(mu));
 	tinfo_key = TINFO_clifford;
 }
 
-clifford::clifford(unsigned char rl, const exvector & v, bool discardable) : inherited(indexed::unknown, v, discardable), representation_label(rl)
+clifford::clifford(unsigned char rl, const ex & metr, const exvector & v, bool discardable) : inherited(not_symmetric(), v, discardable), representation_label(rl), metric(metr)
 {
-	debugmsg("clifford constructor from unsigned char,exvector", LOGLEVEL_CONSTRUCT);
 	tinfo_key = TINFO_clifford;
 }
 
-clifford::clifford(unsigned char rl, exvector * vp) : inherited(indexed::unknown, vp), representation_label(rl)
+clifford::clifford(unsigned char rl, const ex & metr, std::auto_ptr<exvector> vp) : inherited(not_symmetric(), vp), representation_label(rl), metric(metr)
 {
-	debugmsg("clifford constructor from unsigned char,exvector *", LOGLEVEL_CONSTRUCT);
 	tinfo_key = TINFO_clifford;
 }
 
@@ -99,32 +126,51 @@ clifford::clifford(unsigned char rl, exvector * vp) : inherited(indexed::unknown
 // archiving
 //////////
 
-clifford::clifford(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
+clifford::clifford(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
 {
-	debugmsg("clifford constructor from archive_node", LOGLEVEL_CONSTRUCT);
 	unsigned rl;
 	n.find_unsigned("label", rl);
 	representation_label = rl;
+	n.find_ex("metric", metric, sym_lst);
 }
 
 void clifford::archive(archive_node &n) const
 {
 	inherited::archive(n);
 	n.add_unsigned("label", representation_label);
+	n.add_ex("metric", metric);
 }
 
 DEFAULT_UNARCHIVE(clifford)
 DEFAULT_ARCHIVING(diracone)
+DEFAULT_ARCHIVING(cliffordunit)
 DEFAULT_ARCHIVING(diracgamma)
 DEFAULT_ARCHIVING(diracgamma5)
+DEFAULT_ARCHIVING(diracgammaL)
+DEFAULT_ARCHIVING(diracgammaR)
 
 //////////
-// functions overriding virtual functions from bases classes
+// functions overriding virtual functions from base classes
 //////////
 
+ex clifford::get_metric(const ex & i, const ex & j) const
+{
+	return indexed(metric, i, j);
+}
+
+bool clifford::same_metric(const ex & other) const
+{
+	if (is_a<clifford>(other)) {
+		return get_metric().is_equal(ex_to<clifford>(other).get_metric());
+	} else if (is_a<indexed>(other)) {
+		return get_metric(other.op(1), other.op(2)).is_equal(other);
+	} else
+		return false;
+}
+
 int clifford::compare_same_type(const basic & other) const
 {
-	GINAC_ASSERT(other.tinfo() == TINFO_clifford);
+	GINAC_ASSERT(is_a<clifford>(other));
 	const clifford &o = static_cast<const clifford &>(other);
 
 	if (representation_label != o.representation_label) {
@@ -135,79 +181,307 @@ int clifford::compare_same_type(const basic & other) const
 	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) && 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, level);
+		c.s << "\\";
+	} else
+		this->print_dispatch<inherited>(c, level);
+}
+
+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, level);
+		c.s << "\\hspace{-1.0ex}/}";
+	} else {
+		c.s << "\\clifford[" << int(representation_label) << "]";
+		this->print_dispatch<inherited>(c, level);
+	}
+}
+
 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);
+
+	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((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(c.op(1)).get_dim());
+		b = indexed(c.op(0), ix.toggle_variance());
+		i = ix;
+	}
+}
 
-DEFAULT_PRINT(diracone, "ONE")
-DEFAULT_PRINT(diracgamma, "gamma")
-DEFAULT_PRINT(diracgamma5, "gamma5")
+/** Predicate for finding non-clifford objects. */
+struct is_not_a_clifford : public std::unary_function<ex, bool> {
+	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_ex_of_type(*self, clifford));
-	GINAC_ASSERT(is_ex_of_type(*other, indexed));
-	GINAC_ASSERT(is_ex_of_type(self->op(0), diracgamma));
-	unsigned char rl = ex_to_clifford(*self).get_representation_label();
+	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();
 
-	if (is_ex_of_type(other->op(0), diracgamma)) {
+	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());
 
-		ex dim = ex_to_idx(self->op(1)).get_dim();
+	if (is_a<clifford>(*other)) {
 
-		// gamma~mu*gamma.mu = dim*ONE
-		if (other - self == 1) {
+		// Contraction only makes sense if the represenation 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 (other - self == 2
-		        && is_ex_of_type(self[1], clifford)) {
+		// gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
+		} else if (num == 2
+		        && is_a<clifford>(self[1])) {
 			*self = 2 - dim;
-			*other = _ex1();
+			*other = _ex1;
 			return true;
 
-		// gamma~mu*gamma~alpha*gamma~beta*gamma.mu = 4*g~alpha~beta+(dim-4)*gamam~alpha*gamma~beta
-		} else if (other - self == 3
-		        && is_ex_of_type(self[1], clifford)
-		        && is_ex_of_type(self[2], clifford)) {
-			*self = 4 * metric_tensor(self[1].op(1), self[2].op(1)) * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
-			self[1] = _ex1();
-			self[2] = _ex1();
-			*other = _ex1();
+		// 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+(4-dim)*gamma~alpha*gamma~beta*gamma~delta
-		} else if (other - self == 4
-		        && is_ex_of_type(self[1], clifford)
-		        && is_ex_of_type(self[2], clifford)
-		        && is_ex_of_type(self[3], clifford)) {
-			*self = -2 * self[3] * self[2] * self[1] + (4 - dim) * self[1] * self[2] * self[3];
-			self[1] = _ex1();
-			self[2] = _ex1();
-			self[3] = _ex1();
-			*other = _ex1();
+		// 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)), true);
+			std::fill(self + 1, other, _ex1);
+			*other = _ex_2;
+			return true;
+
+		// gamma~mu Sodd gamma~alpha gamma.mu = 2 gamma~alpha Sodd + 2 Sodd_R gamma~alpha
+		// (commutate contracted indices towards each other, then use
+		// Chisholm identity in 4 dimensions)
+		} else if (((other - self) & 1) && dim.is_equal(4)) {
+			if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
+				return false;
+
+			exvector::iterator next_to_last = other - 1;
+			ex S = ncmul(exvector(self + 1, next_to_last), true);
+			ex SR = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(next_to_last), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
+
+			*self = (*next_to_last) * S + SR * (*next_to_last);
+			std::fill(self + 1, other, _ex1);
+			*other = _ex2;
+			return true;
+
+		// gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
+		// (commutate contracted indices towards each other, simplify_indexed()
+		// will re-expand and re-run the simplification)
+		} else {
+			if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
+				return false;
+
+			exvector::iterator next_to_last = other - 1;
+			ex S = ncmul(exvector(self + 1, next_to_last), true);
+
+			*self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
+			std::fill(self + 1, other + 1, _ex1);
 			return true;
 		}
+
+	} else if (is_a<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;
 }
 
+/** An utility function looking for a given metric within an exvector,
+ *  used in cliffordunit::contract_with(). */
+static int find_same_metric(exvector & v, ex & c)
+{
+	for (int i=0; i<v.size();i++) {
+		if (!is_a<clifford>(v[i]) && is_a<indexed>(v[i])
+		    && ex_to<clifford>(c).same_metric(v[i]) 
+		    && (ex_to<varidx>(c.op(1)) == ex_to<indexed>(v[i]).get_indices()[0]
+			|| ex_to<varidx>(c.op(1)).toggle_variance() == ex_to<indexed>(v[i]).get_indices()[0])) {
+			return ++i; // next to found
+		}
+	}
+	return 0; //nothing found
+}
+
+/** Contraction of a Clifford unit with something else. */
+bool cliffordunit::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
+{
+	GINAC_ASSERT(is_a<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 represenation labels are equal
+		// and the metrics are the same
+		if ((ex_to<clifford>(*other).get_representation_label() != rl) 
+		    && unit.same_metric(*other))
+			return false;
+
+		// Find if a previous contraction produces the square of self
+		int prev_square = find_same_metric(v, self[0]);
+		varidx d((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(self->op(1)).get_dim());
+		ex squared_metric = unit.get_metric(self->op(1), d) * unit.get_metric(d.toggle_variance(), other->op(1));
+
+		// e~mu e.mu = Tr ONE
+		if (other - self == 1) {
+			if (prev_square != 0) {
+				*self = squared_metric;
+				v[prev_square-1] = _ex1;
+			} else
+				*self = unit.get_metric(self->op(1), other->op(1));
+			*other = dirac_ONE(rl);
+			return true;
+
+		// e~mu e~alpha e.mu = (2e~alpha^2-Tr) e~alpha
+		} else if (other - self == 2
+		        && is_a<clifford>(self[1])) {
+
+			const ex & ia = self[1].op(1);
+			const ex & ib = self[1].op(1);
+			if (is_a<tensmetric>(unit.get_metric()))
+				*self = 2 - unit.get_metric(self->op(1), other->op(1));
+			else if (prev_square != 0) {
+				*self = 2-squared_metric;
+				v[prev_square-1] = _ex1;
+			} else 
+				*self = 2*unit.get_metric(ia, ib) - unit.get_metric(self->op(1), other->op(1));
+			*other = _ex1;
+			return true;
+
+		// e~mu S e~alpha e.mu = 2 e~alpha^3 S - e~mu S e.mu e~alpha
+		// (commutate contracted indices towards each other, simplify_indexed()
+		// will re-expand and re-run the simplification)
+		} else {
+			exvector::iterator it = self + 1, next_to_last = other - 1;
+			while (it != other) {
+				if (!is_a<clifford>(*it))
+					return false;
+				++it;
+			}
+
+			it = self + 1;
+			ex S = _ex1;
+			while (it != next_to_last) {
+				S *= *it;
+				*it++ = _ex1;
+			}
+
+			const ex & ia = next_to_last->op(1);
+			const ex & ib = next_to_last->op(1);
+			if (is_a<tensmetric>(unit.get_metric()))
+				*self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
+			else if (prev_square != 0) {
+				*self = 2 * (*next_to_last) * S  - (*self) * S * (*other) * (*next_to_last)*unit.get_metric(self->op(1),self->op(1));
+				v[prev_square-1] = _ex1;
+			} else 
+				*self = 2 * (*next_to_last) * S* unit.get_metric(ia,ib) - (*self) * S * (*other) * (*next_to_last);
+			*next_to_last = _ex1;
+			*other = _ex1;
+			return true;
+		}
+
+	} 
+
+	return false;
+}
+
 /** Perform automatic simplification on noncommutative product of clifford
- *  objects. This removes superfluous ONEs, permutes gamma5's to the front
+ *  objects. This removes superfluous ONEs, permutes gamma5/L/R's to the front
  *  and removes squares of gamma objects. */
-ex clifford::simplify_ncmul(const exvector & v) const
+ex clifford::eval_ncmul(const exvector & v) const
 {
 	exvector s;
 	s.reserve(v.size());
-	unsigned rl = ex_to_clifford(v[0]).get_representation_label();
 
 	// Remove superfluous ONEs
 	exvector::const_iterator cit = v.begin(), citend = v.end();
 	while (cit != citend) {
-		if (!is_ex_of_type(cit->op(0), diracone))
+		if (!is_a<clifford>(*cit) || !is_a<diracone>(cit->op(0)))
 			s.push_back(*cit);
 		cit++;
 	}
@@ -215,69 +489,177 @@ ex clifford::simplify_ncmul(const exvector & v) const
 	bool something_changed = false;
 	int sign = 1;
 
-	// Anticommute gamma5's to the front
+	// Anticommutate gamma5/L/R's to the front
 	if (s.size() >= 2) {
 		exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
 		while (true) {
 			exvector::iterator it = next_to_last;
 			while (true) {
 				exvector::iterator it2 = it + 1;
-				if (!is_ex_of_type(it->op(0), diracgamma5) && is_ex_of_type(it2->op(0), diracgamma5)) {
-					it->swap(*it2);
-					sign = -sign;
-					something_changed = true;
+				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--;
+				--it;
 			}
 			if (next_to_last == first)
 				break;
-			next_to_last--;
+			--next_to_last;
 		}
 	}
 
-	// Remove squares of gamma5
-	while (s.size() >= 2 && is_ex_of_type(s[0].op(0), diracgamma5) && is_ex_of_type(s[1].op(0), diracgamma5)) {
-		s.erase(s.begin(), s.begin() + 2);
-		something_changed = true;
-	}
-
 	// Remove equal adjacent gammas
 	if (s.size() >= 2) {
-		exvector::iterator it = s.begin(), itend = s.end() - 1;
-		while (it != itend) {
+		exvector::iterator it, itend = s.end() - 1;
+		for (it = s.begin(); it != itend; ++it) {
 			ex & a = it[0];
 			ex & b = it[1];
-			if (is_ex_of_type(a.op(0), diracgamma) && is_ex_of_type(b.op(0), diracgamma)) {
+			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)) {
+
 				const ex & ia = a.op(1);
 				const ex & ib = b.op(1);
-				if (ia.is_equal(ib)) {
-					a = lorentz_g(ia, ib);
-					b = dirac_ONE(rl);
+				if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
+					a = ex_to<clifford>(a).get_metric(ia, ib);
+					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((new symbol)->setflag(status_flags::dynallocated), 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;
 			}
-			it++;
 		}
 	}
 
-	if (s.size() == 0)
-		return clifford(diracone(), rl) * sign;
+	if (s.empty())
+		return clifford(diracone(), representation_label) * sign;
 	if (something_changed)
-		return nonsimplified_ncmul(s) * sign;
+		return reeval_ncmul(s) * sign;
 	else
-		return simplified_ncmul(s) * sign;
+		return hold_ncmul(s) * sign;
+}
+
+ex clifford::thiscontainer(const exvector & v) const
+{
+	return clifford(representation_label, get_metric(), v);
+}
+
+ex clifford::thiscontainer(std::auto_ptr<exvector> vp) const
+{
+	return clifford(representation_label, get_metric(), vp);
+}
+
+ex diracgamma5::conjugate() const
+{	
+	return _ex_1 * (*this);
 }
 
-ex clifford::thisexprseq(const exvector & v) const
+ex diracgammaL::conjugate() const
 {
-	return clifford(representation_label, v);
+	return (new diracgammaR)->setflag(status_flags::dynallocated);
 }
 
-ex clifford::thisexprseq(exvector * vp) const
+ex diracgammaR::conjugate() const
 {
-	return clifford(representation_label, vp);
+	return (new diracgammaL)->setflag(status_flags::dynallocated);
 }
 
 //////////
@@ -286,26 +668,59 @@ ex clifford::thisexprseq(exvector * vp) const
 
 ex dirac_ONE(unsigned char rl)
 {
-	return clifford(diracone(), rl);
+	static ex ONE = (new diracone)->setflag(status_flags::dynallocated);
+	return clifford(ONE, rl);
+}
+
+ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl)
+{
+	static ex unit = (new cliffordunit)->setflag(status_flags::dynallocated);
+
+	if (!is_a<varidx>(mu))
+		throw(std::invalid_argument("index of Clifford unit must be of type varidx"));
+
+	if (is_a<indexed>(metr))
+		return clifford(unit, mu, metr.op(0), rl);
+	else if(is_a<tensmetric>(metr) || is_a<matrix>(metr)) 
+		return clifford(unit, mu, metr, rl);
+	else
+		throw(std::invalid_argument("metric for Clifford unit must be of type indexed, tensormetric or matrix"));
 }
 
 ex dirac_gamma(const ex & mu, unsigned char rl)
 {
-	if (!is_ex_of_type(mu, varidx))
+	static ex gamma = (new diracgamma)->setflag(status_flags::dynallocated);
+
+	if (!is_a<varidx>(mu))
 		throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
 
-	return clifford(diracgamma(), mu, rl);
+	return clifford(gamma, mu, default_metric(), rl);
 }
 
 ex dirac_gamma5(unsigned char rl)
 {
-	return clifford(diracgamma5(), rl);
+	static ex gamma5 = (new diracgamma5)->setflag(status_flags::dynallocated);
+	return clifford(gamma5, rl);
+}
+
+ex dirac_gammaL(unsigned char rl)
+{
+	static ex gammaL = (new diracgammaL)->setflag(status_flags::dynallocated);
+	return clifford(gammaL, rl);
+}
+
+ex dirac_gammaR(unsigned char rl)
+{
+	static ex gammaR = (new diracgammaR)->setflag(status_flags::dynallocated);
+	return clifford(gammaR, rl);
 }
 
 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
 {
-	varidx mu((new symbol)->setflag(status_flags::dynallocated), dim);
-	return indexed(e, mu.toggle_variance()) * dirac_gamma(mu, 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
+	return clifford(e, varidx(0, dim), default_metric(), rl);
 }
 
 /** Check whether a given tinfo key (as returned by return_type_tinfo()
@@ -315,113 +730,481 @@ static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
 	return ti == (TINFO_clifford + rl);
 }
 
-ex dirac_trace(const ex & e, unsigned char rl)
+/** Check whether a given tinfo key (as returned by return_type_tinfo()
+ *  is that of a clifford object (with an arbitrary representation label). */
+static bool is_clifford_tinfo(unsigned ti)
 {
-	if (is_ex_of_type(e, clifford)) {
+	return (ti & ~0xff) == TINFO_clifford;
+}
 
-		if (ex_to_clifford(e).get_representation_label() == rl
-		 && is_ex_of_type(e.op(0), diracone))
-			return _ex4();
-		else
-			return _ex0();
+/** Extract representation label from tinfo key (as returned by
+ *  return_type_tinfo()). */
+static unsigned char get_representation_label(unsigned ti)
+{
+	return ti & 0xff;
+}
+
+/** Take trace of a string of an even number of Dirac gammas given a vector
+ *  of indices. */
+static ex trace_string(exvector::const_iterator ix, size_t num)
+{
+	// Tr gamma.mu gamma.nu = 4 g.mu.nu
+	if (num == 2)
+		return lorentz_g(ix[0], ix[1]);
+
+	// Tr gamma.mu gamma.nu gamma.rho gamma.sig = 4 (g.mu.nu g.rho.sig + g.nu.rho g.mu.sig - g.mu.rho g.nu.sig )
+	else if (num == 4)
+		return lorentz_g(ix[0], ix[1]) * lorentz_g(ix[2], ix[3])
+		     + lorentz_g(ix[1], ix[2]) * lorentz_g(ix[0], ix[3])
+		     - lorentz_g(ix[0], ix[2]) * lorentz_g(ix[1], ix[3]);
+
+	// Traces of 6 or more gammas are computed recursively:
+	// Tr gamma.mu1 gamma.mu2 ... gamma.mun =
+	//   + g.mu1.mu2 * Tr gamma.mu3 ... gamma.mun
+	//   - g.mu1.mu3 * Tr gamma.mu2 gamma.mu4 ... gamma.mun
+	//   + g.mu1.mu4 * Tr gamma.mu3 gamma.mu3 gamma.mu5 ... gamma.mun
+	//   - ...
+	//   + g.mu1.mun * Tr gamma.mu2 ... gamma.mu(n-1)
+	exvector v(num - 2);
+	int sign = 1;
+	ex result;
+	for (size_t i=1; i<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)) {
 
-	} else if (is_ex_exactly_of_type(e, add)) {
+		unsigned char rl = ex_to<clifford>(e).get_representation_label();
 
-		// Trace of sum = sum of traces
-		ex sum = _ex0();
-		for (unsigned i=0; i<e.nops(); i++)
-			sum += dirac_trace(e.op(i), rl);
-		return sum;
+		// Are we taking the trace over this object's representation label?
+		if (rls.find(rl) == rls.end())
+			return e;
 
-	} else if (is_ex_exactly_of_type(e, mul)) {
+		// 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 (unsigned i=0; i<e.nops(); i++) {
+		ex prod = _ex1;
+		for (size_t i=0; i<e.nops(); i++) {
 			const ex &o = e.op(i);
-			unsigned ti = o.return_type_tinfo();
-			if (is_clifford_tinfo(o.return_type_tinfo(), rl))
-				prod *= dirac_trace(o, rl);
+			if (is_clifford_tinfo(o.return_type_tinfo()))
+				prod *= dirac_trace(o, rls, trONE);
 			else
 				prod *= o;
 		}
 		return prod;
 
-	} else if (is_ex_exactly_of_type(e, ncmul)) {
+	} else if (is_exactly_a<ncmul>(e)) {
+
+		unsigned char rl = get_representation_label(e.return_type_tinfo());
 
-		if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
-			return _ex0();
+		// Are we taking the trace over this string's representation label?
+		if (rls.find(rl) == rls.end())
+			return e;
 
-		// Expand product, if necessary
-		ex e_expanded = e.expand();
-		if (!is_ex_of_type(e_expanded, ncmul))
-			return dirac_trace(e_expanded, rl);
+		// 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_ex_of_type(e.op(0).op(0), diracgamma5);
-		unsigned num = e.nops();
+		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
+			// gamma5 * gamma.mu * gamma.nu are zero
 			if ((num & 1) == 0 || num == 3)
-				return _ex0();
-
-			// Tr gamma5 S_2k =
-			//   epsilon0123_mu1_mu2_mu3_mu4 * Tr gamma_mu1 gamma_mu2 gamma_mu3 gamma_mu4 S_2k
-			ex dim = ex_to_idx(e.op(1).op(1)).get_dim();
-			varidx mu1((new symbol)->setflag(status_flags::dynallocated), dim),
-			       mu2((new symbol)->setflag(status_flags::dynallocated), dim),
-			       mu3((new symbol)->setflag(status_flags::dynallocated), dim),
-			       mu4((new symbol)->setflag(status_flags::dynallocated), dim);
-			exvector v;
-			v.reserve(num + 3);
-			v.push_back(dirac_gamma(mu1, rl));
-			v.push_back(dirac_gamma(mu2, rl));
-			v.push_back(dirac_gamma(mu3, rl));
-			v.push_back(dirac_gamma(mu4, rl));
-			for (int i=1; i<num; i++)
-				v.push_back(e.op(i));
-
-			return (eps0123(mu1.toggle_variance(), mu2.toggle_variance(), mu3.toggle_variance(), mu4.toggle_variance()) *
-			        dirac_trace(ncmul(v), rl)).simplify_indexed() / 24;
+				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)
-				return 4 * lorentz_g(e.op(0).op(1), e.op(1).op(1));
-
-			// Traces of 4 or more gammas are computed recursively:
-			// Tr gamma_mu1 gamma_mu2 ... gamma_mun =
-			//   + eta_mu1_mu2 * Tr gamma_mu3 ... gamma_mun
-			//   - eta_mu1_mu3 * Tr gamma_mu2 gamma_mu4 ... gamma_mun
-			//   + eta_mu1_mu4 * Tr gamma_mu3 gamma_mu3 gamma_mu5 ... gamma_mun
-			//   - ...
-			//   + eta_mu1_mun * Tr gamma_mu2 ... gamma_mu(n-1)
-			exvector v(num - 2);
-			int sign = 1;
-			const ex &ix1 = e.op(0).op(1);
-			ex result;
-			for (int i=1; i<num; i++) {
-				for (int n=1, j=0; n<num; n++) {
-					if (n == i)
-						continue;
-					v[j++] = e.op(n);
+				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 (lst::const_iterator i = rll.begin(); i != rll.end(); ++i) {
+		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)
+		|| is_a<lst>(e_)) {
+		return e_.map(fcn);
+	} else {
+		ex e=simplify_indexed(e_);
+		// Scan for any ncmul objects
+		exmap srl;
+		ex aux = e.to_rational(srl);
+		for (exmap::iterator i = srl.begin(); i != srl.end(); ++i) {
+
+			ex lhs = i->first;
+			ex rhs = i->second;
+
+			if (is_exactly_a<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
+				exvector::iterator 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);
+						it[0] = (ex_to<clifford>(save0).get_metric(i1, i2) * b1 * b2).simplify_indexed();
+						it[1] = v.size() == 2 ? _ex2 * dirac_ONE(ex_to<clifford>(it[1]).get_representation_label()) : _ex2;
+						ex sum = ncmul(v);
+						it[0] = save1;
+						it[1] = save0;
+						sum -= ncmul(v, true);
+						i->second = canonicalize_clifford(sum);
+						goto next_sym;
+					}
+					++it;
 				}
-				result += sign * lorentz_g(ix1, e.op(i).op(1)) * dirac_trace(ncmul(v), rl);
-				sign = -sign;
+next_sym:	;
 			}
-			return result;
 		}
+		return aux.subs(srl, subs_options::no_pattern).simplify_indexed();
 	}
+}
+
+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) || is_a<lst>(e)) {
+		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)
+{
+	pointer_to_map_function_1arg<unsigned char> fcn(remove_dirac_ONE, rl);
+	if (is_a<clifford>(e) && ex_to<clifford>(e).get_representation_label() >= rl) {
+		if (is_a<diracone>(e.op(0)))
+			return 1;
+		else
+			throw(std::invalid_argument("Expression is a non-scalar Clifford number!"));
+	} 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) || is_a<lst>(e)) {
+		return e.map(fcn);
+	} else if (is_a<power>(e)) {
+		return pow(remove_dirac_ONE(e.op(0)), e.op(1));
+	} else
+		return e;
+}
+
+ex clifford_norm(const ex & e)
+{
+	return sqrt(remove_dirac_ONE(canonicalize_clifford(e * clifford_bar(e)).simplify_indexed()));
+}
+
+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("Cannot find inverse of Clifford number with zero norm!"));
+}
 
-	return _ex0();
+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("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)) {
+		varidx mu = ex_to<varidx>(e.op(1));
+		unsigned dim = (ex_to<numeric>(mu.get_dim())).to_int();
+
+		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, ex_to<varidx>(mu).toggle_variance()) * e;
+				else
+					throw(std::invalid_argument("Dimensions of vector and clifford unit mismatch"));
+			} else
+				throw(std::invalid_argument("First argument should be a vector vector"));
+		} else if (is_a<lst>(v)) {
+			if (dim == ex_to<lst>(v).nops())
+				return indexed(matrix(dim, 1, ex_to<lst>(v)), ex_to<varidx>(mu).toggle_variance()) * e;
+			else
+				throw(std::invalid_argument("List length and dimension of clifford unit mismatch"));
+		} else
+			throw(std::invalid_argument("Cannot construct from anything but list or vector"));
+	} else
+		throw(std::invalid_argument("The second argument should be a Clifford unit"));
+}
+
+/** Auxiliary structure to define a function for striping one Clifford unit
+ * from vectors. Used in  clifford_to_lst(). */
+static ex get_clifford_comp(const ex & e, const ex & c) 
+{
+	pointer_to_map_function_1arg<const ex &> fcn(get_clifford_comp, c);
+	int ival = ex_to<numeric>(ex_to<varidx>(c.op(1)).get_value()).to_int();
+		
+	if (is_a<add>(e) || is_a<lst>(e) // || is_a<pseries>(e) || is_a<integral>(e)
+		|| is_a<matrix>(e)) 
+		return e.map(fcn);
+	else if (is_a<ncmul>(e) || is_a<mul>(e)) {
+		// find a Clifford unit with the same metric, delete it and substitute its index
+		size_t ind = e.nops() + 1;
+		for (size_t j = 0; j < e.nops(); j++) 
+			if (is_a<clifford>(e.op(j)) && ex_to<clifford>(c).same_metric(e.op(j)))
+				if (ind > e.nops()) 
+					ind = j;
+				else 
+					throw(std::invalid_argument("Expression is a Clifford multi-vector"));
+		if (ind < e.nops()) {
+			ex S = 1;
+			bool same_value_index, found_dummy;
+			same_value_index = ( ex_to<varidx>(e.op(ind).op(1)).is_numeric()
+								 &&  (ival == ex_to<numeric>(ex_to<varidx>(e.op(ind).op(1)).get_value()).to_int()) );
+			found_dummy = same_value_index;
+			for(size_t j=0; j < e.nops(); j++)
+				if (j != ind) 
+					if (same_value_index) 
+						S = S * e.op(j);
+					else {
+						exvector ind_vec = ex_to<indexed>(e.op(j)).get_dummy_indices(ex_to<indexed>(e.op(ind)));
+						if (ind_vec.size() > 0) {
+							found_dummy = true;
+							exvector::const_iterator it = ind_vec.begin(), itend = ind_vec.end();
+							while (it != itend) {
+								S = S * e.op(j).subs(lst(ex_to<varidx>(*it) == ival, ex_to<varidx>(*it).toggle_variance() == ival), subs_options::no_pattern);
+								++it;
+							}
+						} else
+							S = S * e.op(j);
+					}
+			return (found_dummy ? S : 0);
+		} else
+			throw(std::invalid_argument("Expression is not a Clifford vector to the given units"));
+	} else if (e.is_zero()) 
+		return e;
+	else if (is_a<clifford>(e) && ex_to<clifford>(e).same_metric(c))
+		if ( ex_to<varidx>(e.op(1)).is_numeric() &&
+			 (ival != ex_to<numeric>(ex_to<varidx>(e.op(1)).get_value()).to_int()) )
+			return 0;
+		else 
+			return 1;
+	else
+		throw(std::invalid_argument("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));
+	varidx mu = ex_to<varidx>(c.op(1));
+	if (! mu.is_dim_numeric())
+		throw(std::invalid_argument("Index should have a numeric dimension"));
+	unsigned int D = ex_to<numeric>(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), 2).is_zero() 
+				or (not is_a<numeric>(pow(c.subs(mu == i), 2))))
+				algebraic = false;
+	lst V; 
+	if (algebraic) 
+		for (unsigned int i = 0; i < D; i++) 
+			V.append(remove_dirac_ONE(
+						simplify_indexed(canonicalize_clifford(e * c.subs(mu == i) +  c.subs(mu == i) * e))
+						/ (2*pow(c.subs(mu == i), 2))));
+	else {
+		ex e1 = canonicalize_clifford(e);
+		for (unsigned int i = 0; i < D; i++) 
+			V.append(get_clifford_comp(e1, c.subs(c.op(1) == i)));
+	}
+	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) && ! is_a<lst>(v))
+		throw(std::invalid_argument("parameter v should be either vector or list"));
+	
+	if (is_a<clifford>(G)) {
+		cu = G;
+	} else {
+		if (is_a<indexed>(G)) 
+			D = ex_to<varidx>(G.op(1)).get_dim();
+		else if (is_a<matrix>(G)) 
+			D = ex_to<matrix>(G).rows(); 
+		else throw(std::invalid_argument("metric should be an indexed object, matrix, or a Clifford unit"));
+		
+		varidx mu((new symbol)->setflag(status_flags::dynallocated), D);
+		cu = clifford_unit(mu, G, rl);
+	}
+
+	x = lst_to_clifford(v, cu); 
+	ex e = simplify_indexed(canonicalize_clifford((a * x + b) * clifford_inverse(c * x + d)));
+	return clifford_to_lst(e, cu, false);
+}
+
+ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl)
+{
+	if (is_a<matrix>(M)) 
+		return clifford_moebius_map(ex_to<matrix>(M)(0,0), ex_to<matrix>(M)(0,1), 
+						ex_to<matrix>(M)(1,0), ex_to<matrix>(M)(1,1), v, G, rl);
+	else
+		throw(std::invalid_argument("parameter M should be a matrix"));
 }
 
 } // namespace GiNaC