X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fclifford.cpp;h=ba5d33986a17eb3755a2c72aec33f24ac6429504;hp=8ab70652551a562ef2ccdba5591223645d68781b;hb=56f1beacac549ea8a66d46bffc6e0b3d8047d5c5;hpb=aac27f42467a94f0688964d5fc7db0c23d2486c0

diff --git a/ginac/clifford.cpp b/ginac/clifford.cpp
index 8ab70652..ba5d3398 100644
--- a/ginac/clifford.cpp
+++ b/ginac/clifford.cpp
@@ -1,10 +1,9 @@
 /** @file clifford.cpp
  *
- *  Implementation of GiNaC's clifford objects.
- *  No real implementation yet, to be done.     */
+ *  Implementation of GiNaC's clifford algebra (Dirac gamma) objects. */
 
 /*
- *  GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
+ *  GiNaC Copyright (C) 1999-2004 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
@@ -21,228 +20,985 @@
  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  */
 
-#include <string>
-
 #include "clifford.h"
+
 #include "ex.h"
+#include "idx.h"
 #include "ncmul.h"
+#include "symbol.h"
+#include "numeric.h" // for I
+#include "symmetry.h"
+#include "lst.h"
+#include "relational.h"
+#include "operators.h"
+#include "add.h"
+#include "mul.h"
+#include "power.h"
+#include "matrix.h"
 #include "archive.h"
 #include "utils.h"
-#include "debugmsg.h"
 
-#ifndef NO_NAMESPACE_GINAC
 namespace GiNaC {
-#endif // ndef NO_NAMESPACE_GINAC
 
-GINAC_IMPLEMENT_REGISTERED_CLASS(clifford, lortensor)
+GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(clifford, indexed,
+  print_func<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
 //////////
 
-// public
+static ex default_metric()
+{
+	static ex m = (new minkmetric)->setflag(status_flags::dynallocated);
+	return m;
+}
 
-clifford::clifford()
+clifford::clifford() : representation_label(0), metric(default_metric())
 {
-	debugmsg("clifford default constructor",LOGLEVEL_CONSTRUCT);
 	tinfo_key = TINFO_clifford;
 }
 
-clifford::~clifford()
+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)
 {
-	debugmsg("clifford destructor",LOGLEVEL_DESTRUCT);
-	destroy(false);
+	tinfo_key = TINFO_clifford;
 }
 
-clifford::clifford(const clifford & other)
+/** 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) : inherited(b, mu), representation_label(rl), metric(metr)
 {
-	debugmsg("clifford copy constructor",LOGLEVEL_CONSTRUCT);
-	copy(other);
+	GINAC_ASSERT(is_a<varidx>(mu));
+	tinfo_key = TINFO_clifford;
 }
 
-const clifford & clifford::operator=(const clifford & other)
+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 operator=",LOGLEVEL_ASSIGNMENT);
-	if (this != &other) {
-		destroy(true);
-		copy(other);
-	}
-	return *this;
+	tinfo_key = TINFO_clifford;
 }
 
-// protected
+clifford::clifford(unsigned char rl, const ex & metr, std::auto_ptr<exvector> vp) : inherited(not_symmetric(), vp), representation_label(rl), metric(metr)
+{
+	tinfo_key = TINFO_clifford;
+}
 
-void clifford::copy(const clifford & other)
+//////////
+// archiving
+//////////
+
+clifford::clifford(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
 {
-	inherited::copy(other);
+	unsigned rl;
+	n.find_unsigned("label", rl);
+	representation_label = rl;
+	n.find_ex("metric", metric, sym_lst);
 }
 
-void clifford::destroy(bool call_parent)
+void clifford::archive(archive_node &n) const
 {
-	if (call_parent) inherited::destroy(call_parent);
+	inherited::archive(n);
+	n.add_unsigned("label", representation_label);
+	n.add_ex("metric", metric);
 }
 
+DEFAULT_UNARCHIVE(clifford)
+DEFAULT_ARCHIVING(diracone)
+DEFAULT_ARCHIVING(cliffordunit)
+DEFAULT_ARCHIVING(diracgamma)
+DEFAULT_ARCHIVING(diracgamma5)
+DEFAULT_ARCHIVING(diracgammaL)
+DEFAULT_ARCHIVING(diracgammaR)
+
 //////////
-// other constructors
+// functions overriding virtual functions from base classes
 //////////
 
-// public
+ex clifford::get_metric(const ex & i, const ex & j) const
+{
+	return indexed(metric, symmetric2(), i, j);
+}
 
-clifford::clifford(const std::string & n, const ex & mu) : inherited(lortensor_symbolic, n, mu)
+bool clifford::same_metric(const ex & other) const
 {
-	debugmsg("clifford constructor from string,ex",LOGLEVEL_CONSTRUCT);
-	tinfo_key=TINFO_clifford;
+	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;
 }
 
-clifford::clifford(const std::string & n, const exvector & iv) : inherited(lortensor_symbolic, n, iv)
+int clifford::compare_same_type(const basic & other) const
 {
-	debugmsg("clifford constructor from string,exvector", LOGLEVEL_CONSTRUCT);
-	GINAC_ASSERT(all_of_type_lorentzidx());
-	tinfo_key=TINFO_clifford;
+	GINAC_ASSERT(is_a<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);
 }
 
-clifford::clifford(const std::string & n, exvector *ivp) : inherited(lortensor_symbolic, n, *ivp)
+bool clifford::match_same_type(const basic & other) const
 {
-	debugmsg("clifford constructor from string,exvector", LOGLEVEL_CONSTRUCT);
-	GINAC_ASSERT(all_of_type_lorentzidx());
-	tinfo_key=TINFO_clifford;
+	GINAC_ASSERT(is_a<clifford>(other));
+	const clifford &o = static_cast<const clifford &>(other);
+
+	return (representation_label == o.representation_label) && same_metric(o);
 }
 
-//////////
-// archiving
-//////////
+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);
+}
 
-/** Construct object from archive_node. */
-clifford::clifford(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
+void clifford::do_print_dflt(const print_dflt & c, unsigned level) const
 {
-	debugmsg("clifford constructor from archive_node", LOGLEVEL_CONSTRUCT);
+	// dirac_slash() object is printed differently
+	if (is_dirac_slash(seq[0])) {
+		seq[0].print(c, level);
+		c.s << "\\";
+	} else
+		this->print_dispatch<inherited>(c, level);
 }
 
-/** Unarchive the object. */
-ex clifford::unarchive(const archive_node &n, const lst &sym_lst)
+void clifford::do_print_latex(const print_latex & c, unsigned level) const
 {
-	return (new clifford(n, sym_lst))->setflag(status_flags::dynallocated);
+	// dirac_slash() object is printed differently
+	if (is_dirac_slash(seq[0])) {
+		c.s << "{";
+		seq[0].print(c, level);
+		c.s << "\\hspace{-1.0ex}/}";
+	} else
+		this->print_dispatch<inherited>(c, level);
 }
 
-/** Archive the object. */
-void clifford::archive(archive_node &n) const
+DEFAULT_COMPARE(diracone)
+DEFAULT_COMPARE(cliffordunit)
+DEFAULT_COMPARE(diracgamma)
+DEFAULT_COMPARE(diracgamma5)
+DEFAULT_COMPARE(diracgammaL)
+DEFAULT_COMPARE(diracgammaR)
+
+DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
+DEFAULT_PRINT_LATEX(cliffordunit, "e", "e")
+DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
+DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
+DEFAULT_PRINT_LATEX(diracgammaL, "gammaL", "{\\gamma_L}")
+DEFAULT_PRINT_LATEX(diracgammaR, "gammaR", "{\\gamma_R}")
+
+/** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
+static void base_and_index(const ex & c, ex & b, ex & i)
 {
-	inherited::archive(n);
+	GINAC_ASSERT(is_a<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;
+	}
 }
 
-//////////
-// functions overriding virtual functions from bases classes
-//////////
+/** 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 represenation labels are equal
+		if (ex_to<clifford>(*other).get_representation_label() != rl)
+			return false;
+
+		// gamma~mu gamma.mu = dim ONE
+		if (other - self == 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_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 (other - self == 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 (other - self == 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 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 {
+			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;
+			}
+
+			*self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
+			*next_to_last = _ex1;
+			*other = _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;
+	}
 
-// public
+	return false;
+}
 
-basic * clifford::duplicate() const
+/** 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)
 {
-	debugmsg("clifford duplicate",LOGLEVEL_DUPLICATE);
-	return new clifford(*this);
+	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
 }
 
-void clifford::printraw(std::ostream & os) const
+/** Contraction of a Clifford unit with something else. */
+bool cliffordunit::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 {
-	debugmsg("clifford printraw",LOGLEVEL_PRINT);
-	os << "clifford(" << "indices=";
-	printrawindices(os);
-	os << ",hash=" << hashvalue << ",flags=" << flags << ")";
+	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>(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;
 }
 
-void clifford::printtree(std::ostream & os, unsigned indent) const
+/** 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
 {
-	debugmsg("clifford printtree",LOGLEVEL_PRINT);
-	os << std::string(indent,' ') << " (clifford): "
-	   << seq.size() << "indices=";
-	printtreeindices(os, indent);
-	os << ", hash=" << hashvalue
-	   << " (0x" << std::hex << hashvalue << std::dec << ")"
-	   << ", flags=" << flags << std::endl;
+	exvector s;
+	s.reserve(v.size());
+
+	// Remove superfluous ONEs
+	exvector::const_iterator cit = v.begin(), citend = v.end();
+	while (cit != citend) {
+		if (!is_a<clifford>(*cit) || !is_a<diracone>(cit->op(0)))
+			s.push_back(*cit);
+		cit++;
+	}
+
+	bool something_changed = false;
+	int sign = 1;
+
+	// Anticommute gamma5/L/R's to the front
+	if (s.size() >= 2) {
+		exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
+		while (true) {
+			exvector::iterator it = next_to_last;
+			while (true) {
+				exvector::iterator it2 = it + 1;
+				if (is_a<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)) {
+
+				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);
+					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;
+			}
+		}
+	}
+
+	if (s.empty())
+		return clifford(diracone(), representation_label) * sign;
+	if (something_changed)
+		return reeval_ncmul(s) * sign;
+	else
+		return hold_ncmul(s) * sign;
 }
 
-void clifford::print(std::ostream & os, unsigned upper_precedence) const
+ex clifford::thiscontainer(const exvector & v) const
 {
-	debugmsg("clifford print",LOGLEVEL_PRINT);
-	os << name;
-	printindices(os);
+	return clifford(representation_label, get_metric(), v);
 }
 
-bool clifford::info(unsigned inf) const
+ex clifford::thiscontainer(std::auto_ptr<exvector> vp) const
 {
-	return inherited::info(inf);
+	return clifford(representation_label, get_metric(), vp);
 }
 
-// protected
+ex diracgamma5::conjugate() const
+{	
+	return _ex_1 * (*this);
+}
 
-int clifford::compare_same_type(const basic & other) const
+ex diracgammaL::conjugate() const
 {
-	GINAC_ASSERT(is_of_type(other,clifford));
-	// only compare indices
-	return exprseq::compare_same_type(other);
+	return (new diracgammaR)->setflag(status_flags::dynallocated);
 }
 
-bool clifford::is_equal_same_type(const basic & other) const
+ex diracgammaR::conjugate() const
 {
-	GINAC_ASSERT(is_of_type(other,clifford));
-	// only compare indices
-	return exprseq::is_equal_same_type(other);
+	return (new diracgammaL)->setflag(status_flags::dynallocated);
 }
 
-ex clifford::thisexprseq(const exvector & v) const
+//////////
+// global functions
+//////////
+
+ex dirac_ONE(unsigned char rl)
 {
-	return clifford(name, v);
+	static ex ONE = (new diracone)->setflag(status_flags::dynallocated);
+	return clifford(ONE, rl);
 }
 
-ex clifford::thisexprseq(exvector *vp) const
+ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl)
 {
-	return clifford(name, vp);
+	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"));
+
+	return clifford(unit, mu, metr, rl);
 }
 
-ex clifford::simplify_ncmul(const exvector & v) const
+ex dirac_gamma(const ex & mu, unsigned char rl)
 {
-	return simplified_ncmul(v);
+	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(gamma, mu, default_metric(), rl);
 }
 
-//////////
-// virtual functions which can be overridden by derived classes
-//////////
+ex dirac_gamma5(unsigned char rl)
+{
+	static ex gamma5 = (new diracgamma5)->setflag(status_flags::dynallocated);
+	return clifford(gamma5, rl);
+}
 
-// none
+ex dirac_gammaL(unsigned char rl)
+{
+	static ex gammaL = (new diracgammaL)->setflag(status_flags::dynallocated);
+	return clifford(gammaL, rl);
+}
 
-//////////
-// non-virtual functions in this class
-//////////
+ex dirac_gammaR(unsigned char rl)
+{
+	static ex gammaR = (new diracgammaR)->setflag(status_flags::dynallocated);
+	return clifford(gammaR, rl);
+}
 
-// none
+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
+	return clifford(e, varidx(0, dim), rl);
+}
 
-//////////
-// static member variables
-//////////
+/** Check whether a given tinfo key (as returned by return_type_tinfo()
+ *  is that of a clifford object with the specified representation label. */
+static bool is_clifford_tinfo(unsigned ti, unsigned char rl)
+{
+	return ti == (TINFO_clifford + rl);
+}
 
-// none
+/** Check whether a given tinfo key (as returned by return_type_tinfo()
+ *  is that of a clifford object (with an arbitrary representation label). */
+static bool is_clifford_tinfo(unsigned ti)
+{
+	return (ti & ~0xff) == TINFO_clifford;
+}
 
-//////////
-// friend functions
-//////////
+/** 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;
+}
 
-/** Construct an object representing a Dirac gamma matrix. The index must
- *  be of class lorentzidx.
- *
- *  @param mu Index
- *  @return newly constructed object */
-clifford clifford_gamma(const ex & mu)
+ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
 {
-	return clifford("gamma", mu);
+	if (is_a<clifford>(e)) {
+
+		if (!ex_to<clifford>(e).get_representation_label() == rl)
+			return _ex0;
+		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(), rl))
+				prod *= dirac_trace(o, rl, trONE);
+			else
+				prod *= o;
+		}
+		return prod;
+
+	} else if (is_exactly_a<ncmul>(e)) {
+
+		if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
+			return _ex0;
+
+		// 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, rl, 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<unsigned char, const ex &> fcn(dirac_trace, rl, trONE);
+		return e.map(fcn);
+
+	} else
+		return _ex0;
 }
 
-//////////
-// global constants
-//////////
+ex canonicalize_clifford(const ex & 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;
+			}
+next_sym:	;
+		}
+	}
+	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)) {
+		return e.map(fcn);
+	} else if (is_a<ncmul>(e)) {
+		return e.map(fcn);
+	} else if (is_a<power>(e)) {
+		return pow(clifford_prime(e.op(0)), e.op(1));
+	} else
+		return e;
+}
 
-const clifford some_clifford;
-const std::type_info & typeid_clifford = typeid(some_clifford);
+ex delete_ONE(const ex &e)
+{
+	pointer_to_map_function fcn(delete_ONE);
+	if (is_a<clifford>(e) && is_a<diracone>(e.op(0))) {
+		return 1;
+	} else if (is_a<add>(e)) {
+		return e.map(fcn);
+	} else if (is_a<ncmul>(e)) {
+		return e.map(fcn);
+	} else if (is_a<mul>(e)) {
+		return e.map(fcn);
+	} else if (is_a<power>(e)) {
+		return pow(delete_ONE(e.op(0)), e.op(1));
+	} else
+		return e;
+}
 
-#ifndef NO_NAMESPACE_GINAC
+ex clifford_norm(const ex &e)
+{
+	return sqrt(delete_ONE((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);
+}
+
+ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl)
+{
+	unsigned min, max;
+	if (!ex_to<idx>(mu).is_dim_numeric())
+		throw(std::invalid_argument("Index should have a numeric dimension"));
+	unsigned dim = (ex_to<numeric>(ex_to<idx>(mu).get_dim())).to_int();
+	ex c = clifford_unit(mu, metr, rl);
+
+	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)
+				if (is_a<varidx>(mu)) // need to swap variance
+					return indexed(v, ex_to<varidx>(mu).toggle_variance()) * c;
+				else
+					return indexed(v, mu) * c;
+			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()) * c;
+		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"));
+}
+ 
 } // namespace GiNaC
-#endif // ndef NO_NAMESPACE_GINAC