X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fclifford.cpp;h=f6a511c4a3c8f21002b58daf1adf2b3e269759e7;hp=8d41e92cb96603c84a3f77d31301a5a881a0686e;hb=9546e7613b679b28659b0774d2398ad92db26b61;hpb=2565309dd7c38635c191eacf2a4af9b23fc0d310;ds=inline

diff --git a/ginac/clifford.cpp b/ginac/clifford.cpp
index 8d41e92c..f6a511c4 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) 1999-2000 Johannes Gutenberg University Mainz, Germany
+ *  GiNaC Copyright (C) 1999-2001 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
@@ -21,198 +20,613 @@
  *  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 "utils.h"
+#include "symbol.h"
+#include "numeric.h" // for I
+#include "symmetry.h"
+#include "lst.h"
+#include "relational.h"
+#include "mul.h"
+#include "print.h"
+#include "archive.h"
 #include "debugmsg.h"
+#include "utils.h"
+
+#include <stdexcept>
 
-#ifndef NO_GINAC_NAMESPACE
 namespace GiNaC {
-#endif // ndef NO_GINAC_NAMESPACE
+
+GINAC_IMPLEMENT_REGISTERED_CLASS(clifford, indexed)
+GINAC_IMPLEMENT_REGISTERED_CLASS(diracone, tensor)
+GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma, tensor)
+GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma5, tensor)
 
 //////////
 // default constructor, destructor, copy constructor assignment operator and helpers
 //////////
 
-// public
-
-clifford::clifford()
+clifford::clifford() : representation_label(0)
 {
-    debugmsg("clifford default constructor",LOGLEVEL_CONSTRUCT);
-    serial=next_serial++;
-    name=autoname_prefix()+ToString(serial);
-    tinfo_key=TINFO_clifford;
+	debugmsg("clifford default constructor", LOGLEVEL_CONSTRUCT);
+	tinfo_key = TINFO_clifford;
 }
 
-clifford::~clifford()
+void clifford::copy(const clifford & other)
 {
-    debugmsg("clifford destructor",LOGLEVEL_DESTRUCT);
-    destroy(0);
+	inherited::copy(other);
+	representation_label = other.representation_label;
 }
 
-clifford::clifford(const clifford & other)
+DEFAULT_DESTROY(clifford)
+DEFAULT_CTORS(diracone)
+DEFAULT_CTORS(diracgamma)
+DEFAULT_CTORS(diracgamma5)
+
+//////////
+// 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)
 {
-    debugmsg("clifford copy constructor",LOGLEVEL_CONSTRUCT);
-    copy (other);
+	debugmsg("clifford constructor from ex", LOGLEVEL_CONSTRUCT);
+	tinfo_key = TINFO_clifford;
 }
 
-const clifford & clifford::operator=(const clifford & other)
+/** Construct object with one Lorentz index. This constructor is for internal
+ *  use only. Use the dirac_gamma() function instead.
+ *  @see dirac_gamma */
+clifford::clifford(const ex & b, const ex & mu, unsigned char rl) : inherited(b, mu), representation_label(rl)
 {
-    debugmsg("clifford operator=",LOGLEVEL_ASSIGNMENT);
-    if (this != &other) {
-        destroy(1);
-        copy(other);
-    }
-    return *this;
+	debugmsg("clifford constructor from ex,ex", LOGLEVEL_CONSTRUCT);
+	GINAC_ASSERT(is_a<varidx>(mu));
+	tinfo_key = TINFO_clifford;
 }
 
-// protected
-
-void clifford::copy(const clifford & other)
+clifford::clifford(unsigned char rl, const exvector & v, bool discardable) : inherited(sy_none(), v, discardable), representation_label(rl)
 {
-    indexed::copy(other);
-    name=other.name;
-    serial=other.serial;
+	debugmsg("clifford constructor from unsigned char,exvector", LOGLEVEL_CONSTRUCT);
+	tinfo_key = TINFO_clifford;
 }
 
-void clifford::destroy(bool call_parent)
+clifford::clifford(unsigned char rl, exvector * vp) : inherited(sy_none(), vp), representation_label(rl)
 {
-    if (call_parent) {
-        indexed::destroy(call_parent);
-    }
+	debugmsg("clifford constructor from unsigned char,exvector *", LOGLEVEL_CONSTRUCT);
+	tinfo_key = TINFO_clifford;
 }
 
 //////////
-// other constructors
+// archiving
 //////////
 
-// public
+clifford::clifford(const archive_node &n, const 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;
+}
 
-clifford::clifford(const string & initname)
+void clifford::archive(archive_node &n) const
 {
-    debugmsg("clifford constructor from string",LOGLEVEL_CONSTRUCT);
-    name=initname;
-    serial=next_serial++;
-    tinfo_key=TINFO_clifford;
+	inherited::archive(n);
+	n.add_unsigned("label", representation_label);
 }
 
+DEFAULT_UNARCHIVE(clifford)
+DEFAULT_ARCHIVING(diracone)
+DEFAULT_ARCHIVING(diracgamma)
+DEFAULT_ARCHIVING(diracgamma5)
+
 //////////
-// functions overriding virtual functions from bases classes
+// functions overriding virtual functions from base classes
 //////////
 
-// public
-
-basic * clifford::duplicate() const
+int clifford::compare_same_type(const basic & other) const
 {
-    debugmsg("clifford duplicate",LOGLEVEL_DUPLICATE);
-    return new clifford(*this);
-}
+	GINAC_ASSERT(is_of_type(other, clifford));
+	const clifford &o = static_cast<const clifford &>(other);
 
-void clifford::printraw(ostream & os) const
-{
-    debugmsg("clifford printraw",LOGLEVEL_PRINT);
-    os << "clifford(" << "name=" << name << ",serial=" << serial
-       << ",indices=";
-    printrawindices(os);
-    os << ",hash=" << hashvalue << ",flags=" << flags << ")";
+	if (representation_label != o.representation_label) {
+		// different representation label
+		return representation_label < o.representation_label ? -1 : 1;
+	}
+
+	return inherited::compare_same_type(other);
 }
 
-void clifford::printtree(ostream & os, unsigned indent) const
+bool clifford::match_same_type(const basic & other) const
 {
-    debugmsg("clifford printtree",LOGLEVEL_PRINT);
-    os << string(indent,' ') << name << " (clifford): "
-       << "serial=" << serial << ","
-       << seq.size() << "indices=";
-    printtreeindices(os,indent);
-    os << ", hash=" << hashvalue << " (0x" << hex << hashvalue << dec << ")"
-       << ", flags=" << flags << endl;
+	GINAC_ASSERT(is_of_type(other, clifford));
+	const clifford &o = static_cast<const clifford &>(other);
+
+	return representation_label == o.representation_label;
 }
 
-void clifford::print(ostream & os, unsigned upper_precedence) const
+void clifford::print(const print_context & c, unsigned level = 0) const
 {
-    debugmsg("clifford print",LOGLEVEL_PRINT);
-    os << name;
-    printindices(os);
+	if (!is_a<diracgamma5>(seq[0]) && !is_a<diracgamma>(seq[0]) && !is_a<diracone>(seq[0])) {
+
+		// dirac_slash() object is printed differently
+		if (is_a<print_tree>(c))
+			inherited::print(c, level);
+		else if (is_a<print_latex>(c)) {
+			c.s << "{";
+			seq[0].print(c, level);
+			c.s << "\\hspace{-1.0ex}/}";
+		} else {
+			seq[0].print(c, level);
+			c.s << "\\";
+		}
+
+	} else
+		inherited::print(c, level);
 }
 
-void clifford::printcsrc(ostream & os, unsigned type, unsigned upper_precedence) const
+DEFAULT_COMPARE(diracone)
+DEFAULT_COMPARE(diracgamma)
+DEFAULT_COMPARE(diracgamma5)
+
+DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
+DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
+DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
+
+/** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
+static void base_and_index(const ex & c, ex & b, ex & i)
 {
-    debugmsg("clifford print csrc",LOGLEVEL_PRINT);
-    print(os,upper_precedence);
+	GINAC_ASSERT(is_a<clifford>(c));
+	GINAC_ASSERT(c.nops() == 2);
+
+	if (is_a<diracgamma>(c.op(0))) { // proper dirac gamma object
+		i = c.op(1);
+		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;
+	}
 }
 
-bool clifford::info(unsigned inf) const
+/** Contraction of a gamma matrix with something else. */
+bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 {
-    return indexed::info(inf);
+	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_a<clifford>(*other)) {
+
+		ex dim = ex_to<idx>(self->op(1)).get_dim();
+
+		// 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;
+		}
+	}
+
+	return false;
 }
 
-// protected
-
-int clifford::compare_same_type(const basic & other) const
+/** Perform automatic simplification on noncommutative product of clifford
+ *  objects. This removes superfluous ONEs, permutes gamma5's to the front
+ *  and removes squares of gamma objects. */
+ex clifford::simplify_ncmul(const exvector & v) const
 {
-    GINAC_ASSERT(other.tinfo() == TINFO_clifford);
-    const clifford *o = static_cast<const clifford *>(&other);
-    if (serial==o->serial) {
-        return indexed::compare_same_type(other);
-    }
-    return serial < o->serial ? -1 : 1;
+	exvector s;
+	s.reserve(v.size());
+
+	// Remove superfluous ONEs
+	exvector::const_iterator cit = v.begin(), citend = v.end();
+	while (cit != citend) {
+		if (!is_a<diracone>(cit->op(0)))
+			s.push_back(*cit);
+		cit++;
+	}
+
+	bool something_changed = false;
+	int sign = 1;
+
+	// Anticommute gamma5'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<diracgamma5>(it->op(0)) && is_a<diracgamma5>(it2->op(0))) {
+					it->swap(*it2);
+					sign = -sign;
+					something_changed = true;
+				}
+				if (it == first)
+					break;
+				--it;
+			}
+			if (next_to_last == first)
+				break;
+			--next_to_last;
+		}
+	}
+
+	// Remove squares of gamma5
+	while (s.size() >= 2 && is_a<diracgamma5>(s[0].op(0)) && is_a<diracgamma5>(s[1].op(0))) {
+		s.erase(s.begin(), s.begin() + 2);
+		something_changed = true;
+	}
+
+	// Remove equal adjacent gammas
+	if (s.size() >= 2) {
+		exvector::iterator it, itend = s.end() - 1;
+		for (it = s.begin(); it != itend; ++it) {
+			if (!is_a<clifford>(it[0]))
+				continue;
+			ex & a = it[0];
+			ex & b = it[1];
+			if (is_a<diracgamma>(a.op(0)) && is_a<diracgamma>(b.op(0))) {
+				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 = lorentz_g(ia, ib);
+					b = dirac_ONE(representation_label);
+					something_changed = true;
+				}
+			} else if (!is_a<diracgamma>(a.op(0)) && !is_a<diracgamma>(b.op(0))) {
+				const ex & ba = a.op(0);
+				const ex & bb = b.op(0);
+				if (ba.is_equal(bb)) { // a\ a\ -> a^2
+					varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).get_dim());
+					a = indexed(ba, ix) * indexed(bb, ix.toggle_variance());
+					b = dirac_ONE(representation_label);
+					something_changed = true;
+				}
+			}
+		}
+	}
+
+	if (s.empty())
+		return clifford(diracone(), representation_label) * sign;
+	if (something_changed)
+		return nonsimplified_ncmul(s) * sign;
+	else
+		return simplified_ncmul(s) * sign;
 }
 
-ex clifford::simplify_ncmul(const exvector & v) const
+ex clifford::thisexprseq(const exvector & v) const
 {
-    return simplified_ncmul(v);
+	return clifford(representation_label, v);
 }
 
-unsigned clifford::calchash(void) const
+ex clifford::thisexprseq(exvector * vp) const
 {
-    hashvalue=golden_ratio_hash(golden_ratio_hash(0x55555556U ^
-                                                  golden_ratio_hash(tinfo_key) ^
-                                                  serial));
-    setflag(status_flags::hash_calculated);
-    return hashvalue;
+	return clifford(representation_label, vp);
 }
 
 //////////
-// virtual functions which can be overridden by derived classes
+// global functions
 //////////
 
-// none
+ex dirac_ONE(unsigned char rl)
+{
+	return clifford(diracone(), rl);
+}
 
-//////////
-// non-virtual functions in this class
-//////////
+ex dirac_gamma(const ex & mu, unsigned char rl)
+{
+	if (!is_a<varidx>(mu))
+		throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
 
-void clifford::setname(const string & n)
+	return clifford(diracgamma(), mu, rl);
+}
+
+ex dirac_gamma5(unsigned char rl)
 {
-    name=n;
+	return clifford(diracgamma5(), rl);
 }
 
-// private
+ex dirac_gamma6(unsigned char rl)
+{
+	return clifford(diracone(), rl) + clifford(diracgamma5(), rl);
+}
 
-string & clifford::autoname_prefix(void)
+ex dirac_gamma7(unsigned char rl)
 {
-    static string * s=new string("clifford");
-    return *s;
+	return clifford(diracone(), rl) - clifford(diracgamma5(), rl);
 }
 
-//////////
-// static member variables
-//////////
+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);
+}
 
-// private
+/** 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);
+}
 
-unsigned clifford::next_serial=0;
+/** 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;
+}
 
-//////////
-// global constants
-//////////
+/** 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, unsigned 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 (unsigned i=1; i<num; i++) {
+		for (unsigned 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;
+}
 
-const clifford some_clifford;
-const type_info & typeid_clifford=typeid(some_clifford);
+ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
+{
+	if (is_a<clifford>(e)) {
+
+		if (ex_to<clifford>(e).get_representation_label() == rl
+		 && is_a<diracone>(e.op(0)))
+			return trONE;
+		else
+			return _ex0();
+
+	} else if (is_ex_exactly_of_type(e, mul)) {
+
+		// Trace of product: pull out non-clifford factors
+		ex prod = _ex1();
+		for (unsigned 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_ex_exactly_of_type(e, ncmul)) {
+
+		if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
+			return _ex0();
+
+		// Expand product, if necessary
+		ex e_expanded = e.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));
+		unsigned 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)
+			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 * (eps0123(i1, i2, i3, i4) * 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
+			exvector ix(num-1), bv(num-1);
+			for (unsigned 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 (unsigned i=0; i<num-3; i++) {
+				ex idx1 = ix[i];
+				for (unsigned j=i+1; j<num-2; j++) {
+					ex idx2 = ix[j];
+					for (unsigned k=j+1; k<num-1; k++) {
+						ex idx3 = ix[k];
+						for (unsigned 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 (unsigned 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 * eps0123(idx1, idx2, idx3, idx4)
+							        * 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 (unsigned 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();
+}
+
+ex canonicalize_clifford(const ex & e)
+{
+	// Scan for any ncmul objects
+	lst srl;
+	ex aux = e.to_rational(srl);
+	for (unsigned i=0; i<srl.nops(); i++) {
+
+		ex lhs = srl.op(i).lhs();
+		ex rhs = srl.op(i).rhs();
+
+		if (is_ex_exactly_of_type(rhs, ncmul)
+		 && 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)) {
+				srl.let_op(i) = (lhs == canonicalize_clifford(rhs_expanded));
+				continue;
+
+			} else if (!is_a<clifford>(rhs.op(0)))
+				continue;
+
+			exvector v;
+			v.reserve(rhs.nops());
+			for (unsigned 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)))
+				++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] = (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
+					it[1] = _ex2();
+					ex sum = ncmul(v);
+					it[0] = save1;
+					it[1] = save0;
+					sum -= ncmul(v, true);
+					srl.let_op(i) = (lhs == canonicalize_clifford(sum));
+					goto next_sym;
+				}
+				++it;
+			}
+next_sym:	;
+		}
+	}
+	return aux.subs(srl).simplify_indexed();
+}
 
-#ifndef NO_GINAC_NAMESPACE
 } // namespace GiNaC
-#endif // ndef NO_GINAC_NAMESPACE