X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fclifford.cpp;h=86678ecc6f342e881b3074c524258160287b8917;hp=32096410218b76c5cd4c3e179f902faa12b5920e;hb=9254356cc8381a5c189eb48cd40f316916453172;hpb=6c9a7dfd6b92368b21a2fe5aaf3638bc849099ba

diff --git a/ginac/clifford.cpp b/ginac/clifford.cpp
index 32096410..86678ecc 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) 1999-2002 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
@@ -20,18 +20,23 @@
  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  */
 
+#include <iostream>
+#include <stdexcept>
+
 #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 "mul.h"
 #include "print.h"
 #include "archive.h"
-#include "debugmsg.h"
 #include "utils.h"
 
-#include <stdexcept>
-
 namespace GiNaC {
 
 GINAC_IMPLEMENT_REGISTERED_CLASS(clifford, indexed)
@@ -40,12 +45,11 @@ GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma, tensor)
 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma5, tensor)
 
 //////////
-// default constructor, destructor, copy constructor assignment operator and helpers
+// default ctor, dtor, copy ctor, assignment operator and helpers
 //////////
 
 clifford::clifford() : representation_label(0)
 {
-	debugmsg("clifford default constructor", LOGLEVEL_CONSTRUCT);
 	tinfo_key = TINFO_clifford;
 }
 
@@ -69,7 +73,6 @@ DEFAULT_CTORS(diracgamma5)
  *  @see dirac_ONE */
 clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
 {
-	debugmsg("clifford constructor from ex", LOGLEVEL_CONSTRUCT);
 	tinfo_key = TINFO_clifford;
 }
 
@@ -78,20 +81,17 @@ clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representatio
  *  @see dirac_gamma */
 clifford::clifford(const ex & b, const ex & mu, unsigned char rl) : inherited(b, mu), representation_label(rl)
 {
-	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 exvector & v, bool discardable) : inherited(sy_none(), v, discardable), representation_label(rl)
 {
-	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, exvector * vp) : inherited(sy_none(), vp), representation_label(rl)
 {
-	debugmsg("clifford constructor from unsigned char,exvector *", LOGLEVEL_CONSTRUCT);
 	tinfo_key = TINFO_clifford;
 }
 
@@ -101,7 +101,6 @@ clifford::clifford(unsigned char rl, exvector * vp) : inherited(indexed::unknown
 
 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;
@@ -119,12 +118,12 @@ DEFAULT_ARCHIVING(diracgamma)
 DEFAULT_ARCHIVING(diracgamma5)
 
 //////////
-// functions overriding virtual functions from bases classes
+// functions overriding virtual functions from base classes
 //////////
 
 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) {
@@ -135,6 +134,34 @@ 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;
+}
+
+void clifford::print(const print_context & c, unsigned level) const
+{
+	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);
+}
+
 DEFAULT_COMPARE(diracone)
 DEFAULT_COMPARE(diracgamma)
 DEFAULT_COMPARE(diracgamma5)
@@ -143,17 +170,40 @@ 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)
+{
+	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 if (is_a<diracgamma5>(c.op(0))) { // gamma5
+		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;
+	}
+}
+
 /** 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, clifford)) {
+	if (is_a<clifford>(*other)) {
 
-		ex dim = ex_to_idx(self->op(1)).get_dim();
+		// Contraction only makes sense if the represenation labels are equal
+		if (ex_to<clifford>(*other).get_representation_label() != rl)
+			return false;
+
+		ex dim = ex_to<idx>(self->op(1)).get_dim();
 
 		// gamma~mu gamma.mu = dim ONE
 		if (other - self == 1) {
@@ -163,19 +213,34 @@ bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other
 
 		// gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
 		} else if (other - self == 2
-		        && is_ex_of_type(self[1], clifford)) {
+		        && 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 * lorentz_g(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();
+		        && 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
@@ -184,21 +249,21 @@ bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other
 		} else {
 			exvector::iterator it = self + 1, next_to_last = other - 1;
 			while (it != other) {
-				if (!is_ex_of_type(*it, clifford))
+				if (!is_a<clifford>(*it))
 					return false;
-				it++;
+				++it;
 			}
 
 			it = self + 1;
-			ex S = _ex1();
+			ex S = _ex1;
 			while (it != next_to_last) {
 				S *= *it;
-				*it++ = _ex1();
+				*it++ = _ex1;
 			}
 
 			*self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
-			*next_to_last = _ex1();
-			*other = _ex1();
+			*next_to_last = _ex1;
+			*other = _ex1;
 			return true;
 		}
 	}
@@ -213,12 +278,11 @@ ex clifford::simplify_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++;
 	}
@@ -233,48 +297,59 @@ ex clifford::simplify_ncmul(const exvector & v) const
 			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)) {
+				if (is_a<clifford>(*it) && is_a<clifford>(*it2) && !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--;
+				--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;
+			bool a_is_diracgamma = is_a<diracgamma>(a.op(0));
+			bool b_is_diracgamma = is_a<diracgamma>(b.op(0));
+			if (a_is_diracgamma && b_is_diracgamma) {
 				const ex & ia = a.op(1);
 				const ex & ib = b.op(1);
-				if (ia.is_equal(ib)) {
+				if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
 					a = lorentz_g(ia, ib);
-					b = dirac_ONE(rl);
+					b = dirac_ONE(representation_label);
+					something_changed = true;
+				}
+			} else if (is_a<diracgamma5>(a.op(0)) && is_a<diracgamma5>(b.op(0))) {
+				// Remove squares of gamma5
+				a = dirac_ONE(representation_label);
+				b = dirac_ONE(representation_label);
+				something_changed = true;
+			} else if (!a_is_diracgamma && !b_is_diracgamma) {
+				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;
 				}
 			}
-			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;
 	else
@@ -302,7 +377,7 @@ ex dirac_ONE(unsigned char rl)
 
 ex dirac_gamma(const ex & mu, unsigned char rl)
 {
-	if (!is_ex_of_type(mu, varidx))
+	if (!is_a<varidx>(mu))
 		throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
 
 	return clifford(diracgamma(), mu, rl);
@@ -325,8 +400,10 @@ ex dirac_gamma7(unsigned char 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), rl);
 }
 
 /** Check whether a given tinfo key (as returned by return_type_tinfo()
@@ -336,33 +413,67 @@ 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();
+/** 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;
+}
 
-	} else if (is_ex_exactly_of_type(e, add)) {
+ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
+{
+	if (is_a<clifford>(e)) {
 
-		// 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;
+		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();
+		ex prod = _ex1;
 		for (unsigned 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);
+				prod *= dirac_trace(o, rl, trONE);
 			else
 				prod *= o;
 		}
@@ -371,15 +482,15 @@ ex dirac_trace(const ex & e, unsigned char rl)
 	} else if (is_ex_exactly_of_type(e, ncmul)) {
 
 		if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
-			return _ex0();
+			return _ex0;
 
 		// Expand product, if necessary
 		ex e_expanded = e.expand();
-		if (!is_ex_of_type(e_expanded, ncmul))
-			return dirac_trace(e_expanded, rl);
+		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_ex_of_type(e.op(0).op(0), diracgamma5);
+		bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
 		unsigned num = e.nops();
 
 		if (has_gamma5) {
@@ -387,62 +498,137 @@ ex dirac_trace(const ex & e, unsigned char rl)
 			// 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 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 _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();
+			}
 
-			return (eps0123(mu1.toggle_variance(), mu2.toggle_variance(), mu3.toggle_variance(), mu4.toggle_variance()) *
-			        dirac_trace(ncmul(v), rl)).simplify_indexed() / 24;
+	   		// 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();
+				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 =
-			//   + 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;
-			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);
+			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;
 				}
-				result += sign * lorentz_g(ix1, e.op(i).op(1)) * dirac_trace(ncmul(v), rl);
-				sign = -sign;
+				++it;
 			}
-			return result;
+next_sym:	;
 		}
 	}
-
-	return _ex0();
+	return aux.subs(srl).simplify_indexed();
 }
 
 } // namespace GiNaC