- powers with negative exponents are printed as fractions in the LaTeX output
[ginac.git] / ginac / clifford.cpp
index eb1629d..0a10d45 100644 (file)
@@ -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
  *  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)
 GINAC_IMPLEMENT_REGISTERED_CLASS(diracone, tensor)
 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma, tensor)
 GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma5, tensor)
+GINAC_IMPLEMENT_REGISTERED_CLASS(diracgammaL, tensor)
+GINAC_IMPLEMENT_REGISTERED_CLASS(diracgammaR, 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;
 }
 
@@ -58,6 +65,8 @@ DEFAULT_DESTROY(clifford)
 DEFAULT_CTORS(diracone)
 DEFAULT_CTORS(diracgamma)
 DEFAULT_CTORS(diracgamma5)
+DEFAULT_CTORS(diracgammaL)
+DEFAULT_CTORS(diracgammaR)
 
 //////////
 // other constructors
@@ -68,7 +77,6 @@ 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;
 }
 
@@ -77,20 +85,17 @@ 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;
 }
 
@@ -100,7 +105,6 @@ 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;
@@ -116,14 +120,16 @@ DEFAULT_UNARCHIVE(clifford)
 DEFAULT_ARCHIVING(diracone)
 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
 //////////
 
 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) {
@@ -134,59 +140,142 @@ 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<diracgammaL>(seq[0]) &&
+           !is_a<diracgammaR>(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)
+DEFAULT_COMPARE(diracgammaL)
+DEFAULT_COMPARE(diracgammaR)
+
+DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
+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}")
 
-DEFAULT_PRINT(diracone, "ONE")
-DEFAULT_PRINT(diracgamma, "gamma")
-DEFAULT_PRINT(diracgamma5, "gamma5")
+/** 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)) || 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;
+       }
+}
 
 /** 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_a<clifford>(*other)) {
 
-       if (is_ex_of_type(other->op(0), diracgamma)) {
+               // 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();
+               ex dim = ex_to<idx>(self->op(1)).get_dim();
 
-               // gamma~mu*gamma.mu = dim*ONE
+               // 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
+               // 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
+               // 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();
+                       && 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
+               // 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_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();
+                       && 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;
                }
        }
@@ -195,18 +284,17 @@ bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other
 }
 
 /** 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
 {
        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++;
        }
@@ -214,55 +302,147 @@ ex clifford::simplify_ncmul(const exvector & v) const
        bool something_changed = false;
        int sign = 1;
 
-       // Anticommute gamma5's to the front
+       // 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_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_diracgamma = is_a<diracgamma>(ag);
+                       bool b_is_diracgamma = is_a<diracgamma>(bg);
+
+                       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>(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_diracgamma && !b_is_diracgamma && ag.is_equal(bg)) {
+
+                               // a\ a\ -> a^2
+                               varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).get_dim());
+                               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;
        else
@@ -290,7 +470,7 @@ 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);
@@ -301,37 +481,106 @@ ex dirac_gamma5(unsigned char rl)
        return clifford(diracgamma5(), rl);
 }
 
-ex dirac_trace(const ex & e, unsigned char rl = 0)
+ex dirac_gammaL(unsigned char rl)
 {
-       if (is_ex_of_type(e, clifford)) {
+       return clifford(diracgammaL(), rl);
+}
 
-               if (ex_to_clifford(e).get_representation_label() == rl
-                && is_ex_of_type(e.op(0), diracone))
-                       return _ex4();
-               else
-                       return _ex0();
+ex dirac_gammaR(unsigned char rl)
+{
+       return clifford(diracgammaR(), rl);
+}
+
+ex dirac_gamma6(unsigned char rl)
+{
+       return clifford(diracone(), rl) + clifford(diracgamma5(), rl);
+}
+
+ex dirac_gamma7(unsigned char rl)
+{
+       return clifford(diracone(), rl) - clifford(diracgamma5(), rl);
+}
+
+ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
+{
+       // Slashed vectors are actually stored as a clifford object with the
+       // vector as its base expression and a (dummy) index that just serves
+       // for storing the space dimensionality
+       return clifford(e, varidx(0, dim), rl);
+}
+
+/** 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);
+}
+
+/** 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;
+}
 
-       } else if (is_ex_exactly_of_type(e, add)) {
+/** 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;
+}
 
-               // 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;
+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)
+                       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_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);
-                       if (is_ex_of_type(o, clifford)
-                        && ex_to_clifford(o).get_representation_label() == rl)
-                               prod *= dirac_trace(o, rl);
-                       else if (is_ex_of_type(o, ncmul)
-                        && is_ex_of_type(o.op(0), clifford)
-                        && ex_to_clifford(o.op(0)).get_representation_label() == rl)
-                               prod *= dirac_trace(o, rl);
+                       if (is_clifford_tinfo(o.return_type_tinfo(), rl))
+                               prod *= dirac_trace(o, rl, trONE);
                        else
                                prod *= o;
                }
@@ -339,36 +588,159 @@ ex dirac_trace(const ex & e, unsigned char rl = 0)
 
        } else if (is_ex_exactly_of_type(e, ncmul)) {
 
-               if (!is_ex_of_type(e.op(0), clifford)
-                || ex_to_clifford(e.op(0)).get_representation_label() != rl)
-                       return _ex0();
+               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
+               )).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_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) {
 
                        // Trace of gamma5 * odd number of gammas and trace of
-                       // gamma5 * gamma_mu * gamma_nu are zero
-                       if ((num & 1) == 0 || num == 2)
-                               return _ex0();
+                       // 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 (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 * 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();
+                               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]);
 
-                       // 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));
+                       return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
                }
 
-               throw (std::logic_error("dirac_trace: don't know how to compute trace"));
-       }
+       } 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;
+}
 
-       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();
 }
 
 } // namespace GiNaC