Prettified source code.
[ginac.git] / ginac / clifford.cpp
index 5c7f7c3..3b4684c 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) 1999-2009 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
  *
  *  You should have received a copy of the GNU General Public License
  *  along with this program; if not, write to the Free Software
- *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
+ *  Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
  */
 
 #include "clifford.h"
+
 #include "ex.h"
 #include "idx.h"
 #include "ncmul.h"
 #include "symbol.h"
 #include "numeric.h" // for I
-#include "print.h"
+#include "symmetry.h"
+#include "lst.h"
+#include "relational.h"
+#include "operators.h"
+#include "add.h"
+#include "mul.h"
+#include "power.h"
+#include "matrix.h"
 #include "archive.h"
-#include "debugmsg.h"
 #include "utils.h"
 
 #include <stdexcept>
 
 namespace GiNaC {
 
-GINAC_IMPLEMENT_REGISTERED_CLASS(clifford, indexed)
-GINAC_IMPLEMENT_REGISTERED_CLASS(diracone, tensor)
-GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma, tensor)
-GINAC_IMPLEMENT_REGISTERED_CLASS(diracgamma5, tensor)
+GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(clifford, indexed,
+  print_func<print_dflt>(&clifford::do_print_dflt).
+  print_func<print_latex>(&clifford::do_print_latex))
+
+GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracone, tensor,
+  print_func<print_dflt>(&diracone::do_print).
+  print_func<print_latex>(&diracone::do_print_latex))
+
+GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(cliffordunit, tensor,
+  print_func<print_dflt>(&cliffordunit::do_print).
+  print_func<print_latex>(&cliffordunit::do_print_latex))
+
+GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma, cliffordunit,
+  print_func<print_dflt>(&diracgamma::do_print).
+  print_func<print_latex>(&diracgamma::do_print_latex))
+
+GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgamma5, tensor,
+  print_func<print_dflt>(&diracgamma5::do_print).
+  print_func<print_latex>(&diracgamma5::do_print_latex))
+
+GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaL, tensor,
+  print_func<print_context>(&diracgammaL::do_print).
+  print_func<print_latex>(&diracgammaL::do_print_latex))
+
+GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaR, tensor,
+  print_func<print_context>(&diracgammaR::do_print).
+  print_func<print_latex>(&diracgammaR::do_print_latex))
 
 //////////
-// default constructor, destructor, copy constructor assignment operator and helpers
+// default constructors
 //////////
 
-clifford::clifford() : representation_label(0)
+clifford::clifford() : representation_label(0), metric(0), commutator_sign(-1)
 {
-       debugmsg("clifford default constructor", LOGLEVEL_CONSTRUCT);
-       tinfo_key = TINFO_clifford;
 }
 
-void clifford::copy(const clifford & other)
-{
-       inherited::copy(other);
-       representation_label = other.representation_label;
-}
-
-DEFAULT_DESTROY(clifford)
-DEFAULT_CTORS(diracone)
-DEFAULT_CTORS(diracgamma)
-DEFAULT_CTORS(diracgamma5)
+DEFAULT_CTOR(diracone)
+DEFAULT_CTOR(cliffordunit)
+DEFAULT_CTOR(diracgamma)
+DEFAULT_CTOR(diracgamma5)
+DEFAULT_CTOR(diracgammaL)
+DEFAULT_CTOR(diracgammaR)
 
 //////////
 // other constructors
@@ -68,64 +92,146 @@ DEFAULT_CTORS(diracgamma5)
 /** Construct object without any indices. This constructor is for internal
  *  use only. Use the dirac_ONE() function instead.
  *  @see dirac_ONE */
-clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl)
+clifford::clifford(const ex & b, unsigned char rl) : inherited(b), representation_label(rl), metric(0), commutator_sign(-1)
 {
-       debugmsg("clifford constructor from ex", LOGLEVEL_CONSTRUCT);
-       tinfo_key = TINFO_clifford;
 }
 
 /** Construct object with one Lorentz index. This constructor is for internal
- *  use only. Use the dirac_gamma() function instead.
+ *  use only. Use the clifford_unit() or dirac_gamma() functions instead.
+ *  @see clifford_unit
  *  @see dirac_gamma */
-clifford::clifford(const ex & b, const ex & mu, unsigned char rl) : inherited(b, mu), representation_label(rl)
+clifford::clifford(const ex & b, const ex & mu, const ex & metr, unsigned char rl, int comm_sign) : inherited(b, mu), representation_label(rl), metric(metr), commutator_sign(comm_sign)
 {
-       debugmsg("clifford constructor from ex,ex", LOGLEVEL_CONSTRUCT);
-       GINAC_ASSERT(is_ex_of_type(mu, varidx));
-       tinfo_key = TINFO_clifford;
+       GINAC_ASSERT(is_a<varidx>(mu));
 }
 
-clifford::clifford(unsigned char rl, const exvector & v, bool discardable) : inherited(indexed::unknown, v, discardable), representation_label(rl)
+clifford::clifford(unsigned char rl, const ex & metr, int comm_sign, const exvector & v, bool discardable) : inherited(not_symmetric(), v, discardable), representation_label(rl), metric(metr), commutator_sign(comm_sign)
 {
-       debugmsg("clifford constructor from unsigned char,exvector", LOGLEVEL_CONSTRUCT);
-       tinfo_key = TINFO_clifford;
 }
 
-clifford::clifford(unsigned char rl, exvector * vp) : inherited(indexed::unknown, vp), representation_label(rl)
+clifford::clifford(unsigned char rl, const ex & metr, int comm_sign, std::auto_ptr<exvector> vp) : inherited(not_symmetric(), vp), representation_label(rl), metric(metr), commutator_sign(comm_sign)
 {
-       debugmsg("clifford constructor from unsigned char,exvector *", LOGLEVEL_CONSTRUCT);
-       tinfo_key = TINFO_clifford;
+}
+
+return_type_t clifford::return_type_tinfo() const
+{
+       return make_return_type_t<clifford>(representation_label);
 }
 
 //////////
 // archiving
 //////////
 
-clifford::clifford(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
+void clifford::read_archive(const archive_node& n, lst& sym_lst)
 {
-       debugmsg("clifford constructor from archive_node", LOGLEVEL_CONSTRUCT);
+       inherited::read_archive(n, sym_lst);
        unsigned rl;
        n.find_unsigned("label", rl);
        representation_label = rl;
+       n.find_ex("metric", metric, sym_lst);
+       n.find_unsigned("commutator_sign+1", rl);
+       commutator_sign = rl - 1;
 }
 
-void clifford::archive(archive_node &n) const
+void clifford::archive(archive_node & n) const
 {
        inherited::archive(n);
        n.add_unsigned("label", representation_label);
+       n.add_ex("metric", metric);
+       n.add_unsigned("commutator_sign+1", commutator_sign+1);
+}
+
+GINAC_BIND_UNARCHIVER(clifford);
+GINAC_BIND_UNARCHIVER(diracone);
+GINAC_BIND_UNARCHIVER(diracgamma);
+GINAC_BIND_UNARCHIVER(diracgamma5);
+GINAC_BIND_UNARCHIVER(diracgammaL);
+GINAC_BIND_UNARCHIVER(diracgammaR);
+
+
+ex clifford::get_metric(const ex & i, const ex & j, bool symmetrised) const
+{
+       if (is_a<indexed>(metric)) {
+               if (symmetrised && !(ex_to<symmetry>(ex_to<indexed>(metric).get_symmetry()).has_symmetry())) {
+                       if (is_a<matrix>(metric.op(0))) {
+                               return indexed((ex_to<matrix>(metric.op(0)).add(ex_to<matrix>(metric.op(0)).transpose())).mul(numeric(1, 2)),
+                                              symmetric2(), i, j);
+                       } else {
+                               return simplify_indexed(indexed(metric.op(0)*_ex1_2, i, j) + indexed(metric.op(0)*_ex1_2, j, i));
+                       }
+               } else {
+                       return metric.subs(lst(metric.op(1) == i, metric.op(2) == j), subs_options::no_pattern);
+               }
+       } else {
+               exvector indices = metric.get_free_indices();
+               if (symmetrised)
+                       return _ex1_2*simplify_indexed(metric.subs(lst(indices[0] == i, indices[1] == j), subs_options::no_pattern)
+                                                                       + metric.subs(lst(indices[0] == j, indices[1] == i), subs_options::no_pattern));
+               else
+                       return metric.subs(lst(indices[0] == i, indices[1] == j), subs_options::no_pattern);
+       }
 }
 
-DEFAULT_UNARCHIVE(clifford)
-DEFAULT_ARCHIVING(diracone)
-DEFAULT_ARCHIVING(diracgamma)
-DEFAULT_ARCHIVING(diracgamma5)
+bool clifford::same_metric(const ex & other) const
+{
+       ex metr;
+       if (is_a<clifford>(other)) 
+               metr = ex_to<clifford>(other).get_metric();
+       else 
+               metr = other;
+
+       if (is_a<indexed>(metr))
+               return metr.op(0).is_equal(get_metric().op(0));
+       else {
+               exvector indices = metr.get_free_indices();
+               return  (indices.size() == 2) 
+                       && simplify_indexed(get_metric(indices[0], indices[1])-metr).is_zero();
+       }
+}
 
 //////////
-// functions overriding virtual functions from bases classes
+// functions overriding virtual functions from base classes
 //////////
 
+ex clifford::op(size_t i) const
+{
+       GINAC_ASSERT(i<nops());
+       if (nops()-i == 1)
+               return representation_label;
+       else 
+               return inherited::op(i);
+}
+
+ex & clifford::let_op(size_t i)
+{
+        GINAC_ASSERT(i<nops());
+
+       static ex rl = numeric(representation_label);
+        ensure_if_modifiable();
+       if (nops()-i == 1)
+               return rl;
+       else 
+               return inherited::let_op(i);
+}
+
+ex clifford::subs(const exmap & m, unsigned options) const
+{
+       ex subsed = inherited::subs(m, options);
+       if(is_a<clifford>(subsed)) {
+               ex prevmetric = ex_to<clifford>(subsed).metric;
+               ex newmetric = prevmetric.subs(m, options);
+               if(!are_ex_trivially_equal(prevmetric, newmetric)) {
+                       clifford c = ex_to<clifford>(subsed);
+                       c.metric = newmetric;
+                       subsed = c;
+               }
+       }
+       return subsed;
+}
+
 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) {
@@ -136,90 +242,286 @@ 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) && (commutator_sign == o.get_commutator_sign()) && same_metric(o));
+}
+
+static bool is_dirac_slash(const ex & seq0)
+{
+       return !is_a<diracgamma5>(seq0) && !is_a<diracgammaL>(seq0) &&
+              !is_a<diracgammaR>(seq0) && !is_a<cliffordunit>(seq0) &&
+              !is_a<diracone>(seq0);
+}
+
+void clifford::do_print_dflt(const print_dflt & c, unsigned level) const
+{
+       // dirac_slash() object is printed differently
+       if (is_dirac_slash(seq[0])) {
+               seq[0].print(c, precedence());
+               c.s << "\\";
+       } else { // We do not print representation label if it is 0
+               if (representation_label == 0) {
+                       this->print_dispatch<inherited>(c, level);
+               } else { // otherwise we put it before indices in square brackets; the code is borrowed from indexed.cpp 
+                       if (precedence() <= level) {
+                               c.s << '(';
+                       }
+                       seq[0].print(c, precedence());
+                       c.s << '[' << int(representation_label) << ']';
+                       printindices(c, level);
+                       if (precedence() <= level) {
+                               c.s << ')';
+                       }
+               }
+       }
+}
+
+void clifford::do_print_latex(const print_latex & c, unsigned level) const
+{
+       // dirac_slash() object is printed differently
+       if (is_dirac_slash(seq[0])) {
+               c.s << "{";
+               seq[0].print(c, precedence());
+               c.s << "\\hspace{-1.0ex}/}";
+       } else {
+               c.s << "\\clifford[" << int(representation_label) << "]";
+               this->print_dispatch<inherited>(c, level);
+       }
+}
+
 DEFAULT_COMPARE(diracone)
+DEFAULT_COMPARE(cliffordunit)
 DEFAULT_COMPARE(diracgamma)
 DEFAULT_COMPARE(diracgamma5)
+DEFAULT_COMPARE(diracgammaL)
+DEFAULT_COMPARE(diracgammaR)
 
-DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
+DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbf{1}")
+DEFAULT_PRINT_LATEX(cliffordunit, "e", "e")
 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
+DEFAULT_PRINT_LATEX(diracgammaL, "gammaL", "{\\gamma_L}")
+DEFAULT_PRINT_LATEX(diracgammaR, "gammaR", "{\\gamma_R}")
+
+/** This function decomposes gamma~mu -> (1, mu) and a\ -> (a.ix, ix) */
+static void base_and_index(const ex & c, ex & b, ex & i)
+{
+       GINAC_ASSERT(is_a<clifford>(c));
+       GINAC_ASSERT(c.nops() == 2+1);
+
+       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;
+       }
+}
+
+/** Predicate for finding non-clifford objects. */
+struct is_not_a_clifford : public std::unary_function<ex, bool> {
+       bool operator()(const ex & e)
+       {
+               return !is_a<clifford>(e);
+       }
+};
 
 /** Contraction of a gamma matrix with something else. */
 bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
 {
-       GINAC_ASSERT(is_ex_of_type(*self, clifford));
-       GINAC_ASSERT(is_ex_of_type(*other, indexed));
-       GINAC_ASSERT(is_ex_of_type(self->op(0), diracgamma));
-       unsigned char rl = ex_to_clifford(*self).get_representation_label();
+       GINAC_ASSERT(is_a<clifford>(*self));
+       GINAC_ASSERT(is_a<indexed>(*other));
+       GINAC_ASSERT(is_a<diracgamma>(self->op(0)));
+       unsigned char rl = ex_to<clifford>(*self).get_representation_label();
+
+       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)) {
 
-       if (is_ex_of_type(*other, clifford)) {
+               // 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();
+               size_t num = other - self;
 
                // gamma~mu gamma.mu = dim ONE
-               if (other - self == 1) {
+               if (num == 1) {
                        *self = dim;
                        *other = dirac_ONE(rl);
                        return true;
 
                // gamma~mu gamma~alpha gamma.mu = (2-dim) gamma~alpha
-               } else if (other - self == 2
-                       && is_ex_of_type(self[1], clifford)) {
+               } else if (num == 2
+                       && is_a<clifford>(self[1])) {
                        *self = 2 - dim;
-                       *other = _ex1();
+                       *other = _ex1;
                        return true;
 
                // gamma~mu gamma~alpha gamma~beta gamma.mu = 4 g~alpha~beta + (dim-4) gamam~alpha gamma~beta
-               } else if (other - self == 3
-                       && is_ex_of_type(self[1], clifford)
-                       && is_ex_of_type(self[2], clifford)) {
-                       *self = 4 * 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();
+               } else if (num == 3
+                       && is_a<clifford>(self[1])
+                       && is_a<clifford>(self[2])) {
+                       ex b1, i1, b2, i2;
+                       base_and_index(self[1], b1, i1);
+                       base_and_index(self[2], b2, i2);
+                       *self = 4 * lorentz_g(i1, i2) * b1 * b2 * dirac_ONE(rl) + (dim - 4) * self[1] * self[2];
+                       self[1] = _ex1;
+                       self[2] = _ex1;
+                       *other = _ex1;
+                       return true;
+
+               // gamma~mu gamma~alpha gamma~beta gamma~delta gamma.mu = -2 gamma~delta gamma~beta gamma~alpha - (dim-4) gamam~alpha gamma~beta gamma~delta
+               } else if (num == 4
+                       && is_a<clifford>(self[1])
+                       && is_a<clifford>(self[2])
+                       && is_a<clifford>(self[3])) {
+                       *self = -2 * self[3] * self[2] * self[1] - (dim - 4) * self[1] * self[2] * self[3];
+                       self[1] = _ex1;
+                       self[2] = _ex1;
+                       self[3] = _ex1;
+                       *other = _ex1;
+                       return true;
+
+               // gamma~mu Sodd gamma.mu = -2 Sodd_R
+               // (Chisholm identity in 4 dimensions)
+               } else if (!((other - self) & 1) && dim.is_equal(4)) {
+                       if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
+                               return false;
+
+                       *self = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(other), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
+                       std::fill(self + 1, other, _ex1);
+                       *other = _ex_2;
+                       return true;
+
+               // gamma~mu Sodd gamma~alpha gamma.mu = 2 gamma~alpha Sodd + 2 Sodd_R gamma~alpha
+               // (commutate contracted indices towards each other, then use
+               // Chisholm identity in 4 dimensions)
+               } else if (((other - self) & 1) && dim.is_equal(4)) {
+                       if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
+                               return false;
+
+                       exvector::iterator next_to_last = other - 1;
+                       ex S = ncmul(exvector(self + 1, next_to_last), true);
+                       ex SR = ncmul(exvector(std::reverse_iterator<exvector::const_iterator>(next_to_last), std::reverse_iterator<exvector::const_iterator>(self + 1)), true);
+
+                       *self = (*next_to_last) * S + SR * (*next_to_last);
+                       std::fill(self + 1, other, _ex1);
+                       *other = _ex2;
                        return true;
 
                // gamma~mu S gamma~alpha gamma.mu = 2 gamma~alpha S - gamma~mu S gamma.mu gamma~alpha
                // (commutate contracted indices towards each other, simplify_indexed()
                // will re-expand and re-run the simplification)
                } else {
-                       exvector::iterator it = self + 1, next_to_last = other - 1;
-                       while (it != other) {
-                               if (!is_ex_of_type(*it, clifford))
-                                       return false;
-                               it++;
-                       }
+                       if (std::find_if(self + 1, other, is_not_a_clifford()) != other)
+                               return false;
 
-                       it = self + 1;
-                       ex S = _ex1();
-                       while (it != next_to_last) {
-                               S *= *it;
-                               *it++ = _ex1();
-                       }
+                       exvector::iterator next_to_last = other - 1;
+                       ex S = ncmul(exvector(self + 1, next_to_last), true);
 
                        *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
-                       *next_to_last = _ex1();
-                       *other = _ex1();
+                       std::fill(self + 1, other + 1, _ex1);
                        return true;
                }
+
+       } else if (is_a<symbol>(other->op(0)) && other->nops() == 2) {
+
+               // x.mu gamma~mu -> x-slash
+               *self = dirac_slash(other->op(0), dim, rl);
+               *other = _ex1;
+               return true;
        }
 
        return false;
 }
 
+/** Contraction of a Clifford unit with something else. */
+bool cliffordunit::contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const
+{
+       GINAC_ASSERT(is_a<clifford>(*self));
+       GINAC_ASSERT(is_a<indexed>(*other));
+       GINAC_ASSERT(is_a<cliffordunit>(self->op(0)));
+       clifford unit = ex_to<clifford>(*self);
+       unsigned char rl = unit.get_representation_label();
+
+       if (is_a<clifford>(*other)) {
+               // Contraction only makes sense if the represenation labels are equal
+               // and the metrics are the same
+               if ((ex_to<clifford>(*other).get_representation_label() != rl) 
+                   && unit.same_metric(*other))
+                       return false;
+
+               exvector::iterator before_other = other - 1;
+               ex mu = self->op(1);
+               ex mu_toggle = other->op(1);
+               ex alpha = before_other->op(1);
+
+               // e~mu e.mu = Tr ONE
+               if (other - self == 1) {
+                       *self = unit.get_metric(mu, mu_toggle, true);
+                       *other = dirac_ONE(rl);
+                       return true;
+
+               } else if (other - self == 2) {
+                       if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
+                               // e~mu e~alpha e.mu = 2*e~mu B(alpha, mu.toggle_variance())-Tr(B) e~alpha
+                               *self = 2 * (*self) * unit.get_metric(alpha, mu_toggle, true) - unit.get_metric(mu, mu_toggle, true) * (*before_other);
+                               *before_other = _ex1;
+                               *other = _ex1;
+                               return true;
+
+                       } else {
+                               // e~mu S e.mu = Tr S ONE
+                               *self = unit.get_metric(mu, mu_toggle, true);
+                               *other = dirac_ONE(rl);
+                               return true;
+                       }
+               } else {
+               // e~mu S e~alpha e.mu = 2 e~mu S B(alpha, mu.toggle_variance()) - e~mu S e.mu e~alpha
+               // (commutate contracted indices towards each other, simplify_indexed()
+               // will re-expand and re-run the simplification)
+                       if (std::find_if(self + 1, other, is_not_a_clifford()) != other) {
+                               return false;
+                       }
+                       
+                       ex S = ncmul(exvector(self + 1, before_other), true);
+
+                       if (is_a<clifford>(*before_other) && ex_to<clifford>(*before_other).get_representation_label() == rl) {
+                               *self = 2 * (*self) * S * unit.get_metric(alpha, mu_toggle, true) - (*self) * S * (*other) * (*before_other);
+                       } else {
+                               // simply commutes
+                               *self = (*self) * S * (*other) * (*before_other);
+                       }
+                               
+                       std::fill(self + 1, other + 1, _ex1);
+                       return true;
+               }
+       }
+       return false;
+}
+
 /** Perform automatic simplification on noncommutative product of clifford
- *  objects. This removes superfluous ONEs, permutes gamma5's to the front
+ *  objects. This removes superfluous ONEs, permutes gamma5/L/R's to the front
  *  and removes squares of gamma objects. */
-ex clifford::simplify_ncmul(const exvector & v) const
+ex clifford::eval_ncmul(const exvector & v) const
 {
        exvector s;
        s.reserve(v.size());
-       unsigned rl = ex_to_clifford(v[0]).get_representation_label();
 
        // Remove superfluous ONEs
        exvector::const_iterator cit = v.begin(), citend = v.end();
        while (cit != citend) {
-               if (!is_ex_of_type(cit->op(0), diracone))
+               if (!is_a<clifford>(*cit) || !is_a<diracone>(cit->op(0)))
                        s.push_back(*cit);
                cit++;
        }
@@ -227,69 +529,179 @@ ex clifford::simplify_ncmul(const exvector & v) const
        bool something_changed = false;
        int sign = 1;
 
-       // Anticommute gamma5's to the front
+       // Anticommutate gamma5/L/R's to the front
        if (s.size() >= 2) {
                exvector::iterator first = s.begin(), next_to_last = s.end() - 2;
                while (true) {
                        exvector::iterator it = next_to_last;
                        while (true) {
                                exvector::iterator it2 = it + 1;
-                               if (!is_ex_of_type(it->op(0), diracgamma5) && is_ex_of_type(it2->op(0), diracgamma5)) {
-                                       it->swap(*it2);
-                                       sign = -sign;
-                                       something_changed = true;
+                               if (is_a<clifford>(*it) && is_a<clifford>(*it2)) {
+                                       ex e1 = it->op(0), e2 = it2->op(0);
+
+                                       if (is_a<diracgamma5>(e2)) {
+
+                                               if (is_a<diracgammaL>(e1) || is_a<diracgammaR>(e1)) {
+
+                                                       // gammaL/R gamma5 -> gamma5 gammaL/R
+                                                       it->swap(*it2);
+                                                       something_changed = true;
+
+                                               } else if (!is_a<diracgamma5>(e1)) {
+
+                                                       // gamma5 gamma5 -> gamma5 gamma5 (do nothing)
+                                                       // x gamma5 -> -gamma5 x
+                                                       it->swap(*it2);
+                                                       sign = -sign;
+                                                       something_changed = true;
+                                               }
+
+                                       } else if (is_a<diracgammaL>(e2)) {
+
+                                               if (is_a<diracgammaR>(e1)) {
+
+                                                       // gammaR gammaL -> 0
+                                                       return _ex0;
+
+                                               } else if (!is_a<diracgammaL>(e1) && !is_a<diracgamma5>(e1)) {
+
+                                                       // gammaL gammaL -> gammaL gammaL (do nothing)
+                                                       // gamma5 gammaL -> gamma5 gammaL (do nothing)
+                                                       // x gammaL -> gammaR x
+                                                       it->swap(*it2);
+                                                       *it = clifford(diracgammaR(), ex_to<clifford>(*it).get_representation_label());
+                                                       something_changed = true;
+                                               }
+
+                                       } else if (is_a<diracgammaR>(e2)) {
+
+                                               if (is_a<diracgammaL>(e1)) {
+
+                                                       // gammaL gammaR -> 0
+                                                       return _ex0;
+
+                                               } else if (!is_a<diracgammaR>(e1) && !is_a<diracgamma5>(e1)) {
+
+                                                       // gammaR gammaR -> gammaR gammaR (do nothing)
+                                                       // gamma5 gammaR -> gamma5 gammaR (do nothing)
+                                                       // x gammaR -> gammaL x
+                                                       it->swap(*it2);
+                                                       *it = clifford(diracgammaL(), ex_to<clifford>(*it).get_representation_label());
+                                                       something_changed = true;
+                                               }
+                                       }
                                }
                                if (it == first)
                                        break;
-                               it--;
+                               --it;
                        }
                        if (next_to_last == first)
                                break;
-                       next_to_last--;
+                       --next_to_last;
                }
        }
 
-       // Remove squares of gamma5
-       while (s.size() >= 2 && is_ex_of_type(s[0].op(0), diracgamma5) && is_ex_of_type(s[1].op(0), diracgamma5)) {
-               s.erase(s.begin(), s.begin() + 2);
-               something_changed = true;
-       }
-
        // Remove equal adjacent gammas
        if (s.size() >= 2) {
-               exvector::iterator it = s.begin(), itend = s.end() - 1;
-               while (it != itend) {
+               exvector::iterator it, itend = s.end() - 1;
+               for (it = s.begin(); it != itend; ++it) {
                        ex & a = it[0];
                        ex & b = it[1];
-                       if (is_ex_of_type(a.op(0), diracgamma) && is_ex_of_type(b.op(0), diracgamma)) {
+                       if (!is_a<clifford>(a) || !is_a<clifford>(b))
+                               continue;
+
+                       const ex & ag = a.op(0);
+                       const ex & bg = b.op(0);
+                       bool a_is_cliffordunit = is_a<cliffordunit>(ag);
+                       bool b_is_cliffordunit =  is_a<cliffordunit>(bg);
+
+                       if (a_is_cliffordunit && b_is_cliffordunit && ex_to<clifford>(a).same_metric(b)
+                               && (ex_to<clifford>(a).get_commutator_sign() == -1)) {
+                               // This is done only for Clifford algebras 
+                               
                                const ex & ia = a.op(1);
                                const ex & ib = b.op(1);
-                               if (ia.is_equal(ib)) {
-                                       a = lorentz_g(ia, ib);
-                                       b = dirac_ONE(rl);
+                               if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
+                                       a = ex_to<clifford>(a).get_metric(ia, ib, true);
+                                       b = dirac_ONE(representation_label);
                                        something_changed = true;
                                }
+
+                       } else if ((is_a<diracgamma5>(ag) && is_a<diracgamma5>(bg))) {
+
+                               // Remove squares of gamma5
+                               a = dirac_ONE(representation_label);
+                               b = dirac_ONE(representation_label);
+                               something_changed = true;
+
+                       } else if ((is_a<diracgammaL>(ag) && is_a<diracgammaL>(bg))
+                               || (is_a<diracgammaR>(ag) && is_a<diracgammaR>(bg))) {
+
+                               // Remove squares of gammaL/R
+                               b = dirac_ONE(representation_label);
+                               something_changed = true;
+
+                       } else if (is_a<diracgammaL>(ag) && is_a<diracgammaR>(bg)) {
+
+                               // gammaL and gammaR are orthogonal
+                               return _ex0;
+
+                       } else if (is_a<diracgamma5>(ag) && is_a<diracgammaL>(bg)) {
+
+                               // gamma5 gammaL -> -gammaL
+                               a = dirac_ONE(representation_label);
+                               sign = -sign;
+                               something_changed = true;
+
+                       } else if (is_a<diracgamma5>(ag) && is_a<diracgammaR>(bg)) {
+
+                               // gamma5 gammaR -> gammaR
+                               a = dirac_ONE(representation_label);
+                               something_changed = true;
+
+                       } else if (!a_is_cliffordunit && !b_is_cliffordunit && ag.is_equal(bg)) {
+
+                               // a\ a\ -> a^2
+                               varidx ix((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(a.op(1)).minimal_dim(ex_to<idx>(b.op(1))));
+                               
+                               a = indexed(ag, ix) * indexed(ag, ix.toggle_variance());
+                               b = dirac_ONE(representation_label);
+                               something_changed = true;
                        }
-                       it++;
                }
        }
 
-       if (s.size() == 0)
-               return clifford(diracone(), rl) * sign;
+       if (s.empty())
+               return dirac_ONE(representation_label) * sign;
        if (something_changed)
-               return nonsimplified_ncmul(s) * sign;
+               return reeval_ncmul(s) * sign;
        else
-               return simplified_ncmul(s) * sign;
+               return hold_ncmul(s) * sign;
+}
+
+ex clifford::thiscontainer(const exvector & v) const
+{
+       return clifford(representation_label, metric, commutator_sign, v);
+}
+
+ex clifford::thiscontainer(std::auto_ptr<exvector> vp) const
+{
+       return clifford(representation_label, metric, commutator_sign, vp);
+}
+
+ex diracgamma5::conjugate() const
+{      
+       return _ex_1 * (*this);
 }
 
-ex clifford::thisexprseq(const exvector & v) const
+ex diracgammaL::conjugate() const
 {
-       return clifford(representation_label, v);
+       return (new diracgammaR)->setflag(status_flags::dynallocated);
 }
 
-ex clifford::thisexprseq(exvector * vp) const
+ex diracgammaR::conjugate() const
 {
-       return clifford(representation_label, vp);
+       return (new diracgammaL)->setflag(status_flags::dynallocated);
 }
 
 //////////
@@ -298,152 +710,653 @@ ex clifford::thisexprseq(exvector * vp) const
 
 ex dirac_ONE(unsigned char rl)
 {
-       return clifford(diracone(), rl);
+       static ex ONE = (new diracone)->setflag(status_flags::dynallocated);
+       return clifford(ONE, rl);
+}
+
+static unsigned get_dim_uint(const ex& e)
+{
+       if (!is_a<idx>(e))
+               throw std::invalid_argument("get_dim_uint: argument is not an index");
+       ex dim = ex_to<idx>(e).get_dim();
+       if (!dim.info(info_flags::posint))
+               throw std::invalid_argument("get_dim_uint: dimension of index should be a positive integer");
+       unsigned d = ex_to<numeric>(dim).to_int();
+       return d;
+}
+
+ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl)
+{
+       //static ex unit = (new cliffordunit)->setflag(status_flags::dynallocated);
+       ex unit = (new cliffordunit)->setflag(status_flags::dynallocated);
+
+       if (!is_a<idx>(mu))
+               throw(std::invalid_argument("clifford_unit(): index of Clifford unit must be of type idx or varidx"));
+
+       exvector indices = metr.get_free_indices();
+
+       if (indices.size() == 2) {
+               return clifford(unit, mu, metr, rl);
+       } else if (is_a<matrix>(metr)) {
+               matrix M = ex_to<matrix>(metr);
+               unsigned n = M.rows();
+               bool symmetric = true;
+
+               //static idx xi((new symbol)->setflag(status_flags::dynallocated), n),
+               //      chi((new symbol)->setflag(status_flags::dynallocated), n);
+               idx xi((new symbol)->setflag(status_flags::dynallocated), n),
+                       chi((new symbol)->setflag(status_flags::dynallocated), n);
+               if ((n ==  M.cols()) && (n == get_dim_uint(mu))) {
+                       for (unsigned i = 0; i < n; i++) {
+                               for (unsigned j = i+1; j < n; j++) {
+                                       if (!M(i, j).is_equal(M(j, i))) {
+                                               symmetric = false;
+                                       }
+                               }
+                       }
+                       return clifford(unit, mu, indexed(metr, symmetric?symmetric2():not_symmetric(), xi, chi), rl);
+               } else {
+                       throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be a square matrix with the same dimensions as index"));
+               }
+       } else if (indices.size() == 0) { // a tensor or other expression without indices
+               //static varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim()),
+               //      chi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim());
+               varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim()),
+                       chi((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(mu).get_dim());
+               return clifford(unit, mu, indexed(metr, xi, chi), rl);
+       }  else 
+               throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be of type tensor, matrix or an expression with two free indices"));
 }
 
 ex dirac_gamma(const ex & mu, unsigned char rl)
 {
-       if (!is_ex_of_type(mu, varidx))
-               throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
+       static ex gamma = (new diracgamma)->setflag(status_flags::dynallocated);
 
-       return clifford(diracgamma(), mu, rl);
+       if (!is_a<varidx>(mu))
+               throw(std::invalid_argument("dirac_gamma(): index of Dirac gamma must be of type varidx"));
+
+       static varidx xi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim()),
+               chi((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(mu).get_dim());
+       return clifford(gamma, mu, indexed((new minkmetric)->setflag(status_flags::dynallocated), symmetric2(), xi, chi), rl);
 }
 
 ex dirac_gamma5(unsigned char rl)
 {
-       return clifford(diracgamma5(), rl);
+       static ex gamma5 = (new diracgamma5)->setflag(status_flags::dynallocated);
+       return clifford(gamma5, rl);
 }
 
-ex dirac_gamma6(unsigned char rl)
+ex dirac_gammaL(unsigned char rl)
 {
-       return clifford(diracone(), rl) + clifford(diracgamma5(), rl);
+       static ex gammaL = (new diracgammaL)->setflag(status_flags::dynallocated);
+       return clifford(gammaL, rl);
 }
 
-ex dirac_gamma7(unsigned char rl)
+ex dirac_gammaR(unsigned char rl)
 {
-       return clifford(diracone(), rl) - clifford(diracgamma5(), rl);
+       static ex gammaR = (new diracgammaR)->setflag(status_flags::dynallocated);
+       return clifford(gammaR, rl);
 }
 
 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl)
 {
-       varidx mu((new symbol)->setflag(status_flags::dynallocated), dim);
-       return indexed(e, mu.toggle_variance()) * dirac_gamma(mu, rl);
+       // Slashed vectors are actually stored as a clifford object with the
+       // vector as its base expression and a (dummy) index that just serves
+       // for storing the space dimensionality
+
+       static varidx xi((new symbol)->setflag(status_flags::dynallocated), dim),
+               chi((new symbol)->setflag(status_flags::dynallocated), dim);
+   return clifford(e, varidx(0, dim), indexed((new minkmetric)->setflag(status_flags::dynallocated), symmetric2(), xi, chi), 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)
+/** Extract representation label from tinfo key (as returned by
+ *  return_type_tinfo()). */
+static unsigned char get_representation_label(const return_type_t& ti)
 {
-       return ti == (TINFO_clifford + rl);
+       return (unsigned char)ti.rl;
 }
 
-ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
+/** 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)
 {
-       if (is_ex_of_type(e, clifford)) {
+       // 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;
+}
 
-               if (ex_to_clifford(e).get_representation_label() == rl
-                && is_ex_of_type(e.op(0), diracone))
-                       return trONE;
-               else
-                       return _ex0();
+ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE)
+{
+       if (is_a<clifford>(e)) {
+
+               unsigned char rl = ex_to<clifford>(e).get_representation_label();
 
-       } else if (is_ex_exactly_of_type(e, add)) {
+               // Are we taking the trace over this object's representation label?
+               if (rls.find(rl) == rls.end())
+                       return 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, trONE);
-               return sum;
+               // Yes, all elements are traceless, except for dirac_ONE and dirac_L/R
+               const ex & g = e.op(0);
+               if (is_a<diracone>(g))
+                       return trONE;
+               else if (is_a<diracgammaL>(g) || is_a<diracgammaR>(g))
+                       return trONE/2;
+               else
+                       return _ex0;
 
-       } else if (is_ex_exactly_of_type(e, mul)) {
+       } else if (is_exactly_a<mul>(e)) {
 
                // Trace of product: pull out non-clifford factors
-               ex prod = _ex1();
-               for (unsigned i=0; i<e.nops(); i++) {
+               ex prod = _ex1;
+               for (size_t i=0; i<e.nops(); i++) {
                        const ex &o = e.op(i);
-                       unsigned ti = o.return_type_tinfo();
-                       if (is_clifford_tinfo(o.return_type_tinfo(), rl))
-                               prod *= dirac_trace(o, rl, trONE);
+                       if (is_clifford_tinfo(o.return_type_tinfo()))
+                               prod *= dirac_trace(o, rls, trONE);
                        else
                                prod *= o;
                }
                return prod;
 
-       } else if (is_ex_exactly_of_type(e, ncmul)) {
+       } else if (is_exactly_a<ncmul>(e)) {
 
-               if (!is_clifford_tinfo(e.return_type_tinfo(), rl))
-                       return _ex0();
+               unsigned char rl = get_representation_label(e.return_type_tinfo());
 
-               // Expand product, if necessary
-               ex e_expanded = e.expand();
-               if (!is_ex_of_type(e_expanded, ncmul))
-                       return dirac_trace(e_expanded, rl, trONE);
+               // Are we taking the trace over this string's representation label?
+               if (rls.find(rl) == rls.end())
+                       return e;
+
+               // Substitute gammaL/R and expand product, if necessary
+               ex e_expanded = e.subs(lst(
+                       dirac_gammaL(rl) == (dirac_ONE(rl)-dirac_gamma5(rl))/2,
+                       dirac_gammaR(rl) == (dirac_ONE(rl)+dirac_gamma5(rl))/2
+               ), subs_options::no_pattern).expand();
+               if (!is_a<ncmul>(e_expanded))
+                       return dirac_trace(e_expanded, rls, trONE);
 
                // gamma5 gets moved to the front so this check is enough
-               bool has_gamma5 = is_ex_of_type(e.op(0).op(0), diracgamma5);
-               unsigned num = e.nops();
+               bool has_gamma5 = is_a<diracgamma5>(e.op(0).op(0));
+               size_t num = e.nops();
 
                if (has_gamma5) {
 
                        // Trace of gamma5 * odd number of gammas and trace of
                        // gamma5 * gamma.mu * gamma.nu are zero
                        if ((num & 1) == 0 || num == 3)
-                               return _ex0();
+                               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 =
+                       // Tr gamma5 S_2k =
                        //   I/4! * epsilon0123.mu1.mu2.mu3.mu4 * Tr gamma.mu1 gamma.mu2 gamma.mu3 gamma.mu4 S_2k
-                       ex dim = ex_to_idx(e.op(1).op(1)).get_dim();
-                       varidx mu1((new symbol)->setflag(status_flags::dynallocated), dim),
-                              mu2((new symbol)->setflag(status_flags::dynallocated), dim),
-                              mu3((new symbol)->setflag(status_flags::dynallocated), dim),
-                              mu4((new symbol)->setflag(status_flags::dynallocated), dim);
-                       exvector v;
-                       v.reserve(num + 3);
-                       v.push_back(dirac_gamma(mu1, rl));
-                       v.push_back(dirac_gamma(mu2, rl));
-                       v.push_back(dirac_gamma(mu3, rl));
-                       v.push_back(dirac_gamma(mu4, rl));
-                       for (int i=1; i<num; i++)
-                               v.push_back(e.op(i));
-
-                       return (eps0123(mu1.toggle_variance(), mu2.toggle_variance(), mu3.toggle_variance(), mu4.toggle_variance()) *
-                               dirac_trace(ncmul(v), rl, trONE)).simplify_indexed() * I / 24;
+                       // (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();
+                               return _ex0;
 
                        // Tr gamma.mu gamma.nu = 4 g.mu.nu
-                       if (num == 2)
-                               return trONE * 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 (size_t i=0; i<num; i++)
+                               base_and_index(e.op(i), bv[i], iv[i]);
+
+                       return trONE * (trace_string(iv.begin(), num) * mul(bv)).simplify_indexed();
+               }
+
+       } else if (e.nops() > 0) {
+
+               // Trace maps to all other container classes (this includes sums)
+               pointer_to_map_function_2args<const std::set<unsigned char> &, const ex &> fcn(dirac_trace, rls, trONE);
+               return e.map(fcn);
+
+       } else
+               return _ex0;
+}
+
+ex dirac_trace(const ex & e, const lst & rll, const ex & trONE)
+{
+       // Convert list to set
+       std::set<unsigned char> rls;
+       for (lst::const_iterator i = rll.begin(); i != rll.end(); ++i) {
+               if (i->info(info_flags::nonnegint))
+                       rls.insert(ex_to<numeric>(*i).to_int());
+       }
+
+       return dirac_trace(e, rls, trONE);
+}
+
+ex dirac_trace(const ex & e, unsigned char rl, const ex & trONE)
+{
+       // Convert label to set
+       std::set<unsigned char> rls;
+       rls.insert(rl);
+
+       return dirac_trace(e, rls, trONE);
+}
+
+
+ex canonicalize_clifford(const ex & e_)
+{
+       pointer_to_map_function fcn(canonicalize_clifford);
+
+       if (is_a<matrix>(e_)    // || is_a<pseries>(e) || is_a<integral>(e)
+               || e_.info(info_flags::list)) {
+               return e_.map(fcn);
+       } else {
+               ex e=simplify_indexed(e_);
+               // Scan for any ncmul objects
+               exmap srl;
+               ex aux = e.to_rational(srl);
+               for (exmap::iterator i = srl.begin(); i != srl.end(); ++i) {
+
+                       ex lhs = i->first;
+                       ex rhs = i->second;
+
+                       if (is_exactly_a<ncmul>(rhs)
+                                       && rhs.return_type() == return_types::noncommutative
+                                       && is_clifford_tinfo(rhs.return_type_tinfo())) {
+
+                               // Expand product, if necessary
+                               ex rhs_expanded = rhs.expand();
+                               if (!is_a<ncmul>(rhs_expanded)) {
+                                       i->second = canonicalize_clifford(rhs_expanded);
+                                       continue;
+
+                               } else if (!is_a<clifford>(rhs.op(0)))
+                                       continue;
+
+                               exvector v;
+                               v.reserve(rhs.nops());
+                               for (size_t j=0; j<rhs.nops(); j++)
+                                       v.push_back(rhs.op(j));
+
+                               // Stupid recursive bubble sort because we only want to swap adjacent gammas
+                               exvector::iterator it = v.begin(), next_to_last = v.end() - 1;
+                               if (is_a<diracgamma5>(it->op(0)) || is_a<diracgammaL>(it->op(0)) || is_a<diracgammaR>(it->op(0)))
+                                       ++it;
+
+                               while (it != next_to_last) {
+                                       if (it[0].compare(it[1]) > 0) {
+
+                                               ex save0 = it[0], save1 = it[1];
+                                               ex b1, i1, b2, i2;
+                                               base_and_index(it[0], b1, i1);
+                                               base_and_index(it[1], b2, i2);
+                                               // for Clifford algebras (commutator_sign == -1) metric should be symmetrised
+                                               it[0] = (ex_to<clifford>(save0).get_metric(i1, i2, ex_to<clifford>(save0).get_commutator_sign() == -1) * b1 * b2).simplify_indexed();
+                                               it[1] = v.size() ? _ex2 * dirac_ONE(ex_to<clifford>(save0).get_representation_label()) : _ex2;
+                                               ex sum = ncmul(v);
+                                               it[0] = save1;
+                                               it[1] = save0;
+                                               sum += ex_to<clifford>(save0).get_commutator_sign() * ncmul(v, true);
+                                               i->second = canonicalize_clifford(sum);
+                                               goto next_sym;
+                                       }
+                                       ++it;
                                }
-                               result += sign * lorentz_g(ix1, e.op(i).op(1)) * dirac_trace(ncmul(v), rl, trONE);
-                               sign = -sign;
+next_sym:      ;
                        }
-                       return result;
                }
+               return aux.subs(srl, subs_options::no_pattern).simplify_indexed();
+       }
+}
+
+ex clifford_prime(const ex & e)
+{
+       pointer_to_map_function fcn(clifford_prime);
+       if (is_a<clifford>(e) && is_a<cliffordunit>(e.op(0))) {
+               return -e;
+       } else if (is_a<add>(e) || is_a<ncmul>(e) || is_a<mul>(e) //|| is_a<pseries>(e) || is_a<integral>(e)
+                          || is_a<matrix>(e) || e.info(info_flags::list)) {
+               return e.map(fcn);
+       } else if (is_a<power>(e)) {
+               return pow(clifford_prime(e.op(0)), e.op(1));
+       } else
+               return e;
+}
+
+ex remove_dirac_ONE(const ex & e, unsigned char rl, unsigned options)
+{
+       pointer_to_map_function_2args<unsigned char, unsigned> fcn(remove_dirac_ONE, rl, options | 1);
+       bool need_reevaluation = false;
+       ex e1 = e;
+       if (! (options & 1) )  { // is not a child
+               if (options & 2)
+                       e1 = expand_dummy_sum(e, true);
+               e1 = canonicalize_clifford(e1);
+       }
+       
+       if (is_a<clifford>(e1) && ex_to<clifford>(e1).get_representation_label() >= rl) {
+               if (is_a<diracone>(e1.op(0)))
+                       return 1;
+               else 
+                       throw(std::invalid_argument("remove_dirac_ONE(): expression is a non-scalar Clifford number!"));
+       } else if (is_a<add>(e1) || is_a<ncmul>(e1) || is_a<mul>(e1)  
+                          || is_a<matrix>(e1) || e1.info(info_flags::list)) {
+               if (options & 3) // is a child or was already expanded
+                       return e1.map(fcn);
+               else
+                       try {
+                               return e1.map(fcn);
+                       } catch (std::exception &p) {
+                               need_reevaluation = true;
+                       }
+       } else if (is_a<power>(e1)) {
+               if (options & 3) // is a child or was already expanded
+                       return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
+               else
+                       try {
+                               return pow(remove_dirac_ONE(e1.op(0), rl, options | 1), e1.op(1));
+                       } catch (std::exception &p) {
+                               need_reevaluation = true;
+                       }
+       } 
+       if (need_reevaluation)
+               return remove_dirac_ONE(e, rl, options | 2);
+       return e1;
+}
+
+char clifford_max_label(const ex & e, bool ignore_ONE)
+{
+       if (is_a<clifford>(e))
+               if (ignore_ONE && is_a<diracone>(e.op(0)))
+                       return -1;
+               else
+                       return ex_to<clifford>(e).get_representation_label();
+       else {
+               char rl = -1;
+               for (size_t i=0; i < e.nops(); i++) 
+                       rl = (rl > clifford_max_label(e.op(i), ignore_ONE)) ? rl : clifford_max_label(e.op(i), ignore_ONE);
+               return rl;
        }
+}
 
-       return _ex0();
+ex clifford_norm(const ex & e)
+{
+       return sqrt(remove_dirac_ONE(e * clifford_bar(e)));
+}
+       
+ex clifford_inverse(const ex & e)
+{
+       ex norm = clifford_norm(e);
+       if (!norm.is_zero())
+               return clifford_bar(e) / pow(norm, 2);
+       else 
+               throw(std::invalid_argument("clifford_inverse(): cannot find inverse of Clifford number with zero norm!"));
+}
+
+ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl)
+{
+       if (!ex_to<idx>(mu).is_dim_numeric())
+               throw(std::invalid_argument("lst_to_clifford(): Index should have a numeric dimension"));
+       ex e = clifford_unit(mu, metr, rl);
+       return lst_to_clifford(v, e);
+}
+
+ex lst_to_clifford(const ex & v, const ex & e) {
+       unsigned min, max;
+
+       if (is_a<clifford>(e)) {
+               ex mu = e.op(1);
+               ex mu_toggle
+                       = is_a<varidx>(mu) ? ex_to<varidx>(mu).toggle_variance() : mu;
+               unsigned dim = get_dim_uint(mu);
+
+               if (is_a<matrix>(v)) {
+                       if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows()) {
+                               min = ex_to<matrix>(v).rows();
+                               max = ex_to<matrix>(v).cols();
+                       } else {
+                               min = ex_to<matrix>(v).cols();
+                               max = ex_to<matrix>(v).rows();
+                       }
+                       if (min == 1) {
+                               if (dim == max)
+                                       return indexed(v, mu_toggle) * e;
+                               else if (max - dim == 1) {
+                                       if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows())
+                                               return v.op(0) * dirac_ONE(ex_to<clifford>(e).get_representation_label()) + indexed(sub_matrix(ex_to<matrix>(v), 0, 1, 1, dim), mu_toggle) * e;
+                                       else 
+                                               return v.op(0) * dirac_ONE(ex_to<clifford>(e).get_representation_label()) + indexed(sub_matrix(ex_to<matrix>(v), 1, dim, 0, 1), mu_toggle) * e;
+                               } else
+                                       throw(std::invalid_argument("lst_to_clifford(): dimensions of vector and clifford unit mismatch"));
+                       } else
+                               throw(std::invalid_argument("lst_to_clifford(): first argument should be a vector (nx1 or 1xn matrix)"));
+               } else if (v.info(info_flags::list)) {
+                       if (dim == ex_to<lst>(v).nops())
+                               return indexed(matrix(dim, 1, ex_to<lst>(v)), mu_toggle) * e;
+                       else if (ex_to<lst>(v).nops() - dim == 1)
+                               return v.op(0) * dirac_ONE(ex_to<clifford>(e).get_representation_label()) + indexed(sub_matrix(matrix(dim+1, 1, ex_to<lst>(v)), 1, dim, 0, 1), mu_toggle) * e;
+                       else
+                               throw(std::invalid_argument("lst_to_clifford(): list length and dimension of clifford unit mismatch"));
+               } else
+                       throw(std::invalid_argument("lst_to_clifford(): cannot construct from anything but list or vector"));
+       } else
+               throw(std::invalid_argument("lst_to_clifford(): the second argument should be a Clifford unit"));
+}
+/** Auxiliary structure to define a function for striping one Clifford unit
+ * from vectors. Used in  clifford_to_lst(). */
+static ex get_clifford_comp(const ex & e, const ex & c) 
+{
+       pointer_to_map_function_1arg<const ex &> fcn(get_clifford_comp, c);
+       int ival = ex_to<numeric>(ex_to<idx>(c.op(1)).get_value()).to_int();
+               
+       if (is_a<add>(e) || e.info(info_flags::list) // || is_a<pseries>(e) || is_a<integral>(e)
+               || is_a<matrix>(e)) 
+               return e.map(fcn);
+       else if (is_a<ncmul>(e) || is_a<mul>(e)) {
+               // find a Clifford unit with the same metric, delete it and substitute its index
+               size_t ind = e.nops() + 1;
+               for (size_t j = 0; j < e.nops(); j++) {
+                       if (is_a<clifford>(e.op(j)) && ex_to<clifford>(c).same_metric(e.op(j))) {
+                               if (ind > e.nops()) {
+                                       ind = j;
+                               }
+                               else {
+                                       throw(std::invalid_argument("get_clifford_comp(): expression is a Clifford multi-vector"));
+                               }
+                       }
+               }
+               if (ind < e.nops()) {
+                       ex S = 1;
+                       bool same_value_index, found_dummy;
+                       same_value_index = ( ex_to<idx>(e.op(ind).op(1)).is_numeric()
+                                                                &&  (ival == ex_to<numeric>(ex_to<idx>(e.op(ind).op(1)).get_value()).to_int()) );
+                       found_dummy = same_value_index;
+                       for (size_t j=0; j < e.nops(); j++) {
+                               if (j != ind) {
+                                       if (same_value_index) {
+                                               S = S * e.op(j);
+                                       }
+                                       else {
+                                               exvector ind_vec = ex_to<indexed>(e.op(j)).get_dummy_indices(ex_to<indexed>(e.op(ind)));
+                                               if (ind_vec.size() > 0) {
+                                                       found_dummy = true;
+                                                       exvector::const_iterator it = ind_vec.begin(), itend = ind_vec.end();
+                                                       while (it != itend) {
+                                                               ex curridx = *it;
+                                                               ex curridx_toggle = is_a<varidx>(curridx)
+                                                                       ? ex_to<varidx>(curridx).toggle_variance()
+                                                                       : curridx;
+                                                               S = S * e.op(j).subs(lst(curridx == ival,
+                                                                       curridx_toggle == ival), subs_options::no_pattern);
+                                                               ++it;
+                                                       }
+                                               } else
+                                                       S = S * e.op(j);
+                                       }
+                               }
+                       }
+                       return (found_dummy ? S : 0);
+               } else
+                       throw(std::invalid_argument("get_clifford_comp(): expression is not a Clifford vector to the given units"));
+       } else if (e.is_zero()) 
+               return e;
+       else if (is_a<clifford>(e) && ex_to<clifford>(e).same_metric(c))
+               if ( ex_to<idx>(e.op(1)).is_numeric() &&
+                        (ival != ex_to<numeric>(ex_to<idx>(e.op(1)).get_value()).to_int()) )
+                       return 0;
+               else 
+                       return 1;
+       else
+               throw(std::invalid_argument("get_clifford_comp(): expression is not usable as a Clifford vector"));
+}
+
+
+lst clifford_to_lst(const ex & e, const ex & c, bool algebraic)
+{
+       GINAC_ASSERT(is_a<clifford>(c));
+       ex mu = c.op(1);
+       if (! ex_to<idx>(mu).is_dim_numeric())
+               throw(std::invalid_argument("clifford_to_lst(): index should have a numeric dimension"));
+       unsigned int D = ex_to<numeric>(ex_to<idx>(mu).get_dim()).to_int();
+
+       if (algebraic) // check if algebraic method is applicable
+               for (unsigned int i = 0; i < D; i++) 
+                       if (pow(c.subs(mu == i, subs_options::no_pattern), 2).is_zero() 
+                               or (not is_a<numeric>(pow(c.subs(mu == i, subs_options::no_pattern), 2))))
+                               algebraic = false;
+       lst V; 
+       ex v0 = remove_dirac_ONE(canonicalize_clifford(e+clifford_prime(e)).normal())/2;
+       if (not v0.is_zero())
+               V.append(v0);
+       ex e1 = canonicalize_clifford(e - v0 * dirac_ONE(ex_to<clifford>(c).get_representation_label())); 
+       if (algebraic) {
+               for (unsigned int i = 0; i < D; i++) 
+                       V.append(remove_dirac_ONE(
+                                               simplify_indexed(canonicalize_clifford(e1 * c.subs(mu == i, subs_options::no_pattern) +  c.subs(mu == i, subs_options::no_pattern) * e1))
+                                               / (2*pow(c.subs(mu == i, subs_options::no_pattern), 2))));
+       } else {
+               try {
+                       for (unsigned int i = 0; i < D; i++) 
+                               V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
+               } catch  (std::exception &p) {
+                       /* Try to expand dummy summations to simplify the expression*/
+                       e1 = canonicalize_clifford(expand_dummy_sum(e, true));
+                       V.remove_all();
+                       v0 = remove_dirac_ONE(canonicalize_clifford(e1+clifford_prime(e1)).normal())/2;
+                       if (not v0.is_zero()) {
+                               V.append(v0);
+                               e1 = canonicalize_clifford(e1 - v0 * dirac_ONE(ex_to<clifford>(c).get_representation_label())); 
+                       }
+                       for (unsigned int i = 0; i < D; i++) 
+                               V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
+               }
+       }
+       return V;
+}
+
+
+ex clifford_moebius_map(const ex & a, const ex & b, const ex & c, const ex & d, const ex & v, const ex & G, unsigned char rl)
+{
+       ex x, D, cu;
+       
+       if (! is_a<matrix>(v) && ! v.info(info_flags::list))
+               throw(std::invalid_argument("clifford_moebius_map(): parameter v should be either vector or list"));
+       
+       if (is_a<clifford>(G)) {
+               cu = G;
+       } else {
+               if (is_a<indexed>(G)) {
+                       D = ex_to<idx>(G.op(1)).get_dim();
+                       varidx mu((new symbol)->setflag(status_flags::dynallocated), D);
+                       cu = clifford_unit(mu, G, rl);
+               } else if (is_a<matrix>(G)) {
+                       D = ex_to<matrix>(G).rows(); 
+                       idx mu((new symbol)->setflag(status_flags::dynallocated), D);
+                       cu = clifford_unit(mu, G, rl);
+               } else throw(std::invalid_argument("clifford_moebius_map(): metric should be an indexed object, matrix, or a Clifford unit"));
+               
+       }
+       
+       x = lst_to_clifford(v, cu); 
+       ex e = clifford_to_lst(simplify_indexed(canonicalize_clifford((a * x + b) * clifford_inverse(c * x + d))), cu, false);
+       return (is_a<matrix>(v) ? matrix(ex_to<matrix>(v).rows(), ex_to<matrix>(v).cols(), ex_to<lst>(e)) : e);
+}
+
+ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl)
+{
+       if (is_a<matrix>(M) && (ex_to<matrix>(M).rows() == 2) && (ex_to<matrix>(M).cols() == 2)) 
+               return clifford_moebius_map(M.op(0), M.op(1), M.op(2), M.op(3), v, G, rl);
+       else
+               throw(std::invalid_argument("clifford_moebius_map(): parameter M should be a 2x2 matrix"));
 }
 
 } // namespace GiNaC