Various improvements for clifford class (subs for metric, commutator_sign,

  is_equal with repr labels) [V.Kisil]

diff --git a/check/exam_clifford.cpp b/check/exam_clifford.cpp
index 75110bfa35f90553f66d7cfed705be998f4dd4d7..15daef1750bea9893149fe5ec56337dbe8cd56e4 100644 (file)
--- a/check/exam_clifford.cpp
+++ b/check/exam_clifford.cpp
@@ -236,7 +236,7 @@ static unsigned clifford_check3()
        e = dirac_gamma(mu, 0) * dirac_gamma(mu.toggle_variance(), 1) * dirac_gamma(nu, 0) * dirac_gamma(nu.toggle_variance(), 1);
        result += check_equal_simplify(dirac_trace(e, 0), 4 * dim * dirac_ONE(1));
        result += check_equal_simplify(dirac_trace(e, 1), 4 * dim * dirac_ONE(0));
        e = dirac_gamma(mu, 0) * dirac_gamma(mu.toggle_variance(), 1) * dirac_gamma(nu, 0) * dirac_gamma(nu.toggle_variance(), 1);
        result += check_equal_simplify(dirac_trace(e, 0), 4 * dim * dirac_ONE(1));
        result += check_equal_simplify(dirac_trace(e, 1), 4 * dim * dirac_ONE(0));

-       result += check_equal_simplify(dirac_trace(e, 2), e);
+       result += check_equal_simplify(dirac_trace(e, 2), canonicalize_clifford(e)); // e will be canonicalized by the calculation of the trace

        result += check_equal_simplify(dirac_trace(e, lst(0, 1)), 16 * dim);
 
        return result;
        result += check_equal_simplify(dirac_trace(e, lst(0, 1)), 16 * dim);
 
        return result;

diff --git a/ginac/clifford.cpp b/ginac/clifford.cpp
index 610073899543b4256769bcf50d401d693e5987db..67aa2c97343fe4d643fd36e4b86edd4284036c33 100644 (file)
--- a/ginac/clifford.cpp
+++ b/ginac/clifford.cpp
@@ -74,13 +74,7 @@ GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaR, tensor,
 // default constructors
 //////////
 
 // default constructors
 //////////
 

-static ex default_metric()
-{
-       static ex m = (new minkmetric)->setflag(status_flags::dynallocated);
-       return m;
-}
-
-clifford::clifford() : representation_label(0), metric(default_metric()), anticommuting(false)
+clifford::clifford() : representation_label(0), metric(0), anticommuting(true), commutator_sign(-1)

 {
        tinfo_key = TINFO_clifford;
 }
 {
        tinfo_key = TINFO_clifford;
 }


@@ -99,7 +93,7 @@ DEFAULT_CTOR(diracgammaR)
 /** Construct object without any indices. This constructor is for internal
  *  use only. Use the dirac_ONE() function instead.
  *  @see dirac_ONE */
 /** 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, bool anticommut) : inherited(b), representation_label(rl), metric(0), anticommuting(anticommut)
+clifford::clifford(const ex & b, unsigned char rl, bool anticommut) : inherited(b), representation_label(rl), metric(0), anticommuting(anticommut), commutator_sign(-1)

 {
        tinfo_key = TINFO_clifford;
 }
 {
        tinfo_key = TINFO_clifford;
 }


@@ -108,18 +102,18 @@ clifford::clifford(const ex & b, unsigned char rl, bool anticommut) : inherited(
  *  use only. Use the clifford_unit() or dirac_gamma() functions instead.
  *  @see clifford_unit
  *  @see dirac_gamma */
  *  use only. Use the clifford_unit() or dirac_gamma() functions instead.
  *  @see clifford_unit
  *  @see dirac_gamma */

-clifford::clifford(const ex & b, const ex & mu, const ex & metr, unsigned char rl, bool anticommut) : inherited(b, mu), representation_label(rl), metric(metr), anticommuting(anticommut)
+clifford::clifford(const ex & b, const ex & mu, const ex & metr, unsigned char rl, bool anticommut, int comm_sign) : inherited(b, mu), representation_label(rl), metric(metr), anticommuting(anticommut), commutator_sign(comm_sign)

 {
        GINAC_ASSERT(is_a<varidx>(mu));
        tinfo_key = TINFO_clifford;
 }
 
 {
        GINAC_ASSERT(is_a<varidx>(mu));
        tinfo_key = TINFO_clifford;
 }
 

-clifford::clifford(unsigned char rl, const ex & metr, bool anticommut, const exvector & v, bool discardable) : inherited(not_symmetric(), v, discardable), representation_label(rl), metric(metr), anticommuting(anticommut)
+clifford::clifford(unsigned char rl, const ex & metr, bool anticommut, int comm_sign, const exvector & v, bool discardable) : inherited(not_symmetric(), v, discardable), representation_label(rl), metric(metr), anticommuting(anticommut), commutator_sign(comm_sign)

 {
        tinfo_key = TINFO_clifford;
 }
 
 {
        tinfo_key = TINFO_clifford;
 }
 

-clifford::clifford(unsigned char rl, const ex & metr, bool anticommut, std::auto_ptr<exvector> vp) : inherited(not_symmetric(), vp), representation_label(rl), metric(metr), anticommuting(anticommut)
+clifford::clifford(unsigned char rl, const ex & metr, bool anticommut, int comm_sign, std::auto_ptr<exvector> vp) : inherited(not_symmetric(), vp), representation_label(rl), metric(metr), anticommuting(anticommut), commutator_sign(comm_sign)

 {
        tinfo_key = TINFO_clifford;
 }
 {
        tinfo_key = TINFO_clifford;
 }


@@ -135,6 +129,8 @@ clifford::clifford(const archive_node & n, lst & sym_lst) : inherited(n, sym_lst
        representation_label = rl;
        n.find_ex("metric", metric, sym_lst);
        n.find_bool("anticommuting", anticommuting);
        representation_label = rl;
        n.find_ex("metric", metric, sym_lst);
        n.find_bool("anticommuting", anticommuting);

+       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


@@ -143,6 +139,7 @@ void clifford::archive(archive_node & n) const
        n.add_unsigned("label", representation_label);
        n.add_ex("metric", metric);
        n.add_bool("anticommuting", anticommuting);
        n.add_unsigned("label", representation_label);
        n.add_ex("metric", metric);
        n.add_bool("anticommuting", anticommuting);

+       n.add_unsigned("commutator_sign+1", commutator_sign+1);

 }
 
 DEFAULT_UNARCHIVE(clifford)
 }
 
 DEFAULT_UNARCHIVE(clifford)


@@ -153,9 +150,6 @@ DEFAULT_ARCHIVING(diracgamma5)
 DEFAULT_ARCHIVING(diracgammaL)
 DEFAULT_ARCHIVING(diracgammaR)
 
 DEFAULT_ARCHIVING(diracgammaL)
 DEFAULT_ARCHIVING(diracgammaR)
 

-//////////
-// functions overriding virtual functions from base classes
-//////////

 
 ex clifford::get_metric(const ex & i, const ex & j, bool symmetrised) const
 {
 
 ex clifford::get_metric(const ex & i, const ex & j, bool symmetrised) const
 {


@@ -168,7 +162,8 @@ ex clifford::get_metric(const ex & i, const ex & j, bool symmetrised) const
                                return simplify_indexed(indexed(metric.op(0)*_ex1_2, i, j) + indexed(metric.op(0)*_ex1_2, j, i));
                        }
                } else {
                                return simplify_indexed(indexed(metric.op(0)*_ex1_2, i, j) + indexed(metric.op(0)*_ex1_2, j, i));
                        }
                } else {

-                       return indexed(metric.op(0), ex_to<symmetry>(ex_to<indexed>(metric).get_symmetry()), i, j);
+                       //return indexed(metric.op(0), ex_to<symmetry>(ex_to<indexed>(metric).get_symmetry()), i, j);
+                       return metric.subs(lst(metric.op(1) == i, metric.op(2) == j), subs_options::no_pattern);

                }
        } else {
                // should not really happen since all constructors but clifford() make the metric an indexed object
                }
        } else {
                // should not really happen since all constructors but clifford() make the metric an indexed object


@@ -186,6 +181,31 @@ bool clifford::same_metric(const ex & other) const
                return false;
 }
 
                return false;
 }
 

+//////////
+// 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);
+}
+

 int clifford::compare_same_type(const basic & other) const
 {
        GINAC_ASSERT(is_a<clifford>(other));
 int clifford::compare_same_type(const basic & other) const
 {
        GINAC_ASSERT(is_a<clifford>(other));


@@ -204,7 +224,7 @@ bool clifford::match_same_type(const basic & other) const
        GINAC_ASSERT(is_a<clifford>(other));
        const clifford &o = static_cast<const clifford &>(other);
 
        GINAC_ASSERT(is_a<clifford>(other));
        const clifford &o = static_cast<const clifford &>(other);
 

-       return (representation_label == o.representation_label) && same_metric(o);
+       return ((representation_label == o.representation_label) && (commutator_sign == o.get_commutator_sign()) && same_metric(o));

 }
 
 static bool is_dirac_slash(const ex & seq0)
 }
 
 static bool is_dirac_slash(const ex & seq0)


@@ -255,7 +275,7 @@ DEFAULT_PRINT_LATEX(diracgammaR, "gammaR", "{\\gamma_R}")
 static void base_and_index(const ex & c, ex & b, ex & i)
 {
        GINAC_ASSERT(is_a<clifford>(c));
 static void base_and_index(const ex & c, ex & b, ex & i)
 {
        GINAC_ASSERT(is_a<clifford>(c));

-       GINAC_ASSERT(c.nops() == 2);
+       GINAC_ASSERT(c.nops() == 2+1);

 
        if (is_a<cliffordunit>(c.op(0))) { // proper dirac gamma object or clifford unit
                i = c.op(1);
 
        if (is_a<cliffordunit>(c.op(0))) { // proper dirac gamma object or clifford unit
                i = c.op(1);


@@ -612,8 +632,10 @@ ex clifford::eval_ncmul(const exvector & v) const
                        bool a_is_cliffordunit = is_a<cliffordunit>(ag);
                        bool b_is_cliffordunit =  is_a<cliffordunit>(bg);
 
                        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)) {
-
+                       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)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
                                const ex & ia = a.op(1);
                                const ex & ib = b.op(1);
                                if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha


@@ -676,12 +698,12 @@ ex clifford::eval_ncmul(const exvector & v) const
 
 ex clifford::thiscontainer(const exvector & v) const
 {
 
 ex clifford::thiscontainer(const exvector & v) const
 {

-       return clifford(representation_label, get_metric(), is_anticommuting(), v);
+       return clifford(representation_label, metric, anticommuting, commutator_sign, v);

 }
 
 ex clifford::thiscontainer(std::auto_ptr<exvector> vp) const
 {
 }
 
 ex clifford::thiscontainer(std::auto_ptr<exvector> vp) const
 {

-       return clifford(representation_label, get_metric(), is_anticommuting(), vp);
+       return clifford(representation_label, metric, anticommuting, commutator_sign, vp);

 }
 
 ex diracgamma5::conjugate() const
 }
 
 ex diracgamma5::conjugate() const


@@ -726,7 +748,7 @@ ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl, bool anticomm
                } else {
                        throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be indexed exactly by two indices of same type as the given index"));
                }
                } else {
                        throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be indexed exactly by two indices of same type as the given index"));
                }

-       } else if (is_a<tensmetric>(metr)) {
+       } else if (is_a<tensor>(metr)) {

                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(unit, mu, indexed(metr, xi, chi), rl, anticommuting);
                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(unit, mu, indexed(metr, xi, chi), rl, anticommuting);


@@ -754,7 +776,7 @@ ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl, bool anticomm
                        throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be a square matrix with the same dimensions as index"));
                }
        } else {
                        throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be a square matrix with the same dimensions as index"));
                }
        } else {

-               throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be of type indexed, tensormetric or matrix"));
+               throw(std::invalid_argument("clifford_unit(): metric for Clifford unit must be of type indexed, tensor or matrix"));

        }
 }
 
        }
 }
 


@@ -767,7 +789,7 @@ ex dirac_gamma(const ex & mu, unsigned char rl)
 
        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());
 
        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(default_metric(), symmetric2(), xi, chi), rl, true);
+       return clifford(gamma, mu, indexed((new minkmetric)->setflag(status_flags::dynallocated), symmetric2(), xi, chi), rl, true);

 }
 
 ex dirac_gamma5(unsigned char rl)
 }
 
 ex dirac_gamma5(unsigned char rl)


@@ -793,7 +815,7 @@ 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
        // 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), default_metric(), rl);
+       return clifford(e, varidx(0, dim), 0, rl);

 }
 
 /** Check whether a given tinfo key (as returned by return_type_tinfo()
 }
 
 /** Check whether a given tinfo key (as returned by return_type_tinfo()


@@ -1051,18 +1073,21 @@ ex canonicalize_clifford(const ex & e_)
                                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;
                                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) {
                                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);
                                                ex save0 = it[0], save1 = it[1];
                                                ex b1, i1, b2, i2;
                                                base_and_index(it[0], b1, i1);
                                                base_and_index(it[1], b2, i2);

-                                               it[0] = (ex_to<clifford>(save0).get_metric(i1, i2, true) * b1 * b2).simplify_indexed();
-                                               it[1] = v.size() == 2 ? _ex2 * dirac_ONE(ex_to<clifford>(it[1]).get_representation_label()) : _ex2;
+                                               // 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;
                                                ex sum = ncmul(v);
                                                it[0] = save1;
                                                it[1] = save0;

-                                               sum -= ncmul(v, true);
+                                               sum += ex_to<clifford>(save0).get_commutator_sign() * ncmul(v, true);

                                                i->second = canonicalize_clifford(sum);
                                                goto next_sym;
                                        }
                                                i->second = canonicalize_clifford(sum);
                                                goto next_sym;
                                        }


@@ -1267,25 +1292,25 @@ lst clifford_to_lst(const ex & e, const ex & c, bool algebraic)
 
        if (algebraic) // check if algebraic method is applicable
                for (unsigned int i = 0; i < D; i++) 
 
        if (algebraic) // check if algebraic method is applicable
                for (unsigned int i = 0; i < D; i++) 

-                       if (pow(c.subs(mu == i), 2).is_zero() 
-                               or (not is_a<numeric>(pow(c.subs(mu == i), 2))))
+                       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; 
        if (algebraic) {
                for (unsigned int i = 0; i < D; i++) 
                        V.append(remove_dirac_ONE(
                                algebraic = false;
        lst V; 
        if (algebraic) {
                for (unsigned int i = 0; i < D; i++) 
                        V.append(remove_dirac_ONE(

-                                               simplify_indexed(canonicalize_clifford(e * c.subs(mu == i) +  c.subs(mu == i) * e))
-                                               / (2*pow(c.subs(mu == i), 2))));
+                                               simplify_indexed(canonicalize_clifford(e * c.subs(mu == i, subs_options::no_pattern) +  c.subs(mu == i, subs_options::no_pattern) * e))
+                                               / (2*pow(c.subs(mu == i, subs_options::no_pattern), 2))));

        } else {
                ex e1 = canonicalize_clifford(e);
                try {
                        for (unsigned int i = 0; i < D; i++) 
        } else {
                ex e1 = canonicalize_clifford(e);
                try {
                        for (unsigned int i = 0; i < D; i++) 

-                               V.append(get_clifford_comp(e1, c.subs(c.op(1) == 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(e1, true));
                        for (unsigned int i = 0; i < D; i++) 
                } catch  (std::exception &p) {
                        /* Try to expand dummy summations to simplify the expression*/
                        e1 = canonicalize_clifford(expand_dummy_sum(e1, true));
                        for (unsigned int i = 0; i < D; i++) 

-                               V.append(get_clifford_comp(e1, c.subs(c.op(1) == i)));
+                               V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));

                }
        }
        return V;
                }
        }
        return V;

diff --git a/ginac/clifford.h b/ginac/clifford.h
index 7b120734785a12c0e8f9b255430b1b9cd17b2e99..18ef03c2209fd073778c90b9e9d1644a392e7820 100644 (file)
--- a/ginac/clifford.h
+++ b/ginac/clifford.h
@@ -45,11 +45,11 @@ class clifford : public indexed
        // other constructors
 public:
        clifford(const ex & b, unsigned char rl = 0, bool anticommut = false);
        // other constructors
 public:
        clifford(const ex & b, unsigned char rl = 0, bool anticommut = false);

-       clifford(const ex & b, const ex & mu,  const ex & metr, unsigned char rl = 0, bool anticommut = false);
+       clifford(const ex & b, const ex & mu,  const ex & metr, unsigned char rl = 0, bool anticommut = false, int comm_sign = -1);

 
        // internal constructors
 
        // internal constructors

-       clifford(unsigned char rl, const ex & metr, bool anticommut, const exvector & v, bool discardable = false);
-       clifford(unsigned char rl, const ex & metr, bool anticommut, std::auto_ptr<exvector> vp);
+       clifford(unsigned char rl, const ex & metr, bool anticommut, int comm_sign, const exvector & v, bool discardable = false);
+       clifford(unsigned char rl, const ex & metr, bool anticommut, int comm_sign, std::auto_ptr<exvector> vp);

 
        // functions overriding virtual functions from base classes
 public:
 
        // functions overriding virtual functions from base classes
 public:


@@ -66,22 +66,33 @@ protected:
 public:
        unsigned char get_representation_label() const { return representation_label; }
        ex get_metric() const { return metric; }
 public:
        unsigned char get_representation_label() const { return representation_label; }
        ex get_metric() const { return metric; }

-       ex get_metric(const ex & i, const ex & j, bool symmetrised = false) const;
+       virtual ex get_metric(const ex & i, const ex & j, bool symmetrised = false) const;

        bool same_metric(const ex & other) const;
        bool is_anticommuting() const { return anticommuting; } //**< See the member variable anticommuting */
        bool same_metric(const ex & other) const;
        bool is_anticommuting() const { return anticommuting; } //**< See the member variable anticommuting */

+       int get_commutator_sign() const { return commutator_sign; } //**< See the member variable commutator_sign */
+
+       inline size_t nops() const {return inherited::nops() + 1; }
+       ex op(size_t i) const;
+       ex & let_op(size_t i);
+       ex subs(const exmap & m, unsigned options = 0) const { clifford c = ex_to<clifford>(inherited::subs(m, options)); c.metric_subs(m, options); return c;}
+       ex subs(const lst & ls, const lst & lr, unsigned options = 0) const { clifford c = ex_to<clifford>(ex(*this).subs(ls, lr, options)); c.metric_subs(ls, lr, options); return c;}
+       ex subs(const ex & e, unsigned options = 0) const{ clifford c = ex_to<clifford>(ex(*this).subs(e, options)); c.metric_subs(e, options); return c;};

 
 protected:
        void do_print_dflt(const print_dflt & c, unsigned level) const;
        void do_print_latex(const print_latex & c, unsigned level) const;
 
 protected:
        void do_print_dflt(const print_dflt & c, unsigned level) const;
        void do_print_latex(const print_latex & c, unsigned level) const;

+       void metric_subs(const exmap & m, unsigned options = 0) { metric = metric.subs(m, options); }
+       void metric_subs(const lst & ls, const lst & lr, unsigned options = 0) { metric = metric.subs(ls, lr, options); }
+       void metric_subs(const ex & e, unsigned options = 0) { metric = metric.subs(e, options); }

 
        // member variables
 
        // member variables

-private:
+protected:

        unsigned char representation_label; /**< Representation label to distinguish independent spin lines */
        ex metric; /**< Metric of the space, all constructors make it an indexed object */
        bool anticommuting; /**< Simplifications for anticommuting units is much simpler and we need this info readily available */
        unsigned char representation_label; /**< Representation label to distinguish independent spin lines */
        ex metric; /**< Metric of the space, all constructors make it an indexed object */
        bool anticommuting; /**< Simplifications for anticommuting units is much simpler and we need this info readily available */

+       int commutator_sign; /**< It is the sign in the definition e~i e~j +/- e~j e~i = B(i, j) + B(j, i)*/

 };
 
 };
 

-

 /** This class represents the Clifford algebra unity element. */
 class diracone : public tensor
 {
 /** This class represents the Clifford algebra unity element. */
 class diracone : public tensor
 {