Added contribution from Vladimir Kisil: generalization of clifford algebra:
authorJens Vollinga <vollinga@thep.physik.uni-mainz.de>
Thu, 29 Apr 2004 19:53:12 +0000 (19:53 +0000)
committerJens Vollinga <vollinga@thep.physik.uni-mainz.de>
Thu, 29 Apr 2004 19:53:12 +0000 (19:53 +0000)
* different metrics can now be defined for clifford
* new class "clifford_unit" to represent generators of the algebra

ginac/clifford.cpp
ginac/clifford.h
ginac/tinfos.h

index e6f9c2914be0a638f0e76cabf4879ca2f18fb727..7c77b2a0c2ada4c64e084a7e6608abc65c56a7ff 100644 (file)
  *  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA
  */
 
-#include <iostream>
-#include <stdexcept>
-
 #include "clifford.h"
+
 #include "ex.h"
 #include "idx.h"
 #include "ncmul.h"
 #include "lst.h"
 #include "relational.h"
 #include "operators.h"
+#include "add.h"
 #include "mul.h"
+#include "power.h"
+#include "matrix.h"
 #include "archive.h"
 #include "utils.h"
 
@@ -47,7 +48,11 @@ 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(diracgamma, tensor,
+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))
 
@@ -67,12 +72,13 @@ GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(diracgammaR, tensor,
 // default constructors
 //////////
 
-clifford::clifford() : representation_label(0)
+clifford::clifford() : representation_label(0), metric(lorentz_g(varidx((new symbol)->setflag(status_flags::dynallocated),4),varidx((new symbol)->setflag(status_flags::dynallocated),4)))
 {
        tinfo_key = TINFO_clifford;
 }
 
 DEFAULT_CTOR(diracone)
+DEFAULT_CTOR(cliffordunit)
 DEFAULT_CTOR(diracgamma)
 DEFAULT_CTOR(diracgamma5)
 DEFAULT_CTOR(diracgammaL)
@@ -85,26 +91,27 @@ DEFAULT_CTOR(diracgammaR)
 /** 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)
 {
        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) : inherited(b, mu), representation_label(rl), metric(metr)
 {
        GINAC_ASSERT(is_a<varidx>(mu));
        tinfo_key = TINFO_clifford;
 }
 
-clifford::clifford(unsigned char rl, const exvector & v, bool discardable) : inherited(sy_none(), v, discardable), representation_label(rl)
+clifford::clifford(unsigned char rl, const exvector & v, bool discardable, const ex & metr) : inherited(sy_none(), v, discardable), representation_label(rl), metric(metr)
 {
        tinfo_key = TINFO_clifford;
 }
 
-clifford::clifford(unsigned char rl, std::auto_ptr<exvector> vp) : inherited(sy_none(), vp), representation_label(rl)
+clifford::clifford(unsigned char rl, std::auto_ptr<exvector> vp, const ex & metr) : inherited(sy_none(), vp), representation_label(rl), metric(metr)
 {
        tinfo_key = TINFO_clifford;
 }
@@ -118,16 +125,21 @@ clifford::clifford(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
        unsigned rl;
        n.find_unsigned("label", rl);
        representation_label = rl;
+       ex m;
+       n.find_ex("metric", m, sym_lst);
+       metric = m;
 }
 
 void clifford::archive(archive_node &n) const
 {
        inherited::archive(n);
        n.add_unsigned("label", representation_label);
+       n.add_ex("metric", metric);
 }
 
 DEFAULT_UNARCHIVE(clifford)
 DEFAULT_ARCHIVING(diracone)
+DEFAULT_ARCHIVING(cliffordunit)
 DEFAULT_ARCHIVING(diracgamma)
 DEFAULT_ARCHIVING(diracgamma5)
 DEFAULT_ARCHIVING(diracgammaL)
@@ -137,6 +149,23 @@ DEFAULT_ARCHIVING(diracgammaR)
 // functions overriding virtual functions from base classes
 //////////
 
+ex clifford::get_metric(const ex & i, const ex & j) const 
+{
+       return metric.subs(metric.op(1) == i).subs(metric.op(2) == j);
+}
+
+bool clifford::same_metric(const ex & other) const
+{
+       if (is_a<clifford>(other)) {
+               ex m = get_metric();
+               return m.is_equal(ex_to<clifford>(other).get_metric(m.op(1), m.op(2)));
+       } else if (is_a<indexed>(other)) {
+               ex m = get_metric(other.op(1), other.op(2));
+               return m.is_equal(other);
+       } else
+               return false;
+}
+
 int clifford::compare_same_type(const basic & other) const
 {
        GINAC_ASSERT(is_a<clifford>(other));
@@ -155,13 +184,13 @@ 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;
+       return (representation_label == o.representation_label) && 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<diracgamma>(seq0) &&
+              !is_a<diracgammaR>(seq0) && !is_a<cliffordunit>(seq0) &&
               !is_a<diracone>(seq0);
 }
 
@@ -187,12 +216,14 @@ void clifford::do_print_latex(const print_latex & c, unsigned level) const
 }
 
 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(cliffordunit, "e", "e")
 DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
 DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
 DEFAULT_PRINT_LATEX(diracgammaL, "gammaL", "{\\gamma_L}")
@@ -204,7 +235,7 @@ static void base_and_index(const ex & c, ex & b, ex & i)
        GINAC_ASSERT(is_a<clifford>(c));
        GINAC_ASSERT(c.nops() == 2);
 
-       if (is_a<diracgamma>(c.op(0))) { // proper dirac gamma object
+       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
@@ -308,6 +339,104 @@ bool diracgamma::contract_with(exvector::iterator self, exvector::iterator other
        return false;
 }
 
+/** An utility function looking for given metric within exvector, 
+ * used in cliffordunit::contract_with() */
+int find_same_metric(exvector & v, ex & c){
+       for (int i=0; i<v.size();i++){
+               if (!is_a<clifford>(v[i]) && is_a<indexed>(v[i]) 
+                   && ex_to<clifford>(c).same_metric(v[i]) 
+                   && (ex_to<varidx>(c.op(1)) == ex_to<indexed>(v[i]).get_indices()[0]
+                       || ex_to<varidx>(c.op(1)).toggle_variance() == ex_to<indexed>(v[i]).get_indices()[0])){
+                       return ++i; // next to found
+               }
+       }
+       return 0; //nothing found
+}
+
+/** Contraction of a Clifford  units 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;
+               // Find if a previous contraction produces the square of self
+               int prev_square = find_same_metric(v, self[0]);
+               varidx d((new symbol)->setflag(status_flags::dynallocated),  ex_to<idx>(ex_to<idx>(self->op(1)).get_dim()));
+               ex squared_metric = unit.get_metric(self->op(1), d)*unit.get_metric(d.toggle_variance(), other->op(1));
+
+               // e~mu e.mu = Tr ONE
+               if (other - self == 1) {
+                       if (prev_square != 0) {
+                               *self = squared_metric;
+                               v[prev_square-1] = _ex1;
+                       } else
+                               *self = unit.get_metric(self->op(1), other->op(1));
+                       *other = dirac_ONE(rl);
+                       return true;
+
+               // e~mu e~alpha e.mu = (2e~alpha^2-Tr) e~alpha
+               } else if (other - self == 2
+                       && is_a<clifford>(self[1])) {
+
+                       const ex & ia = self[1].op(1);
+                       const ex & ib = self[1].op(1);
+                       if (is_a<tensmetric>(unit.get_metric().op(0)))
+                               *self = 2 - unit.get_metric(self->op(1), other->op(1));
+                       else if (prev_square != 0) {
+                         *self = 2-squared_metric;
+                               v[prev_square-1] = _ex1;
+                       } else 
+                               *self = 2*unit.get_metric(ia, ib) - unit.get_metric(self->op(1), other->op(1));
+                       *other = _ex1;
+                       return true;
+
+               // e~mu S e~alpha e.mu = 2 e~alpha^3 S - e~mu S e.mu e~alpha
+               // (commutate contracted indices towards each other, simplify_indexed()
+               // will re-expand and re-run the simplification)
+               } else {
+                       exvector::iterator it = self + 1, next_to_last = other - 1;
+                       while (it != other) {
+                               if (!is_a<clifford>(*it))
+                                       return false;
+                               ++it;
+                       }
+
+                       it = self + 1;
+                       ex S = _ex1;
+                       while (it != next_to_last) {
+                               S *= *it;
+                               *it++ = _ex1;
+                       }
+
+                       const ex & ia = next_to_last->op(1);
+                       const ex & ib = next_to_last->op(1);
+                       if (is_a<tensmetric>(unit.get_metric().op(0)))
+                               *self = 2 * (*next_to_last) * S - (*self) * S * (*other) * (*next_to_last);
+                       else if (prev_square != 0) {
+                               *self = 2 * (*next_to_last) * S  - (*self) * S * (*other) * (*next_to_last)*unit.get_metric(self->op(1),self->op(1));
+                       v[prev_square-1] = _ex1;
+                       } else 
+                               *self = 2 * (*next_to_last) * S* unit.get_metric(ia,ib) - (*self) * S * (*other) * (*next_to_last);
+                       *next_to_last = _ex1;
+                       *other = _ex1;
+                       return true;
+               }
+
+       } 
+
+       return false;
+}
+
 /** Perform automatic simplification on noncommutative product of clifford
  *  objects. This removes superfluous ONEs, permutes gamma5/L/R's to the front
  *  and removes squares of gamma objects. */
@@ -410,15 +539,15 @@ ex clifford::eval_ncmul(const exvector & v) const
 
                        const ex & ag = a.op(0);
                        const ex & bg = b.op(0);
-                       bool a_is_diracgamma = is_a<diracgamma>(ag);
-                       bool b_is_diracgamma = is_a<diracgamma>(bg);
+                       bool a_is_cliffordunit = is_a<cliffordunit>(ag);
+                       bool b_is_cliffordunit =  is_a<cliffordunit>(bg);
 
-                       if (a_is_diracgamma && b_is_diracgamma) {
+                       if (a_is_cliffordunit && b_is_cliffordunit && ex_to<clifford>(a).same_metric(b)) {
 
                                const ex & ia = a.op(1);
                                const ex & ib = b.op(1);
                                if (ia.is_equal(ib)) { // gamma~alpha gamma~alpha -> g~alpha~alpha
-                                       a = lorentz_g(ia, ib);
+                                       a = ex_to<clifford>(a).get_metric(ia,ib);
                                        b = dirac_ONE(representation_label);
                                        something_changed = true;
                                }
@@ -455,10 +584,11 @@ ex clifford::eval_ncmul(const exvector & v) const
                                a = dirac_ONE(representation_label);
                                something_changed = true;
 
-                       } else if (!a_is_diracgamma && !b_is_diracgamma && ag.is_equal(bg)) {
+                       } 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;
@@ -476,12 +606,12 @@ ex clifford::eval_ncmul(const exvector & v) const
 
 ex clifford::thiscontainer(const exvector & v) const
 {
-       return clifford(representation_label, v);
+       return clifford(representation_label, v, false, get_metric());
 }
 
 ex clifford::thiscontainer(std::auto_ptr<exvector> vp) const
 {
-       return clifford(representation_label, vp);
+       return clifford(representation_label, vp, get_metric());
 }
 
 ex diracgamma5::conjugate() const
@@ -508,12 +638,28 @@ ex dirac_ONE(unsigned char rl)
        return clifford(diracone(), rl);
 }
 
+ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl)
+{
+       if (!is_a<varidx>(mu))
+               throw(std::invalid_argument("index of Clifford unit must be of type varidx"));
+       if (!is_a<indexed>(metr))
+               throw(std::invalid_argument("metric for Clifford unit must be of type indexed"));
+       exvector d = ex_to<indexed>(metr).get_indices();
+       if (d.size() > 2 || ex_to<idx>(d[0]).get_dim() != ex_to<idx>(d[1]).get_dim())
+         //|| ex_to<idx>(d[0]).get_dim() != ex_to<idx>(mu).get_dim())
+               throw(std::invalid_argument("metric is not square"));
+       else
+               ex_to<idx>(mu).replace_dim(ex_to<idx>(d[0]).get_dim());
+       return clifford(cliffordunit(), mu, metr, rl);
+}
+
 ex dirac_gamma(const ex & mu, unsigned char rl)
 {
        if (!is_a<varidx>(mu))
                throw(std::invalid_argument("index of Dirac gamma must be of type varidx"));
 
-       return clifford(diracgamma(), mu, rl);
+       ex dim = ex_to<idx>(mu).get_dim();
+       return clifford(diracgamma(), mu, lorentz_g(varidx((new symbol)->setflag(status_flags::dynallocated), dim),varidx((new symbol)->setflag(status_flags::dynallocated), dim)), rl);
 }
 
 ex dirac_gamma5(unsigned char rl)
@@ -756,7 +902,7 @@ ex canonicalize_clifford(const ex & e)
                                        ex b1, i1, b2, i2;
                                        base_and_index(it[0], b1, i1);
                                        base_and_index(it[1], b2, i2);
-                                       it[0] = (lorentz_g(i1, i2) * b1 * b2).simplify_indexed();
+                                       it[0] = (ex_to<clifford>(save0).get_metric(i1, i2) * b1 * b2).simplify_indexed();
                                        it[1] = v.size() == 2 ? _ex2 * dirac_ONE(ex_to<clifford>(it[1]).get_representation_label()) : _ex2;
                                        ex sum = ncmul(v);
                                        it[0] = save1;
@@ -773,4 +919,82 @@ next_sym:  ;
        return aux.subs(srl, subs_options::no_pattern).simplify_indexed();
 }
 
-} // namespace GiNaC
+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 (-1)*e;
+       } else if (is_a<add>(e)) {
+               return e.map(fcn);
+       } else if (is_a<ncmul>(e)) {
+                       return e.map(fcn);
+       } else if (is_a<power>(e)) {
+               return pow(clifford_prime(e.op(0)),e.op(1));
+       } 
+       else 
+               return e;
+};
+
+ex delete_ONE (const ex &e)  {
+       pointer_to_map_function fcn(delete_ONE);
+       if (is_a<clifford>(e) && is_a<diracone>(e.op(0))) {
+               return 1;
+       } else if (is_a<add>(e)) {
+               return e.map(fcn);
+       } else if (is_a<ncmul>(e)) {
+                       return e.map(fcn);
+       } else if (is_a<mul>(e)) {
+                       return e.map(fcn);
+       } else if (is_a<power>(e)) {
+               return pow(delete_ONE(e.op(0)),e.op(1));
+       } 
+       else 
+               return e;
+};
+
+ex clifford_norm(const ex &e){
+       return sqrt(delete_ONE((e * clifford_bar(e)).simplify_indexed()));
+} ; 
+
+ex clifford_inverse(const ex &e) {
+       ex norm = clifford_norm(e);
+       if ( !norm.is_zero() )  
+               return  clifford_bar(e)/pow(norm,2);
+}; 
+
+ex lst_to_clifford (const ex &v, const ex &mu, const ex &metr, unsigned char rl) {
+       unsigned min, max;
+       if (!ex_to<idx>(mu).is_dim_numeric())
+               throw(std::invalid_argument("Index should have a numeric dimension"));
+       unsigned dim = (ex_to<numeric>(ex_to<idx>(mu).get_dim())).to_int();
+       ex c = clifford_unit(mu, metr, rl);
+
+       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)
+                               if (is_a<varidx>(mu)) // need to swap variance
+                                       return indexed(v,ex_to<varidx>(mu).toggle_variance())* c; 
+                               else 
+                                       return indexed(v,mu)* c; 
+                       else 
+                               throw(std::invalid_argument("Dimensions of vector and clifford unit mismatch"));
+               else
+                       throw(std::invalid_argument("First argument should be a vector vector"));
+       } else if (is_a<lst>(v)) 
+               if (dim == ex_to<lst>(v).nops())
+                       return indexed(matrix(dim,1,ex_to<lst>(v)),ex_to<varidx>(mu).toggle_variance())* c; 
+               else
+                       throw(std::invalid_argument("List length and dimension of clifford unit mismatch"));
+       else
+               throw(std::invalid_argument("Cannot construct from anything but list or vector"));
+};
+
+ } // namespace GiNaC
+
index 7d6cc80e0240bc4415ac282af7cd70e3336d56be..3fa3627d0dd6b9d131c4d0700ba4e01127650026 100644 (file)
@@ -25,6 +25,8 @@
 
 #include "indexed.h"
 #include "tensor.h"
+#include "symbol.h"
+#include "idx.h"
 
 namespace GiNaC {
 
@@ -41,11 +43,11 @@ class clifford : public indexed
        // other constructors
 public:
        clifford(const ex & b, unsigned char rl = 0);
-       clifford(const ex & b, const ex & mu, unsigned char rl = 0);
+       clifford(const ex & b, const ex & mu,  const ex & metr, unsigned char rl = 0);
 
        // internal constructors
-       clifford(unsigned char rl, const exvector & v, bool discardable = false);
-       clifford(unsigned char rl, std::auto_ptr<exvector> vp);
+       clifford(unsigned char rl, const exvector & v, bool discardable = false,  const ex & metr = lorentz_g(varidx((new symbol)->setflag(status_flags::dynallocated), 4),varidx((new symbol)->setflag(status_flags::dynallocated), 4)));
+       clifford(unsigned char rl, std::auto_ptr<exvector> vp,  const ex & metr = lorentz_g(varidx((new symbol)->setflag(status_flags::dynallocated),4),varidx((new symbol)->setflag(status_flags::dynallocated),4)));
 
        // functions overriding virtual functions from base classes
 protected:
@@ -59,6 +61,9 @@ protected:
        // non-virtual functions in this class
 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) const;
+       bool same_metric(const ex & other) const;
 
 protected:
        void do_print_dflt(const print_dflt & c, unsigned level) const;
@@ -67,6 +72,7 @@ protected:
        // member variables
 private:
        unsigned char representation_label; /**< Representation label to distinguish independent spin lines */
+       ex metric;
 };
 
 
@@ -82,10 +88,29 @@ protected:
 };
 
 
+/** This class represents the Clifford algebra generators (units). */
+class cliffordunit : public tensor
+{
+       GINAC_DECLARE_REGISTERED_CLASS(cliffordunit, tensor)
+
+         // other constructors
+protected:
+        cliffordunit(unsigned ti) : inherited(ti) {}
+                                                                                                    
+       // functions overriding virtual functions from base classes
+public:
+       bool contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const;
+       // non-virtual functions in this class
+protected:
+       void do_print(const print_context & c, unsigned level) const;
+       void do_print_latex(const print_latex & c, unsigned level) const;
+};
+
+
 /** This class represents the Dirac gamma Lorentz vector. */
-class diracgamma : public tensor
+class diracgamma : public cliffordunit
 {
-       GINAC_DECLARE_REGISTERED_CLASS(diracgamma, tensor)
+       GINAC_DECLARE_REGISTERED_CLASS(diracgamma, cliffordunit)
 
        // functions overriding virtual functions from base classes
 public:
@@ -160,6 +185,14 @@ template<> inline bool is_exactly_a<clifford>(const basic & obj)
  *  @return newly constructed object */
 ex dirac_ONE(unsigned char rl = 0);
 
+/** Create a Clifford unit object.
+ *
+ *  @param mu Index (must be of class varidx or a derived class)
+ *  @param metr Metric (must be of class tensor or a derived class)
+ *  @param rl Representation label
+ *  @return newly constructed Clifford unit object */
+ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
+
 /** Create a Dirac gamma object.
  *
  *  @param mu Index (must be of class varidx or a derived class)
@@ -208,6 +241,33 @@ ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
  *  to check two expressions for equality. */
 ex canonicalize_clifford(const ex & e);
 
+/** Automorphism of the Clifford algebra, simply changes signs of all 
+ *  clifford units. */
+ex clifford_prime (const ex &e) ;
+
+/** Main anti-automorphism of the Clifford algebra: make reversion 
+ * and changes signs of all clifford units*/
+inline ex clifford_bar(const ex &e) { return clifford_prime(e.conjugate());};
+
+/** Reversion of the Clifford algebra, coinsides with the conjugate() */
+inline ex clifford_star(const ex &e) {  return e.conjugate();};
+
+ex delete_ONE (const ex &e);
+
+/** Calculation of the norm in the Clifford algebra */
+ex clifford_norm(const ex &e) ; 
+
+/** Calculation of the inverse in the Clifford algebra */
+ex clifford_inverse(const ex &e) ; 
+
+/** List or vector conversion into the Clifford vector 
+ *  @param v List or vector of coordinates
+ *  @param mu Index (must be of class varidx or a derived class)
+ *  @param metr Metric (must be of class tensor or a derived class) 
+ *  @param rl Representation label
+ *  @return Clifford vector with given components */
+ex lst_to_clifford(const ex &v, const ex &mu,  const ex &metr, unsigned char rl = 0) ;
+
 } // namespace GiNaC
 
 #endif // ndef __GINAC_CLIFFORD_H__
index 53e9e84b9b5cea888ae2d40b36d2361be294c153..8bb170a4af69517ef44fb74cff817b603fd14629 100644 (file)
@@ -81,6 +81,7 @@ const unsigned TINFO_su3f          = 0x000e100aU;
 const unsigned TINFO_su3d          = 0x000e100bU;
 const unsigned TINFO_diracone      = 0x000e100cU;
 const unsigned TINFO_diracgamma    = 0x000e100dU;
+const unsigned TINFO_cliffordunit    = 0x000e101dU;
 const unsigned TINFO_diracgamma5   = 0x000e100eU;
 const unsigned TINFO_diracgammaL   = 0x000e100fU;
 const unsigned TINFO_diracgammaR   = 0x000e1010U;