* Interface to GiNaC's clifford algebra (Dirac gamma) objects. */
/*
- * GiNaC Copyright (C) 1999-2004 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2011 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
*/
-#ifndef __GINAC_CLIFFORD_H__
-#define __GINAC_CLIFFORD_H__
+#ifndef GINAC_CLIFFORD_H
+#define GINAC_CLIFFORD_H
#include "indexed.h"
#include "tensor.h"
#include "symbol.h"
#include "idx.h"
-namespace GiNaC {
+#include <set>
+namespace GiNaC {
/** This class holds an object representing an element of the Clifford
* algebra (the Dirac gamma matrices). These objects only carry Lorentz
* indices. Spinor indices are hidden. A representation label (an unsigned
* 8-bit integer) is used to distinguish elements from different Clifford
- * algebras (objects with different labels commute). */
+ * algebras (objects with different labels commutate). */
class clifford : public indexed
{
GINAC_DECLARE_REGISTERED_CLASS(clifford, indexed)
-
// other constructors
public:
clifford(const ex & b, unsigned char rl = 0);
- clifford(const ex & b, const ex & mu, const ex & metr, unsigned char rl = 0);
+ clifford(const ex & b, const ex & mu, const ex & metr, unsigned char rl = 0, int comm_sign = -1);
// internal constructors
- clifford(unsigned char rl, const ex & metr, const exvector & v, bool discardable = false);
- clifford(unsigned char rl, const ex & metr, std::auto_ptr<exvector> vp);
+ clifford(unsigned char rl, const ex & metr, int comm_sign, const exvector & v, bool discardable = false);
+ clifford(unsigned char rl, const ex & metr, int comm_sign, std::auto_ptr<exvector> vp);
// functions overriding virtual functions from base classes
+public:
+ unsigned precedence() const { return 65; }
+ void archive(archive_node& n) const;
+ void read_archive(const archive_node& n, lst& sym_lst);
protected:
ex eval_ncmul(const exvector & v) const;
bool match_same_type(const basic & other) const;
ex thiscontainer(const exvector & v) const;
ex thiscontainer(std::auto_ptr<exvector> vp) const;
unsigned return_type() const { return return_types::noncommutative; }
- unsigned return_type_tinfo() const { return TINFO_clifford + representation_label; }
-
+ return_type_t return_type_tinfo() const;
// 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;
+ virtual ex get_metric(const ex & i, const ex & j, bool symmetrised = false) const;
bool same_metric(const ex & other) const;
+ 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;
protected:
void do_print_dflt(const print_dflt & c, unsigned level) const;
void do_print_latex(const print_latex & c, unsigned level) const;
// member variables
-private:
+protected:
unsigned char representation_label; /**< Representation label to distinguish independent spin lines */
- ex metric;
+ ex metric; /**< Metric of the space, all constructors make it an indexed object */
+ int commutator_sign; /**< It is the sign in the definition e~i e~j +/- e~j e~i = B(i, j) + B(j, i)*/
};
-
+GINAC_DECLARE_UNARCHIVER(clifford);
/** This class represents the Clifford algebra unity element. */
class diracone : public tensor
void do_print(const print_context & c, unsigned level) const;
void do_print_latex(const print_latex & c, unsigned level) const;
};
+GINAC_DECLARE_UNARCHIVER(diracone);
/** This class represents the Clifford algebra generators (units). */
{
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;
void do_print(const print_context & c, unsigned level) const;
void do_print_latex(const print_latex & c, unsigned level) const;
};
+GINAC_DECLARE_UNARCHIVER(diracgamma);
-/** This class represents the Dirac gamma5 object which anticommutes with
+/** This class represents the Dirac gamma5 object which anticommutates with
* all other gammas. */
class diracgamma5 : public tensor
{
void do_print(const print_context & c, unsigned level) const;
void do_print_latex(const print_latex & c, unsigned level) const;
};
+GINAC_DECLARE_UNARCHIVER(diracgamma5);
/** This class represents the Dirac gammaL object which behaves like
void do_print(const print_context & c, unsigned level) const;
void do_print_latex(const print_latex & c, unsigned level) const;
};
+GINAC_DECLARE_UNARCHIVER(diracgammaL);
/** This class represents the Dirac gammaL object which behaves like
void do_print(const print_context & c, unsigned level) const;
void do_print_latex(const print_latex & c, unsigned level) const;
};
+GINAC_DECLARE_UNARCHIVER(diracgammaR);
// global functions
-/** Specialization of is_exactly_a<clifford>(obj) for clifford objects. */
-template<> inline bool is_exactly_a<clifford>(const basic & obj)
+/** Check whether a given return_type_t object (as returned by return_type_tinfo()
+ * is that of a clifford object (with an arbitrary representation label).
+ *
+ * @param ti tinfo key */
+inline bool is_clifford_tinfo(const return_type_t& ti)
{
- return obj.tinfo()==TINFO_clifford;
+ return *(ti.tinfo) == typeid(clifford);
}
/** Create a Clifford unity object.
/** 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 metr Metric (should be indexed, tensmetric or a derived class, or a matrix)
* @param rl Representation label
* @return newly constructed Clifford unit object */
ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
* @param rl Representation label */
ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
+/** Calculate dirac traces over the specified set of representation labels.
+ * The computed trace is a linear functional that is equal to the usual
+ * trace only in D = 4 dimensions. In particular, the functional is not
+ * always cyclic in D != 4 dimensions when gamma5 is involved.
+ *
+ * @param e Expression to take the trace of
+ * @param rls Set of representation labels
+ * @param trONE Expression to be returned as the trace of the unit matrix */
+ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE = 4);
+
+/** Calculate dirac traces over the specified list of representation labels.
+ * The computed trace is a linear functional that is equal to the usual
+ * trace only in D = 4 dimensions. In particular, the functional is not
+ * always cyclic in D != 4 dimensions when gamma5 is involved.
+ *
+ * @param e Expression to take the trace of
+ * @param rll List of representation labels
+ * @param trONE Expression to be returned as the trace of the unit matrix */
+ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
+
/** Calculate the trace of an expression containing gamma objects with
* a specified representation label. The computed trace is a linear
* functional that is equal to the usual trace only in D = 4 dimensions.
/** Reversion of the Clifford algebra, coincides with the conjugate(). */
inline ex clifford_star(const ex & e) { return e.conjugate(); }
-ex delete_ONE(const ex &e);
+/** Replaces dirac_ONE's (with a representation_label no less than rl) in e with 1.
+ * For the default value rl = 0 remove all of them. Aborts if e contains any
+ * clifford_unit with representation_label to be removed.
+ *
+ * @param e Expression to be processed
+ * @param rl Value of representation label
+ * @param options Defines some internal use */
+ex remove_dirac_ONE(const ex & e, unsigned char rl = 0, unsigned options = 0);
+
+/** Returns the maximal representation label of a clifford object
+ * if e contains at least one, otherwise returns -1
+ *
+ * @param e Expression to be processed
+ * @ignore_ONE defines if clifford_ONE should be ignored in the search*/
+int clifford_max_label(const ex & e, bool ignore_ONE = false);
/** Calculation of the norm in the Clifford algebra. */
ex clifford_norm(const ex & e);
*
* @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 metr Metric (should be indexed, tensmetric or a derived class, or a matrix)
* @param rl Representation label
+ * @param e Clifford unit object
* @return Clifford vector with given components */
ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl = 0);
+ex lst_to_clifford(const ex & v, const ex & e);
+
+/** An inverse function to lst_to_clifford(). For given Clifford vector extracts
+ * its components with respect to given Clifford unit. Obtained components may
+ * contain Clifford units with a different metric. Extraction is based on
+ * the algebraic formula (e * c.i + c.i * e)/ pow(e.i, 2) for non-degenerate cases
+ * (i.e. neither pow(e.i, 2) = 0).
+ *
+ * @param e Clifford expression to be decomposed into components
+ * @param c Clifford unit defining the metric for splitting (should have numeric dimension of indices)
+ * @param algebraic Use algebraic or symbolic algorithm for extractions
+ * @return List of components of a Clifford vector*/
+lst clifford_to_lst(const ex & e, const ex & c, bool algebraic=true);
+
+/** Calculations of Moebius transformations (conformal map) defined by a 2x2 Clifford matrix
+ * (a b\\c d) in linear spaces with arbitrary signature. The expression is
+ * (a * x + b)/(c * x + d), where x is a vector build from list v with metric G.
+ * (see Jan Cnops. An introduction to {D}irac operators on manifolds, v.24 of
+ * Progress in Mathematical Physics. Birkhauser Boston Inc., Boston, MA, 2002.)
+ *
+ * @param a (1,1) entry of the defining matrix
+ * @param b (1,2) entry of the defining matrix
+ * @param c (2,1) entry of the defining matrix
+ * @param d (2,2) entry of the defining matrix
+ * @param v Vector to be transformed
+ * @param G Metric of the surrounding space, may be a Clifford unit then the next parameter is ignored
+ * @param rl Representation label
+ * @return List of components of the transformed vector*/
+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 = 0);
+
+/** The second form of Moebius transformations defined by a 2x2 Clifford matrix M
+ * This function takes the transformation matrix M as a single entity.
+ *
+ * @param M the defining matrix
+ * @param v Vector to be transformed
+ * @param G Metric of the surrounding space, may be a Clifford unit then the next parameter is ignored
+ * @param rl Representation label
+ * @return List of components of the transformed vector*/
+ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl = 0);
} // namespace GiNaC
-#endif // ndef __GINAC_CLIFFORD_H__
+#endif // ndef GINAC_CLIFFORD_H
git://www.ginac.de / ginac.git / blobdiff