* Interface to GiNaC's clifford algebra (Dirac gamma) objects. */
/*
- * GiNaC Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2005 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__
#include "indexed.h"
#include "tensor.h"
+#include "symbol.h"
+#include "idx.h"
+
+#include <set>
namespace GiNaC {
* 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, unsigned char rl = 0);
+ 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, int comm_sign = -1);
// 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 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:
+ unsigned precedence() const { return 65; }
protected:
ex eval_ncmul(const exvector & v) const;
bool match_same_type(const basic & other) const;
// non-virtual functions in this class
public:
- unsigned char get_representation_label() const {return representation_label;}
+ unsigned char get_representation_label() const { return representation_label; }
+ ex get_metric() const { return metric; }
+ 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 */
+ 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;
+ 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
-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 */
+ 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 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:
};
-/** 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
{
GINAC_DECLARE_REGISTERED_CLASS(diracgamma5, tensor)
+ // functions overriding virtual functions from base classes
+ ex conjugate() const;
+
// non-virtual functions in this class
protected:
void do_print(const print_context & c, unsigned level) const;
{
GINAC_DECLARE_REGISTERED_CLASS(diracgammaL, tensor)
+ // functions overriding virtual functions from base classes
+ ex conjugate() const;
+
// non-virtual functions in this class
protected:
void do_print(const print_context & c, unsigned level) const;
{
GINAC_DECLARE_REGISTERED_CLASS(diracgammaR, tensor)
+ // functions overriding virtual functions from base classes
+ ex conjugate() const;
+
// non-virtual functions in this class
protected:
void do_print(const print_context & c, unsigned level) const;
* @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 (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, bool anticommuting = false);
+
/** Create a Dirac gamma object.
*
* @param mu Index (must be of class varidx or a derived class)
/** Create a term of the form e_mu * gamma~mu with a unique index mu.
*
+ * @param e Original expression
* @param dim Dimension of index
* @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.
* 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: makes 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, coincides with the conjugate(). */
+inline ex clifford_star(const ex & e) { return e.conjugate(); }
+
+/** 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*/
+char clifford_max_label(const ex & e, bool ignore_ONE = false);
+
+/** 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 (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, bool anticommuting = false);
+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
+ * @param anticommuting indicates if Clifford units anticommutes
+ * @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, bool anticommuting = false);
+
+/** 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
+ * @param anticommuting indicates if Clifford units anticommutes
+ * @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, bool anticommuting = false);
+
} // namespace GiNaC
#endif // ndef __GINAC_CLIFFORD_H__
git://www.ginac.de / ginac.git / blobdiff