X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fclifford.h;h=0f0ebfa42cdffd7c06b5abbb54a57223396ee47f;hp=18d8f0a7e0efcc104550984a35a78fe5f712d159;hb=695f6ae955ec530cded8f21efd5569df39447f76;hpb=bd2d7351b89743eb68ce7d3bfa7ab62c331f9522 diff --git a/ginac/clifford.h b/ginac/clifford.h index 18d8f0a7..0f0ebfa4 100644 --- a/ginac/clifford.h +++ b/ginac/clifford.h @@ -1,9 +1,9 @@ /** @file clifford.h * - * Interface to GiNaC's clifford objects. */ + * Interface to GiNaC's clifford algebra (Dirac gamma) objects. */ /* - * GiNaC Copyright (C) 1999-2000 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 @@ -23,77 +23,311 @@ #ifndef __GINAC_CLIFFORD_H__ #define __GINAC_CLIFFORD_H__ -#include #include "indexed.h" -#include "ex.h" +#include "tensor.h" +#include "symbol.h" +#include "idx.h" + +#include -#ifndef NO_NAMESPACE_GINAC namespace GiNaC { -#endif // ndef NO_NAMESPACE_GINAC -/** Base class for clifford object */ + +/** 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 commutate). */ class clifford : public indexed { -// member functions + GINAC_DECLARE_REGISTERED_CLASS(clifford, indexed) - // default constructor, destructor, copy constructor assignment operator and helpers + // other constructors public: - clifford(); - ~clifford(); - clifford(const clifford & other); - const clifford & operator=(const clifford & other); + clifford(const ex & b, 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 ex & metr, const exvector & v, bool discardable = false); + clifford(unsigned char rl, const ex & metr, std::auto_ptr vp); + + // functions overriding virtual functions from base classes protected: - void copy(const clifford & other); - void destroy(bool call_parent); + 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 vp) const; + unsigned return_type() const { return return_types::noncommutative; } + unsigned return_type_tinfo() const { return TINFO_clifford + representation_label; } - // other constructors + // non-virtual functions in this class public: - explicit clifford(const std::string & initname); + 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; + void do_print_latex(const print_latex & c, unsigned level) const; + + // member variables +private: + unsigned char representation_label; /**< Representation label to distinguish independent spin lines */ + ex metric; +}; + - // functions overriding virtual functions from base classes +/** This class represents the Clifford algebra unity element. */ +class diracone : public tensor +{ + GINAC_DECLARE_REGISTERED_CLASS(diracone, tensor) + + // 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 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: - basic * duplicate() const; - void printraw(std::ostream & os) const; - void printtree(std::ostream & os, unsigned indent) const; - void print(std::ostream & os, unsigned upper_precedence=0) const; - void printcsrc(std::ostream & os, unsigned type, unsigned upper_precedence=0) const; - bool info(unsigned inf) const; + bool contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const; + + // non-virtual functions in this class protected: - int compare_same_type(const basic & other) const; - ex simplify_ncmul(const exvector & v) const; - unsigned calchash(void) const; - - // new virtual functions which can be overridden by derived classes - // none - - // non-virtual functions in this class + 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 cliffordunit +{ + GINAC_DECLARE_REGISTERED_CLASS(diracgamma, cliffordunit) + + // functions overriding virtual functions from base classes public: - void setname(const std::string & n); -private: - std::string & autoname_prefix(void); + 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; +}; + -// member variables +/** 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: - std::string name; - unsigned serial; // unique serial number for comparision -private: - static unsigned next_serial; + void do_print(const print_context & c, unsigned level) const; + void do_print_latex(const print_latex & c, unsigned level) const; }; -// global constants -extern const clifford some_clifford; -extern const type_info & typeid_clifford; +/** This class represents the Dirac gammaL object which behaves like + * 1/2 (1-gamma5). */ +class diracgammaL : public tensor +{ + 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; + void do_print_latex(const print_latex & c, unsigned level) const; +}; -// utility functions -inline const clifford &ex_to_clifford(const ex &e) + +/** This class represents the Dirac gammaL object which behaves like + * 1/2 (1+gamma5). */ +class diracgammaR : public tensor { - return static_cast(*e.bp); + 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; + void do_print_latex(const print_latex & c, unsigned level) const; +}; + + +// global functions + +/** Specialization of is_exactly_a(obj) for clifford objects. */ +template<> inline bool is_exactly_a(const basic & obj) +{ + return obj.tinfo()==TINFO_clifford; } -#ifndef NO_NAMESPACE_GINAC +/** Create a Clifford unity object. + * + * @param rl Representation label + * @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 of class tensmetric or a derived class, or a symmetric matrix) + * @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) + * @param rl Representation label + * @return newly constructed gamma object */ +ex dirac_gamma(const ex & mu, unsigned char rl = 0); + +/** Create a Dirac gamma5 object. + * + * @param rl Representation label + * @return newly constructed object */ +ex dirac_gamma5(unsigned char rl = 0); + +/** Create a Dirac gammaL object. + * + * @param rl Representation label + * @return newly constructed object */ +ex dirac_gammaL(unsigned char rl = 0); + +/** Create a Dirac gammaR object. + * + * @param rl Representation label + * @return newly constructed object */ +ex dirac_gammaR(unsigned char rl = 0); + +/** 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 & 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. + * 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 rl Representation label + * @param trONE Expression to be returned as the trace of the unit matrix */ +ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4); + +/** Bring all products of clifford objects in an expression into a canonical + * order. This is not necessarily the most simple form but it will allow + * 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 all dirac_ONE's in e with 1 (effectively removing them). */ +ex remove_dirac_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 (should be of class tensmetric or a derived class, or a symmetric matrix) + * @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); + +/** 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 */ +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 + * @param rl Representation label */ +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 + * @param rl Representation label */ +ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl = 0); + } // namespace GiNaC -#endif // ndef NO_NAMESPACE_GINAC #endif // ndef __GINAC_CLIFFORD_H__