X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fclifford.h;h=17c70e1bf523250fa706d0ca72520821b0660486;hp=ce5d2acfd50a6b3430d5717c9aa125389d19f3fc;hb=7a9e31b4cf6bbdb07cc7364bcd7903e9fc15a995;hpb=dbd9c306a74f1cb258c0d15a346b973b39deaad2 diff --git a/ginac/clifford.h b/ginac/clifford.h index ce5d2acf..17c70e1b 100644 --- a/ginac/clifford.h +++ b/ginac/clifford.h @@ -3,7 +3,7 @@ * Interface to GiNaC's clifford algebra (Dirac gamma) objects. */ /* - * GiNaC Copyright (C) 1999-2003 Johannes Gutenberg University Mainz, Germany + * GiNaC Copyright (C) 1999-2004 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 @@ -25,6 +25,10 @@ #include "indexed.h" #include "tensor.h" +#include "symbol.h" +#include "idx.h" + +#include namespace GiNaC { @@ -33,7 +37,7 @@ 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) @@ -41,31 +45,36 @@ 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, exvector * vp); // vp will be deleted + 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 -public: - void print(const print_context & c, unsigned level = 0) const; - protected: - ex simplify_ncmul(const exvector & v) const; + ex eval_ncmul(const exvector & v) const; bool match_same_type(const basic & other) const; - ex thisexprseq(const exvector & v) const; - ex thisexprseq(exvector * vp) const; - unsigned return_type(void) const { return return_types::noncommutative; } - unsigned return_type_tinfo(void) const { return TINFO_clifford + representation_label; } + 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; } // non-virtual functions in this class public: - unsigned char get_representation_label(void) const {return representation_label;} + 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; }; @@ -74,33 +83,62 @@ 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: - void print(const print_context & c, unsigned level = 0) const; + 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: - void print(const print_context & c, unsigned level = 0) const; 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 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 -public: - void print(const print_context & c, unsigned level = 0) const; + 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; }; @@ -111,8 +149,12 @@ class diracgammaL : public tensor GINAC_DECLARE_REGISTERED_CLASS(diracgammaL, tensor) // functions overriding virtual functions from base classes -public: - void print(const print_context & c, unsigned level = 0) const; + 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; }; @@ -123,8 +165,12 @@ class diracgammaR : public tensor GINAC_DECLARE_REGISTERED_CLASS(diracgammaR, tensor) // functions overriding virtual functions from base classes -public: - void print(const print_context & c, unsigned level = 0) const; + 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; }; @@ -142,6 +188,14 @@ template<> inline bool is_exactly_a(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 (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) @@ -167,16 +221,33 @@ ex dirac_gammaL(unsigned char rl = 0); * @return newly constructed object */ ex dirac_gammaR(unsigned char rl = 0); -// These functions are deprecated. Use dirac_gammaL/R() instead. -ex dirac_gamma6(unsigned char rl = 0); -ex dirac_gamma7(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. @@ -193,6 +264,68 @@ 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: 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 */ +ex clifford_moebius_map(const ex & a, const ex & b, const ex & c, const ex & d, const ex & v, const ex & G); + +/** 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 */ +ex clifford_moebius_map(const ex & M, const ex & v, const ex & G); + } // namespace GiNaC #endif // ndef __GINAC_CLIFFORD_H__