X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fclifford.h;h=9708b0b6813dd750bf458e2ca26303cf5d5f17e7;hp=f1783396995620d63c1a38f24315675f97476742;hb=ed914545e01d60ecf2544e6141d6c5142c01327f;hpb=53604c3e255c392fb31af8bcecf57f2918495d28 diff --git a/ginac/clifford.h b/ginac/clifford.h index f1783396..9708b0b6 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) 1999-2001 Johannes Gutenberg University Mainz, Germany + * GiNaC Copyright (C) 1999-2016 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 @@ -17,101 +17,199 @@ * * 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 +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, 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 exvector & v, bool discardable = false); - clifford(unsigned char rl, exvector * vp); // vp will be deleted + clifford(unsigned char rl, const ex & metr, int comm_sign, const exvector & v); + clifford(unsigned char rl, const ex & metr, int comm_sign, exvector && v); // functions overriding virtual functions from base classes +public: + unsigned precedence() const override { return 65; } + void archive(archive_node& n) const override; + void read_archive(const archive_node& n, lst& sym_lst) override; protected: - ex simplify_ncmul(const exvector & v) 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 eval_ncmul(const exvector & v) const override; + bool match_same_type(const basic & other) const override; + ex thiscontainer(const exvector & v) const override; + ex thiscontainer(exvector && v) const override; + unsigned return_type() const override { return return_types::noncommutative; } + return_type_t return_type_tinfo() const override; // 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; } + 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 override {return inherited::nops() + 1; } + ex op(size_t i) const override; + ex & let_op(size_t i) override; + ex subs(const exmap & m, unsigned options = 0) const override; + +protected: + void do_print_dflt(const print_dflt & c, unsigned level) const; + void do_print_latex(const print_latex & c, unsigned level) const; + void do_print_tree(const print_tree & c, unsigned level) const; // 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 */ + 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 { GINAC_DECLARE_REGISTERED_CLASS(diracone, tensor) - // functions overriding virtual functions from bases classes + // 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; +}; +GINAC_DECLARE_UNARCHIVER(diracone); + + +/** This class represents the Clifford algebra generators (units). */ +class cliffordunit : public tensor +{ + GINAC_DECLARE_REGISTERED_CLASS(cliffordunit, tensor) + + // 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 override; + + // 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; }; +GINAC_DECLARE_UNARCHIVER(cliffordunit); /** 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 bases classes + // 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; + bool contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const override; + + // 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; }; +GINAC_DECLARE_UNARCHIVER(diracgamma); -/** This class represents the Dirac gamma5 object. */ +/** 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 bases classes -public: - void print(const print_context & c, unsigned level = 0) const; + // functions overriding virtual functions from base classes + ex conjugate() const override; + + // 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; +}; +GINAC_DECLARE_UNARCHIVER(diracgamma5); + + +/** 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 override; + + // 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; }; +GINAC_DECLARE_UNARCHIVER(diracgammaL); + + +/** This class represents the Dirac gammaL object which behaves like + * 1/2 (1+gamma5). */ +class diracgammaR : public tensor +{ + GINAC_DECLARE_REGISTERED_CLASS(diracgammaR, tensor) + + // functions overriding virtual functions from base classes + ex conjugate() const override; + + // 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; +}; +GINAC_DECLARE_UNARCHIVER(diracgammaR); // global functions -inline const clifford &ex_to_clifford(const ex &e) + +/** 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 static_cast(*e.bp); + return *(ti.tinfo) == typeid(clifford); } - /** 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 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); + /** Create a Dirac gamma object. * * @param mu Index (must be of class varidx or a derived class) @@ -125,16 +223,143 @@ ex dirac_gamma(const ex & mu, unsigned char rl = 0); * @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 non-cyclic in D != 4 dimensions when - * gamma5 is involved. + * In particular, the functional is not always cyclic in D != 4 dimensions + * when gamma5 is involved. * - * @param rl Representation label */ -ex dirac_trace(const ex & e, unsigned char rl = 0); + * @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 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); + +/** 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); +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