X-Git-Url: https://www.ginac.de/ginac.git//ginac.git?p=ginac.git;a=blobdiff_plain;f=ginac%2Fclifford.h;h=9708b0b6813dd750bf458e2ca26303cf5d5f17e7;hp=2d02ebcb8df7a4c6bc3db1d12c0fb34ca00aeae2;hb=ed914545e01d60ecf2544e6141d6c5142c01327f;hpb=e5362a33f72613b324b3714524a8c2e5f7b7f46f diff --git a/ginac/clifford.h b/ginac/clifford.h index 2d02ebcb..9708b0b6 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) 1999-2001 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 @@ -17,72 +17,349 @@ * * 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 -#include "lortensor.h" +#include "indexed.h" +#include "tensor.h" +#include "symbol.h" +#include "idx.h" -#ifndef NO_NAMESPACE_GINAC -namespace GiNaC { -#endif // ndef NO_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 always hidden in our implementation. */ -class clifford : public lortensor + * 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 { - GINAC_DECLARE_REGISTERED_CLASS(clifford, lortensor) + 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, int comm_sign = -1); -// friends + // internal constructors + 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); - friend clifford clifford_gamma(const ex & mu); + // 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 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() 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 */ - // other constructors + 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 +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) + + // 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: - clifford(const std::string & n, const ex & mu); - clifford(const std::string & n, const exvector & iv); - clifford(const std::string & n, exvector * ivp); + 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 cliffordunit +{ + GINAC_DECLARE_REGISTERED_CLASS(diracgamma, cliffordunit) // 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; - bool info(unsigned inf) const; - // ex eval(int level=0) const; + bool contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const override; + + // non-virtual functions in this class protected: - int compare_same_type(const basic & other) const; - bool is_equal_same_type(const basic & other) const; - ex simplify_ncmul(const exvector & v) const; - ex thisexprseq(const exvector & v) const; - ex thisexprseq(exvector * vp) const; - - // new virtual functions which can be overridden by derived classes - // none - + 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 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 override; + // non-virtual functions in this class - // none +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); - // member variables - // none + +/** 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); } -clifford clifford_gamma(const ex & mu); +/** 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) + * @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 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); -#ifndef NO_NAMESPACE_GINAC } // namespace GiNaC -#endif // ndef NO_NAMESPACE_GINAC -#endif // ndef __GINAC_CLIFFORD_H__ +#endif // ndef GINAC_CLIFFORD_H