/** @file clifford.h
*
* Interface to GiNaC's clifford algebra (Dirac gamma) objects. */
/*
* GiNaC Copyright (C) 1999-2007 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
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* 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., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
*/
#ifndef __GINAC_CLIFFORD_H__
#define __GINAC_CLIFFORD_H__
#include "indexed.h"
#include "tensor.h"
#include "symbol.h"
#include "idx.h"
#include <set>
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 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, 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<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;
ex thiscontainer(const exvector & v) const;
ex thiscontainer(std::auto_ptr<exvector> 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() 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;
};
/** 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:
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 cliffordunit
{
GINAC_DECLARE_REGISTERED_CLASS(diracgamma, cliffordunit)
// 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 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;
void do_print_latex(const print_latex & c, unsigned level) const;
};
/** 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;
};
/** 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;
// 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<clifford>(obj) for clifford objects. */
template<> inline bool is_exactly_a<clifford>(const basic & obj)
{
return obj.tinfo()==TINFO_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 of class 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<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.
* 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.
* Aborts if e contains any clifford_unit.
*
* @param e Expression to be processed */
ex remove_dirac_ONE(const ex & e);
/** 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 */
ex remove_dirac_ONE(const ex & e, unsigned char rl);
/** 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 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);
/** List or vector conversion into the Clifford vector.
*
* @param v List or vector of coordinates
* @param e Clifford unit object
* @return Clifford vector with given components */
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);
/** Same as clifford_moebius_map(a, b, c, d, v, G, 0). */
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, 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);
/** Same as clifford_moebius_map(M, v, G, 0). */
ex clifford_moebius_map(const ex & M, const ex & v, const ex & G);
} // namespace GiNaC
#endif // ndef __GINAC_CLIFFORD_H__
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