/** @file clifford.h
*
- * Interface to GiNaC's clifford objects.
- *
- * GiNaC Copyright (C) 1999 Johannes Gutenberg University Mainz, Germany
+ * Interface to GiNaC's clifford algebra (Dirac gamma) objects. */
+
+/*
+ * GiNaC Copyright (C) 1999-2011 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
*
* 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"
-#include <string>
-#include <ginac/indexed.h>
+#include <set>
-/** Base class for clifford object */
+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
{
-// member functions
+ 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);
+
+ // internal constructors
+ clifford(unsigned char rl, const ex & metr, int comm_sign, const exvector & v, bool discardable = false);
+ clifford(unsigned char rl, const ex & metr, int comm_sign, std::auto_ptr<exvector> vp);
- // default constructor, destructor, copy constructor assignment operator and helpers
+ // functions overriding virtual functions from base classes
public:
- clifford();
- ~clifford();
- clifford(clifford const & other);
- clifford const & operator=(clifford const & other);
+ unsigned precedence() const { return 65; }
+ void archive(archive_node& n) const;
+ void read_archive(const archive_node& n, lst& sym_lst);
protected:
- void copy(clifford const & other);
- void destroy(bool call_parent);
-
- // other constructors
+ 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; }
+ return_type_t return_type_tinfo() const;
+ // non-virtual functions in this class
public:
- explicit clifford(string const & initname);
+ 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 {return inherited::nops() + 1; }
+ ex op(size_t i) const;
+ ex & let_op(size_t i);
+ ex subs(const exmap & m, unsigned options = 0) 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
+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
+ // functions overriding virtual functions from base classes
public:
- basic * duplicate() const;
- void printraw(ostream & os) const;
- void printtree(ostream & os, unsigned indent) const;
- void print(ostream & os, unsigned upper_precedence=0) const;
- void printcsrc(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(basic const & other) const;
- ex simplify_ncmul(exvector const & 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(string const & n);
-private:
- 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;
+};
+GINAC_DECLARE_UNARCHIVER(diracgamma);
+
-// 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:
+ 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;
+
+ // non-virtual functions in this class
protected:
- 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;
};
+GINAC_DECLARE_UNARCHIVER(diracgammaL);
-// global constants
-extern const clifford some_clifford;
-extern type_info const & typeid_clifford;
+/** 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;
+};
+GINAC_DECLARE_UNARCHIVER(diracgammaR);
+
+
+// global functions
+
+/** 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 *(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)
+ * @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 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);
-// macros
+/** 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);
-#define ex_to_clifford(X) static_cast<clifford const &>(*(X).bp)
+/** 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);
-#endif // ndef __GINAC_CLIFFORD_H__
+/** 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
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