GiNaC/ginac/clifford.h

/** @file clifford.h
 *
 *  Interface to GiNaC's clifford algebra (Dirac gamma) objects. */

/*
 *  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
 *  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., 59 Temple Place, Suite 330, Boston, MA  02111-1307  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
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 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)
 *  @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 (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
 *  @param rl Representation label */
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
 *  @param rl Representation label */
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__
1	cbauer	1.1	/** @file clifford.h
2			*
3	cbauer	1.22	* Interface to GiNaC's clifford algebra (Dirac gamma) objects. */
4	cbauer	1.4
5			/*
6	vollinga	1.46	* GiNaC Copyright (C) 1999-2004 Johannes Gutenberg University Mainz, Germany
7	cbauer	1.2	*
8			* This program is free software; you can redistribute it and/or modify
9			* it under the terms of the GNU General Public License as published by
10			* the Free Software Foundation; either version 2 of the License, or
11			* (at your option) any later version.
12			*
13			* This program is distributed in the hope that it will be useful,
14			* but WITHOUT ANY WARRANTY; without even the implied warranty of
15			* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
16			* GNU General Public License for more details.
17			*
18			* You should have received a copy of the GNU General Public License
19			* along with this program; if not, write to the Free Software
20			* Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
21			*/
22	cbauer	1.1
23	cbauer	1.2	#ifndef __GINAC_CLIFFORD_H__
24			#define __GINAC_CLIFFORD_H__
25	cbauer	1.1
26	cbauer	1.22	#include "indexed.h"
27			#include "tensor.h"
28	vollinga	1.47	#include "symbol.h"
29			#include "idx.h"
30	cbauer	1.5
31	cbauer	1.51	#include <set>
32
33	cbauer	1.5	namespace GiNaC {
34	cbauer	1.1
35	cbauer	1.15
36			/** This class holds an object representing an element of the Clifford
37			* algebra (the Dirac gamma matrices). These objects only carry Lorentz
38	cbauer	1.25	* indices. Spinor indices are hidden. A representation label (an unsigned
39			* 8-bit integer) is used to distinguish elements from different Clifford
40	cbauer	1.50	* algebras (objects with different labels commutate). */
41	cbauer	1.22	class clifford : public indexed
42	cbauer	1.1	{
43	cbauer	1.22	GINAC_DECLARE_REGISTERED_CLASS(clifford, indexed)
44	cbauer	1.15
45	cbauer	1.12	// other constructors
46	cbauer	1.1	public:
47	cbauer	1.25	clifford(const ex & b, unsigned char rl = 0);
48	vollinga	1.47	clifford(const ex & b, const ex & mu, const ex & metr, unsigned char rl = 0);
49	cbauer	1.22
50			// internal constructors
51	cbauer	1.48	clifford(unsigned char rl, const ex & metr, const exvector & v, bool discardable = false);
52			clifford(unsigned char rl, const ex & metr, std::auto_ptr<exvector> vp);
53	cbauer	1.1
54	cbauer	1.12	// functions overriding virtual functions from base classes
55	cbauer	1.1	protected:
56	cbauer	1.42	ex eval_ncmul(const exvector & v) const;
57	cbauer	1.35	bool match_same_type(const basic & other) const;
58	cbauer	1.42	ex thiscontainer(const exvector & v) const;
59	cbauer	1.44	ex thiscontainer(std::auto_ptr<exvector> vp) const;
60	cbauer	1.42	unsigned return_type() const { return return_types::noncommutative; }
61			unsigned return_type_tinfo() const { return TINFO_clifford + representation_label; }
62	cbauer	1.25
63	cbauer	1.26	// non-virtual functions in this class
64			public:
65	cbauer	1.48	unsigned char get_representation_label() const { return representation_label; }
66			ex get_metric() const { return metric; }
67			ex get_metric(const ex & i, const ex & j) const;
68			bool same_metric(const ex & other) const;
69	cbauer	1.26
70	cbauer	1.43	protected:
71			void do_print_dflt(const print_dflt & c, unsigned level) const;
72			void do_print_latex(const print_latex & c, unsigned level) const;
73
74	cbauer	1.25	// member variables
75			private:
76			unsigned char representation_label; /*< Representation label to distinguish independent spin lines /
77	cbauer	1.48	ex metric;
78	cbauer	1.22	};
79
80	cbauer	1.1
81	cbauer	1.23	/** This class represents the Clifford algebra unity element. */
82			class diracone : public tensor
83			{
84			GINAC_DECLARE_REGISTERED_CLASS(diracone, tensor)
85
86	cbauer	1.43	// non-virtual functions in this class
87			protected:
88			void do_print(const print_context & c, unsigned level) const;
89			void do_print_latex(const print_latex & c, unsigned level) const;
90	cbauer	1.23	};
91
92
93	vollinga	1.47	/** This class represents the Clifford algebra generators (units). */
94			class cliffordunit : public tensor
95			{
96			GINAC_DECLARE_REGISTERED_CLASS(cliffordunit, tensor)
97
98	cbauer	1.48	// other constructors
99	vollinga	1.47	protected:
100	cbauer	1.48	cliffordunit(unsigned ti) : inherited(ti) {}
101	vollinga	1.47
102			// functions overriding virtual functions from base classes
103			public:
104			bool contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const;
105	cbauer	1.48
106	vollinga	1.47	// non-virtual functions in this class
107			protected:
108			void do_print(const print_context & c, unsigned level) const;
109			void do_print_latex(const print_latex & c, unsigned level) const;
110			};
111
112
113	cbauer	1.22	/** This class represents the Dirac gamma Lorentz vector. */
114	vollinga	1.47	class diracgamma : public cliffordunit
115	cbauer	1.22	{
116	vollinga	1.47	GINAC_DECLARE_REGISTERED_CLASS(diracgamma, cliffordunit)
117	cbauer	1.1
118	cbauer	1.36	// functions overriding virtual functions from base classes
119	cbauer	1.22	public:
120	cbauer	1.23	bool contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const;
121	cbauer	1.43
122			// non-virtual functions in this class
123			protected:
124			void do_print(const print_context & c, unsigned level) const;
125			void do_print_latex(const print_latex & c, unsigned level) const;
126	cbauer	1.1	};
127
128	cbauer	1.22
129	cbauer	1.50	/** This class represents the Dirac gamma5 object which anticommutates with
130	cbauer	1.28	* all other gammas. */
131	cbauer	1.25	class diracgamma5 : public tensor
132			{
133			GINAC_DECLARE_REGISTERED_CLASS(diracgamma5, tensor)
134
135	vollinga	1.46	// functions overriding virtual functions from base classes
136			ex conjugate() const;
137
138	cbauer	1.43	// non-virtual functions in this class
139			protected:
140			void do_print(const print_context & c, unsigned level) const;
141			void do_print_latex(const print_latex & c, unsigned level) const;
142	cbauer	1.25	};
143
144
145	cbauer	1.40	/** This class represents the Dirac gammaL object which behaves like
146			* 1/2 (1-gamma5). */
147			class diracgammaL : public tensor
148			{
149			GINAC_DECLARE_REGISTERED_CLASS(diracgammaL, tensor)
150
151	vollinga	1.46	// functions overriding virtual functions from base classes
152			ex conjugate() const;
153
154	cbauer	1.43	// non-virtual functions in this class
155			protected:
156			void do_print(const print_context & c, unsigned level) const;
157			void do_print_latex(const print_latex & c, unsigned level) const;
158	cbauer	1.40	};
159
160
161			/** This class represents the Dirac gammaL object which behaves like
162			* 1/2 (1+gamma5). */
163			class diracgammaR : public tensor
164			{
165			GINAC_DECLARE_REGISTERED_CLASS(diracgammaR, tensor)
166	vollinga	1.46
167			// functions overriding virtual functions from base classes
168			ex conjugate() const;
169	cbauer	1.40
170	cbauer	1.43	// non-virtual functions in this class
171			protected:
172			void do_print(const print_context & c, unsigned level) const;
173			void do_print_latex(const print_latex & c, unsigned level) const;
174	cbauer	1.40	};
175
176
177	cbauer	1.15	// global functions
178	kreckel	1.33
179			/** Specialization of is_exactly_a<clifford>(obj) for clifford objects. */
180			template<> inline bool is_exactly_a<clifford>(const basic & obj)
181			{
182			return obj.tinfo()==TINFO_clifford;
183			}
184	cbauer	1.23
185			/** Create a Clifford unity object.
186			*
187	cbauer	1.25	* @param rl Representation label
188	cbauer	1.23	* @return newly constructed object */
189	cbauer	1.25	ex dirac_ONE(unsigned char rl = 0);
190	cbauer	1.22
191	vollinga	1.47	/** Create a Clifford unit object.
192			*
193			* @param mu Index (must be of class varidx or a derived class)
194	cbauer	1.49	* @param metr Metric (should be of class tensmetric or a derived class, or a symmetric matrix)
195	vollinga	1.47	* @param rl Representation label
196			* @return newly constructed Clifford unit object */
197			ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
198
199	cbauer	1.22	/** Create a Dirac gamma object.
200			*
201			* @param mu Index (must be of class varidx or a derived class)
202	cbauer	1.25	* @param rl Representation label
203	cbauer	1.22	* @return newly constructed gamma object */
204	cbauer	1.25	ex dirac_gamma(const ex & mu, unsigned char rl = 0);
205
206			/** Create a Dirac gamma5 object.
207			*
208			* @param rl Representation label
209			* @return newly constructed object */
210			ex dirac_gamma5(unsigned char rl = 0);
211	cbauer	1.26
212	cbauer	1.40	/** Create a Dirac gammaL object.
213			*
214			* @param rl Representation label
215			* @return newly constructed object */
216			ex dirac_gammaL(unsigned char rl = 0);
217
218			/** Create a Dirac gammaR object.
219			*
220			* @param rl Representation label
221			* @return newly constructed object */
222			ex dirac_gammaR(unsigned char rl = 0);
223	cbauer	1.29
224	cbauer	1.28	/** Create a term of the form e_mu * gamma~mu with a unique index mu.
225			*
226	cbauer	1.45	* @param e Original expression
227	cbauer	1.28	* @param dim Dimension of index
228			* @param rl Representation label */
229			ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
230
231	cbauer	1.51	/** Calculate dirac traces over the specified set of representation labels.
232			* The computed trace is a linear functional that is equal to the usual
233			* trace only in D = 4 dimensions. In particular, the functional is not
234			* always cyclic in D != 4 dimensions when gamma5 is involved.
235			*
236			* @param e Expression to take the trace of
237			* @param rls Set of representation labels
238			* @param trONE Expression to be returned as the trace of the unit matrix */
239			ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE = 4);
240
241			/** Calculate dirac traces over the specified list of representation labels.
242			* The computed trace is a linear functional that is equal to the usual
243			* trace only in D = 4 dimensions. In particular, the functional is not
244			* always cyclic in D != 4 dimensions when gamma5 is involved.
245			*
246			* @param e Expression to take the trace of
247			* @param rll List of representation labels
248			* @param trONE Expression to be returned as the trace of the unit matrix */
249			ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
250
251	cbauer	1.26	/** Calculate the trace of an expression containing gamma objects with
252	cbauer	1.27	* a specified representation label. The computed trace is a linear
253			* functional that is equal to the usual trace only in D = 4 dimensions.
254	cbauer	1.28	* In particular, the functional is not always cyclic in D != 4 dimensions
255			* when gamma5 is involved.
256	cbauer	1.26	*
257	cbauer	1.30	* @param e Expression to take the trace of
258			* @param rl Representation label
259			* @param trONE Expression to be returned as the trace of the unit matrix */
260			ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
261	cbauer	1.22
262	cbauer	1.31	/** Bring all products of clifford objects in an expression into a canonical
263			* order. This is not necessarily the most simple form but it will allow
264	cbauer	1.32	* to check two expressions for equality. */
265	cbauer	1.31	ex canonicalize_clifford(const ex & e);
266	vollinga	1.47
267	cbauer	1.48	/** Automorphism of the Clifford algebra, simply changes signs of all
268	vollinga	1.47	* clifford units. */
269	cbauer	1.48	ex clifford_prime(const ex & e);
270	vollinga	1.47
271	cbauer	1.48	/** Main anti-automorphism of the Clifford algebra: makes reversion
272			* and changes signs of all clifford units. */
273			inline ex clifford_bar(const ex & e) { return clifford_prime(e.conjugate()); }
274	vollinga	1.47
275	cbauer	1.48	/** Reversion of the Clifford algebra, coincides with the conjugate(). */
276			inline ex clifford_star(const ex & e) { return e.conjugate(); }
277	vollinga	1.47
278	vollinga	1.52	/** Replaces all dirac_ONE's in e with 1 (effectively removing them). */
279			ex remove_dirac_ONE(const ex & e);
280	vollinga	1.47
281	cbauer	1.48	/** Calculation of the norm in the Clifford algebra. */
282			ex clifford_norm(const ex & e);
283	vollinga	1.47
284	cbauer	1.48	/** Calculation of the inverse in the Clifford algebra. */
285			ex clifford_inverse(const ex & e);
286	vollinga	1.47
287	cbauer	1.48	/** List or vector conversion into the Clifford vector.
288			*
289	vollinga	1.47	* @param v List or vector of coordinates
290			* @param mu Index (must be of class varidx or a derived class)
291	cbauer	1.49	* @param metr Metric (should be of class tensmetric or a derived class, or a symmetric matrix)
292	vollinga	1.47	* @param rl Representation label
293			* @return Clifford vector with given components */
294	cbauer	1.48	ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl = 0);
295	cbauer	1.1
296	vollinga	1.52	/** An inverse function to lst_to_clifford(). For given Clifford vector extracts
297			* its components with respect to given Clifford unit. Obtained components may
298			* contain Clifford units with a different metric. Extraction is based on
299			* the algebraic formula (e * c.i + c.i * e)/ pow(e.i, 2) for non-degenerate cases
300			* (i.e. neither pow(e.i, 2) = 0).
301			*
302			* @param e Clifford expression to be decomposed into components
303			* @param c Clifford unit defining the metric for splitting (should have numeric dimension of indices)
304			* @param algebraic Use algebraic or symbolic algorithm for extractions */
305			lst clifford_to_lst(const ex & e, const ex & c, bool algebraic=true);
306
307			/** Calculations of Moebius transformations (conformal map) defined by a 2x2 Clifford matrix
308			* (a b\\c d) in linear spaces with arbitrary signature. The expression is
309			* (a * x + b)/(c * x + d), where x is a vector build from list v with metric G.
310			* (see Jan Cnops. An introduction to {D}irac operators on manifolds, v.24 of
311			* Progress in Mathematical Physics. Birkhauser Boston Inc., Boston, MA, 2002.)
312			*
313			* @param a (1,1) entry of the defining matrix
314			* @param b (1,2) entry of the defining matrix
315			* @param c (2,1) entry of the defining matrix
316			* @param d (2,2) entry of the defining matrix
317			* @param v Vector to be transformed
318	kreckel	1.53.2.1	* @param G Metric of the surrounding space
319			* @param rl Representation label */
320			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);
321
322			/** Same as clifford_moebius_map(a, b, c, d, v, G, 0). */
323	vollinga	1.52	ex clifford_moebius_map(const ex & a, const ex & b, const ex & c, const ex & d, const ex & v, const ex & G);
324	vollinga	1.53
325			/** The second form of Moebius transformations defined by a 2x2 Clifford matrix M
326			* This function takes the transformation matrix M as a single entity.
327			*
328			* @param M the defining matrix
329			* @param v Vector to be transformed
330	kreckel	1.53.2.1	* @param G Metric of the surrounding space
331			* @param rl Representation label */
332			ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl);
333
334			/** Same as clifford_moebius_map(M, v, G, 0). */
335	vollinga	1.53	ex clifford_moebius_map(const ex & M, const ex & v, const ex & G);
336
337	cbauer	1.5	} // namespace GiNaC
338	cbauer	1.1
339	cbauer	1.2	#endif // ndef __GINAC_CLIFFORD_H__
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