3 * Interface to GiNaC's clifford algebra (Dirac gamma) objects. */
6 * GiNaC Copyright (C) 1999-2006 Johannes Gutenberg University Mainz, Germany
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
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14 * but WITHOUT ANY WARRANTY; without even the implied warranty of
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23 #ifndef __GINAC_CLIFFORD_H__
24 #define __GINAC_CLIFFORD_H__
36 /** This class holds an object representing an element of the Clifford
37 * algebra (the Dirac gamma matrices). These objects only carry Lorentz
38 * indices. Spinor indices are hidden. A representation label (an unsigned
39 * 8-bit integer) is used to distinguish elements from different Clifford
40 * algebras (objects with different labels commutate). */
41 class clifford : public indexed
43 GINAC_DECLARE_REGISTERED_CLASS(clifford, indexed)
45 static const tinfo_static_t return_type_tinfo_static[256];
49 clifford(const ex & b, unsigned char rl = 0, bool anticommut = false);
50 clifford(const ex & b, const ex & mu, const ex & metr, unsigned char rl = 0, bool anticommut = false, int comm_sign = -1);
52 // internal constructors
53 clifford(unsigned char rl, const ex & metr, bool anticommut, int comm_sign, const exvector & v, bool discardable = false);
54 clifford(unsigned char rl, const ex & metr, bool anticommut, int comm_sign, std::auto_ptr<exvector> vp);
56 // functions overriding virtual functions from base classes
58 unsigned precedence() const { return 65; }
60 ex eval_ncmul(const exvector & v) const;
61 bool match_same_type(const basic & other) const;
62 ex thiscontainer(const exvector & v) const;
63 ex thiscontainer(std::auto_ptr<exvector> vp) const;
64 unsigned return_type() const { return return_types::noncommutative; }
65 tinfo_t return_type_tinfo() const { return clifford::return_type_tinfo_static+representation_label; }
67 // non-virtual functions in this class
69 unsigned char get_representation_label() const { return representation_label; }
70 ex get_metric() const { return metric; }
71 virtual ex get_metric(const ex & i, const ex & j, bool symmetrised = false) const;
72 bool same_metric(const ex & other) const;
73 bool is_anticommuting() const { return anticommuting; } //**< See the member variable anticommuting */
74 int get_commutator_sign() const { return commutator_sign; } //**< See the member variable commutator_sign */
76 inline size_t nops() const {return inherited::nops() + 1; }
77 ex op(size_t i) const;
78 ex & let_op(size_t i);
79 ex subs(const exmap & m, unsigned options = 0) const;
82 void do_print_dflt(const print_dflt & c, unsigned level) const;
83 void do_print_latex(const print_latex & c, unsigned level) const;
87 unsigned char representation_label; /**< Representation label to distinguish independent spin lines */
88 ex metric; /**< Metric of the space, all constructors make it an indexed object */
89 bool anticommuting; /**< Simplifications for anticommuting units is much simpler and we need this info readily available */
90 int commutator_sign; /**< It is the sign in the definition e~i e~j +/- e~j e~i = B(i, j) + B(j, i)*/
93 /** This class represents the Clifford algebra unity element. */
94 class diracone : public tensor
96 GINAC_DECLARE_REGISTERED_CLASS(diracone, tensor)
98 // non-virtual functions in this class
100 void do_print(const print_context & c, unsigned level) const;
101 void do_print_latex(const print_latex & c, unsigned level) const;
105 /** This class represents the Clifford algebra generators (units). */
106 class cliffordunit : public tensor
108 GINAC_DECLARE_REGISTERED_CLASS(cliffordunit, tensor)
110 // other constructors
112 cliffordunit(tinfo_t ti) : inherited(ti) {}
114 // functions overriding virtual functions from base classes
116 bool contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const;
118 // non-virtual functions in this class
120 void do_print(const print_context & c, unsigned level) const;
121 void do_print_latex(const print_latex & c, unsigned level) const;
125 /** This class represents the Dirac gamma Lorentz vector. */
126 class diracgamma : public cliffordunit
128 GINAC_DECLARE_REGISTERED_CLASS(diracgamma, cliffordunit)
130 // functions overriding virtual functions from base classes
132 bool contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const;
134 // non-virtual functions in this class
136 void do_print(const print_context & c, unsigned level) const;
137 void do_print_latex(const print_latex & c, unsigned level) const;
141 /** This class represents the Dirac gamma5 object which anticommutates with
142 * all other gammas. */
143 class diracgamma5 : public tensor
145 GINAC_DECLARE_REGISTERED_CLASS(diracgamma5, tensor)
147 // functions overriding virtual functions from base classes
148 ex conjugate() const;
150 // non-virtual functions in this class
152 void do_print(const print_context & c, unsigned level) const;
153 void do_print_latex(const print_latex & c, unsigned level) const;
157 /** This class represents the Dirac gammaL object which behaves like
159 class diracgammaL : public tensor
161 GINAC_DECLARE_REGISTERED_CLASS(diracgammaL, tensor)
163 // functions overriding virtual functions from base classes
164 ex conjugate() const;
166 // non-virtual functions in this class
168 void do_print(const print_context & c, unsigned level) const;
169 void do_print_latex(const print_latex & c, unsigned level) const;
173 /** This class represents the Dirac gammaL object which behaves like
175 class diracgammaR : public tensor
177 GINAC_DECLARE_REGISTERED_CLASS(diracgammaR, tensor)
179 // functions overriding virtual functions from base classes
180 ex conjugate() const;
182 // non-virtual functions in this class
184 void do_print(const print_context & c, unsigned level) const;
185 void do_print_latex(const print_latex & c, unsigned level) const;
191 /** Check whether a given tinfo key (as returned by return_type_tinfo()
192 * is that of a clifford object (with an arbitrary representation label).
194 * @param ti tinfo key */
195 bool is_clifford_tinfo(tinfo_t ti);
197 /** Create a Clifford unity object.
199 * @param rl Representation label
200 * @return newly constructed object */
201 ex dirac_ONE(unsigned char rl = 0);
203 /** Create a Clifford unit object.
205 * @param mu Index (must be of class varidx or a derived class)
206 * @param metr Metric (should be indexed, tensmetric or a derived class, or a matrix)
207 * @param rl Representation label
208 * @return newly constructed Clifford unit object */
209 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0, bool anticommuting = false);
211 /** Create a Dirac gamma object.
213 * @param mu Index (must be of class varidx or a derived class)
214 * @param rl Representation label
215 * @return newly constructed gamma object */
216 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
218 /** Create a Dirac gamma5 object.
220 * @param rl Representation label
221 * @return newly constructed object */
222 ex dirac_gamma5(unsigned char rl = 0);
224 /** Create a Dirac gammaL object.
226 * @param rl Representation label
227 * @return newly constructed object */
228 ex dirac_gammaL(unsigned char rl = 0);
230 /** Create a Dirac gammaR object.
232 * @param rl Representation label
233 * @return newly constructed object */
234 ex dirac_gammaR(unsigned char rl = 0);
236 /** Create a term of the form e_mu * gamma~mu with a unique index mu.
238 * @param e Original expression
239 * @param dim Dimension of index
240 * @param rl Representation label */
241 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
243 /** Calculate dirac traces over the specified set of representation labels.
244 * The computed trace is a linear functional that is equal to the usual
245 * trace only in D = 4 dimensions. In particular, the functional is not
246 * always cyclic in D != 4 dimensions when gamma5 is involved.
248 * @param e Expression to take the trace of
249 * @param rls Set of representation labels
250 * @param trONE Expression to be returned as the trace of the unit matrix */
251 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE = 4);
253 /** Calculate dirac traces over the specified list of representation labels.
254 * The computed trace is a linear functional that is equal to the usual
255 * trace only in D = 4 dimensions. In particular, the functional is not
256 * always cyclic in D != 4 dimensions when gamma5 is involved.
258 * @param e Expression to take the trace of
259 * @param rll List of representation labels
260 * @param trONE Expression to be returned as the trace of the unit matrix */
261 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
263 /** Calculate the trace of an expression containing gamma objects with
264 * a specified representation label. The computed trace is a linear
265 * functional that is equal to the usual trace only in D = 4 dimensions.
266 * In particular, the functional is not always cyclic in D != 4 dimensions
267 * when gamma5 is involved.
269 * @param e Expression to take the trace of
270 * @param rl Representation label
271 * @param trONE Expression to be returned as the trace of the unit matrix */
272 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
274 /** Bring all products of clifford objects in an expression into a canonical
275 * order. This is not necessarily the most simple form but it will allow
276 * to check two expressions for equality. */
277 ex canonicalize_clifford(const ex & e);
279 /** Automorphism of the Clifford algebra, simply changes signs of all
281 ex clifford_prime(const ex & e);
283 /** Main anti-automorphism of the Clifford algebra: makes reversion
284 * and changes signs of all clifford units. */
285 inline ex clifford_bar(const ex & e) { return clifford_prime(e.conjugate()); }
287 /** Reversion of the Clifford algebra, coincides with the conjugate(). */
288 inline ex clifford_star(const ex & e) { return e.conjugate(); }
290 /** Replaces dirac_ONE's (with a representation_label no less than rl) in e with 1.
291 * For the default value rl = 0 remove all of them. Aborts if e contains any
292 * clifford_unit with representation_label to be removed.
294 * @param e Expression to be processed
295 * @param rl Value of representation label
296 * @param options Defines some internal use */
297 ex remove_dirac_ONE(const ex & e, unsigned char rl = 0, unsigned options = 0);
299 /** Returns the maximal representation label of a clifford object
300 * if e contains at least one, otherwise returns -1
302 * @param e Expression to be processed
303 * @ignore_ONE defines if clifford_ONE should be ignored in the search*/
304 char clifford_max_label(const ex & e, bool ignore_ONE = false);
306 /** Calculation of the norm in the Clifford algebra. */
307 ex clifford_norm(const ex & e);
309 /** Calculation of the inverse in the Clifford algebra. */
310 ex clifford_inverse(const ex & e);
312 /** List or vector conversion into the Clifford vector.
314 * @param v List or vector of coordinates
315 * @param mu Index (must be of class varidx or a derived class)
316 * @param metr Metric (should be indexed, tensmetric or a derived class, or a matrix)
317 * @param rl Representation label
318 * @param e Clifford unit object
319 * @return Clifford vector with given components */
320 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl = 0, bool anticommuting = false);
321 ex lst_to_clifford(const ex & v, const ex & e);
323 /** An inverse function to lst_to_clifford(). For given Clifford vector extracts
324 * its components with respect to given Clifford unit. Obtained components may
325 * contain Clifford units with a different metric. Extraction is based on
326 * the algebraic formula (e * c.i + c.i * e)/ pow(e.i, 2) for non-degenerate cases
327 * (i.e. neither pow(e.i, 2) = 0).
329 * @param e Clifford expression to be decomposed into components
330 * @param c Clifford unit defining the metric for splitting (should have numeric dimension of indices)
331 * @param algebraic Use algebraic or symbolic algorithm for extractions
332 * @return List of components of a Clifford vector*/
333 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic=true);
335 /** Calculations of Moebius transformations (conformal map) defined by a 2x2 Clifford matrix
336 * (a b\\c d) in linear spaces with arbitrary signature. The expression is
337 * (a * x + b)/(c * x + d), where x is a vector build from list v with metric G.
338 * (see Jan Cnops. An introduction to {D}irac operators on manifolds, v.24 of
339 * Progress in Mathematical Physics. Birkhauser Boston Inc., Boston, MA, 2002.)
341 * @param a (1,1) entry of the defining matrix
342 * @param b (1,2) entry of the defining matrix
343 * @param c (2,1) entry of the defining matrix
344 * @param d (2,2) entry of the defining matrix
345 * @param v Vector to be transformed
346 * @param G Metric of the surrounding space, may be a Clifford unit then the next parameter is ignored
347 * @param rl Representation label
348 * @param anticommuting indicates if Clifford units anticommutes
349 * @return List of components of the transformed vector*/
350 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, bool anticommuting = false);
352 /** The second form of Moebius transformations defined by a 2x2 Clifford matrix M
353 * This function takes the transformation matrix M as a single entity.
355 * @param M the defining matrix
356 * @param v Vector to be transformed
357 * @param G Metric of the surrounding space, may be a Clifford unit then the next parameter is ignored
358 * @param rl Representation label
359 * @param anticommuting indicates if Clifford units anticommutes
360 * @return List of components of the transformed vector*/
361 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl = 0, bool anticommuting = false);
365 #endif // ndef __GINAC_CLIFFORD_H__