3 * Interface to GiNaC's clifford algebra (Dirac gamma) objects. */
6 * GiNaC Copyright (C) 1999-2008 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)
46 clifford(const ex & b, unsigned char rl = 0);
47 clifford(const ex & b, const ex & mu, const ex & metr, unsigned char rl = 0, int comm_sign = -1);
49 // internal constructors
50 clifford(unsigned char rl, const ex & metr, int comm_sign, const exvector & v, bool discardable = false);
51 clifford(unsigned char rl, const ex & metr, int comm_sign, std::auto_ptr<exvector> vp);
53 // functions overriding virtual functions from base classes
55 unsigned precedence() const { return 65; }
57 ex eval_ncmul(const exvector & v) const;
58 bool match_same_type(const basic & other) const;
59 ex thiscontainer(const exvector & v) const;
60 ex thiscontainer(std::auto_ptr<exvector> vp) const;
61 unsigned return_type() const { return return_types::noncommutative; }
62 return_type_t return_type_tinfo() const;
63 // non-virtual functions in this class
65 unsigned char get_representation_label() const { return representation_label; }
66 ex get_metric() const { return metric; }
67 virtual ex get_metric(const ex & i, const ex & j, bool symmetrised = false) const;
68 bool same_metric(const ex & other) const;
69 int get_commutator_sign() const { return commutator_sign; } //**< See the member variable commutator_sign */
71 inline size_t nops() const {return inherited::nops() + 1; }
72 ex op(size_t i) const;
73 ex & let_op(size_t i);
74 ex subs(const exmap & m, unsigned options = 0) const;
77 void do_print_dflt(const print_dflt & c, unsigned level) const;
78 void do_print_latex(const print_latex & c, unsigned level) const;
82 unsigned char representation_label; /**< Representation label to distinguish independent spin lines */
83 ex metric; /**< Metric of the space, all constructors make it an indexed object */
84 int commutator_sign; /**< It is the sign in the definition e~i e~j +/- e~j e~i = B(i, j) + B(j, i)*/
87 /** This class represents the Clifford algebra unity element. */
88 class diracone : public tensor
90 GINAC_DECLARE_REGISTERED_CLASS(diracone, tensor)
92 // non-virtual functions in this class
94 void do_print(const print_context & c, unsigned level) const;
95 void do_print_latex(const print_latex & c, unsigned level) const;
99 /** This class represents the Clifford algebra generators (units). */
100 class cliffordunit : public tensor
102 GINAC_DECLARE_REGISTERED_CLASS(cliffordunit, tensor)
104 // other constructors
106 cliffordunit(tinfo_t ti) : inherited(ti) {}
108 // functions overriding virtual functions from base classes
110 bool contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const;
112 // non-virtual functions in this class
114 void do_print(const print_context & c, unsigned level) const;
115 void do_print_latex(const print_latex & c, unsigned level) const;
119 /** This class represents the Dirac gamma Lorentz vector. */
120 class diracgamma : public cliffordunit
122 GINAC_DECLARE_REGISTERED_CLASS(diracgamma, cliffordunit)
124 // functions overriding virtual functions from base classes
126 bool contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const;
128 // non-virtual functions in this class
130 void do_print(const print_context & c, unsigned level) const;
131 void do_print_latex(const print_latex & c, unsigned level) const;
135 /** This class represents the Dirac gamma5 object which anticommutates with
136 * all other gammas. */
137 class diracgamma5 : public tensor
139 GINAC_DECLARE_REGISTERED_CLASS(diracgamma5, tensor)
141 // functions overriding virtual functions from base classes
142 ex conjugate() const;
144 // non-virtual functions in this class
146 void do_print(const print_context & c, unsigned level) const;
147 void do_print_latex(const print_latex & c, unsigned level) const;
151 /** This class represents the Dirac gammaL object which behaves like
153 class diracgammaL : public tensor
155 GINAC_DECLARE_REGISTERED_CLASS(diracgammaL, tensor)
157 // functions overriding virtual functions from base classes
158 ex conjugate() const;
160 // non-virtual functions in this class
162 void do_print(const print_context & c, unsigned level) const;
163 void do_print_latex(const print_latex & c, unsigned level) const;
167 /** This class represents the Dirac gammaL object which behaves like
169 class diracgammaR : public tensor
171 GINAC_DECLARE_REGISTERED_CLASS(diracgammaR, tensor)
173 // functions overriding virtual functions from base classes
174 ex conjugate() const;
176 // non-virtual functions in this class
178 void do_print(const print_context & c, unsigned level) const;
179 void do_print_latex(const print_latex & c, unsigned level) const;
185 /** Check whether a given return_type_t object (as returned by return_type_tinfo()
186 * is that of a clifford object (with an arbitrary representation label).
188 * @param ti tinfo key */
189 inline bool is_clifford_tinfo(const return_type_t& ti)
191 return *(ti.tinfo) == typeid(clifford);
194 /** Create a Clifford unity object.
196 * @param rl Representation label
197 * @return newly constructed object */
198 ex dirac_ONE(unsigned char rl = 0);
200 /** Create a Clifford unit object.
202 * @param mu Index (must be of class varidx or a derived class)
203 * @param metr Metric (should be indexed, tensmetric or a derived class, or a matrix)
204 * @param rl Representation label
205 * @return newly constructed Clifford unit object */
206 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
208 /** Create a Dirac gamma object.
210 * @param mu Index (must be of class varidx or a derived class)
211 * @param rl Representation label
212 * @return newly constructed gamma object */
213 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
215 /** Create a Dirac gamma5 object.
217 * @param rl Representation label
218 * @return newly constructed object */
219 ex dirac_gamma5(unsigned char rl = 0);
221 /** Create a Dirac gammaL object.
223 * @param rl Representation label
224 * @return newly constructed object */
225 ex dirac_gammaL(unsigned char rl = 0);
227 /** Create a Dirac gammaR object.
229 * @param rl Representation label
230 * @return newly constructed object */
231 ex dirac_gammaR(unsigned char rl = 0);
233 /** Create a term of the form e_mu * gamma~mu with a unique index mu.
235 * @param e Original expression
236 * @param dim Dimension of index
237 * @param rl Representation label */
238 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
240 /** Calculate dirac traces over the specified set of representation labels.
241 * The computed trace is a linear functional that is equal to the usual
242 * trace only in D = 4 dimensions. In particular, the functional is not
243 * always cyclic in D != 4 dimensions when gamma5 is involved.
245 * @param e Expression to take the trace of
246 * @param rls Set of representation labels
247 * @param trONE Expression to be returned as the trace of the unit matrix */
248 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE = 4);
250 /** Calculate dirac traces over the specified list of representation labels.
251 * The computed trace is a linear functional that is equal to the usual
252 * trace only in D = 4 dimensions. In particular, the functional is not
253 * always cyclic in D != 4 dimensions when gamma5 is involved.
255 * @param e Expression to take the trace of
256 * @param rll List of representation labels
257 * @param trONE Expression to be returned as the trace of the unit matrix */
258 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
260 /** Calculate the trace of an expression containing gamma objects with
261 * a specified representation label. The computed trace is a linear
262 * functional that is equal to the usual trace only in D = 4 dimensions.
263 * In particular, the functional is not always cyclic in D != 4 dimensions
264 * when gamma5 is involved.
266 * @param e Expression to take the trace of
267 * @param rl Representation label
268 * @param trONE Expression to be returned as the trace of the unit matrix */
269 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
271 /** Bring all products of clifford objects in an expression into a canonical
272 * order. This is not necessarily the most simple form but it will allow
273 * to check two expressions for equality. */
274 ex canonicalize_clifford(const ex & e);
276 /** Automorphism of the Clifford algebra, simply changes signs of all
278 ex clifford_prime(const ex & e);
280 /** Main anti-automorphism of the Clifford algebra: makes reversion
281 * and changes signs of all clifford units. */
282 inline ex clifford_bar(const ex & e) { return clifford_prime(e.conjugate()); }
284 /** Reversion of the Clifford algebra, coincides with the conjugate(). */
285 inline ex clifford_star(const ex & e) { return e.conjugate(); }
287 /** Replaces dirac_ONE's (with a representation_label no less than rl) in e with 1.
288 * For the default value rl = 0 remove all of them. Aborts if e contains any
289 * clifford_unit with representation_label to be removed.
291 * @param e Expression to be processed
292 * @param rl Value of representation label
293 * @param options Defines some internal use */
294 ex remove_dirac_ONE(const ex & e, unsigned char rl = 0, unsigned options = 0);
296 /** Returns the maximal representation label of a clifford object
297 * if e contains at least one, otherwise returns -1
299 * @param e Expression to be processed
300 * @ignore_ONE defines if clifford_ONE should be ignored in the search*/
301 char clifford_max_label(const ex & e, bool ignore_ONE = false);
303 /** Calculation of the norm in the Clifford algebra. */
304 ex clifford_norm(const ex & e);
306 /** Calculation of the inverse in the Clifford algebra. */
307 ex clifford_inverse(const ex & e);
309 /** List or vector conversion into the Clifford vector.
311 * @param v List or vector of coordinates
312 * @param mu Index (must be of class varidx or a derived class)
313 * @param metr Metric (should be indexed, tensmetric or a derived class, or a matrix)
314 * @param rl Representation label
315 * @param e Clifford unit object
316 * @return Clifford vector with given components */
317 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl = 0);
318 ex lst_to_clifford(const ex & v, const ex & e);
320 /** An inverse function to lst_to_clifford(). For given Clifford vector extracts
321 * its components with respect to given Clifford unit. Obtained components may
322 * contain Clifford units with a different metric. Extraction is based on
323 * the algebraic formula (e * c.i + c.i * e)/ pow(e.i, 2) for non-degenerate cases
324 * (i.e. neither pow(e.i, 2) = 0).
326 * @param e Clifford expression to be decomposed into components
327 * @param c Clifford unit defining the metric for splitting (should have numeric dimension of indices)
328 * @param algebraic Use algebraic or symbolic algorithm for extractions
329 * @return List of components of a Clifford vector*/
330 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic=true);
332 /** Calculations of Moebius transformations (conformal map) defined by a 2x2 Clifford matrix
333 * (a b\\c d) in linear spaces with arbitrary signature. The expression is
334 * (a * x + b)/(c * x + d), where x is a vector build from list v with metric G.
335 * (see Jan Cnops. An introduction to {D}irac operators on manifolds, v.24 of
336 * Progress in Mathematical Physics. Birkhauser Boston Inc., Boston, MA, 2002.)
338 * @param a (1,1) entry of the defining matrix
339 * @param b (1,2) entry of the defining matrix
340 * @param c (2,1) entry of the defining matrix
341 * @param d (2,2) entry of the defining matrix
342 * @param v Vector to be transformed
343 * @param G Metric of the surrounding space, may be a Clifford unit then the next parameter is ignored
344 * @param rl Representation label
345 * @return List of components of the transformed vector*/
346 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);
348 /** The second form of Moebius transformations defined by a 2x2 Clifford matrix M
349 * This function takes the transformation matrix M as a single entity.
351 * @param M the defining matrix
352 * @param v Vector to be transformed
353 * @param G Metric of the surrounding space, may be a Clifford unit then the next parameter is ignored
354 * @param rl Representation label
355 * @return List of components of the transformed vector*/
356 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl = 0);
360 #endif // ndef __GINAC_CLIFFORD_H__