3 * Interface to GiNaC's clifford algebra (Dirac gamma) objects. */
6 * GiNaC Copyright (C) 1999-2005 Johannes Gutenberg University Mainz, Germany
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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)
47 clifford(const ex & b, unsigned char rl = 0);
48 clifford(const ex & b, const ex & mu, const ex & metr, unsigned char rl = 0);
50 // internal constructors
51 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);
54 // functions overriding virtual functions from base classes
56 unsigned precedence() const { return 65; }
58 ex eval_ncmul(const exvector & v) const;
59 bool match_same_type(const basic & other) const;
60 ex thiscontainer(const exvector & v) const;
61 ex thiscontainer(std::auto_ptr<exvector> vp) const;
62 unsigned return_type() const { return return_types::noncommutative; }
63 unsigned return_type_tinfo() const { return TINFO_clifford + representation_label; }
65 // non-virtual functions in this class
67 unsigned char get_representation_label() const { return representation_label; }
68 ex get_metric() const { return metric; }
69 ex get_metric(const ex & i, const ex & j) const;
70 bool same_metric(const ex & other) const;
73 void do_print_dflt(const print_dflt & c, unsigned level) const;
74 void do_print_latex(const print_latex & c, unsigned level) const;
78 unsigned char representation_label; /**< Representation label to distinguish independent spin lines */
83 /** This class represents the Clifford algebra unity element. */
84 class diracone : public tensor
86 GINAC_DECLARE_REGISTERED_CLASS(diracone, tensor)
88 // non-virtual functions in this class
90 void do_print(const print_context & c, unsigned level) const;
91 void do_print_latex(const print_latex & c, unsigned level) const;
95 /** This class represents the Clifford algebra generators (units). */
96 class cliffordunit : public tensor
98 GINAC_DECLARE_REGISTERED_CLASS(cliffordunit, tensor)
100 // other constructors
102 cliffordunit(unsigned ti) : inherited(ti) {}
104 // functions overriding virtual functions from base classes
106 bool contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const;
108 // non-virtual functions in this class
110 void do_print(const print_context & c, unsigned level) const;
111 void do_print_latex(const print_latex & c, unsigned level) const;
115 /** This class represents the Dirac gamma Lorentz vector. */
116 class diracgamma : public cliffordunit
118 GINAC_DECLARE_REGISTERED_CLASS(diracgamma, cliffordunit)
120 // functions overriding virtual functions from base classes
122 bool contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const;
124 // non-virtual functions in this class
126 void do_print(const print_context & c, unsigned level) const;
127 void do_print_latex(const print_latex & c, unsigned level) const;
131 /** This class represents the Dirac gamma5 object which anticommutates with
132 * all other gammas. */
133 class diracgamma5 : public tensor
135 GINAC_DECLARE_REGISTERED_CLASS(diracgamma5, tensor)
137 // functions overriding virtual functions from base classes
138 ex conjugate() const;
140 // non-virtual functions in this class
142 void do_print(const print_context & c, unsigned level) const;
143 void do_print_latex(const print_latex & c, unsigned level) const;
147 /** This class represents the Dirac gammaL object which behaves like
149 class diracgammaL : public tensor
151 GINAC_DECLARE_REGISTERED_CLASS(diracgammaL, tensor)
153 // functions overriding virtual functions from base classes
154 ex conjugate() const;
156 // non-virtual functions in this class
158 void do_print(const print_context & c, unsigned level) const;
159 void do_print_latex(const print_latex & c, unsigned level) const;
163 /** This class represents the Dirac gammaL object which behaves like
165 class diracgammaR : public tensor
167 GINAC_DECLARE_REGISTERED_CLASS(diracgammaR, tensor)
169 // functions overriding virtual functions from base classes
170 ex conjugate() const;
172 // non-virtual functions in this class
174 void do_print(const print_context & c, unsigned level) const;
175 void do_print_latex(const print_latex & c, unsigned level) const;
181 /** Specialization of is_exactly_a<clifford>(obj) for clifford objects. */
182 template<> inline bool is_exactly_a<clifford>(const basic & obj)
184 return obj.tinfo()==TINFO_clifford;
187 /** Create a Clifford unity object.
189 * @param rl Representation label
190 * @return newly constructed object */
191 ex dirac_ONE(unsigned char rl = 0);
193 /** Create a Clifford unit object.
195 * @param mu Index (must be of class varidx or a derived class)
196 * @param metr Metric (should be of class tensmetric or a derived class, or a matrix)
197 * @param rl Representation label
198 * @return newly constructed Clifford unit object */
199 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
201 /** Create a Dirac gamma object.
203 * @param mu Index (must be of class varidx or a derived class)
204 * @param rl Representation label
205 * @return newly constructed gamma object */
206 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
208 /** Create a Dirac gamma5 object.
210 * @param rl Representation label
211 * @return newly constructed object */
212 ex dirac_gamma5(unsigned char rl = 0);
214 /** Create a Dirac gammaL object.
216 * @param rl Representation label
217 * @return newly constructed object */
218 ex dirac_gammaL(unsigned char rl = 0);
220 /** Create a Dirac gammaR object.
222 * @param rl Representation label
223 * @return newly constructed object */
224 ex dirac_gammaR(unsigned char rl = 0);
226 /** Create a term of the form e_mu * gamma~mu with a unique index mu.
228 * @param e Original expression
229 * @param dim Dimension of index
230 * @param rl Representation label */
231 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
233 /** Calculate dirac traces over the specified set of representation labels.
234 * The computed trace is a linear functional that is equal to the usual
235 * trace only in D = 4 dimensions. In particular, the functional is not
236 * always cyclic in D != 4 dimensions when gamma5 is involved.
238 * @param e Expression to take the trace of
239 * @param rls Set of representation labels
240 * @param trONE Expression to be returned as the trace of the unit matrix */
241 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE = 4);
243 /** Calculate dirac traces over the specified list 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 rll List 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 lst & rll, const ex & trONE = 4);
253 /** Calculate the trace of an expression containing gamma objects with
254 * a specified representation label. The computed trace is a linear
255 * functional that is equal to the usual trace only in D = 4 dimensions.
256 * In particular, the functional is not always cyclic in D != 4 dimensions
257 * when gamma5 is involved.
259 * @param e Expression to take the trace of
260 * @param rl Representation label
261 * @param trONE Expression to be returned as the trace of the unit matrix */
262 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
264 /** Bring all products of clifford objects in an expression into a canonical
265 * order. This is not necessarily the most simple form but it will allow
266 * to check two expressions for equality. */
267 ex canonicalize_clifford(const ex & e);
269 /** Automorphism of the Clifford algebra, simply changes signs of all
271 ex clifford_prime(const ex & e);
273 /** Main anti-automorphism of the Clifford algebra: makes reversion
274 * and changes signs of all clifford units. */
275 inline ex clifford_bar(const ex & e) { return clifford_prime(e.conjugate()); }
277 /** Reversion of the Clifford algebra, coincides with the conjugate(). */
278 inline ex clifford_star(const ex & e) { return e.conjugate(); }
280 /** Replaces dirac_ONE's (with a representation_label no less than rl) in e with 1.
281 * For the default value rl = 0 remove all of them. Aborts if e contains any
282 * clifford_unit with representation_label to be removed.
284 * @param e Expression to be processed
285 * @param rl Value of representation label */
286 ex remove_dirac_ONE(const ex & e, unsigned char rl = 0);
288 /** Calculation of the norm in the Clifford algebra. */
289 ex clifford_norm(const ex & e);
291 /** Calculation of the inverse in the Clifford algebra. */
292 ex clifford_inverse(const ex & e);
294 /** List or vector conversion into the Clifford vector.
296 * @param v List or vector of coordinates
297 * @param mu Index (must be of class varidx or a derived class)
298 * @param metr Metric (should be of class tensmetric or a derived class, or a matrix)
299 * @param rl Representation label
300 * @param e Clifford unit object
301 * @return Clifford vector with given components */
302 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl = 0);
303 ex lst_to_clifford(const ex & v, const ex & e);
305 /** An inverse function to lst_to_clifford(). For given Clifford vector extracts
306 * its components with respect to given Clifford unit. Obtained components may
307 * contain Clifford units with a different metric. Extraction is based on
308 * the algebraic formula (e * c.i + c.i * e)/ pow(e.i, 2) for non-degenerate cases
309 * (i.e. neither pow(e.i, 2) = 0).
311 * @param e Clifford expression to be decomposed into components
312 * @param c Clifford unit defining the metric for splitting (should have numeric dimension of indices)
313 * @param algebraic Use algebraic or symbolic algorithm for extractions
314 * @return List of components of a Clifford vector*/
315 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic=true);
317 /** Calculations of Moebius transformations (conformal map) defined by a 2x2 Clifford matrix
318 * (a b\\c d) in linear spaces with arbitrary signature. The expression is
319 * (a * x + b)/(c * x + d), where x is a vector build from list v with metric G.
320 * (see Jan Cnops. An introduction to {D}irac operators on manifolds, v.24 of
321 * Progress in Mathematical Physics. Birkhauser Boston Inc., Boston, MA, 2002.)
323 * @param a (1,1) entry of the defining matrix
324 * @param b (1,2) entry of the defining matrix
325 * @param c (2,1) entry of the defining matrix
326 * @param d (2,2) entry of the defining matrix
327 * @param v Vector to be transformed
328 * @param G Metric of the surrounding space, may be a Clifford unit then the next parameter is ignored
329 * @param rl Representation label
330 * @return List of components of the transformed vector*/
331 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);
333 /** The second form of Moebius transformations defined by a 2x2 Clifford matrix M
334 * This function takes the transformation matrix M as a single entity.
336 * @param M the defining matrix
337 * @param v Vector to be transformed
338 * @param G Metric of the surrounding space, may be a Clifford unit then the next parameter is ignored
339 * @param rl Representation label
340 * @return List of components of the transformed vector*/
341 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl = 0);
345 #endif // ndef __GINAC_CLIFFORD_H__