3 * Interface to GiNaC's clifford algebra (Dirac gamma) objects. */
6 * GiNaC Copyright (C) 1999-2006 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, bool anticommut = false);
48 clifford(const ex & b, const ex & mu, const ex & metr, unsigned char rl = 0, bool anticommut = false, int comm_sign = -1);
50 // internal constructors
51 clifford(unsigned char rl, const ex & metr, bool anticommut, int comm_sign, const exvector & v, bool discardable = false);
52 clifford(unsigned char rl, const ex & metr, bool anticommut, int comm_sign, 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 const basic* return_type_tinfo() const { return this; }
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 virtual ex get_metric(const ex & i, const ex & j, bool symmetrised = false) const;
70 bool same_metric(const ex & other) const;
71 bool is_anticommuting() const { return anticommuting; } //**< See the member variable anticommuting */
72 int get_commutator_sign() const { return commutator_sign; } //**< See the member variable commutator_sign */
74 inline size_t nops() const {return inherited::nops() + 1; }
75 ex op(size_t i) const;
76 ex & let_op(size_t i);
77 ex subs(const exmap & m, unsigned options = 0) const;
80 void do_print_dflt(const print_dflt & c, unsigned level) const;
81 void do_print_latex(const print_latex & c, unsigned level) const;
85 unsigned char representation_label; /**< Representation label to distinguish independent spin lines */
86 ex metric; /**< Metric of the space, all constructors make it an indexed object */
87 bool anticommuting; /**< Simplifications for anticommuting units is much simpler and we need this info readily available */
88 int commutator_sign; /**< It is the sign in the definition e~i e~j +/- e~j e~i = B(i, j) + B(j, i)*/
91 /** This class represents the Clifford algebra unity element. */
92 class diracone : public tensor
94 GINAC_DECLARE_REGISTERED_CLASS(diracone, tensor)
96 // non-virtual functions in this class
98 void do_print(const print_context & c, unsigned level) const;
99 void do_print_latex(const print_latex & c, unsigned level) const;
103 /** This class represents the Clifford algebra generators (units). */
104 class cliffordunit : public tensor
106 GINAC_DECLARE_REGISTERED_CLASS(cliffordunit, tensor)
108 // other constructors
110 cliffordunit(tinfo_t ti) : inherited(ti) {}
112 // functions overriding virtual functions from base classes
114 bool contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const;
116 // non-virtual functions in this class
118 void do_print(const print_context & c, unsigned level) const;
119 void do_print_latex(const print_latex & c, unsigned level) const;
123 /** This class represents the Dirac gamma Lorentz vector. */
124 class diracgamma : public cliffordunit
126 GINAC_DECLARE_REGISTERED_CLASS(diracgamma, cliffordunit)
128 // functions overriding virtual functions from base classes
130 bool contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const;
132 // non-virtual functions in this class
134 void do_print(const print_context & c, unsigned level) const;
135 void do_print_latex(const print_latex & c, unsigned level) const;
139 /** This class represents the Dirac gamma5 object which anticommutates with
140 * all other gammas. */
141 class diracgamma5 : public tensor
143 GINAC_DECLARE_REGISTERED_CLASS(diracgamma5, tensor)
145 // functions overriding virtual functions from base classes
146 ex conjugate() const;
148 // non-virtual functions in this class
150 void do_print(const print_context & c, unsigned level) const;
151 void do_print_latex(const print_latex & c, unsigned level) const;
155 /** This class represents the Dirac gammaL object which behaves like
157 class diracgammaL : public tensor
159 GINAC_DECLARE_REGISTERED_CLASS(diracgammaL, tensor)
161 // functions overriding virtual functions from base classes
162 ex conjugate() const;
164 // non-virtual functions in this class
166 void do_print(const print_context & c, unsigned level) const;
167 void do_print_latex(const print_latex & c, unsigned level) const;
171 /** This class represents the Dirac gammaL object which behaves like
173 class diracgammaR : public tensor
175 GINAC_DECLARE_REGISTERED_CLASS(diracgammaR, tensor)
177 // functions overriding virtual functions from base classes
178 ex conjugate() const;
180 // non-virtual functions in this class
182 void do_print(const print_context & c, unsigned level) const;
183 void do_print_latex(const print_latex & c, unsigned level) const;
189 /** Create a Clifford unity object.
191 * @param rl Representation label
192 * @return newly constructed object */
193 ex dirac_ONE(unsigned char rl = 0);
195 /** Create a Clifford unit object.
197 * @param mu Index (must be of class varidx or a derived class)
198 * @param metr Metric (should be indexed, tensmetric or a derived class, or a matrix)
199 * @param rl Representation label
200 * @return newly constructed Clifford unit object */
201 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0, bool anticommuting = false);
203 /** Create a Dirac gamma object.
205 * @param mu Index (must be of class varidx or a derived class)
206 * @param rl Representation label
207 * @return newly constructed gamma object */
208 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
210 /** Create a Dirac gamma5 object.
212 * @param rl Representation label
213 * @return newly constructed object */
214 ex dirac_gamma5(unsigned char rl = 0);
216 /** Create a Dirac gammaL object.
218 * @param rl Representation label
219 * @return newly constructed object */
220 ex dirac_gammaL(unsigned char rl = 0);
222 /** Create a Dirac gammaR object.
224 * @param rl Representation label
225 * @return newly constructed object */
226 ex dirac_gammaR(unsigned char rl = 0);
228 /** Create a term of the form e_mu * gamma~mu with a unique index mu.
230 * @param e Original expression
231 * @param dim Dimension of index
232 * @param rl Representation label */
233 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
235 /** Calculate dirac traces over the specified set of representation labels.
236 * The computed trace is a linear functional that is equal to the usual
237 * trace only in D = 4 dimensions. In particular, the functional is not
238 * always cyclic in D != 4 dimensions when gamma5 is involved.
240 * @param e Expression to take the trace of
241 * @param rls Set of representation labels
242 * @param trONE Expression to be returned as the trace of the unit matrix */
243 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE = 4);
245 /** Calculate dirac traces over the specified list of representation labels.
246 * The computed trace is a linear functional that is equal to the usual
247 * trace only in D = 4 dimensions. In particular, the functional is not
248 * always cyclic in D != 4 dimensions when gamma5 is involved.
250 * @param e Expression to take the trace of
251 * @param rll List of representation labels
252 * @param trONE Expression to be returned as the trace of the unit matrix */
253 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
255 /** Calculate the trace of an expression containing gamma objects with
256 * a specified representation label. The computed trace is a linear
257 * functional that is equal to the usual trace only in D = 4 dimensions.
258 * In particular, the functional is not always cyclic in D != 4 dimensions
259 * when gamma5 is involved.
261 * @param e Expression to take the trace of
262 * @param rl Representation label
263 * @param trONE Expression to be returned as the trace of the unit matrix */
264 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
266 /** Bring all products of clifford objects in an expression into a canonical
267 * order. This is not necessarily the most simple form but it will allow
268 * to check two expressions for equality. */
269 ex canonicalize_clifford(const ex & e);
271 /** Automorphism of the Clifford algebra, simply changes signs of all
273 ex clifford_prime(const ex & e);
275 /** Main anti-automorphism of the Clifford algebra: makes reversion
276 * and changes signs of all clifford units. */
277 inline ex clifford_bar(const ex & e) { return clifford_prime(e.conjugate()); }
279 /** Reversion of the Clifford algebra, coincides with the conjugate(). */
280 inline ex clifford_star(const ex & e) { return e.conjugate(); }
282 /** Replaces dirac_ONE's (with a representation_label no less than rl) in e with 1.
283 * For the default value rl = 0 remove all of them. Aborts if e contains any
284 * clifford_unit with representation_label to be removed.
286 * @param e Expression to be processed
287 * @param rl Value of representation label
288 * @param options Defines some internal use */
289 ex remove_dirac_ONE(const ex & e, unsigned char rl = 0, unsigned options = 0);
291 /** Returns the maximal representation label of a clifford object
292 * if e contains at least one, otherwise returns -1
294 * @param e Expression to be processed
295 * @ignore_ONE defines if clifford_ONE should be ignored in the search*/
296 char clifford_max_label(const ex & e, bool ignore_ONE = false);
298 /** Calculation of the norm in the Clifford algebra. */
299 ex clifford_norm(const ex & e);
301 /** Calculation of the inverse in the Clifford algebra. */
302 ex clifford_inverse(const ex & e);
304 /** List or vector conversion into the Clifford vector.
306 * @param v List or vector of coordinates
307 * @param mu Index (must be of class varidx or a derived class)
308 * @param metr Metric (should be indexed, tensmetric or a derived class, or a matrix)
309 * @param rl Representation label
310 * @param e Clifford unit object
311 * @return Clifford vector with given components */
312 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl = 0, bool anticommuting = false);
313 ex lst_to_clifford(const ex & v, const ex & e);
315 /** An inverse function to lst_to_clifford(). For given Clifford vector extracts
316 * its components with respect to given Clifford unit. Obtained components may
317 * contain Clifford units with a different metric. Extraction is based on
318 * the algebraic formula (e * c.i + c.i * e)/ pow(e.i, 2) for non-degenerate cases
319 * (i.e. neither pow(e.i, 2) = 0).
321 * @param e Clifford expression to be decomposed into components
322 * @param c Clifford unit defining the metric for splitting (should have numeric dimension of indices)
323 * @param algebraic Use algebraic or symbolic algorithm for extractions
324 * @return List of components of a Clifford vector*/
325 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic=true);
327 /** Calculations of Moebius transformations (conformal map) defined by a 2x2 Clifford matrix
328 * (a b\\c d) in linear spaces with arbitrary signature. The expression is
329 * (a * x + b)/(c * x + d), where x is a vector build from list v with metric G.
330 * (see Jan Cnops. An introduction to {D}irac operators on manifolds, v.24 of
331 * Progress in Mathematical Physics. Birkhauser Boston Inc., Boston, MA, 2002.)
333 * @param a (1,1) entry of the defining matrix
334 * @param b (1,2) entry of the defining matrix
335 * @param c (2,1) entry of the defining matrix
336 * @param d (2,2) entry of 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 * @param anticommuting indicates if Clifford units anticommutes
341 * @return List of components of the transformed vector*/
342 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);
344 /** The second form of Moebius transformations defined by a 2x2 Clifford matrix M
345 * This function takes the transformation matrix M as a single entity.
347 * @param M the defining matrix
348 * @param v Vector to be transformed
349 * @param G Metric of the surrounding space, may be a Clifford unit then the next parameter is ignored
350 * @param rl Representation label
351 * @param anticommuting indicates if Clifford units anticommutes
352 * @return List of components of the transformed vector*/
353 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl = 0, bool anticommuting = false);
357 #endif // ndef __GINAC_CLIFFORD_H__