3 * Interface to GiNaC's clifford algebra (Dirac gamma) objects. */
6 * GiNaC Copyright (C) 1999-2010 Johannes Gutenberg University Mainz, Germany
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23 #ifndef GINAC_CLIFFORD_H
24 #define GINAC_CLIFFORD_H
35 /** This class holds an object representing an element of the Clifford
36 * algebra (the Dirac gamma matrices). These objects only carry Lorentz
37 * indices. Spinor indices are hidden. A representation label (an unsigned
38 * 8-bit integer) is used to distinguish elements from different Clifford
39 * algebras (objects with different labels commutate). */
40 class clifford : public indexed
42 GINAC_DECLARE_REGISTERED_CLASS(clifford, indexed)
45 clifford(const ex & b, unsigned char rl = 0);
46 clifford(const ex & b, const ex & mu, const ex & metr, unsigned char rl = 0, int comm_sign = -1);
48 // internal constructors
49 clifford(unsigned char rl, const ex & metr, int comm_sign, const exvector & v, bool discardable = false);
50 clifford(unsigned char rl, const ex & metr, int comm_sign, std::auto_ptr<exvector> vp);
52 // functions overriding virtual functions from base classes
54 unsigned precedence() const { return 65; }
55 void archive(archive_node& n) const;
56 void read_archive(const archive_node& n, lst& sym_lst);
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 return_type_t return_type_tinfo() const;
64 // non-virtual functions in this class
66 unsigned char get_representation_label() const { return representation_label; }
67 ex get_metric() const { return metric; }
68 virtual ex get_metric(const ex & i, const ex & j, bool symmetrised = false) const;
69 bool same_metric(const ex & other) const;
70 int get_commutator_sign() const { return commutator_sign; } //**< See the member variable commutator_sign */
72 inline size_t nops() const {return inherited::nops() + 1; }
73 ex op(size_t i) const;
74 ex & let_op(size_t i);
75 ex subs(const exmap & m, unsigned options = 0) const;
78 void do_print_dflt(const print_dflt & c, unsigned level) const;
79 void do_print_latex(const print_latex & c, unsigned level) const;
83 unsigned char representation_label; /**< Representation label to distinguish independent spin lines */
84 ex metric; /**< Metric of the space, all constructors make it an indexed object */
85 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 GINAC_DECLARE_UNARCHIVER(clifford);
89 /** This class represents the Clifford algebra unity element. */
90 class diracone : public tensor
92 GINAC_DECLARE_REGISTERED_CLASS(diracone, tensor)
94 // non-virtual functions in this class
96 void do_print(const print_context & c, unsigned level) const;
97 void do_print_latex(const print_latex & c, unsigned level) const;
99 GINAC_DECLARE_UNARCHIVER(diracone);
102 /** This class represents the Clifford algebra generators (units). */
103 class cliffordunit : public tensor
105 GINAC_DECLARE_REGISTERED_CLASS(cliffordunit, tensor)
107 // functions overriding virtual functions from base classes
109 bool contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const;
111 // non-virtual functions in this class
113 void do_print(const print_context & c, unsigned level) const;
114 void do_print_latex(const print_latex & c, unsigned level) const;
118 /** This class represents the Dirac gamma Lorentz vector. */
119 class diracgamma : public cliffordunit
121 GINAC_DECLARE_REGISTERED_CLASS(diracgamma, cliffordunit)
123 // functions overriding virtual functions from base classes
125 bool contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const;
127 // non-virtual functions in this class
129 void do_print(const print_context & c, unsigned level) const;
130 void do_print_latex(const print_latex & c, unsigned level) const;
132 GINAC_DECLARE_UNARCHIVER(diracgamma);
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;
149 GINAC_DECLARE_UNARCHIVER(diracgamma5);
152 /** This class represents the Dirac gammaL object which behaves like
154 class diracgammaL : public tensor
156 GINAC_DECLARE_REGISTERED_CLASS(diracgammaL, tensor)
158 // functions overriding virtual functions from base classes
159 ex conjugate() const;
161 // non-virtual functions in this class
163 void do_print(const print_context & c, unsigned level) const;
164 void do_print_latex(const print_latex & c, unsigned level) const;
166 GINAC_DECLARE_UNARCHIVER(diracgammaL);
169 /** This class represents the Dirac gammaL object which behaves like
171 class diracgammaR : public tensor
173 GINAC_DECLARE_REGISTERED_CLASS(diracgammaR, tensor)
175 // functions overriding virtual functions from base classes
176 ex conjugate() const;
178 // non-virtual functions in this class
180 void do_print(const print_context & c, unsigned level) const;
181 void do_print_latex(const print_latex & c, unsigned level) const;
183 GINAC_DECLARE_UNARCHIVER(diracgammaR);
188 /** Check whether a given return_type_t object (as returned by return_type_tinfo()
189 * is that of a clifford object (with an arbitrary representation label).
191 * @param ti tinfo key */
192 inline bool is_clifford_tinfo(const return_type_t& ti)
194 return *(ti.tinfo) == typeid(clifford);
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);
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);
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 * @return List of components of the transformed vector*/
349 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);
351 /** The second form of Moebius transformations defined by a 2x2 Clifford matrix M
352 * This function takes the transformation matrix M as a single entity.
354 * @param M the defining matrix
355 * @param v Vector to be transformed
356 * @param G Metric of the surrounding space, may be a Clifford unit then the next parameter is ignored
357 * @param rl Representation label
358 * @return List of components of the transformed vector*/
359 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl = 0);
363 #endif // ndef GINAC_CLIFFORD_H