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, 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 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 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 { clifford c = ex_to<clifford>(inherited::subs(m, options)); c.metric_subs(m, options); return c;}
78 ex subs(const lst & ls, const lst & lr, unsigned options = 0) const { clifford c = ex_to<clifford>(ex(*this).subs(ls, lr, options)); c.metric_subs(ls, lr, options); return c;}
79 ex subs(const ex & e, unsigned options = 0) const{ clifford c = ex_to<clifford>(ex(*this).subs(e, options)); c.metric_subs(e, options); return c;};
82 void do_print_dflt(const print_dflt & c, unsigned level) const;
83 void do_print_latex(const print_latex & c, unsigned level) const;
84 void metric_subs(const exmap & m, unsigned options = 0) { metric = metric.subs(m, options); }
85 void metric_subs(const lst & ls, const lst & lr, unsigned options = 0) { metric = metric.subs(ls, lr, options); }
86 void metric_subs(const ex & e, unsigned options = 0) { metric = metric.subs(e, options); }
90 unsigned char representation_label; /**< Representation label to distinguish independent spin lines */
91 ex metric; /**< Metric of the space, all constructors make it an indexed object */
92 bool anticommuting; /**< Simplifications for anticommuting units is much simpler and we need this info readily available */
93 int commutator_sign; /**< It is the sign in the definition e~i e~j +/- e~j e~i = B(i, j) + B(j, i)*/
96 /** This class represents the Clifford algebra unity element. */
97 class diracone : public tensor
99 GINAC_DECLARE_REGISTERED_CLASS(diracone, tensor)
101 // non-virtual functions in this class
103 void do_print(const print_context & c, unsigned level) const;
104 void do_print_latex(const print_latex & c, unsigned level) const;
108 /** This class represents the Clifford algebra generators (units). */
109 class cliffordunit : public tensor
111 GINAC_DECLARE_REGISTERED_CLASS(cliffordunit, tensor)
113 // other constructors
115 cliffordunit(unsigned ti) : inherited(ti) {}
117 // functions overriding virtual functions from base classes
119 bool contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const;
121 // non-virtual functions in this class
123 void do_print(const print_context & c, unsigned level) const;
124 void do_print_latex(const print_latex & c, unsigned level) const;
128 /** This class represents the Dirac gamma Lorentz vector. */
129 class diracgamma : public cliffordunit
131 GINAC_DECLARE_REGISTERED_CLASS(diracgamma, cliffordunit)
133 // functions overriding virtual functions from base classes
135 bool contract_with(exvector::iterator self, exvector::iterator other, exvector & v) const;
137 // non-virtual functions in this class
139 void do_print(const print_context & c, unsigned level) const;
140 void do_print_latex(const print_latex & c, unsigned level) const;
144 /** This class represents the Dirac gamma5 object which anticommutates with
145 * all other gammas. */
146 class diracgamma5 : public tensor
148 GINAC_DECLARE_REGISTERED_CLASS(diracgamma5, tensor)
150 // functions overriding virtual functions from base classes
151 ex conjugate() const;
153 // non-virtual functions in this class
155 void do_print(const print_context & c, unsigned level) const;
156 void do_print_latex(const print_latex & c, unsigned level) const;
160 /** This class represents the Dirac gammaL object which behaves like
162 class diracgammaL : public tensor
164 GINAC_DECLARE_REGISTERED_CLASS(diracgammaL, tensor)
166 // functions overriding virtual functions from base classes
167 ex conjugate() const;
169 // non-virtual functions in this class
171 void do_print(const print_context & c, unsigned level) const;
172 void do_print_latex(const print_latex & c, unsigned level) const;
176 /** This class represents the Dirac gammaL object which behaves like
178 class diracgammaR : public tensor
180 GINAC_DECLARE_REGISTERED_CLASS(diracgammaR, tensor)
182 // functions overriding virtual functions from base classes
183 ex conjugate() const;
185 // non-virtual functions in this class
187 void do_print(const print_context & c, unsigned level) const;
188 void do_print_latex(const print_latex & c, unsigned level) const;
194 /** Specialization of is_exactly_a<clifford>(obj) for clifford objects. */
195 template<> inline bool is_exactly_a<clifford>(const basic & obj)
197 return obj.tinfo()==TINFO_clifford;
200 /** Create a Clifford unity object.
202 * @param rl Representation label
203 * @return newly constructed object */
204 ex dirac_ONE(unsigned char rl = 0);
206 /** Create a Clifford unit object.
208 * @param mu Index (must be of class varidx or a derived class)
209 * @param metr Metric (should be indexed, tensmetric or a derived class, or a matrix)
210 * @param rl Representation label
211 * @return newly constructed Clifford unit object */
212 ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0, bool anticommuting = false);
214 /** Create a Dirac gamma object.
216 * @param mu Index (must be of class varidx or a derived class)
217 * @param rl Representation label
218 * @return newly constructed gamma object */
219 ex dirac_gamma(const ex & mu, unsigned char rl = 0);
221 /** Create a Dirac gamma5 object.
223 * @param rl Representation label
224 * @return newly constructed object */
225 ex dirac_gamma5(unsigned char rl = 0);
227 /** Create a Dirac gammaL object.
229 * @param rl Representation label
230 * @return newly constructed object */
231 ex dirac_gammaL(unsigned char rl = 0);
233 /** Create a Dirac gammaR object.
235 * @param rl Representation label
236 * @return newly constructed object */
237 ex dirac_gammaR(unsigned char rl = 0);
239 /** Create a term of the form e_mu * gamma~mu with a unique index mu.
241 * @param e Original expression
242 * @param dim Dimension of index
243 * @param rl Representation label */
244 ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
246 /** Calculate dirac traces over the specified set of representation labels.
247 * The computed trace is a linear functional that is equal to the usual
248 * trace only in D = 4 dimensions. In particular, the functional is not
249 * always cyclic in D != 4 dimensions when gamma5 is involved.
251 * @param e Expression to take the trace of
252 * @param rls Set of representation labels
253 * @param trONE Expression to be returned as the trace of the unit matrix */
254 ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE = 4);
256 /** Calculate dirac traces over the specified list of representation labels.
257 * The computed trace is a linear functional that is equal to the usual
258 * trace only in D = 4 dimensions. In particular, the functional is not
259 * always cyclic in D != 4 dimensions when gamma5 is involved.
261 * @param e Expression to take the trace of
262 * @param rll List of representation labels
263 * @param trONE Expression to be returned as the trace of the unit matrix */
264 ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
266 /** Calculate the trace of an expression containing gamma objects with
267 * a specified representation label. The computed trace is a linear
268 * functional that is equal to the usual trace only in D = 4 dimensions.
269 * In particular, the functional is not always cyclic in D != 4 dimensions
270 * when gamma5 is involved.
272 * @param e Expression to take the trace of
273 * @param rl Representation label
274 * @param trONE Expression to be returned as the trace of the unit matrix */
275 ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
277 /** Bring all products of clifford objects in an expression into a canonical
278 * order. This is not necessarily the most simple form but it will allow
279 * to check two expressions for equality. */
280 ex canonicalize_clifford(const ex & e);
282 /** Automorphism of the Clifford algebra, simply changes signs of all
284 ex clifford_prime(const ex & e);
286 /** Main anti-automorphism of the Clifford algebra: makes reversion
287 * and changes signs of all clifford units. */
288 inline ex clifford_bar(const ex & e) { return clifford_prime(e.conjugate()); }
290 /** Reversion of the Clifford algebra, coincides with the conjugate(). */
291 inline ex clifford_star(const ex & e) { return e.conjugate(); }
293 /** Replaces dirac_ONE's (with a representation_label no less than rl) in e with 1.
294 * For the default value rl = 0 remove all of them. Aborts if e contains any
295 * clifford_unit with representation_label to be removed.
297 * @param e Expression to be processed
298 * @param rl Value of representation label
299 * @param options Defines some internal use */
300 ex remove_dirac_ONE(const ex & e, unsigned char rl = 0, unsigned options = 0);
302 /** Returns the maximal representation label of a clifford object
303 * if e contains at least one, otherwise returns -1
305 * @param e Expression to be processed
306 * @ignore_ONE defines if clifford_ONE should be ignored in the search*/
307 char clifford_max_label(const ex & e, bool ignore_ONE = false);
309 /** Calculation of the norm in the Clifford algebra. */
310 ex clifford_norm(const ex & e);
312 /** Calculation of the inverse in the Clifford algebra. */
313 ex clifford_inverse(const ex & e);
315 /** List or vector conversion into the Clifford vector.
317 * @param v List or vector of coordinates
318 * @param mu Index (must be of class varidx or a derived class)
319 * @param metr Metric (should be indexed, tensmetric or a derived class, or a matrix)
320 * @param rl Representation label
321 * @param e Clifford unit object
322 * @return Clifford vector with given components */
323 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl = 0, bool anticommuting = false);
324 ex lst_to_clifford(const ex & v, const ex & e);
326 /** An inverse function to lst_to_clifford(). For given Clifford vector extracts
327 * its components with respect to given Clifford unit. Obtained components may
328 * contain Clifford units with a different metric. Extraction is based on
329 * the algebraic formula (e * c.i + c.i * e)/ pow(e.i, 2) for non-degenerate cases
330 * (i.e. neither pow(e.i, 2) = 0).
332 * @param e Clifford expression to be decomposed into components
333 * @param c Clifford unit defining the metric for splitting (should have numeric dimension of indices)
334 * @param algebraic Use algebraic or symbolic algorithm for extractions
335 * @return List of components of a Clifford vector*/
336 lst clifford_to_lst(const ex & e, const ex & c, bool algebraic=true);
338 /** Calculations of Moebius transformations (conformal map) defined by a 2x2 Clifford matrix
339 * (a b\\c d) in linear spaces with arbitrary signature. The expression is
340 * (a * x + b)/(c * x + d), where x is a vector build from list v with metric G.
341 * (see Jan Cnops. An introduction to {D}irac operators on manifolds, v.24 of
342 * Progress in Mathematical Physics. Birkhauser Boston Inc., Boston, MA, 2002.)
344 * @param a (1,1) entry of the defining matrix
345 * @param b (1,2) entry of the defining matrix
346 * @param c (2,1) entry of the defining matrix
347 * @param d (2,2) entry of 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 & a, const ex & b, const ex & c, const ex & d, const ex & v, const ex & G, unsigned char rl = 0, bool anticommuting = false);
355 /** The second form of Moebius transformations defined by a 2x2 Clifford matrix M
356 * This function takes the transformation matrix M as a single entity.
358 * @param M the defining matrix
359 * @param v Vector to be transformed
360 * @param G Metric of the surrounding space, may be a Clifford unit then the next parameter is ignored
361 * @param rl Representation label
362 * @param anticommuting indicates if Clifford units anticommutes
363 * @return List of components of the transformed vector*/
364 ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl = 0, bool anticommuting = false);
368 #endif // ndef __GINAC_CLIFFORD_H__