* Interface to GiNaC's clifford algebra (Dirac gamma) objects. */
/*
- * GiNaC Copyright (C) 1999-2004 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2005 Johannes Gutenberg University Mainz, Germany
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
#include "symbol.h"
#include "idx.h"
+#include <set>
+
namespace GiNaC {
* algebra (the Dirac gamma matrices). These objects only carry Lorentz
* indices. Spinor indices are hidden. A representation label (an unsigned
* 8-bit integer) is used to distinguish elements from different Clifford
- * algebras (objects with different labels commute). */
+ * algebras (objects with different labels commutate). */
class clifford : public indexed
{
GINAC_DECLARE_REGISTERED_CLASS(clifford, indexed)
};
-/** This class represents the Dirac gamma5 object which anticommutes with
+/** This class represents the Dirac gamma5 object which anticommutates with
* all other gammas. */
class diracgamma5 : public tensor
{
/** Create a Clifford unit object.
*
* @param mu Index (must be of class varidx or a derived class)
- * @param metr Metric (must be of class tensor or a derived class)
+ * @param metr Metric (should be of class tensmetric or a derived class, or a symmetric matrix)
* @param rl Representation label
* @return newly constructed Clifford unit object */
ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
* @param rl Representation label */
ex dirac_slash(const ex & e, const ex & dim, unsigned char rl = 0);
+/** Calculate dirac traces over the specified set of representation labels.
+ * The computed trace is a linear functional that is equal to the usual
+ * trace only in D = 4 dimensions. In particular, the functional is not
+ * always cyclic in D != 4 dimensions when gamma5 is involved.
+ *
+ * @param e Expression to take the trace of
+ * @param rls Set of representation labels
+ * @param trONE Expression to be returned as the trace of the unit matrix */
+ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE = 4);
+
+/** Calculate dirac traces over the specified list of representation labels.
+ * The computed trace is a linear functional that is equal to the usual
+ * trace only in D = 4 dimensions. In particular, the functional is not
+ * always cyclic in D != 4 dimensions when gamma5 is involved.
+ *
+ * @param e Expression to take the trace of
+ * @param rll List of representation labels
+ * @param trONE Expression to be returned as the trace of the unit matrix */
+ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
+
/** Calculate the trace of an expression containing gamma objects with
* a specified representation label. The computed trace is a linear
* functional that is equal to the usual trace only in D = 4 dimensions.
/** Reversion of the Clifford algebra, coincides with the conjugate(). */
inline ex clifford_star(const ex & e) { return e.conjugate(); }
-ex delete_ONE(const ex &e);
+/** Replaces all dirac_ONE's in e with 1 (effectively removing them). */
+ex remove_dirac_ONE(const ex & e);
/** Calculation of the norm in the Clifford algebra. */
ex clifford_norm(const ex & e);
*
* @param v List or vector of coordinates
* @param mu Index (must be of class varidx or a derived class)
- * @param metr Metric (must be of class tensor or a derived class)
+ * @param metr Metric (should be of class tensmetric or a derived class, or a symmetric matrix)
* @param rl Representation label
* @return Clifford vector with given components */
ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl = 0);
+/** An inverse function to lst_to_clifford(). For given Clifford vector extracts
+ * its components with respect to given Clifford unit. Obtained components may
+ * contain Clifford units with a different metric. Extraction is based on
+ * the algebraic formula (e * c.i + c.i * e)/ pow(e.i, 2) for non-degenerate cases
+ * (i.e. neither pow(e.i, 2) = 0).
+ *
+ * @param e Clifford expression to be decomposed into components
+ * @param c Clifford unit defining the metric for splitting (should have numeric dimension of indices)
+ * @param algebraic Use algebraic or symbolic algorithm for extractions */
+lst clifford_to_lst(const ex & e, const ex & c, bool algebraic=true);
+
+/** Calculations of Moebius transformations (conformal map) defined by a 2x2 Clifford matrix
+ * (a b\\c d) in linear spaces with arbitrary signature. The expression is
+ * (a * x + b)/(c * x + d), where x is a vector build from list v with metric G.
+ * (see Jan Cnops. An introduction to {D}irac operators on manifolds, v.24 of
+ * Progress in Mathematical Physics. Birkhauser Boston Inc., Boston, MA, 2002.)
+ *
+ * @param a (1,1) entry of the defining matrix
+ * @param b (1,2) entry of the defining matrix
+ * @param c (2,1) entry of the defining matrix
+ * @param d (2,2) entry of the defining matrix
+ * @param v Vector to be transformed
+ * @param G Metric of the surrounding space
+ * @param rl Representation label */
+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);
+
+/** The second form of Moebius transformations defined by a 2x2 Clifford matrix M
+ * This function takes the transformation matrix M as a single entity.
+ *
+ * @param M the defining matrix
+ * @param v Vector to be transformed
+ * @param G Metric of the surrounding space
+ * @param rl Representation label */
+ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl = 0);
+
} // namespace GiNaC
#endif // ndef __GINAC_CLIFFORD_H__
git://www.ginac.de / ginac.git / blobdiff