Various bug-fixes and enhancements (new moebius transformation).

diff --git a/ginac/clifford.cpp b/ginac/clifford.cpp
index b2e56224e11ee9518ccfe87fdffea1968686591c..da638b2c15c75063040442e229ab3c72b7f5367a 100644 (file)
--- a/ginac/clifford.cpp
+++ b/ginac/clifford.cpp
@@ -402,7 +402,7 @@ bool cliffordunit::contract_with(exvector::iterator self, exvector::iterator oth
 
                // Find if a previous contraction produces the square of self
                int prev_square = find_same_metric(v, self[0]);
 
                // Find if a previous contraction produces the square of self
                int prev_square = find_same_metric(v, self[0]);

-               varidx d((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(ex_to<idx>(self->op(1)).get_dim()));
+               varidx d((new symbol)->setflag(status_flags::dynallocated), ex_to<idx>(self->op(1)).get_dim());

                ex squared_metric = unit.get_metric(self->op(1), d) * unit.get_metric(d.toggle_variance(), other->op(1));
 
                // e~mu e.mu = Tr ONE
                ex squared_metric = unit.get_metric(self->op(1), d) * unit.get_metric(d.toggle_variance(), other->op(1));
 
                // e~mu e.mu = Tr ONE


@@ -677,7 +677,12 @@ ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl)
        if (!is_a<varidx>(mu))
                throw(std::invalid_argument("index of Clifford unit must be of type varidx"));
 
        if (!is_a<varidx>(mu))
                throw(std::invalid_argument("index of Clifford unit must be of type varidx"));
 

-       return clifford(unit, mu, metr, rl);
+       if (is_a<indexed>(metr))
+               return clifford(unit, mu, metr.op(0), rl);
+       else if(is_a<tensmetric>(metr) || is_a<matrix>(metr)) 
+               return clifford(unit, mu, metr, rl);
+       else
+               throw(std::invalid_argument("metric for Clifford unit must be of type indexed, tensormetric or matrix"));

 }
 
 ex dirac_gamma(const ex & mu, unsigned char rl)
 }
 
 ex dirac_gamma(const ex & mu, unsigned char rl)


@@ -987,7 +992,7 @@ next_sym:   ;
        return aux.subs(srl, subs_options::no_pattern).simplify_indexed();
 }
 
        return aux.subs(srl, subs_options::no_pattern).simplify_indexed();
 }
 

-ex clifford_prime(const ex &e)
+ex clifford_prime(const ex & e)

 {
        pointer_to_map_function fcn(clifford_prime);
        if (is_a<clifford>(e) && is_a<cliffordunit>(e.op(0))) {
 {
        pointer_to_map_function fcn(clifford_prime);
        if (is_a<clifford>(e) && is_a<cliffordunit>(e.op(0))) {


@@ -1002,9 +1007,9 @@ ex clifford_prime(const ex &e)
                return e;
 }
 
                return e;
 }
 

-ex delete_ONE(const ex &e)
+ex remove_dirac_ONE(const ex & e)

 {
 {

-       pointer_to_map_function fcn(delete_ONE);
+       pointer_to_map_function fcn(remove_dirac_ONE);

        if (is_a<clifford>(e) && is_a<diracone>(e.op(0))) {
                return 1;
        } else if (is_a<add>(e)) {
        if (is_a<clifford>(e) && is_a<diracone>(e.op(0))) {
                return 1;
        } else if (is_a<add>(e)) {


@@ -1014,21 +1019,23 @@ ex delete_ONE(const ex &e)
        } else if (is_a<mul>(e)) {
                return e.map(fcn);
        } else if (is_a<power>(e)) {
        } else if (is_a<mul>(e)) {
                return e.map(fcn);
        } else if (is_a<power>(e)) {

-               return pow(delete_ONE(e.op(0)), e.op(1));
+               return pow(remove_dirac_ONE(e.op(0)), e.op(1));

        } else
                return e;
 }
 
        } else
                return e;
 }
 

-ex clifford_norm(const ex &e)
+ex clifford_norm(const ex & e)

 {
 {

-       return sqrt(delete_ONE((e * clifford_bar(e)).simplify_indexed()));
+       return sqrt(remove_dirac_ONE(canonicalize_clifford(e * clifford_bar(e)).simplify_indexed()));

 }
 
 }
 

-ex clifford_inverse(const ex &e)
+ex clifford_inverse(const ex & e)

 {
        ex norm = clifford_norm(e);
        if (!norm.is_zero())
                return clifford_bar(e) / pow(norm, 2);
 {
        ex norm = clifford_norm(e);
        if (!norm.is_zero())
                return clifford_bar(e) / pow(norm, 2);

+       else 
+               throw(std::invalid_argument("Cannot find inverse of Clifford number with zero norm!"));

 }
 
 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl)
 }
 
 ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl)


@@ -1066,4 +1073,92 @@ ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char r
                throw(std::invalid_argument("Cannot construct from anything but list or vector"));
 }
  
                throw(std::invalid_argument("Cannot construct from anything but list or vector"));
 }
  

+/** Auxiliary structure to define a function for striping one Clifford unit
+ * from vectors. Used in  clifford_to_lst(). */
+static ex get_clifford_comp(const ex & e, const ex & c) 
+{
+       pointer_to_map_function_1arg<const ex &> fcn(get_clifford_comp, c);
+               
+       if (is_a<add>(e)) 
+               return e.map(fcn);
+       else if (is_a<ncmul>(e) || is_a<mul>(e)) {
+               //find a Clifford unit with the same metric, delete it and substitute its index
+               int ival = ex_to<numeric>(ex_to<varidx>(c.op(1)).get_value()).to_int();
+               size_t ind = e.nops() + 1;
+               for (size_t j = 0; j < e.nops(); j++) 
+                       if (is_a<clifford>(e.op(j)) && ex_to<clifford>(c).same_metric(e.op(j)))
+                               if (ind > e.nops()) 
+                                       ind = j;
+                               else 
+                                       throw(std::invalid_argument("Expression is a Clifford multi-vector"));
+               if (ind < e.nops()) {
+                       ex S = 1;
+                       for(size_t j=0; j < e.nops(); j++)
+                               if (j != ind) {
+                                       exvector ind_vec = ex_to<indexed>(e.op(j)).get_dummy_indices(ex_to<indexed>(e.op(ind)));
+                                       if (ind_vec.size() > 0) {
+                                               exvector::const_iterator it = ind_vec.begin(), itend = ind_vec.end();
+                                               while (it != itend) {
+                                                       S = S * e.op(j).subs(lst(ex_to<varidx>(*it) == ival, ex_to<varidx>(*it).toggle_variance() == ival), subs_options::no_pattern);
+                                                       it++;
+                                               }
+                                       } else
+                                               S = S * e.op(j);
+                               }
+                       return S;
+               } else
+                       throw(std::invalid_argument("Expression is not a Clifford vector to the given units"));
+       } else if (e.is_zero()) 
+               return e;
+       else 
+               throw(std::invalid_argument("Expression is not handlable as a Clifford vector"));
+       
+}
+
+
+lst clifford_to_lst (const ex & e, const ex & c, bool algebraic)
+{
+       GINAC_ASSERT(is_a<clifford>(c));
+       varidx mu = ex_to<varidx>(c.op(1));
+       if (! mu.is_dim_numeric())
+               throw(std::invalid_argument("Index should have a numeric dimension"));
+       unsigned int D = ex_to<numeric>(mu.get_dim()).to_int();
+
+       if (algebraic) // check if algebraic method is applicable
+               for (unsigned int i = 0; i < D; i++) 
+                       if (pow(c.subs(mu == i), 2) == 0)
+                               algebraic = false;
+       lst V; 
+       if (algebraic) 
+               for (unsigned int i = 0; i < D; i++) 
+                       V.append(remove_dirac_ONE(
+                                               simplify_indexed(canonicalize_clifford(e * c.subs(mu == i) +  c.subs(mu == i) * e))
+                                               / (2*pow(c.subs(mu == i), 2))));
+       else {
+               ex e1 = canonicalize_clifford(e);
+               for (unsigned int i = 0; i < D; i++) 
+                       V.append(get_clifford_comp(e1, c.subs(c.op(1) == i)));
+       }
+       return V;
+}
+
+
+ex clifford_moebius_map(const ex & a, const ex & b, const ex & c, const ex & d, const ex & v, const ex & G)
+{
+       ex x, D;
+       if (is_a<indexed>(G)) 
+               D = ex_to<varidx>(G.op(1));
+       else
+               throw(std::invalid_argument("metric should be an indexed object"));
+       
+       varidx mu ((new symbol)->setflag(status_flags::dynallocated), ex_to<varidx>(D).get_dim());
+                  
+       if (! is_a<matrix>(v) && ! is_a<lst>(v))
+               throw(std::invalid_argument("parameter v should be either vector or list"));
+
+       x = lst_to_clifford(v, mu, G);
+       ex e = simplify_indexed(canonicalize_clifford((a * x + b) * clifford_inverse(c * x + d)));
+       ex cu = clifford_unit(mu, G);
+       return clifford_to_lst(e, cu, false);
+}

 } // namespace GiNaC
 } // namespace GiNaC

diff --git a/ginac/clifford.h b/ginac/clifford.h
index 81abd9f5c879c956da862f009fc5f24f6b7bcbc4..d68a18bfde1e6ac5706d936c5bc94f83ad731dda 100644 (file)
--- a/ginac/clifford.h
+++ b/ginac/clifford.h
@@ -275,7 +275,8 @@ inline ex clifford_bar(const ex & e) { return clifford_prime(e.conjugate()); }
 /** Reversion of the Clifford algebra, coincides with the conjugate(). */
 inline ex clifford_star(const ex & e) { return e.conjugate(); }
 
 /** 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);
 
 /** Calculation of the norm in the Clifford algebra. */
 ex clifford_norm(const ex & e);


@@ -292,6 +293,30 @@ ex clifford_inverse(const ex & e);
  *  @return Clifford vector with given components */
 ex lst_to_clifford(const ex & v, const ex & mu,  const ex & metr, unsigned char rl = 0);
 
  *  @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 */
+ex clifford_moebius_map(const ex & a, const ex & b, const ex & c, const ex & d, const ex & v, const ex & G);

 } // namespace GiNaC
 
 #endif // ndef __GINAC_CLIFFORD_H__
 } // namespace GiNaC
 
 #endif // ndef __GINAC_CLIFFORD_H__