// 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
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)
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))) {
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)) {
} 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;
}
-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);
+ 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)
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
/** 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);
* @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__