ex unit_matrix(unsigned x);
ex unit_matrix(unsigned r, unsigned c);
ex symbolic_matrix(unsigned r, unsigned c, const string & base_name);
-ex symbolic_matrix(unsigned r, unsigned c, const string & base_name, const string & tex_base_name);
+ex symbolic_matrix(unsigned r, unsigned c, const string & base_name,
+ const string & tex_base_name);
@end example
@code{diag_matrix()} constructs a diagonal matrix given the list of diagonal
method and linear systems may be solved with:
@example
-matrix matrix::solve(const matrix & vars, const matrix & rhs, unsigned algo=solve_algo::automatic) const;
+matrix matrix::solve(const matrix & vars, const matrix & rhs,
+ unsigned algo=solve_algo::automatic) const;
@end example
Assuming the matrix object this method is applied on is an @code{m}
@example
ex epsilon_tensor(const ex & i1, const ex & i2);
ex epsilon_tensor(const ex & i1, const ex & i2, const ex & i3);
-ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4, bool pos_sig = false);
+ex lorentz_eps(const ex & i1, const ex & i2, const ex & i3, const ex & i4,
+ bool pos_sig = false);
@end example
The first two functions create an epsilon tensor in 2 or 3 Euclidean
you use one of the functions
@example
-ex dirac_trace(const ex & e, const std::set<unsigned char> & rls, const ex & trONE = 4);
+ex dirac_trace(const ex & e, const std::set<unsigned char> & rls,
+ const ex & trONE = 4);
ex dirac_trace(const ex & e, const lst & rll, const ex & trONE = 4);
ex dirac_trace(const ex & e, unsigned char rl = 0, const ex & trONE = 4);
@end example
The @code{dirac_trace()} function is a linear functional that is equal to the
ordinary matrix trace only in @math{D = 4} dimensions. In particular, the
-functional is not cyclic in @math{D != 4} dimensions when acting on
+functional is not cyclic in
+@tex $D \ne 4$
+@end tex
+dimensions when acting on
expressions containing @samp{gamma5}, so it's not a proper trace. This
@samp{gamma5} scheme is described in greater detail in
@cite{The Role of gamma5 in Dimensional Regularization}.
The value of the trace itself is also usually different in 4 and in
-@math{D != 4} dimensions:
+@tex $D \ne 4$
+@end tex
+dimensions:
@example
@{
$2^n$
@end tex
dimensional algebra with
-generators @samp{e~k} satisfying the identities
-@samp{e~i e~j + e~j e~i = B(i, j)} for some matrix (@code{metric})
-@math{B(i, j)}, which may be non-symmetric. Such generators are created
-by the function
+generators
+@tex $e_k$
+@end tex
+satisfying the identities
+@tex
+$e_i e_j + e_j e_i = M(i, j) $
+@end tex
+@ifnottex
+e~i e~j + e~j e~i = M(i, j)
+@end ifnottex
+for some matrix (@code{metric})
+@math{M(i, j)}, which may be non-symmetric and containing symbolic
+entries. Such generators are created by the function
@example
ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
@end example
where @code{mu} should be a @code{varidx} class object indexing the
-generators, @code{metr} defines the metric @math{B(i, j)} and can be
+generators, @code{metr} defines the metric @math{M(i, j)} and can be
represented by a square @code{matrix}, @code{tensormetric} or @code{indexed} class
object, optional parameter @code{rl} allows to distinguish different
Clifford algebras (which will commute with each other). Note that the call
@code{dirac_gamma(mu)}. The method @code{clifford::get_metric()} returns a
metric defining this Clifford number.
-If the matrix @math{B(i, j)} is in fact symmetric you may prefer to create
+If the matrix @math{M(i, j)} is in fact symmetric you may prefer to create
the Clifford algebra units with a call like that
@example
- ex e = clifford_unit(mu, indexed(B, sy_symm(), i, j));
+ ex e = clifford_unit(mu, indexed(M, sy_symm(), i, j));
@end example
since this may yield some further automatic simplifications.
@example
@{
...
- varidx nu(symbol("nu"), 3);
- matrix M(3, 3) = 1, 0, 0,
- 0,-1, 0,
- 0, 0, 0;
+ varidx nu(symbol("nu"), 4);
+ realsymbol s("s");
+ ex M = diag_matrix(lst(1, -1, 0, s));
ex e = clifford_unit(nu, M);
ex e0 = e.subs(nu == 0);
ex e1 = e.subs(nu == 1);
ex e2 = e.subs(nu == 2);
+ ex e3 = e.subs(nu == 3);
...
@}
@end example
-will produce three generators of a Clifford algebra with properties
-@code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1} and @code{pow(e2, 2) = 0}.
+will produce four anti-commuting generators of a Clifford algebra with properties
+@tex
+$e_0^2=1 $, $e_1^2=-1$, $e_2^2=0$ and $e_3^2=s$.
+@end tex
+@ifnottex
+@code{pow(e0, 2) = 1}, @code{pow(e1, 2) = -1}, @code{pow(e2, 2) = 0} and @code{pow(e3, 2) = s}.
+@end ifnottex
@cindex @code{lst_to_clifford()}
A similar effect can be achieved from the function
@example
ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr,
unsigned char rl = 0);
+ ex lst_to_clifford(const ex & v, const ex & e);
@end example
-which converts a list or vector @samp{v = (v~0, v~1, ..., v~n)} into
-the Clifford number @samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n} with @samp{e.k}
-being created by @code{clifford_unit(mu, metr, rl)}. The previous code
-may be rewritten with the help of @code{lst_to_clifford()} as follows
+which converts a list or vector
+@tex
+$v = (v^0, v^1, ..., v^n)$
+@end tex
+@ifnottex
+@samp{v = (v~0, v~1, ..., v~n)}
+@end ifnottex
+into the
+Clifford number
+@tex
+$v^0 e_0 + v^1 e_1 + ... + v^n e_n$
+@end tex
+@ifnottex
+@samp{v~0 e.0 + v~1 e.1 + ... + v~n e.n}
+@end ifnottex
+with @samp{e.k}
+directly supplied in the second form of the procedure. In the first form
+the Clifford unit @samp{e.k} is generated by the call of
+@code{clifford_unit(mu, metr, rl)}. The previous code may be rewritten
+with the help of @code{lst_to_clifford()} as follows
@example
@{
...
- varidx nu(symbol("nu"), 3);
- matrix M(3, 3) = 1, 0, 0,
- 0,-1, 0,
- 0, 0, 0;
- ex e0 = lst_to_clifford(lst(1, 0, 0), nu, M);
- ex e1 = lst_to_clifford(lst(0, 1, 0), nu, M);
- ex e2 = lst_to_clifford(lst(0, 0, 1), nu, M);
+ varidx nu(symbol("nu"), 4);
+ realsymbol s("s");
+ ex M = diag_matrix(lst(1, -1, 0, s));
+ ex e0 = lst_to_clifford(lst(1, 0, 0, 0), nu, M);
+ ex e1 = lst_to_clifford(lst(0, 1, 0, 0), nu, M);
+ ex e2 = lst_to_clifford(lst(0, 0, 1, 0), nu, M);
+ ex e3 = lst_to_clifford(lst(0, 0, 0, 1), nu, M);
...
@}
@end example
@end example
which takes an expression @code{e} and tries to find a list
-@samp{v = (v~0, v~1, ..., v~n)} such that @samp{e = v~0 c.0 + v~1 c.1 + ...
-+ v~n c.n} with respect to the given Clifford units @code{c} and none of
-@samp{v~k} contains the Clifford units @code{c} (of course, this
+@tex
+$v = (v^0, v^1, ..., v^n)$
+@end tex
+@ifnottex
+@samp{v = (v~0, v~1, ..., v~n)}
+@end ifnottex
+such that
+@tex
+$e = v^0 c_0 + v^1 c_1 + ... + v^n c_n$
+@end tex
+@ifnottex
+@samp{e = v~0 c.0 + v~1 c.1 + ... + v~n c.n}
+@end ifnottex
+with respect to the given Clifford units @code{c} and with none of the
+@samp{v~k} containing Clifford units @code{c} (of course, this
may be impossible). This function can use an @code{algebraic} method
-(default) or a symbolic one. With the @code{algebraic} method @samp{v~k} are calculated as
-@samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2) = 0} for some @samp{k}
+(default) or a symbolic one. With the @code{algebraic} method the @samp{v~k} are calculated as
+@tex
+$(e c_k + c_k e)/c_k^2$. If $c_k^2$
+@end tex
+@ifnottex
+@samp{(e c.k + c.k e)/pow(c.k, 2)}. If @samp{pow(c.k, 2)}
+@end ifnottex
+is zero or is not a @code{numeric} for some @samp{k}
then the method will be automatically changed to symbolic. The same effect
is obtained by the assignment (@code{algebraic = false}) in the procedure call.
@tex
$e^*$
@end tex
+@ifnottex
+e*
+@end ifnottex
and
@tex
$\overline{e}$
@end tex
+@ifnottex
+@code{\bar@{e@}}
+@end ifnottex
used in Clifford algebra textbooks.
@cindex @code{clifford_norm()}
@cindex @code{clifford_inverse()}
calculates the norm of a Clifford number from the expression
@tex
-$||e||^2 = e\overline{e}$
+$||e||^2 = e\overline{e}$.
@end tex
-. The inverse of a Clifford expression is returned
-by the function
+@ifnottex
+@code{||e||^2 = e \bar@{e@}}
+@end ifnottex
+ The inverse of a Clifford expression is returned by the function
@example
ex clifford_inverse(const ex & e);
which calculates it as
@tex
-$e^{-1} = e/||e||^2$
+$e^{-1} = \overline{e}/||e||^2$.
@end tex
-. If
+@ifnottex
+@math{e^@{-1@} = \bar@{e@}/||e||^2}
+@end ifnottex
+ If
@tex
$||e|| = 0$
@end tex
+@ifnottex
+@math{||e||=0}
+@end ifnottex
then an exception is raised.
@cindex @code{remove_dirac_ONE()}
@cindex @code{clifford_moebius_map()}
@example
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);
- ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl = 0);
+ const ex & d, const ex & v, const ex & G,
+ unsigned char rl = 0);
+ ex clifford_moebius_map(const ex & M, const ex & v, const ex & G,
+ unsigned char rl = 0);
@end example
-It takes a list or vector @code{v} and makes the Moebius
-(conformal or linear-fractional) transformation @samp{v ->
-(av+b)/(cv+d)} defined by the matrix @samp{M = [[a, b], [c, d]]}. The
-parameter @code{G} defines the metric of the surrounding
-(pseudo-)Euclidean space. The returned value of this function is a list
+It takes a list or vector @code{v} and makes the Moebius (conformal or
+linear-fractional) transformation @samp{v -> (av+b)/(cv+d)} defined by
+the matrix @samp{M = [[a, b], [c, d]]}. The parameter @code{G} defines
+the metric of the surrounding (pseudo-)Euclidean space. This can be a
+matrix or a Clifford unit, in the later case the parameter @code{rl} is
+ignored even if supplied. The returned value of this function is a list
of components of the resulting vector.
+LaTeX output for Clifford units looks like @code{\clifford[1]@{e@}^@{@{\nu@}@}},
+where @code{1} is the @code{representation_label} and @code{\nu} is the
+index of the corresponding unit. This provides a flexible typesetting
+with a suitable defintion of the @code{\clifford} command. For example, the
+definition
+@example
+ \newcommand@{\clifford@}[1][]@{@}
+@end example
+typesets all Clifford units identically, while the alternative definition
+@example
+ \newcommand@{\clifford@}[2][]@{\ifcase #1 #2\or \tilde@{#2@} \or \breve@{#2@} \fi@}
+@end example
+prints units with @code{representation_label=0} as
+@tex
+$e$,
+@end tex
+@ifnottex
+@code{e},
+@end ifnottex
+with @code{representation_label=1} as
+@tex
+$\tilde{e}$
+@end tex
+@ifnottex
+@code{\tilde@{e@}}
+@end ifnottex
+ and with @code{representation_label=2} as
+@tex
+$\breve{e}$.
+@end tex
+@ifnottex
+@code{\breve@{e@}}.
+@end ifnottex
@cindex @code{color} (class)
@subsection Color algebra
@example
> a=expand((sin(x)+sin(y))*(1+p+q)*(1+d));
-d*p*sin(x)+p*sin(x)+q*d*sin(x)+q*sin(y)+d*sin(x)+q*d*sin(y)+sin(y)+d*sin(y)+q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
+d*p*sin(x)+p*sin(x)+q*d*sin(x)+q*sin(y)+d*sin(x)+q*d*sin(y)+sin(y)+d*sin(y)
++q*sin(x)+d*sin(y)*p+sin(x)+sin(y)*p
> collect(a,@{p,q@});
-d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
+d*sin(x)+(d*sin(x)+sin(y)+d*sin(y)+sin(x))*p
++(d*sin(x)+sin(y)+d*sin(y)+sin(x))*q+sin(y)+d*sin(y)+sin(x)
> collect(a,find(a,sin($1)));
(1+q+d+q*d+d*p+p)*sin(y)+(1+q+d+q*d+d*p+p)*sin(x)
> collect(a,@{find(a,sin($1)),p,q@});
needs to be solved:
@example
-ex lsolve(const ex &eqns, const ex &symbols, unsigned options=solve_algo::automatic);
+ex lsolve(const ex & eqns, const ex & symbols,
+ unsigned options = solve_algo::automatic);
@end example
Here, @code{eqns} is a @code{lst} of equalities (i.e. class
@example
// ...
- cout << latex; // all output to cout will be in LaTeX format from now on
+ cout << latex; // all output to cout will be in LaTeX format from
+ // now on
cout << e << endl; // prints '4.5 i+\frac@{3@}@{2@} x^@{2@}'
cout << sin(x/2) << endl; // prints '\sin(\frac@{1@}@{2@} x)'
cout << dflt; // revert to default output format
will print
@example
- @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}+\mathcal@{O@}(\circ^@{3@})
+ @{(-\ln(\circ))@}+@{(-\gamma_E)@} \circ+@{(\frac@{1@}@{12@} \pi^@{2@})@} \circ^@{2@}
+ +\mathcal@{O@}(\circ^@{3@})
@end example
@cindex @code{index_dimensions}
// our own one
set_print_func<power, print_latex>(my_print_power_as_latex);
- // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}\uparrow@{2@}@}"
+ // this prints "-1+@{@{(y+x)@}@}\uparrow@{2@}-3 \frac@{@{x@}\uparrow@{3@}@}@{@{y@}
+ // \uparrow@{2@}@}"
cout << e << endl;
@}
@end example
inline bool operator<(const sprod_s & lhs, const sprod_s & rhs)
@{
- return lhs.left.compare(rhs.left) < 0 ? true : lhs.right.compare(rhs.right) < 0;
+ return lhs.left.compare(rhs.left) < 0
+ ? true : lhs.right.compare(rhs.right) < 0;
@}
@end example
a macro to automate the process of checking for GiNaC.
@example
-AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND} [, @var{ACTION-IF-NOT-FOUND}]]])
+AM_PATH_GINAC([@var{MINIMUM-VERSION}, [@var{ACTION-IF-FOUND}
+ [, @var{ACTION-IF-NOT-FOUND}]]])
@end example
This macro:
ex clifford::get_metric(const ex & i, const ex & j) const
{
- return indexed(metric, symmetric2(), i, j);
+ return indexed(metric, i, j);
}
bool clifford::same_metric(const ex & other) const
c.s << "{";
seq[0].print(c, level);
c.s << "\\hspace{-1.0ex}/}";
- } else
+ } else {
+ c.s << "\\clifford[" << int(representation_label) << "]";
this->print_dispatch<inherited>(c, level);
+ }
}
DEFAULT_COMPARE(diracone)
DEFAULT_COMPARE(diracgammaL)
DEFAULT_COMPARE(diracgammaR)
-DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbb{1}")
+DEFAULT_PRINT_LATEX(diracone, "ONE", "\\mathbf{1}")
DEFAULT_PRINT_LATEX(cliffordunit, "e", "e")
DEFAULT_PRINT_LATEX(diracgamma, "gamma", "\\gamma")
DEFAULT_PRINT_LATEX(diracgamma5, "gamma5", "{\\gamma^5}")
}
-ex canonicalize_clifford(const ex & e)
+ex canonicalize_clifford(const ex & e_)
{
pointer_to_map_function fcn(canonicalize_clifford);
- if (is_a<matrix>(e) // || is_a<pseries>(e) || is_a<integral>(e)
- || is_a<lst>(e)) {
- return e.map(fcn);
+ if (is_a<matrix>(e
_) // || is_a<pseries>(e) || is_a<integral>(e)
+ || is_a<lst>(e_)) {
+ return e_.map(fcn);
} else {
+ ex e=simplify_indexed(e_);
// Scan for any ncmul objects
exmap srl;
ex aux = e.to_rational(srl);
pointer_to_map_function fcn(clifford_prime);
if (is_a<clifford>(e) && is_a<cliffordunit>(e.op(0))) {
return -e;
- } else if (is_a<add>(e) || is_a<ncmul>(e)
// || is_a<pseries>(e) || is_a<integral>(e)
- || is_a<matrix>(e) || is_a<lst>(e)) {
+ } else if (is_a<add>(e) || is_a<ncmul>(e)
|| is_a<mul>(e) //|| is_a<pseries>(e) || is_a<integral>(e)
+ || is_a<matrix>(e) || is_a<lst>(e)) {
return e.map(fcn);
} else if (is_a<power>(e)) {
return pow(clifford_prime(e.op(0)), e.op(1));
return e;
}
-ex remove_dirac_ONE(const ex & e)
+ex remove_dirac_ONE(const ex & e, unsigned char rl)
{
- pointer_to_map_function fcn(remove_dirac_ONE);
- if (is_a<clifford>(e) && is_a<diracone>(e.op(0))) {
- return 1;
+ pointer_to_map_function_1arg<unsigned char> fcn(remove_dirac_ONE, rl);
+ if (is_a<clifford>(e) && ex_to<clifford>(e).get_representation_label() >= rl) {
+ if (is_a<diracone>(e.op(0)))
+ return 1;
+ else
+ throw(std::invalid_argument("Expression is a non-scalar Clifford number!"));
} else if (is_a<add>(e) || is_a<ncmul>(e) || is_a<mul>(e) // || is_a<pseries>(e) || is_a<integral>(e)
- || is_a<matrix>(e) || is_a<lst>(e)) {
+ || is_a<matrix>(e) || is_a<lst>(e)) {
return e.map(fcn);
} else if (is_a<power>(e)) {
return pow(remove_dirac_ONE(e.op(0)), e.op(1));
ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl)
{
- unsigned min, max;
if (!ex_to<idx>(mu).is_dim_numeric())
throw(std::invalid_argument("Index should have a numeric dimension"));
- unsigned dim = (ex_to<numeric>(ex_to<idx>(mu).get_dim())).to_int();
- ex c = clifford_unit(mu, metr, rl);
+ ex e = clifford_unit(mu, metr, rl);
+ return lst_to_clifford(v, e);
+}
- if (is_a<matrix>(v)) {
- if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows()) {
- min = ex_to<matrix>(v).rows();
- max = ex_to<matrix>(v).cols();
- } else {
- min = ex_to<matrix>(v).cols();
- max = ex_to<matrix>(v).rows();
- }
- if (min == 1) {
- if (dim == max)
- if (is_a<varidx>(mu)) // need to swap variance
- return indexed(v, ex_to<varidx>(mu).toggle_variance()) * c;
+ex lst_to_clifford(const ex & v, const ex & e) {
+ unsigned min, max;
+
+ if (is_a<clifford>(e)) {
+ varidx mu = ex_to<varidx>(e.op(1));
+ unsigned dim = (ex_to<numeric>(mu.get_dim())).to_int();
+
+ if (is_a<matrix>(v)) {
+ if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows()) {
+ min = ex_to<matrix>(v).rows();
+ max = ex_to<matrix>(v).cols();
+ } else {
+ min = ex_to<matrix>(v).cols();
+ max = ex_to<matrix>(v).rows();
+ }
+ if (min == 1) {
+ if (dim == max)
+ return indexed(v, ex_to<varidx>(mu).toggle_variance()) * e;
else
- return indexed(v, mu) * c;
+ throw(std::invalid_argument("Dimensions of vector and clifford unit mismatch"));
+ } else
+ throw(std::invalid_argument("First argument should be a vector vector"));
+ } else if (is_a<lst>(v)) {
+ if (dim == ex_to<lst>(v).nops())
+ return indexed(matrix(dim, 1, ex_to<lst>(v)), ex_to<varidx>(mu).toggle_variance()) * e;
else
- throw(std::invalid_argument("Dimensions of vector and clifford unit mismatch"));
+ throw(std::invalid_argument("List length and dimension of clifford unit mismatch"));
} else
- throw(std::invalid_argument("First argument should be a vector vector"));
- } else if (is_a<lst>(v)) {
- if (dim == ex_to<lst>(v).nops())
- return indexed(matrix(dim, 1, ex_to<lst>(v)), ex_to<varidx>(mu).toggle_variance()) * c;
- else
- throw(std::invalid_argument("List length and dimension of clifford unit mismatch"));
+ throw(std::invalid_argument("Cannot construct from anything but list or vector"));
} else
- throw(std::invalid_argument("Cannot construct from anything but list or vector"));
+ throw(std::invalid_argument("The second argument should be a Clifford unit"));
}
/** Auxiliary structure to define a function for striping one Clifford unit
}
-lst clifford_to_lst (const ex & e, const ex & c, bool algebraic)
+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 (algebraic) // check if algebraic method is applicable
for (unsigned int i = 0; i < D; i++)
- if (pow(c.subs(mu == i), 2) == 0)
+ if (pow(c.subs(mu == i), 2).is_zero()
+ or (not is_a<numeric>(pow(c.subs(mu == i), 2))))
algebraic = false;
lst V;
if (algebraic)
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)
{
- ex x, D;
- if (is_a<indexed>(G))
- D = ex_to<varidx>(G.op(1)).get_dim();
- else if (is_a<matrix>(G))
- D = ex_to<matrix>(G).rows();
- else
- throw(std::invalid_argument("metric should be an indexed object or matrix"));
+ ex x, D, cu;
- varidx mu((new symbol)->setflag(status_flags::dynallocated), D);
-
if (! is_a<matrix>(v) && ! is_a<lst>(v))
throw(std::invalid_argument("parameter v should be either vector or list"));
+
+ if (is_a<clifford>(G)) {
+ cu = G;
+ } else {
+ if (is_a<indexed>(G))
+ D = ex_to<varidx>(G.op(1)).get_dim();
+ else if (is_a<matrix>(G))
+ D = ex_to<matrix>(G).rows();
+ else throw(std::invalid_argument("metric should be an indexed object, matrix, or a Clifford unit"));
+
+ varidx mu((new symbol)->setflag(status_flags::dynallocated), D);
+ cu = clifford_unit(mu, G, rl);
+ }
- x = lst_to_clifford(v, mu, G, rl);
+ x = lst_to_clifford(v, cu);
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);
}
/** Create a Clifford unit object.
*
* @param mu Index (must be of class varidx or a derived class)
- * @param metr Metric (should be of class tensmetric or a derived class, or a symmetric matrix)
+ * @param metr Metric (should be of class tensmetric or a derived class, or a matrix)
* @param rl Representation label
* @return newly constructed Clifford unit object */
ex clifford_unit(const ex & mu, const ex & metr, unsigned char rl = 0);
/** Reversion of the Clifford algebra, coincides with the conjugate(). */
inline ex clifford_star(const ex & e) { return e.conjugate(); }
-/** Replaces all dirac_ONE's in e with 1 (effectively removing them). */
-ex remove_dirac_ONE(const ex & e);
+/** Replaces dirac_ONE's (with a representation_label no less than rl) in e with 1.
+ * For the default value rl = 0 remove all of them. Aborts if e contains any
+ * clifford_unit with representation_label to be removed.
+ *
+ * @param e Expression to be processed
+ * @param rl Value of representation label */
+ex remove_dirac_ONE(const ex & e, unsigned char rl = 0);
/** 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 (should be of class tensmetric or a derived class, or a symmetric matrix)
+ * @param metr Metric (should be of class tensmetric or a derived class, or a matrix)
* @param rl Representation label
+ * @param e Clifford unit object
* @return Clifford vector with given components */
ex lst_to_clifford(const ex & v, const ex & mu, const ex & metr, unsigned char rl = 0);
+ex lst_to_clifford(const ex & v, const ex & e);
/** An inverse function to lst_to_clifford(). For given Clifford vector extracts
* its components with respect to given Clifford unit. Obtained components may
*
* @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 */
+ * @param algebraic Use algebraic or symbolic algorithm for extractions
+ * @return List of components of a Clifford vector*/
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
* @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 */
+ * @param G Metric of the surrounding space, may be a Clifford unit then the next parameter is ignored
+ * @param rl Representation label
+ * @return List of components of the transformed vector*/
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
*
* @param M the defining matrix
* @param v Vector to be transformed
- * @param G Metric of the surrounding space
- * @param rl Representation label */
+ * @param G Metric of the surrounding space, may be a Clifford unit then the next parameter is ignored
+ * @param rl Representation label
+ * @return List of components of the transformed vector*/
ex clifford_moebius_map(const ex & M, const ex & v, const ex & G, unsigned char rl = 0);
} // namespace GiNaC
git://www.ginac.de / ginac.git / commitdiff