[GiNaC-devel] lst_to_clifford() and clifford_to_lst() patch
Vladimir Kisil
kisilv at maths.leeds.ac.uk
Mon Aug 13 15:23:14 CEST 2007
Apologies, for resubmission---the correct patch should amend one
more line. It is attached to this mail.
--
Vladimir V. Kisil email: kisilv at maths.leeds.ac.uk
-- www: http://maths.leeds.ac.uk/~kisilv/
-------------- next part --------------
Index: doc/tutorial/ginac.texi
===================================================================
RCS file: /home/cvs/GiNaC/doc/tutorial/ginac.texi,v
retrieving revision 1.197
diff -u -r1.197 ginac.texi
--- doc/tutorial/ginac.texi 2 Aug 2007 14:56:44 -0000 1.197
+++ doc/tutorial/ginac.texi 13 Aug 2007 13:20:46 -0000
@@ -3401,8 +3401,20 @@
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
- at code{clifford_unit(mu, metr, rl)}. The previous code may be rewritten
-with the help of @code{lst_to_clifford()} as follows
+ at code{clifford_unit(mu, metr, rl)}.
+ at cindex pseudo-vector
+If the number of components supplied
+by @code{v} exceeds the dimensionality of the Clifford unit @code{e} by
+1 then function @code{lst_to_clifford()} uses the following
+pseudo-vector representation:
+ at tex
+$v^0 {\bf 1} + v^1 e_0 + v^2 e_1 + ... + v^{n+1} e_n$
+ at end tex
+ at ifnottex
+ at samp{v~0 ONE + v~1 e.0 + v~2 e.1 + ... + v~[n+1] e.n}
+ at end ifnottex
+
+The previous code may be rewritten with the help of @code{lst_to_clifford()} as follows
@example
@{
@@ -3432,17 +3444,25 @@
@ifnottex
@samp{v = (v~0, v~1, ..., v~n)}
@end ifnottex
-such that
+such that the expression is either vector
@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
- at samp{v~k} containing Clifford units @code{c} (of course, this
+or pseudo-vector
+ at tex
+$v^0 {\bf 1} + v^1 e_0 + v^2 e_1 + ... + v^{n+1} e_n$
+ at end tex
+ at ifnottex
+ at samp{v~0 ONE + v~1 e.0 + v~2 e.1 + ... + v~[n+1] e.n}
+ at end ifnottex
+with respect to the given Clifford units @code{c}. Here none of the
+ at samp{v~k} should contain 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 the @samp{v~k} are calculated as
+(default) or a symbolic one. With the @code{algebraic} method the
+ at samp{v~k} are calculated as
@tex
$(e c_k + c_k e)/c_k^2$. If $c_k^2$
@end tex
Index: ginac/clifford.cpp
===================================================================
RCS file: /home/cvs/GiNaC/ginac/clifford.cpp,v
retrieving revision 1.102
diff -u -r1.102 clifford.cpp
--- ginac/clifford.cpp 1 Aug 2007 14:27:25 -0000 1.102
+++ ginac/clifford.cpp 13 Aug 2007 13:20:46 -0000
@@ -1183,13 +1183,20 @@
if (min == 1) {
if (dim == max)
return indexed(v, mu_toggle) * e;
- else
+ else if (max - dim == 1) {
+ if (ex_to<matrix>(v).cols() > ex_to<matrix>(v).rows())
+ return v.op(0) * dirac_ONE(ex_to<clifford>(e).get_representation_label()) + indexed(sub_matrix(ex_to<matrix>(v), 0, 1, 1, dim), mu_toggle) * e;
+ else
+ return v.op(0) * dirac_ONE(ex_to<clifford>(e).get_representation_label()) + indexed(sub_matrix(ex_to<matrix>(v), 1, dim, 0, 1), mu_toggle) * e;
+ } else
throw(std::invalid_argument("lst_to_clifford(): dimensions of vector and clifford unit mismatch"));
} else
throw(std::invalid_argument("lst_to_clifford(): first argument should be a vector (nx1 or 1xn matrix)"));
} else if (v.info(info_flags::list)) {
if (dim == ex_to<lst>(v).nops())
return indexed(matrix(dim, 1, ex_to<lst>(v)), mu_toggle) * e;
+ else if (ex_to<lst>(v).nops() - dim == 1)
+ return v.op(0) * dirac_ONE(ex_to<clifford>(e).get_representation_label()) + indexed(sub_matrix(matrix(dim+1, 1, ex_to<lst>(v)), 1, dim, 0, 1), mu_toggle) * e;
else
throw(std::invalid_argument("lst_to_clifford(): list length and dimension of clifford unit mismatch"));
} else
@@ -1274,19 +1281,28 @@
or (not is_a<numeric>(pow(c.subs(mu == i, subs_options::no_pattern), 2))))
algebraic = false;
lst V;
+ ex v0 = remove_dirac_ONE(canonicalize_clifford(e+clifford_prime(e)).normal())/2;
+ if (not v0.is_zero())
+ V.append(v0);
+ ex e1 = canonicalize_clifford(e - v0 * dirac_ONE(ex_to<clifford>(c).get_representation_label()));
if (algebraic) {
for (unsigned int i = 0; i < D; i++)
V.append(remove_dirac_ONE(
- simplify_indexed(canonicalize_clifford(e * c.subs(mu == i, subs_options::no_pattern) + c.subs(mu == i, subs_options::no_pattern) * e))
+ simplify_indexed(canonicalize_clifford(e1 * c.subs(mu == i, subs_options::no_pattern) + c.subs(mu == i, subs_options::no_pattern) * e1))
/ (2*pow(c.subs(mu == i, subs_options::no_pattern), 2))));
} else {
- ex e1 = canonicalize_clifford(e);
try {
for (unsigned int i = 0; i < D; i++)
V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
} catch (std::exception &p) {
/* Try to expand dummy summations to simplify the expression*/
- e1 = canonicalize_clifford(expand_dummy_sum(e1, true));
+ e1 = canonicalize_clifford(expand_dummy_sum(e, true));
+ V.remove_all();
+ v0 = remove_dirac_ONE(canonicalize_clifford(e1+clifford_prime(e1)).normal())/2;
+ if (not v0.is_zero()) {
+ V.append(v0);
+ e1 = canonicalize_clifford(e1 - v0 * dirac_ONE(ex_to<clifford>(c).get_representation_label()));
+ }
for (unsigned int i = 0; i < D; i++)
V.append(get_clifford_comp(e1, c.subs(c.op(1) == i, subs_options::no_pattern)));
}
Index: check/exam_clifford.cpp
===================================================================
RCS file: /home/cvs/GiNaC/check/exam_clifford.cpp,v
retrieving revision 1.36
diff -u -r1.36 exam_clifford.cpp
--- check/exam_clifford.cpp 31 Jan 2007 22:29:19 -0000 1.36
+++ check/exam_clifford.cpp 13 Aug 2007 13:20:46 -0000
@@ -410,13 +410,23 @@
result += check_equal(canonicalize_clifford(e), 0);
/* lst_to_clifford() and clifford_inverse() check*/
- realsymbol x("x"), y("y"), t("t"), z("z");
+ realsymbol s("s"), t("t"), x("x"), y("y"), z("z");
ex c = clifford_unit(nu, A, 1);
e = lst_to_clifford(lst(t, x, y, z), mu, A, 1) * lst_to_clifford(lst(1, 2, 3, 4), c);
e1 = clifford_inverse(e);
result += check_equal_simplify_term2((e*e1).simplify_indexed(), dirac_ONE(1));
+/* lst_to_clifford() and clifford_to_lst() check for vectors*/
+ e = lst(t, x, y, z);
+ result += check_equal_lst(clifford_to_lst(lst_to_clifford(e, c), c, false), e);
+ result += check_equal_lst(clifford_to_lst(lst_to_clifford(e, c), c, true), e);
+
+/* lst_to_clifford() and clifford_to_lst() check for pseudovectors*/
+ e = lst(s, t, x, y, z);
+ result += check_equal_lst(clifford_to_lst(lst_to_clifford(e, c), c, false), e);
+ result += check_equal_lst(clifford_to_lst(lst_to_clifford(e, c), c, true), e);
+
/* Moebius map (both forms) checks for symmetric metrics only */
matrix M1(2, 2), M2(2, 2);
c = clifford_unit(nu, A);
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