D. Hestenes, G. Sobczyk: Clifford Algebra to Geometric Calculus. Reidel,
1984/1987/1992.
On pages 8-9, the reasoning 1.27 => 1.30 => 1.34 => 1.33 => 1.27 is circular.
[Admittedly, 1.27 => 1.30 could be regarded as a comment for later deductions.]
On page 21, line 21, the formula (2.14b), P_{BA} = P_A+P_B, is incorrect.
Counterexample in R^4: A = e12, B = e34, AB = e1234, u = e23, P_{BA}(u) = e23,
P_A(u) = 0, P_B(u) = 0. [This counterexample is due to Yvon Siret, Grenoble.]
On page 23, lines 2-3, the formulas (2.17a) and (2.17b) are wrong. Consider
R^1: (e1.e1).e1 = 0 =/= e1^(e1.e1) = e1, (e1^1).e1 = 1 =/= e1.(1.e1) = 0.
On page 97 line -6 is wrong. Counterexample in R^4: The decomposition is not
unique for e12+e34 = (e1+e3)(e2+e4)/2+(e1-e3)(e2-e4)/2. [Correct version on
p. 83, where the counterexample is attributed to Bernard Jancewicz, Wroclaw.]
On page 106 line -2 is wrong. Counterexample in R^8: If p = (1+w)(1+e12...8),
where w is a 4-vector such that ww = 7+6w, then for all x in R^8, pxp~ = 0,
which is in R^8. However, p is not of the form Aa+Bb, where a,b in Spin(8) and
A,B in R. This can be seen by considering q = p-Bb and observing that qxq~ is
not a vector, but contains a non-zero 5-vector part, for a non-zero x in R^8.
[See P. Lounesto: "Counterexample to Hestenes & Sobczyk 1984, 1987", Advances
in Applied Clifford Algebras 6 (1996), 75-77.]
On page 218 line 13 formula (5.50) is wrong. Counterexample in R^{3,1}: The
conformal transformation x -> ((1-e14)x-e1+e4)/((e1+e4)x+1+e14) is represented
by a Möbius matrix, whose entries are non-invertible. The transformation
cannot be expressed as a composition of just one rotation, one translation,
one dilation and one transversion (in any order). [This counterexample is
due to Johannes Maks, Delft. See his thesis 1989, page 41, Theorem 2.12.]
On page 297 lines 12-13 the statement "Every Lie algebra is isomorphic to a
bivector algebra" is called the "Most Interesting Conjecture", MIC. The MIC
conceives that in finite dimensions "every Lie algebra is a subalgebra of
so(n,n)". This was proved before 1984 as a corollary to Ado's theorem, without
reference to matrices or Clifford algebras.
On page 297 lines 32-35 are wrong. "Every element s in Spin+(p,q), which is
continuously connected to the identity, can be represented in the form exp(B),
where B is a bivector, if and only if p or q = 0 or 1". Counterexample in
R^{3,1}: The element -exp(F) in Spin+(3,1), F a non-zero bivector such that
FF = 0, is not of the form exp(B), B bivector. [Correct version on p. 108.]
On page 299 line -9 should end as: ... = (1/2)(n-2)(n-7)+1.
On page 301 line -1 should read: ... to SO+(p+1,q+1)/Z_2, hence ...
On page 304 line 8 should read: ... Lie algebra for Spin+(p+1,q+1).
==========================================================================
P. Budinich, A. Trautman: Spinorial Chessboard. Springer, 1988.
This is a good book, but the topic of the title of the book is just an essay
on trying to understand my paper
P. Lounesto: "Scalar products on spinors and an extension of Brauer-Wall
groups", Found. Phys. 11 (1981), 721-740,
which Budinich&Trautman have understood so poorly, that it was necessary
for me to write a rectification of their corruption, in the Section
"Discussion of Budinich&Trautman 1988" on pages 82-87 of
P. Lounesto: "Counterexamples in Clifford algebras",
Advances in Applied Clifford Algebras 6 (1996), 69-104.
==========================================================================
I.M. Benn, R.W. Tucker: An Introduction of Spinors and Geometry with
Applications in Physics. Adam Hilger, 1987.
Benn & Tucker lost their way on the same topic, spinorial chessboard, and made
mistakes, in Table 2.14 on page 76. The correction can be found in the
Section "Benn&Tucker 1987" on pages 10-11 of
P. Lounesto: "Counterexamples in Clifford algebras with CLICAL",
pages 3-30 of R. Ablamowicz et al. (eds.): "Clifford algebras with
numeric and symbolic computations". Birkhäuser, 1996.