I'm looking at a book called "Theory of Holors: A Generalization of
Tensors" by Parry Moon and Domina Eberly Spencer (1986) and wondering if
holors are isomorphic to Clifford's geometric algebra.
Let me quote for a few pages and ask for comments from the GA experts in
the group (and I'm wondering if we should start a separate e-mai list for
GA).
xiii: Examples of holors are complex numbers, vectors, matrices, tensor,
and other hypernumbers.... More complicated holors were developed by
Hamilton (1843), Grassmann (1844),...Einstein (1916), and many others....
We propose a single notation that applies to all holors, be the tensors or
nontensors. A tensor is a holor that transforms in a particularly simple
way. But many holors are not tensors.
page 2: Benjamin Pierce invented a whole new branch of mathematics--the
study of linear associative algebras.
page 3: In the same year (1844) that Hamilton published his first paper on
quaternions, a much more general treatment of holors was poublished by
Herman Grassmann.
page 4: It was not until 40 years later that Willard Gibbs ... worked out a
special case of quaternions called vector analysis...but it applied only to
3-space and was usually expressed in rectangular coordinates.... but it
ignores the subject of invariance. Even with Gibbs, a vector was more than
a set of numbers representing rectangular coordinates: it was a geometric
entity that could be expressed equally well in other coordinate systems.
The merates [elements] were changed when the coordinates were changed, but
the geometric object maintained its identity.
page 5: Physicists in general were not interested in invariants and had
never heard of Ricci's work. It was not until 1916 that Albert Einstein
applied tensors in forumulating his general relativity. Because of the
extraordinary popular appeal of this theory, tensors became famous
overnight.... And even teh subject of vectors took on new life when
presented in tensor form.
Index notation was designed particularly to handle behavior under
coordinate transformations. But the basic notion is applicable even when
no coordinate transformations are involved. Thus, index notation can be
employed advantageously to unify a great field of holors of various
valences and dimensionalities.
page 12: A matrix is a bivalent holor
page 36: The uncontracted product of two holors is defined as a holor whose
valence is the sum of the valences of its factors.
page 37: The factors need not have the same valence. For instance, the
product of a bivalent holor and a nilvalent holor is a bivalent holor.
Anybody heard of holors before? Are they isomorphic to GA?
Thanks,
Larry
This posting is the position of the writer, not that of SUNY-BSC, NAU or the AAPT.