23 research outputs found
Idempotents of Clifford Algebras
A classification of idempotents in Clifford algebras C(p,q) is presented. It
is shown that using isomorphisms between Clifford algebras C(p,q) and
appropriate matrix rings, it is possible to classify idempotents in any
Clifford algebra into continuous families. These families include primitive
idempotents used to generate minimal one sided ideals in Clifford algebras.
Some low dimensional examples are discussed
On the Decomposition of Clifford Algebras of Arbitrary Bilinear Form
Clifford algebras are naturally associated with quadratic forms. These
algebras are Z_2-graded by construction. However, only a Z_n-gradation induced
by a choice of a basis, or even better, by a Chevalley vector space isomorphism
Cl(V) \bigwedge V and an ordering, guarantees a multi-vector decomposition
into scalars, vectors, tensors, and so on, mandatory in physics. We show that
the Chevalley isomorphism theorem cannot be generalized to algebras if the
Z_n-grading or other structures are added, e.g., a linear form. We work with
pairs consisting of a Clifford algebra and a linear form or a Z_n-grading which
we now call 'Clifford algebras of multi-vectors' or 'quantum Clifford
algebras'. It turns out, that in this sense, all multi-vector Clifford algebras
of the same quadratic but different bilinear forms are non-isomorphic. The
usefulness of such algebras in quantum field theory and superconductivity was
shown elsewhere. Allowing for arbitrary bilinear forms however spoils their
diagonalizability which has a considerable effect on the tensor decomposition
of the Clifford algebras governed by the periodicity theorems, including the
Atiyah-Bott-Shapiro mod 8 periodicity. We consider real algebras Cl_{p,q} which
can be decomposed in the symmetric case into a tensor product Cl_{p-1,q-1}
\otimes Cl_{1,1}. The general case used in quantum field theory lacks this
feature. Theories with non-symmetric bilinear forms are however needed in the
analysis of multi-particle states in interacting theories. A connection to
q-deformed structures through nontrivial vacuum states in quantum theories is
outlined.Comment: 25 pages, 1 figure, LaTeX, {Paper presented at the 5th International
Conference on Clifford Algebras and their Applications in Mathematical
Physics, Ixtapa, Mexico, June 27 - July 4, 199
Visualising text co-occurrence networks
We present a tool for automatically generating a visual summary of unstructured text data retrieved from documents, web sites or social media feeds. Unlike tools such as word clouds, we are able to visualise structures and topic relationships occurring in a document. These relationships are determined by a unique approach to co-occurrence analysis. The algorithm applies a decaying function to the distance between word pairs found in the original text such that words regularly occurring close to each other score highly, but even words occurring some distance apart will make a small contribution to the overall co-occurrence score. This is in contrast to other algorithms which simply count adjacent words or use a sliding window of fixed size. We show, with examples, how the network generated can be presented in tree or graph format. The tree format allows for the user to interact with the visualisation and expand or contract the data to a preferred level of detail. The tool is available as a web application and can be viewed using any modern web browse
Dirac-Hestenes spinor fields in Riemann-Cartan spacetime
In this paper we study Dirac-Hestenes spinor fields (DHSF) on a
four-dimensional Riemann-Cartan spacetime (RCST). We prove that these fields
must be defined as certain equivalence classes of even sections of the Clifford
bundle (over the RCST), thereby being certain particular sections of a new
bundle named Spin-Clifford bundle (SCB). The conditions for the existence of
the SCB are studied and are shown to be equivalent to the famous Geroch's
theorem concerning to the existence of spinor structures in a Lorentzian
spacetime. We introduce also the covariant and algebraic Dirac spinor fields
and compare these with DHSF, showing that all the three kinds of spinor fields
contain the same mathematical and physical information. We clarify also the
notion of (Crumeyrolle's) amorphous spinors (Dirac-K\"ahler spinor fields are
of this type), showing that they cannot be used to describe fermionic fields.
We develop a rigorous theory for the covariant derivatives of Clifford fields
(sections of the Clifford bundle (CB)) and of Dirac-Hestenes spinor fields. We
show how to generalize the original Dirac-Hestenes equation in Minkowski
spacetime for the case of a RCST. Our results are obtained from a variational
principle formulated through the multiform derivative approach to Lagrangian
field theory in the Clifford bundle.Comment: 45 pages, special macros kapproc.sty and makro822.te
Quantum field theory and Hopf algebra cohomology
We exhibit a Hopf superalgebra structure of the algebra of field operators of
quantum field theory (QFT) with the normal product. Based on this we construct
the operator product and the time-ordered product as a twist deformation in the
sense of Drinfeld. Our approach yields formulas for (perturbative) products and
expectation values that allow for a significant enhancement in computational
efficiency as compared to traditional methods. Employing Hopf algebra
cohomology sheds new light on the structure of QFT and allows the extension to
interacting (not necessarily perturbative) QFT. We give a reconstruction
theorem for time-ordered products in the spirit of Streater and Wightman and
recover the distinction between free and interacting theory from a property of
the underlying cocycle. We also demonstrate how non-trivial vacua are described
in our approach solving a problem in quantum chemistry.Comment: 39 pages, no figures, LaTeX + AMS macros; title changed, minor
corrections, references update