267 research outputs found
Hypercomplex Algebras and their application to the mathematical formulation of Quantum Theory
Quantum theory (QT), namely in terms of Schr\"odinger's 1926 wave functions
in general requires complex numbers to be formulated. However, it soon turned
out to even require some hypercomplex algebra. Incorporating Special Relativity
leads to an equation (Dirac 1928) requiring pairwise anti-commuting
coefficients, usually matrices. A unitary ring of square matrices
is an associative hypercomplex algebra by definition. Since only the algebraic
properties and relations of the elements matter, we replace the matrices by
biquaternions. In this paper, we first consider the basics of non-relativistic
and relativistic QT. Then we introduce general hypercomplex algebras and also
show how a relativistic quantum equation like Dirac's one can be formulated
using biquaternions. Subsequently, some algebraic preconditions for operations
within hypercomplex algebras and their subalgebras will be examined. For our
purpose equations akin to Schr\"odinger's should be able to be set up and
solved. Functions of complementary variables should be Fourier transforms of
each other. This should hold within a purely non-real subspace which must hence
be a subalgebra. Furthermore, it is an ideal denoted by . It must
be isomorphic to , hence containing an internal identity element.
The bicomplex numbers will turn out to fulfil these preconditions, and
therefore, the formalism of QT can be developed within its subalgebras. We also
show that bicomplex numbers encourage the definition of several different kinds
of conjugates. One of these treats the elements of like the usual
conjugate treats complex numbers. This defines a quantity what we call a
modulus which, in contrast to the complex absolute square, remains non-real
(but may be called `pseudo-real'). However, we do not conduct an explicit
physical interpretation here but we leave this to future examinations.Comment: 21 pages (without titlepage), 14 without titlepage and appendi
Fundamental representations and algebraic properties of biquaternions or complexified quaternions
The fundamental properties of biquaternions (complexified quaternions) are
presented including several different representations, some of them new, and
definitions of fundamental operations such as the scalar and vector parts,
conjugates, semi-norms, polar forms, and inner and outer products. The notation
is consistent throughout, even between representations, providing a clear
account of the many ways in which the component parts of a biquaternion may be
manipulated algebraically
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William Kingdon Clifford (1845-1879)
William Kingdon Clifford was an English mathematician and philosopher who worked extensively in many branches of pure mathematics and classical mechanics. Although he died young, he left a deep and long-lasting legacy, particularly in geometry. One of the main achievements that he is remembered for is his pioneering work on integrating Hamilton’s Elements of Quaternions with Grassmann’s Theory of Extension into a more general coherent corpus, now referred to eponymously as Clifford algebras. These geometric algebras are utilised in engineering mechanics (especially in robotics) as well as in mathematical physics (especially in quantum mechanics) for representing spatial relationships, motions, and dynamics within systems of particles and rigid bodies. Clifford’s study of geometric algebras in both Euclidean and non-Euclidean spaces led to his invention of the biquaternion, now used as an efficient representation for twists and wrenches in the same context as that of Ball’s Theory of Screws
Toy models of a non-associative quantum mechanics
Toy models of a non-associative quantum mechanics are presented. The
Heisenberg equation of motion is modified using a non-associative commutator.
Possible physical applications of a non-associative quantum mechanics are
considered. The idea is discussed that a non-associative algebra could be the
operator language for the non-perturbative quantum theory. In such approach the
non-perturbative quantum theory has observables and unobservables quantities.Comment: main formulas are colore
Eigenbundles, Quaternions, and Berry's Phase
Given a parameterized space of square matrices, the associated set of
eigenvectors forms some kind of a structure over the parameter space. When is
that structure a vector bundle? When is there a vector field of eigenvectors?
We answer those questions in terms of three obstructions, using a Homotopy
Theory approach. We illustrate our obstructions with five examples. One of
those examples gives rise to a 4 by 4 matrix representation of the Complex
Quaternions. This representation shows the relationship of the Biquaternions
with low dimensional Lie groups and algebras, Electro-magnetism, and Relativity
Theory. The eigenstructure of this representation is very interesting, and our
choice of notation produces important mathematical expressions found in those
fields and in Quantum Mechanics. In particular, we show that the Doppler shift
factor is analogous to Berry's Phase.Comment: 22 pages, also found on http://math.purdue.edu/~gottlie
C. S. Peirce and the Square Root of Minus One: Quaternions and a Complex Approach to Classes of Signs and Categorical Degeneration
The beginning for C. S. Peirce was the reduction of the traditional categories in a list composed of a fundamental triad: quality, respect and representation. Thus, these three would be named as Firstness, Secondness and Thirdness, as well given the ability to degeneration. Here we show how this degeneration categorical is related to mathematical revolution which Peirce family, especially his father Benjamin Peirce, took part: the advent of quaternions by William Rowan Hamilton, a number system that extends the complex numbers, i.e. those numbers which consists of an imaginary unit built by the square root of minus one. This is a debate that can, and should, have contributions that take into account the role that mathematical analysis and linear algebra had in C. S. Peirce’s past
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