Related papers: Fourvector algebra
Natural linear and coalgebra transformations of tensor algebras are studied. The representations of certain combinatorial groups are given. These representations are connected to natural transformations of tensor algebras and to the groups…
The purpose of this paper is to show that the mathematical treatment of three dimensional rotations can be simplified, and its geometrical understanding improved, by using the Rodrigues' vector representation. We present a novel geometrical…
The algebra of the relativistic composition of velocities is shown to be isomorphic to an algebraic loop defined on division algebras. This makes calculations in special relativity effortless and straightforward, unlike the standard…
A particle which lives in a d-dimensional ordinary and a d-dimensional Grassmann space manifests itself in an ordinary four-dimensional subspace as a spinor, a scalar or a vector with charges. Operators of the Lorentz transformations and…
We construct a large family of evidently integrable Hamiltonian systems which are generalizations of the KM system. The Hamiltonian vector field is homogeneous cubic but in a number of cases a simple change of variables transforms such a…
We consider the class of algebras of rank 4 equipped with a standard involution over an arbitrary base ring. In particular, we characterize quaternion rings, those algebras defined by the construction of the even Clifford algebra.
A novel approach is introduced for obtaining precise solutions of the pairing Hamiltonian for tetraquarks, which utilizes an algebraic technique in infinite dimensions. The parameters involved in the transition phase are calibrated based on…
We present here a product between vectors and scalars that {\it mixes} them within their own space, using imaginaries to describe geometric products between vectors as complex vectors, rather than introducing higher order/dimensional vector…
We set up an Einstein-Gauss-Bonnet theory in four dimensions, based on the recent formulation of pure gravity with extra dimensions of vanishing metrical length [1]. In absence of torsion, the effective field equations depend only on the…
Due to the existence of incompatible observables, the propositional calculus of a quantum system does not form a Boolean algebra but an orthomodular lattice. Such lattice can be realised as a lattice of subspaces on a real, complex or…
Applications of the three-dimensional transformation for rotating coordinate systems to quantum mechanics, general theory relativity and optics are considered.
The notion of hidden symmetry algebra used in the context of exactly solvable systems is re-examined from the purely algebraic way, analyzing subspaces of commuting polynomials that generate finite-dimensional quadratic algebras. By…
In this work, we use real quaternions and the basic concept of the final speed of light in an attempt to enhance the standard description of special relativity. First, we demonstrate that it is possible to introduce a quaternion time domain…
The quantum constraint equations for a relativistic three-dimensional harmonic oscillator are shown to find concise expression in terms of Lorentz covariant ladder operators. These ladder operators consist of two conjugate 4-vectors that…
This paper describes the passage of light through a system of waveplates mathematically in terms of quaternions, an extension of the complex numbers, instead of the more usual Jones vectors and Jones matrices. Both the light beam and the…
Recent developments of Baxter algebras have lead to applications to combinatorics, number theory and mathematical physics. We relate Baxter algebras to Stirling numbers of the first kind and the second kind, partitions and multinomial…
Some comments are made on the matrices which serve as the basis of a quaternionic algebra. We show that these matrices are related with the quaternionic action of the imaginary units from the left and from the right.
This article provides a simple geometric interpretation of the quadratic formula. The geometry helps to demystify the formula's complex appearance and casts it into a much simpler existence, thus potentially benefits early algebra students.
Arithmetic operations can be defined in various ways, even if one assumes commutativity and associativity of addition and multiplication, and distributivity of multiplication with respect to addition. In consequence, whenever one encounters…
In special relativity, spacetime algebra (STA) provides a powerful and insightful approach to an invariant formulation of physics. However, in this geometric algebra of spacetime, relativistic physics is usually considered a misnomer: STA…