Quaternion Field Research Articles

Quaternionic Variational Formalism for General Relativity in Riemann and Riemann-Cartan Space-Times

It is shown that there exists a 2-dimensional matrix representation of complex quaternions over real quaternions, which allows to define Pauli matrix in 4 dimensions over the quaternionic field and leads to the quaternionic spinor group previously proposed. It is also attempted to apply complex quaternions to general relativity at the level of the variational formalism. Linear gravitational Lagrangian in Riemann-Cartan space-time U4 is derived using quaternion caluculus; namely scalar curvature in U4 is put into a quaternionic form. Consequently, Einstein-Hilbert Lagrangian in Riemann space R4 is also defined over quaternions, as first shown by Sachs. The matter fields coupled to gravity are assumed to be the scalar and the Dirac fields. The quaternionic variational formalism corresponds to the first-order formalism but with a limited pattern of allowed fields such that the quaternionic fields carry only coordinate tensor indices but no local Lorentz indices which are contracted with that possessed by the basis of complex quaternions. In particular, both the quaternionic vierbein field and Lorentz gauge field (corresponding to the spin connection) are regarded as coordinate vectors which are independently varied, obtaining Einstein and Cartan equations, respectively. It is incidentally shown that the consistent condition of Einstein equation in U4 is proved via the variational formalism and the anti-symmetric part of Einstein equation together with Cartan equation in U4 leads to an identity which expresses the anti-symmetric part of the enegy-momentum tensor by means of the covariant divergence of the spin angular momentum tensor, both of Dirac field. We also present pedagogical proofs of Bianchi and Bach-Lanczos identities in U4 using the quaternionic formalism.

RANDOM RIGHT EIGENVALUES OF GAUSSIAN QUATERNIONIC MATRICES

We consider a random matrix whose entries are independent Gaussian variables taking values in the field of quaternions with variance 1/n. Using logarithmic potential theory, we prove the almost sure convergence, as the dimension n goes to infinity, of the empirical distribution of the right eigenvalues towards some measure supported on the unit ball of the quaternions field. Some comments on more general Gaussian quaternionic random matrix models are also made.

Voronovskaja-Type Results in Compact Disks for Quaternion q-Bernstein Operators, q ≥ 1

Attaching to a compact disk \({\overline{\mathbb{D}_{r}}}\) in the quaternion field \({\mathbb{H}}\) and to some analytic function in Weierstrass sense on \({\overline{\mathbb{D}_{r}}}\) the so-called q-Bernstein operators with q ≥ 1, Voronovskaja-type results with quantitative upper estimates are proved. As applications, the exact orders of approximation in \({\overline{\mathbb{D}_{r}}}\) for these operators, namely \({\frac{1}{n}}\) if q = 1 and \({\frac{1}{q^{n}}}\) if q > 1, are obtained. The results extend those in the case of approximation of analytic functions of a complex variable in disks by q-Bernstein operators of complex variable in Gal (Mediterr J Math 5(3):253–272, 2008) and complete the upper estimates obtained for q-Bernstein operators of quaternionic variable in Gal (Approximation by Complex Bernstein and Convolution-Type Operators, 2009; Adv Appl Clifford Alg, doi:10.1007/s00006-011-0310-8, 2011).

Note on a New Quaternionic Approach to Relativity and the Dirac Theory

It is shown that there exists an extra imaginary unit commuting with Hamilton's imaginary units of quaternions. The proof is based upon the fact that the Dirac representation of the Lorentz group can be obtained not only over the complex field but also over the field of quaternions.

The algebraic structure of quaternionic analysis

The regularity of a quaternionic function is reinterpreted through a new canonical decomposition of the real differential, giving new insights into the algebraic properties of the regularity itself. The result comes from a somewhat unusual point of view on the automorphisms of the quaternionic field: a general notion of quaternionic linearity is associated to them, and some unnoticed metric properties of their inner representation are used to build up the theory.

Approximation by Quaternion q-Bernstein Polynomials, q > 1

Defining for q > 1 the q-Bernstein polynomials of degree n of a quaternion variable, attached to a function f defined on a ball in the field of quaternions, the order of approximation \({\frac{1} {q^n}}\) is obtained when f is in some classes of analytic functions in the sense of Weierstrass. The result extends that in the case of approximation of analytic functions of a complex variable in disks, by q-Bernstein polynomials of complex variable.

Least squares solution of the quaternion matrix equation with the least norm

For any , is called the j-conjugate matrix of A. If , A is called a j-self-conjugate matrix. If , A is called an anti j-self-conjugate matrix. By using the complex representation of quaternion matrices, the Moore–Penrose generalized inverse and the Kronecker product of matrices, we derive the expressions of the least squares solution with the least norm, the least squares j-self-conjugate solution with the least norm, and the least squares anti j-self-conjugate solution with the least norm of the matrix equation over the skew field of quaternions, respectively.

Schwarzschild solution of the generally covariant quaternionic field equations of Sachs

Sachs has derived quaternion field equations that fully exploit the underlying symmetry of the principle of general relativity, one in which the fundamental 10 component metric field is replaced by a 16 component four-vector quaternion. Instead of the 10 field equations of Einstein's tensor formulation, these equations are 16 in number corresponding to the 16 analytic parametric functions {\partial}x^{{\mu}'}/{\partial}x^{{\nu}} of the Einstein Lie Group. The difference from the Einstein equations is that these equations are not covariant with respect to reflections in space-time, as a consequence of their underlying quaternionic structure. These equations can be combined into a part that is even and a part that is odd with respect to spatial or temporal reflections. This paper constructs a four-vector quaternion solution of the quaternionic field equation of Sachs that corresponds to a spherically symmetric static metric. We show that the equations for this four-vector quaternion corresponding to a vacuum solution lead to differential equations that are identical to the corresponding Schwarzschild equations for the metric tensor components.

Global Structure of Quaternion Polynomial Differential Equations

In this paper we mainly study the global structure of the quaternion Bernoulli equations $\dot q=aq+bq^n$ for $q\in \mathbb H$ the quaternion field and also some other form of cubic quaternion differential equations. By using the Liouvillian theorem of integrability and the topological characterization of $2$--dimensional torus: orientable compact connected surface of genus one, we prove that the quaternion Bernoulli equations may have invariant tori, which possesses a full Lebesgue measure subset of $\mathbb H$. Moreover, if $n=2$ all the invariant tori are full of periodic orbits; if $n=3$ there are nfiinitely many invariant tori fulfilling periodic orbits and also infinitely many invariant ones fulfilling dense orbits.

The Nonexistence of Pseudoquaternions in $${\mathbb{C}^{2\times 2}}$$

The field of quaternions, denoted by $${\mathbb{H}}$$ can be represented as an isomorphic four dimensional subspace of $${\mathbb{R}^{4\times 4}}$$ , the space of real matrices with four rows and columns. In addition to the quaternions there is another four dimensional subspace in $${\mathbb{R}^{4\times 4}}$$ which is also a field and which has – in connection with the quaternions – many pleasant properties. This field is called field of pseudoquaternions. It exists in $${\mathbb{R}^{4\times 4}}$$ but not in $${\mathbb{H}}$$ . It allows to write the quaternionic linear term axb in matrix form as Mx where x is the same as the quaternion x only written as a column vector in $${\mathbb{R}^4}$$ . And M is the product of the matrix associated with the quaternion a with the matrix associated with the pseudoquaternion b. Now, the field of quaternions can also be represented as an isomorphic four dimensional subspace of $${\mathbb{C}^{2\times 2}}$$ over $${\mathbb{R}}$$ , the space of complex matrices with two rows and columns. We show that in this space pseudoquaternions with all the properties known from $${\mathbb{R}^{4\times 4}}$$ do not exist. However, there is a subset of $${\mathbb{C}^{2\times 2}}$$ for which some of the properties are still valid. By means of the Kronecker product we show that there is a matrix in $${\mathbb{C}^{4\times 4}}$$ which has the properties of the pseudoquaternionic matrix.

A Quaternion Widely Linear Adaptive Filter

A quaternion widely linear (QWL) model for quaternion valued mean-square-error (MSE) estimation is proposed. The augmented statistics are first introduced into the field of quaternions, and it is demonstrated that this allows for capturing the complete second order statistics available. The QWL model is next incorporated into the quaternion least mean-square (QLMS) algorithm to yield the widely linear QLMS (WL-QLMS). This allows for a unified approach to adaptive filtering of both <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\BBQ$</tex> </formula> -proper and <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\BBQ$</tex> </formula> -improper signals, leading to improved accuracies compared to the QLMS class of algorithms. Simulations on both benchmark and real world data support the analysis.

On solutions of matrix equation XF−AX=C and $XF-A\widetilde{X}=C$ over quaternion field

By means of Kronecker map and complex representation of a quaternion matrix, some explicit solutions to the quaternion matrix equations XF−AX=C and \(XF-A\widetilde{X}=C\) are established. One of the solutions is neatly expressed by a symmetric matrix, a controllability matrix and an observability matrix. In addition, two practical algorithms for these two equations are given.

Algorithm Q-LSQR for the least squares problem in quaternionic quantum theory

In this paper, an iterative algorithm for the standard quaternionic least squares problem is proposed without using the real (complex) representation. Our algorithm is implemented in the quaternion field and by means of direct quaternion arithmetic and is a natural generalization of the LSQR algorithm for the real least squares problem.

Spontaneous $$ \mathcal{N} $$ = 2 → $$ \mathcal{N} $$ = 1 supersymmetry breaking in supergravity and type II string theory

Using the embedding tensor formalism we give the general conditions for the existence of N=1 vacua in spontaneously broken N=2 supergravities. Our results confirm the necessity of having both electrically and magnetically charged multiplets in the spectrum, but also show that no further constraints on the special Kahler geometry of the vector multiplets arise. The quaternionic field space of the hypermultiplets must have two commuting isometries, and as an example we discuss the special quaternionic-Kahler geometries which appear in the low-energy limit of type II string theories. For these cases we find the general solution for stable Minkowski and AdS N=1 vacua, and determine the charges in terms of the holomorphic prepotentials. We find that the string theory realisation of the N=1 Minkowski vacua requires the presence of non-geometric fluxes, whereas they are not needed for the AdS vacua. We also argue that our results should hold in the presence of spacetime and worldsheet instanton corrections.

Systems of two iterated functions over skew field of quaternions

Systems of linear iterated functions f0(z) = qz + a, f1(z) = qz + b over the field of complex numbers have been investigated since 1985 (Barnsley and Harrington). Much attention is paid to the question of the connection of their attractors. We consider systems of iterated functions f0(z) = qzp + a, f1(z) = qzp + b over the skew field of quaternions. We simplify the form of such systems and study the structure of their attractors.

The two-sided Laplace transformation over the Cayley-Dickson algebras and its applications

This paper is devoted to the noncommutative version of the Laplace transformation. New types of direct and inverse transformations of the Laplace type over general Cayley-Dickson algebras, in particular, the skew field of quaternions and the octonion algebra are investigated. Examples are given. Theorems about properties of such transformations and also theorems about images and originals in conjunction with the operations of multiplication, differentiation, integration, convolution, shift, and homothety are proved. Applications are given to a solution of differential equations over the Cayley-Dickson algebras.

Orthogonality preserving bijective maps on finite dimensional projective spaces over division rings

Let be a finite dimensional right vector space over a division ring 𝔽, equipped with a nondegenerate semi-sesquilinear inner product ⟨·,·⟩. Let D and E be bijective linear operators on . The operators D and E induce indefinite inner products on , by the formulas ⟨Dx,y⟩ and ⟨Ex,y ⟩, where . We study maps on the projective space over that send D-orthogonal one-dimensional subspaces (elements of the projective space) to E-orthogonal one-dimensional subspaces. We prove that under the assumption of bijectivity of the map, and assuming n ≥ 3, such a map T preserves (D, E )-orthogonality if and only if it preserves (D, E )-orthogonality in both directions. In this case, it is induced by a semilinear transformation on that is (D, E )-unitary up to a multiplicative constant. For n = 3, it turns out the injectivity assumption alone suffices to reach the same conclusion. The main results are specialized to the particular cases when 𝔽 is the (commutative) complex field and the (noncommutative) skew field of real quaternions. In particular, the existence of (D, E )-unitary semilinear maps is discussed in these cases.

Quaternion normed space with isometry group ℤ2

In a finite-dimensional linear space over the quaternion field, we construct a norm with the following property. Any linear isometry is one of the two transformations x ↦ x and x ↦ −x.

An iterative algorithm for least squares problem in quaternionic quantum theory

Quaternionic least squares (QLS) problem is one method of solving overdetermined sets of quaternion linear equations A X B = E that is appropriate when there is error in the matrix E. In this paper, by means of real representation of a quaternion matrix, we introduce a concept of norm of quaternion matrices, which is different from that in [T. Jiang, L. Chen, Algebraic algorithms for least squares problem in quaternionic quantum theory, Comput. Phys. Comm. 176 (2007) 481–485; T. Jiang, M. Wei, Equality constrained least squares problem over quaternion field, Appl. Math. Lett. 16 (2003) 883–888], and derive an iterative method for finding the minimum-norm solution of the QLS problem in quaternionic quantum theory.

Distribution and estimation for eigenvalues of real quaternion matrices

In this paper, the concept of generalized spherical neighborhood on the real quaternion field is introduced. Then, Schwartz’ s inequality and the Gerschgorin’s theorem are proved on the real quaternion field and the problems of the distribution and estimation of real quaternion matrix eigenvalues are solved. Finally, some upper bound and lower bound estimation theorems for the moment, the real and imaginary part of the real quaternion matrices eigenvalues are obtained.