arXiv subjects

Adolfas Dargys

Publications and source records attributed to Adolfas Dargys.

7 recordsLinked to original sources

arXiv2026

Square roots of complexified quaternions

Square roots of complexified (complex) quaternions, namely, the Hamilton quaternion, coquaternion, nectorine, and conectorine are investigated. The isomorphisms between the complex quaternions and 3-dimensional multivectors of Clifford algebras is employed for this purpose. Root examples for all named quaternions are presented from which follows that the complex quaternionic roots may assume discrete or continuous form, or there may be no roots at all.

math-ph↗

arXiv2025

Spectral square roots of the multivector

The problem of multivector (MV) multiple square roots in real geometric Clifford algebras Cl(p,q) with symbolic coefficients is considered. The method to find multiple MV square roots that is based on R.Bott's periodicity table and matrix eigensystem in Cl(p,q) is proposed. The method can be applied to MV having both numerical and symbolic coefficients. In addition, method allows to determine the domain of the existence of thus obtained spectral square roots. A number of examples is presented for multivectors in low, p+q<= 3, and higher dimensional Clifford algebras, including 4D (anti)-Euclidean space and relativistic Cl(1,3) and Cl(3,1) algebras. Tables of the required basis vectors for conversion of MV to Bott's matrix representation have been found from respective algebra idempotents using ideal theory and presented for real Clifford algebras in Appendix.

math-ph↗

arXiv2022

The characteristic polynomial in calculation of exponential and elementary functions in Clifford algebras

Formulas to calculate multivector exponentials in a base-free representation and in a orthonormal basis are presented for an arbitrary Clifford geometric algebra Cl(p,q). The formulas are based on the analysis of roots of characteristic polynomial of a multivector. Elaborate examples how to use the formulas in practice are presented. The results may be useful in the quantum circuits or in the problems of analysis of evolution of the entangled quantum states.

quant-ph↗

arXiv2022

Exponential and logarithm of multivector in low dimensional (n=p+q<3) Clifford algebras

Closed form expressions for a multivector exponential and logarithm are presented in real Clifford geometric algebras Cl(p,q)when n=p+q=1 (complex and hyperbolic numbers) and n=2 (Hamilton, split and conectorine quaternions). Starting from Cl(0,1) and Cl(1,0) algebras wherein square of a basis vector is either -1 or +1, we have generalized exponential and logarithm formulas to 2D quaternionic algebras, Cl(0,2), Cl(1,1), and Cl(2,0). The sectors in the multivector coefficient space where 2D logarithm exists are found. They are related with a square root of the multivector.

math-ph↗

arXiv2021

Exponentials of general multivector (MV) in 3D Clifford algebras

Closed form expressions to calculate the exponential of a general multivector (MV) in Clifford geometric algebras (GAs) Cl(p,q) are presented for n=p+q=3. The obtained exponential formulas were applied to find exact GA trigonometric and hyperbolic functions of MV argument. We have verified that the presented exact formulas are in accord with series expansion of MV hyperbolic and trigonometric functions. The exponentials may be applied to solve GA differential equations, in signal and image processing, automatic control and robotics.

math.RA↗

arXiv2015

Calculation of quantum eigens with geometrical algebra rotors

A practical computation method to find the eigenvalues and eigenspinors of quantum mechanical Hamiltonian is presented. The method is based on reduction of the eigenvalue equation to well known geometric algebra rotor equation and, therefore, allows to replace the usual det(H-E)=0 quantization condition by much simple vector norm preserving requirement. In order to show how it works in practice a number of examples are worked out in Cl_{3,0} (monolayer graphene and spin in the quantum well) and in Cl_{3,1} (two coupled two-level atoms and bilayer graphene) algebras.

math-ph↗

arXiv2014

Pseudospin, velocity and Berry phase in a bilayer graphene

Hamiltonian and eigenstate problem is formulated for a bilayer graphene in terms of Clifford's geometric algebra \textit{Cl}$_{3,1}$. It is shown that such approach allows to perform analytical calculations in a simple way if geometrical algebra rotors are used. The measured quantities are express through spectrum and rotation half-angle of the pseudospin that appears in geometric algebra rotors. Properties of free charge carriers -- pseudospin, velocity and Berry phase -- in a bilayer graphene are investigated in the presence of the external voltage applied between the two layers.

cond-mat.mes-hall↗