# Artūras AcusVilnius University · Institute of Theoretical Physics and Astronomy

Artūras Acus

Doctor of Philosophy

Geometric (Clifford) algebra ant its application in theoretical physics

## About

44

Publications

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134

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Introduction

Geometric algebra and its applications in physics

Additional affiliations

January 2019 - present

January 2006 - August 2017

**Vilnius Pedagogical University**

Position

- Professor (Assistant)

## Publications

Publications (44)

Closed form expressions in real Clifford geometric algebras Cl0,3,Cl3,0,Cl1,2$$ C{l}_{0,3},C{l}_{3,0},C{l}_{1,2} $$, and Cl2,1$$ C{l}_{2,1} $$ are presented in a coordinate‐free form for exponential function when the exponent is a general multivector. The main difficulty in solving the problem is connected with an entanglement (or mixing) of vector...

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 ma...

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 exponent...

Closed form expressions to calculate the exponential of a general multivector (MV) in Clifford geometric algebras (GAs) Clp;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 se...

Closed form expressions in a coordinate-free form in real Clifford geometric algebras (GAs) Cl(0,3), Cl(3,0)$, Cl(1,2) and Cl(2,1) are found for exponential function when the exponent is the most general multivector (MV). The main difficulty in solving the problem is connected with an entanglement or mixing of vector and bivector components. After...

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...

A classical system, which is analogous to the quantum one with a backﬂow of probability, is proposed. The system consists of a chain of masses interconnected by springs and attached by other springs to ﬁxed supports. Thanks to the last springs the cutoﬀ frequency and dispersion appears in the spectrum of waves propagating along the chain. It is sho...

A classical system, which is analogous to the quantum one with a backflow of probability, is proposed. The system consists of a chain of masses interconnected by springs, as well attached by other springs to fixed supports. Thanks to the last springs the cutoff frequency and dispersion appears in the spectrum of waves propagating along the chain. I...

An algorithm to extract the square root in radicals from a multivector (MV) in real Clifford algebras Cl(p,q) for n=p+q <=3 is presented. We show that in the algebras Cl(3,0), Cl(1,2) and Cl(0,3) there are up to four isolated roots in a case of the most general (generic) MV. The algebra Cl(2,1) makes up an exception and the MV here can have up to 1...

The problem of square root of multivector (MV) in real 3D (n = 3) Clifford algebras Cl3;0, Cl2;1, Cl1;2 and Cl0;3 is considered. It is shown that the square root of general 3D MV can be extracted in radicals. Also, the article presents basis-free roots of MV grades such as scalars, vectors, bivectors, pseudoscalars and their combinations, which may...

The solution of the Liouville equation for the ensemble of free particles is presented and the classical analog to the quantum accelerating Airy wave packet is constructed and discussed. Considering the motion of various classical packets – with an infinite and restricted distribution of velocities of particles – and also the motion of their fronts...

The solution of the Liouville equation for the ensemble of free particles is presented and the classical analog to the quantum accelerating Airy wave packet is constructed and discussed. Considering the motion of various classical packets -- with an infinite and restricted distribution of velocities of particles -- and also the motion of their fron...

The coordinate-free solutions of the multivector equation $ax+xb=c$ are discussed and presented for the Clifford algebras $Cl_{p,q}$ when $p+q\le 3$.

The algorithm of finding an inverse multivector (MV) numerically and symbolically is of paramount importance in the applied Clifford geometric algebra (GA) \( Cl _{p,q}\). The first general MV inversion algorithm was based on matrix representation of MV. The complexity of calculations and the size of the answer in a symbolic form grow exponentially...

The algorithm of finding inverse multivector (MV) in a symbolic form is of paramount importance in computational Clifford or geometric algebra (GA) $Cl_{p,q}$. The first attempts of inversion of general MV were based on matrix representation of MV basis elements. However, the complexity of such calculations grows exponentially with the dimension $n...

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 require...

We continue the discussion on the interaction energy of the axially symmetric
Hopfions evaluated directly from the product anzsatz. The Hopfions are given by
the projection of Skyrme model solutions onto the coset space SU(2)/U(1). Our
results show that if the separation between the constituents is not small, the
product ansatz can be considered as...

We theoretically explore atomic Bose-Einstein condensates (BECs) subject to
position-dependent spin-orbit coupling (SOC). This SOC can be produced by
cyclically Raman coupling four internal atomic ground (or metastable) states in
an environment where the detuning from two-photon Raman resonance depends on
position. The resulting spin-orbit coupled...

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 pse...

A simple method of temperature and electron density measurement in quasi-equilibrium plasma at temperatures below 10 kK from the ratio of mass spectrometric signals of the doubly and singly ionized ions of Ba and Pb has been proposed. High masses of the ions and possibility to select the odd isotopes makes it possible to perform measurements at a l...

We discuss the relation between the solutions of the Skyrme model of lower
degrees and the corresponding axially symmetric Hopfions which is given by the
projection onto the coset space SU(2)/U(1). The interaction energy of the
Hopfions is evaluated directly from the product ansatz. Our results show that
if the separation between the constituents i...

We study the effect of the canonical quantization of the rotational mode of
the charge Q=1 and Q=2 spinning Hopfions. The axially-symmetric solutions are
constructed numerically, it is shown the quantum corrections to the mass of the
configurations are relatively large.

We introduce a one-dimensional two-component system with the self-focusing
cubic nonlinearity concentrated at a symmetric set of two spots. Effects of the
spontaneous symmetry breaking (SSB) of localized modes were previously studied
in the single-component version of this system. In this work, we study the
evolution (in the configuration space of...

The rational map approximation provides an opportunity to describe light nuclei as classical solitons with baryon number B > 1 in the framework of the Skyrme model. The rational map ansatz yields a possibility of factorization of S3 baryon charge into S1 and S2 parts, the phenomenology of the model being strongly affected by the chosen factorizatio...

Analytical expressions for spectrum, eigenfunctions and dipole matrix
elements of a square double quantum well (DQW) are presented for a general case
when the potential in different regions of the DQW has different heights and
effective masses are different. This was achieved by Gr\"obner basis algorithm
which allows to disentangle the resulting co...

The knowledge of exact wave functions is required in calculating physical parameters such as optical dipole moments, scattering matrix elements, or in wave function engineering. In this report we describe how a system of algebraic equations that follows from the Schrödinger equation can be reduced to a Gröbner basis from which the exact wave functi...

Essentially higher ionization degree of small concentrations of elements in inductively coupled plasma in comparison to the ionization of pure elements is emphasized. This conclusion is used to determine the relative dependence of the sensitivity of the inductively coupled plasma mass spectrometer on the atomic mass. The possibility of evaluation o...

We analyse the effect of the canonical quantization of the rotational mode of the O(3)σ-model which includes the Skyrme term. Numerical evidence is presented that the quantum correction to the mass of the rotationally-invariant charge n=1,2 configurations may stabilize the solution even in the limit of vanishing potential. The corresponding range o...

Analysis of the multivariate data distributions can be helpful or directly applicable in pattern recognition tests. Estimate of the volume of the critical region of overlapping distributions is essential in determination of the condence level of classica- tion. Mathematical tools for analysis of the multivariate distributions (included probability,...

The rational map approximation to the solution to the SU(2) Skyrme model with baryon number B=4 is canonically quantized. The quantization procedure leads to anomalous breaking of the chiral symmetry, and exponential fall-off of the energy density of the soliton at large distances. The model is extended to SU(2) representations of arbitrary dimensi...

The ground state configurations of the solution to Skyrme's topological soliton model for systems with baryon number larger than 1 are well approximated with rational map ans\"atze, without individual baryon coordinates. Here canonical quantization of the baryon number 2 system, which represents the deuteron, is carried out in the rational map appr...

The characteristic feature of the ground state configuration of the Skyrme model description of nuclei is the absence of recognizable individual nucleons. The classical skyrmion with baryon number 2 is axially symmetric, and can be approximated by a simple rational map anzats. Quantized version of biskyrmion then can be identified with dibarion if...

The explicit expressions for the electric, magnetic, axial and induced pseudoscalar form factors of the nucleons are derived in the {\it ab initio} quantized Skyrme model. The canonical quantization procedure ensures the existence of stable soliton solutions with good quantum numbers. The form factors are derived for representations of arbitrary di...

Spontaneous symmetry breaking occurs when the symmetry that a physical system possesses, is not preserved for the ground state of the system. Although the procedure of symmetry breaking is quite clear from the mathematical point of view, the physical interpretation of the phenomenon is worth to be better understood. In this note we present a simple...

This paper is a PhD thesis defended at Institute of Theoretical Physics and Astronomy on 18 December, 1998. The following (abbreviated) statements represent the main results of the work: 1.Each of SU(2) representation j yields the different quantum Lagrangian density. As a consequence, theoretical observables depend on representation j which can be...

A constructive realization of Skyrme's conjecture that an effective pion mass ``may arise as a self consistent quantal effect'' based on an ab initio quantum treatment of the Skyrme model is presented. In this quantum mechanical Skyrme model the spectrum of states with $I=J$, which appears in the collective quantization, terminates without any infi...

The Skyrme model is considered quantum mechanically ab initio in various irreducible representations of the SU(2) group. The canonical quantization procedure yields negative mass correction ensuring existence of stabile soliton solution even in chiral limit. The evaluated static properties of nucleons (masses, magnetic moments, radii etc.) are in a...

The representations of general dimension are constructed for the $SU(2)$ Skyrme model, treated quantum mechanically {\it ab initio. } This quantum Skyrme model has a negative mass term correction, that is not present in the classical Hamiltonian. The magnitude of the quantum mechanical mass correction increases with the dimension of the representat...

For the distance learning and scientific cooperation the possibility to run software by Internet is essential. Ideas, algorithms, models and other software can be tested and investigated by everyone. Computer experimentation on the web becomes a natural approach to research and to scientific collaboration. The new field of complexity exhibits the l...

The Skyrme model (1) has been extended to describe classical solitons with baryon number B > 1 by rational map approximation (2). The canon- ical quantization procedure (3) applied to rotational degrees of freedom of 4 He and 16 O nuclei leads to anomalous breaking of the chiral symmetry and exponential falloff of the energy density of the soliton...