Ramaswamy Jagannathan

• PhD
• Professor at Institute of Mathematical Sciences

Quantum mechanics of bending of a charged particle beam by a dipole magnet is studied in the paraxial approximation. Nonrelativistic quantum mechanics of the system is studied using the nonrelativistic Schroedinger equation ignoring the spin of the particle. To study the relativistic quantum mechanics of the system, ignoring the spin of the particl...

Classical and quantum theories of the Wien velocity filter are presented. The classical theory is based on the beam optical Hamiltonian and the Lie operator method, which readily leads to the results obtained through the widely used differential equations approach. So far, the quantum mechanics of the Wien filter has not been studied. We explain th...

The integrals $\int x^n e^{-x}\,dx$, $\int x^n e^x\,dx$, $\int x^n \sinh{x}\,dx$, and $\int x^n \sin{x}\,dx$ are seen to give rise to sets of Appell polynomials with the exponential generating functions $e^{xt}/(1 - t)$, $e^{xt}/(1 + t)$, $e^{xt}/\left( 1 - t^2 \right)$, and $e^{xt}/\left( 1 + t^2 \right)$, respectively, and these are special cases...

An eight-dimensional matrix representation of the Maxwell equations for a linear inhomogeneous medium has been obtained earlier based on the Riemann-Silberstein-Weber vector starting from the equations satisfied by it. A new eight-dimensional matrix representation, related to the earlier one by a similarity transformation, is derived starting ab in...

Scalar theory of quantum electron beam optics, at the single-particle level, derived from the Dirac equation using a Foldy-Wouthuysen-like transformation technique is considered. Round magnetic electron lenses with Glaser and power law models for the axial magnetic field B(z) are studied. Paraxial quantum propagator for the Glaser model lens is obt...

Scalar theory of quantum electron beam optics, at the single-particle level, derived from the Dirac equation using a Foldy-Wouthuysen-like transformation technique is considered. Round magnetic electron lenses with Glaser and power law models for the axial magnetic field $B(z)$ are studied. Paraxial quantum propagator for the Glaser model lens is o...

The Tsallis q-exponential function \(e_{q}(x) = (1+(1-q)x)^{\frac {1}{1-q}}\) is found to be associated with the deformed oscillator defined by the relations \(\left [N,a^{\dagger }\right ] = a^{\dagger }\), [N,a] = −a, and \(\left [a,a^{\dagger }\right ] = \phi _{T}(N+1)-\phi _{T}(N)\), with ϕT(N) = N/(1 + (q − 1)(N − 1)). In a Bargmann-like repre...

Quantum mechanics of bending of a nonrelativistic monoenergetic charged particle beam by a dipole magnet is studied in the paraxial approximation. The transfer map for the position and momentum components of a particle of the beam between two transverse planes at different points on the curved optic axis of the system is derived starting with the n...

Quantum mechanics of bending of a nonrelativistic monoenergetic charged particle beam by a dipole magnet is studied in the paraxial approximation. The transfer map for the position and momentum components of a particle of the beam between two transverse planes at different points on the curved optic axis of the system is derived starting with the n...

The Tsallis $q$-exponential function $e_q(x) = (1+(1-q)x)^{1/(1-q)}$ is found to be associated with the deformed oscillator defined by the relations $\left[N,a^\dagger\right] = a^\dagger$, $\left[N,a\right] = -a$, and $\left[a,a^\dagger\right] = \phi_T(N+1)-\phi_T(N)$, with $\phi_T(N) = N/(1+(1-q)(1-N))$. In a Bargmann-like representation of this d...

Quantum Mechanics of Charged Particle Beam Optics in Devices from Electron Microscopes to Particle Accelerators Classical Charged Particle Beam Optics used in the design and operation of all present-day charged particle beam devices, from low energy electron microscopes to high energy particle accelerators, is entirely based on classical mechanic...

This chapter reviews quantum mechanics required to pursue the quantum charged particle beam optics, the central theme of this book. All physical phenomena are quantum mechanical at the fundamental level. Classical mechanics observed in the macroscopic world is an approximation. That is why it raises a curiosity when classical mechanics is found to...

This chapter provides a basic introduction to classical charged particle beam optics before proceeding to the development of quantum charged particle beam optics in later chapters. Charged particle beam optics is the study of transport of charged particle beams through electromagnetic optical systems. Such systems, like different types of electroma...

This Chapter develops the scalar quantum charged particle beam optics, applicable to spin-$0$ and spinless particles ({\em i.e.}, particles for which the spin is ignored), based on the relativistic Klein-Gordon equation and the nonrelativistic Schr\"{o}dinger equation. Normal magnetic quadrupole which focuses and defocuses the charged particle beam...

This chapter points to the synergy between optics and mechanics citing the relevant example of the early days of electron optics when the analogy with light optics provided much guidance. The inspiration for the spinor theory of charged particle beam optics also came from the studies on the optics of polarized light! It is recalled that the formali...

This chapter provides the essential concepts of classical mechanics required for understanding the classical charged particle beam optics. All charged particle beam devices, from low energy electron microscopes to high energy particle accelerators, are designed and operated very successfully on the basis of classical mechanics though the beams prop...

This chapter considers the quantum charged particle beam optics for spin-$\frac{1}{2}$ particles. For electrons, or for any spin-$\frac{1}{2}$ particle, the proper equation to be the basis for the theory of quantum beam optics is the Dirac equation. This chapter presents a general formalism of spinor quantum charged particle beam optics based on t...

The main aim of this chapter is to provide the motivation for studying the quantum mechanics of charged particle beam optics. It summarizes briefly the contents of the later chapters on classical mechanics, quantum mechanics, classical charged particle beam optics, and the scalar and spinor theories of quantum charged particle beam optics. http:/...

In [Arch. Math. 7, 28 (1956), Utilitas Math. 15, 51 (1979)] Carlitz introduced the degenerate Bernoulli numbers and polynomials by replacing the exponential factors in the corresponding classical generating functions with their deformed analogs: $\exp(t) \rightarrow (1+\lambda t)^{1/\lambda}$, and $\exp(tx) \rightarrow (1+\lambda t)^{x/\lambda}$. T...

Inspired by Reshetikhin's twisting procedure to obtain multiparametric extensions of a Hopf algebra, a general "symmetry transformation" of the "particle conserving" R-matrix is found such that the resulting multiparametric R-matrix, with a spectral parameter as well as a color parameter, is also a solution of the Yang–Baxter equation (YBE). The co...

A q-deformation of the parasupersymmetric quantum mechanics of one boson and one parafermion of an arbitrary order p is constructed. The deformed commutators in the algebra generate braid-type relations which lift the degeneracy originally present in the undeformed case. The connection between the deformed algebra and a class of self-similar potent...

The Tamm-Dancoff (TD) deformation of the boson oscillator incorporates a high energy cutoff in its spectrum. It is found that one can obtain a similar deformation of any generalized bosonic oscillator algebra. The Hopf (or ‘quantum’) algebraic aspects of the TD-deformation are discussed. Examples are given.

A three-dimensional polynomial algebra of order m is defined by the commutation relations [P0,P±] = ± P±, [P+,P-] = ϕ(m)(P0) where ϕ(m) (P0) is an mth order polynomial in P0 with the coefficients being constants or central elements of the algebra. It is shown that two given mutually commuting polynomial algebras of orders l and m can be combined to...

In view of the possible relevance of the q-calculus based on Jackson's q-derivative operator in the phenomenological applications of quantum algebras it is pointed out that for real q(> 1) such that (q − 1) ≈ 0 there is a formal correspondence, up to first order in (q − 1), between the q-calculus and the calculus that can be developed assuming that...

A (p,q)-analog of the classical Rogers-Szegö polynomial is defined by replacing the q-binomial coefficient in it by the (p,q)-binomial coefficient corresponding to the definition of (p,q)-number as [n]p,q = (pn - qn)/(p - q){[n]}_{p,q} = ({p}^{n} - {q}^{n})/(p - q) . Exactly like the Rogers-Szegö polynomial is associated with the q-oscillator alge...

Generalized Clifford algebras (GCAs) and their physical applications were extensively studied for about a decade from 1967 by Alladi Ramakrishnan and his collaborators under the name of L-matrix theory. Some aspects of GCAs and their physical applications are outlined here. The topics dealt with include: GCAs and projective representations of finit...

Using a realization of the q-exponential function as an infinite multiplicative sereis of the ordinary exponential functions we obtain new nonlinear connection formulae of the q-orthogonal polynomials such as q-Hermite, q-Laguerre and q-Gegenbauer polynomials in terms of their respective classical analogs. Comment: 14 pages

We give a method to embed the q-series in a (p,q)-series and derive the corresponding (p,q)-extensions of the known q-identities. The (p,q)-hypergeometric series, or twin-basic hypergeometric series (diferent from the usual bibasic hypergeometric series), is based on the concept of twin-basic number [n]_{p,q} = (p^n - q^n)/(p-q). This twin-basic nu...

A scheme of q-deformation of nonlinear maps is introduced. As a specific example, a q-deformation procedure related to the Tsallis q-exponential function is applied to the logistic map. Compared to the canonical logistic map, the resulting family of q-logistic maps is shown to have a wider spectrum of interesting behaviours, including the co-existe...

Motivated by studies onq-deformed physical systems related to quantum group structures, and by the elements of Tsallis statistical mechanics, the concept ofq-deformed nonlinear maps is introduced. As a specific example, aq-deformation procedure is applied to the logistic map. Compared to the canonical logistic map, the resulting family ofq-logistic...

Katriel, Rasetti and Solomon introduced a q-analogue of the Zassenhaus formula written as where A and B are two generally noncommuting operators and eqz is the Jackson q-exponential, and derived the expressions for c2, c3 and c4. It is shown that one can also write Explicit expressions for , and are given.

An analogue of Euler's partition identity: "The number of partitions of a positive integer v into odd parts equals the number of its partitions into distinct parts" is obtained for ordered partitions. The ideas developed are then used in obtaining ...

Classical mechanical treatment of charged particle beam optics is so far very satisfactory from a practical point of view in applications ranging from electron microscopy to accelerator technology. However, it is desirable to understand the underlying quantum mechanics since the classical treatment is only an approximation. Quantum mechanical treat...

The Foldy-Wouthuysen iterative diagonalization technique is applied to the Helmholtz equation to obtain a Hamiltonian description of the propagation of a monochromatic quasiparaxial light beam through a system in which the refractive index $n(x,y,z)$ varies about a background value $n_0$ such that $|n(x,y,z)-n_0| \ll n_0$. This technique is present...

Four classes of three-dimensional quadratic algebra of the type [Q0,Q?] = ?Q?, [Q+,Q-] = aQ02 + bQ0 + c, where (a,b,c) are constants or central elements of the algebra, are constructed using a generalization of the well known two-mode bosonic realizations of su(2) and su(1,1). The resulting matrix representations and single variable differential op...

Some very elementary ideas about quantum groups and quantum algebras are introduced and a few examples of their physical applications are mentioned.

It has been found that quantum corrections can substantially affect the classical results of tracking for trajectories close to the separatrix. Hence the development of a basic formalism for obtaining the quantum maps for any particle beam optical system is called for. To this end, it is observed that several aspects of quantum maps for the beam op...

Quadratic algebras of the type $\lsb Q_0, Q_\pm \rsb$ $=$ $\pm Q_\pm$, $\lsb Q_+, Q_- \rsb$ $=$ $aQ_0^2 + bQ_0 + c$ are studied using three-mode bosonic realizations. Matrix representations and single variable differential operator realizations are obtained. Examples of physical relevance of such algebras are given.

We present a general unified approach for finding the coherent states (CSs) of polynomially deformed algebras such as the quadratic and Higgs algebras, which are relevant for various multiphoton processes in quantum optics. We give a general procedure to map these deformed algebras to appropriate Lie algebras. This is used, for the noncompact cases...

A very elementary introduction to quantum algebras is presented and a few examples of their physical applications are mentioned.

We present a general unified approach for finding the coherent states of polynomially deformed algebras such as the quadratic and Higgs algebras, which are relevant for various multiphoton processes in quantum optics. We give a general procedure to map these deformed algebras to appropriate Lie algebras. This is used, for the non compact cases, to...

A two-parameter (p, q) deformation of the Jaynes-Cummings model is obtained using a recently developed (p, q)-deformed oscillator. In the rotating wave approximation (RWA) the dynamical symmetry of the model is the quantum superalgebra up,q(1 mod 1). The partition function of the model is obtained as a path integral over generalized Perelomov coher...

For multimode systems of deformed oscillators covariant under the actions of the quantum groups SUq(n), SUq(n mod m), GLp,q(n) and GLp,q(n mod m) the number operators are constructed explicitly in terms of the creation and annihilation operators. The relation between the various kinds of deformed oscillator systems, representations of these oscilla...

A large class of bosonic coherent states known in the literature have been constructed in a unified way by Shanta et. al. (1994). It is shown that this method can be easily extended to generalized bosonic-oscillator systems.

The quantum algebra Up,q(gl(2)), with two independent deformation parameters (p, q), is studied and, in particular, its universal R-matrix is constructed using Reshetikhin's method. A contraction procedure then leads to the (p, q)-deformed Heisenberg algebra Up, q(h(1)) and its universal R-matrix. Using a Sugawara construction employing an infinite...

Presents the available solutions to the problem of expressing the number operator in terms of the creation and annihilation operators in the case of the various single-mode q-oscillators. This study reveals interesting number theoretic aspects of the problem.

The algebra of q-fermion operators, developed earlier by two of the present authors is re-examined. It is shown that these operators represent particles that are distinct from usual spacetime fermions except in the limit q=1. It is shown that it is possible to introduce generalized q-oscillators defined for - infinity (q<or=1. In the range - infini...

A conformal dimension ( Delta ) dependent (p,q)-deformed Virasoro ((p,q-Virasoro) algebra with two independent deformation parameters (p,q) is constructed. The comultiplication rule for the generating functional for the Delta =0, 1 case is established and found to be depending on p and q individually. The central charge term for the (p,q)-Virasoro...

It is shown that a recursive use of the transformation for a terminating 3F2(1) series used by Weber and Erdelyi (1952), which belongs, as shown by Whipple (1925), to a set of equivalent 3F2(1) functions obtained by Thomae (1979), results in a 72-element group associated with 18 terminating series. The generators, conjugacy classes, invariant subgr...

It is noted that the study of a quantum algebra sup,q(2), with two independent deformation parameters (p, q), leads to a '(p, q)-oscillator' realization for it. The analysis is extended to the (p, q)-analogues of su(1, 1), osp(2/1) and the centreless Virasoro algebra. The standard single-parameter (q) deformations are obtained in the limit p=q.

Explicit realizations of the quantum groups GLp,q(2) and GLp,q(1 mod 1) corresponding to unimodular values of the deformation parameters p and q are given in terms of the canonically conjugate (X, P) operators, using the Heisenberg-Weyl commutation relations. Matrix representations are also discussed. Some observations are made on similar realizati...

Non-relativistic quantum theory of a particle in a non-commutative space is formulated within the framework of the usual quantum mechanics itself. This is accomplished by showing explicitly that the Wess-Zumino formalism of deformed quantum mechanical phase space corresponding to a non-commutative space can be realized, in principle, in terms of th...

Two-parameter (p,q) deformation of a parabosonic algebra underlying the two-particle Calogero model is considered. The Fock-space representation, Bargmann-Fock representation, coherent states, spectrum-generating algebra and a hidden supersymmetry are discussed. The paraboson algebra induces a centreless Virasoro-type algebra with a doubling of the...

Using the representations of the Heisenberg-Weyl relations the authors develop a systematic scheme for constructing finite and infinite dimensional representations of the elements of the quantum groups GLq(n), where the deformation parameter q is a primitive root of unity. Explicit results for the examples GLq(2), GLq(3) and GLq(4) are discussed.

The dioptric power of an optical system can be expressed as a four-component dioptric power matrix. We generalize and reformulate the standard matrix approach by utilizing the methods of Lie algebra. This generalization helps one deal with nonlinear problems (such as aberrations) and further extends the standard matrix formulation. Explicit formula...

Positive discrete series representations of the Lie algebra $su(1,1)$ and the quantum algebra $U_q(su(1,1))$ are considered. The diagonalization of a self-adjoint operator (the Hamiltonian) in these representations and in tensor products of such representations is determined, and the generalized eigenvectors are constructed in terms of orthogonal p...

Using the technique developed by Fronsdal and Galindo (Lett. Math. Phys. 27 (1993) 57) for studying the Hopf duality between the quantum algebras Fun p;q (GL(2)) and U p;q (gl(2)), the Hopf structure of U p;q (gl(1j1)), dual to Fun p;q (GL(1j1)), is derived and the corresponding universal T -matrix of Fun p;q (GL(1j1)), embodying the suitably modif...

It is suggested that the (p,q)-hypergeometric series studied by Burban and Klimyk (in Integral Transforms and Special Functions, 2 (1994) 15 - 36) can be considered as a special case of a more general (P,Q)-hypergeometric series.

The traditional approach to accelerator optics, based mainly on classical mechanics, is working excellently from the practical point of view. However, from the point of view of curiosity, as well as with a view to explore quantitatively the consequences of possible small quantum corrections to the classical theory, a quantum mechanical formalism of...

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Quantum Theory of the optics of charged particles is discussed. The formalism of spinor theory, using essentially an algebraic approach, has been extended to the scalar theory, using two-component formalism, suitable for treating the forward and backward propagating beams separately, analogous to the Feshbach-Villars representation of the Klein-Gor...

The dually conjugate Hopf algebras Fun p,q(R) and U p,q(R) associated with the two-parametric (p, q)-Alexander-Conway solution (R) of the Yang-Baxter equation are studied. Using the Hopf duality construction, the full Hopf structure of the quasitriangular enveloping algebra U p,q(R) is extracted. The universal script T sign-matrix for Fun p,q(R) is...

The dually conjugate Hopf algebras F unp,q(R) and Up,q(R) associated with the two-parametric (p, q)-Alexander-Conway solution (R) of the Yang-Baxter equation are studied. Using the Hopf duality construction, the full Hopf structure of the quasitriangular enveloping algebra Up,q(R) is extracted. The universal T - matrix for F unp,q(R) is derived. Wh...

The dually conjugate Hopf superalgebras Fun p,q (GL(11)) and U p,q (gl(11)) are studied using the Frnsdal-Galindo approach and the full Hopf structure of U p,q (gl(11)) is extracted. A finite expression for the universal T-matrix, identified with the dual form and expressing the generalization of the exponential map of the classical groups, is obta...

The dually conjugate Hopf algebras $Fun_{p,q}(R)$ and $U_{p,q}(R)$ associated with the two-parametric $(p,q)$-Alexander-Conway solution $(R)$ of the Yang-Baxter equation are studied. Using the Hopf duality construction, the full Hopf structure of the quasitriangular enveloping algebra $U_{p,q}(R)$ is extracted. The universal ${\cal T}$-matrix for $...

For the quantum group $GL_{p,q}(2)$ and the corresponding quantum algebra $U_{p,q}(gl(2))$ Fronsdal and Galindo explicitly constructed the so-called universal $T$-matrix. In a previous paper we showed how this universal $T$-matrix can be used to exponentiate representations from the quantum algebra to get representations (left comodules) for the...

Beam optics of a spin-1/2 particle with anomalous magnetic moment is studied in the monoenergetic and paraxial approximations based on the Dirac equation; the treatment is at the level of single-particle dynamics, considers the electromagnetic field as classical and disregards radiation aspects. The general theory, developed for any magnetic optica...

A quantum algebraU p, q (,H,X ) associated with a nonstandardR-matrix with two deformation parameters (p, q) is studied and, in particular, its universal -matrix is derived using Reshetikhin's method. Explicit construction of the (p, q)-dependent nonstandardR-matrix is obtained through a coloured generalized boson realization of the universal -matr...

A finite-dimensional matrix representation of the Jackson $q$-differential operator $D_q$, defined by $D_qf(x)$ $=$ $(f(qx)-f(x))/(x(q-1))$, is written down following Calogero. Such a representation of $D_q$ should have applications in $q$-analysis leading to the corresponding extensions of the numerous results of Calogero's work.

Using the Fronsdal-Galindo formula for the exponential mapping from the quantum algebra $U_{p,q}(gl(2))$ to the quantum group $GL_{p,q}(2)$, we show how the $(2j+1)$-dimensional representations of $GL_{p,q}(2)$ can be obtained by `exponentiating' the well-known $(2j+1)$-dimensional representations of $U_{p,q}(gl(2))$ for $j$ $=$ $1,{3/2},... $; $j$...

The quantum theory of charged-particle beam transport through a magnetic lens system with a straight optic axis, at the level of single-particle dynamics and disregarding spin (or, when nonzero, assuming it to be an independent spectator degree of freedom), is presented, based on the Schroedinger and Klein-Gordon equations in a form suitable for an...

Application of the Wigner phase space distribution for studying the quantum mechanics of charged particle beam transport through electromagnetic optical systems should provide a natural link between the classical and the quantum descriptions. In this context, the relation between the transformation of the Wigner function of a charged particle opti...

Using the technique developed by Fronsdal and Galindo (Lett. Math. Phys. 27 (1993) 57) for studying the Hopf duality between the quantum algebras $Fun_{p,q}(GL(2))$ and $U_{p,q}(gl(2))$, the Hopf structure of $U_{p,q}(gl(1|1))$, dual to $Fun_{p,q}(GL(1|1))$, is derived and the corresponding universal ${\cal T}$-matrix of $Fun_{p,q}(GL(1|1))$, embod...

A quantum algebra $U_{p,q}(\zeta ,H,X_\pm )$ associated with a nonstandard $R$-matrix with two deformation parameters$(p,q)$ is studied and, in particular, its universal ${\cal R}$-matrix is derived using Reshetikhin's method. Explicit construction of the $(p,q)$-dependent nonstandard $R$-matrix is obtained through a coloured generalized boson real...

We consider the algebra $R$ generated by three elements $A,B,H$ subject to three relations $[H,A]=A$, $[H,B]=-B$ and $\{A,B\}=f(H)$. When $f(H)=H$ this coincides with the Lie superalgebra $osp(1/2)$; when $f$ is a polynomial we speak of polynomial deformations of $osp(1/2)$. Irreducible representations of $R$ are described, and in the case $\deg(f)...

We consider the algebra $R$ generated by three elements $A,B,H$ subject to three relations $[H,A]=A$, $[H,B]=-B$ and $\{A,B\}=f(H)$. When $f(H)=H$ this coincides with the Lie superalgebra $osp(1/2)$; when $f$ is a polynomial we speak of polynomial deformations of $osp(1/2)$. Irreducible representations of $R$ are described, and in the case $\deg(f)...

Using the technique developed by Fronsdal and Galindo for studying the Hopf duality between the quantum algebras Fun(sub p,q)(GL(2)) and U(sub p,q)(gl(2)), the Hopf structure of U(sub p,q)(gl(1/1)), dual to Fun(sub p,q)(GL(1/1)), is derived and the corresponding universal (Tau)-matrix of Fun(sub p,q)(GL(1/1)), embodying the suitably modified expone...

Lie algebraic approach to the classical theory of charged particle beam dynamics is well known (see,e.g., E. Forest and K. Hirata, KEK Report 92-12 and references therein). We formulate the quantum theory of charged particle beam dynamics, at the single particle level, using a similar Lie algebraic framework. Examples of applications to magnetic le...

A general frame-work for the study of relativistic electron beam transport through electromagnetic lens systems based on the Dirac equation, at the level of single particle dynamics, has been under development recently (R. Jagannathan, R. Simon, E.C.G. Sudarshan and N. Mukunda, Phys. Lett. A134 (1989) 457; R. Jagannathan, Phys. Rev. A42 (1990) 667...

DOI:https://doi.org/10.1103/PhysRevA.44.7856

The quantum theory of a magnetic electron lens with rotational symmetry about its straight optic axis has recently been studied entirely on the basis of the Dirac equation [R. Jagannathan, R. Simon, E. C. G. Sudarshan, and N. Mukunda, Phys. Lett. A 134, 457 (1989)]. Following the same type of algebraic approach, the present paper elaborates on the...

A quantum theory of magnetic electron lenses based on a convenient formulation of the Dirac theory is outlined. It is shown that the passage from the conventional scalar theory to the spinor theory can be accomplished through a simple algebraic rule in analogy with the passage from scalar to vector light optics.

Using the formalism of dynamical maps it is shown that if a quantum measurement process is to be described in terms of a non-negative phase-space distribution obtained by smoothing the Wigner distribution of the quantum state, then the smoothing kernel characterizing the measuring apparatus cannot be an arbitrary Wigner distribution.

Without Abstract

A finite-dimensional analog of Weyl's formulation of quantum kinematics of a physical system through irreducible Abelian groups of unitary ray rotations in system space offers many possibilities for the quantum mechanics of confined particles. This paper is devoted to the expansion of the recently developed framework of such Weylian finite-dimensio...

The finite-dimensional quantum mechanics (FDQM) based on Weyl’s form of Heisenberg’s canonical commutation relations, developed for the case of one-dimensional space, is extended to three-dimensional space. This FDQM is applicable to the physics of particles confined to move within finite regions of space and is significantly different from the cur...

As an example of the important role of number theory in certain physical problems, the number theoretical aspects of the socalled self-duality conditions for spin-systems defined on a group manifold are analysed in the special case when the underlying group is a generalised Clifford group.

Some graded Lie algebras with the Lie algebras so(3), so(2,1), so(4), so(3,1), and so(2,2) as their Bose sectors are realized in terms of bosons, para‐bosons and certain bilinear combinations of bosons and fermions.

This paper analyzes the possible implications of interpreting the finitedimensional representations of canonically conjugate quantum mechanical position, and momentum operators of a particle consistent with Weyl's form of Heisenberg's commutation relation as the actual position, and momentum operators of the particle when it is confined to move wit...

Some graded Lie algebras with the Lie algebras so(3), so(2,1 ), so(4), so(3,1), and so(2,2) as their Bose sectors are realized in terms of bosons, para-bosons and certain bilinear combinations of bosons and fermions.