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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 $\phi_T(N) = N/(1+(q-1)(N-1))$. In a Bargmann-like representation of this deformed oscillator the annihilation operator $a$ corresponds to a deformed derivative with the Tsallis $q$-exponential functions as its eigenfunctions, and the Tsallis $q$-exponential functions become the coherent states of the deformed oscillator. When $q = 2$ these deformed oscillator coherent states correspond to states known variously as phase coherent states, harmonious states, or pseudothermal states. Further, when $q = 1$ this deformed oscillator is a canonical boson oscillator, when $1 < q < 2$ its ground state energy is same as for a boson and the excited energy levels lie in a band of finite width, and when $q \longrightarrow 2$ it becomes a two-level system with a nondegenerate ground state and an infinitely degenerate excited state., Comment: Third version in which three more references have been added. To appear in International Journal of Theoretical Physics

It is possible to apply quantum methodologies to the matrix-representation of Maxwell’s equations. This results in a unified formalism of light beam optics and light polarization, accompanied with wavelength-dependent contributions.

The scalar optics is based on a Hamiltonian derived using the Fermat's principle of least time. The same Hamiltonian can now be derived from the Maxwell equations. The Helmholtz equations has a striking mathematical similarity to the Klein–Gordon equation for the relativistic spin-0 particle. It is possible to make use of this strong similarity through quantum techniques and develop an alternative to the traditional approaches. The non-traditional formalism of Helmholtz optics reproduces the traditional results as expected. Moreover, it leads to the wavelength-dependent modifications of the paraxial as well as the aberrating behavior. To illustrate the formalism, we consider the propagation of light in optical fibers in substantial detail. We also consider the propagation of light through a graded index slab.

A unified formalism of light beam optics and light polarization is presented. The starting point of our formalism is an exact eight-dimensional matrix-representation of Maxwell's equations in an inhomogeneous medium, which is presented in detail. The beam-optical Hamiltonians are derived without any specifications on the varying refractive index. The new formalism generalizes the traditional and non-traditional treatments of Helmholtz optics. As for the light polarization, the elegant Mukunda–Simon–Sudarshan rule for transition from scalar optics to vector wave optics is obtained as the paraxial limit of the general formalism presented here. The new formalism is a suitable candidate to extend the traditional theory of polarization beyond the paraxial approximation. The unified formalism light beam-optics and light polarization advances the Hamilton's optical-mechanical analogy into the wavelength-dependent regime.

We explore the ways to linearize the wave equations. Special emphasis is paid to the Klein–Gordon equation for a spin-0 relativistic particle and the Helmholtz equation governing scalar optics. Owing to the mathematical similarity, both of these equations are linearized using the Feshbach–Villars procedure. Maxwell's equations are linear but coupled and constrained. So, a matrix representation is presented. New formalisms of beam optics are presented using the linearized form of the Helmholtz equations and the Dirac- like matrix form of Maxwell's equations respectively. It is shown that the matrix formulation of the Maxwell optics naturally leads to a unified treatment of beam optics (including aberrations to all orders) and light polarization, from a single parent Hamiltonian. The non-traditional treatments using quantum methodologies lead to wavelength-dependent modifications of the traditional prescriptions. In the limit of low wavelength, the non-traditional prescriptions of both Helmholtz optics and Maxwell optics presented here reproduce the ‘Lie algebraic formalism of light beam optics’. The accompanying machinery of the Foldy–Wouthuysen transformation technique is described. From the new prescriptions of light beam optics, it is seen that the Hamilton's optical-mechanical analogy persists in the wavelength-dependent regime.

Hamilton's optical–mechanical analogy between the trajectory of material particles in potential fields and the path of light rays in media with continuously variable refractive index played an important role in the development of Schrodinger's wave mechanics. We shall examine how the Hamilton's analogy between charged-particle beam optics and light beam optics is extended to the wavelength-dependent regime. Brief accounts of the quantum prescriptions of charged-particle beam optics along with the non-traditional prescriptions of light beam optics are also presented. These prescriptions have been instrumental in seeing the analogy in the wavelength-dependent regime.

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 nonrelativistic Schr\"{o}dinger equation. It is found that the quantum transfer map contains the classical transfer map as the main part and there are tiny quantum correction terms. The negligibly small quantum corrections explain the remarkable success of classical mechanics in charged particle beam optics., Comment: 21 pages

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 obtained in terms of the well known fundamental solutions of its paraxial equation of motion. In the case of lenses with the power law model for $B(z)$ the well known fundamental solutions of the paraxial equations, obtained by solving the differential equation, are constructed using the Peano-Baker series also. Quantum mechanics of aberrations is discussed briefly. Role of quantum uncertainties in aberrations, and in the nonlinear part of the equations of motion for a nonparaxial beam, is pointed out. The main purpose of this article is to understand the quantum mechanics of electron beam optics though the influence of quantum effects on the optics of present-day electron beam devices might be negligible., 40 pages. Two equations, (24) and (25), have been added. To appear in Optik

It is well-known that the Helmholtz equation describing scalar optics has a striking mathematical similarity with the Klein–Gordon equation of relativistic quantum mechanics. Exploiting this similarity, quantum methodologies are applied to the Helmholtz equation leading to a new formalism of scalar optics. Paraxial and aberrating Hamiltonians are derived for a general spatially varying refractive index. The beam-optical Hamiltonians thus derived give rise to wavelength-dependent contributions. Example of the graded-index medium is covered in detail. Explicit formulae relating the incident and emergent light rays for the general case of a thick lens are also presented.

The way to generalize scalar to wave optics, thus including polarization in the treatment consistent with the Maxwell equations was shown by Mukunda, Simon and Sudarshan for paraxial systems, based on a group theoretical analysis. Here, the Mukunda–Simon–Sudarshan (MSS) theory for the passage from scalar to vector optics is derived by casting the basic formalism in a framework very similar to the Dirac electron theory. The resulting formalism is suitable for extending the MSS-theory beyond the paraxial approximation.

A unified formalism of light beam optics and light polarization is presented. The starting point of our formalism is an exact matrix-representation of Maxwell's equations in an inhomogeneous medium, which is presented in detail. The beam-optical Hamiltonians are derived without any specifications on the form of the varying refractive index. The new formalism generalizes the traditional and non-traditional prescriptions of Helmholtz optics. As for the light polarization, the elegant Mukunda–Simon–Sudarshan rule for transition from scalar to vector wave optics is obtained as the paraxial limit of the general formalism presented here. The new formalism is a suitable candidate to extend traditional theory of polarization beyond the paraxial approximation. The unified formalism of light beam-optics and light polarization further strengthens the Hamilton's optical-mechanical analogy, particularly in the wavelength-dependent regime.

An exact treatment of beam optics, starting ab initio from the Maxwell's equations is presented. The starting point of this approach is a matrix representation of the Maxwell's equation in a medium with varying permittivity and permeability. Formal expressions are obtained for the paraxial and leading order aberrating Hamiltonians, without making any assumptions on the form of the varying refractive index. We derive the wavelength-dependent contributions at each order, starting with the lowest-order paraxial Hamiltonian. To illustrate the general theory, we consider the computations of the transfer maps for an axially symmetric graded-index medium. For this system, in the traditional approaches, one gets only six aberrations. In our formalism, we get all the nine aberrations permitted by the axial symmetry. The six aberrations coefficients of the traditional approaches get modified by the wavelength-dependent contributions and the remaining three are pure wavelength-dependent. It is very interesting to note that apart from the wavelength-dependent modifications of the aberrations, this approach also gives rise to the image rotation. The present study is the generalization of the traditional and non-traditional prescription of Helmholtz optics. In the low wavelength limit our formalism reproduces the Lie algebraic formalism of optics. The present study further strengthens the close analogies between the various prescriptions of light optics and charged-particle optics. The new formalism presented here, provides a natural framework to study beam-optics and polarization in a unified manner.

The classical treatments have been successful in designing numerous charged-particle devices. It is natural to develop a quantum prescription, since all systems are fundamentally quantum mechanical in nature. The quantum theory leads to new insights accompanied with wavelength-dependent contributions. The action of a magnetic quadrupole is derived from the Dirac equation.

The order of the set of equivalent resistances, A(n) of n equal resistors combined in series and in parallel has been traditionally addressed computationally, for n up to 22. For larger n there have been constraints of computer memory. Here, we present an analytical approach using the Farey sequence with Fibonacci numbers as its argument. The approximate formula, A(n) ~ 2.55^n, obtained from the computational data up to n = 22 is consistent with the strict upper bound, A(n) ~ 2.618^n, presented here. It is further shown that the Farey sequence approach, developed for the A(n) is applicable to configurations other than the series and/or parallel, namely the bridge circuits and non-planar circuits. Expressions describing set theoretic relations among the sets A(n) are presented in detail. For completeness, programs to generate the various integer sequences occurring in this study, using the symbolic computer language MATHEMATCA, are also presented.

I read your Japan special report with a keen interest. My first visit to Japan was in March 1994 to attend the JSPS-KEK International Spring School: High Energy Ion Beams – Novel Beam Techniques and their Applications.

Matrix representations of Maxwell's equations have a striking resemblance to the Dirac equation. We exploit this resemblance to build a beam optics formalism from an exact eight-dimensional matrix representation of the Maxwell equations. The Foldy–Wouthuysen iterative diagonalization technique is employed to obtain a Hamiltonian description for a system with varying refractive index. The beam-optical Hamiltonian is shown to have a wavelength-dependent part, resulting in the wavelength-dependent modifications of light beam optics. The present study is the generalization of the traditional and non-traditional prescription of Helmholtz optics.

The Helmholtz wave equation is linearized using the Feshbach–Villars procedure used for linearizing the Klein–Gordon equation, based on the close algebraic analogy between the Helmholtz equation and the Klein–Gordon equation for a spin-0 particle. The Foldy–Wouthuysen iterative diagonalization technique is then applied to the linearized Helmholtz equation to obtain a Hamiltonian description for a system with varying refractive index. The Hamiltonian has a wavelength-dependent part absent in the traditional descriptions. Besides reproducing all the traditional quasi-paraxial terms, our method leads to additional contributions dependent on the wavelength. Applied to the axially symmetric graded-index fiber, this method results in wavelength-dependent modifications of the paraxial behavior and the aberration coefficients to all orders. Explicit expression for the modified aberration coefficients to the third order are presented. Sixth- and eighth-order Hamiltonians are also presented.

The understanding and manipulation of beam optics and light polarization are cornerstones of optical technology. During 2016, we outlined new quantum formalisms for both the Helmholtz equation and for Maxwell’s equations that enable the description of beam optics and light polarization in a unified framework.

The International Large Detector (ILD) is a concept for a detector at the International Linear Collider, ILC. The ILC will collide electrons and positrons at energies of initially 500 GeV, upgradeable to 1 TeV. The ILC has an ambitious physics program, which will extend and complement that of the Large Hadron Collider (LHC). A hallmark of physics at the ILC is precision. The clean initial state and the comparatively benign environment of a lepton collider are ideally suited to high precision measurements. To take full advantage of the physics potential of ILC places great demands on the detector performance. The design of ILD is driven by these requirements. Excellent calorimetry and tracking are combined to obtain the best possible overall event reconstruction, including the capability to reconstruct individual particles within jets for particle ow calorimetry. This requires excellent spatial resolution for all detector systems. A highly granular calorimeter system is combined with a central tracker which stresses redundancy and efficiency. In addition, efficient reconstruction of secondary vertices and excellent momentum resolution for charged particles are essential for an ILC detector. The interaction region of the ILC is designed to host two detectors, which can be moved into the beam position with a push-pull scheme. The mechanical design of ILD and the overall integration of subdetectors takes these operational condition s into account. The International Large Detector (ILD) is a concept for a detector at the International Linear Collider, ILC. The ILC will collide electrons and positrons at energies of initially 500 GeV, upgradeable to 1 TeV. The ILC has an ambitious physics program, which will extend and complement that of the Large Hadron Collider (LHC). A hallmark of physics at the ILC is precision. The clean initial state and the comparatively benign environment of a lepton collider are ideally suited to high precision measurements. To take full advantage of the physics potential of ILC places great demands on the detector performance. The design of ILD is driven by these requirements. Excellent calorimetry and tracking are combined to obtain the best possible overall event reconstruction, including the capability to reconstruct individual particles within jets for particle ow calorimetry. This requires excellent spatial resolution for all detector systems. A highly granular calorimeter system is combined with a central tracker which stresses redundancy and efficiency. In addition, efficient reconstruction of secondary vertices and excellent momentum resolution for charged particles are essential for an ILC detector. The interaction region of the ILC is designed to host two detectors, which can be moved into the beam position with a push-pull scheme. The mechanical design of ILD and the overall integration of subdetectors takes these operational conditions into account.

An analysis of the decay τ-→π-π0vτ has been performed using the ARGUS detector at the DORIS II storage ring. The branching ratio has been determined to be Br(τ-→π-π0vτ=(22.6±0.4±0.9)%. The shape of the π-π0 invariant mass spectrum is found to be in good agreement with the predictions obtained using the conserved vector current (CVC) hypothesis, suggesting that the π-π0 system is produced in aJP=1− state. An analysis of the measured decay angular distribution of the pions with respect to the flight direction of the π-π0 system demonstrates the vector nature of the coupling at the τvτ vertex. With the assumption of zerovτ mass thevτ spin has been shown to be\(J_{v_\tau}= \tfrac{1}{2} \).

In the paraxial regime of three-dimensional optics, two evolution Hamiltonians are equivalent when one can be transformed to the other modulo scale by similarity through an optical system. To determine the equivalence sets of paraxial optical Hamiltonians one requires the orbit analysis of the algebra sp(4, R) of 4 x 4 real Hamiltonian matrices. Our strategy uses instead the isomorphic algebra so(3, 2) of 5 x 5 matrices with metric (+1, +1, +1, -1, -1) to find four orbit regions (strata), six isolated orbits at their boundaries, and six degenerate orbits at their common point. We thus resolve the degeneracies of the eigenvalue classification.

In their most recent article, Grado-Caffaro et al. have addressed the question of the ‘photon velocity’. They have expressed the photon velocity in terms of the wavefunctions of the Klein–Gordon equation (Grado-Caffaro and Grado-Caffaro [4] ). In this note, we closely follow their work and explicitly obtain the photon velocity using the free solutions of the Klein–Gordon equation. It is shown that the plane wave solutions give rise to six possible values of the photon velocity. Two of these solutions are the most expected ( v = ± c ). The remaining four solutions, the real pair ±0.786c and the imaginary pair ±1.272ic are difficult to comprehend.

A general procedure for construction of the formalism of quantum beam optics for any particle is reviewed. The quantum formalism of spin-1/2 particle beam optics is presented starting {\em ab initio} with the Dirac equation. As an example of application the case of normal magnetic quadrupole lens is discussed. In the classical limit the quantum formalism leads to the well-known Lie algebraic formalism of classical particle beam optics., ReVTeX 06 pages, in Proc. of the 18th Advanced ICFA Beam Dynamics Workshop on Quantum Aspects of Beam Physics (QABP) (15-20 October 2000, Capri, ITALY) Editor: Pisin Chen (World Scientific, Singapore, 2002) http://www.pd.infn.it/~khan/ and http://www.imsc.ernet.in/~jagan/

An interpretation of the formation of halo in accelerators based on quantum-like theory by a diffraction model is given in terms of the transversal beam motion. Physical implications of the longitudinal dynamics are also examined, LaTeX2e, 4 pages, Nucl. Instrum. Methods. A 464 (2001) 461-464. http://www.pd.infn.it/~khan/ and http://www.pd.infn.it/~pusterla/

A summary of results on synchrotron radiation is presented along with notes on its properties and applications. Quantum aspects are briefly mentioned. Synchrotron radiation facilities are described briefly with a detailed coverage to the accelerator programmes in India. The relocated and other upcoming synchrotrons are also described in some detail., Comment: ReVTeX, 07 pages, in Resonance, 6, No. 11, pp. 77-86 (November 2001). http://www.pd.infn.it/~khan/ and http://www.imsc.ernet.in/~jagan/

An interpretation of the formation of halo in accelerators based on quantum-like theory by a diffraction model is given in terms of the transversal beam motion. Physical implications of the longitudinal dynamics are also examined., 9 pages. (Communicated) http://www.pd.infn.it/~khan/ and http://www.pd.infn.it/~pusterla/

An interpretation of the ``halo puzzle'' in accelerators based on quantum-like diffraction is given. Comparison between this approach and the others based on classical mechanics equations is exhibited., Comment: 6 pages. To appear in Proceedings of the 1999 Particle Accelerator Conference (PAC99), 29 March -- 02 April 1999, New York City, Eds: A. Luccio and W. MacKay. http://www.pd.infn.it/~khan/ and http://www.pd.infn.it/~pusterla/

Present understanding of accelerator optics is based mainly on classical mechanics and electrodynamics. In recent years quantum theory of charged-particle beam optics has been under development. In this paper the newly developed formalism is outlined., 7 pages. To appear in Proceedings of the 1999 Particle Accelerator Conference (PAC99), 29 March -- 02 April 1999, New York City, Eds: A. Luccio and W. MacKay. http://www.pd.infn.it/~khan/

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 Schr\"odinger and Klein-Gordon equations in a form suitable for analyzing the paraxial and aberration aspects in a systematic way using a Lie algebraic approach. In the classical limit, the well known Lie algebraic treatment of the corresponding classical theory is obtained. As examples, quadrupole and axially symmetric magnetic lenses are considered. An extension of the theory to the cases of electrostatic and other electromagnetic lens systems is outlined. This work is complementary to an already known similar approach to the spinor electron optics based on the Dirac equation and provides the corresponding framework when the optics of charged particles, with or without spin, is described with scalar wave functions in the nonrelativistic and relativistic situations.