$\mathfrak{D}$ -Differentiation in Hilbert Space and the Structure of Quantum Mechanics

An appropriate kind of curved Hilbert space is developed in such a manner that it admits operators of - and -differentiation, which are the analogues of the familiar covariant and D-differentiation available in a manifold. These tools are then employed to shed light on the space-time structure of Quantum Mechanics, from the points of view of the Feynman ‘path integral’ and of canonical quantisation. (The latter contains, as a special case, quantisation in arbitrary curvilinear coordinates when space is flat.) The influence of curvature is emphasised throughout, with an illustration provided by the Aharonov-Bohm effect.     

\mathcal{A} -invariance: An Axiomatic Approach to Quantum Relativity

The sort of approach claimed by the title of this article is realizable, at least, within the framework of ADG where we do not assume any “spacetime” supplying the dynamics we employ. The latter classical type of argument can naturally be included herewith along with its concomitant impediments that are emanated therefrom and are essentially “absorbed”, technically speaking, by the proposed mechanism. So our approach, being “manifoldless” (thence, no smoothness, in the standard sense) does not contain any such issue, as before, according to the very definitions, being thus “singularities”-free. As a consequence, the equations that one would be able to formulate within the present set-up will be, by the very essence of the matter, already the quantized ones. Dedicated to Professor Rafael D. Sorkin on the occasion of his 60th birthday with much friendship and recognition of his creative pursuit in theoretical physics.     

Improving anti-reflectivity and laser damage threshold of \hbox {SiO}_{2}/\hbox {ZrO}_{2} thin films by laser shock peening at 1064 nm

In this study, anti-reflection (AR) \(\hbox {SiO}_{2}/ \hbox {ZrO}_{2}\) thin films with 3-layers were designed and fabricated by the essential Macleod software and physical vapor deposition, respectively. In order to improve the optical and physical properties of the prepared samples, laser shock peening (LSP) technique was applied. For this purpose, an Argon Fluoride Excimer laser \((\lambda =193 \,\text {nm})\) with 110 and 240 mJ energies and 1 Hz frequency at different pulses was used. The effect of LSP method in improving transmissions and laser damage thresholds of the prepared samples was proved by using UV–Vis–IR spectroscopy in the wavelength range of 400–1200 nm and international standard ISO11254 at 1064 nm. In addition, scanning electron microscopy was used to check the effect of applying LSP.     

Quantum Codes from Repeated-Root Cyclic and Negacyclic Codes of Length 4ps Over �� p m $\mathbb {F}_{p^{m}}$

International Journal of Theoretical Physics - Let $\mathbb {F}_{p^{m}}$ be a finite field with pm elements for an odd prime p and a positive integer m. In this paper, we aim to construct...     

$\mathfrak{D}$-Differentiation in Hilbert Space and the Structure of Quantum Mechanics Part II: Accelerated Observers and Fictitious Forces

We investigate a possible form of Schrödinger’s equation as it appears to moving observers. It is shown that, in this framework, accelerated motion requires fictitious potentials to be added to the original equation. The gauge invariance of the formulation is established. The example of accelerated Euclidean transformations is treated explicitly, which contain Galilean transformations as special cases. The relationship between an acceleration and a gravitational field is found to be compatible with the picture of the ‘Einstein elevator’. The physical effects of an acceleration are illustrated by the problem of the uniformly-accelerated harmonic oscillator.     

$$\mathfrak{s}\mathfrak{u} $$ q(2)-Invariant Schrödinger Equation of the Three-Dimensional Harmonic Oscillator

We propose a q-deformation of the -invariant Schrödinger equation of a spinless particle in a central potential, which allows us not only to determine a deformed spectrum and the corresponding eigenstates, as in other approaches, but also to calculate the expectation values of some physically-relevant operators. Here we consider the case of the isotropic harmonic oscillator and of the quadrupole operator governing its interaction with an external field. We obtain the spectrum and wave functions both for and generic , and study the effects of the q-value range and of the arbitrariness in the Casimir operator choice. We then show that the quadrupole operator in l=0 states provides a good measure of the deformation influence on the wave functions and on the Hilbert space spanned by them.     

The Quantum Superalgebra {\mathfrak{osp}_{q}(1|2)} and a q-Generalization of the Bannai–Ito Polynomials

The Racah problem for the quantum superalgebra \({\mathfrak{osp}_{q}(1|2)}\) is considered. The intermediate Casimir operators are shown to realize a q-deformation of the Bannai–Ito algebra. The Racah coefficients of \({\mathfrak{osp}_q(1|2)}\) are calculated explicitly in terms of basic orthogonal polynomials that q-generalize the Bannai–Ito polynomials. The relation between these q-deformed Bannai–Ito polynomials and the q-Racah/Askey–Wilson polynomials is discussed.     

Hidden local symmetry and the reaction e+e- → \pi^{+}_{} \pi^{-}_{} \pi^{+}_{} \pi^{-}_{} at energies \sqrt{{s}} ⩽ 1 GeV

Based on the generalized hidden local symmetry as the chiral model of pseudoscalar, vector, and axial vector mesons, the excitation curve of the reaction e ⁺ e ^- → $ \pi^{+}_{}$ $ \pi^{-}_{}$ $ \pi^{+}_{}$ $ \pi^{-}_{}$ is calculated for energies in the interval 0.65 ? $ \sqrt{{s}}$ ? 1 GeV. The theoretical predictions are compared to available data of CMD-2 and BaBaR. It is shown that the inclusion of heavy isovector resonances ρ(1450) and ρ(1700) is necessary for reconciling calculations with the data. It is found that at $ \sqrt{{s}}$ ≈ 1 GeV the contributions of the above resonances are much larger, by a factor of 30, than the ρ(770) one, and amount to a considerable fraction ～ 0.3-0.6 of the latter at $ \sqrt{{s}}$ ～ m _ρ .     

A 480 ms isomeric 29/2−-state of the(p_{1/2}^{ - 2} )_{0^ + } f_{5/2}^{ - 1} (i_{13/2}^{ - 2} )_{12^ + } configuration in203Pb

A new, 480 ms, 29/2^? isomeric level has been found in²⁰³Pb at an excitation energy of 2950.1 keV by bombarding²⁰⁴Hg with α-particles in the energy range 45–55 MeV using the Stockholm 225-cm cyclotron. This 29/2^? state is suggested to be mainly due to the configuration (p ₁ ^2/?2 f ₅ ^2/?1 i ₁₃ ^2/?2 )₁₂. The 29/2^? state decays predominantly by a 153.4 keV M2 transition to a 23/2^? level and by a 1027.5 keV M4 transition to a 21/2⁺ level, followed by two E2 transitions of energies 258.6 keV and 838.7 keV, respectively, to the previously known 13/2⁺, 6.4 s isomeric level. The decay scheme of the 29/2^? isomeric state is based on experimental information obtained from total and delayedγ-ray intensities,γγ-coincidences, excitation functions, lifetime and delayed conversion electron measurements. The presence of the 29/2^? level confirms an essential and expected feature of the shell model for five neutron holes added to the²⁰⁸Pb-core.     

Optical properties of $$\hbox {TlGa}(\hbox {S}_{{x}}\hbox {Se}_{1-x})_{2}$$ layered mixed crystals $$(0 \le x \le 1)$$(0≤x≤1): Absorption edge and photoluminescence study at $$T = 10\hbox { K}$$T=10K

Industrial demand for neutrons constrains careful energy measurements. Elastic scattering of monoenergetic \(\alpha \)-particles from neutron collision enables neutron energy measurement by calculating the amount of deviation from the position where collision takes place. The neutron numbers with specific energy is obtained by counting the number of \(\alpha \)-particles in the corresponding location on the charged particle detector. Monte Carlo simulation and COMSOL Multiphysics5.2 are used to account for one-to-one collision of neutrons with \(\alpha \)-particles.     

The structures and properties of FeSin/FeSi\hbox{$_{\mathsf{n}}^{+}$}+n/FeSi\hbox{$_{\mathsf{n}}^{-}$}−n (n = 1 ~ 8) clusters

The geometry, stability, and electronic properties of iron-doped silicon clusters FeSi _n /FeSi\hbox{$_{n}^{+}$}+n/FeSi\hbox{$_{n}^{-}$}?n (n = 1 ~ 8) have been systematically investigated using the density functional theory (DFT) approach at the B3LYP/6-311+G* level. Our results show that the ground state structures of FeSi _n /FeSi\hbox{$_{n}^{+}$}+n/FeSi\hbox{$_{n}^{-}$}?n change from planar to three-dimensional for n > 3. Bipyramidal structures, or their face-capped isomers, are favored for the larger clusters. For neutral FeSi _n clusters, their ground state structures are the trigonal, tetragonal, capped tetragonal, capped pentagonal, and combined tetragonal bipyramids for n = 4 ~ 8, respectively. The lowest-energy structures of the anionic FeSi\hbox{$_{n}^{-}$}?n clusters essentially retain similar frameworks to their neutral counterparts, while those of the cationic FeSi\hbox{$_{n}^{+}$}+n clusters are significantly deformed; this is confirmed by their calculated ionization potential and electronic affinity values. For most of the stable structures, the spin electronic configurations are s = 1 or 2 for neutral FeSi _n , s = 3/2 or 5/2 for ionic FeSi\hbox{$_{n}^{+}$}+n/FeSi\hbox{$_{n}^{-}$}?n. The average binding energy values generally increase with increasing cluster size, indicating the clusters can continue to gain energy during the growth process. Fragmentation and second-order energy peaks (maxima) are found at n = 2, 5, and 7 for FeSi _n /FeSi\hbox{$_{n}^{-}$}?n, n = 4 and 6 for FeSi\hbox{$_{n}^{+}$}+n, suggesting that these clusters possess higher relative stability. Furthermore, the HOMO-LUMO gap values show that anionic FeSi\hbox{$_{n}^{-}$}?n have greater chemical reactivity than cationic FeSi\hbox{$_{n}^{+}$}+n and neutral FeSi _n , except when n = 7.     

Exact calculation of O(α) QED corrections for f_1 \overline {f_1 } \uparrow f_2 \overline {f_2 } processes induced by polarized incident particles

For $f_1 \overline {f_1 } \uparrow f_2 \overline {f_2 }$ (γ) processes induced by polarized incident particles, exact covariant expressions for the energy spectrum of one of the emitted fermions, for the total cross section, and for the polarization asymmetry are obtained in theα³ order of QED.     

Cool white light emission on Ca3−(l+n)MgSi2O8:\mathrm{Ce}_{\mathrm{l}}^{3+}, \mathrm{Eu}_{n}^{2+} phosphors and analysis of energy transfer mechanism

Rare earth ions (Ce³⁺, Eu²⁺) activated Ca₃MgSi₂O₈ (CMSO) phosphors have been synthesized using solid-state reaction method in 95%N₂+5%H₂ reduction atmosphere at elevated temperatures by varying Eu²⁺ concentration from 0.0075 to 0.0300 at the fixed Ce_0.03 concentration to study their photoluminescence (PL) properties. An energy transfer occurs from Ce³⁺ to Eu²⁺ through a significant overlap of Eu²⁺ excitation spectrum with Ce³⁺ emission spectrum in CMSO, together with the systematic relative decrease and increase in emission bands of Ce³⁺ and Eu²⁺, respectively, have been observed. To support the phenomenon, diffuse reflectance spectra show various absorption levels corresponding to Ce³⁺, Eu²⁺, and/or mixture of both rare earth ions. An optimum emission was realized at 0.0150 of Eu²⁺ via. energy transfer from Ce³⁺ ion. By utilizing the principle of energy transfer, the critical distance (R _c ) between activator ions was found to be 18.64 Å. The CIE chromaticity coordinates measured on the Ca₃MgSi₂O₈:Ce³⁺, Eu²⁺ phosphors excited under ultraviolet (365 nm) source shows the values lie in cool white light region could be applied to solid state lighting.     

In memory of Alekse $$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l} $$ Sergeevich Nikiforov (March 28, 1927–October 1, 2002)

Chronicle

In memory of Alekse $\overset{\lower0.5em\hbox{$\overset{\lower0.5em\hbox{ Sergeevich Nikiforov (March 28, 1927–October 1, 2002)     

O(α s ) corrections to the polar angle dependence of the longitudinal spin–spin correlation asymmetry in e^{+}e^{-}\to q\bar{q}

We provide analytical results for the O(α _s ) corrections to the polar angle dependence of the longitudinal spin–spin correlation asymmetry in $e^{+}e^{-}\to q\bar{q}$ . For top quark pair production the O(α _s ) corrections to the longitudinal spin–spin asymmetry are strongly polar angle dependent and can amount up to ?4% in the q ²-range from above $t\bar{t}$ threshold up to $\sqrt{q^{2}}=1000\ \mbox {GeV}$ . The O(α _s ) radiative corrections to the correlation asymmetry are below ≈1% in the forward direction where the cross section is largest. In the $e^{+}e^{-}\to b\bar{b}$ case the O(α _s ) corrections reduce the asymmetry value from its m _b =0 value of ?100% to approximately ?96% for q ²-values around the Z peak and are practically independent of the value of the polar angle θ. This reduction can be traced to finite anomalous contributions from residual mass effects which survive the m _b →0 limit. We discuss the role of the anomalous contributions and the pattern of how they contribute to spin-flip and non-flip terms.     

Bannai–Ito algebras and the universal R-matrix of $$\pmb {\mathfrak {osp}}(1|2)$$osp(1|2)

Letters in Mathematical Physics - The Bannai–Ito algebra BI(n) is viewed as the centralizer of the action of $$\mathfrak {osp}(1|2)$$ in the n-fold tensor product of the universal algebra of...     

Coset space dimensional reduction and Wilson flux breaking of ten-dimensional $\mathcal{N}=1$ , E8 gauge theory

We consider a supersymmetric E ₈ gauge theory, defined in ten dimensions and we determine all four-dimensional gauge theories resulting from the generalized dimensional reduction à la Forgacs–Manton over coset spaces, followed by a subsequent application of the Wilson flux spontaneous symmetry-breaking mechanism. Our investigation is constrained only by the requirements that (i) the dimensional reduction leads to the potentially phenomenologically interesting, anomaly-free, four-dimensional E ₆, SO₁₀ and SU₅ GUTs and (ii) the Wilson flux mechanism makes use only of the freely acting discrete symmetries of all possible six-dimensional coset spaces. Supported by the EPEAEK II programme IRAKLEITOS. Partially supported by the NTUA programme for basic research “Karatheodoris” and the European Union’s RTN programme under contract MRTN-CT-2006-035505.     

The 3–3–1 Model with RH Neutrinos and Associated ZH Production at High-Energy ${\bf \textit{e}}^{+}{\bf \textit{e}}^{-}$ Collider

In the context of SU(3) _C ?? SU(3) _L ?? U(1) _X (3–3–1) model with right-handed neutrinos, we study the Higgsstrahlung process e ^?+? e ^???→ZH and calculate the cross section of this process at leading order. Our numerical results showed that the production cross sections for this process can be significantly large as \(M_{Z'}\approx \sqrt{s}\). With reasonable values of the Z′ mass M _Z′, Z′ exchange can generate large corrections to the cross sections of this process, which might be detected in the future high-energy linear e ^?+? e ^??? collider experiments.