Conjugate Pairs Research Articles

CONSTRUCTING QUANTUM ANGULAR MOMENTUM L3 IN SPECIFIC DIRECTION BY USING U(1) GROUP

The purpose of this work is to investigate the mathematical structure of finite quantum angular momentum in a specific direction L3 which can be constructed from the representation of the U(1) group. The angular momentum eigenstate is invariant under Abelian rotation symmetry. The character group of U(1) is constructed to show that there exists an additive unitary operator for the angular momentum eigenstate, and the rotation eigenstate is invariant under it. The Weyl relation is proved by showing that the angle and angular momentum L3 are the canonical conjugate pair of observables.

Algebraic fundamental groups of fake projective planes

Fundamental groups of fake projective planes fall into fifty distinct isomorphism classes, one for each complex conjugate pair. We prove that this is not the case for their algebraic fundamental groups: there are only forty-six isomorphism classes. We show that there are four pairs of complex conjugate pairs of fake projective planes that are \operatorname{Aut}(\mathbb{C}) -equivalent and hence have mutually isomorphic algebraic fundamental groups. All other pairs of algebraic fundamental groups are shown to be distinct through explicit finite étale covers. As a by-product, this provides the first examples of commensurable but nonisomorphic lattices in a rank one semisimple Lie group that have isomorphic profinite completions.

ВЛИЯНИЕ ТОЧЕК ПОВОРОТА НА ЗАДЕРЖКУ РЕШЕНИЯ ВБЛИЗИ НЕУСТОЙЧИВОГО ПОЛОЖЕНИЯ РАВНОВЕСИЯ

This paper considers a system of equations consisting of three first-order equations of fast variables and one equation of a slow variable. The first approximation matrix of the system of fast variables has different eigenvalues, with one pair of complex conjugates. The eigenvalues have zeros and they are turning points. The system has an equilibrium position, and the stability of the equilibrium position is lost at a certain value of the slow variable. The problem is set to study the delay in solving a system of fast variables near an unstable equilibrium position and the influence of turning points on the delay. The delay of the solution near an unstable position is proven and the influence of turning points is clarified. The proof used geometric constructions using the level line of harmonic functions, methods of successive approximations, Laplace, stationary phase, integration by parts and comparison of infinitesimals.

Variance-based uncertainty relations and entanglement amplification for particles constrained on a torus

We formulate the variance-based uncertainty relations (URs) via the Robertson’s inequality, for a 2-particle entangled system constrained on a torus and subject to a stationary magnetic field B→\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {\\vec {B}}$$\\end{document}. We explore the system’s parameter space and show that these new URs have field-tunable uncertainty bounds. Our analysis reveals that A→\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {\\vec {A}}$$\\end{document} (vector potential) induces a phase shift in the state, due to the Aharonov–Bohm effect, leading to a perturbed system dynamics which results in asymmetric product of variance (POV\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {POV}$$\\end{document}). Additionally, we give the critical range of A→\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {\\vec {A}}$$\\end{document} and B→\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {\\vec {B}}$$\\end{document} where the system acts as an entanglement amplifier; this amplification is also discussed under various geometric parameters. The possibility of reducing the POV\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {POV}$$\\end{document} of the conjugate pair [q, p] below the known benchmark value by the Generalized Uncertainty Relation (GUR) is also demonstrated. Finally, we check the susceptibility of state coherence to B→\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {\\vec {B}}$$\\end{document} by saturating the angular momentum uncertainty relation and identify the critical coherence value Bc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {B}}_c$$\\end{document} such that when B≠Bc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {B}}\ e {\\mathcal {B}}_c$$\\end{document}, the state decoheres.

Enhancing computation performance of rational approximation for frequency-dependent network equivalents with parallelism and complex vector fitting

This work examines two strategies for enhancing the rational approximation of Frequency-Dependent Network Equivalents (FDNE) using an 8-port FDNE featuring a frequency response marked by numerous peaks and valleys. Firstly, we employ Complex Vector Fitting (CVF), an alternative to the Vector Fitting (VF). CVF eliminates the constraint of complex conjugate pairs and was originally conceived for modeling baseband equivalents through scattering parameters. The implications of CVF for admittance (or impedance) matrix synthesis have not yet been previously reported in specialized literature. To enhance code performance and remove dependence on commercial software such as MATLAB®, VF and CVF were implemented in the C-language, utilizing a low-level linear algebra package and exploiting parallelism. We evaluated performance by varying the model order, number of ports, and frequency samples. The results confirm the feasibility of our approach, prompting a more in-depth exploration of the potential benefits regarding FDNE realization.

A Direct Entropic Approach to the Thermal Balance of Spontaneous Chemical Reactions.

When working with, and learning about, the thermal balance of a chemical reaction, we need to consider two overlapping but conceptually distinct aspects: one relates to the process of reallocating entropy between reactants and products (because of different specific entropies of the new substances compared to those of the old), and the other to dissipative processes. Together, they determine how much entropy is exchanged between the chemicals and their environment (i.e., in heating and cooling). By making explicit use of (a) the two conjugate pairs chemical amount (i.e., amount of substance) and chemical potential, and entropy and temperature, respectively, (b) the laws of balance of amount of substance on the one hand and entropy on the other, and (c) a generalized approach to the energy principle, it is possible to create both imaginative and formal conceptual tools for modeling thermal balances associated with chemical transformations in general and exothermic and endothermic reactions in particular. In this paper, we outline the concepts and relations needed for a direct approach to chemical and thermal dynamics, create a model of exothermic and endothermic reactions, including numerical examples, and discuss how to relate the direct entropic approach to traditional models of these phenomena.

Finite Groups All of Whose Subgroups are P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {P}$$\\end{document}-Subnormal or TI\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{TI}\\,}}$$\\end{document}-Subgroups

Let P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {P}$$\\end{document} be the set of all prime numbers. A subgroup H of a finite group G is said to be P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {P}$$\\end{document}-subnormal in G if there exists a chain of subgroups H=H0⊆H1⊆⋯⊆Hn-1⊆Hn=G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} H = H_0 \\subseteq H_1 \\subseteq \\cdots \\subseteq H_{n-1} \\subseteq H_n = G \\end{aligned}$$\\end{document}such that either Hi-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{i-1}$$\\end{document} is normal in Hi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_i$$\\end{document} or |Hi:Hi-1|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|H_i{:}\\, H_{i-1}|$$\\end{document} is a prime number for every i=1,2,…,n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$i = 1, 2, \\ldots , n$$\\end{document}. A subgroup H of G is called a TI\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{TI}\\,}}$$\\end{document}-subgroup if every pair of distinct conjugates of H has trivial intersection. The aim of this paper is to give a complete description of all finite groups in which every non-P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {P}$$\\end{document}-subnormal subgroup is a TI\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{TI}\\,}}$$\\end{document}-subgroup.

Spectral clustering of Markov chain transition matrices with complex eigenvalues

The Robust Perron Cluster Analysis (PCCA+) has become a popular spectral clustering algorithm for coarse-graining transition matrices of nearly decomposable Markov chains with transition states. Originally developed for reversible Markov chains, the algorithm only worked for transition matrices with real eigenvalues. In this paper, we therefore extend the theoretical framework of PCCA+ to Markov chains with a complex eigen-decomposition. We show that by replacing a complex conjugate pair of eigenvectors by their real and imaginary components, a real representation of the same subspace is obtained, which is suitable for the cluster analysis. We show that our approach leads to the same results as the generalized PCCA+ (GPCCA), which replaces the complex eigen-decomposition by a conceptually more difficult real Schur decomposition. We apply the method on non-reversible Markov chains, including circular chains, and demonstrate its efficiency compared to GPCCA. The experiments are performed in the Matlab programming language and codes are provided.

Effect of impregnation strategy on structural characteristics of Ce-Mn/Al2O3 and its catalytic ozonation of benzoic acid.

Ce-Mn binary oxides supported on Al2O3 (Ce-Mn/Al2O3), with enhanced activity and stability for catalytic ozonation of benzoic acid, were synthesized using a facile impregnation method. The competitive synergetic effects between cerium and manganese significantly influenced the structural characteristics and catalytic performance of the catalysts depending on the impregnation sequence. Catalysts prepared via the one-step impregnation process exhibited a higher concentration of homogeneous Ce3+ species on the catalyst surface. This led to an increase in surface oxygen vacancies, thereby enhancing catalytic activity. In contrast, the two-step impregnation process resulted in fewer oxygen vacancies due to reduced competitive effects between cerium and manganese. Overall, the optimized Ce-Mn/Al2O3 catalysts demonstrated improved catalytic performance in ozonation reactions, highlighting the importance of impregnation method and calcination conditions in tailoring catalyst properties for enhanced activity and stability. Oxygen vacancies play a crucial role as active sites for ozone adsorption and dissociation into *O2 and *O, facilitated by the reduction of Mn4+ to Mn3+ and the oxidation of Ce3+ to Ce4+. This process forms an electron closed loop that maintains electron balance. The synergistic interactions between cerium and manganese enable rapid electron transfer between Ce4+ and Mn3+, facilitating the regeneration of Ce3+ and Mn4+. Due to the increase of the dual redox conjugate pairs and the surface reactive oxygen species, the catalytic ozonation activity and stability of Ce-Mn/Al2O3 was enhanced.

25-26 Ağustos 2018 Jeomanyetik Fırtına Sırasında Afrika Bölgesi Üzerindeki Manyetik Eşlenik Çiftleri Üzerindeki TEC Değişikliklerinin İncelenmesi

In this study, the electron transport process resulting from electromagnetic drift between two magnetic conjugate pairs over the African region during the August 25-26 2018 geomagnetic storm was investigated. The effects of geomagnetic conditions presented with Dst index and IMF Bz values on Total Electron Content (TEC) values at conjugate stations were compared separately for stormy and quiet periods. During the storm period, the effect of TEC values at stations in the northern hemisphere (Haifa and Djibouti) on the TEC values at stations in the southern hemisphere (Ambalavao and Malindi) is greater than the effect of TEC values in the southern hemisphere on TEC values in the northern hemisphere. According to this result, it can be said that the south-directed electromagnetic convection was more than the north-directed convection in the examined dates. When the coefficients are examined, it can be said that the interaction is more in the magnetic conjugate pair that is closer to the equator during the storm period, and the interaction is more in the magnetic conjugate pair that is far from the equator during the silent period. Considering the coefficients calculated for Dst and IMF Bz, it is seen that the TEC values are very small compared to their coefficients. From this it can be concluded that the effect of Dst and IMF Bz is much smaller than the effect of TEC values at a station on TEC values at its magnetic conjugates.

A Nonlinear Dynamical View of Kleiber's Law on the Metabolism of Plants and Animals.

Kleiber's empirical law, which describes that metabolism increases as the mass to the power 3/4, has arguably remained life sciences' enigma since its formal uncovering in 1930. Why is this behavior sustained over many orders of magnitude? There have been quantitative rationalizations put forward for both plants and animals based on realistic mechanisms. However, universality in scaling laws of this kind, like in critical phenomena, has not yet received substantiation. Here, we provide an account, with quantitative reproduction of the available data, of the metabolism for these two biology kingdoms by means of broad arguments based on statistical mechanics and nonlinear dynamics. We consider iterated renormalization group (RG) fixed-point maps that are associated with an extensive generalized (Tsallis) entropy. We find two unique universality classes that satisfy the 3/4 power law. One corresponds to preferential attachment processes-rich gets richer-and the other to critical processes that suppress the effort for motion. We discuss and generalize our findings to other empirical laws that exhibit similar situations, using data based on general but different concepts that form a conjugate pair that gives rise to the same power-law exponents.

Preparation of Li3V2(PO4)3@C composites using aluminum chloride catalyzed fermentation residue of bioethanol production with reed as carbon source for long life Li-ion batteries

The capability and longevity of lithium-ion batteries (LIBs) are heavily influenced by simultaneously increasing the lithium-ion extraction/insertion dynamics and enhancing the construction stabilization of Li3V2(PO4)3 (LVP) positive electrode material. With the goal to improve the battery's stability, fermentation residues rich in functional groups that contain oxygen might be wrapped around LVP. This is the first time, to our knowledge, that high-performance LVP cathode materials have been created utilizing leftover fermentation waste as a carbon source. More crucially, due to the promoted invertible V3+/V4+ redox conjugate pair created by oxygen groups coinciding with the improved Li+ dynamics properties, the top-performing material generated an excellent primal discharging capability of 131.9 mA h/g (approaching the projected performance) at 0.5 C. Additionally, after 500 cycles, the improved battery exhibits higher capacity retention of 95%.

Conjugate hemispheric response of Earth’s ionosphere due to 2019 Antarctic minor Sudden Stratospheric Warming (SSW): Comparison with 2013 Arctic major SSW

In this paper, we attempt to compare the Antarctic SSW of 2019 with the Arctic SSW of 2013. Despite being a minor SSW, the rare Antarctic event of September 2019 caused a significant enhancement of stratospheric temperature at high latitudes with more temperature deviation than the Arctic major SSW of 2013. On the basis of the duration and extent of stratospheric zonal mean zonal wind, the 2019 SSW is comparable to major SSWs. An exceptionally strong wave 1 planetary wave was observed during the 2019 SSW. We have investigated the response of the ionospheric total electron content (TEC) during both the SSWs over two low mid-latitude locations Okinawa and Darwin, which form magnetic conjugate pair. The extent of maximum ionospheric deviations during the 2019 SSW is almost comparable to the 2013 SSW in their respective hemispheres. We have observed mostly night time enhancement in TEC during the Arctic SSW, while daytime enhancement is noted during the Antarctic SSW. Hemispheric asymmetry is observed during both the SSWs. Minor geomagnetic storm has been noticed coinciding with each SSW during different phases of the warming. The SSW, solar flux, and storm contribution on the TEC have been calculated during both the SSWs, which shows clear contribution of SSW on the total TEC despite the presence of storm and during moderately high solar activity. We have also investigated the foF2 variations from ionosonde and IRI-16 at the same two locations. The peak electron density from ionosonde responds to the warming in a manner similar to TEC. The foF2 of IRI-16 exhibits overestimation during daytime and underestimation during nighttime on quiet days during low solar activity (2019), while during moderate solar activity (2013), it overestimates during morning hours. However, it can not predict the enhancements of foF2 during SSWs.

P-330 - Chemically Identical Pair of Theranostic Small-Molecule Drug Conjugates for Prostate Cancer Imaging and Chemotherapy

P-330 - Chemically Identical Pair of Theranostic Small-Molecule Drug Conjugates for Prostate Cancer Imaging and Chemotherapy

Observer-predictor based on a PD controller for high-order delayed systems with one unstable pole and n pairs of complex conjugate poles

This paper considers the stability of continuous linear systems with time delay. In particular, systems with one unstable pole and n pairs of complex conjugate and/or real poles are analyzed. The proposed solution consists of using an estimated state observer with a closed-loop PD type controller to access to the internal variables of the system. A partition of the time delay is proposed in the observer to addmit the double of the maximum size that when a simple PD controller is considered. The necessary and sufficient conditions for the stability of the proposed control scheme are presented and an example is presented with simulations to demostrate the usefulness of the proposed control strategy.

Antisunward Streaming Westward Sub‐Auroral Ion Drifts (SAID) Developed in the Postmidnight (1–4) Magnetic Local Time Sector During 2013

AbstractIn this study, we have surveyed the Defense Meteorological Satellite Program F15 plots of horizontal ion velocity measurements for the calendar year of 2013 and specified 15 antisunward (westward) sub‐auroral ion drifts (SAID) detections taken in the 1–4 magnetic local time (MLT) sector based on their respective underlying polar convections. We find that during these 15 SAID events, the strong duskside westward polar plasma convection extended across the midnight meridian into the postmidnight 1–4 MLT sector making there the westward SAID flows streaming antisunward. These antisunward (westward) SAID flows were observed only once as a northern‐southern conjugate pair over the Mid Pacific and repeatedly in the South American longitude sector's wider region. We could correlate six antisunward (westward) SAID flows with their respective nearby outward SAID E fields observed by one of the Van Allen Probe (VAP) satellites in the inner magnetosphere. These provided two strong correlations depicting inner‐magnetosphere—topside‐ionosphere conjugate observations and four weak correlations. The VAP observations illustrate (a) the newly‐formed outward SAID E field and (b) its plasma environment still showing (c) the signatures of short‐circuiting across (d) the newly‐formed plasmapause appearing with (e) the hot zone on the earthward/tailward side and sometimes with (f) a localized heat source. Implying different underlying heating mechanisms, the hot zone and the heat source were associated with energetic plasma particles accelerated by plasma turbulence, low‐frequency waves including electromagnetic ion cyclotron waves, and/or auroral kilometric radiation bursts.

Modeling the Stackelberg game with a boundedly rational follower in deterioration supply chain-based interaction with the leader’s hybrid pricing strategy

In the conventional Stackelberg game, all players are perfect optimizers, able to choose the best responses to their competitors' decisions. In contrast, in the Stackelberg game with bounded rationality, players are unable to optimize their individual payoffs and face strategic uncertainty due to the performance of competitors. In real games, players are usually unaware of the performance of their competitors. Therefore, to survive in the competitive market, they are forced to make decisions under conditions of bounded rationality. Due to the importance of decision-making in this situation, in this study, the Stackelberg game of deteriorating inventory within a two-level supply chain was developed considering the concept of bounded rationality. The follower in the Stackelberg game (manufacturer) is poorly aware of the leader's performance in terms of order size. Therefore, she/he manufactures the products at rates with normal distribution. For the first time, an algorithm based on Bayesian conjugate pair was proposed to solve the Stackelberg game with bounded rationality for deteriorating products. In addition, a comparison was made between the above conditions and a mode in which the leader's strategy is fully observable to the follower. The approach used in the proposed algorithm to adjust the production rate was also compared with the adjustment based on the partial adjustment model, and the obtained results prove the convergence of the proposed algorithm.

Switching quantum reference frames in the N-body problem and the absence of global relational perspectives

Given the importance of quantum reference frames (QRFs) to both quantum and gravitational physics, it is pertinent to develop a systematic method for switching between the descriptions of physics relative to different choices of QRFs, which is valid in both fields. Here we continue with such a unifying approach, begun in [Quantum 4, 225 (2020)], whose key ingredient is a symmetry principle, which enforces physics to be relational. Thanks to gauge related redundancies, this leads to a perspective-neutral structure which contains all frame choices at once and via which frame perspectives can be consistently switched. Formulated in the language of constrained systems, the perspective-neutral structure is the constraint surface classically and the gauge invariant Hilbert space in the Dirac quantized theory. By contrast, a perspective relative to a specific frame corresponds to a gauge choice and the associated reduced phase and Hilbert space. QRF changes thus amount to a gauge transformation. We show that they take the form of `quantum coordinate changes'. We illustrate this in a general mechanical model, namely the relational N-body problem in 3D space with rotational and translational symmetry. This model is especially interesting because it features the Gribov problem so that globally valid gauge fixing conditions, and hence relational frame perspectives, are absent. The constraint surface is topologically non-trivial and foliated by 3-, 5- and 6-dimensional gauge orbits, where the lower dimensional orbits are a set of measure zero. The N-body problem also does not admit globally valid canonically conjugate pairs of Dirac observables. These challenges notwithstanding, we exhibit how one can construct the QRF transformations for the 3-body problem. Our construction also sheds new light on the generic inequivalence of Dirac and reduced quantization through its interplay with QRF perspectives.

Quasi-Hermitian Formulation of Quantum Mechanics Using Two Conjugate Schrödinger Equations

To the existing list of alternative formulations of quantum mechanics, a new version of the non-Hermitian interaction picture is added. What is new is that, in contrast to the more conventional non-Hermitian model-building recipes, the primary information about the observable phenomena is provided not only by the Hamiltonian but also by an additional operator with a real spectrum (say, R(t)) representing another observable. In the language of physics, the information carried by R(t)≠R†(t) opens the possibility of reaching the exceptional-point degeneracy of the real eigenvalues, i.e., a specific quantum phase transition. In parallel, the unitarity of the system remains guaranteed, as usual, via a time-dependent inner-product metric Θ(t). From the point of view of mathematics, the control of evolution is provided by a pair of conjugate Schrödiner equations. This opens the possibility od an innovative dyadic representation of pure states, by which the direct use of Θ(t) is made redundant. The implementation of the formalism is illustrated via a schematic cosmological toy model in which the canonical quantization leads to the necessity of working with two conjugate Wheeler-DeWitt equations. From the point of view of physics, the “kinematical input” operator R(t) may represent either the radius of a homogeneous and isotropic expanding empty Universe or, if you wish, its Hubble radius, or the scale factor a(t) emerging in the popular Lemaitre-Friedmann-Robertson-Walker classical solutions, with the exceptional-point singularity of the spectrum of R(t) mimicking the birth of the Universe (“Big Bang”) at t=0.

The Frequency Spectrum Analysis of Wideband Oscillations of Grid-Connected Voltage Source Converter

With the widespread application of voltage source converters (VSCs) in power systems with high renewable resource penetration, the problem of wideband oscillations caused by interactions between power converters and the power grid has drawn great attention. This paper presents a new methodology for simplifying the mathematical derivations of the impedance model of a VSC. Using the new simplified model, the stability and spectrum characteristics of the performance of the VSC can be clearly analyzed. The proposed second-order d-q impedance model of a grid-connected VSC in the presence of a PLL not only mathematically demonstrates that wideband oscillatory modes always occur in conjugate pairs, but also delivers a clear vision of the physical essence of these wideband oscillations. A classic modal analysis is executed in this paper to demonstrate the stability and frequency spectral characteristics of wideband oscillations in a VSC integrated system. The impact of the control parameters of the VSC on wideband oscillatory stability is thoroughly analyzed by modal analysis using ‘DIgSILENT PowerFactory’ as well as hardware-in-the-loop experiments. This methodology can be considered a powerful tool for the control design of VSCs in the modern power system.