Conjugation Operator Research Articles

Quasi-Compactness of Operators for General Markov Chains and Finitely Additive Measures

We study Markov operators T, A, and T* of general Markov chains on an arbitrary measurable space. The operator, T, is defined on the Banach space of all bounded measurable functions. The operator A is defined on the Banach space of all bounded countably additive measures. We construct an operator T*, topologically conjugate to the operator T, acting in the space of all bounded finitely additive measures. We prove the main result of the paper that, in general, a Markov operator T* is quasi-compact if and only if T is quasi-compact. It is proved that the conjugate operator T* is quasi-compact if and only if the Doeblin condition (D) is satisfied. It is shown that the quasi-compactness conditions for all three Markov operators T, A, and T* are equivalent to each other. In addition, we obtain that, for an operator T* to be quasi-compact, it is necessary and sufficient that it does not have invariant purely finitely additive measures. A strong uniform reversible ergodic theorem is proved for the quasi-compact Markov operator T* in the space of finitely additive measures. We give all the proofs for the most general case. A detailed analysis of Lin’s counterexample is provided.

On the quadratic closeness of eigenfunctions of «unperturbed» and «perturbed» differentiation operators on an interval

The article considers the eigenvalue problem of the differentiation operator, when the spectral param- eter is also present in the boundary condition with an integral perturbation, where the integrand has the property of limited variation and has a value of unity at the ends of the segment [-1, 1]. The derivatives with respect to the time variable included in the boundary condition naturally arise when solving (by the Fourier method) initial boundary value problems for evolution equations. The conjugate operator is constructed. It is shown that the spectral questions of the conjugate operator have a similar structure. The characteristic determinant of the original direct spectral problem with an integral perturbation of the boundary condition and for the eigenvalue problem of a loaded first-order differential equation on a segment with a periodic boundary condition, which is an entire analytical function of the spectral parameter, is constructed. Based on the formula of the characteristic determinant, conclusions are drawn about the asymptotic behavior of the spectrum of the original «perturbed» differentiation operator and the loaded first-order differential equation on the segment. A special feature of the operator under consideration is the non-self-adjointness of the operator in L2(–1,1). The quadratic proximity of the systems of eigen functions of the «unperturbed» and «perturbed» differentiation operators and the Riesz basis property of these systems are proved. In this case, the system of eigenfunctions of the «perturbed» operator is not orthonormal.

Mourre Theory and Asymptotic Observables in Local Relativistic Quantum Field Theory

We prove the convergence of Araki–Haag detectors in any Haag–Kastler quantum field theory with an upper and lower mass gap. We cover the case of a single Araki–Haag detector on states of bounded energy, which are selected from the absolutely continuous part of the energy-momentum spectrum sufficiently close to the lower boundary of the multi-particle spectrum. These states essentially encompass those states in the multi-particle spectrum lying below the three-particle threshold. In our proof, we draw on insights from proofs of asymptotic completeness in quantum mechanics. Notably, we apply Mourre’s conjugate operator method for the first time within the framework of Haag–Kastler quantum field theory. Furthermore, we discuss applications of our findings for the problem of asymptotic completeness in local relativistic quantum field theory.

Mourre theory and spectral analysis of energy-momentum operators in relativistic quantum field theory

A central task of theoretical physics is to analyse spectral properties of quantum mechanical observables. In this endeavour, Mourre’s conjugate operator method emerged as an effective tool in the spectral theory of Schrödinger operators. This paper introduces a novel class of examples from relativistic quantum field theory that are amenable to Mourre’s method. By assuming Lorentz covariance and the spectrum condition, we derive a limiting absorption principle for the energy-momentum operators and provide new proofs of the absolute continuity of the energy-momentum spectra. Moreover, under the assumption of dilation covariance, we show that the spectrum of the relativistic mass operator is purely absolutely continuous in (0,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(0,\\infty )$$\\end{document}.

On Bicon-Numbers With Their Basic Properties and Applications in Quantum Systems.

Motivated by the fact that there exists the operation of conjugation in quantum systems, the concept of bicon-numbers is proposed in this article. The bicon-numbers are defined by introducing two symbolic parameters into the set of complex numbers. The basic functions of these two symbolic parameters are specified by an axiom which abstracts the operation of complex conjugation. Basic properties are developed for the operations of addition and multiplication in the bicon-number set. In addition, several different forms are given for bicon-numbers, and the corresponding operation rules are established. By exploring the relations of the vensors in the bicon-number set, the structure of the bicon-number set is depicted, and real matrix representations of bicon-numbers are also presented. Besides, bicomplex matrix representations for bicon-numbers are also investigated in view that the operation of multiplication for bicomplex numbers possesses commutativity property. In addition, the matrices with bicon-numbers as entries are investigated, and state responses of some quantum systems are given within the framework of bicon-numbers.

Sedenionic matrices and their properties

In this article, the matrix algebra is well-known concept in mathematics, has been extended to sedenionic-coefficient matrices using sedenions, which have many applications in recent years. Subsequently, sedenionic-coefficient matrices have been obtained in their real, complex, quaternionic and octonionic forms. Based on these definitions, the arithmetic operations of addition, multiplication, conjugation, transpose and conjugate transpose for sedenionic matrices and their complex, quaternionic and octonionic variations have been established and their algebraic properties scrutinized. Lastly, vector space over real and complex numbers and module structure over quaternions of sedenionic matrices has been searched.

Classes of Operators Related to a Complex Conjugation Operator

Classes of Operators Related to a Complex Conjugation Operator

R7-branes as charge conjugation operators

R7-branes are a class of recently discovered nonsupersymmetric real codimension-two duality defects in type IIB string theory predicted by the swampland cobordism conjecture. For type IIB realizations of 6D SCFTs with N=(2,0) supersymmetry, wrapping an R7-brane “at infinity” leads to a topological operator associated with a zero-form charge conjugation symmetry that squares to the identity. Similar considerations hold for those theories obtained from further toroidal compactification, but this can be obstructed by bundle curvature effects. Using some minimal data on the topological sector of the R7-branes, we extract the associated fusion rules for these charge conjugation operators. More broadly, we sketch a top down realization of various topological operators/interfaces associated with C, R, and T transformations. We also use holography to provide strong evidence for the existence of the R7-brane which is complementary to the cobordism conjecture. Similar considerations apply to other string-realized QFTs with symmetry operators constructed via nonsupersymmetric branes which carry a conserved charge. Published by the American Physical Society 2024

Lagrange formula conjugate third order differential equation

The paper considers a boundary value problem for a third order with no smooth coefficients and pure derivatives. Odds. This is due to the fact to introduce the concept of the conjugate Green's function. It is very difficult to write the form of the conjugate differential operator corresponding to equation in the Lagrange sense. Therefore, in this work, without using strict conditions smoothness under the conditions and boundedness, an explicit form is found conjugate operator since the initial-boundary value problem for integral-differential equations has been studied based on the introduction special conjugate systems in the form of an integral-algebraic equations’ system. In this article, it can be said that Green's function is considered based on Lagrange's formula for the third-order differential equation with boundary conditions and its conjugate.

The conjugate operator on variable harmonic Bergman spaces

We prove that the harmonic conjugation operator on variable exponent harmonic Bergman spaces in the unit disc is bounded when the exponent has positive minimum and finite maximum and satisfies the log-Hölder condition. Under these conditions on the exponent, we also give a characterization of variable exponent Bergman spaces in terms of the derivatives of functions in them. Several key lemmas involve the boundedness of various maximal operators in these variable spaces.

ALGORITHM FOR FINDING THE INVERSE ELEMENTS AND SOLUTION OF THE SILVESTER EQUATION IN THE CLIFFORD ALGEBRAS R4,0, R1,3, R5,0

The purpose of the work is to find an algorithm for finding inverse elements in the Clifford algebras R4,0, R1,3, R5,0 and to solve the nonlinear Sylvester equation .
 
 Materials and methods. Using the basic conjugation operations in Clifford algebras, finding an algorithm for finding inverse elements. Application of this algorithm to solve the Sylvester equation.
 
 Results of the work. In Clifford algebras R4,0, R1,3, R5,0, which have a great application in physics, a method for finding inverse elements and equations for finding zero divisors were found. The found algorithm is used to solve the Sylvester equation. For Clifford algebras of even dimension R4,0, R1,3 an algorithm for finding inverse elements is given. Finding inverse elements is closely related to the concept of zero divisors in these algebras. The inverse element method is used to solve the Sylvester equation, using even conjugation, reverse conjugation and Clifford conjugation. For the odd Clifford algebra R5,0, a conjugation is found that can be used to apply the algorithm for finding the inverse element. The method of finding the inverse element is used to solve the Sylvester equation, which, in particular, is used to ensure the robustness of the piezodrive using the controlled relative interval method.
 
 Findings. An algorithm for finding inverse elements is constructed and the Sylvester equation is solved in the Clifford algebras R4,0, R1,3, R5,0.

Asymmetric particle-antiparticle Dirac equation: first quantization

We derive a Dirac-like equation, the asymmetric Dirac equation, where particles and antiparticles sharing the same wave number have different energies and momenta. We show that this equation is Lorentz covariant under proper Lorentz transformations (boosts and spatial rotations) and also determine the corresponding transformation law for its wave function. We obtain a formal connection between the asymmetric Dirac equation and the standard Dirac equation and we show that by properly adjusting the free parameters of the present wave equation we can make it reproduce the predictions of the usual Dirac equation. We show that the rest mass of a particle in the theoretical framework of the asymmetric Dirac equation is a function of a set of four parameters, which are relativistic invariants under proper Lorentz transformations. These four parameters are the analog to the mass that appears in the standard Dirac equation. We prove that in order to guarantee the covariance of the asymmetric Dirac equation under parity and time reversal operations (improper Lorentz transformations) as well as under the charge conjugation operation, these four parameters change sign in exactly the same way as the four components of a four-vector. The mass, though, being a function of the square of those parameters remains an invariant. We also extensively study the free particle plane wave solutions to the asymmetric Dirac equation and derive its energy, helicity, and spin projection operators as well as several Gordon’s identities. The hydrogen atom is solved in the present context after applying the minimal coupling prescription to the asymmetric Dirac equation, which also allows us to appropriately obtain its non-relativistic limit.

Asymmetric particle-antiparticle Dirac equation: second quantization

We build the fully relativistic quantum field theory related to the asymmetric Dirac fields first presented in a prequel to this work. These fields are solutions of the asymmetric Dirac equation, a Lorentz covariant Dirac-like equation whose positive and ‘negative’ frequency plane wave solutions’ dispersion relations are no longer degenerate. At the second quantization level, we show that this implies that particles and antiparticles sharing the same wave number have different energies and momenta. In spite of that, we prove that by properly fixing the values of the relativistic invariants that define the asymmetric Dirac free field Lagrangian density, we can build a consistent, fully relativistic, and renormalizable quantum electrodynamics (QED) that is empirically equivalent to the standard QED. We discuss the reasons and implications of this non-trivial equivalence, exploring qualitatively other scenarios in which the asymmetric Dirac fields may lead to beyond the standard model predictions. We give a complete account of how the asymmetric Dirac fields and the corresponding annihilation and creation operators transform under improper Lorentz transformations (parity and time reversal operations) and under the charge conjugation operation. We also prove that the present theory respects the CPT theorem.

The discrete canonical commutation relationship

We use non-standard finite differences to propose a quantum momentum operator to be used when the spectrum of the operator is discrete. The defined discrete operator complies with the discrete versions of the properties that the continuous variable operator has. The discrete derivative is exact for its eigenfunction, that is, exponential functions. We obtain the discrete adjoint of the momentum operator. The canonical commutation relationship between conjugate operators for discrete variables is diagonal along a particular direction.

Non-self-adjoint model statement of boundary problem of acoustics. Part 1

Based on the mathematical definition of a non-self-adjoint operator and the physical meaning of a specific boundary value problem, a non-self-adjoint model statement of boundary value problems in acoustics is formulated. As an example, boundary value problems for the reflection of a plane wave and a spherical wave at the interface between two liquid media are considered. A new definition of the reflection coefficient of a spherical wave is introduced. In the region of subcritical angles of incidence, the new definition takes into account the appearance in the total sound field of converging recoil waves corresponding to the eigenfunctions of the adjoint operator. In the region of supercritical angles of incidence, the participation of the eigenfunctions of two conjugate operators in the total sound field forms a nonzero power flux through the interface and its transformation into the vortex component of the intensity vector at the total internal reflection horizon. The definition of the total internal reflection horizon is given. Experimental data are presented that confirm the new definition of the reflection coefficient and the physical correctness of the non-self-adjoint model statement.

5D Elko spinor field non-minimally coupled to nonmetricity in f(Q) gravity

This paper aims to investigate the localization of the five-dimensional spinor field known as Elko (dual-helicity eigenspinors of the charge conjugation operator) by employing a Yukawa-like geometrical coupling in which the Elko field is non-minimally coupled to nonmetricity scalar Q. We adopt the braneworld scenarios in which the first-order formalism with sine-Gordon and linear superpotentials is employed to obtain the warp factors. A linear function supports the zero-mode trapping within the geometric coupling, leading to the same effective potential as the scalar field. Moreover, an exotic term must be added to obtain real-valued massive modes. Such modes are investigated through the Schrödinger-like approach.

Scattering theory for Dirac fields near an extreme Kerr–de Sitter black hole

In this paper, we construct a scattering theory for classical massive Dirac fields near the horizon of an extreme Kerr-de Sitter blackhole. Our main tool is the existence of a conjugate operator in the sense of Mourre theory. Additionally, despite the fact that effects of the rotation are 'amplified' near the double horizon, we show that one can still reduce our study to a 1-dimensional problem through an appropriate decomposition of the Hilbert space.

Fermionic dark matter-photon quantum interaction: A mechanism for darkness

Mass dimension one fermionic fields are prime candidates to describe dark matter, due to their intrinsic neutral nature, as they are constructed as eigenstates of the charge conjugation operator with dual helicity. To formulate the meaning of the darkness, the fermion-photon coupling is scrutinized with a Pauli-like interaction, and the path integral is then formulated from the phase space constraint structure. Ward–Takahashi-like identities and Schwinger–Dyson equations, together with renormalizability, are employed to investigate a phenomenological mechanism to avoid external light signals. Accordingly, the non-polarized pair annihilation and Compton-like processes are shown to vanish at the limit of small scattering angles even if considering 1-loop radiative corrections, reinforcing the dark matter interpretation. However, dark matter interactions with nucleons are still possible. Motivated by recent nucleon-recoil experiments to detect dark matter, we furnish a consistent theoretical setup to describe interaction with the photon compatible with the prevalence of darkness.

Lattice quantum Villain Hamiltonians: compact scalars, U(1) gauge theories, fracton models and quantum Ising model dualities

We construct Villain Hamiltonians for compact scalars and abelian gauge theories. The Villain integers are promoted to integral spectrum operators, whose canonical conjugates are naturally compact scalars. Further, depending on the theory, these conjugate operators can be interpreted as (higher-form) gauge fields. If a gauge symmetry is imposed on these dual gauge fields, a natural constraint on the Villain operator leads to the absence of defects (e.g. vortices, monopoles,…). These lattice models therefore have the same symmetry and anomaly structure as their corresponding continuum models. Moreover they can be formulated in a way that makes the well-know dualities look manifest, e.g. a compact scalar in 2d has a T-duality, in 3d is dual to a U(1) gauge theory, etc. We further discuss the gauged version of compact scalars on the lattice, its anomalies and solution, as well as a particular limit of the gauged XY model at strong coupling which reduces to the transverse-field Ising model. The construction for higher-form gauge theories is similar. We apply these ideas to the constructions of some models which are of interest to fracton physics, in particular the XY-plaquette model and the tensor gauge field model. The XY-plaquette model in 2+1d coupled to a tensor gauge fields at strong gauge coupling is also exactly described by a transverse field quantum J1 − J2 Ising model with J1 = 2J2, and discuss the phase structure of such models.

Conjugate complex function coupling strategy of interferometric field: Synthetic-wavelength interferometric fringe extraction in simultaneous dual-wavelength interferometry

Simultaneous dual-wavelength interferometry (SDWI) supplies a better solution to extend the measurement range of traditional single-wavelength interferometry with higher efficiency. To decompose the needed longer beat-frequency synthetic-wavelength (λS) information from the generated moiré fringe pattern in SDWI, a conjugate complex function coupling strategy (CCFC) of interferometric field is proposed in this study. By the Fourier transform and the half-band filter in one side of the spectrum, the positive half spectrum is obtained. The inverse Fourier transform and conjugate operation are performed on the half spectrums to obtain the two conjugate complex functions of the interferometric field. Coupled by the multiplication for the conjugate complex functions, the lower beat-frequency synthetic-wavelength interferometric fringe pattern is derived directly. And if the SDWI is implemented with the π/2 phase shift at λS, the obtained phase-shift interferogram for λS could be demodulated by the conventional phase-shift algorithm. Compared with other spatial-domain Fourier transform demodulation theory, the half-band filter used in CCFC method is appropriate for the different spectral width and carrier spatial frequency. The necessary introduced linear carrier is merely about 0.0947 times of the carrier in the former numerically. And it is only used to pursue the separation between the ±1 order phase lobes instead of the separation between the different wavelength phase lobes. The feasibility and applicability of the CCFC method are verified using simulation and experimental results.