Hyperfunction Quantum Field Theory: Localized Fields Without Localized Test Functions

This note addresses the problem of localization in quantum field theory; more specifically we contribute to the ongoing discussion about the most appropriate concept of localization which one should use in relativistic quantum field theory: through localized test functions or through the fields directly without localized test functions. In standard quantum field theory, i.e., in relativistic quantum field theory in terms of tempered distributions according to Gårding and Wightman, this is done through localized test functions. In hyperfunction quantum field theory (HFQFT), i.e., relativistic quantum field theory in terms of Fourier hyperfunctions this is done through the fields themselves. In support of the second approach we show here that it has a much wider range of applicability. It can even be applied to relativistic quantum field theories which do not admit compactly supported test functions at all. In our construction of explicit models we rely on basic results from the theory of quasi-analytic functions.     

Quantum fluctuations in quantum lattice systems with continuous symmetry

We discuss conditions for the absence of spontaneous breakdown of continuous symmetries in quantum lattice systems atT=0. Our analysis is based on Pitaevskii and Stringari's idea that the uncertainty relation can be employed to show quantum fluctuations. For one-dimensional systems, it is shown that the ground state is invariant under a continuous transformation if a certain uniform susceptibility is finite. For the two- and three-dimensional systems, it is shown that truncated correlation functions cannot decay any more rapidly than|r| ^–d+1 whenever the continuous symmetry is spontaneously broken. Both of these phenomena occur owing to quantum fluctuations. Our theorems cover a wide class of quantum lattice systems having not-too-long-range interactions.     

Quantum Monodromy in the Spectrum of Schrödinger Equation with a Decatic Potential

In this study the spectral problem of the two-dimensional Schrödinger equation with the cylindrically symmetrical decatic potential is carried out. The concept of quantum monodromy is introduced to give insight into the energy levels of system with this potential. It is shown that quantum monodromy occurs at = 0 in the distribution of eigenstates around a critical point on the spectrum at E = 0 with zero angular momentum, such that there can be no smoothly valid assignment of quantum number. Cases with the three-well and four-well potentials are presented to give rise to the double degeneracies with respect to energy except for the angular momentum m = 0.     

FRW minisuperspace with local N=4 supersymmetry and self‐interacting scalar field

A supersymmetric FRW model with a scalar supermultiplet and generic superpotential is analysed from a quantum cosmological perspective. The corresponding Lorentz and supersymmetry constraints allow to establish a system of first order partial differential equations from which solutions can be obtained. We show that this is possible when the superpotential is expanded in powers of a parameter λ?1. At order λ⁰ we find the general class of solutions, which include in particular quantum states reported in the current literature. New solutions are partially obtained at order λ¹, where the dependence on the superpotential is manifest. These classes of solutions can be employed to find states for higher orders in λ. Our analysis further points to the following: (i) supersymmetric wave functions can only be found when the superpotential has either an exponential behaviour, an effective cosmological constant form or is zero; (ii) If the superpotential behaves differently during other periods, the wave function is trivial ( = 0, i.e., no supersymmetric states). We conclude this paper discussing how our FRW minisuperspace (with N = 4 supersymmetry and invariance under time‐reparametrization) can be relevant concerning the issue of supersymmetry breaking.     

Cosmological Wave Function and Wormhole Wave Function with a Conformal Complex Scalar Field

Quantum cosmology and the quantum wormhole witha conformal complex scalar field are discussed, thecorresponding Wheeler-DeWitt equations are obtained, andthe cosmological wave functions and wormhole wave functions are calculated, respectively,with different boundary conditions. From thecosmological wave function it is found that theprobability density of the universe is zero at a = 0,while at the ground state the most probable radius is aboutthe Planck scale. It is also shown that there exist twodifferent types of universes, which can be connected bythe quantum tunneling effect, transiting from one region to another. It follows from thewormhole wave function that the most probable radius ofthe wormholes is about the Planck scale, which impliesthat the wormhole is steady due to the quantumeffect.     

Simple classical mapping of the spin-polarized quantum electron gas: distribution functions and local-field corrections

We use the now well known spin unpolarized exchange-correlation energy E(xc) of the uniform electron gas as the basic "many-body" input to determine the temperature T(q) of a classical Coulomb fluid having the same correlation energy as the quantum system. It is shown that the spin-polarized pair distribution functions (SPDFs) of the classical fluid at T(q), obtained using the hypernetted chain equation, are in excellent agreement with those of the T = 0 quantum fluid obtained by quantum Monte Carlo (QMC) simulations. These methods are computationally simple and easily applied to problems which are currently beyond QMC simulations. Results are presented for the SPDFs and the local-field corrections to the response functions of the electron fluid at T = 0 and finite T.     

Quantum Secure Direct Communication with W State

A new theoretical scheme for quantum secure direct communication is proposed, where four-qubit symmetric W state functions as quantum channel. It is shown that two legitimate users can directly transmit the secret messages by using Bell-basis measurements and classical communication. The scheme is completely secure if the quantum channel is perfect. Even if the quantum channel is unsecured, it is still possible for two users to perform their secure communication. One bit secret message can be transmitted by sending a bit classical information.     

New Quantization Method for Evaluation of Eigenenergies

A quantum dynamical equation is constructed as the limit of a sequence of functions (called Semiquantum momentum functions or SQMF). The quantum action variable J is defined as the limit of the sequence of contour integrals of SQMFs such that the quantization condition is J = n, where n is a nonnegative integer for eigenvalues and a noninteger for off eigenvalues. This quantization condition is exact and J is an analytic function of energy. Based on new definitions, an accurate numerical method is developed for obtaining eigenenergies. The method can be applied to both real and PT symmetric complex potentials. The validity and the accuracy of this new method is demonstrated with three illustrations.     

Functional integral representations of partition function without limiting procedure. Techniques of calculation of moments

A definition of the Feynman path integral which does not rest on a limiting procedure based on time-slicing has been given by DeWitt-Morette. We present in this paper a discussion of real Gaussian measures and formulate expressions for the quantum statistical partition function directly in terms of measures of integration on the topological vector space ø₀ of continuous functions defined on the time intervalT = (t _a ,t _b ), such thatx(t _a ,t _b )=0 for allx ? ø₀. We give a definition of a measure for the space ø₀ equivalent to the path integral based on the Uhlenbeck-Ornstein probability distribution. We give expressions for the partition function using the Wiener-Feynman measure and the Uhlenbeck-Ornstein measure. As an exercise in the use of the new techniques, we present calculations of moments of potential functions. The techniques will enable one to solve in a rigorous manner practical problems in quantum statistical mechanics.     

Fractional Fourier Transformation for Quantum Mechanical Wave Functions Studied by Virtue of IWOP Technique

Starting from the optical fractional Fourier transform (FFT) and using the technique of integration withinan ordered product of operators we establish a formalism of FFT for quantum mechanical wave functions. In doing so, theessence of FFT can be seen more clearly, and the FFT of some wave functions can be derived more directly and concisely.We also point out that different FFT integral kernels correspond to different quantum mechanical representations. Theyare generalized FFT. The relationship between the FFT and the rotated Wigner operator is studied by virtue of theWeyl ordered form of the Wigner operator.     

Quantum Noether method

We present a general method for constructing perturbative quantum field theories with global symmetries. We start from a free non-interacting quantum field theory with given global symmetries and we determine all perturbative quantum deformations assuming the construction is not obstructed by anomalies. The method is established within the causal Bogoliubov-Shirkov-Epstein-Glaser approach to perturbative quantum field theory (which leads directly to a finite perturbative series and does not rely on an intermediate regularization). Our construction can be regarded as a direct implementation of Noether's method at the quantum level. We illustrate the method by constructing the pure Yang-Mills theory (where the relevant global symmetry is BRST symmetry), and the N = 1 supersymmetric model of Wess and Zumino. The whole construction is done before the so-called adiabatic limit is taken. Thus, all considerations regarding symmetry, unitarity and anomalies are well defined even for massless theories.     

Ab initio dynamics of the He + H+ 2 → HeH+ +H reaction: a new potential energy surface and quantum mechanical cross-sections

The reaction He + H⁺ ₂(v,j = 0) → HeH⁺(v′ = 0, j′) for v = 0, 1,2 and 3 and for scattering energies near the threshold (0.95–1.15 eV) has been studied by calculating ab initio points at MRCI level and ‘exact’ integral quantum reactive cross-sections. More than 1400 nuclear geometries have been chosen to cover the most important regions for the dynamics, an extended set of points being taken directly on a hyperspherical coordinate grid. A many-body expansion with a large number of terms permits an accurate analytical representation of the potential energy surface with a root-mean-square deviation <12meV. The hyperquantization algorithm has been extended to obtain quantum mechanical integral cross-sections which are compared with previous calculations and with experimental results.     

键长-键角和Radau坐标下哈密顿算符在傅里叶基组表象下的厄米性

在量子动力学计算中,有时候为了规避奇点问题或者节省计算量,我们经常需要对哈密顿量进行变换. 然而,在使用傅里叶基矢计算时,哈密顿量的变换形式容易导致哈密顿矩阵失去厄米性,进而有些情况下使数值计算变得不稳定. 本文主要讨论构建具有厄米性的哈密顿算符的方法. 以三原子分子为例,构建了键长—键角和Radau坐标下描述分子运动的各种形式的哈密顿量. 基于这些哈密顿量,采用含时波包方法计算了OClO分子的吸收光谱,讨论了非厄米性矩阵对计算结果的影响. 本文所得到的结论对基于基函数展开的量子动力学计算都是适用的.     

对数饱和非线性介质中的自洽多模高斯孤子解

通过解对数饱和非线性介质中光场满足的非线性薛定谔方程,得到一组厄米-高斯型的自洽多模解.在借鉴了R.G.Glauber的相干态理论的基础上,合理假设这组解在非线性介质中呈泊松分布,进而得到了在对数饱和非线性介质中存在高斯孤子的结论,并获得高斯孤子解、非线性系数与泊松参量三者之间的关系.该关系说明,若在介质中存在高斯孤子解,其非线性系数必须满足条件a≥1.当α=1,在介质中仅存在单模高斯孤子,其光斑尺寸必须满足的条件W=1/k n0/n2.在该条件下,以"束腰"注入介质中的高斯光束才可以在保持其光斑尺寸不变的情况下传输,否则光斑尺寸会存在一定的振荡.而振荡方式、振荡范围(表示光斑尺寸展宽或压缩及大小)与输入光束光斑尺寸及其一次导数有直接关系.     

Ortho-para H₂ conversion by proton exchange at low temperature: an accurate quantum mechanical study

We report extensive, accurate fully quantum, time-independent calculations of cross sections at low collision energies, and rate coefficients at low temperatures for the H? + H?(v = 0, j) → H? + H?(v = 0, j') reaction. Different transitions are considered, especially the ortho-para conversion (j = 1 → j' = 0) which is of key importance in astrophysics. This conversion process appears to be very efficient and dominant at low temperature, with a rate coefficient of 4.15 × 10?1? cm3 molecule?1 s?1 at 10 K. The quantum mechanical results are also compared with statistical quantum predictions and the reaction is found to be statistical in the low temperature regime (T < 100 K).     

A theoretical model for quantum nanostructures electronic wave functions, magnetic field effects

Analytical solutions of electronic wave functions in symmetric quantum ring (QR), quantum wire (QWR) and quantum dots (QD) structures are given using a parabolic coordinates system. The solutions for low-energy states are combinations of Bessel functions. The density of states of perfect 1D QR and QWR are shown to be equivalent. The continuous evolution from a 0D QD to a perfect 1D QR can be precisely described. The sharp variation of electronic properties, related to the build up of a potential energy barrier at the early stage of the QR formation, is studied analytically. Paramagnetic and diamagnetic couplings to a magnetic field are computed for QR and QD. It is shown theoretically that magnetic field induces an oscillation of the magnetization in QR.     

Quantum Teleportation and Resonance Information Transfer from One Atom to Another at Arbitrary Interatomic Distances

The feasibility of resonance transfer of quantum information from one double-level atom to another that is at an arbitrary distance from the former one has been proved. Symmetric and antisymmetric combinations of the wave functions of individual atoms are considered. When taking into account the interatomic dipole–dipole interaction, a certain energy corresponds to each wave function. A solution has been found to a system of equations for the amplitudes of the probability that a resonance photon will be absorbed by one of the system atoms, and it has been shown that the interaction of the system with actual photons has the result that the wave function of the final state of the system can be represented as a linear combination of the functions < 00|, < 0n|, and < n0| corresponding to the ground and excited states of individual atoms. The amplitude of the probability of each of these states depends on the interatomic distance and on the parameters of the action of actual photons on atoms. Three types of solution to the system of equations have been investigated for the resonance and nonresonance absorption of photons and different interatomic distances. It has been shown that when atoms are at an infinite distance from one another, so that there is no dipole–dipole interaction of atoms, quantum information can be transferred from one atom to another with a characteristic time considerably shorter than the time it takes for a photon to cover the interatomic distance. This effect is referred to as the effect of quantum teleportation in a system of resonance atoms.     

具有Dzyaloshinskii-Moriya相互作用的XY模型的量子相干性

研究了具有Dzyaloshinskii-Moriya(DM)相互作用的一维横场XY自旋链的量子相变和量子相干性.采用约旦-维格纳变换严格求解了哈密顿量,并描绘了体系的关联函数和相图,相图包含反铁磁相、顺磁相和螺旋相.利用相对熵和Jensen-Shannon熵讨论了XY模型的量子相干性.研究发现,相对熵与Jensen-Shannon熵所表现的行为都可以很好地表征该模型的量子相变.非螺旋相中量子相干性不依赖DM相互作用,而在螺旋相DM相互作用对量子相干性有显著影响.此外,指出了在带有DM相互作用的这一类反射对称破缺体系中关联函数计算的常见问题.