Hyperfunction quantum field theory II. Euclidean Green's functions

The axioms for Euclidean Green's functions are extended to hyperfunction fields without being supplemented by any condition like the linear growth condition of Osterwalder and Schrader.     

Theory of fourier hyperfunctions and its applications to quantum field theory

A new type of Fourier hyperfunctions is introduced. The axiomatic quantum field theory in terms of Fourier hyperfunctions is shown to include Wightman's formulation of tempered fields and its generalizations. The complete equivalence is established between the axioms for Wightman Fourier hyperfunctions and those for Green's functions by eliminating from the latter the linear growth condition of Osterwalder and Schrader.     

Euclidean Green's functions for Jaffe fields

We extend the axioms for Euclidean Green's functions recently proposed by Osterwalder and Schrader to Jaffe fields.     

On the equivalence of the Euclidean and Wightman formulation of field theory

A mistake in the paper [1] on the “Axioms for Euclidean Green's Functions” is corrected in the following sense: thanks to these axioms the Euclidean Schwinger functionsS _n can be analytically continued to the corresponding Wightman functionsW _n possessing all the correct analyticity properties and satisfying a generalized positivity condition in the complex domain. It is however suggested by the proof that their tempered behaviour near the Minkowski points cannot be guaranteed without additional assumptions.     

Remarks on some new models of interacting quantum fields with indefinite metric

We study quantum field models in indefinite metric. We introduce the modified Wightman axioms of Morchio and Strocchi as a general framework of indefinite metric quantum field theory (QFT) and present concrete interacting relativistic models obtained by analytical continuation from some stochastic processes with Euclidean invariance. As a first step towards scattering theory in indefinite metric QFT, we give a proof of the spectral condition on the translation group for the relativistic models.     

Convolution of Lorentz Invariant Ultradistributions and Field Theory

A general definition of convolution between two arbitrary four-dimensional Lorentz invariant (fdLi) tempered ultradistributions is given, in both Minkowski and Euclidean space (spherically symmetric tempered Ultradistributions). The product of two arbitrary fdLi distributions of exponential type is defined via the convolution of its corresponding Fourier transforms. Several examples of convolution of two fdLi tempered ultadisrtibutions are given. In particular, we calculate exactly the convolution of two Feynman's massless prapagators. An expression for the Fourier transform of a Lorentz invariant tempered ultradistribution in terms of modified Bessel distributions is obtained in this work (generalization of Bochner's formula to Minkowski space). From the deduction of the convoltion formula, we obtain the generalization to the Minkowski space, of the dimensional regularization of the perturbation theory of Green functions in the Euclidean configuration space given in Erdelyi (Higher Transcendental Functions, 1953). As an example we evaluate the convolution of two n-dimensional complex-mass Wheeler propagators.     

Closed-form solutions for continuous time random walks on finite chains

Continuous time random walks (CTRWs) on finite arbitrarily inhomogeneous chains are studied. By introducing a technique of counting all possible trajectories, we derive closed-form solutions in Laplace space for the Green's function (propagator) and for the first passage time probability density function (PDF) for nearest neighbor CTRWs in terms of the input waiting time PDFs. These solutions are also the Laplace space solutions of the generalized master equation. Moreover, based on our counting technique, we introduce the adaptor function for expressing higher order propagators (joint PDFs of time-position variables) for CTRWs in terms of Green's functions. Using the derived formula, an escape problem from a biased chain is considered.     

Localized partial traps in diffusion processes and random walks

Reaction-diffusion equations, in which the reaction is described by a sink term consisting of a sum of delta functions, are studied. It is shown that the Laplace transform of the reactive Green's function can be analytically expressed in terms of the Green's function for diffusion in the absence of reaction. Moreover, a simple relation between the Green's functions satisfying the radiation boundary condition and the reflecting boundary condition is obtained. Several applications are presented and the formalism is used to establish the relationship between the time-dependent geminate recombination yield and the bimolecular reaction rate for diffusion-influenced reactions. Finally, an analogous development for lattice random walks is presented.     

On the decomposition of wightman functionals in the euclidean framework

We consider Wightman functionals having the property that the Schwinger functions can be represented by a Euclidean invariant measure on the space of tempered distributions. The strong form of the Osterwalder-Schrader positivity condition is shown to imply that the measure is positive and some restrictions on the Schwinger functions are discussed which guarantee that this condition holds. The ergodic decomposition of the measure leads to a decomposition of the Wightman functional into such with cluster property. We discuss also the role of the positivity condition in connection with a general criterion for the existence of a decomposition.     

基于动态散斑时频特性的合作目标微动参数反演

基于激光动态散斑的时频信号,建立了一套适用于角反射器阵列的微动参数反演算法。首先利用物理光学方法推导了角反射器及角反射器阵列激光散斑的实时强度公式,然后基于短时傅里叶变换研究了散斑功率谱形成机制及其数字特征,最后提出了频谱相关法及时频-相幅变换算法,并采用该方法提取了动态散斑时频谱线周期及振幅分布,反演了三种典型运动状态下目标的自旋周期及旋转轴指向。结果表明:基于几个周期的散斑强度序列,所提反演算法可以得到高精度的旋转周期及视线角,但对旋转轴方位角的反演精度相对较差,需要更多的观测数据才能得到满意的结果。     

基于约束优化的多光谱辐射真温反演算法

多光谱辐射测温是通过测量待测物某点的多个光谱辐射强度信息,通过普朗克公式反演获得真实温度。但是,通过普朗克公式获得的多光谱辐射测温方程组,是欠定方程组,即N个方程,N+1个未知数(N个未知的光谱发射率ε_λi和1个待求真温T)。目前,多采用事先假设一组发射率模型(发射率-波长或发射率-温度模型),假设模型与实际情况如果相符,则反演结果能够满足要求,如果假设模型与实际情况不符,则反演结果误差很大。但是,发射率模型受温度、表面状态、波长等诸多因素影响,难以事先确定发射率模型。因此受未知光谱发射率的制约一直是多光谱辐射测温理论面临的主要障碍,能否在无需任何光谱发射率假设模型的情况下,实现真温和光谱发射率的直接反演一直是多光谱辐射测温理论研究的热点和难点。通过对参考温度模型的分析表明,多光谱辐射测温反演过程的实质是寻找一组光谱发射率,使得每个通道方程解得的真温都相同,如不相同则继续寻找合适的光谱发射率,直到每个通道解得的真温都相等。为此,提出将多光谱辐射测温参考温度模型的求解过程转换为约束优化问题,即在光谱发射率0≤ε_λi≤1的约束条件下,通过梯度投影算法不断寻找光谱发射率,带入多光谱辐射测温参考温度模型方程组后,计算温度反演值的方差,直到每个光谱通道方程获得的温度值应该近似相等,此时各个光谱通道的温度反演值方差最小,这样就把多光谱辐射真温和发射率的反演问题转换为约束优化问题。约束优化算法是解决这一类问题的主要方法,但为了满足Ax≥b的约束条件,将0≤ε_λi≤1分解为ε_λi≥0和-ε_λi≥-1的两个约束条件,从而满足了约束优化问题Ax≥b的约束条件。这样就可以通过约束优化算法在无需任何光谱发射率假设模型的条件下,直接求解真温和光谱发射率。实验采用六种不同光谱发射率分布模式(随波长递增、递减、凸波动、凹波动、“M”型波动、“W”型波动)的材料为研究对象,以验证新算法对不同材料光谱发射率分布反演的适应性,利用Matlab的minRosen函数,选择光谱发射率的初始值均为0.5(取中间值,提高计算效率)。针对六种不同光谱发射率模型的仿真结果表明,新算法无需任何有关发射率的先验知识,对不同发射率模型反演结果均表现较好,在真温1 800 K的情况下,绝对误差均小于20 K,相对误差均小于1.2%,新算法具有无需考虑任何光谱发射率先验知识、反演精度较高及适合于各种发射率模型等优点,进一步完善了多光谱辐射测温理论,在高温测量领域具有良好的应用前景。     

Scalar Green's functions in an Euclidean space with a conical-type line singularity

In an Euclidean space with a conical-type line singularity, we determine the Green's function for a charged massive scalar field interacting with a magnetic flux running through the line singularity. We give an integral expression of the Green's function and a local form in the neighbourhood of the point source, where it is the sum of the usual Green's function in Euclidean space and a regular term. As an application, we derive the vacuum energy-momentum tensor in the massless case for an arbitrary magnetic flux.Supported by a grant from CNPq (Brazilian government agency FA)     

Optimal Filtering Applied to the Inversion of the Laplace Transform

Inversion of the Laplace transform, used in the laser scattering measurement of colloidal particle size distributions, presents severe numerical difficulties. In the presence of noise the variance of the inversion integral is infinite, indicating maximum uncertainty in the inversion. This paper applies the method of minimum variance, or “optimal”, filtering to the eigenfunction spectrum of the Laplace transform, giving an inversion which has finite variance. Spectral decomposition using the eigenfunctions of the Laplace transform gives a representation of the noise and desired signals analogous to the Fourier spectrum used in linear system theory. It is possible to obtain a filtered estimate of the unknown linewidth distribution. The requirement that the variance of this filtered estimate is minimum leads to a Wiener-Hopf integral equation defining the optimal filter. The results of this paper provide a basis of comparison of all methods of inversion of the Laplace transform, including the extensive literature of colloidal particle sizing by laser scattering or photon correlation.     

Determination of Particle Size Distribution by the analysis of intensity-constrained multi-angle photon correlation spectroscopic data

Laser light scattering (LLS), especially dynamic laser light scattering (DLS), also known as photon correlation spectroscopy (PCS), is a well established method for particle size distribution analysis. It usually involves a Laplace inversion of the field autocorrelation function. However, the resolution is limited because of the ill-conditioned nature of this Laplace inversion. No unique solution exists when noise is present on the data. In contrast with this ill-conditioned nature, the angular dependence of scattered (static) intensities is precisely not ill-conditioned, which allows the resolution of the ill-conditioned inversion of DLS data to be improved. In order to characterize samples with more complicated size distributions, an intensityconstrained multi-angle PCS data analysis program has been developed, which is an alternative way of normalizing the field correlation function to that reported by Cummins and Staples [12]. In this program, the field autocorrelation function is normalized to the scattering intensity by using a predetermined coherent factor at each angle, which provides an additional constraint on the Laplace inversion of multi-angle PCS data analysis. The alternative analysis improves the resolution of PCS and provides a more reliable particle size distribution than single-angle data analysis. Both simulated and measured LLS data are used to illustrate its application, resolution and limitations.     

Euclidean Fermi fields with a hermitean Feynman-Kac-Nelson formula. I

We construct free, Euclidean, spin one-half, quantum fields with the following properties: (i) CAR; (ii) Symanzik positivity; (iii) Osterwalder-Schrader positivity; (iv) no doubling of particle or spin states. They admit the recovery of the relativistic Dirac field by the Osterwalder-Schrader technique. We then formally parametrize interacting theories by a natural class of Hermitean, Euclidean actions, and obtain a simple, Hermitean, Feynman-Kac-Nelson formula. The interacting theory formally obeys all the properties (i)–(iv), and admits the reconstruction of a physical Hilbert space, including a Hermitean, contraction semigroup for the Wick rotated time evolution. We propose a system of axioms for the interacting theory.     

Retrieval of transverse relaxation time distribution from spin-echo data by recurrent neural network

Inversion of transverse relaxation time decay curve from spin-echo experiments was carried out using Hopfield neural network, to obtain the transverse relaxation time distribution. The performance of this approach was tested against simulated and experimental data. The initial guess, necessary for the integration procedure, was established as the analytical Laplace inversion. Together with errors in the simulated data, inversion was also carried out with errors in this initial guess. The probability density function, calculated by the neural network, is used in multiple sclerosis diagnostics.     

Scattering formalism for non-localizable fields

We consider the theory of a non-localizable relativistic quantum field. Nonlocalizability means that the field is not a tempered distribution, but increases strongly for large momenta. Local commutativity can then not be satisfied. Instead we assume the existence of Green's functions with the usual analyticity properties. We show that in such a theory theS-matrix can be defined, and its elements can be expressed in terms of the fields by the usual reduction formulae.     

Construction of the Green's function for the pseudoharmonical oscillator

In the present paper we have constructed the Green's function for the pseudoharmonical potential, which is considered as an intermediate potential between the harmonic and anharmonic potentials. We have used a hybrid method, by combining the Laplace transformation method and the Green's function technique. The Green's function is used for obtaining the density matrix for a quantum-statistical system, in coordinate representation. Even if this is not a new result, the method can be applied to a class of exactly solvable potentials.     

Adiabatic pumping through interacting quantum dots

We present a general formalism to study adiabatic pumping through interacting quantum dots. We derive a formula that relates the pumped charge to the local, instantaneous Green's function of the dot. This formula is then applied to the infinite-U Anderson model for both weak and strong tunnel-coupling strengths.     

Kinetic equation for electrons of a semiconductor in a strong electric field: Green's resolvent

Using the Laplace transform of Boltzmann's equation, a system of linking equations is obtained for the series expansion of the mapping of the distribution function in powers of the field. Since the form of the equations is identical, instead of direct integration it is possible to use the method of Green's functions for the solution of the system. Using the method of perturbances Green's resolvent is constructed in the form of an infinite series. Using the example of the evaluation of the distribution function in a high-frequency field, the convergence of the proposed method is demonstrated.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, Vol. 16, No. 7, pp. 22–25, July, 1973.