Renormalization group and scale invariance in terms of asymptotic fields

The expansion of the massive renormalized field operator in terms of asymptotic fields is studied. We derive the renormalization group equation for the renormalized field operator. We obtain the renormalized scale transformations with Callan-Symanzik corrections as generated by canonical scale transformations of asymptotic fields.     

Statistical behaviors for renormalization of correlated permeability field

In this article, the statistical properties for the renormalized permeability obtained from the renormalization of the correlated permeability field are investigated. In contrast to the uncorrelated porous media, the scaling of the variance of the renormalized permeability field exhibits a crossover behavior. When the correlation lengths are larger compared with the domain scale covered by the renormalization procedure, the variance of the renormalized permeability will decrease slowly and the scaling exponent will be close to zero. As the renormalization number increases, the covered domain scale will eventually become larger than the correlation lengths, and then the scaling property will transit to the uncorrelated case. The convergent values of the renormalized permeability for isotropic and anisotropic correlated media are also investigated. Both the theoretical analysis and the simulation results show that larger correlation length in one direction will lead to a larger convergent value in the corresponding direction. For the log-normal permeability field, numerical simulations show that the crossover scaling and also the convergent value for the renormalized permeability can be fitted very well by simple mathematical functions.     

On the Connections between the Non-linearity of Equations of Motion in Quantum Theory,Asymptotic Behavior and Renormalization

It is shown that the ordinary perturbation expressions used in quantum mechanics lead to the wrong asymptotic behavior of the Heisenberg observables as function of time. This difficulty is traced to the non-linearity of the Heisenberg equations of motion and is studied in the context of a one-dimensional non-linear oscillator problem. It is found that the correct asymptotic behavior can be obtained by a process of renormalization analogous to renormalization theory in quantum field theory. It turns out that the renormalized parameters analogous to mass and wave-function renormalization are not c-numbers but are instead q-numbers. It is suggested that the renormalization parameters of quantum field theory are also q-numbers.     

Reduction of radiative transfer problems in semi-infinite media to linear Fredholm integral equations

Linear Fredholm integral equations are derived for the Stokes vector of polarized radiation, emergent from a scattering plane parallel semi-infinite medium, by means of the full range orthogonality and completeness properties of Case's eigensolutions. A renormalization concerning the eigenmode with the greatest discrete eigenvalue is applied, which permits us to obtain a new integral equation for the zeroth Fourier component of the radiation field. The kernel of the integral equations is given in terms of Case's eigenfunctions or of the Green's function matrix for an infinite medium. For isotropic scattering, it is shown that the integral equation can be solved by means of a very rapidly convergent Neumann series. Physical arguments lead to the conclusion that the renormalized Fredholm integral equations are well suited also for arbitrary phase matrices.     

基于有限元法的光子并矢格林函数重整化及其在自发辐射率和能级移动研究中的应用

任意微纳结构中量子点的自发辐射率和能级移动均可用并矢格林函数表达.当源点和场点在同一位置时,格林函数的实部是发散的.为解决这一发散问题,可采用重整化格林函数方法.本文提出一种计算重整化格林函数和散射格林函数的方法.该方法利用有限元,计算点电偶极子的辐射场,将其在量子点体积内做平均得到重整化的并矢格林函数,减去均匀空间中解析的重整化格林函数,得到重整化的散射格林函数.在均匀空间情况下,本方法所得数值结果与解析解一致.将该方法应用到银纳米球系统,以解析的散射格林函数作为参考,结果表明该方法能准确处理散射格林函数的重整化问题.将该方法应用到表面等离激元纳米腔中,发现有极大的自发辐射增强和能级移动,且该结果不依赖于量子点的体积.这些研究在光与物质相互作用领域具有积极的意义.     

Renormalization group and Fermi liquid theory

We give a Hamiltonian-based interpretation of microscopic Fermi liquid theory within a renormalization group framework. The Fermi liquid fixed-point Hamiltonian with its leading-order corrections is identified and we show that the mean field calculations for this model correspond to the Landau phenomenological approach. This is illustrated first of all for the Kondo and Anderson models of magnetic impurities which display Fermi liquid behaviour at low temperatures. We then show how these results can be deduced by a reorganization of perturbation theory, in close parallel to that for the renormalized φ⁴ field theory. The Fermi liquid results follow from the two lowest order diagrams of the renormalized perturbation expansion. The calculations for the impurity models are simpler than for the general case because the self-energy depends on frequency only. We show, however, that a similar renormalized expansion can be derived also for the case of a translationally invariant system. The parameters specifying the Fermi liquid fixed-point Hamiltonian are related to the renormalized vertices appearing in the perturbation theory. The collective zero sound modes appear in the quasiparticle-quasihole ladder sum of the renormalized perturbation expansion. The renormalized perturbation expansion can in principle be used beyond the Fermi liquid regime to higher temperatures. This approach should be particularly useful for heavy fermions and other strongly correlated electron systems, where the renormalization of the single-particle excitations are particularly large.

We briefly look at the breakdown of Fermi liquid theory from a renormalized perturbation theory point of view. We show how a modified version of the approach can be used in some situations, such as the spinless Luttinger model, where Fermi liquid theory is not applicable. Other examples of systems where the Fermi liquid theory breaks down are also briefly discussed.     

Wilsonian RG and redundant operators in nonrelativistic effective field theory

In a Wilsonian renormalization group (RG) analysis, redundant operators, which may be eliminated by using field redefinitions, emerge naturally. It is therefore important to include them. We consider a nonrelativistic effective theory (the so-called “pionless” nuclear effective field theory) as a concrete example and show that the off-shell amplitudes cannot be renormalized if the redundant operators are not included. The relation between the theories with and without such redundant operators is established in the low-energy expansion. We perform a Wilsonian RG analysis for the off-shell scattering amplitude in the theory with the redundant operator.     

高频波激发低频磁场和离子声波的重整化强湍动理论

本文给出了高温等离子体中高频波激发低频磁场和离子声波强湍动过程的重整化理论,以便改善通常的弱非线性处理方法,从Vlasov-Maxwell方程组出发,在Fourier表象中得到了包含“最发散”和“次发散”效应互相耦合的高频和低频传播于重整化方程组,从而获得了高、低频振荡粒子重整化分布函数和场的耦合关系。在“最发散”重整化近似下,我们求解了高低频传播子方程组,得到了展开到v₄(高频湍动场能密度与等离子体热能密度之比)一次方的近似解和重整化介电函数等表达式。然后,在Fourier逆变换下导得了我们所要的时空表用中重整化强湍动方程组。最后,作为一个说明重整化作用的例子,在一维稳态下求解了孤立子的形式。 关键词：     

Renormalization of massless fields with hard-soft infrared decomposition

The decomposition of Feynman integrals with massless propagators into hard and soft contributions is systematically effected in renormalized field theory. It is shown that the decomposition leads to an elegant method of renormalizing massless field theories. Ultraviolet and infrared finite composite fields (normal products) are defined and renormalized field equations are derived. Exploiting a gauge principle, scalar ghosts arising in the hard-soft decomposition are eliminated and a renormalization group equation is derived to describe the effects of changes in the mass scale.     

无碰撞等离子体的相干重整化理论

本文提出对Власов-Poisson方程进行微扰处理的一种重整化方案。利用图形展开方法证明了该理论到任意阶微扰的可重整化性质。给出了重整化传播量的一般形式。分析了相干项和绝对非相干项的物理意义。给出了重整化介电函数的正确表示,并对它的意义做了讨论。通过和以往重整化理论的比较,指出这种重整化方案是一种真正的完全重整化。 关键词：     

Improved renormalization group equations with dimensional regularization and the ε expansions

Due to the absence of dimensional cut-off parameters in the dimensional regularization scheme, vanishing of the renormalized mass of the scalar boson implies vanishing of its renormalized mass; thus the masses of both bosons and fermions in renormalizable field theories can be made finite by multiplicative mass renormalizations. The improved renormalization group equations in D dimensions are derived in such a way that both the large (or the small) momentum limits and the Wilson ? expansions can be uniformly treated for the fermion as well as the boson cases. We discuss the improved equations for φ₆³ theory, φ₄⁴ theory, quantumelectrodynamics, massive vector-gluon model, and non-Abelian guage theories incorporating fermions. For the latter three classes of theories, the gauge dependent problem of the coefficient functions in the improved renormalization group equations is discussed.     

Study of a two-dimensional model with axial-current-pseudoscalar derivative interaction

A detailed study is made of a massive pseudoscalar field interacting via derivative coupling with massless fermions in two-dimensional space-time. The model provides an example of a soluble renormalizable theory with an anomalous axial-vector current and a zero-mass particle interpretation for the fermion. Except for a finite mass and wavefunction renormalization, the boson remains free in the presence of the interaction. The canonical fermion field exhibits an anomalous dimension that is found to be in agreement with the asymptotic Callan-Symanzik equation. The connection between the Wilson expansion for defining operator products in this model and the Dyson equations of renormalized perturbation theory is discussed, and agreement with second-order perturbation theory is verified by explicit calculation.     

The renormalized σ model

A renormalization procedure of the boson σ model based on the finite field equations of Brandt-Wilson is given. We first show that the current operators appearing in the field equations, which are finite local limit of sums of nonlocal field products and suitable subtraction terms, can be chosen to be the same form as the one given for the symmetric limit except for the symmetry breaking constant source term itself. The set of integral equations derived from the field equations is shown to be equivalent to the usual Bogoliubov-Parasiuk-Hepp renormalization theory, and gives us immediately all the renormalized Green's functions in each order of perturbation theory in clear and straightforward fashion. We then analyze the structures of the model in detail. In particular, Ward identities are shown to be satisfied to all orders of perturbation theory. The Goldstone theorem is a particular consequence of these identities.     

Field Equations in Quantum Electrodynamics

A formulation of quantum electrodynamics is presented, based on finite local field equations. These Dirac and Maxwell equations have the usual form except that the current operators f(x) and j^μ (x) are explicitly expressed as local limits of sums of non-local field products and suitable subtraction terms. These limits are shown to exist and to yield finite operators in the sense that the iterative solutions to the field equations are equivalent to conventional renormalized perturbation theory. The various invariance properties of the theory, including Lorentz invariance, gauge invariance, charge conjugation invariance, and renormalization invariance, are discussed and related directly to the field equations and current definitions. Initially only the general forms of the currents, based on dimensional arguments, are given. The electric current, for example, contains the (suitably defined) term :A³(x) :.The corresponding field equations are used to derive renormalized Dyson-Schwinger-type integral equations for the renormalized proper part functions ∑, II^μν, Λ^μ, and X^αβγδ (the four-photon vertex function), etc. Application of the boundary conditions ∑(p̀ = m) = ∑′(p̀ = m) = II(O) = II′(O) = II″(O) = Λ(p̀ = m, o) = X(O, O, O, O) = O completely specifies the current operators. Consistency is established by deriving the same equations from rigorous renormalization theory so that their iterative solutions are proved to reproduce the correct renormalized perturbation expansion. The electric current operator is exhibited in a manifestly gauge invariant form and in a form which is manifestly negative under charge conjugation. It is shown, in fact, that much of j^μ (x) can be determined directly from the requirements of gauge invariance and charge conjugation covariance, without recourse to the integral equations. It is suggested that equal time commutation relations can serve to similarly specify the rest of the current.     

非对称Anderson模型的重整微扰展开

利用Yamada微扰论结合重整微扰方法来计算非对称Anderson模型,得到了局域电子占据数、重整化因子、重整化的局域能级以及重整化参数关于裸参数的展开式.计算了局域电子态密度和低温杂质电导,还计算了磁场对它们的影响,这些结果适用于从弱耦合到强耦合的整个耦合强度区域.由于在哈密顿量中已经将局域能级进行了初步重整,采用的重整微扰方法比Hewson的重整微扰方法更适合于研究非对称Anderson模型. 关键词： 非对称Anderson模型 重整化 磁场     

An exact,finite, gauge-invariant,non-perturbative approach to QCD renormalization

A particular choice of renormalization, within the simplifications provided by the non-perturbative property of Effective Locality, leads to a completely finite, non-perturbative approach to renormalized QCD, in which all correlation functions can, in principle, be defined and calculated. In this Model of renormalization, only the Bundle chain-Graphs of the cluster expansion are non-zero. All Bundle graphs connecting to closed quark loops of whatever complexity, and attached to a single quark line, provided no ‘self-energy’ to that quark line, and hence no effective renormalization. However, the exchange of momentum between one quark line and another, involves only the cluster-expansion’s chain graphs, and yields a set of contributions which can be summed and provide a finite color-charge renormalization that can be incorporated into all other QCD processes. An application to High Energy elastic pp scattering is now underway.     

Renormalized cluster expansion for multiple scattering in disordered systems

We study wave propagation in a disordered system of scatterers and derive a renormalized cluster expansion for the optical potential or self-energy of the average wave. We show that in the problem of multiple scattering a repetitive structure of Ornstein-Zernike type may be detected. We derive exact expressions for two elementary constituents of the renormalized scattering series, called the reaction field operator and the short-range connector. These expressions involve sums of integrals of a product of a chain correlation function and a nodal connector. We expect that approximate calculation of the reaction field operator and the short-range connector allows one to find a good approximation to the self-energy, even for high density of scatterers. The theory applies to a wide variety of systems.     

Renormalized field theory of critical dynamics

We formulate a Gell'Mann-Low-type renormalization group approach to the critical dynamics of stochastic models described by Langevin or Fokker-Planck equations including mode-coupling terms.Dynamical correlation and response functions are expressed in terms of path integrals, which are investigated by well-known methods of renormalized perturbation theory.Dynamical scaling laws and relations between static and dynamic critical exponents are derived. The leading temperature-dependence of correlation and response functions is obtained from the Kadanoff-Wilson short-distance expansion. We also consider corrections to dynamic scaling which are due to a finite lattice constant.     

Infrared problems in quantum electrodynamics

We present a self-contained treatment of the infrared problem in Quantum Electrodynamics. Our program includes a derivation and proof of finiteness of modified reduction formulae for scattering in Coulomb potentials and unitary extensions of the relativistic Coulomb amplitudes in the forward direction. The renormalization structure of the theory is discussed in connection with the infrared problem and the renormalization group is reconsidered and shown to be inadequate for the “improvement” of perturbation theoretic results. However, simple forms of the renormalization group equations are easily established, which allow for a simple discussion of the renormalization structure and the extraction of physical quantities out of Green functions normalized at an arbitrary mass μ < m (m is the fermion mass). As an example of such a quantity we consider the construction of a renormalized and infrared finite mass-operator in presence of external fields. Scattering theory in Quantum Electrodynamics is elaborated in the context of the coherent state formulation of the asymptotic condition. Dimensional regularization techniques are systematically used for the reduction of coherent states and the construction of S-matrix elements and the cross-section formulae. The latter are obtained in a relatively simple form, which allows for a direct comparison with the exact cross-section formulae derived in the traditional context. This establishes the equivalence of the two approaches at the cross-section level. Various applications illustrate the techniques presented here and relative topics are discussed.     

Small distance behaviour in field theory and power counting

For infinitesimal changes of vertex functions under infinitesimal variation of all renormalized parameters, linear combinations are found such that the net infinitesimal changes of all vertex functions are negligible relative to those functions themselves at large momenta in all orders of renormalized perturbation theory. The resulting linear first order partial differential equations for the asymptotic forms of the vertex functions are, in quantum electrodynamics, solved in terms of one universal function of one variable and one function of one variable for each vertex function whereby, in contrast to the renormalization group treatment of this problem, the universal function is obtained from nonasymptotic considerations. A relation to the breaking of scale invariance in renormalizable theories is described.