Continuum regularization of quantum field theory

As an extension of our earlier one-loop renormalization studies at the regularized Schwinger-Dyson level, we report here on equivalent renormalization programs for regularized Langevin systems. Proper structure is discussed, and proper one-loop renormalizations of the Green functions of φ ₆ ³ and QCD₄ are given. An optional apparent?-renormalization is discussed as a technical simplificaiton for gauge theories with Zwanziger's gauge-fixing.     

Ward identity for gauge field in stochastic quantization with nonlocal form factors

In the framework of the stochastic quantization method with nonlocal form factors, two-, three- and four-point correlation functions for the Yang-Mills field and function are calculated. A Schwinger-Dyson renormalization program is formulated for regularized QCD₄. It is shown that the gauge fixing due to Zwanziger does not break gauge invariance and that the Ward identity is also fulfilled.     

Implications of Analyticity to the Solution of Schwinger-Dyson Equations in Minkowski Space

We discuss some recent developments in nonperturbative studies of quantum field theory (QFT) using the Schwinger-Dyson equations formulated directly in Minkowski space. We begin with the introduction of essential ideas of the integral representation in QFT and a discussion of the renormalization in this approach. The technique based on the integral representation of Green’s functions is exploited to solve Schwinger-Dyson equations in several models of quantum field theory, e.g., in scalar models and in strong coupling QED₃₊₁ in the quenched and in the unquenched approximation. The phenomenon of dynamical chiral symmetry breaking in regularized theory is touched. In QCD, the analyticity of the gluon propagator on the complex momentum square plane is exploited to continue some recent lattice data to the timelike momentum axis. We find a contribution to the non-positive absorptive part in the Landau-gauge gluon propagator which is in agreement with some other new recent analyses.     

Lectures on the functional renormalization group method

These introductory notes are about functional renormalization group equations and some of their applications. It is emphasised that the applicability of this method extends well beyond critical systems, it actually provides us a general purpose algorithm to solve strongly coupled quantum field theories. The renormalization group equation of F. Wegner and A. Houghton is shown to resum the loop-expansion. Another version, due to J. Polchinski, is obtained by the method of collective coordinates and can be used for the resummation of the perturbation series. The genuinely non-perturbative evolution equation is obtained by a manner reminiscent of the Schwinger-Dyson equations. Two variants of this scheme are presented where the scale which determines the order of the successive elimination of the modes is extracted from external and internal spaces. The renormalization of composite operators is discussed briefly as an alternative way to arrive at the renormalization group equation. The scaling laws and fixed points are considered from local and global points of view. Instability induced renormalization and new scaling laws are shown to occur in the symmetry broken phase of the scaler theory. The flattening of the effective potential of a compact variable is demonstrated in case of the sine-Gordon model. Finally, a manifestly gauge invariant evolution equation is given for QED.     

THE FOKKER-PLANCK EQUATiON AND THE SCHWI NGER-DYSON EQUATION IN LATTICE GAUGE THEORY

The stochastic quantization in Lattice Gauge Theory (LGT) is discussed by using Langevin equations and Fokker-Planck equations. It is shown that the evolution equation in stochastic process reduces to Schwinger-Dyson equation when lattice system reaches equilibrium.     

The Fokker-Planck equations in lattice gauge theories

The stochastic quantization in lattice gauge theories (LGT) is discussed by using Langevin equations and Fokker-Planck equations. It is shown that the evolution equations in the stochastic process reduce to the Schwinger-Dyson equation when the lattice system reaches equilibrium.     

格点规范场理论中Wilson圈变量的Schwinger-Dyson方程和普适性问题

本文引入了相当广泛的几类Wilson作用量,推导了与这些作用量相应的Wilson圈变量的Schwinger-Dyson方程,并证明在连续极限下,这些方程有着相同的形式,从而说明这些作用量是等价的。 关键词：     

Computing the couplings of Ising systems from Schwinger-Dyson equations

We study the problem of determining the couplings of Ising systems in terms of expectation values. We find that all previously known methods can be based on Schwinger-Dyson equations. Indeed, new methods can be derived with relative advantages with respect to all of them. Application for the renormalization group flow of the Ising model on a square lattice is presented.     

Generalized Wilson expansions in dimensional renormalization

Operator product expansions in the framework of dimensional regularization and renormalization are discussed. Following the definition of a subtraction operator for dimensionally regularized and points-split Green's functions, a generalized Wilson expansion pansion is proved. The terms of the expansion are normal products defined via dimensional renormalization, and the coefficients are doubly regularized with singularities in the physical dimension as the spacial separations of the product fields vanish, or at zero separations as the dimension of space-time becomes physical.     

Nonlocal stochastic quantization of scalar electrodynamics

Quantization of the electromagnetic interactions of scalar charged particles is considered within the stochastic Langevin and Schwinger-Dyson equations with nonlocal white noise. Fulfillment of the gauge-invariant condition in such a scheme is studied in detail. Matrix elements of the vacuum polarization and self-energy diagrams of the scalar electrodynamics are calculated explicitly, which reduce to usual nonlocal scalar electrodynamic results.     

Quasiparticle renormalization of the memory function approach for response functions

The problem of renormalization of the double-time Green function method for response functions is treated in the framework of the Mori-theory for the special case of Fermi liquids and finite Fermi systems. It is shown that the quasiparticle-quasihole renormalization of the response function can be carried out under conditions which are physically equivalent to the conditions under which the quasiparticle-quasihole renormalization is performed in the many-time Green function theory of Fermi liquids.     

On dynamical gluon mass generation

The effective gluon propagator constructed with the pinch technique is governed by a Schwinger-Dyson equation with special structure and gauge properties, that can be deduced from the correspondence with the background field method. Most importantly the non-perturbative gluon self-energy is transverse order-by-order in the dressed loop expansion, and separately for gluonic and ghost contributions, a property which allows for a meanigfull truncation. A linearized version of the truncated Schwinger-Dyson equation is derived, using a vertex that satisfies the required Ward identity and contains massless poles. The resulting integral equation, subject to a properly regularized constraint, is solved numerically, and the main features of the solutions are briefly discussed.     

A renormalization procedure for the correlation function theory of finite Fermi systems

The quasiparticle renormalization of symmetrized correlation functions is treated in the framework of the double-time Green function theory of many body systems. The work is based on the Mori-theory of response functions transcribed for symmetrized correlation functions. For the specific example of finite Fermi systems it is shown that the physical situation assumed in the quasiparticle-quasihole renormalization of the many-time Green function theory allows to define an equivalent renormalization procedure for correlation functions. This procedure uses projection operator techniques and is therefore of purely algebraic nature.     

Continuum-regularized quantum gravity

The recent continuum regularization ofd-dimensional Euclidean gravity is generalized to arbitrary power-law measure and studied in some detail as a representative example of coordinate-invariant regularization. The weak-coupling expansion of the theory illustrates a generic geometrization of regularized Schwinger-Dyson rules, generalizing previous rules in flat space and flat superspace. The rules are applied in a non-trivial explicit check of Einstein invariance at one loop: The cosmological counterterm is computed and its contribution is included in a verification that the graviton mass is zero.     

IR renormalization of general effective actions and Hawking flux in 2D gravity theories

The infrared problem of the effective action in 2D is discussed in the framework of the covariant perturbation theory. The divergences are regularized by a mass and the leading term is evaluated up to the third order of perturbation theory. A summation scheme is proposed which isolates the divergences from the finite part of the series and results in a single term. The latter turns out to be equivalent to the coupling to a certain classical external field. This suggests renormalization by factorization. PACS 4.60.-m; 4.60.Kz; 4.70.Dy     

The renormalization flow in the adjoint SU(2) lattice Higgs model

Applying the Monte Carlo renormalization group method we investigate the flow of coupling constants for the 8⁴ and 16⁴ SU(2) lattice Higgs model with triplet Higgs fields. The couplings of the renormalized actions are determined using Schwinger-Dyson equations. From the flow we find new evidence for the existence of a tricritical point at finite values of the inverse gauge coupling β but no indication for a discontinuity fixed point.     

On the equivalence of generalized Langevin equation and generalized master equation

It is shown that the generalized Langevin equation derived by Mori is equivalent to the generalized master equation of Nakajima and Zwanzig. Both equations of motion are related, just in the same way as Heisenberg and Schrödinger picture are related.     

What can we learn from the finite temperature renormalization group?

The renormalization group for finite temperature quantum field theories is studied, in particular for λ?⁴. It is shown that the “high” temperature limit can only be discussed perturbatively ifT dependent renormalization schemes are implemented. Zero temperature renormalization schemes or renormalization at some fixed reference temperatureT _o are both inadequate as they imply perturbative expansions about fixed points of the renormalization group which are associated with a zero temperature system and a system at temperatureT _o respectively.T dependent schemes give rise to an expansion about the true fixed point of the system, the resulting renormalization group allows the entire crossover between high and low temperature behaviour to be investigated.