Anomaly Problem in the N=1 Supersymmetric Electrodynamics as a Consequence of the Inconsistency of the Dimensional Reduction Method

We compare the calculation of the two-loop -function in the N=1 supersymmetric electrodynamics regularized via higher derivatives and via dimensional reduction. We show that the renormalized effective action is the same for both regularizations. But in the method of higher derivatives, unlike in the dimensional reduction, the -function defined as the derivative of the renormalized coupling constant with respect to log turns out to be purely one-loop. The anomaly problem therefore does not occur in this regularization, because in the method of higher derivatives, the diagrams with counterterm insertions make a nonzero contribution, which is evaluated exactly in all orders of the perturbation theory. When dimensional reduction is used, this contribution is zero. We argue that this result is a consequence of the mathematical inconsistency of the dimensional reduction method and that just this inconsistency leads to the anomaly problem.     

Verifying a new identity for Green’s functions of the n=1 supersymmetric non-Abelian Yang-Mills theory with matter fields

We use the higher covariant derivative regularization to investigate a new identity for Green’s functions. It relates certain coefficients of the matter superfield vertex function for which one of the external matter legs is not chiral. Calculations in the first nontrivial order (for the two-loop vertex function) demonstrate that the new identity also holds in the non-Abelian Yang-Mills theory with matter fields. We demonstrate that the new identity follows because the three-loop integrals determining the Gell-Mann-Low function are integrals of total derivatives. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 270–281, August, 2008.     

Relativistic dynamics for spin-zero and spin-half particles

The classical and quantum mechanics of a system of directly interacting relativistic particles is discussed. We first discuss the spin-zero case, where we basically follow Rohrlich in introducing a set of covariant centre of mass (CM) and relative variables. The relation of these to the classic formulation of Bakamjian and Thomas is also discussed. We also discuss the important case of relativistic potentials which may depend on total four-momentum squared. We then consider the quantum mechanical case of spin-half particles. The negative energy difficulty is solved by introducing a number of first class constraints which fix the spinor structure of physical solutions and ensure the existence of proper CM and relative variables. We derive the form of interactions consistent with Lorentz invariance, space inversion, time reversal and charge conjugation and with the above mentioned first class constraints and find that it is analogous to that for the non-relativistic case. Finally the relationship of the present work with some previous work is briefly discussed.     

Two-loop Gell-Mann-Low function of the <Emphasis Type="Italic">N</Emphasis>=1 supersymmetric Yang-Mills theory regularized by higher covariant derivatives

We calculate the two-loop Gell-Mann-Low function for the N=1 supersymmetric Yang-Mills theory regularized by higher covariant derivatives. We show that the integrals determining this function reduce to total derivatives and can be easily calculated analytically. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 3, pp. 398–414, June, 2008.     

Second-Order Covariant Tensor Decomposition in Curved Spacetime

Local four-dimensional tensor decomposition formulae for generic vectors and 2-tensors in spacetime, in terms of scalar and antisymmetric covariant tensor potentials, are studied within the framework of tensor distributions. Earlier first-order decompositions are extended to include the case of four-dimensional symmetric 2-tensors and new second-order decompositions are introduced.     

Wick-Ordered Entire Functions of the Indefinite Metric Free Field

We present a simple and general method for constructing Wick-ordered entire functions of free fields with an indefinite metric, based on using an appropriate generalization of the Paley–Wiener–Schwartz theorem.     

Functors of the category of abelian sheaves on regular affine scheme

In this short paper, we prove that ifR is a regular local ring of unequal characteristic, then there exists an additive covariant functorG from the category of abelian sheaves on SpecR to the category of abelian groups such that id _R (G(R))>dimG(R). This result shows that the answer to the question 3.8 (ii) in [3] may be negative.     

Twisted Semigroup Crossed Products and Twisted Toeplitz Algebras of Ordered Groups

Let г^＋ be the positive cone of a totally ordered abelian group г, and σa cocycle in г. We study the twisted crossed products by actions of г＋ as endomorphisms of C^＊-algebras, and use this to generalize the theorem of Ji.