The widely used nonperturbative wave functions and distribution functions of QCD are determined as matrix elements of light-ray operators. These operators appear as large momentum limit of nonlocal hadron operators or as summed up local operators in light-cone expansions. Nonforward one-particle matrix elements of such operators lead to new distribution amplitudes describing both hadrons simultaneously. These distribution functions depend besides other variables on two scaling variables. They are applied for the description of exclusive virtual Compton scattering in the Bjorken region near forward direction and the two meson production process. The evolution equations for these distribution amplitudes are derived on the basis of the renormalization group equation of the considered operators. This includes that also the evolution kernels follow from the anomalous dimensions of these operators. Relations between different evolution kernels (especially the Altarelli-Parisi and the Brodsky-Lepage) kernels are derived and explicitly checked for the existing two-loop calculations of QCD. Technical basis of these results are support and analytically properties of the anomalous dimensions of light-ray operators obtained with the help of the α-representation of Green's functions.
Bordag, Geyer, Klimchitskaya, and Mostepanenko Reply: We confirm the conclusions of [1] that the results of [2] are incorrect. The preceding Comment [3] on [1] in support of [2] is invalid for the reasons discussed below.In the beginning of the Comment the well-known equations of the Lifshitz theory of the Casimir force between dielectrics at nonzero temperature are restated [Eqs.(1)- (3)]. The problem under discussion is connected with their zero-frequency contribution to the Casimir force. According to [3] at zero frequency the function f͑k, v͒ characterizing the contribution of the TE mode has the value f͑k, 0͒ 0. This is correct only on the basis of Eq. (3) of [3] if one assumes that k fi 0 and
We present a direct approach for the calculation of functional determinants of the Laplace operator on balls. Dirichlet and Robin boundary conditions are considered. Using this approach, formulas for any value of the dimension, D, of the ball, can be obtained quite easily. Explicit results are presented here for dimensions D = 2, 3, 4, 5 and 6. *
The Casimir and van der Waals interaction between two dissimilar thick dielectric plates is reconsidered on the basis of thermal quantum field theory in Matsubara formulation. We briefly review two main derivations of the Lifshitz formula in the framework of thermal quantum field
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