Presenting Different Time Steps, at the Start of Inflation, Using Kiefer Density Matrix, for the Use of an Inflaton, in Determining Different Conceivable Time Intervals for Time Flow Analysis

We are using the book “Towards Quantum Gravity with an article by Claus Kiefer as to a quantum gravity interpretation of the density matrix in the early universe. The density matrix we are using is a one loop approximation, with inflaton value and potential terms, like V(phi) using the Padmanabhan values one can expect if the scale factor is a ~ a(Initial) times t ^ gamma , from early times . In doing so, we isolate out presuming a very small initial time step candidates initial time values which are from a polynomial for time values due to the Kiefer Density value.

CPT-violating z = 3 Horava–Lifshitz QED and generation of the Carroll–Field–Jackiw term

We study the new extension of the [Formula: see text] Horava–Lifshitz QED involving a CPT-breaking term, characterized by the axial vector [Formula: see text], and calculate the Carroll–Field–Jackiw (CFJ) term in the one-loop approximation. Explicitly, we use two regularization schemes and demonstrate that in our case, the CFJ term is finite but ambiguous, so that its exact coefficient depends on the used regularization.

The renormalization group in perturbative quantum gravity

This is a short chapter summarizing the main results concerning the renormalization group in models of pure quantum gravity, without matter fields. The chapter starts with a critical analysis of non-perturbative renormalization group approaches, such as the asymptotic safety hypothesis. After that, it presents solid one-loop results based on the minimal subtraction scheme in the one-loop approximation. The polynomial models that are briefly reviewed include the on-shell renormalization group in quantum general relativity, and renormalization group equations in fourth-derivative quantum gravity and superrenormalizable models. Special attention is paid to the gauge-fixing dependence of the renormalization group trajectories.

Boundary conformal field theory at the extraordinary transition: The layer susceptibility to O(ε)

We present an analytic calculation of the layer (parallel) susceptibility at the extraordinary transition in a semi-infinite system with a flat boundary. Using the method of integral transforms put forward by McAvity and Osborn [Nucl. Phys. B455 (1995) 522] in the boundary CFT, we derive the coordinate-space representation of the mean-field propagator at the transition point. The simple algebraic structure of this function provides a practical possibility of higher-order calculations. Thus we calculate the explicit expression for the layer susceptibility at the extraordinary transition in the one-loop approximation. Our result is correct up to order O(ε) of the ε = 4 − d expansion and holds for arbitrary width of the layer and its position in the half-space. We discuss the general structure of our result and consider the limiting cases related to the boundary operator expansion and (bulk) operator product expansion. We compare our findings with previously known results and less complicated formulas in the case of the ordinary transition. We believe that analytic results for layer susceptibilities could be a good starting point for efficient calculations of two-point correlation functions. This possibility would be of great importance given the recent breakthrough in bulk and boundary conformal field theories in general dimensions.

Application of a computer algebra systems to the calculation of the \(\pi\pi\)-scattering amplitude

The aim of this work is to develop a set of programs for calculation the scattering amplitudes of the elementary particles, as well as automating the calculation of amplitudes using the appropriate computer algebra systems (Mathematica, Form, Cadabra). The paper considers the process of pion-pion scattering in the framework of the effective Nambu-Iona-Lasinio model with two quark flavours. The Package-X for Mathematica is used to calculate the scattering amplitude (starting with the calculation of Feynman diagrams and ending with the calculation of Feynman integrals in the one-loop approximation). The loop integrals are calculated in general kinematics in Package-X using the Feynman parametrization technique. A simple check of the program is made: for the case with zero temperature, the scattering lengths \(a_0 = 0.147\) and \(a_2 = -0.0475\) are calculated and the total cross section is constructed. The results are compared with other models as well as with experimental data.

Identification of Chaboche–Lemaitre combined isotropic–kinematic hardening model parameters assisted by the fuzzy logic analysis

A very good knowledge of material properties is required in the analysis of severe plastic deformation problems in which the classical material processing methods are accelerated by the application of the additional cyclic load. A general fuzzy logic-based approach is proposed for the analysis of experimental and numerical data in this paper. As an application of the fuzzy analysis, the calibration of Chaboche–Lemaitre model hardening parameters of PA6 aluminum is considered here. The experimental data obtained in a symmetrical strain-controlled cyclic tension–compression test were used to estimate the material’s hardening parameters. The numerically generated curves were compared to the experimental ones. For better fitting of numerical and experimental results, the optimization approach using the least-square method was applied. Unfortunately, commonly accepted calibration methods can provide various sets of hardening parameters. In order to choose the most reliable set, the fuzzy analysis was used. Primarily selected values of hardening parameters were assumed to be fuzzy input parameters. The error of the hysteresis loop approximation for each set was used to compute its membership function. The discrete value of this error was obtained in the defuzzification step. The correct selections of hardening parameters were verified in ratcheting and mean stress relaxation tests. The application of the fuzzy analysis has improved the convergence between experimental and numerical stress–strain curves. The fuzzy logic allows analyzing the variation of elastic–plastic material response when some imprecisions or uncertainties of input parameters are taken into consideration.

On the creation of charged massless fermion pair by a photon in an external constant uniform magnetic field in 2+1 dimensions

Creation of charged massless fermion pair by a photon in a constant uniform magnetic field is considered in the one-loop approximation of the [Formula: see text]-dimensional quantum electrodynamics (QED[Formula: see text]). We calculate the elastic scattering amplitude (EAS) of photon using the polarization operator of photon in the above magnetic field obtained earlier in our work. We analyze the elastic scattering amplitude of photon at various values of the photon energy and magnetic field strength. Very simple analytical formulas for EAS of photon are obtained in the so-called quasi-classical region. In the massive quantum electrodynamics the elastic scattering amplitude of photon defines its “mass” squared in the presence of external electromagnetic field and the imaginary part of EAS describes the total probability of charged massive fermion pair creation in the electromagnetic field; we assume that it is the case and in the massless quantum electrodynamics.

The all-loop perturbative derivation of the NSVZ $$\beta $$-function and the NSVZ scheme in the non-Abelian case by summing singular contributions

The perturbative all-loop derivation of the NSVZ $$\beta $$ β -function for $${{\mathcal {N}}}=1$$ N = 1 supersymmetric gauge theories regularized by higher covariant derivatives is finalized by calculating the sum of singularities produced by quantum superfields. These singularities originate from integrals of double total derivatives and determine all contributions to the $$\beta $$ β -function starting from the two-loop approximation. Their sum is expressed in terms of the anomalous dimensions of the quantum gauge superfield, of the Faddeev–Popov ghosts, and of the matter superfields. This allows obtaining the NSVZ equation in the form of a relation between the $$\beta $$ β -function and these anomalous dimensions for the renormalization group functions defined in terms of the bare couplings. It holds for an arbitrary renormalization prescription supplementing the higher covariant derivative regularization. For the renormalization group functions defined in terms of the renormalized couplings we prove that in all loops one of the NSVZ schemes is given by the HD + MSL prescription.

Off-shell renormalization in the presence of dimension 6 derivative operators. II. Ultraviolet coefficients

The full off-shell one loop renormalization for all divergent amplitudes up to dimension 6 in the Abelian Higgs-Kibble model, supplemented with a maximally power counting violating higher-dimensional gauge-invariant derivative interaction $$\sim g ~ \phi ^\dagger \phi (D^\mu \phi )^\dagger D_\mu \phi $$ ∼ g ϕ † ϕ ( D μ ϕ ) † D μ ϕ , is presented. This allows one to perform the complete renormalization of radiatively generated dimension 6 operators in the model at hand. We describe in details the technical tools required in order to disentangle the contribution to ultraviolet divergences parameterized by (generalized) non-polynomial field redefinitions. We also discuss how to extract the dependence of the $$\beta $$ β -function coefficients on the non-renormalizable coupling g in one loop approximation, as well as the cohomological techniques (contractible pairs) required to efficiently separate the mixing of contributions associated to different higher-dimensional operators in a spontaneously broken effective field theory.