Becchi–Rouet–Stora–Tyutin (BRST) symmetry. Gauge theories: Zinn-Justin equation (ZJ) and renormalization

The first part of the chapter describes Faddeev–Popov's quantization method, nd the resulting Slavnov–Taylor (ST) identities, in a simple context. This construction automatically implies, after introduction of Faddeev–Popov ‘ghost’ fermions, a Becchi–Rouet–Stora–Tyutin (BRST) symmetry, whose properties are derived. The differential operator, of fermionic type, representing the BRST symmetry, with a proper choice of variables, has the form of a cohomology operator, and a simple form in terms of Grassmann coordinates. The second part of the chapter is devoted to the quantization and renormalization of non-Abelian gauge theories. Quantization of gauge theories require a gauge-fixing procedure. Starting from the non-covariant temporal gauge, and using a simple identity, one shows the equivalence with a quantization in a general class of gauges, including relativistic covariant gauges. Adapting the formalism developed in the first part, ST identities, and the corresponding BRST symmetry are derived. However, the explicit form of the BRST symmetry is not stable under renormalization. The BRST symmetry implies a more general, quadratic master equation, also called Zinn-Justin (ZJ) equation, satisfied by the quantized action, equation in which gauge and BRST symmetries are no longer explicit. By contrast, in the case of renormalizable gauges, the ZJ equation is stable under renormalization, and its solution yields the general form of the renormalized gauge action.

Non-Abelian gauge theories: Introduction

To be able to describe the other fundamental interactions, beyond quantum electrodynamics (QED), weak and strong interactions, it is necessary to generalize the concept of gauge symmetry to non-Abelian groups. Therefore, in this chapter, a quantum field theory (QFT)-invariant under local, that is, space-time-dependent, transformations of matrix representations of a general compact Lie groups are constructed. Inspired by the Abelian example, the geometric concept of parallel transport is introduced, a concept discussed more extensively later in the framework of Riemannian manifolds. All the required mathematical quantities for gauge theories then appear naturally. Gauge theories are quantized in the temporal gauge. The equivalence with covariant gauges is then established. Some formal properties of the quantized theory, like the Becchi–Rouet–Stora–Tyutin (BRST) symmetry, are derived. Feynman rules of perturbation theory are derived, the regularization of perturbation theory is discussed, a somewhat non-trivial problem. Some general properties of the non-Abelian Higgs mechanism are described.

Fermionic degeneracy and non-local contributions in flag-dipole spinors and mass dimension one fermions

We construct a mass dimension one fermionic field associated with flag-dipole spinors. These spinors are related to Elko (flag-pole spinors) by a one-parameter matrix transformation $${\mathcal {Z}}(z)$$ Z ( z ) where z is a complex number. The theory is non-local and non-covariant. While it is possible to obtain a Lorentz-invariant theory via $$\tau $$ τ -deformation, we choose to study the effects of non-locality and non-covariance. Our motivation for doing so is explained. We show that a fermionic field with $$|z|\ne 1$$ | z | ≠ 1 and $$|z|=1$$ | z | = 1 are physically equivalent. But for fermionic fields with more than one value of z, their interactions are z-dependent thus introducing an additional fermionic degeneracy that is absent in the Lorentz-invariant theory. We study the fermionic self-interaction and the local U(1) interaction. In the process, we obtained non-local contributions for fermionic self-interaction that have previously been neglected. For the local U(1) theory, the interactions contain time derivatives that renders the interacting density non-commutative at space-like separation. We show that this problem can be resolved by working in the temporal gauge. This issue is also discussed in the context of gravity.

On plasmon contribution to the hot A0 condensate

In SU(2) gluodynamics, the Debye gluon contribution WD(A0) to the effective action of the temporal gauge field component (we consider A0 = const) in the background Rξext is calculated at high temperaturegauge. It is shown that at A0 ≠ 0 the standard definition k0 = 0 , |k| → 0 corresponds to long distance correlations for the longitudinal in internal space gluons. The transversal gluons become screened by the A0 background field. Therefore, they give zero contributions and have to be excluded from the correlation corrections. The total effective action accounting for the one-loop, two-loop, and correct WD(A0) satisfies Nielsen’s identity that proves gauge invariance of the A0 condensation phenomenon.

ALTERAÇÕES DEMOGRÁFICAS NO ESPAÇO MONTANHOSO DOS CÁRPATOS: UM ESTUDO DE CASO SOBRE AS MONTANHAS APUSENI (ROMÊNIA), ENTRE 1850 E 2011

Through this study we wanted to make an emphasis on demographic changes in the Apuseni Mountains, analysing the numerical evolution of the population. The period under consideration is between 1850 and 2011, and due to the high temporal gauge we decided to divide into six smaller periods. The necessary data were obtained from the consultation of population censuses and on the website of the National Institute of Statistics. Microsoft Excel 2013 was used to calculate the demographic growth and decrease values. After obtaining the values, they were processed through Geographic Information Systems (ArcGis 10.3 program), finally a series of maps were made. During the study, we found that demographic evolution was closely related to certain decisions, phenomena. Activities in the mining sector, administrative reforms and changing political regimes from socialist to capitalist were the main causes leading to demographic growth and decrease in the Apuseni Mountains from 1850-2011. Towards the end of the 20th century, the demographic component began to face risk phenomena such as the migration of the young population, demographic ageing and depopulation.

The Cauchy Problem for Space-Time Monopole Equations in Temporal and Spatial Gauge

We prove global existence of solution to space-time monopole equations in one space dimension under the spatial gauge condition A1=0 and the temporal gauge condition A0=0.