scholarly journals A modified gravity model coupled to a Dirac field in 2D space–times with quadratic nonmetricity and curvature

2021 ◽  
Author(s):  
Caglar Pala ◽  
Ertan Kok ◽  
Ozcan Sert ◽  
Muzaffer Adak
Keyword(s):  
Two Dimensions ◽  
Exterior Algebra ◽  
Dirac Field ◽  
Dirac Spinor ◽  
Field Equations ◽  
Independent Variables ◽  
Basic Concepts ◽  
Gauge Structure ◽  
2D Space ◽  
Vacuum Theory

After summarizing the basic concepts for the exterior algebra, we first discuss the gauge structure of the bundle over base manifold for deciding the form of the gravitational sector of the total Lagrangian in any dimensions. Then we couple minimally a Dirac spinor field to our gravitational Lagrangian 2-form which is quadratic in the nonmetricity and both linear and quadratic in the curvature in two dimensions. Subsequently, we obtain field equations by varying the total Lagrangian with respect to the independent variables. Finally, we find some classes of solutions of the vacuum theory and then a solution of the Dirac equation in a specific background and analyze them.

scholarly journals Some nozzle flows found by the hodograph method

1959 ◽  
Vol 1 (1) ◽  
pp. 80-94 ◽  
Author(s):  
T. M. Cherry
Keyword(s):  
Exact Solutions ◽  
Two Dimensions ◽  
Nozzle Design ◽  
Perfect Gas ◽  
Field Equations ◽  
Rigid Boundary ◽  
Hodograph Method ◽  
Isentropic Flow ◽  
Fixed Boundaries ◽  
Nozzle Flows

For investigating the steady irrotational isentropic flow of a perfect gas in two dimensions, the hodograph method is to determine in the first instance the position coordinates x, y and the stream function ψ as functions of velocity compoments, conveniently taken as q (the speed) and θ (direction angle). Inversion then gives ψ, q, θ as functions of x, y. The method has the great advantage that its field equations are linear, so that it is practicable to obtain exact solutions, and from any two solutions an infinity of others are obtainable by superposition. For problems of flow past fixed boundaries the linearity of the field equations is usually offset by non-linearity in the boundary conditions, but this objection does not arise in problems of transsonic nozzle design, where the rigid boundary is the end-point of the investigation.

Green function formulation of the Dirac field in curved space

1966 ◽  
Vol 294 (1439) ◽  
pp. 437-448 ◽  
Keyword(s):  
Green Function ◽  
Dirac Field ◽  
Green Functions ◽  
Field Equations ◽  
Curved Space ◽  
Particle Field ◽  
Function Formulation ◽  
Essential Idea ◽  
Mass Field

A Green function formulation of the Dirac field in curved space is considered in the cases where the mass is constant and where it is regarded as a direct particle field in the manner of Hoyle & Narlikar (1964 c ). This description is equivalent to, and in some ways more satisfactory than, that given in terms of a suitable Lagrangian, in which the Dirac or the mass field is regarded as independent of the geometry. The essential idea is to define the Dirac or the mass field in terms of certain Green functions and sources so that the field equations are satisfied identically, and then to obtain the contribution of these fields to the metric field equations from the variation of a suitable action that is defined in terms of the Green functions and sources.

Generalized non-linear σ-models in two dimensions

2021 ◽  
pp. 692-720
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Symmetric Spaces ◽  
Homogeneous Spaces ◽  
Two Dimensions ◽  
Asymptotic Freedom ◽  
Quantum Field Theories ◽  
Field Equations ◽  
Local Conservation ◽  
Non Linear ◽  
Coset Spaces ◽  
Formal Properties

This chapter describes the formal properties, and discusses the renormalization, of quantum field theories (QFT) based on homogeneous spaces: coset spaces of the form G/H, where G is a compact Lie group and H a Lie subgroup. In physics, they appear naturally in the case of spontaneous symmetry breaking, and describe the interaction between Goldstone modes. Homogeneous spaces are associated with non-linear realizations of group representations. There exist natural ways to embed these manifolds in flat Euclidean spaces, spaces in which the symmetry group acts linearly. As in the example of the non-linear σ-model, this embedding is first used, because the renormalization properties are simpler, and the physical interpretation of the more direct correlation functions. Then, in a generic parametrization, the renormalization problem is solved by the introduction of a Becchi–Rouet–Stora–Tyutin (BRST)-like symmetry with anticommuting (Grassmann) parameters, which also plays an essential role in quantized gauge theories. The more specific properties of models corresponding to a special class of homogeneous spaces, symmetric spaces (like the non-linear σ-model), are studied. These models are characterized by the uniqueness of the metric and thus, of the classical action. In two dimensions, from the classical field equations an infinite number of non-local conservation laws can be derived. The field and the unique coupling renormalization group (RG) functions are calculated at one-loop order, in two dimensions, and shown to imply asymptotic freedom.

scholarly journals ON THE HAMILTONIAN FORMULATION OF THE EINSTEIN–HILBERT ACTION IN TWO DIMENSIONS

2006 ◽  
Vol 21 (11) ◽  
pp. 899-905 ◽  
Author(s):  
N. KIRIUSHCHEVA ◽  
S. V. KUZMIN
Keyword(s):  
Hamiltonian Formulation ◽  
Two Dimensions ◽  
General Covariance ◽  
Sufficient Condition ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Metric Tensor ◽  
Gauge Transformations ◽  
Total Divergence ◽  
Hilbert Action

It is shown that if general covariance is to be preserved (i.e. a coordinate system is not fixed) the well-known triviality of the Einstein field equations in two dimensions is not a sufficient condition for the Einstein–Hilbert action to be a total divergence. Consequently, a Hamiltonian formulation is possible without any modification of the two-dimensional Einstein–Hilbert action. We find the resulting constraints and the corresponding gauge transformations of the metric tensor.

scholarly journals A generally relativistic gauge classification of the Dirac fields

2016 ◽  
Vol 13 (06) ◽  
pp. 1650078 ◽  
Author(s):  
Luca Fabbri
Keyword(s):  
Axial Vector ◽  
Supplementary Information ◽  
Dirac Field ◽  
Field Equations ◽  
Momentum Vector ◽  
Nonrelativistic Approximation ◽  
Gauge Transformations ◽  
Dirac Fields

We consider generally relativistic gauge transformations for the spinorial fields finding two mutually exclusive but together exhaustive classes in which fermions are placed adding supplementary information to the results obtained by Lounesto, and identifying quantities analogous to the momentum vector and the Pauli–Lubanski axial vector. We discuss how our results are similar to those obtained by Wigner by taking into account the system of Dirac field equations. We will investigate the consequences for the dynamics and in particular we shall address the problem of getting the nonrelativistic approximation in a consistent way. We are going to comment on extensions.

scholarly journals Noether Gauge Symmetry of Dirac Field in (2 + 1)-Dimensional Gravity

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Ganim Gecim ◽  
Yusuf Kucukakca ◽  
Yusuf Sucu
Keyword(s):  
Gauge Symmetry ◽  
Early Time ◽  
Gravitational Theory ◽  
Dirac Field ◽  
Late Time ◽  
Field Equations ◽  
Dynamical Equations ◽  
Symmetry Approach ◽  
Cosmological Solutions ◽  
Dimensional Gravity

We consider a gravitational theory including a Dirac field that is nonminimally coupled to gravity in 2 + 1 dimensions. Noether gauge symmetry approach can be used to fix the form of coupling functionF(Ψ)and the potentialV(Ψ)of the Dirac field and to obtain a constant of motion for the dynamical equations. In the context of (2 + 1)-dimensional gravity, we investigate cosmological solutions of the field equations using these forms obtained by the existence of Noether gauge symmetry. In this picture, it is shown that, for the nonminimal coupling case, the cosmological solutions indicate both an early-time inflation and late-time acceleration for the universe.

scholarly journals Unification of Gravity and Electromagnetism

2021 ◽  
Keyword(s):  
Gravitational Field ◽  
Energy Momentum Tensor ◽  
Mathematical Structure ◽  
Momentum Tensor ◽  
Dirac Field ◽  
Field Equations ◽  
Mathematical Structures ◽  
Gravitational Field Equations ◽  
Definition Of ◽  
Two Sides

Gravity and electromagnetism are two sides of the same coin, which is the clue of this unification. Gravity and electromagnetism are representing by two mathematical structures, symmetric and antisymmetric respectively. Einstein gravitational field equation is the symmetric mathematical structure. Electrodynamics Lagrangian is three parts, for electromagnetic field, Dirac field and interaction term. The definition of canonical energy momentum tensor was used for each term in Electrodynamics Lagrangian to construct the antisymmetric mathematical structure. Symmetric and antisymmetric gravitational field equations are two sides of the same Lagrangian

scholarly journals Multinational qEEG developmental surfaces

2019 ◽  
Author(s):  
Shiang Hu ◽  
Ally Ngulugulu ◽  
Jorge Bosch-Bayard ◽  
Maria L. Bringas-Vega ◽  
Pedro A. Valdes-Sosa
Keyword(s):  
Power Spectra ◽  
Two Dimensions ◽  
Spectral Features ◽  
Linear Mixed Effects ◽  
Independent Variables ◽  
Eeg Power Spectra ◽  
The Usa ◽  
Specific Factors ◽  
Quantitative Electroencephalogram ◽  
The Individual

AbstractThe quantitative electroencephalogram (qEEG) is a diagnostic method based on the spectral features of the resting state EEG. The departure of spectral features from normality is gauged by the z transform with respect to the age-adjusted mean and deviation of normative databases – known as the developmental equations/surfaces. However, the extent to which the data collected from different countries with various equipment require separate developmental equations remains unanswered. Here, we analyzed the EEG of 535 subjects from 3 countries, Switzerland, the USA and Cuba. The EEG power spectra of all samples were log transformed and their relations to the covariables (‘age’, ‘frequency’, ‘country’ and ‘individual’) were analyzed using the linear mixed effects model. We found that the origin ‘country’ of the subjects did not play a significant effect on the log spectra, even without interactions with other independent variables, whereas, ‘age’ and ‘frequency’ were highly significant. To estimate the developmental surfaces in greater detail, we carried out kernel regression (lowess) in two dimensions of log-age and frequency. We found two main phenomena: 1) slow rhythms (δ, θ) predominated in the lower ages and then decreased with a tendency to disappear at higher ages; 2) α rhythm was absent at lower ages, but gradually appeared more relevant in occipital and parietal regions, and increased with aging with an increasing centering frequency of α rhythm. We consider both phenomena as an expression of healthy neurodevelopmental and maturation related to age. It is the first study of multinational qEEG developmental surfaces accounting for ‘country’. The results demonstrate the possibility of creating international qEEG norms since the ‘individual’ and ‘age’ variability are much larger than the specific factors like ‘country’, or the technology employed ‘device’.

A non-linear theory of strong interactions

1958 ◽  
Vol 247 (1249) ◽  
pp. 260-278 ◽  
Keyword(s):  
Linear Theory ◽  
Two Dimensions ◽  
Strong Interactions ◽  
Isotopic Spin ◽  
Meson Field ◽  
Field Equations ◽  
Constant Length ◽  
Cubic Symmetry ◽  
Continuous Rotation ◽  
Non Linear

A non-linear theory of mesons, nucleons and hyperons is proposed. The three independent fields of the usual symmetrical pseudo-scalar pion field are replaced by the three directions of a four-component field vector of constant length, conceived in an Euclidean four-dimensional isotopic spin space. This length provides the universal scaling factor, all other constants being dimensionless; the mass of the meson field is generated by a ϕ 4 term; this destroys the continuous rotation group in the iso-space, leaving a ‘cubic’ symmetry group. Classification of states by this group introduces quantum numbers corresponding to isotopic spin and to ‘strangeness’; one consequence is that, at least in elementary interactions, charge is only conserved modulo 4. Furthermore, particle states have not a well-defined parity, but parity is effectively conserved for meson-nucleon interactions. A simplified model, using only two dimensions of space and iso-space, is considered further; the non-linear meson field has solutions with particle character, and an indication is given of the way in which the particle field variables might be introduced as collective co-ordinates describing the dynamics of these particular solutions of the meson field equations, suggesting a unified theory based on the meson field alone.

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