scholarly journals An analysis of the first-order form of gauge theories

2012 ◽  
Vol 90 (2) ◽  
pp. 165-174 ◽  
Author(s):  
N. Kiriushcheva ◽  
S.V. Kuzmin ◽  
D.G.C. McKeon
Keyword(s):  
Gauge Theory ◽  
Vector Fields ◽  
Three Dimensional ◽  
Mass Term ◽  
Maxwell Theory ◽  
First Order ◽  
Order Form ◽  
Constraint Structure ◽  
The Relationship ◽  
Hilbert Action

The first-order form of a Maxwell theory and U(1) gauge theory in which a gauge invariant mass term appears is analyzed using the Dirac procedure. The form of the gauge transformation that leaves the action invariant is derived from the constraints present. A nonabelian generalization is similarly analyzed. This first-order three dimensional massive gauge theory is rewritten in terms of two interacting vector fields. The constraint structure when using light-cone coordinates is considered. The relationship between first- and second-order forms of the two-dimensional Einstein–Hilbert action is explored where a Lagrange multiplier is used to ensure their equivalence.

The Theoretical Approach to the Improvement of the Interpolation Error

2009 ◽  
pp. 58-71
Author(s):  
Carlo Ciulla
Keyword(s):  
Three Dimensional ◽  
Interpolation Error ◽  
The Novel ◽  
Interpolation Function ◽  
Linear Quadratic ◽  
Partial Derivatives ◽  
First Order ◽  
Sampling Locations ◽  
Theory Equation ◽  
The Relationship

In the sections of this chapter the reader will be introduced to the sequence of mathematical processes which, starting from a model interpolation function yield to the corresponding SRE-based paradigm. Particularly, this chapter addresses the development of the SRE-based bivariate interpolation function. The mathematical procedure is consistently iterated in the rest of the book for all the other model functions that the unifying theory embraces. The first step of the procedure is that of the calculation of the intensity-curvature terms and through their ratio the Intensity-Curvature Functional is calculated for the model function. The second step is that of calculating the first order partial derivatives of the Intensity-Curvature Functional. Thirdly, the polynomial consisting of the first order partial derivatives is solved to obtain the Sub-pixel Efficacy Region. At this point, the formula of the unifying theory (equation [21]) sets the stage to obtain the novel re-sampling locations. Worth noting that this formula can be adapted to cover cases of one, two, and three dimensional interpolation functions and it is also consistently employed for linear quadratic cubic and trigonometric (Sinc) models. This shall be manifest throughout the remainder of the book. The remainder of this chapter discusses on the nature of the SRE, also makes a connection with Chapter XX of the book within the context of the relationship existing between resolution and interpolation error, and in the last section, the concept of resilient interpolation is introduced and the relevant math is illustrated for the case of the bivariate linear function.

STUDY OF Z(N) GAUGE THEORIES ON A THREE-DIMENSIONAL PSEUDORANDOM LATTICE

1988 ◽  
Vol 03 (06) ◽  
pp. 1499-1518
Author(s):  
D. PERTERMANN ◽  
J. RANFT
Keyword(s):  
Phase Transitions ◽  
Gauge Theory ◽  
Gauge Theories ◽  
Lattice Gauge Theory ◽  
Three Dimensional ◽  
Expectation Values ◽  
First Order ◽  
Gauge Models ◽  
Lattice Gauge

Using the simplicial pseudorandom version of lattice gauge theory we study simple Z(n) gauge models in D=3 dimensions. In this formulation it is possible to interpolate continuously between a regular simplicial lattice and a pseudorandom lattice. Calculating average plaquette expectation values we look for the phase transitions of the Z(n) gauge models with n=2 and 3. We find all the phase transitions to be of first order, also in the case of the Z(2) model. The critical couplings increase with the irregularity of the lattice.

scholarly journals CANONICAL FORMULATION OF A BOSONIC MATTER FIELD IN (1 + 1)-DIMENSIONAL CURVED SPACE

2007 ◽  
Vol 22 (26) ◽  
pp. 4833-4848 ◽  
Author(s):  
R. N. GHALATI ◽  
D. G. C. MCKEON ◽  
T. N. SHERRY
Keyword(s):  
Scalar Field ◽  
Affine Connection ◽  
Matter Field ◽  
Degree Of Freedom ◽  
Curved Space ◽  
First Order ◽  
Order Form ◽  
Canonical Formulation ◽  
Dynamical Degree ◽  
Hilbert Action

We study a bosonic scalar in (1 + 1)-dimensional curved space that is coupled to a dynamical metric field. This metric, along with the affine connection, also appears in the Einstein–Hilbert action [Formula: see text] when written in first-order form. After applying the Dirac constraint formalism to the Einstein–Hilbert action and the action of the bosonic scalar field separately, we apply it to these actions when they are combined. Only in the latter case does a dynamical degree of freedom emerge.

scholarly journals THE CANONICAL STRUCTURE OF THE FIRST-ORDER EINSTEIN–HILBERT ACTION

2010 ◽  
Vol 25 (17) ◽  
pp. 3453-3480 ◽  
Author(s):  
D. G. C. MCKEON
Keyword(s):  
Phase Space ◽  
Gauge Invariance ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Affine Connection ◽  
Canonical Structure ◽  
First Order ◽  
Order Form ◽  
Solving Equations ◽  
Hilbert Action

The Dirac constraint formalism is used to analyze the first-order form of the Einstein–Hilbert action in d > 2 dimensions. Unlike previous treatments, this is done without eliminating fields at the outset by solving equations of motion that are independent of time derivatives when they correspond to first class constraints. As anticipated by the way in which the affine connection transforms under a diffeomorphism, not only primary and secondary but also tertiary first class constraints arise. These leave d(d-3) degrees of freedom in phase space. The gauge invariance of the action is discussed, with special attention being paid to the gauge generators of Henneaux, Teitelboim and Zanelli and of Castellani.

scholarly journals Holographic Projection of Electromagnetic Maxwell Theory

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1134
Author(s):  
Erica Bertolini ◽  
Nicola Maggiore
Keyword(s):  
Gauge Theory ◽  
Degrees Of Freedom ◽  
Vector Fields ◽  
Central Charge ◽  
Momentum Tensor ◽  
Boundary Theory ◽  
Coupling Constant ◽  
Maxwell Theory ◽  
Chern Simons

The 4D Maxwell theory with single-sided planar boundary is considered. As a consequence of the presence of the boundary, two broken Ward identities are recovered, which, on-shell, give rise to two conserved currents living on the edge. A Kaç-Moody algebra formed by a subset of the bulk fields is obtained with central charge proportional to the inverse of the Maxwell coupling constant, and the degrees of freedom of the boundary theory are identified as two vector fields, also suggesting that the 3D theory should be a gauge theory. Finally the holographic contact between bulk and boundary theory is reached in two inequivalent ways, both leading to a unique 3D action describing a new gauge theory of two coupled vector fields with a topological Chern-Simons term with massive coefficient. In order to check that the 3D projection of 4D Maxwell theory is well defined, we computed the energy-momentum tensor and the propagators. The role of discrete symmetries is briefly discussed.

scholarly journals A CANONICAL ANALYSIS OF THE EINSTEIN–HILBERT IN FIRST ORDER FORM

2006 ◽  
Vol 21 (16) ◽  
pp. 3401-3420 ◽  
Author(s):  
N. KIRIUSHCHEVA ◽  
S. V. KUZMIN ◽  
D. G. C. MCKEON
Keyword(s):  
Degrees Of Freedom ◽  
Symmetric Matrix ◽  
Affine Connection ◽  
Symmetric Tensor ◽  
Dynamical Variable ◽  
Canonical Analysis ◽  
First Order ◽  
Independent Variables ◽  
Order Form ◽  
Hilbert Action

Using the Dirac constraint formalism, we examine the canonical structure of the Einstein–Hilbert action [Formula: see text], treating the metric gαβ and the symmetric affine connection [Formula: see text] as independent variables. For d>2 tertiary constraints naturally arise; if these are all first class, there are d(d-3) independent variables in phase space, the same number that a symmetric tensor gauge field ϕμν possesses. If d = 2, the Hamiltonian becomes a linear combination of first class constraints obeying an SO (2, 1) algebra. These constraints ensure that there are no independent degrees of freedom. The transformation associated with the first class constraints is not a diffeomorphism when d = 2; it is characterized by a symmetric matrix ξμν. We also show that the canonical analysis is different if [Formula: see text] is used in place of gαβ as a dynamical variable when d = 2, as in d dimensions, [Formula: see text]. A comparison with the formalism used in the ADM analysis of the Einstein–Hilbert action in first order form is made by applying this approach in the two-dimensional case with hαβ and [Formula: see text] taken to be independent variables.

scholarly journals Ghosts from ghosts in the BRST formalism

2015 ◽  
Vol 93 (11) ◽  
pp. 1292-1295 ◽  
Author(s):  
D.G.C. McKeon
Keyword(s):  
Gauge Theory ◽  
Spinning Particle ◽  
Gauge Invariant ◽  
First Order ◽  
Hilbert Action

We show that the Hamiltonian HQ, introduced in the course of BRST analysis of a gauge theory, may in fact be associated with an action that itself is gauge invariant. This action can then be treated using the BRST formalism. We illustrate this by considering the spinning particle and the first-order Einstein–Hilbert action in 1 + 1 dimensions.

scholarly journals Burgers’ Equations in the Riemannian Geometry Associated with First-Order Differential Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-8 ◽  
Author(s):  
Z. Ok Bayrakdar ◽  
T. Bayrakdar
Keyword(s):  
Vector Fields ◽  
Integral Manifold ◽  
Integrability Condition ◽  
Three Dimensional ◽  
Dimensional Manifold ◽  
Jet Bundle ◽  
First Order ◽  
Burgers Equations ◽  
Integral Curves ◽  
Metric Connection

We construct metric connection associated with a first-order differential equation by means of the generator set of a Pfaffian system on a submanifold of an appropriate first-order jet bundle. We firstly show that the inviscid and viscous Burgers’ equations describe surfaces attached to an ODE of the form dx/dt=u(t,x) with certain Gaussian curvatures. In the case of PDEs, we show that the scalar curvature of a three-dimensional manifold encoding a system of first-order PDEs is determined in terms of the integrability condition and the Gaussian curvatures of the surfaces corresponding to the integral curves of the vector fields which are annihilated by the contact form. We see that an integral manifold of any PDE defines intrinsically flat and totally geodesic submanifold.

scholarly journals Approximate Ad Hoc Parametric Solutions for Nonlinear First-Order PDEs Governing Two-Dimensional Steady Vector Fields

2010 ◽  
Vol 2010 ◽  
pp. 1-23
Author(s):  
M. P. Markakis
Keyword(s):  
Vector Fields ◽  
Ad Hoc ◽  
Three Dimensional ◽  
Solid Surfaces ◽  
Two Dimensional ◽  
Nonlinear Odes ◽  
First Order ◽  
Parabolic Geometry ◽  
Abel Differential Equation ◽  
Analytical Tools

Through a suitable ad hoc assumption, a nonlinear PDE governing a three-dimensional weak, irrotational, steady vector field is reduced to a system of two nonlinear ODEs: the first of which corresponds to the two-dimensional case, while the second involves also the third field component. By using several analytical tools as well as linear approximations based on the weakness of the field, the first equation is transformed to an Abel differential equation which is solved parametrically. Thus, we obtain the two components of the field as explicit functions of a parameter. The derived solution is applied to the two-dimensional small perturbation frictionless flow past solid surfaces with either sinusoidal or parabolic geometry, where the plane velocities are evaluated over the body's surface in the case of a subsonic flow.

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