ANATOMY OF EINSTEIN'S THEORY: ABELIAN DECOMPOSITION OF GENERAL RELATIVITY

2012 ◽  
Vol 07 ◽  
pp. 116-147 ◽  
Author(s):  
Y. M. CHO
Keyword(s):  
Gauge Theory ◽  
Lorentz Group ◽  
Degrees Of Freedom ◽  
Gauge Potential ◽  
Abelian Gauge ◽  
Maximal Abelian Subgroup ◽  
Metric Tensor ◽  
Lorentz Gauge ◽  
Abelian Decomposition

Treating Einstein's theory as a gauge theory of Lorentz group, we decompose the gravitational connection (the gauge potential of Lorentz group) Γμ into the restricted connection of the maximal Abelian subgroup of Lorentz group and the valence connection which transforms covariantly under Lorentz gauge transformation. With this decomposition we show that the Einstein's theory can be decomposed into the restricted part made of the restricted connection which has the full Lorentz gauge invariance and the valence part made of the valence connection which plays the role of gravitational source of the restricted gravity. We show that there are two different Abelian decomposition of Einstein's theory, the light-like (or null) decomposition and the non light-like (or non-null) decomposition. In this decomposition the role of the metric gμν is replaced by a four-index metric tensor gμν which transforms covariantly under the Lorentz group, and the metric-compatibility condition ∇αgμν = 0 of the connection is replaced by the gauge and generally covariant condition [Formula: see text]. The decomposition shows the existence of a restricted theory of gravitation which has the full general invariance but is much simpler and has less physical degrees of freedom than Einstein's theory. Moreover, it tells that the restricted gravity can be written as an Abelian gauge theory, which implies that the graviton can be described by a massless spin-one field.

scholarly journals ABELIAN DECOMPOSITION OF GENERAL RELATIVITY

2009 ◽  
Vol 24 (18n19) ◽  
pp. 3327-3341 ◽  
Author(s):  
Y. M. CHO
Keyword(s):  
General Relativity ◽  
Gauge Theory ◽  
Quantum Gravity ◽  
Lorentz Group ◽  
Lorentz Invariance ◽  
Gauge Potential ◽  
Abelian Decomposition ◽  
Gravitational Source ◽  
Einstein’S General Relativity ◽  
Local Lorentz Invariance

We present an Abelian decomposition of Einstein's general relativity, viewing Einstein's theory as a gauge theory of Lorentz group and identifying the gravitational connection as the gauge potential of Lorentz group. The decomposition confirms the existence of the restricted gravity which is much simpler than Einstein's theory but which has the full local Lorentz invariance (and thus the full general invariance). Moreover, it tells that Einstein's theory can be viewed as the restricted gravity which has the Lorentz covariant valence connection as the gravitational source. With the Abelian decomposition we show how to construct all possible vacuum gravitational connections, which can be classified by the knot topology π3(S3) = π3(S2). We discuss the physical implications of our result in quantum gravity.

scholarly journals Dirac equation based on the vector representation of the Lorentz group

2020 ◽  
Vol 135 (10) ◽  
Author(s):  
Eckart Marsch ◽  
Yasuhito Narita
Keyword(s):  
Gauge Theory ◽  
Dirac Equation ◽  
Lorentz Group ◽  
Degrees Of Freedom ◽  
Lagrangian Density ◽  
Degree Of Freedom ◽  
Vector Representation ◽  
Massive Fermion

AbstractIn this paper, we derive an expanded Dirac equation for a massive fermion doublet, which has in addition to the particle/antiparticle and spin-up/spin-down degrees of freedom explicity an isospin-type degree of freedom. We begin with revisiting the four-vector Lorentz group generators, define the corresponding gamma matrices and then write a Dirac equation for the fermion doublet with eight spinor components. The appropriate Lagrangian density is established, and the related chiral and SU(2) symmetry is discussed in detail, as well as applications to an electroweak-style gauge theory. In “Appendix,” we present some of the relevant matrices.

scholarly journals TOPOLOGICALLY MASSIVE GAUGE THEORY: A LORENTZIAN SOLUTION

2007 ◽  
Vol 22 (16n17) ◽  
pp. 2961-2976 ◽  
Author(s):  
K. SAYGILI
Keyword(s):  
Field Equation ◽  
Gauge Theory ◽  
Field Strength ◽  
Gauge Transformation ◽  
Global Section ◽  
Gauge Potential ◽  
Abelian Gauge ◽  
Abelian Field ◽  
Hopf Map ◽  
Topological Mass

We obtain a Lorentzian solution for the topologically massive non-Abelian gauge theory on AdS space [Formula: see text] by means of an SU (1, 1) gauge transformation of the previously found Abelian solution. There exists a natural scale of length which is determined by the inverse topological mass ν ~ ng2. In the topologically massive electrodynamics the field strength locally determines the gauge potential up to a closed 1-form via the (anti-)self-duality equation. We introduce a transformation of the gauge potential using the dual field strength which can be identified with an Abelian gauge transformation. Then we present map [Formula: see text] including the topological mass which is the Lorentzian analog of the Hopf map. This map yields a global decomposition of [Formula: see text] as a trivial [Formula: see text] bundle over the upper portion of the pseudosphere [Formula: see text] which is the Hyperboloid model for the Lobachevski geometry. This leads to a reduction of the Abelian field equation onto [Formula: see text] using a global section of the solution on [Formula: see text]. Then we discuss the integration of the field equation using the Archimedes map [Formula: see text]. We also present a brief discussion of the holonomy of the gauge potential and the dual field strength on [Formula: see text].

scholarly journals On Spinor Representations in the Weyl Gauge Theory

1997 ◽  
Vol 12 (26) ◽  
pp. 1957-1968 ◽  
Author(s):  
B. M. Barbashov ◽  
A. B. Pestov
Keyword(s):  
Gauge Theory ◽  
Differential Forms ◽  
Current Source ◽  
Gauge Potential ◽  
Spinor Representation ◽  
Abelian Gauge ◽  
Cartan Torsion ◽  
Gauge Space ◽  
Spinor Current ◽  
Spinor Representations

A spinor current-source is found in the Weyl non-Abelian gauge theory which does not contain the abstract gauge space. It is shown that the searched spinor representation can be constructed in the space of external differential forms and it is a 16-component quantity for which a gauge-invariant Lagrangian is determined. The connection between the Weyl non-Abelian gauge potential and the Cartan torsion field, and the problem of a possible manifestation of the considered interactions are considered.

scholarly journals FEYNMAN PERTURBATION THEORY FOR GAUGE THEORY ON TRANSVERSE LATTICE

2010 ◽  
Vol 25 (18n19) ◽  
pp. 3621-3640
Author(s):  
M. S. KARNEVSKIY ◽  
S. A. PASTON
Keyword(s):  
Perturbation Theory ◽  
Gauge Theory ◽  
Degrees Of Freedom ◽  
Rotational Invariance ◽  
Abelian Gauge ◽  
Gauge Invariant ◽  
Transverse Lattice ◽  
Abelian Gauge Theory ◽  
Independent Variables ◽  
Ultraviolet Regularization

Feynman perturbation theory for non-Abelian gauge theory in light-like gauge is investigated. A lattice along two spacelike directions is used as a gauge invariant ultraviolet regularization. For preservation of the polynomiality of action, we use as independent variables arbitrary (nonunitary) matrices related to the link of the lattice. The action of the theory is selected in such a way to preserve as much as possible the rotational invariance, which remains after an introduction of the lattice, as well as to make superfluous degrees of freedom vanish in the limit of removing the regularization. Feynman perturbation theory is constructed and diagrams which does not contain ultraviolet divergences are analyzed. The scheme of renormalization of this theory is discussed.

scholarly journals LORENTZ GAUGE THEORY AND SPINOR INTERACTION

2008 ◽  
Vol 23 (08) ◽  
pp. 1282-1285 ◽  
Author(s):  
NAKIA CARLEVARO ◽  
ORCHIDEA MARIA LECIAN ◽  
GIOVANNI MONTANI
Keyword(s):  
Gauge Theory ◽  
Gauge Field ◽  
Lorentz Group ◽  
Flat Space ◽  
Stationary Solutions ◽  
Space Time ◽  
Pauli Equation ◽  
Relativistic Limit ◽  
Curved Space

A gauge theory of the Lorentz group, based on the different behavior of spinors and vectors under local transformations, is formulated in a flat space-time and the role of the torsion field within the generalization to curved space-time is briefly discussed. The spinor interaction with the new gauge field is then analyzed assuming the time gauge and stationary solutions, in the non-relativistic limit, are treated to generalize the Pauli equation.

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 MINIMAL MODEL OF LORENTZ GAUGE GRAVITY WITH DYNAMICAL TORSION

2010 ◽  
Vol 25 (14) ◽  
pp. 2867-2882 ◽  
Author(s):  
Y. M. CHO ◽  
D. G. PAK ◽  
B. S. PARK
Keyword(s):  
Constant Curvature ◽  
Quantum Dynamics ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Space Time ◽  
Effective Theory ◽  
Metric Tensor ◽  
Lorentz Gauge ◽  
Constant Curvature Space ◽  
Gauge Gravity

A new Lorentz gauge gravity model with R2-type Lagrangian is proposed. In the absence of classical torsion, the model admits a topological phase with an arbitrary metric. We analyze the equations of motion in constant curvature space–time background using the Lagrange formalism and demonstrate that the model possesses a minimal set of dynamic degrees of freedom for the torsion. Surprisingly, the number of torsion dynamic degrees of freedom equals the number of physical degrees of freedom for the metric tensor. An interesting feature of the model is that the spin-2 mode of torsion becomes dynamical essentially due to the nonlinear structure of the theory. We perform covariant one-loop quantization of the model for a special case of constant curvature space–time background. We treat the contortion as a quantum field variable whereas the metric tensor is kept as a classical object. We discuss a possible mechanism of an emergent Einstein gravity as a part of the effective theory induced due to quantum dynamics of torsion.

scholarly journals SOLITONIC ASPECTS OF q-FIELD THEORIES

2003 ◽  
Vol 18 (04) ◽  
pp. 627-650 ◽  
Author(s):  
R. J. FINKELSTEIN
Keyword(s):  
Gauge Theory ◽  
Lie Algebra ◽  
Lie Group ◽  
Degrees Of Freedom ◽  
Field Theories ◽  
Standard Theory ◽  
Abelian Gauge ◽  
Abelian Gauge Theory ◽  
Point Particles ◽  
Non Locality

We have examined the deformation of a generic non-Abelian gauge theory obtained by replacing its Lie group by the corresponding quantum group. This deformed gauge theory has more degrees of freedom than the theory from which it is derived. By going over from point particles in the standard theory to solitonic particles in the deformed theory, it is proposed that we interpret the new degrees of freedom as being descriptive of the non-locality of the deformed theory. It also turns out that the original Lie algebra gets replaced by two dual algebras, one of which lies close to and approaches the original Lie algebra in a correspondence limit, while the second algebra is new and disappears in this same correspondence limit. The exotic field particles associated with the second algebra can be interpreted as quark-like constituents of the solitons, which are themselves described as point particles in the first algebra. These ideas are explored for q-deformed SU(2) and GL q(3).

The quark confinement in extended gauge theory

1991 ◽  
Vol 69 (12) ◽  
pp. 1441-1446 ◽  
Author(s):  
J. M. S. Rana ◽  
H. C. Chandola ◽  
B. S. Rajput
Keyword(s):  
Gauge Theory ◽  
Topological Structure ◽  
Degrees Of Freedom ◽  
Gauge Potential ◽  
Collective Modes ◽  
Color Singlet ◽  
Quark Confinement ◽  
Gauge Groups ◽  
Physical Spectrum ◽  
Dynamical Degrees

To investigate the possible physical implications of the topological structure of non-Abelian dyons in connection with the issue of quark confinement in quantum chromodynamics (QCD), extended gauge theory is formulated in SU(2) and SU(3) gauge groups from the corresponding restricted chromodynamics (RCD) by reactivating the suppressed dynamical degrees of freedom and constructing the gauge potential in terms of the binding gluons (the RCD piece) and the valence gluons (the reactivated piece). It is shown that in this extended QCD, the confinement mechanism of the corresponding RCD remains intact. The physical spectrum contains color-singlet generalized electric glueballs made of valence gluon pairs as well as the generalized magnetic glueballs as massive collective modes of the condensed vacuum.

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