22,081 research outputs found
BRST cohomology and Hodge decomposition theorem in Abelian gauge theory
We discuss the Becchi-Rouet-Stora-Tyutin (BRST) cohomology and Hodge
decomposition theorem for the two dimensional free U(1) gauge theory. In
addition to the usual BRST charge, we derive a local, conserved and nilpotent
co(dual)-BRST charge under which the gauge-fixing term remains invariant. We
express the Hodge decomposition theorem in terms of these charges and the
Laplacian operator. We take a single photon state in the quantum Hilbert space
and demonstrate the notion of gauge invariance, no-(anti)ghost theorem,
transversality of photon and establish the topological nature of this theory by
exploiting the concepts of BRST cohomology and Hodge decomposition theorem. In
fact, the topological nature of this theory is encoded in the vanishing of the
Laplacian operator when equations of motion are exploited. On the two
dimensional compact manifold, we derive two sets of topological invariants with
respect to the conserved and nilpotent BRST- and co-BRST charges and express
the Lagrangian density of the theory as the sum of terms that are BRST- and
co-BRST invariants. Mathematically, this theory captures together some of the
key features of both Witten- and Schwarz type of topological field theories.Comment: 20 pages, LaTeX, no figures, Title and text have been changed,
Journal reference is given, some references have been adde
A Concise Introduction to Perturbation Theory in Cosmology
We give a concise, self-contained introduction to perturbation theory in
cosmology at linear and second order, striking a balance between mathematical
rigour and usability. In particular we discuss gauge issues and the active and
passive approach to calculating gauge transformations. We also construct
gauge-invariant variables, including the second order tensor perturbation on
uniform curvature hypersurfaces.Comment: revtex4, 16 pages, 3 figures; v2: minor changes, typos corrected,
reference added, version accepted by CQ
Gauge Transformations, BRST Cohomology and Wigner's Little Group
We discuss the (dual-)gauge transformations and BRST cohomology for the two
(1 + 1)-dimensional (2D) free Abelian one-form and four (3 + 1)-dimensional
(4D) free Abelian 2-form gauge theories by exploiting the (co-)BRST symmetries
(and their corresponding generators) for the Lagrangian densities of these
theories. For the 4D free 2-form gauge theory, we show that the changes on the
antisymmetric polarization tensor e^{\mu\nu} (k) due to (i) the (dual-)gauge
transformations corresponding to the internal symmetry group, and (ii) the
translation subgroup T(2) of the Wigner's little group, are connected with
each-other for the specific relationships among the parameters of these
transformation groups. In the language of BRST cohomology defined w.r.t. the
conserved and nilpotent (co-)BRST charges, the (dual-)gauge transformed states
turn out to be the sum of the original state and the (co-)BRST exact states. We
comment on (i) the quasi-topological nature of the 4D free 2-form gauge theory
from the degrees of freedom count on e^{\mu\nu} (k), and (ii) the Wigner's
little group and the BRST cohomology for the 2D one-form gauge theory {\it
vis-{\`a}-vis} our analysis for the 4D 2-form gauge theory.Comment: LaTeX file, 29 pages, misprints in (3.7), (3.8), (3.9), (3.13) and
(4.14)corrected and communicated to IJMPA as ``Erratum'
Wigner's little group and BRST cohomology for one-form Abelian gauge theory
We discuss the (dual-)gauge transformations for the gauge-fixed Lagrangian
density and establish their intimate connection with the translation subgroup
T(2) of the Wigner's little group for the free one-form Abelian gauge theory in
four -dimensions (4D) of spacetime. Though the relationship between
the usual gauge transformation for the Abelian massless gauge field and T(2)
subgroup of the little group is quite well-known, such a connection between the
dual-gauge transformation and the little group is a new observation. The above
connections are further elaborated and demonstrated in the framework of
Becchi-Rouet-Stora-Tyutin (BRST) cohomology defined in the quantum Hilbert
space of states where the Hodge decomposition theorem (HDT) plays a very
decisive role.Comment: LaTeX file, 17 pages, Journal-ref. give
Supersymmetric Oscillator: Novel Symmetries
We discuss various continuous and discrete symmetries of the supersymmetric
simple harmonic oscillator (SHO) in one (0 + 1)-dimension of spacetime and show
their relevance in the context of mathematics of differential geometry. We show
the existence of a novel set of discrete symmetries in the theory which has,
hitherto, not been discussed in the literature on theoretical aspects of SHO.
We also point out the physical relevance of our present investigation.Comment: REVTeX file, 5 pages, minor changes in title, text and abstract,
references expanded, version to appear in EP
Geometrical Aspects Of BRST Cohomology In Augmented Superfield Formalism
In the framework of augmented superfield approach, we provide the geometrical
origin and interpretation for the nilpotent (anti-)BRST charges, (anti-)co-BRST
charges and a non-nilpotent bosonic charge. Together, these local and conserved
charges turn out to be responsible for a clear and cogent definition of the
Hodge decomposition theorem in the quantum Hilbert space of states. The above
charges owe their origin to the de Rham cohomological operators of differential
geometry which are found to be at the heart of some of the key concepts
associated with the interacting gauge theories. For our present review, we
choose the two -dimensional (2D) quantum electrodynamics (QED) as a
prototype field theoretical model to derive all the nilpotent symmetries for
all the fields present in this interacting gauge theory in the framework of
augmented superfield formulation and show that this theory is a {\it unique}
example of an interacting gauge theory which provides a tractable field
theoretical model for the Hodge theory.Comment: LaTeX file, 25 pages, Ref. [49] updated, correct page numbers of the
Journal are give
Superfield Approach To Nilpotent Symmetries For QED From A Single Restriction: An Alternative To The Horizontality Condition
We derive together the exact local, covariant, continuous and off-shell
nilpotent Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry
transformations for the U(1) gauge field (A_\mu), the (anti-)ghost fields
((\bar C)C) and the Dirac fields (\psi, \bar\psi) of the Lagrangian density of
a four (3 + 1)-dimensional QED by exploiting a single restriction on the six
(4, 2)-dimensional supermanifold. A set of four even spacetime coordinates
x^\mu (\mu = 0, 1, 2, 3) and two odd Grassmannian variables \theta and
\bar\theta parametrize this six dimensional supermanifold. The new gauge
invariant restriction on the above supermanifold owes its origin to the (super)
covariant derivatives and their intimate relations with the (super) 2-form
curvatures (\tilde F^{(2)})F^{(2)} constructed with the help of (super) 1-form
gauge connections (\tilde A^{(1)})A^{(1)} and (super) exterior derivatives
(\tilde d)d. The results obtained separately by exploiting (i) the
horizontality condition, and (ii) one of its consistent extensions, are shown
to be a simple consequence of this new single restriction on the above
supermanifold. Thus, our present endeavour provides an alternative to (and, in
some sense, generalization of) the horizontality condition of the usual
superfield formalism applied to the derivation of BRST symmetries.Comment: LaTeX file, 15 pages, journal-versio
Nilpotent Symmetries For Matter Fields In Non-Abelian Gauge Theory: Augmented Superfield Formalism
In the framework of superfield approach to Becchi-Rouet-Stora-Tyutin (BRST)
formalism, the derivation of the (anti-)BRST nilpotent symmetries for the
matter fields, present in any arbitrary interacting gauge theory, has been a
long-standing problem. In our present investigation, the local, covariant,
continuous and off-shell nilpotent (anti-)BRST symmetry transformations for the
Dirac fields are derived in the framework of the augmented
superfield formulation where the four -dimensional (4D) interacting
non-Abelian gauge theory is considered on the six -dimensional
supermanifold parametrized by the four even spacetime coordinates and a
couple of odd elements ( and ) of the Grassmann algebra.
The requirement of the invariance of the matter (super)currents and the
horizontality condition on the (super)manifolds leads to the derivation of the
nilpotent symmetries for the matter fields as well as the gauge- and the
(anti-)ghost fields of the theory in the general scheme of the augmented
superfield formalism.Comment: LaTeX file, 16 pages, printing mistakes in the second paragraph of
`Introduction' corrected, a footnote added, these modifications submitted as
``erratum'' to IJMPA in the final for
Augmented Superfield Approach to Nilpotent Symmetries in the Modified Version of 2D Proca Theory
We derive the complete set of off-shell nilpotent and absolutely
anticommuting Becchi-Rouet-Stora-Tyutin (BRST), anti-BRST and (anti-)co-BRST
symmetry transformations for all the fields of the modified version of two
(1+1)-dimensional (2D) Proca theory by exploiting the "augmented" superfield
formalism where the (dual-)horizontality conditions and (dual-)gauge-invariant
restrictions are exploited together. We capture the (anti-)BRST and
(anti-)co-BRST invariance of the Lagrangian density in the language of
superfield formalism. We also express the nilpotency and absolute
anticommutativity of the (anti-)BRST and (anti-)co-BRST charges within the
framework of augmented superfield formalism. This exercise leads to some novel
observations which have, hitherto, not been pointed out in the literature
within the framework of superfield approach to BRST formalism. For the sake of
completeness, we also mention, very briefly, a unique bosonic symmetry, the
ghost-scale symmetry and discrete symmetries of the theory and show that the
algebra of conserved charges captures the cohomological aspects of differential
geometry. Thus, our present modified 2D Proca theory is a model for the Hodge
Theory.Comment: LaTeX file, 32 pages, journal reference give
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