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'

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Superfield approach to symmetry invariance in QED with complex scalar fields

We show that the Grassmannian independence of the super Lagrangian density, expressed in terms of the superfields defined on a (4, 2)-dimensional supermanifold, is a clear-cut proof for the Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST invariance of the corresoponding four (3 + 1)-dimensional (4D) Lagrangian density that describes the interaction between the U(1) gauge field and the charged complex scalar fields. The above 4D field theoretical model is considered on a (4, 2)-dimensional supermanifold parametrized by the ordinary four spacetime variables x^\mu (with \mu = 0, 1, 2, 3) and a pair of Grassmannian variables \theta and \bar\theta (with \theta^2 = \bar\theta^2 = 0, \theta \bar\theta + \bar\theta \theta = 0). Geometrically, the (anti-)BRST invariance is encoded in the translation of the super Lagrangian density along the Grassmannian directions of the above supermanifold such that the outcome of this shift operation is zero.Comment: LaTeX file, 14 pages, minor changes in the title and text, version to appear in ``Pramana - Journal of Physics'

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 $(3 + 1)$-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

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 $(1 + 1)$-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

Abelian 2-form gauge theory: superfield formalism

We derive the off-shell nilpotent Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry transformations for {\it all} the fields of a free Abelian 2-form gauge theory by exploiting the geometrical superfield approach to BRST formalism. The above four (3 + 1)-dimensional (4D) theory is considered on a (4, 2)-dimensional supermanifold parameterized by the four even spacetime variables x^\mu (with \mu = 0, 1, 2, 3) and a pair of odd Grassmannian variables \theta and \bar\theta (with \theta^2 = \bar\theta^2 = 0, \theta \bar\theta + \bar\theta \theta = 0). One of the salient features of our present investigation is that the above nilpotent (anti-)BRST symmetry transformations turn out to be absolutely anticommuting due to the presence of a Curci-Ferrari (CF) type of restriction. The latter condition emerges due to the application of our present superfield formalism. The actual CF condition, as is well-known, is the hallmark of a 4D non-Abelian 1-form gauge theory. We demonstrate that our present 4D Abelian 2-form gauge theory imbibes some of the key signatures of the 4D non-Abelian 1-form gauge theory. We briefly comment on the generalization of our supperfield approach to the case of Abelian 3-form gauge theory in four (3 + 1)-dimensions of spacetime.Comment: LaTeX file, 23 pages, journal versio

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