scholarly journals Universal Superspace Unitary Operator and Nilpotent (Anti-)Dual-BRST Symmetries: Superfield Formalism

2016 ◽  
Vol 2016 ◽  
pp. 1-17
Author(s):  
T. Bhanja ◽  
N. Srinivas ◽  
R. P. Malik
Keyword(s):  
Gauge Theories ◽  
Unitary Operator ◽  
Group Structure ◽  
Brst Symmetry ◽  
Rigid Rotor ◽  
Unitary Operators ◽  
Bosonic Field ◽  
Hermitian Conjugate ◽  
Symmetry Transformations ◽  
The One

We exploit the key concepts of the augmented version of superfield approach to Becchi-Rouet-Stora-Tyutin (BRST) formalism to derive the superspace (SUSP)dualunitary operator and its Hermitian conjugate and demonstrate their utility in the derivation of the nilpotent and absolutely anticommuting (anti-)dual-BRST symmetry transformations for a set of interesting models of theAbelian1-form gauge theories. These models are the one (0+1)-dimensional (1D) rigid rotor and modified versions of the two (1+1)-dimensional (2D) Proca as well as anomalous gauge theories and 2D model of a self-dual bosonic field theory. We show theuniversalityof the SUSPdualunitary operator and its Hermitian conjugate in the cases ofallthe Abelian models under consideration. These SUSP dual unitary operators, besides maintaining the explicit group structure, provide the alternatives to the dual horizontality condition (DHC) and dual gauge invariant restrictions (DGIRs) of the superfield formalism. The derivations of thedualunitary operators and corresponding (anti-)dual-BRST symmetries are completelynovelresults in our present investigation.

scholarly journals Universal Superspace Unitary Operator for Some Interesting Abelian Models: Superfield Approach

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
T. Bhanja ◽  
N. Srinivas ◽  
R. P. Malik
Keyword(s):  
Gauge Theories ◽  
Unitary Operator ◽  
Brst Symmetry ◽  
Scalar Fields ◽  
Gauge Field Theory ◽  
Rigid Rotor ◽  
Complex Scalar ◽  
Gauge Invariant ◽  
Hermitian Conjugate ◽  
Symmetry Transformations

Within the framework of augmented version of superfield formalism, we derive the superspace unitary operator and show its usefulness in the derivation of Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry transformations for a set of interesting models for the Abelian 1-form gauge theories. These models are (i) a one (0+1)-dimensional (1D) toy model of a rigid rotor, (ii) the two (1+1)-dimensional (2D) modified versions of the Proca and anomalous Abelian 1-form gauge theories, and (iii) the 2D self-dual bosonic gauge field theory. We provide, in some sense, the alternatives to the horizontality condition (HC) and the gauge invariant restrictions (GIRs) in the language of the above superspace (SUSP) unitary operator. One of the key observations of our present endeavor is the result that the SUSP unitary operator and its Hermitian conjugate are found to be thesameforallthe Abelian models under consideration (including the 4DinteractingAbelian 1-form gauge theories with Dirac and complex scalar fields which have been discussed earlier). Thus, we establish theuniversalityof the SUSP operator for the above Abelian theories.

scholarly journals Understanding mathematics of Grover’s algorithm

2021 ◽  
Vol 20 (5) ◽  
Author(s):  
Paweł J. Szabłowski
Keyword(s):  
Phase Change ◽  
Unitary Operator ◽  
Mathematical Structure ◽  
Complex Numbers ◽  
Unitary Operators ◽  
Grover’S Algorithm ◽  
Great Probability ◽  
Grover's Algorithm ◽  
One Step ◽  
The One

AbstractWe analyze the mathematical structure of the classical Grover’s algorithm and put it within the framework of linear algebra over the complex numbers. We also generalize it in the sense, that we are seeking not the one ‘chosen’ element (sometimes called a ‘solution’) of the dataset, but a set of m such ‘chosen’ elements (out of $$n>m)$$ n > m ) . Besides, we do not assume that the so-called initial superposition is uniform. We assume also that we have at our disposal an oracle that ‘marks,’ by a suitable phase change $$\varphi $$ φ , all these ‘chosen’ elements. In the first part of the paper, we construct a unique unitary operator that selects all ‘chosen’ elements in one step. The constructed operator is uniquely defined by the numbers $$\varphi $$ φ and $$\alpha $$ α which is a certain function of the coefficients of the initial superposition. Moreover, it is in the form of a composition of two so-called reflections. The result is purely theoretical since the phase change required to reach this heavily depends on $$\alpha $$ α . In the second part, we construct unitary operators having a form of composition of two or more reflections (generalizing the constructed operator) given the set of orthogonal versors. We find properties of these operations, in particular, their compositions. Further, by considering a fixed, ‘convenient’ phase change $$\varphi ,$$ φ , and by sequentially applying the so-constructed operator, we find the number of steps to find these ‘chosen’ elements with great probability. We apply this knowledge to study the generalizations of Grover’s algorithm ($$m=1,\phi =\pi $$ m = 1 , ϕ = π ), which are of the form, the found previously, unitary operators.

scholarly journals Nilpotent and absolutely anticommuting symmetries in the Freedman–Townsend model: Augmented superfield formalism

2014 ◽  
Vol 29 (31) ◽  
pp. 1450183 ◽  
Author(s):  
A. Shukla ◽  
S. Krishna ◽  
R. P. Malik
Keyword(s):  
Gauge Theories ◽  
Brst Symmetry ◽  
Kinetic Term ◽  
The Novel ◽  
Gauge Invariant ◽  
Yang Mills ◽  
Gauge Symmetries ◽  
Form Field ◽  
Symmetry Transformations ◽  
Augmented Superfield Formalism

We derive the off-shell nilpotent and absolutely anticommuting Becchi–Rouet–Stora–Tyutin (BRST) and anti-BRST symmetry transformations, corresponding to the (1-form) Yang–Mills (YM) and (2-form) tensorial gauge symmetries of the four (3+1)-dimensional (4D) Freedman–Townsend (FT) model, by exploiting the augmented version of Bonora–Tonin's (BT) superfield approach to BRST formalism where the 4D flat Minkowskian theory is generalized onto the (4, 2)-dimensional supermanifold. One of the novel observations is the fact that we are theoretically compelled to go beyond the horizontality condition (HC) to invoke an additional set of gauge-invariant restrictions (GIRs) for the derivation of the full set of proper (anti-)BRST symmetries. To obtain the (anti-)BRST symmetry transformations, corresponding to the tensorial (2-form) gauge symmetries within the framework of augmented version of BT-superfield approach, we are logically forced to modify the FT-model to incorporate an auxiliary 1-form field and the kinetic term for the antisymmetric (2-form) gauge field. This is also a new observation in our present investigation. We point out some of the key differences between the modified FT-model and Lahiri-model (LM) of the dynamical non-Abelian 2-form gauge theories. We also briefly mention a few similarities.

scholarly journals NONCOMMUTATIVE GAUGE THEORIES: MODEL FOR HODGE THEORY

2013 ◽  
Vol 28 (25) ◽  
pp. 1350122 ◽  
Author(s):  
SUDHAKER UPADHYAY ◽  
BHABANI PRASAD MANDAL
Keyword(s):  
Differential Geometry ◽  
Compact Manifold ◽  
Gauge Theories ◽  
Brst Symmetry ◽  
Decomposition Theorem ◽  
Hodge Theory ◽  
De Rham ◽  
Symmetry Transformations ◽  
Moyal Plane ◽  
Noncommutative Gauge Theories

The nilpotent Becchi–Rouet–Stora–Tyutin (BRST), anti-BRST, dual-BRST and anti-dual-BRST symmetry transformations are constructed in the context of noncommutative (NC) 1-form as well as 2-form gauge theories. The corresponding Noether's charges for these symmetries on the Moyal plane are shown to satisfy the same algebra, as by the de Rham cohomological operators of differential geometry. The Hodge decomposition theorem on compact manifold is also studied. We show that noncommutative gauge theories are field theoretic models for Hodge theory.

scholarly journals AUGMENTED SUPERFIELD APPROACH TO NON-YANG–MILLS SYMMETRIES OF JACKIW–PI MODEL: NOVEL OBSERVATIONS

2013 ◽  
Vol 28 (06) ◽  
pp. 1350011 ◽  
Author(s):  
SAURABH GUPTA ◽  
R. KUMAR
Keyword(s):  
Gauge Theories ◽  
Brst Symmetry ◽  
Auxiliary Field ◽  
The Novel ◽  
Yang Mills ◽  
Symmetry Transformations ◽  
Augmented Superfield Formalism

We derive the off-shell nilpotent and absolutely anti-commuting Becchi–Rouet–Stora–Tyutin (BRST) as well as anti-BRST symmetry transformations corresponding to the non-Yang–Mills (NYM) symmetry transformations of (2+1)-dimensional Jackiw–Pi (JP) model within the framework of "augmented" superfield formalism. The Curci–Ferrari (CF) restriction, which is a hallmark of non-Abelian one-form gauge theories, does not appear in this case. One of the novel features of our present investigation is the derivation of proper (anti-)BRST symmetry transformations corresponding to the auxiliary field ρ that cannot be derived by any conventional means.

scholarly journals Reparameterization Invariant Model of a Supersymmetric System: BRST and Supervariable Approaches

2021 ◽  
Vol 2021 ◽  
pp. 1-24
Author(s):  
A. Tripathi ◽  
B. Chauhan ◽  
A. K. Rao ◽  
R. P. Malik
Keyword(s):  
Brst Quantization ◽  
Relativistic Particle ◽  
Brst Symmetry ◽  
Target Space ◽  
Spinning Particle ◽  
1D Model ◽  
Symmetry Transformations ◽  
The Absolute ◽  
The One ◽  
Relativistic Particles

We carry out the Becchi-Rouet-Stora-Tyutin (BRST) quantization of the one 0 + 1 -dimensional (1D) model of a free massive spinning relativistic particle (i.e., a supersymmetric system) by exploiting its classical infinitesimal and continuous reparameterization symmetry transformations. We use the modified Bonora-Tonin (BT) supervariable approach (MBTSA) to BRST formalism to obtain the nilpotent (anti-)BRST symmetry transformations of the target space variables and the (anti-)BRST invariant Curci-Ferrari- (CF-) type restriction for the 1D model of our supersymmetric (SUSY) system. The nilpotent (anti-)BRST symmetry transformations for other variables of our model are derived by using the (anti-)chiral supervariable approach (ACSA) to BRST formalism. Within the framework of the latter, we have shown the existence of the CF-type restriction by proving the (i) symmetry invariance of the coupled Lagrangians and (ii) the absolute anticommutativity property of the conserved (anti-)BRST charges. The application of the MBTSA to a physical SUSY system (i.e., a 1D model of a massive spinning particle) is a novel result in our present endeavor. In the application of ACSA, we have considered only the (anti-)chiral super expansions of the supervariables. Hence, the observation of the absolute anticommutativity of the (anti-)BRST charges is a novel result. The CF-type restriction is universal in nature as it turns out to be the same for the SUSY and non-SUSY reparameterization (i.e., 1D diffeomorphism) invariant models of the (non-)relativistic particles.

scholarly journals Supervariable Approach to the Nilpotent Symmetries for a Toy Model of the Hodge Theory

2016 ◽  
Vol 2016 ◽  
pp. 1-13 ◽  
Author(s):  
D. Shukla ◽  
T. Bhanja ◽  
R. P. Malik
Keyword(s):  
Brst Symmetry ◽  
Hodge Theory ◽  
Present Theory ◽  
Rigid Rotor ◽  
Geometrical Meaning ◽  
Horizontality Condition ◽  
Symmetry Transformations ◽  
Brst Invariance ◽  
Standard Techniques

We exploit the standard techniques of the supervariable approach to derive the nilpotent Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry transformations for a toy model of the Hodge theory (i.e., a rigid rotor) and provide the geometrical meaning and interpretation to them. Furthermore, wealsoderive the nilpotent (anti-)co-BRST symmetry transformations for this theory within the framework of the above supervariable approach. We capture the (anti-)BRST and (anti-)co-BRST invariance of the Lagrangian of our present theory within the framework of augmented supervariable formalism. We also express the (anti-)BRST and (anti-)co-BRST charges in terms of the supervariables (obtained after the application of the (dual-)horizontality conditions and (anti-)BRST and (anti-)co-BRST invariant restrictions) to provide the geometrical interpretations for their nilpotency and anticommutativity properties. The application of the dual-horizontality condition and ensuing proper (i.e., nilpotent and absolutely anticommuting) fermionic (anti-)co-BRST symmetries are completelynovelresults in our present investigation.

scholarly journals (Anti-)chiral superfield approach to interacting Abelian 1-form gauge theories: Nilpotent and absolutely anticommuting charges

2018 ◽  
Vol 33 (04) ◽  
pp. 1850026 ◽  
Author(s):  
B. Chauhan ◽  
S. Kumar ◽  
R. P. Malik
Keyword(s):  
Gauge Theories ◽  
Brst Symmetry ◽  
Scalar Fields ◽  
The Novel ◽  
Complex Scalar ◽  
Symmetry Transformations ◽  
Brst Invariance ◽  
Lagrangian Densities ◽  
Chiral Superfield ◽  
Chiral Superfields

We derive the off-shell nilpotent (fermionic) (anti-)BRST symmetry transformations by exploiting the (anti-)chiral superfield approach (ACSA) to Becchi–Rouet–Stora–Tyutin (BRST) formalism for the interacting Abelian 1-form gauge theories where there is a coupling between the U(1) Abelian 1-form gauge field and Dirac as well as complex scalar fields. We exploit the (anti-)BRST invariant restrictions on the (anti-)chiral superfields to derive the fermionic symmetries of our present D-dimensional Abelian 1-form gauge theories. The novel observation of our present investigation is the derivation of the absolute anticommutativity of the nilpotent (anti-)BRST charges despite the fact that our ordinary D-dimensional theories are generalized onto the (D,[Formula: see text]1)-dimensional (anti-) chiral super-submanifolds (of the general (D,[Formula: see text]2)-dimensional supermanifold) where only the (anti-)chiral super expansions of the (anti-)chiral superfields have been taken into account. We also discuss the nilpotency of the (anti-)BRST charges and (anti-)BRST invariance of the Lagrangian densities of our present theories within the framework of ACSA to BRST formalism.

scholarly journals COMMENTS ON THE DUAL-BRST SYMMETRY

2011 ◽  
Vol 26 (36) ◽  
pp. 2739-2744 ◽  
Author(s):  
S. KRISHNA ◽  
A. SHUKLA ◽  
R. P. MALIK
Keyword(s):  
Differential Geometry ◽  
Gauge Theories ◽  
Short Note ◽  
Brst Symmetry ◽  
Hodge Theory ◽  
De Rham ◽  
Symmetry Transformations ◽  
Original Motivation

In view of a raging controversy on the topic of dual-Becchi–Rouet–Stora–Tyutin (dual-BRST/co-BRST) and anti-co-BRST symmetry transformations in the context of four (3+1)-dimensional (4D) Abelian two-form and 2D (non-)Abelian one-form gauge theories, we attempt, in our present short note, to settle the dust by taking the help of mathematics of differential geometry, connected with the Hodge theory, which was the original motivation for the nomenclature of "dual-BRST symmetry" in our earlier set of works. It has been claimed, in a recent set of papers, that the co-BRST symmetries are not independent of the BRST symmetries. We show that the BRST and co-BRST symmetries are independent symmetries in the same fashion as the exterior and co-exterior derivatives are independent entities belonging to the set of de Rham cohomological operators of differential geometry.

scholarly journals (Anti)chiral Superfield Approach to Nilpotent Symmetries: Self-Dual Chiral Bosonic Theory

2017 ◽  
Vol 2017 ◽  
pp. 1-14 ◽  
Author(s):  
N. Srinivas ◽  
T. Bhanja ◽  
R. P. Malik
Keyword(s):  
Field Theory ◽  
Lagrangian Density ◽  
Brst Symmetry ◽  
Bosonic Field ◽  
Symmetry Transformations ◽  
The Absolute ◽  
Chiral Superfield ◽  
Chiral Superfields

We exploit the beauty and strength of the symmetry invariant restrictions on the (anti)chiral superfields to derive the Becchi-Rouet-Stora-Tyutin (BRST), anti-BRST, and (anti-)co-BRST symmetry transformations in the case of a two (1+1)-dimensional (2D) self-dual chiral bosonic field theory within the framework of augmented (anti)chiral superfield formalism. Our 2D ordinary theory is generalized onto a (2,2)-dimensional supermanifold which is parameterized by the superspace variable ZM=xμ,θ,θ¯, where xμ (with μ=0,1) are the ordinary 2D bosonic coordinates and (θ,θ¯) are a pair of Grassmannian variables with their standard relationships: θ2=θ¯2=0, θθ¯+θ¯θ=0. We impose the (anti-)BRST and (anti-)co-BRST invariant restrictions on the (anti)chiral superfields (defined on the (anti)chiral (2,1)-dimensional supersubmanifolds of the above general (2,2)-dimensional supermanifold) to derive the above nilpotent symmetries. We do not exploit the mathematical strength of the (dual-)horizontality conditions anywhere in our present investigation. We also discuss the properties of nilpotency, absolute anticommutativity, and (anti-)BRST and (anti-)co-BRST symmetry invariance of the Lagrangian density within the framework of our augmented (anti)chiral superfield formalism. Our observation of the absolute anticommutativity property is a completely novel result in view of the fact that we have considered only the (anti)chiral superfields in our present endeavor.

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