non-Abelian 1-form Gauge Theory Research Articles

How many quantum gates do gauge theories require?

We discuss the implementation of lattice gauge theories on digital quantum computers, focusing primarily on the number of quantum gates required to simulate their time evolution. We find that to compile quantum circuits, using available state-of-the-art methods with our own augmentations, the cost of a single time step of an elementary plaquette is beyond what is reasonably practical in the current era of quantum hardware. However, we observe that such costs are highly sensitive to the truncation scheme used to derive different Hamiltonian formulations of non-Abelian gauge theories, emphasizing the need for low-dimensional truncations of such models in the same universality class as the desired theories.

3D Jackiw–Pi model: (anti-)chiral superfield approach to BRST formalism

We discuss and derive the continuous Becchi-Rouet-Stora-Tyutin (BRST)and anti-BRST symmetry transformations for the Jackiw-Pi (JP) model of three (2 + 1)-dimensional (3D) massive non-Abelian 1-form gauge theory by exploiting the standard technique of (anti-)chiral superfield approach (ACSA) to BRST formalism where a few appropriate and specific sets of (anti-)BRST invariant quantities (i. e. physical quantities at quantum level) play a very important role. We provide the explicit derivation of the nilpotency and absolute anticommutativity properties of (anti-)BRST conserved charges and the existence of the Curci-Ferrari (CF) condition within the realm of ACSA to BRST formalism where we take only a single Grassmannian variable into account. We also provide clear proof of (anti-) BRST invariances of the coupled (but equivalent) Lagrangian densities within the framework of ACSA to BRST approach where the emergence of the CF-condition is observed.

Asymptotic symmetries of Yang-Mills fields in Hamiltonian formulation

We investigate the asymptotic symmetry group of the free SU(N )-Yang-Mills theory using the Hamiltonian formalism. We closely follow the strategy of Henneaux and Troessaert who successfully applied the Hamiltonian formalism to the case of gravity and electrodynamics, thereby deriving the respective asymptotic symmetry groups of these theories from clear-cut first principles. These principles include the minimal assumptions that are necessary to ensure the existence of Hamiltonian structures (phase space, symplectic form, differentiable Hamiltonian) and, in case of Poincaré invariant theories, a canonical action of the Poincaré group. In the first part of the paper we show how these requirements can be met in the non-abelian SU(N )-Yang-Mills case by imposing suitable fall-off and parity conditions on the fields. We observe that these conditions admit neither non-trivial asymptotic symmetries nor non-zero global charges. In the second part of the paper we discuss possible gradual relaxations of these conditions by following the same strategy that Henneaux and Troessaert had employed to remedy a similar situation in the electromagnetic case. Contrary to our expectation and the findings of Henneaux and Troessaert for the abelian case, there seems to be no relaxation that meets the requirements of a Hamiltonian formalism and allows for non-trivial asymptotic symmetries and charges. Non-trivial asymptotic symmetries and charges are only possible if either the Poincaré group fails to act canonically or if the formal expression for the symplectic form diverges, i.e. the form does not exist. This seems to hint at a kind of colour-confinement built into the classical Hamiltonian formulation of non-abelian gauge theories.

Modified 2D Proca Theory: Revisited under BRST and (Anti-)chiral Superfield Formalisms

Within the framework of Becchi-Rouet-Stora-Tyutin (BRST) approach, we discuss mainly the fermionic (i.e., off-shell nilpotent) (anti-)BRST, (anti-)co-BRST, and some discrete dual symmetries of the appropriate Lagrangian densities for a two (1+1)-dimensional (2D) modified Proca (i.e., a massive Abelian 1-form) theory without any interaction with matter fields. One of the novel observations of our present investigation is the existence of some kinds of restrictions in the case of our present Stückelberg-modified version of the 2D Proca theory which is not like the standard Curci-Ferrari (CF) condition of a non-Abelian 1-form gauge theory. Some kinds of similarities and a few differences between them have been pointed out in our present investigation. To establish the sanctity of the above off-shell nilpotent (anti-)BRST and (anti-)co-BRST symmetries, we derive them by using our newly proposed (anti-)chiral superfield formalism where a few specific and appropriate sets of invariant quantities play a decisive role. We express the (anti-)BRST and (anti-)co-BRST conserved charges in terms of the superfields that are obtained after the applications of (anti-)BRST and (anti-)co-BRST invariant restrictions and prove their off-shell nilpotency and absolute anticommutativity properties, too. Finally, we make some comments on (i) the novelty of our restrictions/obstructions and (ii) the physics behind the negative kinetic term associated with the pseudoscalar field of our present theory.

Non-Abelian basis tensor gauge theory

Basis tensor gauge theory is a vierbein analog reformulation of ordinary gauge theories in which the difference of local field degrees of freedom has the interpretation of an object similar to a Wilson line. Here we present a non-Abelian basis tensor gauge theory formalism. Unlike in the Abelian case, the map between the ordinary gauge field and the basis tensor gauge field is nonlinear. To test the formalism, we compute the beta function and the two-point function at the one-loop level in non-Abelian basis tensor gauge theory and show that it reproduces the well-known results from the usual formulation of non-Abelian gauge theory.

Nilpotent charges in an interacting gauge theory and an 𝒩 = 2 SUSY quantum mechanical model: (Anti-)chiral superfield approach

We exploit the power and potential of the (anti-)chiral superfield approach (ACSA) to Becchi–Rouet–Stora–Tyutin (BRST) formalism to derive the nilpotent (anti-)BRST symmetry transformations for any arbitrary [Formula: see text]-dimensional interacting non-Abelian 1-form gauge theory where there is an [Formula: see text] gauge invariant coupling between the gauge field and the Dirac fields. We derive the conserved and nilpotent (anti-)BRST charges and establish their nilpotency and absolute anticommutativity properties within the framework of ACSA to BRST formalism. The clinching proof of the absolute anticommutativity property of the conserved and nilpotent (anti-)BRST charges is a novel result in view of the fact that we consider, in our endeavor, only the (anti-)chiral super expansions of the superfields that are defined on the [Formula: see text]-dimensional super-submanifolds of the general [Formula: see text]-dimensional supermanifold on which our [Formula: see text]-dimensional ordinary interacting non-Abelian 1-form gauge theory is generalized. To corroborate the novelty of the above result, we apply the ACSA to an [Formula: see text] supersymmetric (SUSY) quantum mechanical (QM) model of a harmonic oscillator and show that the nilpotent and conserved [Formula: see text] supercharges of this system do not absolutely anticommute.

Casimir effect in ( 2+1 )-dimensional Yang-Mills theory as a probe of the magnetic mass

We consider the Casimir effect in a gauge-invariant Hamiltonian formulation of nonabelian gauge theories in (2+1) dimensions, for an arbitrary gauge group. We show that the result is in good agreement with recent lattice simulations. We also argue that the Casimir effect may be viewed as a good probe of magnetic screening effects in (3+1)-dimensional gauge theories at high temperatures.

Superfield Approach to Nilpotency and Absolute Anticommutativity of Conserved Charges: 2D Non-Abelian 1-Form Gauge Theory

We exploit the theoretical strength of augmented version of superfield approach (AVSA) to Becchi-Rouet-Stora-Tyutin (BRST) formalism to express the nilpotency and absolute anticommutativity properties of the (anti-)BRST and (anti-)co-BRST conserved charges for the two (1+1)-dimensional (2D) non-Abelian 1-form gauge theory (without any interaction with matter fields) in the language of superspace variables, their derivatives, and suitable superfields. In the proof of absolute anticommutativity property, we invoke the strength of Curci-Ferrari (CF) condition for the (anti-)BRST charges. No such outside condition/restriction is required in the proof of absolute anticommutativity of the (anti-)co-BRST conserved charges. The latter observation (as well as other observations) connected with (anti-)co-BRST symmetries and corresponding conserved charges are novel results of our present investigation. We also discuss the (anti-)BRST and (anti-)co-BRST symmetry invariance of the appropriate Lagrangian densities within the framework of AVSA. In addition, we dwell a bit on the derivation of the above fermionic (nilpotent) symmetries by applying the AVSA to BRST formalism, where only the (anti)chiral superfields are used.

Nilpotent symmetries and Curci–Ferrari-type restrictions in 2D non-Abelian gauge theory: Superfield approach

We derive the off-shell nilpotent symmetries of the two [Formula: see text]-dimensional (2D) non-Abelian 1-form gauge theory by using the theoretical techniques of the geometrical superfield approach to Becchi–Rouet–Stora–Tyutin (BRST) formalism. For this purpose, we exploit the augmented version of superfield approach (AVSA) and derive theoretically useful nilpotent (anti-)BRST, (anti-)co-BRST symmetries and Curci–Ferrari (CF)-type restrictions for the self-interacting 2D non-Abelian 1-form gauge theory (where there is no interaction with matter fields). The derivation of the (anti-)co-BRST symmetries and all possible CF-type restrictions are completely novel results within the framework of AVSA to BRST formalism where the ordinary 2D non-Abelian theory is generalized onto an appropriately chosen [Formula: see text]-dimensional supermanifold. The latter is parametrized by the superspace coordinates [Formula: see text] where [Formula: see text] (with [Formula: see text]) are the bosonic coordinates and a pair of Grassmannian variables [Formula: see text] obey the relationships: [Formula: see text], [Formula: see text]. The topological nature of our 2D theory allows the existence of a tower of CF-type restrictions.

Some novel features in 2D non-Abelian theory: BRST approach

Within the framework of Becchi–Rouet–Stora–Tyutin (BRST) formalism, we discuss some novel features of a two (1[Formula: see text]+[Formula: see text]1)-dimensional (2D) non-Abelian 1-form gauge theory (without any interaction with matter fields). Besides the usual off-shell nilpotent and absolutely anticommutating (anti-)BRST symmetry transformations, we discuss the off-shell nilpotent and absolutely anticommutating (anti-)co-BRST symmetry transformations. Particularly, we lay emphasis on the existence of the coupled (but equivalent) Lagrangian densities of the 2D non-Abelian theory in view of the presence of (anti-)co-BRST symmetry transformations where we pin-point some novel features associated with the Curci–Ferrari (CF-)type restrictions. We demonstrate that these CF-type restrictions can be incorporated into the (anti-)co-BRST invariant Lagrangian densities through the fermionic Lagrange multipliers which carry specific ghost numbers. The modified versions of the Lagrangian densities (where we get rid of the new CF-type restrictions) respect some precise symmetries as well as a couple of symmetries with CF-type constraints. These observations are completely novel as far as the BRST formalism, with proper (anti-)co-BRST symmetries, is concerned.

Superspace Unitary Operator in Superfield Approach to Non-Abelian Gauge Theory with Dirac Fields

Within the framework of augmented version of the superfield approach to Becchi-Rouet-Stora-Tyutin (BRST) formalism, we derive the superspace unitary operator (and its Hermitian conjugate) in the context of four (3 + 1)-dimensional (4D) interacting non-Abelian 1-form gauge theory with Dirac fields. The ordinary 4D non-Abelian theory, defined on the flat 4D Minkowski spacetime manifold, is generalized onto a (4, 2)-dimensional supermanifold which is parameterized by the spacetime bosonic coordinatesxμ(withμ=0,1,2,3) and a pair of Grassmannian variables (θ,θ-) which satisfy the standard relationships:θ2=θ-2=0and θθ-+θ-θ=0. Various consequences of the application of the above superspace (SUSP) unitary operator (and its Hermitian conjugate) are discussed. In particular, we obtain the results of the application of horizontality condition (HC) and gauge-invariant restriction (GIR) in the language of the above SUSP operators. One of the novel results of our present investigation is the derivation of explicit expressions for the SUSP unitary operator (and its Hermitian conjugate) without imposing any Hermitian conjugation condition fromoutsideon the parameters and (super)fields of the supersymmetric version of our 4D interacting non-Abelian 1-form theory with Dirac fields.

Superfield approach to nilpotent symmetries in 3D Jackiw–Pi model of massive non-Abelian theory

In the available literature, only the Becchi–Rouet–Stora–Tyutin (BRST) symmetries are known for the Jackiw–Pi model of the three (2 + 1)-dimensional (3D) massive non-Abelian gauge theory. We derive the off-shell nilpotent [Formula: see text] and absolutely anticommuting (sbsab + sabsb = 0) (anti-)BRST transformations s(a)b corresponding to the usual Yang–Mills gauge transformations of this model by exploiting the “augmented” superfield formalism where the horizontality condition and gauge invariant restrictions blend together in a meaningful manner. There is a non-Yang–Mills (NYM) symmetry in this theory, too. However, we do not touch the NYM symmetry in our present endeavor. This superfield formalism leads to the derivation of an (anti-)BRST invariant Curci–Ferrari restriction, which plays a key role in the proof of absolute anticommutativity of s(a)b. The derivation of the proper anti-BRST symmetry transformations is important from the point of view of geometrical objects called gerbes. A novel feature of our present investigation is the derivation of the (anti-)BRST transformations for the auxiliary field ρ from our superfield formalism, which is neither generated by the (anti-)BRST charges nor obtained from the requirements of nilpotency and (or) absolute anticommutativity of the (anti-)BRST symmetries for our present 3D non-Abelian 1-form gauge theory.

TOPOLOGICALLY MASSIVE NON-ABELIAN THEORY: SUPERFIELD APPROACH

We apply the well-established techniques of geometrical superfield approach to Becchi–Rouet–Stora–Tyutin (BRST) formalism in the context of four (3+1)-dimensional (4D) dynamical non-Abelian 2-form gauge theory by exploiting its inherent "scalar" and "vector" gauge symmetry transformations and derive the corresponding off-shell nilpotent and absolutely anticommuting BRST and anti-BRST symmetry transformations. Our approach leads to the derivation of three (anti-)BRST invariant Curci–Ferrari (CF)-type restrictions that are found to be responsible for the absolute anticommutativity of the BRST and anti-BRST symmetry transformations. We derive the coupled Lagrangian densities that respect the (anti-)BRST symmetry transformations corresponding to the "vector" gauge transformations. We also capture the (anti-)BRST invariance of the CF-type restrictions and coupled Lagrangian densities within the framework of our superfield approach. We obtain, furthermore, the off-shell nilpotent (anti-)BRST symmetry transformations when the (anti-)BRST symmetry transformations corresponding to the "scalar" and "vector" gauge symmetries are merged together. These off-shell nilpotent "merged" (anti-)BRST symmetry transformations are, however, found to be non-anticommuting in nature.

BRST analysis of topologically massive gauge theory: novel observations

A dynamical non-Abelian 2-form gauge theory (with B∧F term) is endowed with the “scalar” and “vector” gauge symmetry transformations. In our present endeavor, we exploit the latter gauge symmetry transformations and perform the Becchi–Rouet–Stora–Tyutin (BRST) analysis of the four (3+1)-dimensional (4D) topologically massive non-Abelian 2-form gauge theory. We demonstrate the existence of some novel features that have, hitherto, not been observed in the context of BRST approach to 4D (non-)Abelian 1-form as well as Abelian 2-form and 3-form gauge theories. We comment on the differences between the novel features that emerge in the BRST analysis of the “scalar” and “vector” gauge symmetries.

A NOTE ON THE (ANTI-)BRST INVARIANT LAGRANGIAN DENSITIES FOR THE FREE ABELIAN 2-FORM GAUGE THEORY

We show that the previously known off-shell nilpotent [Formula: see text] and absolutely anticommuting (sb sab + sab sb = 0) Becchi–Rouet–Stora–Tyutin (BRST) transformations (sb) and anti-BRST transformations (sab) are the symmetry transformations of the appropriate Lagrangian densities of a four (3+1)-dimensional (4D) free Abelian 2-form gauge theory which do not explicitly incorporate a very specific constrained field condition through a Lagrange multiplier 4D vector field. The above condition, which is the analogue of the Curci–Ferrari restriction of the non-Abelian 1-form gauge theory, emerges from the Euler–Lagrange equations of motion of our present theory and ensures the absolute anticommutativity of the transformations s(a)b. Thus, the coupled Lagrangian densities, proposed in our present investigation, are aesthetically more appealing and more economical.

Nilpotent symmetry invariance in the non-Abelian 1-form gauge theory: Superfield formalism

We demonstrate that the nilpotent Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST symmetry invariance of the Lagrangian density of a four (3 + 1)-dimensional (4D) non-Abelian 1-form gauge theory with Dirac fields can be captured within the framework of the superfield approach to BRST formalism. The above 4D theory, where there is an explicit coupling between the non-Abelian 1-form gauge field and the Dirac fields, is considered on a (4, 2)-dimensional supermanifold, parameterized by the bosonic 4D space-time variables and a pair of Grassmannian variables. We show that the Grassmannian independence of the super Lagrangian density, expressed in terms of the (4, 2)-dimensional superfields, is a clear signature of the presence of the (anti-)BRST invariance in the original 4D theory.

Abelian 2-form gauge theory: superfield formalism

We derive the off-shell nilpotent Becchi–Rouet–Stora–Tyutin (BRST) and anti-BRST symmetry transformations for all the fields of a free Abelian 2-form gauge theory by exploiting the geometrical superfield approach to the 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 μ (with μ=0,1,2,3) and a pair of odd Grassmannian variables θ 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 superfield approach to the case of Abelian 3-form gauge theory in four, (3+1), dimensions of spacetime.

Free Abelian 2-form gauge theory: BRST approach

We discuss various symmetry properties of the Lagrangian density of a four- (3+1)-dimensional (4D) free Abelian 2-form gauge theory within the framework of Becchi–Rouet–Stora–Tyutin (BRST) formalism. The present free Abelian gauge theory is endowed with a Curci–Ferrari type condition, which happens to be a key signature of the 4D non-Abelian 1-form gauge theory. In fact, it is due to the above condition that the nilpotent BRST and anti-BRST symmetries of our present theory are found to be absolutely anticommuting in nature. For the present 2-form theory, we discuss the BRST, anti-BRST, ghost and discrete symmetry properties of the Lagrangian densities and derive the corresponding conserved charges. The algebraic structure, obeyed by the above conserved charges, is deduced and the constraint analysis is performed with the help of physicality criteria, where the conserved and nilpotent (anti-) BRST charges play completely independent roles. These physicality conditions lead to the derivation of the above Curci–Ferrari type restriction, within the framework of the BRST formalism, from the constraint analysis.

Gauge theories on A(dS) space and Killing vectors

We provide a general technique for collectively analysing a manifestly covariant formulation of non-abelian gauge theories on both anti-de Sitter as well as de Sitter spaces. This is done by stereographically projecting the corresponding theories, defined on a flat Minkowski space, onto the surface of the A(dS) hyperboloid. The gauge and matter fields in the two descriptions are mapped by conformal Killing vectors and conformal Killing spinors, respectively. A bilinear map connecting the spinors with the vector is established. Different forms of gauge fixing conditions and their equivalence are discussed. The U(1) axial anomaly as well as the non-abelian covariant and consistent chiral anomalies on A(dS) space are obtained. Electric-magnetic duality is demonstrated. The zero curvature limit is shown to yield consistent findings.

Gauge Freedom in the Path Integral Formalism

We investigate ’t Hooft’s technique of changing the gauge parameter of the linear covariant gauge from the point of view of the path integral with respect to the gauge freedom. Extension of the degrees of freedom allows us to formulate a system with extended gauge symmetry. The gauge fixing for this extended symmetry yields the ’t Hooft averaging as a path integral over the additional degrees of freedom. Another gauge fixing is found as a non-abelian analogue of the type II gaugeon formalism of Yokoyama and Kubo. In this connection, the ’t Hooft average can be viewed as the analogue of the type I gaugeon formalism. As a result, we obtain gauge covariant formulations of non-abelian gauge theories, which allow us to understand ’t Hoot’s technique also from the canonical fromalism.