We have previously developed the BRST quantization on the hypersurface $V_{N-1}$ embedded in N dimensional Euclidean space $R_N$ in both Hamiltonian and Lagrangian formulation. We generalize the formalism in the case of L dimensional manifold $V_L$ embedded in $R_N$ with $1\leq L < N$. The result is essentially the same as the previous one. We have also verified the results obtained here using a simple example of particle motion on a torus knot.
We study some reparametrization invariant theories in context of the BRST-co-BRST quantization method. The method imposes restrictions on the possible gauge fixing conditions and leads to well defined inner product states through a gauge regularisation procedure. Two explicit examples are also treated in detail.
BRST quantization of the one-dimensional constrained matrix model which describes two-dimensional Yang-Mills theory on the cylinder is performed. Classical and quantum BRST generators and BRST-invariant hamiltonians are constructed. Evolution operator is expressed in terms of BRST path integral. Advantages of the BRST quantization over the reduced phase space approach leading to the theory of $N$ free fermions are discussed.
One Lagrangian BRST quantization principle is that of imposing correct Schwinger-Dyson equations through the BRST Ward identities. In this paper we show how to derive the analogous $Sp(2)$-symmetric quantization condition in flat coordinates from an underlying $Sp(2)$-symmetric Schwinger-Dyson BRST symmetry. We also show under what conditions this can be recast in the language of triplectic quantization.
The BRST quantization of matrix Chern-Simons theory is carried out, the symmetries of the theory are analysed and used to constrain the form of the effective action.
The Hamiltonian BRST quantization of the null spinning string for different number of supersymmetries is given. A null spinning string with manifest space-time conformal invariance is constructed. Its Brst quantization gives negative critical dimension for $N\neq 0$ and reproduces previous results for $N=0$.
Recent results of BRST quantization on inner product spaces are reviewed. It is shown how relativistic particle models may be quantized with finite norms and that the relation between the operator method and the conventional path integral treatments is nontrivial.
We propose an extended BRST invariant Lagrangian quantization scheme of general gauge theories based on explicit realization of "modified triplectic algebra" in general coordinates. All the known Lagrangian quantization schemes based on the extended BRST symmetry are obtained by specifying the (free) parameters of that method.
The BFV-BRST Hamiltonian quantization method is presented for the theories where the gauge parameters are restricted by differential equations. The general formalism is exemplified by the Maxwell-like theory of symmetric tensor field.
A short rewiev of covariant quantization methods based on BRST-antiBRST symmetry is given. In particular problems of correct definition of Sp(2) symmetric quantization scheme known as triplectic quantization are considered.
The quantization of SU(2) Yang-Mills theory reduced to 0+1 space-time dimensions is performed in the BRST framework. We show that in the unitary gauge $A_0 = 0$ the BRST procedure has difficulties which can be solved by introduction of additional singlet ghost variables. In the Lorenz gauge $\dot{A}_0 = 0$ one has additional unphysical degrees of freedom, but the BRST quantization is free of the problems in the unitary gauge.
We generalize the method of superfield Lagrangian BRST quantization in the part of the gauge-fixing procedure and obtain a quantization method that can be considered as an alternative to the Batalin - Vilkovisky formalism.
We propose a superfield formalism of Lagrangian BRST-antiBRST quantization of arbitrary gauge theories in general coordinates with the base manifold of fields and antifields desribed in terms of both bosonic and fermionic variables.
We propose a modification of the gauge-fixing procedure in the Lagrangian method of superfield BRST quantization for general gauge theories which simultaneously provides a natural generalization of the well-known BV quantization scheme as far as gauge-fixing is concerned. A superfield form of BRST symmetry for the vacuum functional is found. The gauge-independence of the S-matrix is established.
We perform the antifield BRST quantization of duality-symmetric Maxwell theory and show explicitly the quantum equivalence of the different formulations (covariant and non-covariant). The non-covariant gauge-fixed action is used in the computation of propagators for this model.
We give a formal proof of the equivalence of Hamiltonian and Lagrangian BRST quantization. This is done for a generic $Sp(2)$-symmetric theory using flat (Darboux) coordinates. A new quantum master equation is derived in a Hamiltonian setting which contains all the Hamiltonian fields and momenta of a given theory.
Spin-5/2 gauge fields are quantized in an irreducible way within both the BRST and BRST-anti-BRST manners. To this end, we transform the reducible generating set into an irreducible one, such that the physical observables corresponding to these two formulations coincide. The gauge-fixing procedure emphasizes on the one hand the differences among our procedure and the results obtained in the literature, and on the other hand the equivalence between our BRST and BRST-anti-BRST approaches.
We perform the BFV-BRST quantization of the fourth-order Pais-Uhlenbeck oscillator (PUO). We show that although the PUO is not naturally constrained in the sense of Dirac-Bergmann, it is possible to profit from the introduction of suitable constraints in phase space in order to obtain a proper BRST invariant quantum system. Starting from its second-class constrained system description, we use the BFFT conversional approach to obtain first-class constraints as gauge symmetry generators. After the Abelianization of the constraints, we obtain the conserved BRST charge, the corresponding BRST tran…
Lagragian quantization rules for general gauge theories are proposed on a basis of a superfield formulation of the standard BRST symmetry. Independence of the $S$-matrix on a choice of the gauge is proved. The Ward identities in terms of superfields are derived.
New BRST-invariant states for SU(3) gauge field theory are presented. The states have finite norms and unlike the states that are usually used to derive path integrals, they break SU(3) symmetry by choosing preferred gauge directions. This symmetry breaking may also give effective masses to some of the gauge bosons of the theory.
We propose an extended BRST invariant Lagrangian quantization scheme of general gauge theories based on an explicit realization of the modified triplectic algebra that was announced in our previous investigation (hep-th/0104189). The algebra includes, besides the odd operators $V^a$ appearing in the triplectic formalism, also the odd operators $U^a$ introduced within modified triplectic quantization, both of which being anti-Hamiltonian vector fields. We show that some even supersymplectic structure defined on the space of fields and antifields provides the extended BRST path integral with a w…
We construct the BRST operator for the nonlinear $WB_2$ and $W_4$ algebras. Contrary to the general belief, the nilpotent condition of the BRST operator doesn't determine all the coefficients. We find a three and seven parameter family of nilpotent BRST operator for $WB_2$ and $W_4$ respectively. These free parameters are related to the canonical transformation of the ghost antighost fields.
An analysis of the state space in the BRST--quantization in the Schroedinger representation is performed on the basis of the results obtained earlier in the framework of the Fock space representation. It is shown that to get satisfactory results it is necessary to have from the very beginning a meaningful definition of the total state space.
The complete structure of the $WG_2$ algebra is obtained from an explicit realization by an abstract Virasoro algebra and a free boson field. We then construct its BRST operator and find a seven-parameter family of nilpotnt BRST operators. These free parameters are related to the canonical transformations of the ghost, antighost fields which leave the total stress-energy tensor and the antighost field $b$ invariant.
BRST-methods provide elegant and powerful tools for the construction and analysis of constrained systems, including models of particles, strings and fields. These lectures provide an elementary introduction to the ideas, illustrated with some important physical applications.
Working from first principles, quantization of a class of Hamiltonian systems with reducible symmetry is carried out by constructing first the appropriate reduced phase space and then the BRST cohomology. The constraints of this system correspond to a first class set for a group G and a second class set for a subgroup H. The BRST operator constructed is equivariant with respect to H. Using algebraic techniques analogous to those of equivariant de Rham theory, the BRST operator is shown to correspond to that obtained by BV quantization of a class of systems with reducible symmetry. The 'ghosts …
We study the BRST quantization of bosonic and NSR strings propagating in AdS(3) x N backgrounds. The no-ghost theorem is proved using the Frenkel-Garland-Zuckerman method. Regular and spectrally-flowed representations of affine SL(2,R) appear on an equal footing. Possible generalizations to related curved backgrounds are discussed.
The BRST quantization of particle motion on the hypersurface $V_{(N-1)}$ embedded in Euclidean space $R_N$ is carried out both in Hamiltonian and Lagrangian formalism. Using Batalin-Fradkin-Fradkina-Tyutin (BFFT) formalism, the second class constrained obtained using Hamiltonian analysis are converted into first class constraints. Then using BFV analysis the BRST symmetry is constructed. We have given a simple example of these kind of system. In the end we have discussed Batalin-Vilkovisky formalism in the context of this (BFFT modified) system.
It is shown how the BRST quantization can be applied to a gauge invariant sector of theories with anomalously broken symmetries. This result is used to show that shifting the anomalies to a classically trivial sector of fields (Wess-Zumino mechanism) makes it possible to quantize the physical sector using a standard BRST procedure, as for a non anomalous theory. The trivial sector plays the role of a topological sector if the system is quantized without shifting the anomalies.
We give a simple geometrical picture of the basic structures of the covariant $Sp(2)$ symmetric quantization formalism -- triplectic quantization -- recently suggested by Batalin, Marnelius and Semikhatov. In particular, we show that the appearance of an even Poisson bracket is not a particular property of triplectic quantization. Rather, any solution of the classical master equation generates on a Lagrangian surface of the antibracket an even Poisson bracket. Also other features of triplectic quantization can be identified with aspects of conventional Lagrangian BRST quantization without exte…
It has recently been shown that the Field Antifield quantization of anomalous irreducible gauge theories with closed algebra can be represented in a BRST superspace where the quantum action at one loop order, including the Wess Zumino term, and the anomalies show up as components of the same superfield. We show here how the Chiral Schwinger model can be represented in this formulation.
We study a sixth order derivative scalar field model in Minkowski spacetime as a toy model of higher-derivative critical gravity theories. This model is consistently quantized when using the Becchi-Rouet-Stora-Tyutin (BRST) quantization scheme even though it does not show gauge symmetry manifestly. Imposing a BRST quartet generated by two scalars and ghosts, there remains a non-trivial subspace with positive norm. This might be interpreted as a Minkowskian dual version of the unitary truncation in the logarithmic conformal field theory.
The physical state condition in the BRST quantization of Chern-Simons field theory is used to derive Gauss law constraints in the presence of Wilson loops, which play an important role in explicitly establishing the connection of Chern-Simons field theory with 2-dimensional conformal field theory.
We show that the harmonic Becchi-Rouet-Stora-Tyutin method of quantizing bosonic systems with second-class constraints or first-class holomorphic constraints extends to systems having both bosonic and fermionic second-class or first-class holomorphic constraints. Using a limit argument, we show that the harmonic BRST modified path integral reproduces the correct Senjanovic measure.
The BRST quantization of a gauge theory in noncommutative geometry is carried out in the ``matrix derivative" approach. BRST/anti-BRST transformation rules are obtained by applying the horizontality condition, in the superconnection formalism. A BRST/anti-BRST invariant quantum action is then constructed, using an adaptation of the method devised by Baulieu and Thierry-Mieg for the Yang-Mills case. The resulting quantum action turns out to be the same as that of a gauge theory in the 't Hooft gauge with spontaneously broken symmetry. Our result shows that only the even part of the supergroup a…
The correspondence between BRST-BFV, Dirac and projection operator approaches to quantize constrained systems is analyzed. It is shown that the component of the BFV wave function with maximal number of ghosts and antighosts in the Schrodinger representation may be viewed as a wave function in the projection operator approach. It is shown by using the relationship between different quantization techniques that the Marnelius inner product for BRST-BFV systems should be in general modified in order to take into account the topology of the group; the Giulini-Marolf group averaging prescription for…
The BRST quantization of strings is revisited and the derivation of the path integral measure for scattering amplitudes is streamlined. Gauge invariances due to zero modes in the ghost sector are taken into account by using the Batalin-Vilkovisky formalism. This involves promoting the moduli of Riemann surfaces to quantum mechanical variables on which BRST transformations act. The familiar ghost and antighost zero mode insertions are recovered upon integrating out auxiliary fields. In contrast to the usual treatment, the gauge-fixed action including all zero mode insertions is BRST invariant. …
We consider the Hamiltonian BRST quantization of a noncommutative non abelian gauge theory. The Seiberg-Witten map of all phase-space variables, including multipliers, ghosts and their momenta, is given in first order in the noncommutative parameter $θ$. We show that there exists a complete consistence between the gauge structures of the original and of the mapped theories, derived in a canonical way, once we appropriately choose the map solutions.
Recent formal solutions of BRST quantization on inner product spaces within the operator method are shown to lead to an unexpected interpretation of the conventional path integral formulation. The relation between the Hamiltonians in the two formulations is nontrivial. For the operator method the correspondence requires certain quantum rules which make the formal solutions exact, and for the path integral the correspondence yields a precise connection between boundary conditions and the choice of gauge fixing.
Some general formulas are derived for the solutions of a BRST quantization on inner product spaces of finite dimensional bosonic gauge theories invariant under arbitrary Lie groups. A detailed analysis is then performed of SL(2,R) invariant models and some possible geometries of the Lagrange multipliers are derived together with explicit results for a class of SL(2,R) models. Gauge models invariant under a nonunimodular gauge group are also studied in some detail.
In our previous papers, we prove the no-ghost theorem without light-cone directions (hep-th/0005002, hep-th/0303051). We point out that our results are valid for more general backgrounds. In particular, we prove the no-ghost theorem for AdS_3 in the context of the BRST quantization (with the standard restriction on the spin). We compare our BRST proof with the OCQ proof and establish the BRST-OCQ equivalence for AdS_3. The key in both approaches lies in the certain structure of the matter Hilbert space as a product of two Verma modules. We also present the no-ghost theorem in the most general …
In this paper, the BRST symmetry transformation is presented for the noncommutative U(N) gauge theory. The nilpotency of the charge associated to this symmetry is then proved. As a consequence for the space-like non-commutativity parameter, the Hilbert space of physical states is determined by the cohomology space of the BRST operator as in the commutative case. Further, the unitarity of the S-matrix elements projected onto the subspace of physical states is deduced.
We introduce an operator version of the BRST-BFV effective action for arbitrary systems with first-class constraints. Using the Schwinger action principle we calculate the propagators corresponding to: (i) the parametrized non-relativistic free particle, (ii) the relativistic free particle and (iii) the spining relativistic free particle. Our calculation correctly imposes the BRST-invariance at the end-points. The precise use of the additional boundary terms required in the description of fermionic variables is also incorporated.
A new generalization of the vector Schwinger model is considered where gauge symmetry is broken at the quantum mechanical level. By proper extension of the phase space this broken symmetry has been restored. Also an equivalent first class theory is reformulated in the actual phase space using Mitra and Rajaraman's prescription \cite{mr1,mr2}. A BRST invariant effective action is also formulated. The new dynamical fields introduced, turn into Wess-Zumino scalar.
The off-shell nilpotent BRST charge and the BRST invariant effective action for non-abelian BF topological theories over D-dimensional manifolds are explicitly constructed. These theories have the feature of being reducible with exactly D-3 stages of reducibility. The adequate extended phase space including the different levels of ghosts for ghosts is explicitly obtained. Using the structure of the resulting BRST charge we show that for topological BF theories the semi-classical approximation completely describes the quantum theory. The independence of the partition function on the metric also…
The Batalin-Fradkin-Tyutin (BFT) scheme, which is an improved version of Dirac quantization, is applied to the $CP^1$ model, and the compact form of a nontrivial first-class Hamiltonian is directly obtained by introducing the BFT physical fields. We also derive a BRST-invariant gauge fixed Lagrangian through the standard path-integral procedure. Furthermore, performing collective coordinate quantization we obtain energy spectrum of rigid rotator in the $CP^1$ model. Exploiting the Hopf bundle, we also show that the $CP^1$ model is exactly equivalent to the O(3) nonlinear sigma model at the can…
We consider possible superfield representations of extended BRST symmetry for general gauge theories within the principle of gauge-fixing based on a generating equation for the gauge functional. We examine admissible superfield choices for an extended antibracket and delta-operator with given algebraic properties and show that only one of these choices is compatible with the requirement of extended BRST symmetry realized in terms of supertranslations along Grassmann coordinates. We demonstrate that this realization leads to the gauge-independence of the S-matrix.
It is shown that for a large class of non-holonomic quantum mechanical systems one can make the computation of BRST charge fully algorithmic. Two computer algebra programs written in the language of {\tt REDUCE} are described. They are able to realize the complex calculations needed to determine the charge for general nonlinear algebras. Some interesting specific solutions are discussed.
Recently derived general formal solutions of a BRST quantization on inner product spaces of irreducible Lie group gauge theories are applied to trivial models and relativistic particle models for particles with spin 0, 1/2 and 1. In the process general quantization rules are discovered which make the formal solutions exact. The treatment also give evidence that the formal solutions are directly generalizable to theories with graded gauge symmetries. For relativistic particles reasonable results are obtained although there exists no completely Lorentz covariant quantization of the coordinate an…
There is an elaborated abstract form of BRST quantization on inner product spaces within the operator formalism which leads to BRST invariant states of the form |ph>=e^{[Q,ψ]} |ϕ> where ψis a gauge fixing fermion, and where |ϕ> is a BRST invariant state determined by simple hermitian conditions. These state representations are closely related to the path integral formulation. Here we analyse the basics of this approach in detail. The freedom in the choice of ψand |ϕ> as well as their properties under gauge transformations are explicitly determined for simple abelian models. In all considered c…
A previously proposed generalized BRST quantization on inner product spaces for second class constraints is further developed through applications. This BRST method involves a conserved generalized BRST charge Q which is not nilpotent but which satisfies Q=δ+δ^{\dagger}, δ^2=0, and by means of which physical states are obtained from the projection δ|ph>=δ^{\dagger}|ph>=0. A simple model is analyzed in detail from which some basic properties and necessary ingredients are extracted. The method is then applied to a massive vector field. An effective theory is derived which is close to the one of …
We apply the Batalin-Tyutin Hamiltonian method to the Abelian Proca model in order to convert a second class constraint system into a first class one systematically by introducing the new fields. Then, according to the BFV formalism we obtain that the desired resulting Lagrangian preserving standard BRST symmetry naturally includes the well-known Stükelberg scalar related to the explicit gauge-breaking effect due to the presence of the mass term. Furthermore, we also discuss the nonlocal symmetry structure of this model in the context of the nonstandard BRST symmetry.
Correspondence between BRST-BFV, Dirac and refined algebraic (group averaging, projection operator) approaches to quantize constrained systems is analyzed. For the closed-algebra case, it is shown that the component of the BFV wave function with maximal (minimal) number of ghosts and antighosts in the Schrodinger representation may be viewed as a wave function in the refined algebraic (Dirac) quantization approach. The Giulini-Marolf group averaging formula for the inner product in the refined algebraic quantization approach is obtained from the Batalin-Marnelius prescription for the BRST-BFV …
We propose a construction for nonlinear off-shell gauge field theories based on a constrained system quantized in the sense of deformation quantization. The key idea is to consider the star-product BFV--BRST master equation as an equation of motion. The construction is formulated in terms of the BRST extention of the unfolded formalism that can also be understood as an appropriate generalization of the AKSZ procedure. As an application, we consider a very simple constrained system, a quantized scalar particle, and show that it gives rise to an off-shell higher-spin gauge theory that automatica…
In this paper we discuss the BRST-quantization of anomalous 2d-Yang Mills (YM) theory. Since we use an oscillator basis for the YM-Fock-space the anomaly appears already for a pure YM-system and the constraints form a Kac-Moody algebra with negative central charge. We also discuss the coupling of chiral fermions and find that the BRST-cohomology for systems with chiral fermions in a sufficiently large representation of the gauge group is completely equivalent to the cohomology of the finite dimensional gauge group. For pure YM theory or YM theory coupled to chiral fermions in small representat…
We consider the general gauge theory with a closed irreducible gauge algebra possessing the non-anomalous global (super)symmetry in the case when the gauge fixing procedure violates the global invariance of classical action. The theory is quantized in the framework of BRST-BV approach in the form of functional integral over all fields of the configuration space. It is shown that the global symmetry transformations are deformed in the process of quantization and the full quantum action is invariant under such deformed global transformations in the configuration space. The deformed global transf…
The basic BRST cohomological properties of a free, massless tensor field with the mixed symmetry of the Riemann tensor are studied in detail. It is shown that any non-trivial co-cycle from the local BRST cohomology group can be taken to stop at antighost number three, its last component belonging to the cohomology of the exterior longitudinal derivative and containing non-trivial elements from the (invariant) characteristic cohomology.
We develop the basic ideas and equations for the BRST quantization of Yang-Mills theories in an explicit Hamiltonian approach, without any reference to the Lagrangian approach at any stage of the development. We present a new representation of ghost fields that combines desirable self-adjointness properties with canonical anticommutation relations for ghost creation and annihilation operators, thus enabling us to characterize the physical states on a well-defined Fock space. The Hamiltonian is constructed by piecing together simple BRST invariant operators to obtain a minimal invariant extensi…
We analyze the problems with the so called gauge invariant quantization of the anomalous gauge field theories originary due to Faddeev and Shatashvili (FS). Our analysis bring to a generalization of FS method which allows to construct a series of classically equivalent theories which are non equivalent at quantum level. We prove that these classical theories are all consistent with the BRST invariance of the original gauge symmetry with suitably augmented field content. As an example of such a scenario, we discuss the class of physically distinct models of two dimensional induced gravity which…
