Towards Lagrangian formulations of mixed-symmetry Higher Spin Fields on AdS-space within BFV-BRST formalism
Alexander A. Reshetnyak
备注：
Slightly enlarged contribution to the Proceedings of the International Bogolyubov Conference-2009 "Problems of Theoretical and Mathematical Physics", Moscow - Dubna, August 21 - 27, 2009, 9 pages, no figures
The spectrum of superstring theory on the $AdS_5 \times S_5$ Ramond-Ramond background in tensionless limit contains integer and half-integer higher-spin fields subject at most to two-rows Young tableaux $Y(s_1,s_2)$. We review the details of a gauge-invariant Lagrangian description of such massive and massless higher-spin fields in anti-de-Sitter spaces with arbitrary dimensions. The procedure is based on the construction of Verma modules, its oscillator realizations and of a BFV-BRST operator for non-linear algebras encoding unitary irreducible representations of AdS group.
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.
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 general BRST-BFV analysis of anomaly in the string theory coupled to background fields is carried out. An exact equation for c-valued symbol of anomaly operator is found and structure of its solutions is studied.
Starting from an associated reparametrization-invariant action, the generalization of the BRST-BFV method for the case of nonstationary systems is constructed. The extension of the Batalin-Tyutin conversional approach is also considered in the nonstationary case. In order to illustrate these ideas, the propagator for the time-dependent two-dimensional rotor is calculated by reformulating the problem as a system with only first class constraints and subsequently using the BRST-BFV prescription previously obtained.
BFV--BRST charge for q-deformed algebras is not unique. Different constructions of it in the classical as well as in the quantum phase space for the $q$-deformed algebra sl_q(2) are discussed. Moreover, deformation of the phase space without deforming the generators of sl(2) is considered. $\hbar$-q-deformation of the phase space is shown to yield the Witten's second deformation. To study the BFV--BRST cohomology problem when both the quantum phase space and the group are deformed, a two parameter deformation of sl(2) is proposed, and its BFV-BRST charge is given.
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…
We study finite field dependent BRST-BFV transformations for dynamical systems with first- and second-class constraints within the generalized Hamiltonian formalism. We find explicitly their Jacobians and the form of a solution to the compensation equation necessary for generating an arbitrary finite change of gauge-fixing functionals in the path integral.
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.
We study systematically finite BRST-BFV transformations in the generalized Hamiltonian formalism. We present explicitly their Jacobians and the form of a solution to the compensation equation determining the functional field dependence of finite Fermionic parameters, necessary to generate arbitrary finite change of gauge-fixing functions in the path integral.
We study systematically finite BRST-BFV transformations in $Sp(2)$-extended generalized Hamiltonian formalism. We present explicitly their Jacobians and the form of a solution to the compensation equation determining the functional field dependence of finite Fermionic parameters, necessary to generate arbitrary finite change of gauge-fixing functions in the path integral.
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.
The transition amplitude is obtained for a free massive particle of arbitrary spin by calculating the path integral in the index-spinor formulation within the BFV-BRST approach. None renormalizations of the path integral measure were applied. The calculation has given the Weinberg propagator written in the index-free form with the use of index spinor. The choice of boundary conditions on the index spinor determines holomorphic or antiholomorphic representation for the canonical description of particle/antiparticle spin.
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 …
The key aspects of a gauge-invariant Lagrangian description of massive and massless half-integer higher-spin fields in AdS spaces with a two-row Young tableaux $Y(s_1,s_2)$ are presented in an unconstrained description, as well as in off-shell formulations with algebraic constraints, on the basis of BFV-BRST operators for non-linear operator superalgebras, encoding the initial conditions realized by constraints in a Fock space and extracting the higher-spin fields from unitary representations of the AdS group.
Two-dimensional supergravity theory is quantized as an anomalous gauge theory. In the Batalin-Fradkin (BF) formalism, the anomaly-canceling super-Liouville fields are introduced to identify the original second-class constrained system with a gauge-fixed version of a first-class system. The BFV-BRST quantization applies to formulate the theory in the most general class of gauges. A local effective action constructed in the configuration space contains two super-Liouville actions; one is a noncovariant but local functional written only in terms of 2D supergravity fields, and the other contains t…
Gauge-invariant systems of a general form with higher order derivatives of gauge parameters are investigated within the framework of the BFV formalism. Higher order terms of the BRST charge and BRST-invariant Hamiltonian are obtained. It is shown that the identification rules for Lagrangian and Hamiltonian ghost variables depend on the choice of the extension of constraints from the primary constraint surface.
Constrained hamiltonian structure of noncommutative gauge theory for the gauge group U(1) is discussed. Constraints are shown to be first class, although, they do not give an Abelian algebra in terms of Poisson brackets. The related BFV-BRST charge gives a vanishing generalized Poisson bracket by itself due to the associativity of *-product. Equivalence of noncommutative and ordinary gauge theories is formulated in generalized phase space by using BFV-BRST charge and a solution is obtained. Gauge fixing is discussed.
The regularization scheme is proposed for the constrained Hamiltonian formulation of the gauge fields coupled to the chiral or axial fermions. The Schwinger terms in the regularized operator first-class constraint algebra are shown to be consistent with the covariant divergence anomaly of the corresponding current. Regularized quantum master equations are studied, and the Schwinger terms are found out to break down both nilpotency of the BRST-charge and its conservation law. Wess-Zumino consistency conditions are studied for the BRST anomaly and they are shown to contradict to the covariant Sc…
I point out an unexpected relation between the BV (Batalin-Vilkovisky) and the BFV (Batalin-Fradkin-Vilkovisky) formulations of the same pure gauge (topological) theory. The nonminimal sector in the BV formulation of the topological theory allows one to construct the Poisson bracket and the BRST charge on some Lagrangian submanifold of the BV configuration space; this Lagrangian submanifold can be identified with the phase space of the BFV formulation of the same theory in the minimal sector of ghost variables. The BFV Poisson bracket is induced by a natural even Poisson bracket on the station…
The BRST quantization of the Abelian Proca model is performed using the Batalin-Fradkin-Tyutin and the Batalin-Fradkin-Vilkovisky formalism. First, the BFT Hamiltonian method is applied in order to systematically convert a second class constraint system of the model into an effectively first class one by introducing new fields. In finding the involutive Hamiltonian we adopt a new approach which is more simpler than the usual one. We also show that in our model the Dirac brackets of the phase space variables in the original second class constraint system are exactly the same as the Poisson brac…
We introduce and study a new discrete basis of gravity constraints by making use of harmonic expansion for closed cosmological models. The full set of constraints is splitted into area-preserving spatial diffeomorphisms, forming closed subalgebra, and Virasoro-like generators. Operatorial Hamiltonian BFV-BRST quantization is performed in the framework of perturbative expansion in the dimensionless parameter which is a positive power of the ratio of Planckian volume to the volume of the Universe. For the (N+1) - dimensional generalization of stationary closed Bianchi-I cosmology the nilpotency …
Quantum localization of classical mechanics within the BRST-BFV and BV (or field-antifield) quantization methods are studied. It is shown that a special choice of gauge fixing functions (or BRST-BFV charge) together with the unitary limit leads to Hamiltonian localization in the path integral of the BRST-BFV formalism. In turn, we find that a special choice of gauge fixing functions being proportional to extremals of an initial non-degenerate classical action together with a very special solution of the classical master equation result in Lagrangian localization in the partition function of th…
We identify a strong similarity among several distinct originally second-class systems, including both mechanical and field theory models, which can be naturally described in a gauge-invariant way. The canonical structure of such related systems is encoded into a gauge-invariant generalization of the quantum rigid rotor. We pursue the BRST symmetry analysis and the BFV functional quantization for the mentioned gauge-invariant version of the generalized quantum rigid rotor. We obtain different equivalent effective actions according to specific gauge-fixing choices, showing explicitly their BRST…
Gauge-invariant descriptions for a free bosonic scalar field of continuous spin in a $d$-dimensional Minkowski space-time using a metric-like formulation are constructed on the basis of a constrained BRST-BFV approach we propose. The resulting BRST-BFV equations of motion for a scalar field augmented by ghost operators contains different sets of auxiliary fields, depending on the manner of a partial gauge-fixing and a resolution of some of the equations of motion for a BRST-unfolded first-stage reducible gauge theory. To achieve an equivalence of the resulting BRST-unfolded constrained equatio…
Gauge invariant Lagrangian descriptions of irreducible and reducible half-integer higher-spin mixed-symmetric massless and massive representations of the Poincare group with off-shell algebraic constraints are constructed within a metric-like formulation in a $d$-dimensional flat space-time on the basis of a suggested constrained BRST approach. A Lorentz-invariant resolution of the BRST complex within the constrained BRST formulations produces a gauge-invariant Fang-Fronsdal Lagrangian entirely in terms of the initial triple gamma-traceless spin-tensor field $Ψ_{(μ)_{n}}$ with gamma-traceless …
In the BRST-BFV scheme for noncommutative D-branes with constant NS $B$-field, introducing ghost degrees of freedom we construct the gauge fixed Hamiltonian and corresponding effective Lagrangian invariant under nilpotent BRST charge. It is also shown that the presence of auxiliary variables introduced via the improved Dirac formalism plays a crucial role in the construction of the BRST invariant Lagrangian.
BRST-BFV method for constrained Lagrangian formulations (LFs) for (ir)reducible half-integer HS Poincare group representations in Minkowski space is suggested. The procedure is derived by 2 ways: from the unconstrained BRST-BFV method for mixed-symmetry HS fermionic fields subject to an arbitrary Young tableaux with k rows (suggested in arXiv:1211.1273[hep-th]) by extracting the second-class constraints, $\widehat{O}_α=(\widehat{O}_a, \widehat{O}^+_a)$, from a total superalgebra of constraints, second, in self-consistent way by means of finding BRST-extended initial off-shell algebraic constra…
The four dimensional SU(3) WZW model coupled to electromagnetism is treated as a constrained system in the context of Batalin-Fradkin- Vilkovisky formalism. It is shown that this treatment is equivalent to the Faddeev-Jackiw (FJ) approach. It is also shown that the field redefinitions that transform the fields of the model into BRST and $σ$ closed are actually the Darboux's transformations used in the FJ formalism.
We consider the Hamiltonian and Lagrangian embedding of a first-order, massive spin-one, gauge non-invariant theory involving anti-symmetric tensor field. We apply the BFV-BRST generalised canonical approach to convert the model to a first class system and construct nil-potent BFV-BRST charge and an unitarising Hamiltonian. The canonical analysis of the Stückelberg formulation of this model is presented. We bring out the contrasting feature in the constraint structure, specifically with respect to the reducibility aspect, of the Hamiltonian and the Lagrangian embedded model. We show that to ob…
The recently proposed generalized field method for solving the master equation of Batalin and Vilkovisky is applied to a gauge theory of quadratic Lie algebras in 2-dimensions. The charge corresponding to BRST symmetry derived from this solution in terms of the phase space variables by using the Noether procedure, and the one found due to the BFV-method are compared and found to coincide. $W_3$ algebra, formulated in terms of a continuous variable is emploied in the mentioned gauge theory to construct a $W_3$ topological gravity. Moreover, its gauge fixing is briefly discussed.
The details of Lagrangian description of irreducible integer higher-spin representations of the Poincare group with an Young tableaux $Y[\hat{s}_1,\hat{s}_2]$ having $2$ columns are considered for Bose particles propagated on an arbitrary dimensional Minkowski space-time. The procedure is based, first, on using of an auxiliary Fock space generated by Fermi oscillators (antisymmetric basis), second, on construction of the Verma module and finding auxiliary oscillator realization for $sl(2)\oplus sl(2)$ algebra which encodes the second-class operator constraints subsystem in the HS symmetry supe…
For a wide class of mechanical systems, invariant under gauge transformations with higher (arbitrary) order time derivatives of gauge parameters, the equivalence of Lagrangian and Hamiltonian BRST formalisms is proved. It is shown that the Ostrogradsky formalism establishes the natural rules to relate the BFV ghost canonical pairs with the ghosts and antighosts introduced by the Lagrangian approach. Explicit relation between corresponding gauge-fixing terms is obtained.
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.
A supersymmetric gauge invariant action is constructed over any 4-dimensional Riemannian manifold describing Witten's theory of 4-monopoles. The topological supersymmetric algebra closes off-shell. The multiplets include the auxiliary fields and the Wess-Zumino fields in an unusual way, arising naturally from BRST gauge fixing. A new canonical approach over Riemann manifolds is followed, using a Morse function as an euclidean time and taking into account the BRST boundary conditions that come from the BFV formulation. This allows a construction of the effective action starting from gauge princ…
We present an analysis on the BRST symmetry transformations of the Horava theory under the BFV quantization, both in the nonprojectable and projectable cases. We obtain that the BRST transformations are intimately related to a particular spatial diffeomorphism along one of the ghost vector fields. We show explicitly the invariance of the quantum action and the nilpotence of the BRST transformations, using this diffeomorphism largely. The BRST symmetry is verified in the whole phase space, inside and outside the constrained surface. When restricted to the constrained surface, the BRST transform…
A recent progress in a gauge-invariant Lagrangian description of massive and massless half-integer higher-spin fields in AdS and Minkowski spaces is presented. The procedure is based on a BFV-BRST operator, encoding the initial conditions realized by constraints in a Fock space and extracting the higher-spin fields from unitary irreducible representations of the AdS (Poincare) group. The construction is applicable to higher-spin tensor fields with a multi-row Young tableaux.
The microcanonical density matrix in closed cosmology has a natural definition as a projector on the space of solutions of Wheeler-DeWitt equations, which is motivated by the absence of global non-vanishing charges and energy in spatially closed gravitational systems. Using the BRST/BFV formalism in relativistic phase space of gauge and ghost variables we derive the path integral representation for this projector and the relevant statistical sum. This derivation circumvents the difficulties associated with the open algebra of noncommutative quantum Dirac constraints and the construction/regula…
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 the BRST quantization of gauge theories, the zero locus $Z_Q$ of the BRST differential $Q$ carries an (anti)bracket whose parity is opposite to that of the fundamental bracket. We show that the on-shell BFV/BV gauge symmetries are in a 1:1 correspondence with Hamiltonian vector fields on $Z_Q$, and observables of the BRST theory are in a 1:1 correspondence with characteristic functions of the bracket on $Z_Q$. By reduction to the zero locus, we obtain relations between bracket operations and differentials arising in different complexes (the Gerstenhaber, Schouten, Berezin-Kirillov, and Skly…
A constrained BRST-BV Lagrangian formulation for totally symmetric massless HS fields in a $d$-dimensional Minkowski space is extended to a non-minimal constrained BRST-BV Lagrangian formulation by using a non-minimal BRST operator $Q_{c|\mathrm{tot}}$ with non-minimal Hamiltonian BFV oscillators $\overline{C}, \overline{\mathcal{P}}, λ, π$, as well as antighost and Nakanishi-Lautrup tensor fields, in order to introduce an admissible self-consistent gauge condition. The gauge-fixing procedure involves an operator gauge-fixing BRST-BFV Fermion $Ψ_H$ as a kernel of the gauge-fixing BRST-BV Fermi…
We show that the BFV quantization scheme can be implemented in the nonprojectable 2+1 Horava theory. This opens the possibility of imposing more general gauge conditions in the quantization of this theory. The BFV quantization is based on the canonical formalism, which is suitable to incorporate the measure associated to the second-class constraints that the theory has. Special features of the Hamiltonian density and the matrix of second-class constraints allow that the system be involutive in terms of Dirac brackets, which is a nontrivial requisite for implementing the BFV formalism. We prese…
Using the generalized Hamiltonian method of Batalin, Fradkin and Vilkovsky we develop the BRST formalism for the bosonic string on AdS(5)xS(5) formulated as principal chiral model. Then we show that the monodromy matrix and non-local charges are BRST invariant.
Using BRST-BV formulation of relativistic dynamics, arbitrary spin massless and massive field propagating in flat space and arbitrary spin massless fields propagating in AdS space are considered. For such fields, BRST-BV Lagrangians invariant under gauge transformations are obtained. The Lagrangians and gauge transformations are built in terms of traceless gauge fields and traceless gauge transformation parameters. The use of the Poincare parametrization of AdS space allows us to get simple BRST-BV Lagrangian for AdS fields. By imposing the Siegel gauge condition, we get gauge-fixed Lagrangian…
By means of a generalized quartet mechanism we show in a model independent way that a BRST quantization on an inner product space leads to physical states of the form |ph>=e^{[Q, ψ]} |ph>_0 where Q is the nilpotent BRST operator, ψa hermitian fermionic gauge fixing operator, and |ph>_0 BRST invariant states determined by a hermitian set of BRST doublets in involution. |ph>_0 does not belong to an inner product space although |ph> does. Since the BRST quartets are split into two sets of hermitian BRST doublets there are two choices for |ph>_0 and the corresponding ψ. When applied to general, bo…
The constrained Hamiltonian formalism is worked out for the theories where the gauge symmetry parameters are unfree, being restricted by differential equations. The Hamiltonian BFV-BRST embedding is elaborated for this class of gauge theories. The general formalism is exemplified by the linearized unimodular gravity.
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.
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 …
The detailed description of the method of the construction of the nilpotent BRST charges for nonlinear algebras of constraints appearing in the description of the massless higher spin fields on the $AdS_D$ background is presented. It is shown that the corresponding BRST charge is not uniquely defined, but this ambiguity has no impact on the physical content of the theory.
Totally symmetric arbitrary spin massless and massive fields in AdS space are studied. For such fields, we obtain Lagrangians which are invariant under global BRST transformations. The Lagrangians are used for computation of partition functions and effective actions. We demonstrate that BRST invariant bulk action for massless field evaluated on the solution of Dirichlet problem for gauge massless fields and Faddeev-Popov fields leads to BRST invariant effective action for canonical shadow gauge fields and shadow Faddeev-Popov fields, while the BRST invariant bulk action for massive field evalu…
We develop BRST symmetry for the first time for a particle on the surface of a torus knot by analyzing the constraints of the system. The theory contains $2^{nd}$ class constraints and has been extended by introducing Wess-Zumino term to the convert it into a theory with first class constraints. BFV analysis of the extended theory is performed to construct BRST/anti-BRST symmetries for the particle on torus knot. The nilpotent BRST/anti-BRST charges which generate such symmetries are constructed explicitly. The states annihilated by these nilpotent charges consist the physical Hilbert space. W…
A concise formulation for mixed-symmetry gauge fields on AdS space is proposed. It is explicitly local, gauge invariant, and has manifest AdS symmetry. Various other known formulations (including the original formulation of Metsaev and the unfolded formulation) can be derived through the appropriate reductions and gauge fixing. As a byproduct, we also identify some new useful formulations of the theory that can be interesting for further developments. The formulation is presented in the BRST terms and extensively uses Howe duality. In particular, the BRST operator is a sum of the term associat…
We perform the quantization of a massive particle propagating on AdS_5. We use the twistor formulation in which the action can be brought into a quadratic form. We construct the BRST operator which commutes with AdS_5 isometries forming SU(2,2). The condition of a consistent BRST quantization requires that the AdS energy E is quantized in units of the AdS_5 radius R, E=\frac{1}{2R}(N_a +N_b+4), with N_a, N_b being some non-negative integers. We also argue that the mass operator will be identified with the moduli of the U(1) central extension Z of the SU(2,2|4) algebra in the supersymmetric cas…
We construct a concise gauge invariant formulation for massless, partially massless, and massive bosonic AdS fields of arbitrary symmetry type at the level of equations of motion. Our formulation admits two equivalent descriptions: in terms of the ambient space and in terms of an appropriate vector bundle, as an explicitly local first-order BRST formalism. The second version is a parent-like formulation that can be used to generate various other formulations via equivalent reductions. In particular, we demonstrate a relation to the unfolded description of massless and partially massless fields…
A geometric multisymplectic formulation of the classical BRST symmetry of constrained first-order classical field theories is described. To effect this we introduce graded analogues of the bundles and manifolds of the multisymplectic formulation of first-order field theories. The Lagrange-d'Alembert formalism is also developed in terms of the multisymplectic framework. The result is a covariant Hamiltonian BFV formalism.
