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Register a new userBRST-BFV and BRST-BV Descriptions for Bosonic Fields with Continuous Spin on $R^{1,d-1}$
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Alexander Reshetnyak
Publication date
2019
fields
Physics
and research's language is
English
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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 equations of motion with the initial irreducible Poincare group conditions of a Bargmann--Wigner type, it is demonstrated that one should replace the field in these conditions by a class of gauge-equivalent configurations. Triplet-like, doublet-like constrained descriptions, as well as an unconstrained quartet-like non-Lagrangian and Lagrangian formulations, are derived using both Fronsdal-like and new tensor fields. In particular, the BRST--BV equations of motion and Lagrangian using an appropriate set of Lagrangian multipliers in the minimal sector of the respective field and antifield configurations are constructed in a manifest way.
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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 $overline{C}, overline{mathcal{P}}, lambda, pi$, 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 $Psi_H$ as a kernel of the gauge-fixing BRST-BV Fermion functional $Psi$, manifesting the concept of BFV-BV duality. A Fock-space quantum action with non-minimal BRST-extended off-shell constraints is constructed as a shift of the total generalized field-antifield vector by a variational derivative of the gauge-fixing Fermion $Psi$ in a total BRST-BV action $S^{Psi}_{0|s} = int d eta_0 langle chi^{Psi{} 0}_{mathrm{tot}|c} big| Q_{c|mathrm{tot}}big| chi^{Psi{} 0}_{mathrm{tot}|c}rangle$. We use a gauge condition which depends on two gauge parameters, thereby extending the case of $R_xi$-gauges. For triplet and duplet formulations we explored the representations with only traceless field-antifield and source variables. For the generating functionals of Greens functions, BRST symmetry transformations are suggested and Ward identities are obtained.
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 superalgebra. Application of an BRST-BFV receipt permits to reproduce gauge-invariant Lagrangians with reducible gauge symmetries describing the free dynamics of both massless and massive mixed-antisymmetric bosonic fields of any spin with appropriate number of gauge and Stueckelberg fields. The general prescription possesses by the possibility to derive constrained Lagrangians with only BRST-invariant extended algebraic constraints which describes the Poincare group irreducible representations in terms of mixed-antisymmetric tensor fields with 2 group indices.
We study the properties of nonlinear superalgebras $mathcal{A}$ and algebras $mathcal{A}_b$ arising from a one-to-one correspondence between the sets of relations that extract AdS-group irreducible representations $D(E_0,s_1,s_2)$ in AdS$_d$-spaces and the sets of operators that form $mathcal{A}$ and $mathcal{A}_b$, respectively, for fermionic, $s_i=n_i+{1/2}$, and bosonic, $s_i=n_i$, $n_i in mathbb{N}_0$, $i=1,2$, HS fields characterized by a Young tableaux with two rows. We consider a method of constructing the Verma modules $V_mathcal{A}$, $V_{mathcal{A}_b}$ for $mathcal{A}$, $mathcal{A}_b$ and establish a possibility of their Fock-space realizations in terms of formal power series in oscillator operators which serve to realize an additive conversion of the above (super)algebra ($mathcal{A}$) $mathcal{A}_b$, containing a set of 2nd-class constraints, into a converted (super)algebra $mathcal{A}_{b{}c}$ = $mathcal{A}_{b}$ + $mathcal{A}_b$ ($mathcal{A}_c$ = $mathcal{A}$ + $mathcal{A}$), containing a set of 1st-class constraints only. For the algebra $mathcal{A}_{b{}c}$, we construct an exact nilpotent BFV--BRST operator $Q$ having nonvanishing terms of 3rd degree in the powers of ghost coordinates and use $Q$ to construct a gauge-invariant Lagrangian formulation (LF) for HS fields with a given mass $m$ (energy $E_0(m)$) and generalized spin $mathbf{s}$=$(s_1,s_2)$. LFs with off-shell algebraic constraints are examined as well.
We compute partition functions of Chern-Simons type theories for cylindrical spacetimes $I times Sigma$, with $I$ an interval and $dim Sigma = 4l+2$, in the BV-BFV formalism (a refinement of the Batalin-Vilkovisky formalism adapted to manifolds with boundary and cutting-gluing). The case $dim Sigma = 0$ is considered as a toy example. We show that one can identify - for certain choices of residual fields - the physical part (restriction to degree zero fields) of the BV-BFV effective action with the Hamilton-Jacobi action computed in the companion paper [arXiv:2012.13270], without any quantum corrections. This Hamilton-Jacobi action is the action functional of a conformal field theory on $Sigma$. For $dim Sigma = 2$, this implies a version of the CS-WZW correspondence. For $dim Sigma = 6$, using a particular polarization on one end of the cylinder, the Chern-Simons partition function is related to Kodaira-Spencer gravity (a.k.a. BCOV theory); this provides a BV-BFV quantum perspective on the semiclassical result by Gerasimov and Shatashvili.
We construct a Lagrangian description of irreducible half-integer higher-spin representations of the Poincare group with the corresponding Young tableaux having two rows, on a basis of the BRST approach. Starting with a description of fermionic higher-spin fields in a flat space of any dimension in terms of an auxiliary Fock space, we realize a conversion of the initial operator constraint system (constructed with respect to the relations extracting irreducible Poincare-group representations) into a first-class constraint system. For this purpose, we find auxiliary representations of the constraint subsuperalgebra containing the subsystem of second-class constraints in terms of Verma modules. We propose a universal procedure of constructing gauge-invariant Lagrangians with reducible gauge symmetries describing the dynamics of both massless and massive fermionic fields of any spin. No off-shell constraints for the fields and gauge parameters are used from the very beginning. It is shown that the space of BRST cohomologies with a vanishing ghost number is determined only by the constraints corresponding to an irreducible Poincare-group representation. To illustrate the general construction, we obtain a Lagrangian description of fermionic fields with generalized spin (3/2,1/2) and (3/2,3/2) on a flat background containing the complete set of auxiliary fields and gauge symmetries.
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