General Superfield Quantization Method. III. Construction of Quantization Scheme

Extension procedure for supermanifold ${\cal M}_{cl}$ of superfields ${\cal A}^{\imath}(\theta)$, ghost number construction are considered. Classical and $\hbar$-deformed generating (master) equations, existence theorems for their solutions are formulated in $T^{\ast}_{odd}{\cal M}_{min}$, $T^{\ast}_{odd}{\cal M}_{ext}$. Analogous scheme is realized for BV similar generating equations. Master equations versions for GSQM and BV similar scheme are deformed in powers of superfields ${\stackrel{\circ}{\Gamma}}{}^p(\theta)$ = $\bigl({\stackrel{\circ}{\Phi}}{}^B(\theta)$, ${\stackrel{\circ}{\Phi}}{}^{\ast}_B(\theta)\bigr)$ into supermanifold $T_{odd}(T^{\ast}_{odd}{\cal M}_{ext})$. Arbitrariness in a choice of solutions for these equations is described. Investigation of formal Hamiltonian systems for II class theories [2] defined via corresponding master equations solutions is conducted. Gauge fixing for those theories is described by two ways. Functional integral of superfunctions on $T_{odd}(T^{\ast}_{odd}{\cal M}_{ext})$ is defined. Properties for generating functionals of Green's superfunctions are studied. $\theta$-component quantization formulation, connection with BV method and superfield quantization [3] are established. Quantization scheme realization is demonstrated on 6 models.Comment: 59 pages, Latex, no figure

On Lagrangian formulations for mixed-symmetry HS fields on AdS spaces within BFV-BRST approach

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.Comment: SLightly enlarged contribution to the Proceedings of the XIII International Conference "Selected Problems of Modern Theoretical Physics", Dubna, Russia, June 23-27, 2008; 4 pages, no figures; v2: two references adde

General Superfield Quantization Method. II. General Superfield Theory of Fields: Hamiltonian Formalism

In the framework of started in Ref.[1] construction procedure of the general superfield quantization method for gauge theories in Lagrangian formalism the rules for Hamiltonian formulation of general superfield theory of fields (GSTF) are introduced and are on the whole considered. Mathematical means developed in [1] for Lagrangian formulation of GSTF are extended to use in Hamiltonian one. Hamiltonization for Lagrangian formulation of GSTF via Legendre transform of superfunction $S_{L}\bigl({\cal A}(\theta),{\stackrel{\circ}{\cal A}}(\theta),\theta\bigr)$ with respect to ${\stackrel{\circ}{{\cal A}^{\imath}}}(\theta)$ is considered. As result on the space $T^{\ast}_{odd}{\cal M}_{cl}\times \{\theta\}$ parametrized by classical superfields ${\cal A}^{\imath}(\theta)$, superantifields ${\cal A}^{\ast}_{\imath}(\theta)$ and odd Grassmann variable $\theta$ the superfunction $S_{H}({\cal A}(\theta),{\cal A}^{\ast}(\theta),\theta)$ is defined. Being equivalent to different types of Euler-Lagrange equations the distinct Hamiltonian systems are investigated. Translations along $\theta$ for superfunctions on $T^{\ast}_{odd}{\cal M}_{cl}\times \{\theta\}$ being associated with these systems are studied. Various types of antibrackets and differential operators acting on $C^{k}\bigl(T^{\ast}_{odd}{\cal M}_{cl} \times \{\theta\} \bigr)$ are considered. Component (on $\theta$)formulation for GSTF quantities and operations is produced. Analogy between ordinary Hamiltonian classical mechanics and GSTF in Hamiltonian formulation is proposed. Realization of the GSTF general scheme is demonstrated on 6 models.Comment: 55 pages, LaTeX, abstract and introduction are extended, 2 references are adde

Nonlinear Operator Superalgebras and BFV-BRST Operators for Lagrangian Description of Mixed-symmetry HS Fields in AdS Spaces

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.Comment: Enlarged contribution to the Proceedings of the "XXVII International Colloquium on Group Theoretical Methods in Physics", Yerevan, Armenia, August 13-19, 2008; 15 pages, no figures; v2: minor changes; v3: misprints in Eqs. (6), (15) removed; v4: typos corrected in the footnote 1, in Eq. (34), formula (35) adde

Basic features of General Superfield Quantization Method for gauge theories in Lagrangian formalism

The rules for superfield Lagrangian quantization method for general gauge theories on a basis of their generalization to special superfield models within a so-called $\theta$-superfield theory of fields ($\theta$-STF) are formulated. The $\theta$-superfield generating functionals of Green's functions together with effective action are constructed. Their properties including new interpretation and superfield realization of BRST transformations, Ward identities are studied.Comment: v1: 6p., no figures, talk given at International Workshop "Supersymmetries and Quantum Symmetries" - SQS'03 (Dedicated to the 75th anniversary of the birth of V.I.Ogievetsky); JINR, Dubna; July 24-29, 2003; v2: final version, typos corrected, formulae (1), (5), (8), (14)-(16), (21), (26), (27), (30), (31), formulations and statements improve

Programming Realization of Symbolic Computations for Non-linear Commutator Superalgebras over the Heisenberg--Weyl Superalgebra: Data Structures and Processing Methods

We suggest a programming realization of an algorithm for verifying a given set of algebraic relations in the form of a supercommutator multiplication table for the Verma module, which is constructed according to a generalized Cartan procedure for a quadratic superalgebra and whose elements are realized as a formal power series with respect to non-commuting elements. To this end, we propose an algebraic procedure of Verma module construction and its realization in terms of non-commuting creation and annihilation operators of a given Heisenberg--Weyl superalgebra. In doing so, we set up a problem which naturally arises within a Lagrangian description of higher-spin fields in anti-de-Sitter (AdS) spaces: to verify the fact that the resulting Verma module elements obey the given commutator multiplication for the original non-linear superalgebra. The problem setting is based on a restricted principle of mathematical induction, in powers of inverse squared radius of the AdS-space. For a construction of an algorithm resolving this problem, we use a two-level data model within the object-oriented approach, which is realized on a basis of the programming language C#. The program allows one to consider objects (of a less general nature than non-linear commutator superalgebras) that fall under the class of so-called $GR$-algebras, for whose treatment one widely uses the module \emph{Plural} of the system \emph{Singular} of symbolic computations for polynomials.Comment: 35 pages, 2 figures in eps-format, corrected typos, added reference

Peculiarities of temperature dependence for generalized Hall-Petch law and two-phase model for deformable polycrystalline materials

In the framework of the suggested in [arxiv:1803.08247 [cond-mat.mtrl-sci]] statistical theory of the equilibrium flow stress, including yield strength, ${\sigma}_y$, of polycrystalline materials under quasi-static (in case of tensile strain) plastic deformation in dependence on average size, d, of the crystallites (grains) in the range, $10^{-8}$ m - $10^{-2}$ m. it is found the coincidences of the theoretical and experimental data of ${\sigma}_y$ for the materials with BCC (${\alpha}$- Fe), FCC (Cu, Al, Ni) and HCP (${\alpha}$-Ti, Zr) crystal lattice at T=300K. The temperature dependence of the strength characteristics is studied. It is shown on the example of Al, that the yield strength grows with decreasing of the temperature for all grains with d greater than $3*d_0$ (with $d_0$ being extremal size of the grain for maximal ${\sigma}_y$) and then ${\sigma}_y$ decreases in the nano-crystalline region, thus determining a temperature-dimension effect. Stress-strain curves, ${\sigma}={\sigma}({\epsilon})$, are constructed for the pure crystalline phase of ${\alpha}$-Fe with Backofen-Consid\'ere fracture criterion validity. The single-phase model of polycrystalline material is augmented by means of inclusion of a softening grain boundary phase.Comment: 11 pages, 3 figures, 4 tables, pdf-version only, bad term "vacancy" is changed on "nanopore", definition of temperature-dimension effect introduced, footnotes 2,4,6 added, description of twinning in terms of partial dislocations, equilibrium flow stress and comments added, typo in Eq. 2 corrected, Eq.3 extended, 3 references adde

Gauge-invariant Lagrangians for mixed-antisymmetric higher spin fields

Lagrangian descriptions of irreducible and reducible integer higher-spin representations of the Poincare group subject to a Young tableaux $Y[\hat{s}_1,\hat{s}_2]$ with two columns are constructed within a metric-like formulation in a $d$-dimensional flat space-time on the basis of a BRST approach extending the results of [arXiv:1412.0200[hep-th]]. A Lorentz-invariant resolution of the BRST complex within both the constrained and unconstrained BRST formulations produces a gauge-invariant Lagrangian entirely in terms of the initial tensor field $\Phi_{[\mu]_{\hat{s}_1}, [\mu]_{\hat{s}_2}}$ subject to $Y[\hat{s}_1,\hat{s}_2]$ with an additional tower of gauge parameters realizing the $(\hat{s}_1-1)$-th stage of reducibility with a specific dependence on the value $(\hat{s}_1-\hat{s}_2)=0,1,...,\hat{s}_1$. Minimal BRST--BV action is suggested, being proper solution to the master equation in the minimal sector and providing objects appropriate to construct interacting Lagrangian formulations with mixed-antisymmetric fields in a general framework.Comment: 8 pages, extended Contribution to the Proceedings of the International Workshop "Supersymmetry and Quantum Symmetries" (SQS'2015, August 3 - August 8, 2015, Dubna, Russia); corrections in (4),(19) were made; ref.25, footnote 1 added; acknowledgements, footnote 2 with comments on gauge transformations in [28, 29] updated; resulting actions (18), (19) correcte

General Lagrangian Formulation for Higher Spin Fields with Arbitrary Index Symmetry. 2. Fermionic fields

We continue the construction of a Lagrangian description of irreducible half-integer higher-spin representations of the Poincare group with an arbitrary Young tableaux having $k$ rows, on a basis of the BRST--BFV approach suggested for bosonic fields in our first article (Nucl. Phys. B862 (2012) 270, [arXiv:1110.5044[hep-th]). Starting from a description of fermionic mixed-symmetry higher-spin fields in a flat space of any dimension in terms of an auxiliary Fock space associated with a special Poincare module, we realize a conversion of the initial operator constraint system (constructed with respect to the relations extracting irreducible Poincare-group representations) into a system of first-class constraints. To do this, we find, in first time, by means of generalized Verma module the auxiliary representations of the constraint subsuperalgebra, to be isomorphic due to Howe duality to $osp(k|2k)$ superalgebra, and containing the subsystem of second-class constraints in terms of new oscillator variables. We suggest a universal procedure of finding unconstrained gauge-invariant Lagrangians with reducible gauge symmetries, describing the dynamics of both massless and massive fermionic fields of any spin. It is shown that the space of BRST cohomologies with a vanishing ghost number is determined only by constraints corresponding to an irreducible Poincare-group representation. As examples of the general approach, we propose a method of Lagrangian construction for fermionic fields subject to an arbitrary Young tableaux having 3 rows, and obtain a gauge-invariant Lagrangian for a new model of a massless rank-3 spin-tensor field of spin (5/2,3/2) with first-stage reducible gauge symmetries and a non-gauge Lagrangian for a massive rank-3 spin-tensor field of spin (5/2,3/2).Comment: 69 pages, no figures, published version, misprints with HS subsuperalgebra osp(1|2k) correcte

On Lagrangian Formulation for Half-integer HS Fields within Hamiltonian BRST Approach

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.Comment: 5 pages, Contribution to Proceedings of the International Workshop "Supersymmetries and Quantum Symmetries", Dubna, July, 30 - August, 4, 2007, added referenc