General Superfield Quantization Method. I. Lagrangian Formalism of $\theta$-Superfield Theory of Fields

Abstract

The rules to construct Lagrangian formulation for $\theta$-superfield theory of fields ($\theta$-STF) are introduced and considered on the whole in the framework of new superfield quantization method for general gauge theories. Algebraic, group-theoretic and analytic description aspects for supervariables over (Grassmann) algebras containing anticommuting generating element $\theta$ and interpreted further in particular as an "odd" time are examined. Superfunction $S_L(\theta )$, its global symmetries are defined on the extended space $T_{odd}{\mathcal{M}}_{cl}\times \{\theta \}$ parameterized by superfields ${\mathcal{A}}^{ı}(\theta ),\frac{d{\mathcal{A}}^{ı}(\theta )}{d\theta },\theta$. Extremality properties of superfunctional $Z[\mathcal{A}]=\int d\theta S_L(\theta )$ and ones of corresponding Euler-Lagrange equations are analyzed. The direct and inverse problems of zero locus reduction for extended (anti)symplectic manifolds over ${\mathcal{M}}_{min}$ = $\{({\mathcal{A}}^{ı}(\theta ),C^{\alpha }(\theta ))\}$, with (odd) even brackets, corresponding to initial $\theta$-superfield models are employed to construct iteratively the new interconnected models both embedded into manifolds above with reduced brackets and enlarging them with continued ones. Component (on $\theta$) formulation for $\theta$-STF variables and operations is produced providing the connection with standard gauge field theory. Realization of $\theta$-STF constructions is demonstrated on models of scalar, spinor, vector superfields which are used to formulate the $\theta$-superfield model with abelian two-parametric gauge supergroup.

50 pages, LaTeX, no figures, title changed, introduction updated, section with zero-locus reduction problems and methods of new superfield models construction added, section with differential equations study with odd operator transferred into appendix, new superfield gauge model, 8 references added, typos corrected