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

Abstract

The rules to construct Lagrangian formulation for $θ$-superfield theory of fields ($θ$-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 $θ$ and interpreted further in particular as an "odd" time are examined. Superfunction $S_{L}(θ)$, its global symmetries are defined on the extended space $T_{odd}{\cal M}_{cl} \times \{θ\}$ parameterized by superfields ${\cal A}^{\imath}(θ), \frac{d{\cal A}^{\imath}(θ)}{dθ\phantom{xxx}}, θ$. Extremality properties of superfunctional $Z[{\cal A}]=\int dθS_{L}(θ)$ and ones of corresponding Euler-Lagrange equations are analyzed. The direct and inverse problems of zero locus reduction for extended (anti)symplectic manifolds over ${\cal M}_{min}$ = $\{({\cal A}^{\imath}(θ), C^α(θ))\}$, with (odd) even brackets, corresponding to initial $θ$-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 $θ$) formulation for $θ$-STF variables and operations is produced providing the connection with standard gauge field theory. Realization of $θ$-STF constructions is demonstrated on models of scalar, spinor, vector superfields which are used to formulate the $θ$-superfield model with abelian two-parametric gauge supergroup.