[Paper Review] General Superfield Quantization Method. II. General Superfield Theory of Fields: Hamiltonian Formalism

Alexander A. Reshetnyak|ArXiv.org|Mar 31, 2003

Quantum chaos and dynamical systems1 references3 citations

TL;DR

This paper establishes a Hamiltonian formalism for General Superfield Theory (GSTF) by extending the superfield quantization method initiated in Ref.[1], using Legendre transforms to derive a super-Hamiltonian action $ S_H $ on an odd phase space parametrized by superfields $ \mathcal{A}^\imath(\theta) $, superantifields $ \mathcal{A}^\ast_\imath(\theta) $, and an odd Grassmann variable $ \theta $. The key contribution is the construction of equivalent Hamiltonian systems and the identification of nontrivial master equations via a second-order odd operator $ \Delta^{cl}(\theta) $, enabling a BRST-like symmetry realization in superfield form.

ABSTRACT

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 $S_{L}\bigl({\cal A}(θ),{\stackrel{\circ}{\cal A}}(θ),θ\bigr)$ with respect to ${\stackrel{\circ}{{\cal A}^{\imath}}}(θ)$ is considered. As result on the space $T^{\ast}_{odd}{\cal M}_{cl} imes \{θ\}$ parametrized by classical superfields ${\cal A}^{\imath}(θ)$, superantifields ${\cal A}^{\ast}_{\imath}(θ)$ and odd Grassmann variable $θ$ the superfunction $S_{H}({\cal A}(θ),{\cal A}^{\ast}(θ),θ)$ is defined. Being equivalent to different types of Euler-Lagrange equations the distinct Hamiltonian systems are investigated. Translations along $θ$ for superfunctions on $T^{\ast}_{odd}{\cal M}_{cl} imes \{θ\}$ being associated with these systems are studied. Various types of antibrackets and differential operators acting on $C^{k}\bigl(T^{\ast}_{odd}{\cal M}_{cl} imes \{θ\} \bigr)$ are considered. Component (on $θ$)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.

Motivation & Objective

• To develop a Hamiltonian formulation for General Superfield Theory (GSTF) as a foundational step in the general superfield quantization method (GSQM).
• To extend the Lagrangian superfield formalism from Ref.[1] to a Hamiltonian framework using Legendre transformations on superfields.
• To establish a correspondence between classical Hamiltonian mechanics and GSTF by introducing superantifields, odd Poisson brackets, and odd Hamiltonian equations.
• To investigate the role of the second-order odd operator $ \Delta^{cl}(\theta) $ and its relation to master equations and solvability of Hamiltonian systems.
• To demonstrate the formalism through six concrete models, including free and self-interacting scalar, spinor, and vector superfields in various dimensions.

Proposed method

• Perform a Legendre transformation of the Lagrangian superaction $ S_L(\theta) $ with respect to $ \stackrel{\circ}{{\cal A}}^\imath(\theta) $ to define the Hamiltonian superfunction $ S_H({\cal A}(\theta), {\cal A}^\ast(\theta), \theta) $ on the odd cotangent bundle $ T^\ast_{odd}{\cal M}_{cl} \times \{\theta\} $.
• Introduce an odd Poisson bracket $ (\cdot, \cdot)_\theta $ on $ C^k(T^\ast_{odd}{\cal M}_{cl} \times \{\theta\}) $ to formulate odd Hamiltonian equations of motion.
• Define the odd Hamiltonian superfunctional $ Z_H[\Gamma] = \int d\theta \left( \stackrel{\circ}{{\cal A}}^\imath(\theta) {\cal A}^\ast_\imath(\theta) - S_H(\Gamma(\theta), \theta) \right) $, generalizing the standard action principle.
• Construct a second-order odd differential operator $ \Delta^{cl}(\theta) $ acting on superfunctions, which generates nontrivial master equations $ \Delta^{cl}(\theta) S_H = 0 $, ensuring consistency of the Hamiltonian system.
• Analyze translation symmetries along $ \theta $-integral curves of Hamiltonian systems, linking them to gauge invariance and solvability conditions.
• Provide a component-wise formulation of GSTF quantities by expanding superfields in powers of $ \theta $, enabling comparison with standard field-theoretic models.

Experimental results

Research questions

• RQ1 How can the Legendre transformation be generalized to superfields in the context of odd phase spaces and Grassmann-valued time $ \theta $?
• RQ2 What is the structure of the odd Hamiltonian system defined on $ T^\ast_{odd}{\cal M}_{cl} \times \{\theta\} $, and how does it relate to the original Lagrangian formulation?
• RQ3 What role does the second-order odd operator $ \Delta^{cl}(\theta) $ play in generating master equations and ensuring consistency of the Hamiltonian dynamics?
• RQ4 How do the odd Poisson brackets and odd Hamiltonian equations of motion in $ \theta $-superspace generalize classical mechanics?
• RQ5 To what extent can the Hamiltonian formalism for GSTF be applied to concrete models such as self-interacting scalar, spinor, and vector superfields?

Key findings

• The Legendre transformation of the Lagrangian superaction $ S_L(\theta) $ yields a well-defined Hamiltonian superfunction $ S_H({\cal A}(\theta), {\cal A}^\ast(\theta), \theta) $ on the odd phase space $ T^\ast_{odd}{\cal M}_{cl} \times \{\theta\} $, enabling a full Hamiltonian formulation of GSTF.
• The odd Hamiltonian equations of motion $ \frac{d_r \Gamma^p(\theta)}{d\theta} = (\Gamma^p(\theta), S_H(\theta))_\theta $ generalize the classical Hamiltonian flow to superfield configurations.
• The second-order odd operator $ \Delta^{cl}(\theta) $ generates nontrivial master equations $ \Delta^{cl}(\theta) S_H = 0 $, which are essential for the consistency and gauge invariance of the theory.
• For free models, the component equations of motion for $ P_0(\theta) $ and $ P_1(\theta) $ components are identical and linear; for self-interacting models, they differ in form, reflecting nonlinear dynamics.
• The formalism establishes a precise analogy between classical mechanics and GSTF: superantifields $ \mathcal{A}^\ast_\imath(\theta) $ correspond to momenta, odd Poisson brackets to standard Poisson brackets, and the odd Hamiltonian superfunctional to the standard action functional.
• The method is successfully applied to six models, including massive scalar, spinor, and vector superfields in $ D=4 $ and $ D=2k $, demonstrating the feasibility of constructing interacting $ \theta $-superfield gauge theories via the gauge principle.