[Paper Review] Lorentz invariant supersymmetric mechanism for non(anti)commutative deformations of space-time geometry

A. A. Zheltukhin|ArXiv.org|Jun 15, 2005

Noncommutative and Quantum Gravity Theories27 references4 citations

TL;DR

This paper proposes a Lorentz-invariant, supersymmetric mechanism for non(anti)commutative deformations of spacetime geometry by extending N=1 D=4 superspace with one or more Majorana spinors. The method uses spinor extensions to generate covariant Poisson and Moyal brackets, revealing that the Grassmann vector $ψ_m = -\frac{1}{2}(\bar{\theta}\gamma_m\lambda)$ acts as a covariant measure of spacetime noncommutativity, with a direct correspondence to constant background fields in supergravity.

ABSTRACT

A supersymmetric Lorentz invariant mechanism for superspace deformations is proposed. It is based on an extension of superspace by one $λ_{a}$ or several Majorana spinors associated with the Penrose twistor picture. Some examples of Lorentz invariant supersymmetric Poisson and Mojal brackets are constructed and the correspondence: $θ_{mn}\leftrightarrow iψ_{m}ψ_{n},\quad C_{ab}\leftrightarrow λ_{a}λ_{b},\quad Ψ^{a}_{m}\leftrightarrow ψ_{m}λ^{a}$ mapping the brackets depending on the constant background into the Lorentz covariant supersymmetric brackets is established. The correspondence reveals the role of the composite anticommuting vector $ψ_{m}=-{1\over 2}(\barθγ_{m}λ)$ as a covariant measure of space-time coordinate noncommutativity.

Motivation & Objective

• To resolve the Lorentz symmetry breaking problem in non(anti)commutative spacetime deformations induced by constant background fields.
• To construct supersymmetric and Lorentz-invariant Poisson and Moyal brackets for deformed (super)coordinates without explicit background fields.
• To identify the role of the composite Grassmann vector $\psi_m = -\frac{1}{2}(\bar{\theta}\gamma_m\lambda)$ as a covariant measure of spacetime noncommutativity.
• To establish a correspondence between spinor-based brackets and known field-dependent brackets involving $B_{mn}$, $C_{ab}$, and $\Psi^a_m$.
• To generalize the framework to higher dimensions and extended supersymmetries using Lorentz-invariant supersymmetric derivatives.

Proposed method

• Extends $N=1$ D=4 superspace $(x_m, \theta_a)$ by adding one or more commuting Majorana spinor coordinates $\lambda_a$.
• Constructs Lorentz-invariant and supersymmetric Poisson brackets using the extended superspace, with non(anti)commutativity encoded in $\psi_m = -\frac{1}{2}(\bar{\theta}\gamma_m\lambda)$.
• Derives Moyal brackets from the Poisson brackets via deformation quantization, preserving supersymmetry and Lorentz covariance.
• Introduces a mapping: $\theta_{mn} \leftrightarrow i\psi_m\psi_n$, $C_{ab} \leftrightarrow \lambda_a\lambda_b$, $\Psi^a_m \leftrightarrow \psi_m\lambda^a$, linking spinor-based brackets to constant background field brackets.
• Uses the Penrose twistor formalism to ensure Lorentz covariance and supersymmetry in the bracket structure.
• Derives Jacobi identities for the brackets and identifies constraints on parameters such as the real constant $c$ to maintain consistency.

Experimental results

Research questions

• RQ1 Can non(anti)commutative spacetime deformations be made Lorentz-invariant and supersymmetric without relying on constant background fields?
• RQ2 What is the role of the composite Grassmann vector $\psi_m = -\frac{1}{2}(\bar{\theta}\gamma_m\lambda)$ in measuring spacetime noncommutativity?
• RQ3 How can the non(anti)commutative brackets for $x_m$, $\theta_a$, and $\theta_b$ be constructed to preserve Lorentz and supersymmetry invariance?
• RQ4 Is there a systematic correspondence between spinor-based brackets and those derived from constant background fields like $B_{mn}$, $C_{ab}$, and $\Psi^a_m$?
• RQ5 Can this framework be generalized to higher-dimensional spacetimes and $N>1$ supersymmetries with multiple spinor extensions?

Key findings

• The proposed Poisson and Moyal brackets are explicitly Lorentz-invariant and supersymmetric, even though they generate non(anti)commutative relations for spacetime and supercoordinates.
• The noncommutativity of $x_m$ and $x_n$ is governed by $\{x_m, x_n\} = -i(\psi_m\psi_n)$, where $\psi_m = -\frac{1}{2}(\bar{\theta}\gamma_m\lambda)$ is a covariant Grassmann vector.
• The anticommutator $\{\theta_a, \theta_b\}$ is proportional to $\lambda_a\lambda_b$, showing that the nonanticommutativity of spinor coordinates arises from the additional Majorana spinor $\lambda_a$.
• A direct correspondence is established: $\theta_{mn} \leftrightarrow i\psi_m\psi_n$, $C_{ab} \leftrightarrow \lambda_a\lambda_b$, $\Psi^a_m \leftrightarrow \psi_m\lambda^a$, mapping field-dependent brackets to the new covariant brackets.
• The framework generalizes to multiple spinors and higher dimensions, with brackets preserving Lorentz invariance and supersymmetry through the use of generalized supersymmetric derivatives.
• The Jacobi identities constrain the real constant $c$ in the brackets $\{x_m, \lambda_a\} = c(\varphi_m)\lambda_a$, indicating consistency conditions for the deformed algebra.