[Paper Review] Odd Nambu bracket on Grassmann algebra

D. Soroka, Vyacheslav Soroka|ArXiv.org|Aug 8, 2006

Advanced Topics in Algebra7 references3 citations

TL;DR

This paper introduces a Grassmann-odd Nambu bracket on the Grassmann algebra, defined via a third-order differential operator Δ₋₃, which generalizes the linear odd bracket to three-ary operations. The key contribution is the construction of a consistent odd Nambu bracket satisfying a generalized Jacobi identity and a modified Leibniz rule, with applications to BRST symmetry and odd Poisson structures in theoretical physics.

ABSTRACT

The Grassmann-odd Nambu bracket on the Grassmann algebra is proposed.

Motivation & Objective

• To extend the concept of odd brackets from binary to ternary operations on Grassmann algebras.
• To define a Grassmann-odd Nambu bracket that generalizes the linear odd bracket (1) to three-ary operations.
• To establish the algebraic consistency of the new bracket through generalized Leibniz rules and Jacobi-type identities.
• To relate the Nambu bracket to the operator Δ₋₃ and its action on products of three Grassmann functions.
• To explore connections with BRST symmetry and odd differential operators in field-theoretic contexts.

Proposed method

• The odd Nambu bracket is defined as {A,B,C}₁ = εαβγ ∂θαA ∂θβB ∂θγC, using the Levi-Civita tensor and Grassmann derivatives.
• The operator Δ₋₃ = 1/3! εαβγ ∂θα∂θβ∂θγ is used to derive the bracket's properties via its action on products of functions.
• The bracket's deviation from the Leibniz rule is captured by Δ(A,B) = 1/2 εαβγ [(∂θα∂θβA)∂θγB + (−1)^p(A)(∂θαA)∂θβ∂θγB], which measures non-associativity.
• The generalized Leibniz rule for Δ₋₃ acting on {A,B,C}₁ is derived, involving multiple terms with sign factors depending on Grassmann parities.
• The Jacobi-type identity for the bracket is derived explicitly, involving ten cyclic terms with Grassmann sign factors.
• The bracket's Grassmann parity is shown to be p({A,B,C}₁) = p(A)+p(B)+p(C)+1 mod 2, confirming its odd nature.

Experimental results

Research questions

• RQ1 How can the linear odd bracket be generalized to a ternary operation on Grassmann algebras?
• RQ2 What is the algebraic structure of a Grassmann-odd Nambu bracket, and how does it satisfy generalized Jacobi identities?
• RQ3 How does the third-order differential operator Δ₋₃ relate to the Nambu bracket and its Leibniz rule?
• RQ4 What is the role of the quantity Δ(A,B) in measuring the deviation of the Nambu bracket from standard Leibniz rules?
• RQ5 How do Grassmann parities affect the symmetry and consistency of the Nambu bracket?

Key findings

• The Grassmann-odd Nambu bracket is defined as {A,B,C}₁ = εαβγ ∂θαA ∂θβB ∂θγC, which is Grassmann-odd with parity p({A,B,C}₁) = p(A)+p(B)+p(C)+1 mod 2.
• The operator Δ₋₃ satisfies a generalized Leibniz rule with respect to the Nambu bracket, involving multiple sign-corrected terms with Δ(A,B), Δ(B,C), and Δ(A,C).
• The Nambu bracket satisfies a complex Jacobi-type identity with ten cyclic terms, each with Grassmann sign factors depending on the parities of the functions involved.
• The bracket is related to the linear odd bracket via {A,B}₁ = θα{θα,A,B}₁, showing a consistent hierarchy between binary and ternary structures.
• The quantity Δ(A,B) is shown to be equal to 1/2 ∂θα{θα,A,B}₁, linking the divergence of the Nambu bracket to the bilinear deviation operator.
• The operator Δ₋₃ is expressed as Δ₋₃A = 1/3! ∂θα∂θβ{θα,θβ,A}₁, providing a closed-form expression for the differential operator in terms of the Nambu bracket.