On the Quantization of Nambu Brackets

TL;DR

This work addresses the long-standing problem of quantizing the three-dimensional Nambu bracket by constructing explicit quantum realizations that preserve key classical properties. It provides a skew-symmetric, FI-satisfying quantum Nambu bracket using square matrices and, separately, three-index cubic matrices, extending the strategy to higher dimensions. The results establish concrete algebraic tools and reveal a gauge-structure induced by triple brackets, with potential implications for M-theory and covariant Matrix theory. By outlining a general n-dimensional construction, the paper lays groundwork for broader applications in non-Abelian tensor gauge theories and spacetime uncertainty relations in high-energy theory.

Abstract

We present several non-trivial examples of the three-dimensional quantum Nambu bracket which involve square matrices or three-index objects. Our examples satisfy two fundamental properties of the classical Nambu bracket: they are skew-symmetric and they obey the Fundamental Identity. We contrast our approach to the existing literature on the quantum deformations of Nambu mechanics. We also discuss possible applications of our results in M-theory.