Map of Witten's * to Moyal's *

TL;DR

The paper establishes a precise map between Witten's open-string star product and the Moyal star product by recasting string fields into a half-Fourier, phase-space framework. Through the split-string formalism and Fourierization of odd modes, it shows that Witten's product is equivalent to a Moyal product acting on the phase-space of the even string modes $x_{2n}$ with conjugate momenta $p_{2n}$, while either fixing the center-of-mass mode $x_0$ or the midpoint $ar{x}$ depending on the modding convention. This yields a Lorentz-invariant noncommutative geometry that mirrors noncommutative field theory and enables transferring techniques between the two domains, including a projector/Wigner-function perspective for string states (e.g., the sliver). The framework promises to broaden computational methods in string field theory and to clarify midpoint ambiguities, with potential applications to D-brane solutions and the ghost/BRST structure.

Abstract

It is shown that Witten's star product in string field theory, defined as the overlap of half strings, is equivalent to the Moyal star product involving the relativistic phase space of even string modes. The string field A(x[σ]) can be rewritten as a phase space field of the even modes $x_{2n},x_{0}, p_{2n}$ where $x_{2n}$ are the positions of the even string modes, and $p_{2n}$ are related to the Fourier space of the odd modes $x_{2n+1}$ up to a linear transformation. The $p_{2n}$ play the role of conjugate momenta for the even modes $x_{2n}$ under the string star product. The split string formalism is used in the intermediate steps to establish the map from Witten's star-product to Moyal's star-product. An ambiguity related to the midpoint in the split string formalism is clarified by considering odd or even modding for the split string modes, and its effect in the Moyal star product formalism is discussed. The noncommutative geometry defined in this way is technically similar to the one that occurs in noncommutative field theory, but it includes the timelike components of the string modes, and is Lorentz invariant. This map could be useful to extend the computational methods and concepts from noncommutative field theory to string field theory and vice-versa.