Map of Witten's star to Moyal's star

It is shown that Witten's star product in string field theory, defined as t he overlap of half strings, is equivalent to the Moyal star product involvi ng the relativistic phase space of even string modes. The string field psi (A)(x(mu)[sigma]) can be rewritten as a phase space field of the even modes A(x(mu)(2n), x(0), p(2n)(mu)), where x(2n)(mu) are the positions of the ev en string modes, and p(2n)(mu) are related to the Fourier space of the mode s 2(n+1)(mu) up to a linear transformation. The p(2n)(mu) play the role of conjugate momenta for the even modes x(2n)(mu) under the string star produc t. The split string formalism is used in the intermediate steps to establis h the map from Witten's *-product to Moyal's *-star product. An ambiguity r elated to the midpoint in the split string formalism is clarified by consid ering odd or even modding for the split string modes, and its effect in the Moyal star product formalism is discussed. The noncommutative geometry def ined in this way is technically similar to the one that occurs in noncommut ative field theory, but it includes the timelike components of the string m odes, and is Lorentz invariant. This map could be useful to extend the comp utational methods and concepts from noncommutative field theory to string f ield theory and vice versa. (C) 2001 Published by Elsevier Science B.V.