Moyal's Star Product in Half Fourier Space

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 ψ A (x µ [σ]) can be rewritten as a phase space field of the even modes A (x µ 2n , x 0 , p µ 2n) , where x µ 2n are the positions of the even string modes, and p µ 2n are related to the Fourier space of the odd modes x µ 2n+1 up to a linear transformation. The p µ 2n play the role of conjugate momenta for the even modes x µ 2n under the string star product. The split string formalism is used in the intermediate steps to establish the map from Witten's ⋆-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. A long time ago deformation quantization was developed as a method for studying quantum mechanics. The methods for establishing the correspondence between deformation quantiza-tion and the traditional formulation of quantum mechanics was developed by Weyl, Wigner, Moyal and many others [1]-[3], leading the way to the modern ideas of noncommutative geometry [4]. Using Weyl's correspondence [1] one establishes a map between functions of operatorsˆAˆx M , ˆ p M acting in a Hilbert space and their image in phase space A x M , p M .