Map of Witten's * to Moyal's *

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[\sigma]) 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.

To preserve the quantum rules, phase space functions must be multiplied with each other by using the associative, noncommutative, Moyal star product (or its deformed generalizations). Note that a spacetime metric does not enter in this expression because all positions have upper spacetime indices and all momenta have lower spacetime indices. If we use the notation X m ≡ x M , p M with a single index m that takes D ×2 values, the Moyal star product takes a form which is more familiar in the recent physics literature on noncommutative geometry where θ mn = δ M N ε ij , with i = (1, 2) referring to (x, p) respectively, and ε ij the antisymmetric Sp(2, R) invariant metric. Henceforth we will set = 1 for simplicity. The star commutator between any two phase space fields is defined by [A, B] ⋆ ≡ A ⋆ B − B ⋆ A. The phase space coordinates satisfy [X m , X n ] ⋆ = iθ mn , which is equivalent to the Heisenberg algebra for Let us now consider the Fourier transform in the momentum variables p M . We will call this "half-Fourier space" since only one of the noncommutative variables is being Fourier transformed. So, the transform of A x M , p M is a bi-local function ψ A x M , y M in position space, but we will write it in the formÃ l M , r M ≡ ψ A x M , y M where Thus, we define The Moyal star product of two functions (A ⋆ B) (x, p) = C (x, p) may now be evaluated in terms of these integral representations. The result can be written in terms of the Fourier One finds thatC l M , r M is related toÃ l M , r M andB l M , r M by a matrix-like multiplication with continuous indices This is verified explicitly by the following steps where, in the second line the derivatives with respect to p are evaluated, in the third line the translation operators e ∂x are applied on the x coordinates on the left and right respectively, in the fourth line one defines y + = y + y ′ and y − = x − (y − y ′ ) /2, and finally the y − integration is performed. Hence, in our notation, the Moyal star product in Fourier space is equivalent to infinite matrix multiplication with the rules of Eq.(7): the right variable ofÃ is identified with the left variable ofB and then integrated. Eq. (7) is the key observation for establishing a direct relation between Witten's star-product and Moyal's star-product as we will see below.
The fact that the Moyal star-product is related to some version of matrix multiplication is no surprise, as by now a few versions of matrix representations have been used in the physics literature. The one used here is straightforward: after using the Weyl correspondence to derive an operatorÂ from the function A (x, p) , the matrix representation A (l, r) is nothing but the matrix elements of the operatorÂ in position space: A (l, r) = l|Â|r .

Witten's star product in split string space
We will show that the continuous matrix representation of the Moyal star product of the previous section is in detail related to Witten's star product in string field theory. The rough idea is to replace the points l M , r M by left and right sides of a string x µ [σ] = l µ [σ] ⊕ r µ [σ] (with l, r defined relative to the midpoint at σ = π/2). If we consider the fields of two strings , then Witten's string star product is formally given by the functional integral [5] C (l µ where z µ [σ] = r µ 1 [σ] = l µ 2 [σ] corresponds to the overlap of half of the first string with half of the second string, andC (l µ 1 [σ] , r µ 2 (σ)) is the field describing the joined half strings as a new full string x µ 3 [σ] = l µ 1 [σ] ⊕ r µ 2 (σ). Considering the close analogy to Eq.(7), morally we anticipate to be able to rewrite Witten's star product as a Moyal star-product of the form (1) in a larger space. In the remainder of this paper the details of this map will be clarified, and will be shown that Witten's ⋆ is indeed Moyal's ⋆ in half of a relativistic phase space of the full string, involving only the even modes (x 2n , p 2n ) or only the odd modes (x 2n−1 , p 2n−1 ). Our result may be summarized as follows: Define the Fourier transform of the string field in the odd modes only as follows where T 2k,2l−1 is a matrix to be defined below. Then Witten's star product (9) for two string fields (ψ A ⋆ witten ψ B )(x 0 , x 2n , x 2n−1 ) is equivalent to the Moyal star product for their Fourier transformed fields (A ⋆ B)(x 2n , x 0 , p 2n ) with the usual definition of the Moyal star product involving the phase space of only the even modes Furthermore, the definition of trace is the phase space integration In the definitions of both the star (A ⋆ B)(x 2n , x 0 , p 2n ) and trace T rA either the center of mass mode x 0 or the midpoint modex ≡ x( π 2 ) = x 0 + √ 2 ∞ n=1 x 2n (−1) n is held fixed while taking derivatives or doing integration with respect to the x 2n modes. This is precisely related to the midpoint ambiguity in the split string formalism which will be clarified in the next section.
Witten's star product (9) is more carefully defined in the split string formalism which was developed sometime ago [6] [7] and was used in recent studies of string field theory [8] [9] [10]. As mentioned in the previous paragraph there is a dilemma involving the midpoint which so far has remained obscure in the literature. We will address this issue in the next section by considering the options that are available in the formulation of the split string formalism, namely odd versus even modding of the split string modes [6] [7]. The choice affects the definition of the star product within the split string formalism. In turn this choice is related to whether the center of mass mode x 0 or the midpoint modex is held fixed as described in the previous paragraph. In this section we begin with the odd modding that has been used in the recent literature.
The open string position modes are identified as usual by the expansion , where x 0 is the center of mass position of the string. We will omit the spacetime index µ on all vectors whenever there is no confusion. The position of the midpoint is given byx ≡ x( π 2 ) = x 0 + √ 2 ∞ n=1 x n cos nπ 2 . Therefore, instead of x 0 we may usex as the independent degree of freedom and write the following mode expansions for the full string x(σ), as well as for the left side l (σ) ≡ x(σ) | 0 ≤ σ ≤ π 2 and the right side These mode expansions are obtained by imposing Neumann boundary conditions at the ends of the string ∂ σ x| 0,π = ∂ σ l| 0 = ∂ σ r| 0 = 0, and Dirichlet boundary conditions at the midpoint x( π 2 ) = l( π 2 ) = r( π 2 ) =x. Using the completeness and orthogonality of the trigonometric functions in these expansions one can easily extract the relationship between the left/right modes and the full string modes The result is where The inverse relations are We note that T 0,2m−1 is given by Eq. (21), but it also satisfies T 0, In the split string notation the string field is . Note that the midpointx is treated as the independent degree of freedom rather than the center of mass mode x 0 . Witten's star product takes the form (no integration overx) By analogy to section-1 we see that we should compare l µ 2m−1 , r µ 2m−1 to l M , r M and therefore via Eqs. (4,22,24) we should establish the following correspondence This suggests that we define a Fourier transform in twice the odd modes 2x µ 2m−1 ∼ y M to obtain the string field in phase space. The Fourier parameters would play the role of conjugate momenta to the following combination of even modes Rx e ≡ ∞ n=1 R 2m−1,2n x µ 2n ∼ x M . We will use the symbol Rx e as a short hand notation to indicate that the even modes (denoted by the subscript e) are transformed by the matrix R. Therefore it is convenient to choose the Fourier parameters in the combination p e T ≡ { ∞ n=1 p µ 2n T 2n,2m−1 ; m = 1, 2, · · · } ∼ p M . We will also use x odd as a short hand notation for x odd = x µ 2n−1 . Then we define the string field in phase space by complete analogy to Eq.
The right hand side is the same expression given in (10) sinceÃ (Rx e + x odd ,x, Rx e − x odd ) = ψ A (x 0 , x 2n , x 2n−1 ). This construction guarantees that Witten's star product in the split string notation of Eq.(29) will be precisely reproduced by a Moyal star product in which Rx e and p e T are the conjugate phase space variables. This is seen by retracing the steps of the computations that lead to Eq. (8). Therefore, the Moyal star should satisfy [Rx e , p e T ] ⋆ ∼ i, or in more detail However, using the fact that R and T are each other's inverses we see that this is equivalent to the simple Heisenberg commutation relations under the Moyal star product given in Eq.(11) Therefore, the Witten star-product reduces just to the usual Moyal product in the phase space of only the even modes. Since the Moyal ⋆ and ψ A (x 0 , x 2n , x 2n−1 ) are both independent of R and T we see that R and T can be removed from the phase space string fields in comparing the left hand sides of Eqs.(10,31). Therefore we may write the string field in (31) simply as A (x e ,x, p e ) and define string field theory using the Moyal star product given in Eq.(11) The net effect of the intermediate steps involving the split string formalism with odd modes l 2n−1 , r 2n−1 is to keep the midpointx fixed while evaluating string overlaps in Eq.(29). Therefore, in the Moyal basis x 2n , p 2n , the star product of Eq.(11) must be evaluated by first writing all string fields ψ A,B (x 0 , x 2n , x 2n−1 ) in terms ofx instead of x 0 , and then applying the derivatives with respect to x 2n . Other than this relic of split strings, the relation between the original string field ψ A (x 0 , x 2n , x 2n−1 ) and its Fourier transform A(x 2n , x 0 , p 2n ) given in Eq.(10), or the computation of star products, do not involve the split string formalism.

Split strings with even modes
It seems puzzling thatx was distinguished since x 0 appears to be more natural in the Moyal basis. Furthermore, x 0 is gauge invariant under world sheet reparametrizations, unlikex. In fact, there is another split string formalism [7] that favors fixing x 0 rather thanx as explained below. First we note the following properties of trigonometric functions when 0 ≤ σ ≤ π for integers m, n ≥ 1 Both sides of these equations satisfy Neumann boundary conditions at σ = 0 and Dirichlet boundary conditions at σ = π 2 , and both are equivalent complete sets of trigonometric functions for the range 0 ≤ σ ≤ π 2 . In the previous section we made the choice of expanding l(σ), r(σ) in terms of the odd modes. Now we see that we could also expand them in terms of the even modes as follows and similarly for r(σ). Comparing to the expressions in the previous section, and using (36) we can find the relation between the odd modes (l 2n−1 , r 2n−1 ) and the even modes (l 2n , r 2n ) Furthermore, by using the relation between the odd string modes (l 2n−1 ,x, r 2n−1 ) and the full string modes (x 0 , x 2n , x 2n−1 ) in Eqs.(22-24) or by direct comparison to x(σ), we derive the relation between the even split string modes and the full string modes.
The inverse relation is Furthermore, the relations between the zero modes are Note that the matching condition at the midpoint l(π/2) = r(π/2) = x(π/2) =x is satisfied by the even modes, because l 0 , r 0 automatically obey the relation thanks to the property T 0,2n−1 + 2 ∞ n=1 (−1) m T 2m,2n−1 = 0. This setup allows us to define split string fields that distinguish the center of mass mode x 0 rather than the midpointx as followŝ where (l 2n , r 2n ) are given above in terms of the full string modes. For this case we define the Witten star product in the form (no integration over x 0 ) Evidently, this product is different than the one in Eq.(29) since the mode that is held fixed during integration is different (i.e. x 0 rather thanx). We now repeat the arguments that made analogies to section-1 to rewrite this overlap of half strings in terms of a Moyal product. We find again that the even modes in phase space x 2n , p 2n are the relevant ones. Furthermore, the expressions for the string field in phase space A(x 2n , x 0 , p 2n ) in terms of the original ψ A (x 0 , x 2n , x 2n−1 ) is identical to the one given in Eq. (10), and the expression for the star product is also the same as Eq. (11). The only difference is that now x 0 must be held fixed while evaluating the derivatives with respect to x 2n . We see that the relic of the split string formalism with even modes is to hold x 0 fixed (rather thanx) while performing computations in the Moyal basis.
More work is required in order to decide which of these procedures is the correct definition of string field theory. In particular, the symmetries of the full action, including ghosts, will be relevant in distinguishing them from each other. Of course, the computation of string amplitudes will also play a role. We hope to report on further work along these lines in a future publication.

Remarks
It seems puzzling that only the even modes appear in the Moyal star product. Although the theory in position space contains both even and odd position modes x 2n , x 2n−1 , the mapping of the Witten ⋆ to the Moyal ⋆ necessarily requires that the Fourier space for the odd positions be named as the even momenta since (x e .p e ) are canonical under the Moyal star-product. Likewise, the Fourier space for the even position should be named as the odd momenta. Therefore the double Fourier transform of the string field A (x e ,x, p e ) , with Fourier kernels of the form (10) (with T or R as needed) that mix odd-even phase space variables, would be written purely in terms of odd phase space variables a (p odd ,x, x odd ) .
In usual phase space quantization all positions and all momenta enter directly in the Moyal product, however in the present case, which is designed to be equivalent to Witten's open string field theory, only half of the phase space variables enter into the definition of the Moyal star product in Eq. (11). 2 It is straightforward to extend the star product to the ghost sector in the bosonized ghost formalism. Then the ghost field φ(σ) plays the role of one extra dimension. If we follow the standard wisdom, our Moyal star product, including ghosts, would be modified only by inserting a phase at the midpoint, exp( 3i 2 φ( π 2 )), after evaluating the Moyal product. In view of the discussion in the previous sections one should analyse this phase insertion more carefully. To construct a string field theory one would also need to define a BRST operator with the usual properties. The study of string field theory takes a new form with the new star product. It would be interesting to see where this leads.
The new Moyal star product in string field theory Eq. (11) is Lorentz invariant, in contrast to the Moyal product used in recent studies of noncommutative field theory (in the presence of a Magnetic field). The string field formalism includes gauge symmetries that remove ghosts. Since string theory makes sense, and is ghost free (unitary), it implies that there is at least one way of making sense of noncommutative field theory when timelike components of coordinates are included (see also [11] in this respect).
In the presence of a large background B-field the midpoint coordinatesx µ become noncommuting as well. In that case, the Moyal star product in Eq.(11) is easily modified to accomodate the midpoint noncommutativity in the usual way. If the B-field is small the noncommutative structure is considerably more complicated.
For the study of D p -brane solutions in the vacuum string field theory approach of [12][13][8] [9] one seeks solutions (independent of x 0 orx) to the projector equation in the pure matter sector. Such projectors, involving the Moyal product in phase space, have been studied for a long time in the literature; they are known as Wigner functions [2] and they have applications in various branches of physics. It would be simple to generalize the known Wigner functions to the multi-dimensional string mode space needed in string theory, and then study their interpretation in string theory. In particular, the recent solutions obtained in the split string formalism can be easily Fourier transformed to the phase space formalism. For example, the solution for the sliver state in our formalism becomes a Gaussian of the form where d is the number of spacetime dimensions, M is a matrix in even mode space, 1 e is the identity matrix in that space, and det (2 × 1 e ) = ( ∞ n=1 2). The phase space integral over 2 A recent application of deformation quantization produced the proper approach for discussing two time physics in a field theory setting. This has conceptual and technical similarities to string field theory, especially with the new form of string theory based on the Moyal product in phase space [11].
this function gives the rank of the projector, and this is easily seen to be rank one for any matrix M.
In more general computations we anticipate that it would be useful to evaluate the star product for phase space functions of the form where in even mode space, (a, d) are symmetric matrices, b is a general square matrix with b T its transpose, (v µ , w µ ) are column matrices, and N is a normalization factor. In general these parameters are complex numbers. With the definition of trace given in Eq. (12) where σ is the purely imaginary matrix that results from the star commutation rules of (x e , p e ) in even mode space [x µ e , p ν e ] ⋆ = iη µν 1 e and [x µ e , x ν e ] ⋆ = 0, [p µ e , p ν e ] ⋆ = 0 Eq.(52) may be used as a generating function for the star products of more general functions. For example, the star products of more general functions, such as polynomials multiplied by exponentials of the form (49), can be obtained by taking derivatives of both sides of (52) with respect to the parameters in A M 1 ,λ 1 ,N 1 , A M 2 ,λ 2 ,N 2 . We see that Eqs.(46,47) for the projector follow from the more general multiplication rule (52). We also see that a more general projector is given by Eq.(49) when M satisfies σMσ = M −1 and N = (det (2 × 1 e )) 26 exp (−λ µ M −1 λ µ /4) , since according to (53-56), one gets M = M 1 = M 2 = M 3 , and λ = λ 1 = λ 2 = λ 12 and N = N 1 = N 2 = N 12 . The trace of the more general projector is still 1, according to (51). It has long been known that Witten's star product defines a noncommutative geometry for strings, but its relation to other forms of noncommutative geometry has remained obscure. By making the present bridge to the Moyal star product one may expect new progress, as well as cross fertilization between studies in string field theory and noncommutative field theory, and perhaps even other fields of physics that utilize Wigner functions.