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Open String Fields As Matrices

We present a new representation of the string vertices of the cubic open string field theory. By using this three-string vertex, we attempt to identify open string fields as huge-sized matrices by following Witten's idea. By using these huge matrices, we obtain some results about the construction of partial isometries in the algebra of open string fields.


Introduction
In the paper [1], Rastelli, Sen and Zwiebach have proposed the cubic open string field theory [2] around the tachyon vacuum, where it is supposed that only closed strings exist with no open strings. In the subsequent work [3], they found D-branes as classical solutions in their proposed open string field theory by using the projection fields obtained in [4] for the matter sector. The analogous technique where projection operators are used to find classical solutions has also been seen in noncommutative field theories [5,6]. Especially in the latter paper [6], this solution generating technique has been used to find solutions in noncommutative gauge theory. This solution generating technique needs the system under consideration to have gauge invariance. Since the cubic open string field theory is invariant under huge gauge symmetry, it seems to be very useful in finding classical solutions analytically. In fact, the authors of [7,6] have argued the possibility that such techniques can be applied to the cubic string field theory and the boundary string field theory (See also [4] for another approach).
Let us briefly review their discussion about the application of the solution generating technique to the cubic string field theory. The variation of the action [2]  and its finite form can be seen to be (1.5) Now we turn to the solution generating technique with two operators, U and V , satisfying U ⋆ V = I, V ⋆ U = I − P, (1.6) where P is an operator such that QP = 0. Then we consider the field redefinition in the equation of motion (1.2) and obtain δS δA If D-brane solutions could be obtained in this way, we could gain a deeper understanding of the tachyon condensation and the D-brane dynamics in the cubic string field theory.
In the cubic string field theory, our present technology does not seem enough to find such partial isometries U , V , let alone the nontrivial classical solutions to (1.2). In the case of the field theories [5,6], under the Weyl transformation, fields are transformed to operators acting on the ordinary Hilbert space. Such operators can also be given as matrices in terms of states in the Hilbert space. Therefore, the Moyal product of fields is transformed to the usual matrix multiplication, which seems to make it easy to find projection operators and partial isometries.  [2] such that the right half of the string of A is identified with the left half of B and summed over all the possible configurations to give the third string A ⋆ B, we are very tempted to consider the right half of the string field A as the column of some huge matrix and the left of B as the row. In fact, Witten have already discussed the above idea in his paper [2]. In this paper, we will try to give an explicit realization of this idea by introducing a new representation of string vertices. Although we believe that our attempt is successful to some extent, there might exist some subtleties in our treatment of the midpoint of string vertices, as will be discussed in section 5.
In section 2, we will present the new string vertices for the matter sector, which have a simpler form compared to the usual oscillator representation [8][9][10][11][12]. The kinetic term (the Virasoro operator) L 0 in a flat Minkowski spacetime becomes diagonal in terms of the usual oscillators of the string coordinates. In this paper, we introduce another set of oscillators.
These oscillators do not any longer diagonalize the above Virasoro operator, but make string vertices very simple. Since our attention will be paid mainly to the open string product ⋆, the new oscillators will be adopted for our study in this paper. In addition, we have not found such simple vertices for the ghost sector. Since the open string product does not mix the ghost operators with ones in the matter sector, we will be content with the study of the open string product only in the matter sector.
In section 3, we will obtain the mapping from string fields to huge-sized matrices by using the above-mentioned three-string vertex. In section 4, taking advantage of the matrix representation of string fields, we will construct some projectors and partial isometries, and discuss the relation of these operators to D-branes. Most of section 5 is devoted to the discussion about potential subtleties in this work.

A New Representation of String Vertices
In this section, we shall introduce a new representation of the three-string vertex, which is given in terms of unusual oscillator modes of string coordinates. Although the BRS charge in a flat Minkowski spacetime is not diagonalized by the new oscillators, the three-string vertex becomes simple in terms of them. The simplicity of our string vertices turns out to give us more insights into the structure of the open string product.

Preliminary
To discuss the new representation of string vertices, we introduce unconventional oscillators in the Hilbert space of a single open string. Let us consider an open string parametrized by σ ∈ [0, π] with the Neumann boundary condition at σ = 0, π. The ordinary mode expansion of the coordinate X i (σ) and the momentum P j (σ) is given by η jk α k n cos nσ   . (2.1) The canonical commutation relation is where δ(σ, σ ′ ) is the δ-function satisfying the Neumann boundary condition: From (2.2), the commutation relations of the modes are found to be The SL(2, R) vacuum |0 is defined by Now, let us introduce a new annihilation operator a(σ) by where t is an arbitrary parameter 1 . Then, the commutator of a i (σ) and its hermitian Using a i (σ), we define the "a-vacuum" |Ω by a i (σ)|Ω = 0, for ∀σ ∈ [0, π]. and are summed over to give the δ-function The annihilation operator a i (σ) can thus be expanded by ϕ n as (2.14) By introducing the oscillators satisfying [b i n , b † mj ] = δ i j δ n,m , the relation (2.14) can be rewritten as where θ n is given by Therefore, a i n and b i n are related by the Bogoliubov transformation where U is a unitary operator For the zero-mode part, the zero momentum eigenstate |p = 0 and a-vacuum are related by Note that the transformation in the zero-mode part is not unitary. After all, the relation between |Ω and |0 is found to be

The String Vertices
A string vertex is determined up to an overall normalization by an overlap condition. For the midpoint interaction as in Witten's open string field theory [2], the overlap condition is given by for σ ∈ L, where L denotes the interval [0, π/2] and R will be used to denote [π/2, π]. From (2.6), we find the overlap condition For the notational simplicity, we henceforth omit vector indices i, j of the Lorentz group.
Before going further, let us make a small digression on a few computations to show how our vertices facilitate the computations which have been supposed to be very tedious.
The next computation is concerned with the 'identity' field satisfying the overlap condition for the K-th string. The use of the identity field allows us to obtain In order to prove this identity, the coherent state will be useful. Here z(σ) is expanded as z(σ) = ∞ r=0 z n ϕ n (σ). Also, its hermitian conjugate z| satisfies where the measure is given by [dzdz] = ∞ n=0 dx n dy n if z n = x n + iy n . By making use of the coherent state, we can prove the formula and ξ(σ) = (λ(σ), µ(σ)).
The formula (2.37) helps us to calculate For two coherent states |z and |w , a small computation shows that the product |z ⋆ w is given by If the notation |z L , z R denotes |z with the functions z L,R (σ) = θ L,R (σ)z(σ), the above equation (2.41) has a simple form

String Field Oscillators
In this section, we construct the Fock space in which all the states are obtained by acting on the Fock vacuum |Ω str with string fields A(σ) and A † (σ). These string fields play the same role as the annihilation and creation operators, and satisfy the commutation where [ , ] ⋆ denotes the commutator with the open string product: and I denotes the identity field. We henceforth call the operators A(σ), A † (σ) the "string field oscillators".
To seek the string field oscillators, let us begin with the open string product of fields I(z) given by Making use of (2.37), we find that Taking the derivatives with respect to z(σ) and w(σ) in (3.4), we obtain Since the L.H.S. of (3.5) is antisymmetric under the exchange of σ and σ ′ , we can see that The string field oscillators can thus be identified as Here we notice that the identity I maps the right part of the worldsheet oscillator a † (σ) into the string field annihilation operator A(σ) and the left part into the creation operator A † (σ). Note that the form of A(σ) is somehow reminiscent of the nilpotent string field After the substitution z(σ) → z(σ)θ R (σ) and w(σ) → w(σ)θ R (σ) in (3.4), the derivatives at z(σ) = w(σ) = 0 with respect to both of z(σ) and w(σ) give The recursive use of (3.8) provides Here we have defined the exponential of string field M by (3.10) Taking a step further, we look for the Fock vacuum |Ω str of the string field oscillators; A(σ)|Ω str = 0. (3.11) To this end, it is appropriate to consider the open string product In (3.12), replacing z(σ) with z(π − σ)θ L (σ) in the first equation and with z(σ)θ R (σ) in the second, we take the first derivative with respect to z(σ) at z(σ) = w(σ) = 0. Then we find that which, together with (3.11), indicates that recalling here that |Ω is the a-vacuum in (2.8). Furthermore, combining (3.9) and (3.12), we find From these equations (3.15), we understand that the coherent field |z can be written as It is convenient to introduce the string field coherent state |z and its conjugate such that satisfying that A(σ)|z = θ R (σ)z(σ)|z , w|A † (σ) = w|θ R (σ)w(σ). Therefore, the coherent state in terms of these states becomes 19) and (3.17) turns into z| − r(w) = exp(− R r(w) · z).
We have so far seen that the open string product can be rephrased by using the ordinary matrix product. The 'integration' of the open string field theory is defined by using the identity field |I as |z L , w R = I|z L , w R , which is especially used in defining the action of the string field theory. In our matrix language, this integration can be seen to correspond to the trace Tr |z w| ≡ w|z = I | − r(z), w . (3.20) Moreover, the inner product R(1, 2)||z 1 |w 2 can be given by the trace as
The mapping of the modes a † n I to the modes A n and A † n can be obtained by using the relations with the substitution of (3.23) and inversely In fact, we obtain where the definition of the matrices R nm and L nm is See appendix A for the properties of these matrices. The inverse mapping turns out to be Thus, all the modes A n and A † n are almost in one-to-one correspondence with all the modes a † n 2 .

30)
2 In order to prove the statement, we would need to estimate the Jacobian associated with the change of the variables (3.27). From the condition A(σ) = A(π − σ), we can see that A(σ) satisfies the extra boundary condition ∂ σ A(σ = π/2) = 0. Therefore, the mapping (3.29): A n , A † n → a † n probably is not completely surjective. This subtlety will be discussed in section 5. and the string field coherent state is thus rewritten as (3.31) in terms of the modes A † n . Making use of the modes A † n , we can define states with definite occupation numbers of the modes: where the infinitely dimensional vector n denotes (n 0 , n 1 , · · ·). In terms of such states |n , we have Since (3.25) and the commutation relation (3.1) provide the commutation relations among the modes A n and A † n as the states |n satisfy the orthonormality condition m|n = δ m 0 ,n 0 δ m 1 ,n 1 · · · ≡ δ m,n . Then for the identity field I, since I ⋆ Ψ = Ψ ⋆ I = Ψ up to some potential anomalies [14,15,16], we expect that the formula I = n |n n| (3.39) holds. In order to prove (3.39), we have to precisely estimate the measure [dzdz] in (3.37) under the change of the variables: z n → z L,n , z R,n , which we can perform in a similar way to the mapping (3.29). To this end, we need to compute the Jacobian associated with the change of the variables, but we have not done such a computation. Instead, we will hereafter assume (3.39).

Some Applications of the String Field Algebra
In the last section, we have seen that string fields can be expanded in terms of the operators |n m|. As explained in the lecture [17] (and related references therein), in the case of noncommutative (NC) R 2 , where we have the creation and annihilation operators a † , a satisfying a, a † = 1, the generating function of the operators |n m| is a key ingredient to find projection operators and shift operators, of which the latter give a field theoretical analog of the partial isometries U and V mentioned in the introduction. Therefore, before proceeding to the string field theory, let us briefly recall some facts we shall need later.
The defining property of G(z, z) is and we impose on G(z, z) the boundary condition The generating function P (u) of projection operators P n = |n n|: can be given by using G(z, z) as In particular, since the generating function P (u) satisfies P (0) = P 0 , P (1) = I, the completeness condition ∞ n=0 P n = I (4.6) can also be expressed by (4.5) at u = 1; Moreover, using the function G(z, z), we can construct the shift operator (4.8)

The Generating Function of the Operators |n m|
We can see from the previous section that the generating function G(z, w) of the string field theory is given by where θ R (σ)z(σ) = √ 2 ∞ n=0z R,n ϕ 2n (σ) and similarly for w(σ). The equation (3.36) provides the counterpart of (4.7) for u = 1; (4.11)

Projection States and Partial Isometries
In the open string field theory around the tachyon vacuum [1], projection operators found in [4] have been used to describe D-branes as soliton solutions [3]. Before that, we had seen a somehow analogous usage of projection operators in the noncommutative field theory [5,18,19]. Therefore, it is natural to study projections in the algebra of string fields.
The defining property of the projection state |P is P ⋆ P = P ; more precisely, In general, for orthogonal projectors P n satisfying P n ⋆ P m = δ n,m P m , ∞ n=0 P n = I, (4.13) the generating function P n u n (4.14) can be seen to satisfy the equation P (u) ⋆ P (v) = P (uv), to which one of the solutions is found to be Note that P (u) interpolates between the a-vacuum and the identity The projectors P n can thus be read as However, unfortunately these projection operators P n in (4.17) do not have finite norms; P n |P n = ∞. Incidentally, P (u) in the usual oscillator formalism [8][9][10][11][12] is presented in appendix B, and its norm also turns out to be ill-defined.
Besides the obvious example of the projectors P n = |n n|, an alternative projector is given by P (z) = e − 1 2 z·r(z) |z , where the reality condition P | = r(P )| enforces z(σ) = ∞ n=0 z n ϕ n (σ) to satisfy thatz n = (−) n+1 z n . This projection operator is associated with the coherent state |z as The projection operator P 0 = |Ω str Ω str | = |Ω satisfies a(σ)|Ω = 0, which suggests that P 0 does not correspond to any extended objects 3 . Indeed, P 0 = P (z = 0) is an instanton-like object, since its profile is found to be gaussian in all the directions, including the 'time' direction. Furthermore, the projector P (z) can also be seen to be an instanton.
Among the parameters z n , the zero mode z 0 represents the location of the instanton P (z) in spacetime. As in [1], if the ghost part of these solutions is universally given, the ratio of the norms of these solutions is equal to the ratio of their tensions in the tachyon string field theory [1]. Since the norm of the projector P (z) is the same as that of P 0 , the moduli parameters z(σ) may correspond to collective coordinates or the gauge redundancy. We shall leave a closer inspection of this problem to the future. The other projections P n are also instanton-like. Thus, we may regard the projection operators N k=1 P n k as Ninstanton solutions, though we have not fully uncovered the physics of such objects.
Finally, we shall discuss partial isometries in the cubic string field theory in our matrix language. As in [6,17], our experience with noncommutative solitons on NC R 2 shows one obvious example of partial isometries The commutation relation [A n , A † m ] ⋆ = δ n,m I ensures that U ⋆ V = I, by which we can verify that V ⋆ U is a projector. Since A n |Ω str = 0, we can see that V ⋆ U = I. These operators U , V are thus the shift operators associated with the n th string field oscillator.
A further example can be given by |n 0 , 0 n 0 + 1, 0|, with the defining property In this case, the pair (U, V ) is an analogue of the one used in the discussion of noncommutative U (1) instantons on R 4 [21].
Before concluding this section, we should point out that, in order for these partial isometries to give classical solutions, The action of the BRS charge Q on the resulting projector V ⋆ U must vanish, as explained in the introduction. For the BRS charge Q in a usual flat Minkowski spacetime, however, this turns out not to be the case.

Conclusion and Discussion
In this paper, we have introduced a new presentation of the string vertices in the cubic string field theory. We have seen that the new string vertices have very simple structures and facilitate some computations which have been very complicated. By making use of these new vertices, we have tried to reformulate the open string field product or the ⋆product as the usual matrix multiplication 4 . Indeed, we believe that to some extent in this paper, we have been successful in obtaining huge-sized matrices as open string fields.
However, we have also been cavalier on two points: our treatment of the midpoint and some anomalies [14,15,16] associated with singularities of the ⋆-product.
Since our representation of the string vertices satisfies the overlap condition even at the midpoint, we believe there is nothing wrong with our string vertices. However, our discussion in section 3 about splitting the operator a † (σ) I into A n and A † n is subtle, because of the usage of the functions θ L,R (σ) as projectors; θ L,R (σ) 2 = θ L,R (σ). In addition, The string oscillators are defined by using the identity state I. It is known [16,14] that states like Q L I in [13] have anomalies in calculations with the string vertices. Since the modes A n and A † n are also defined by acting on I with the integration of a † (σ) over half of the open strings, there might exist some potential anomalies in the matrix multiplication in our formulation.
In this paper, we have restricted our attention to only the matter part of open strings.
As for the ghost part, we have not been successful in finding a simple representation of the string vertices by using the fermionic ghosts b(σ) and c(σ). However, the ghost part of the vertices might be able to be simplified by using the bosonized ghost in a similar way to the matter part of our vertices. Since the construction of the ghost part requires the insertion of the ghosts at the midpoint, we would have more serious problems concerning the midpoint, as we have discussed in the above paragraphs.
We could have obtained our matrix formulation of the cubic open string field theory in the background B-field [27,28]. It would be interesting to explore what is happening in our matrix formulation in the large B limit and to compare that with the discussion in [29,7].
As in the paper [3], it is found that the projectors of the matter part give the ratios between the D-brane solutions in the tachyon string field theory [1] with the ansatz of the universality of the ghost parts by making use of the method in [4]. It is possible to apply an analogous method to our vertices, and to find the projectors which are supposed to be D-branes. However, we can analytically calculate the norm of those solutions and find no finite tensions of those solutions except the instanton-like solutions given in section 4.
By the tensions we mean the norms of the solution divided by the volume factor of the corresponding D-brane worldvolume 5 . So far, we have not succeeded in finding any brane solutions with finite norms by using our three-string vertex. Since it is likely that the norm of such D-brane solutions can be computed analytically by using our vertex, the discovery of the solutions gives a nontrivial check on the interpretation of them as D-branes.
As discussed in [30], the charges of D-branes are classified by K-theory and it is closely related to the topological configurations of tachyon fields. Although D-branes in bosonic string theory do not carry any charges, it may be interesting to study K-theory of our matrix algebra [31] of string fields.
Note added: after completing this work, we received the papers [25,26] concerning a similar idea to ours: splitting open strings into their left and right halves. However, on the whole, our approach is slightly different from theirs [25,26], especially on the introduction of the new representation of the string vertices. As a technical point, upon splitting open strings, we used a different Fourier expansion of the 'left' and 'right' halves from theirs [25,26]. It is at present unclear to us how these expansions are precisely related to each other. Furthermore, from [25,26] we learnt of the references [22,23,24], where earlier discussions (also even earlier references therein) about the idea of splitting open strings are available, besides the original discussion in [2].
After submitting the original version of this paper to the arXiv.org e-print archive, we were informed of a paper [32] arguing another approach to formulate string fields as matrices. where C nm = (−1) n δ nm . By making use of the completeness condition (2.10), (2.11) of ϕ n , we can see that P L and P R are orthogonal projections: P L + P R = 1, P 2 L,R = P L,R , P L P R = P R P L = 0. In this appendix, we consider the generating function of orthogonal projections using the Gross-Jevicki vertex. We follow the notation in [3]. The function P (u) satisfying P (u) ⋆ P (v) = P (uv) is found to be P (u) = det This can be checked by using the formula Here we assumed that C, T 1 , T 2 and X commute with each other. Note that P (u) interpolates between the matter part of "sliver" [15,4,3] and the identity: