Here we are going to give a brief derivation of the transformation matrices between the half string coordinates and the full string coordinates needed for the construction of the half string interacting vertex in terms of the oscillator representation of the full string. For this we shall follow closely the discussion of reference     . To make this more concrete, we recall the standard mode expansion for the open bosonic string coordinate
where and correspond to the ghost part . The half string coordinates and for the left and right halves of the string are defined in the usual way
where both and satisfy the usual Neumann boundary conditions at and a Dirichlet boundary conditions . Thus they have expansions of the form
Comparing Equation (1) and Equation (3) we obtain an expression for the half string modes in terms of the full string modes
where the change of representation matrices are given by
Since the transformation in (4) is non singular, one may invert the relation in (4). Inverting (4) we find
In the decomposition of the string into right and left pieces in (2), we singled out the midpoint coordinate. Consequently the relationship between and does not involve the zero mode of . At , we have
and so the center of mass may be related to the half string coordinates and the midpoint coordinate
Equations (8) and (9) with Equations (4) and (7) complete the equivalence between , , and , .
For later use we also need the relationships between , the half string conjugate momenta and , the full string conjugate momenta. Using Dirac quantization procedure
we find (thereafter; the space-time index is suppressed),
To obtain the full string conjugate momenta in terms of the half string conjugate momenta, we need to invert the above relations; skipping the technical details we find
We notice that the existence of the one-to-one correspondence between the half string and the full string degrees of freedom guarantees the existence of the identification
where stands for the completion of the full string Hilbert space and , , in the tensor product stand for the two half-string Hilbert spaces and the Hilbert space of functions of the mid-point, respectively.
2. The Half-String Overlaps
The half string three interaction vertex of the open bosonic string ( ) have been constructed in the half-string oscillator representation  . Here we are interested in constructing the comma three interaction vertex in terms of the oscillator representation of the full string. Here we shall only consider the coordinate piece of the comma three interaction vertex. The ghost part of the vertex ( ) in the bosonic representation is identical to the coordinate piece apart from the ghost mid-point insertions required for ghost number conservation at the mid-point. To simplify the calculation we introduce a new set of coordinates and momenta based on a Fourier transform1
where and r refers to the left (L) and right (R) parts of the string. The superscripts 1, 2 and 3 refers to string 1, string 2 and string 3, respectively. Similarly one obtains a new set for the conjugate momenta , and as well as a new set for the creation-annihilation operators . In the Fourier space the degrees of freedom in the function overlaps equations decouple which result in a considerable reduction of the amount of algebra involved in such calculations as we shall see shortly. Notice that in the Fourier space the commutation relations are
Since , then and are no longer canonical variables. The canonical variables in this case are and . Thus the Fourier transform does not conserve the original commutation relations. The variables and are still canonical however. This is a small price to pay for decoupling string three in the Fourier space from the other two strings as we shall see in the construction of the comma three interaction vertex. Recall that the overlap equations for the comma three interacting vertex are given by
for the coordinates (where the mid-point coordinate and the identifications and are understood). The comma coordinates are defined in the usual way 
The overlaps for the canonical momenta are given by
where the mid-point momentum is defined in the usual way . The comma coordinates and their canonical momenta obey the usual commutation relations
In Fourier space of the comma, the overlap equations for the half string coordinates read
where Equation (29) is to be understood as an overlap Equation (i.e., its action on the three vertex is zero). Similarly the canonical momenta of the half string in the Fourier space of the comma translate into
The overlap conditions on and determine the form of the comma three interaction vertex. Thus in the Fourier space of the comma the overlap equations separate into two sets. The half string three vertex
therefore separates into a product of two pieces one depending on
and the other one depending on
Notice that in this notation we have and (where the usual convention applies). Observe that the first of these equations is identical to the overlap equation for the identity vertex. Hence, the comma 3-Vertex takes the form
where C and H are infinite dimensional matrices computed in  and the integration over gives . However and so is the statements of conservation of momentum at the center of mass of the three strings. Notice that the comma three interaction vertex separates into a product of two pieces as anticipated. The vacuum of the three strings, i.e., , is however invariant under the -Fourier transformation. Thus we have . If we choose to substitute the explicit values of the matrices, the above expression reduces to the simple form
where denotes the vacuum in the left (right) product of the Hilbert space of the three strings. Here denotes oscillators in the jth string Hilbert space. For simplicity the Lorentz index ( ) and the Minkowski metric used to contract the Lorentz indices, have been suppressed in Equation (40). We shall follow this convention throughout this paper.
Though the form of the comma 3-Vertex given in Equation (40) is quite elegant, it is very cumbersome to relate it directly to the SCSV 3-Vertex due to the fact that connection between the vacuum in the comma theory and the vacuum in the SCSV is quite involved. One also needs to use the change of representation formulas  to recast the quadratic form in the half string creation operators in terms of the full string creation-annihilation operators which adds more complications to an already difficult problem. On the other hand the task could be greatly simplified if we express the comma vertex in the full string basis. This may be achieved simply by re expressing the comma overlaps in terms of overlaps in the full string basis. Moreover, the proof of the Ward-like identities will also simplify a great deal if the comma 3-Vertex is expressed in the full string basis. Before we express the half-string 3-Vertex is expressed in the full string basis, we need first to solve the comma overlap equations in (27), (30) and (32), (34) for the Fourier modes of the comma coordinates and momenta, respectively. The modes in the Fourier space are given by
where , and a similar set for the conjugate momenta. The overlap equations for the coordinates in (27) and (28) and the properties imposed in the Fourier expansion of the comma coordinates
where , imply that their Fourier modes in the comma basis satisfy
From the overlap in (30) we obtain
For the Fourier modes of the conjugate momenta one obtains
where . We see that the comma overlaps in the full string basis separates into a product of two pieces depending on
respectively, where the creation and annihilation operators and in the -Fourier space are defined in the usual way
and similarly for , and , . Notice that in the -Fourier space, , . For the matter sector, the comma 3-Vertex would be represented as exponential of quadratic form in the creation operators , and . Thus the comma 3-Vertex in the full string -Fourier space takes the form
where denotes the matter part of the vacuum in the Hilbert space of the three strings and
The ghost piece of the 3-Vertex in the bosonized form has the same structure as the coordinate piece apart from the mid point insertions. In the -Fourier space and only . Thus the mid-point insertion is given by . The effect of the insertion is to inject the ghost number into the vertex at its mid-point to conserve the ghost number at the string mid-point, where the conservation of ghost number is violated due to the concentration of the curvature at the mid-point. Thus the ghost part of the 3-Vertex takes the form
where denotes the ghost part of the vacuum in the Hilbert space of the three strings and has the exact structure as the coordinate piece . The mid-point insertion in (62) may be written in terms of the creation annihilation operators
If we now commute the annihilation operators in the mid-point insertion through the exponential of the quadratic form in the creation operators in the three-string ghost vertex ( ), the three-string ghost vertex in (62) takes the form
We note that commuting the annihilation operators in the mid-point insertion through results in the doubling of the creation operator in the mid-point insertion.
3. The Half-String 3-Vertex in the Full String Basis
We now proceed to express the half-string overlaps in the Hilbert space of the full string theory. The change of representation between the half-string modes and the full string modes derived in  is given by
where ; ; and the matrices and are given by
Now the overlap equations in (47), (50) and (29) become
respectively. The overlaps for the complex conjugate of the first two equations could be obtained simply by taking the complex conjugation. Similarly from the overlaps in (49), (52) and (35) we obtain
We have seen in reference  the and so the overlap conditions in (72) and (73) reduce to
It is important to keep in mind that the equality sign appearing in Equations (68) through (75) is an equality between action of the operators when acting on the comma vertex except for Equation (75) which is the conservation of the momentum carried by the third string in the Fourier space.
The comma vertex in the full string basis now satisfies the comma overlaps in (68), (69), (70), (71), (74) and (74). First let us consider the overlaps in (68), (69) and (70), i.e.,
(as well as their complex conjugates), where . For the remaining overlaps, i.e., equations in (71) and (74), we have
where . We notice that these overlaps are identical to the overlap equations for the identity vertex    . Thus
The explicit form of the matrix F, may be obtained from the overlap equations given by (76), (77) and (78) as well as their complex conjugates. It will turn out that the matrix F has the following properties
which are consistent with the properties of the coupling matrices in Witten’s theory of open bosonic strings  . This indeed is a nontrivial check on the validity of the comma approach to the theory of open bosonic strings.
Now substituting (61) into (76) and writing in terms of and , we obtain the first equation for the matrix F
where . Next from the overlap equation in (77) we obtain a second condition on the F matrix
where . The overlaps for the mid-point in (78) give
Solving Equations (84) and (85), we have
where all . Finally Equation (86) leads to
Now the explicit form of the F matrix is completely given by the set of Equations (87), (88), (89), (90), (91), (92), (93), (94) and (95) provided that the
inverse of the exist. Now we proceed to compute the required inverse.
4. Finding the Inverse
In the half string formulation the combination is a special case of the more general expression, , where
and N is the number of strings2. For the case of interest, N corresponds to 3 and . It is however more constructive to consider the generic combination . Again for the case of interest one has and . For the inverse of , we propose the Ansatz
The coefficients and are the modes appearing in the Taylor expansion of the functions and respectively. For the
three interaction vertex and the Taylor modes and reduce to and found in references  . These coefficients are treated in details in appendix A. The free parameters , and p are to be determined by demanding that (96) satisfies the identities
which implies that is left inverse and the identity
which implies that is right inverse. Here I is the identity matrix in the space of N strings.
Before we proceed to fix the constants , and p, there are two special cases where the inverse could be obtained with ease with the help of the commutation relations of the half string creation annihilation operators . They are given by and .
For , the combination reduces to and the inverse . To see this we only need to verify that . We first consider
Using the commutation relations
(where ) for the half string creation annihilation modes , one can show that the the combination inside the first bracket is the identity matrix I and the combination inside the second bracket is identically zero. To see this recall that the change of representation between the full string creation annihilation modes and the half string creation annihilation modes is given by
and is given by the same expression with . Substituting (101) into (100) one obtains
for and . Since for , then the above equation does not yield any information about the combination for . However, for , Equation (102) yields
Similarly one has
for . In this case, the above expression gives the following identity
for all possible values of r and s. Substituting Equations (103) and (105) into Equation (99) we arrive at
Thus is the right inverse of . To complete the proof one must show that is also a left inverse; that is we need to establish the following identity
The proof of the above identity follows at once from the change of representation between the half string creation annihilation modes and the full string creation annihilation modes given by
(where ) and the commutation relations
Using Equations (108) and (109) and skipping the algebraic details, one obtains the following identities
needed to prove that the combination is also a left inverse. This completes the proof.
For , the combination reduces to and the inverse . The proof that is the right inverse follows at once simply by taking the transpose of the already established identity in (106). To show that the combination is also the left inverse of the combination one only needs to take the transpose of (107); thus leading to the desired result.
Now we proceed to fix the constants in (96) for . From Equations (66) and (67), we have
respectively. First we proceed with the identity in (97). If we could solve for the free parameters , , and p in terms of the known parameters and then the Ansatz in (96) is the left inverse of the matrix . For the off diagonal elements; that is , the identity in (97), yields, after much use of the identities in ,
where the quantities
have been considered in . The quantities are related to through the identity . The quantity has the value . In order for the right hand side of the above expression to vanish, the coefficients of , and must vanish separately. The vanishing of the coefficient of the can be established explicitly by substituting the explicit values for . The vanishing of the coefficient of term leads to the following conditions on the free parameters
The vanishing of the coefficient of does not lead to new conditions on the free parameters but it provides a consistency condition. The equivalence between the half-string field theory and Witten’s theory of open bosonic strings will guarantee that this consistency condition will be met. In fact we have verified this requirement explicitly.
For the diagonal elements ( ), the identity in (97), after much use of the various identities in , yields
has been considered in . Using the explicit values of , , and which are given in  and imposing the conditions obtained in (116), the above expression reduces, after a lengthy exercise, otherwise a straight forward algebra, to
where the quantities were introduced in . The above expression may be reduced further by expressing in terms of through the relations
which have been established in . Hence
In arriving at the above expression we used the fact that
Further simplification of (122) may be achieved by substituting the explicit values of and found in . Thus Equation (122) reduces to
To compute the right-hand side of the above expression we need to evaluate the expression inside the square bracket. We will show that this expression has the explicit value 2/2n. Consider the matrix element defined by
The matrix element satisfies the following recursion relationship, which may be verified by direct substitution
for integer. Letting , in (126), we obtain
Summing both sides of (127) over m, we have
Substituting the explicit values for , and into (128) we obtain
Recalling the recursion relations for the Taylor modes established in 
If we now set in the recursion relations in (130) and (131) and then rearrange terms, we have
Substituting (132) and (133) in the above equations into (129), we find
Repeated application of the above identity implies that
Substituting the explicit form of into the above identity we have
To complete the proof, it remains to show that the expression inside the square bracket on the right hand side of Equation (136) is equal to unity. This we do by explicit computation. Consider
Using the summation formulas for and , which are given in , the above expression reduces to
and so Equation (136) yields
Substituting this result for the expression in the square bracket in (124) leads to one more condition on the parameters and p
Collecting all the conditions on the free parameters, and then solving for the parameters and , and p in terms of the known parameters and , we find
The desired expression for the inverse of , is therefore given by substituting the values of and given by the above expressions into the Ansatz for in Equation (96). Hence,
This shows that the above expression is the left inverse. To complete the proof we need to check that the identity in (98) is also satisfied and leads to the same conditions as in Equations (143). This in fact we did verify. The special cases of and have been treated earlier. This completes the construction of the inverse for the general case of the matrix.
In the particular case of , the parameters and respectively, and the above relations in (143) become
For the particular case of and , the relations in (145) yield
If we choose , then we have
For the case of interest, that is, the three interaction vertex and so that . This implies that the Taylor modes and in the expansion of and are and in the expansion of and encountered in reference  . Thus the inverse of now reads
This is the required inverse needed to finish the construction of the half-string three interaction vertex in terms of the full-string basis. The expression in (149) is indeed the right and left inverse of as can be checked explicitly. See ref. .
5. Computing the Explicit Values of the Matrix Elements of the F Matrix
To complete the construction of the comma 3-Vertex
in the -Fourier space of the full string, we need the explicit form of the F matrix. Here we shall give the steps involved in the computation of the matrix elements of F and relegate many of the technical details to appendix A. For the purpose of illustration consider . Substituting the explicit value of
obtained in (149) into Equation (87) gives
where . Using partial fractions, the above expression becomes
where the quantities appearing in the above expression are defined have been evaluated in . Thus substituting the explicit values of these quantities into (152) and combining terms we find
The explicit value of the may be computed by substituting (153) into (93). Doing that and rearranging terms we get
The sum appearing on the right-hand side has the value , so we obtain
which gives the explicit value of at once. This result is consistent with that given in  . To obtain the explicit value of , we first need to evaluate the sum over k in Equation (94), i.e.,
where the explicit expression for in terms of the change of representation matrices is given by Equation (88). Thus substituting (88) into the above expression we have
If we commute3 the sums over k and l, we get
Substituting Equation (149) for the inverse of the combination into the above expression and summing over k from 1 to , we obtain
In arriving at the above result we made use of the identities
which were derived in . Similar expressions hold for the sums over ; see ref. . Now substituting Equation (159) into (157) gives
Using the explicit value of and and rewriting in terms of , the above expression becomes
Substituting this result into (94), we find
which has the same form as given in (153). Thus in this case we see that the property holds.
Next we consider the evaluation of . If we replace and in (92) by their explicit values, given respectively by Equation (66), (67) and (153), we have
In order to benefit from the results obtained in  to help carry out the sums we first need to extend the range of the sums to include . Hence adding zero in the form , the above expression becomes
The sums in the square brackets have been evaluated in . Thus one finds
where . To check if the property continue to hold, we need to compute explicitly the value of . It is important to verify that the matrix F is self adjoint for the consistency of our formulation. The matrix element involves the matrix element which in turn is expressed
in terms of the combination and the matrix element
itself. To carry out the calculation, unfortunately we first need to compute the explicit value of . The matrix element is given by (89)
Substituting the explicit value of into the above equation and summing over m, we find
where . Combining Equation (168) with Equation (95), leads to
To evaluate the sums appearing in (169) we first need to extend their range to include . Doing so and making use of the result of already established identities in appendix A, Equation (169) reduces to
Solving the above equation for , we obtain
which is precisely the adjoint of ; see Equation (166). Thus we have
The result obtained in (171) may be now used to find the explicit value of . Thus substituting Equation (171) back into Equation (168), we find
The computation of the matrix element is indeed quite cumbersome. The difficulty arises from the fact that the defining equation of , which is given by (91), involves this summing over the matrix which is potentially divergent when the summing index m takes the k value. The limiting procedures involved in smoothing out the divergence are quite delicate and require careful consideration. Thus here we shall only give the final result; the details may be found in ,
Comparing Equations (173) and (174), we see that
To complete fixing the comma interaction vertex in the full-string basis we still need to compute the remaining elements, namely and . The computation of the matrices and involve two distinct cases. The off diagonal case is given by and the diagonal case is given by . Though the off diagonal elements are not difficult to compute, the diagonal elements are indeed quite involved and they can be evaluated by setting in the defining equations for and and then explicitly performing the sums with the help of the various identities we have established in . An alternative way of computing the diagonal elements is to take the limit of in the explicit expressions for the off diagonal elements. We have computed the diagonal elements both ways and obtained the same result which is a non trivial consistency check on our formalism. For illustration, here we shall compute the diagonal elements by the limiting process we spoke of as we shall see shortly. But first let us compute the off diagonal elements. We first consider . From Equation (88), we have
Substituting the explicit value of and into the above equation, we have
The difficulty in evaluating the sums arises from the fact in performing these sums one usually make use of partial fraction to reduce them to the standard sums treated in  ; however partial fraction in this case fails due to a divergence arising from the particular case when . Thus to carry our program through, we first consider the case for which . For , partial fraction can be used to reduce the sums in the above expression to the standard results obtained in . Skipping some rather straight forward algebra, we find
where , and . Substituting the value of , which is given by Equation (163), we have
which is the desired result valid for ,and subject to the condition . Note that in this case we have
as expected. As we pointed earlier the diagonal element may be obtained by taking the limit of in Equation (177). Hence
This result may be simplified further with the help of the following identities derived in 
The generalization of the plus combination in the square bracket has been considered before; its value is given explicitly by setting in Equation (141)
Using this identity, we obtain
Using Equation (163) to eliminate , the above expression becomes
which satisfies the property
Finally we consider the matrix elements . From Equation (90), we have
The values of and are given by Equations (173) and (171) respectively. Hence, substituting the explicit value of and in (189) and skipping some rather straightforward algebra, we find
Now there are two cases to consider and . For , Equation (190) becomes
where and we have made use of the results in  to evaluate the various sums. Thus for , we see that
For , Equation (190) becomes
where we have made use of the results in  to evaluate the various sums appearing in the steps leading to the above result. Using the identities
derived in , the above expression becomes
which is clearly self adjoint. Thus from Equations (191) and (196) it follows that
as expected. With this result we, establish that as anticipated.
In the original variables, the comma three-string in (60) can be written in the form
The matrix elements may be obtained by comparing (199) to (60). For example consider the terms involving and in (60)
Comparing this result with the terms and , we obtain
where we have used the fact that . Likewise one expresses the remaining matrix element in terms of the matrix elements and and their complex conjugates. All in all we have
which is the same result obtained in ref. . Equation (203) gives completely the comma interaction three vertex in the full string basis in the representation with oscillator zero modes.
Sometimes it is useful to express the comma vertex in the momentum representation. For a single oscillator with momentum p and creation operator , the change of basis is accomplished by
with being the oscillator ground state. Thus using the above identity and Equation (61) one finds the following representation for the Vertex in the momentum space
where the prime matrices are related to the unprimed matrices by
The property in Equation (83) implies that in the momentum representation, the matrix satisfies
For , we have , and so Equation (205) may be written as
where the matrix G is defined through the relation
The ghost part of the comma vertex in the full string basis has the same structure as the coordinate one apart from the mid-point insertions
where the s are the bosonic oscillators defined by the expansion of the bosonized ghost fields and is the exponential of the quadratic form in the ghost creation operators with the same structure as the coordinate piece of the vertex.
We have successfully constructed the comma three interaction vertex of the open bosonic string in terms of the oscillator representation of the full open bosonic string. The form of the vertex we have obtained for both the matter and ghost sectors are those obtained in ref.   . This establishes the equivalence between Witten’s 3-interaction vertex of open bosonic strings and the half string 3-vertex directly without the need for the coherent state methods employed in ref. .
1This technique was first used by D. Gross and A. Jevicki in 1986.
2The reason we are considering this more general expression is that this combination appears in computing the N-interaction vertex which will probe useful in future work and it does not add to the level of difficulty in finding the inverse.
3Since both the sums over l and k are uniformly convergent, one may perform the sums in any order. We have carried the sums in the two different orders and found that the result is the same. However, it is much easier to perform the sum over k first followed by the sum over l rather than the reverse. Here we shall follow the former.
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