Abstract
The interacting comma 3-vertex for the bosonic open string in the full string basis is derived using the half string overlap relations directly. Thus avoiding the coherent states technique employed in earlier derivations. The resulting form of the interacting 3-vertex turns out to be precisely the desired expression obtained in terms of the full string oscillator modes. This derivation establishes that the comma 3-vertex and Witten's 3-vertex are identical and therefore are interchangeable.
Highlights
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
Avoiding the coherent states technique employed in earlier derivations
The half string coordinates xL,μ (σ ) and xR,μ (σ ) for the left and right halves of the string are defined in the usual way xL,μ
Summary
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 [1] [2] [3] [4] [5]. 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 p2n=−1 pnL − pnR ,. Where H stands for the completion of the full string Hilbert space and HL , HR , HM in the tensor product stand for the two half-string Hilbert spaces and the Hilbert space of functions of the mid-point, respectively
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