Editing String field theory (section)

===Quantization===
To consistently quantize <math> S(\Psi) </math> one has to fix a gauge.  The traditional choice has been Feynman–Siegel gauge,
:: <math> b_0 \Psi = 0 \left.\right. \ .</math>
Because the gauge transformations are themselves redundant (there are gauge transformations of the gauge transformations), the gauge fixing procedure requires introducing an infinite number of ghosts via the [[Batalin–Vilkovisky formalism|BV formalism]].<ref>{{cite journal | last=Thorn | first=Charles B. | title=String field theory | journal=Physics Reports | publisher=Elsevier BV | volume=175 | issue=1–2 | year=1989 | issn=0370-1573 | doi=10.1016/0370-1573(89)90015-x | bibcode=1989PhR...175....1T | pages=1–101}}</ref>  The complete gauge fixed action is given by
:: <math> S_{\text{gauge-fixed}} = \tfrac{1}{2} \langle \Psi | c_0 L_0 |\Psi\rangle + \tfrac{1}{3} \langle \Psi,\Psi,\Psi \rangle \ , </math>
where the field <math> \Psi </math> is now allowed to be of ''arbitrary ghostnumber''.  In this gauge, the [[Feynman diagrams]] are constructed from a single propagator and vertex. The propagator takes the form of a strip of worldsheet of width <math> \pi </math> and length <math> T </math>

:: [[Image:OSFT propagator.svg]]

There is also an insertion of an integral of the <math> b </math>-ghost along the red line.  The modulus, <math> T </math> is integrated from 0 to <math> \infty </math>.

The three vertex can be described as a way of gluing three propagators together, as shown in the following picture:

:: [[Image:OSFT three vertex.svg]]

In order to represent the vertex embedded in three dimensions, the propagators have been folded in half along their midpoints.  The resulting geometry is completely flat except for a single curvature singularity where the midpoints of the three propagators meet.

These Feynman diagrams generate a complete cover of the moduli space of open string scattering diagrams.  It follows that, for on-shell amplitudes, the ''n''-point open string amplitudes computed using Witten's open string field theory are identical to those computed using standard worldsheet methods.<ref>{{cite journal|author1-link=Steven Giddings | last1=Giddings | first1=Steven B. | last2=Martinec | first2=Emil | last3=Witten | first3=Edward | title=Modular invariance in string field theory | journal=Physics Letters B | publisher=Elsevier BV | volume=176 | issue=3–4 | year=1986 | issn=0370-2693 | doi=10.1016/0370-2693(86)90179-6 | bibcode=1986PhLB..176..362G | pages=362–368}}</ref><ref>{{cite journal | last=Zwiebach | first=Barton | title=A proof that Witten's open string theory gives a single cover of moduli space | journal=Communications in Mathematical Physics | publisher=Springer Science and Business Media LLC | volume=142 | issue=1 | year=1991 | issn=0010-3616 | doi=10.1007/bf02099176 | bibcode=1991CMaPh.142..193Z | pages=193–216| s2cid=121798009 | url=http://projecteuclid.org/euclid.cmp/1104248494 | url-access=subscription }}</ref>