Editing String field theory (section)

== Free covariant string field theory ==
An important step in the construction of covariant string field theories (preserving manifest [[Lorentz invariance]]) was the construction of a covariant [[kinetic term]].  This kinetic term can be considered a string field theory in its own right: the string field theory of free strings.  Since the work of [[Warren Siegel]],<ref>W. Siegel, "String Field Theory Via BRST", in Santa Barbara 1985, Proceedings, Unified String Theories, 593; <br> W. Siegel, "Introduction to string field theory", Adv. Ser. Math. Phys. '''8'''.  Reprinted as hep-th/0107094</ref> it has been standard to ''first'' BRST-quantize the free string theory and ''then'' second quantize so that the classical fields of the string field theory include ghosts as well as matter fields.  For example, in the case of the bosonic open string theory in 26-dimensional flat spacetime, a general element of the Fock-space of the BRST quantized string takes the form (in radial quantization in the upper half plane),

::<math> |\Psi\rangle = \int d^{26} p  \left (T(p) c_1 e^{i p\cdot X} |0\rangle 
+ A_\mu (p) \partial X^\mu c_1 e^{i p \cdot X} |0 \rangle + \chi (p) c_0 e^{i p \cdot X}|0\rangle + \ldots
\right),</math>
where <math> |0\rangle </math> is the free string vacuum and the dots represent more massive fields.  In the language of worldsheet string theory, <math> T(p) </math>, <math> A_\mu(p) </math>, and <math> \chi(p) </math> represent the amplitudes for the string to be found in the various basis states.  After second quantization, they are interpreted instead as classical fields representing the tachyon <math> T </math>, gauge field <math> A_\mu </math> and a ghost field <math> \chi </math>.

In the worldsheet string theory, the unphysical elements of the Fock space are removed by imposing the condition <math> Q_B |\Psi \rangle = 0 </math> as well as the equivalence relation <math> |\Psi \rangle \sim |\Psi\rangle + Q_B |\Lambda \rangle </math>.  After second quantization, the equivalence relation is interpreted as a [[gauge invariance]], whereas the condition that <math> |\Psi \rangle </math> is physical is interpreted as an [[equation of motion]].  Because the physical fields live at ghostnumber one, it is also assumed that the string field <math> |\Psi \rangle </math> is a ghostnumber one element of the Fock space.

In the case of the open bosonic string a gauge-unfixed action with the appropriate symmetries and equations of motion was originally obtained by [[André Neveu]], Hermann Nicolai and [[Peter C. West]].<ref>{{cite journal | last1=Neveu | first1=A. | last2=Nicolai | first2=H. | last3=West | first3=P. | title=New symmetries and ghost structure of covariant string theories | journal=Physics Letters B | publisher=Elsevier BV | volume=167 | issue=3 | year=1986 | issn=0370-2693 | doi=10.1016/0370-2693(86)90351-5 | bibcode=1986PhLB..167..307N | pages=307–314| url=https://cds.cern.ch/record/164099 }}</ref> It is given by
:: <math>
   S_{\text{free open}} (\Psi) = \tfrac{1}{2} \langle \Psi | Q_B |\Psi\rangle \ ,
</math>
where <math> \langle \Psi | </math> is the [[Belavin–Polyakov–Zamolodchikov equations|BPZ]]-dual of <math> |\Psi \rangle </math>.<ref>{{cite journal | last1=Belavin | first1=A.A. | last2=Polyakov | first2=A.M. | last3=Zamolodchikov | first3=A.B. | title=Infinite conformal symmetry in two-dimensional quantum field theory | journal=Nuclear Physics B | publisher=Elsevier BV | volume=241 | issue=2 | year=1984 | issn=0550-3213 | doi=10.1016/0550-3213(84)90052-x | bibcode=1984NuPhB.241..333B | pages=333–380| url=https://cds.cern.ch/record/152341 }}</ref>

For the bosonic closed string, construction of a BRST-invariant kinetic term requires additionally that one impose <math> (L_0 - \tilde{L}_0) |\Psi\rangle = 0 </math> and <math> (b_0 - \tilde{b}_0) |\Psi\rangle = 0 </math>.  The kinetic term is then
:: <math> S_{\text{free closed}} = \tfrac{1}{2} \langle \Psi | (c_0 - \tilde{c}_0) Q_B |\Psi \rangle  \ .</math>

Additional considerations are required for the superstrings to deal with the superghost zero-modes.