Editing String field theory (section)

{{Short description|Formalism in string theory}}
{{String theory|cTopic=Theory}}
'''String field theory''' ('''SFT''') is a formalism in [[string theory]] in which the dynamics of [[special relativity|relativistic]] strings is reformulated in the language of [[quantum field theory]].  This is accomplished at the level of [[perturbation theory (quantum mechanics)|perturbation theory]] by finding a collection of vertices for joining and splitting strings, as well as string [[propagator]]s, that give a [[Feynman diagram]]-like expansion for string scattering amplitudes.  In most string field theories, this expansion is encoded by a [[Action (physics)|classical action]] found by [[second-quantized|second-quantizing]] the free string and adding interaction terms.  As is usually the case in second quantization, a [[classical field]] configuration of the second-quantized theory is given by a wave function in the original theory.  In the case of string field theory, this implies that a classical configuration, usually called the '''string field''', is given by an element of the free string [[Fock space]].

The principal advantages of the formalism are that it allows the computation of [[off-shell]] [[probability amplitude|amplitudes]] and, when a classical action is available, gives non-perturbative information that cannot be seen directly from the standard genus expansion of string scattering.  In particular, following the work of [[Ashoke Sen]],<ref>{{cite journal | last=Sen | first=Ashoke | title=Universality of the tachyon potential | journal=Journal of High Energy Physics | volume=1999 | issue=12 | date=1999-12-29 | issn=1029-8479 | doi=10.1088/1126-6708/1999/12/027 | pages=027|arxiv=hep-th/9911116| bibcode=1999JHEP...12..027S | s2cid=1506387 }}</ref> it has been useful in the study of [[tachyon condensation]] on unstable [[D-branes]]. It has also had applications to [[topological string theory]],<ref>E. Witten, "Chern–Simons gauge theory as a string theory", Prog. Math. ''' 133''' 637, (1995)</ref> non-commutative geometry,<ref>E. Witten, "Noncommutative tachyons and string field theory", hep-th/0006071</ref> and strings in low dimensions.<ref>{{cite journal | last1=Gaiotto | first1=Davide | last2=Rastelli | first2=Leonardo | title=A paradigm of open/closed duality Liouville D-branes and the Kontsevich model | journal=Journal of High Energy Physics | volume=2005 | issue=7 | date=2005-07-25 | issn=1029-8479 | doi=10.1088/1126-6708/2005/07/053 | pages=053|arxiv=hep-th/0312196| bibcode=2005JHEP...07..053G | s2cid=15225459 }}</ref>

String field theories come in a number of varieties depending on which type of string is second quantized:  ''Open string field theories'' describe the scattering of open strings, ''closed string field theories'' describe closed strings, while ''open-closed string field theories'' include both open and closed strings.

In addition, depending on the method used to fix the worldsheet [[diffeomorphisms]] and [[conformal transformation]]s in the original free string theory, the resulting string field theories can be very different.  Using [[light cone gauge]], yields ''light-cone string field theories'' whereas using [[BRST quantization]], one finds ''covariant string field theories''.  There are also hybrid string field theories, known as ''covariantized light-cone string field theories'' which use elements of both light-cone and BRST gauge-fixed string field theories.<ref>{{cite journal | last1=Hata | first1=Hiroyuki | last2=Itoh | first2=Katsumi | last3=Kugo | first3=Taichiro | last4=Kunitomo | first4=Hiroshi | last5=Ogawa | first5=Kaku | title=Manifestly covariant field theory of interacting string I | journal=Physics Letters B | publisher=Elsevier BV | volume=172 | issue=2 | year=1986 | issn=0370-2693 | doi=10.1016/0370-2693(86)90834-8 | bibcode=1986PhLB..172..186H | pages=186–194}}</ref>

A final form of string field theory, known as ''background independent open string field theory'', takes a very different form; instead of second quantizing the worldsheet string theory, it second quantizes the space of two-dimensional quantum field theories.<ref>{{cite journal | last=Witten | first=Edward | title=On background-independent open-string field theory | journal=Physical Review D | volume=46 | issue=12 | date=1992-12-15 | issn=0556-2821 | doi=10.1103/physrevd.46.5467 | pmid=10014938 | pages=5467–5473|arxiv=hep-th/9208027| bibcode=1992PhRvD..46.5467W | s2cid=1135319 }}</ref>