String field theory

In theoretical physics, string field theory is a proposal to define string theory in such a way that the background independence is respected. String field theory can be understood as a quantum field theory with infinitely many fields which are unified into one master "string field". In perturbative string theory, scattering amplitudes are found by summing a genus expansion of Feynman diagrams, in analogy with the loop expansion in quantum field theory. However, this procedure does not follow from first principles, but rather from symmetry arguments and intuition. When quantized, the action governing the string field would, in principle, reproduce all the Feynman diagrams of splitting and joining perturbative strings, but also encode non-perturbative effects.

String field theory did not turn out to be helpful in the second superstring revolution because this revolution has revealed that other objects such as branes are as fundamental as the strings themselves. String field theory is based on the assumption that the strings are the fundamental objects, and it makes it more difficult (or impossible) to understand dualities within its framework.

There are several versions of string field theory—for example the boundary string field theory or the cubic (Chern-Simons-like) string field theory constructed by Edward Witten. In the late 1990s, both of them turned out to be very useful to understand tachyon condensation.

An important tool in formulating string field theory is the BRST formalism.

Mathematics

The BRST (covariant) form of the action for bosonic open string field theory is given by the functional integral:

$S=int\{Phi\; [X]\; left(\; int^pi\_\{-pi\}\{left(frac\{delta\}\{delta\; X\_mu(sigma)\}+X\text{'}\_mu(sigma)\; ight)^2dsigma\}\; -1\; ight)Phi\; [X]\; \}D\; [X]\; ^\{26\}\; +int\{left(\; V\; [X,Y,Z]\; Phi\; [X]\; Phi\; [Y]\; Phi\; [Z]\; ight)\}D\; [X]\; ^\{26\}D\; [Y]\; ^\{26\}D\; [Z]\; ^\{26\}$where V is a combination of delta functions which 'sew up' three strings in an interaction and ensure locality. Closed string theory is more complicated and involves a non-polynomial action. These actions have not been particularly useful in string theory in deriving new results. However some things can easily be seen from the action such as the fact that the length of the string increases the mass of the string which is seen from the X' term which is not found in point particle field theory. Also, the fact that there is a -1 in the action shows that this (non-supersymmetric) bosonic action contains tachyons.

:::$V\; [X,Y,Z]\; =\; prod\_\{0lesigmale\; i/2\}\; delta\; [X(sigma)-Y(pi-sigma)]\; delta\; [Y(sigma)-Z(pi-sigma)]\; delta\; [Z(sigma)-X(pi-sigma)]$

This action actually has too many degrees of freedom and we need to introduce constraints or ghost fields.

References

* [http://xstructure.inr.ac.ru/x-bin/theme2.py?arxiv=hep-th&level=2&index1=3877000 String field theory on arxiv.org]