String Field Theory - Free Covariant String Field Theory

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, 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),

$|\Psi\rangle = \int d^{26} p \left (T(p) c_1 e^{i p\cdot X} |0\rangle + A_\mu (p) \partial X^\mu c_1 e^{i p \cdot X} |0 \rangle + \chi (p) c_0 e^{i p \cdot X}|0\rangle + \ldots \right),$

where is the free string vacuum and the dots represent more massive fields. In the language of worldsheet string theory, and 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, gauge field and a ghost field .

In the worldsheet string theory, the unphysical elements of the Fock space are removed by imposing the condition as well as the equivalence relation . After second quantization, the equivalence relation is interpreted as a gauge invariance, whereas the condition that 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 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. It is given by

$S_{\text{free open}} (\Psi) = \tfrac{1}{2} \langle \Psi | Q_B |\Psi\rangle \ ,$

where is the BPZ-dual of .

For the bosonic closed string, construction of a BRST-invariant kinetic term requires additionally that one impose and . The kinetic term is then

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

Read more about this topic: String Field Theory

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