9
String Field
Abstract
In this chapter, we introduce general concepts about the string field. The goal
is to give an idea of which type of object it is and the different possibilities for
describing it. We will see that the string field is a functional and, for this reason,
it is more convenient to work with the associated ket field, which can itself be
represented in momentum space. We focus on what to expect from a free field,
taking inspiration from the worldsheet theory. The interpretation becomes more
difficult when taking into account the interactions.
9.1
Field Functional
A string field, after quantization, is an operator that creates or destroys a string at a
given time. Since a string is a 1-dimensional extended object, the string field must
depend on the spatial positions of each point of the string denoted collectively as X μ .
Hence, the string field is a functional μ ]. The fact that it is a functional rather
than a function makes the construction of a field theory much more challenging:
it asks for revisiting all concepts we know in point-particle QFT without any prior
experience with a simple model. 1
It is important that the dependence is only on the shape and not on the
parametrization. However, it is simpler to first work with a specific parametrization
X(σ ) and make sure that nothing depends on it at the end (equivalent to imposing
the invariance under reparametrization of the worldsheet). This leads to work with
a functional )] of fields on the worldsheet (at fixed time). To proceed, one
1 The problem is not in working with the wordline formalism and writing a BRST field theory,
but really to take into account the spatial extension of the objects. In fact, generalizing further to
functionals of extended (p > 1)-dimensional objects—branes—shows that SFT is the simplest of
such field theories.
© Springer Nature Switzerland AG 2021
H. Erbin, String Field Theory, Lecture Notes in Physics 980,
https://doi.org/10.1007/978-3-030-65321-7_9
205
String Field
Abstract
In this chapter, we introduce general concepts about the string field. The goal
is to give an idea of which type of object it is and the different possibilities for
describing it. We will see that the string field is a functional and, for this reason,
it is more convenient to work with the associated ket field, which can itself be
represented in momentum space. We focus on what to expect from a free field,
taking inspiration from the worldsheet theory. The interpretation becomes more
difficult when taking into account the interactions.
9.1
Field Functional
A string field, after quantization, is an operator that creates or destroys a string at a
given time. Since a string is a 1-dimensional extended object, the string field must
depend on the spatial positions of each point of the string denoted collectively as X μ .
Hence, the string field is a functional μ ]. The fact that it is a functional rather
than a function makes the construction of a field theory much more challenging:
it asks for revisiting all concepts we know in point-particle QFT without any prior
experience with a simple model. 1
It is important that the dependence is only on the shape and not on the
parametrization. However, it is simpler to first work with a specific parametrization
X(σ ) and make sure that nothing depends on it at the end (equivalent to imposing
the invariance under reparametrization of the worldsheet). This leads to work with
a functional )] of fields on the worldsheet (at fixed time). To proceed, one
1 The problem is not in working with the wordline formalism and writing a BRST field theory,
but really to take into account the spatial extension of the objects. In fact, generalizing further to
functionals of extended (p > 1)-dimensional objects—branes—shows that SFT is the simplest of
such field theories.
© Springer Nature Switzerland AG 2021
H. Erbin, String Field Theory, Lecture Notes in Physics 980,
https://doi.org/10.1007/978-3-030-65321-7_9
205