208
9 String Field
fields, (2) in the coupling between the modes and (3) as a complicated momentum
dependence of the action.
When we are not interested in the spacetime properties, we will write a generic
basis of the Hilbert space H as {φ r }:
| =
r
ψ r |φ r .
(9.4)
The sum over r includes discrete and continuous labels.
Example 9.1: Scalar Field
In order to illustrate the notations for a point-particle, consider a scalar field φ(x).
It can be expanded in Fourier modes as
φ(x) =
d D k
(2π) D φ(k)e
ik·x .
(9.5)
The corresponding ket |φ is found by expanding on a basis {|k}:
|φ =
d D k
(2π) D φ(k) |k ,
φ(k)= =k|φ .
(9.6)
Similarly, the position space field is defined from the basis {|x} such that
φ(x) = =x|φ =
d D k
(2π) D x|k k|φ ,
x|k = e
ik·x .
(9.7)
9.3
Summary
In this chapter, we introduced general ideas about what a string field is. We now
need to write an action. In general, one proceeds in two steps:
1. build the kinetic term (free theory):
(a) equations of motion → physical states
(b) equivalence relation → gauge symmetry
2. add interactions and deform the gauge transformation.
We consider the first point in the next chapter, but we will have to introduce more
machinery in order to discuss interactions.
9 String Field
fields, (2) in the coupling between the modes and (3) as a complicated momentum
dependence of the action.
When we are not interested in the spacetime properties, we will write a generic
basis of the Hilbert space H as {φ r }:
| =
r
ψ r |φ r .
(9.4)
The sum over r includes discrete and continuous labels.
Example 9.1: Scalar Field
In order to illustrate the notations for a point-particle, consider a scalar field φ(x).
It can be expanded in Fourier modes as
φ(x) =
d D k
(2π) D φ(k)e
ik·x .
(9.5)
The corresponding ket |φ is found by expanding on a basis {|k}:
|φ =
d D k
(2π) D φ(k) |k ,
φ(k)= =k|φ .
(9.6)
Similarly, the position space field is defined from the basis {|x} such that
φ(x) = =x|φ =
d D k
(2π) D x|k k|φ ,
x|k = e
ik·x .
(9.7)
9.3
Summary
In this chapter, we introduced general ideas about what a string field is. We now
need to write an action. In general, one proceeds in two steps:
1. build the kinetic term (free theory):
(a) equations of motion → physical states
(b) equivalence relation → gauge symmetry
2. add interactions and deform the gauge transformation.
We consider the first point in the next chapter, but we will have to introduce more
machinery in order to discuss interactions.