18
Momentum-Space SFT
Abstract
In this chapter, we describe the general properties of SFT actions in the
momentum space. This allows to make SFT more intuitive, but also to use
standard QFT methods to prove various properties of string theory. We explain
how the Wick rotation is generalized for theories with vertices diverging at
infinite real energies (Lorentzian signature). This allows to prove important
properties of string theory, such as unitarity or crossing symmetry.
18.1 General Form
Since the explicit expressions of the string vertices are not known, it is not possible
to write explicitly the SFT action. However, the general properties of the vertices
are known: then, one can write a general QFT that contains SFT as a subcase. This
is sufficient to already extract a lot of information. The other advantage is that the
QFT language is more familiar and intuitive in many situations. Hence, one can use
this general form to built intuition before translating the results in a more stringy
language. In a nutshell, SFT is a QFT:
• with an infinite number of fields (of all spins);
• with an infinite number of interactions;
• with non-local interactions ∝ e −#k 2 ;
• that reproduces the worldsheet amplitudes (if the latter are well-defined).
© 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_18
361
Momentum-Space SFT
Abstract
In this chapter, we describe the general properties of SFT actions in the
momentum space. This allows to make SFT more intuitive, but also to use
standard QFT methods to prove various properties of string theory. We explain
how the Wick rotation is generalized for theories with vertices diverging at
infinite real energies (Lorentzian signature). This allows to prove important
properties of string theory, such as unitarity or crossing symmetry.
18.1 General Form
Since the explicit expressions of the string vertices are not known, it is not possible
to write explicitly the SFT action. However, the general properties of the vertices
are known: then, one can write a general QFT that contains SFT as a subcase. This
is sufficient to already extract a lot of information. The other advantage is that the
QFT language is more familiar and intuitive in many situations. Hence, one can use
this general form to built intuition before translating the results in a more stringy
language. In a nutshell, SFT is a QFT:
• with an infinite number of fields (of all spins);
• with an infinite number of interactions;
• with non-local interactions ∝ e −#k 2 ;
• that reproduces the worldsheet amplitudes (if the latter are well-defined).
© 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_18
361