1.3 String Field Theory
19
explicitly. Finally, the advantage of the first formulation is that it provides a general
formulation of SFT at the quantum level, from which the general structure can be
studied.
1.3.3 Expression with Spacetime Fields
To obtain a more intuitive picture and to make contact with the spacetime fields, the
field is expanded in terms of 1-particle states in the momentum representation
| =
n
d D k
(2π) D ψ α (k) |k, α ,
(1.31)
where α denotes collectively the discrete labels of the CFT eigenstates. The
coefficients ψ α (k) of the CFT states |k, α are spacetime fields, the first ones being
the same as those found in the first-quantized picture (Sect. 1.2.1)
ψ α = {T , G μν , B μν , ,, . . .}.
(1.32)
Then, inserting this expansion in the action gives an expression like S[T , G μν , . . .].
The exact expression of this action is out of reach and only the lowest terms are
explicitly computable for a given CFT background. Nonetheless, examining the
string field action indicates what is the generic form of the action in terms of the
spacetime fields. One can then study the properties of such a general QFT: since
it is more general than the SFT (expanded) action, any result derived for it will
also be valid for SFT. This approach is very fruitful for studying properties related
to consistency of QFT (unitarity, soft theorems. . . ) and this can provide helpful
phenomenological models.
In conclusion, SFT can be seen as a regular QFT with the following properties:
• infinite number of fields;
• non-local interaction (proportional to e −k 2 # );
• the amplitudes agree with the worldsheet amplitudes (when the latter can be
defined);
• genuine (IR) divergences agree but can be handled with the usual QFT tools.
1.3.4 Applications
The first aspect is the possibility to use standard QFT techniques (such as renormalization) to study—and to make sense of—string amplitudes. In this sense, SFT can
be viewed as providing recipes for computing quantities in the worldsheet theory
which are otherwise not defined. This program has been pushed quite far in the last
years.
19
explicitly. Finally, the advantage of the first formulation is that it provides a general
formulation of SFT at the quantum level, from which the general structure can be
studied.
1.3.3 Expression with Spacetime Fields
To obtain a more intuitive picture and to make contact with the spacetime fields, the
field is expanded in terms of 1-particle states in the momentum representation
| =
n
d D k
(2π) D ψ α (k) |k, α ,
(1.31)
where α denotes collectively the discrete labels of the CFT eigenstates. The
coefficients ψ α (k) of the CFT states |k, α are spacetime fields, the first ones being
the same as those found in the first-quantized picture (Sect. 1.2.1)
ψ α = {T , G μν , B μν , ,, . . .}.
(1.32)
Then, inserting this expansion in the action gives an expression like S[T , G μν , . . .].
The exact expression of this action is out of reach and only the lowest terms are
explicitly computable for a given CFT background. Nonetheless, examining the
string field action indicates what is the generic form of the action in terms of the
spacetime fields. One can then study the properties of such a general QFT: since
it is more general than the SFT (expanded) action, any result derived for it will
also be valid for SFT. This approach is very fruitful for studying properties related
to consistency of QFT (unitarity, soft theorems. . . ) and this can provide helpful
phenomenological models.
In conclusion, SFT can be seen as a regular QFT with the following properties:
• infinite number of fields;
• non-local interaction (proportional to e −k 2 # );
• the amplitudes agree with the worldsheet amplitudes (when the latter can be
defined);
• genuine (IR) divergences agree but can be handled with the usual QFT tools.
1.3.4 Applications
The first aspect is the possibility to use standard QFT techniques (such as renormalization) to study—and to make sense of—string amplitudes. In this sense, SFT can
be viewed as providing recipes for computing quantities in the worldsheet theory
which are otherwise not defined. This program has been pushed quite far in the last
years.