15.2 Gauge Fixed Theory
317
The action (15.24) looks overly more complicated than a typical QFT theory:
instead of few interaction terms for low n (n ≤ 4 in d = 4 renormalizable theories),
it has contact interactions of all orders n ∈ N. The terms with g ≥ 1 are associated
to quantum corrections as they indicate the power of ¯
h, which means that they can
be interpreted as counter-terms. But, how is it that one needs counter-terms despite
the claim that every Feynman graphs (including the fundamental vertices) in SFT
are finite? The role of renormalization is not only to cure UV divergences, but also
IR divergences (due to vacuum shift and mass renormalization). Equivalently, this
can be understood by the necessity to correct the asymptotic states of the theory,
or to consider renormalized instead of bare quantities. Indeed, the asymptotic states
obtained from the linearized classical equations of motion are idealization: turning
on interactions modifies the states. In typical QFTs, these corrections are infinite,
and renormalization is crucial to extract a number; however, even if the effect is
finite, it is needed to describe correctly the physical quantities [16, p. 411]. There is
a second reason for these additional terms: when relaxing the gauge fixing condition,
the path integral is anomalous under the gauge symmetry, and the terms with
g > 0 are necessary to cancel the anomaly (this will be discussed more precisely in
Sect. 15.4). It may thus seem that SFT cannot be predictive because of the infinite
number of counter-terms. Fortunately, this is not the case: the main reason for
the loss of predictability in non-renormalizable theory is that the renormalization
procedure introduces an infinite number 2 of arbitrary parameters (and thus making
a prediction would require to have already made an infinite number of observations
to determine all the parameters). These parameters come from the subtraction of
two infinities: there is no unique way to perform it, and thus one needs to introduce
a new parameter. The case of SFT is different: since every quantity is finite, the
renormalization has no ambiguity because one subtracts two finite numbers, and the
result is unambiguous. As a consequence, renormalization does not introduce any
new parameter, and there is a unique coupling constant g s in the theory, which is
determined by the tree-level cubic interaction. The coupling constants of higherorder and higher-loop interactions are all determined by powers of g s , and thus a
unique measurement is sufficient to make predictions.
Another important point is that the action (15.24) is not uniquely defined. The
definition of the vertices depends on the choice of the local coordinates and of the
stub parameter s 0 . Changing them modifies the vertices, and thus the action. But,
one can show that the different theories are related by field redefinitions and are
thus equivalent.
2 In practice, this number does not need to be infinite to wreck predictability, and it is sufficient that
it is very large.
317
The action (15.24) looks overly more complicated than a typical QFT theory:
instead of few interaction terms for low n (n ≤ 4 in d = 4 renormalizable theories),
it has contact interactions of all orders n ∈ N. The terms with g ≥ 1 are associated
to quantum corrections as they indicate the power of ¯
h, which means that they can
be interpreted as counter-terms. But, how is it that one needs counter-terms despite
the claim that every Feynman graphs (including the fundamental vertices) in SFT
are finite? The role of renormalization is not only to cure UV divergences, but also
IR divergences (due to vacuum shift and mass renormalization). Equivalently, this
can be understood by the necessity to correct the asymptotic states of the theory,
or to consider renormalized instead of bare quantities. Indeed, the asymptotic states
obtained from the linearized classical equations of motion are idealization: turning
on interactions modifies the states. In typical QFTs, these corrections are infinite,
and renormalization is crucial to extract a number; however, even if the effect is
finite, it is needed to describe correctly the physical quantities [16, p. 411]. There is
a second reason for these additional terms: when relaxing the gauge fixing condition,
the path integral is anomalous under the gauge symmetry, and the terms with
g > 0 are necessary to cancel the anomaly (this will be discussed more precisely in
Sect. 15.4). It may thus seem that SFT cannot be predictive because of the infinite
number of counter-terms. Fortunately, this is not the case: the main reason for
the loss of predictability in non-renormalizable theory is that the renormalization
procedure introduces an infinite number 2 of arbitrary parameters (and thus making
a prediction would require to have already made an infinite number of observations
to determine all the parameters). These parameters come from the subtraction of
two infinities: there is no unique way to perform it, and thus one needs to introduce
a new parameter. The case of SFT is different: since every quantity is finite, the
renormalization has no ambiguity because one subtracts two finite numbers, and the
result is unambiguous. As a consequence, renormalization does not introduce any
new parameter, and there is a unique coupling constant g s in the theory, which is
determined by the tree-level cubic interaction. The coupling constants of higherorder and higher-loop interactions are all determined by powers of g s , and thus a
unique measurement is sufficient to make predictions.
Another important point is that the action (15.24) is not uniquely defined. The
definition of the vertices depends on the choice of the local coordinates and of the
stub parameter s 0 . Changing them modifies the vertices, and thus the action. But,
one can show that the different theories are related by field redefinitions and are
thus equivalent.
2 In practice, this number does not need to be infinite to wreck predictability, and it is sufficient that
it is very large.