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Gauge invariance in Regge/String Theory and its observations
Studying the scattering amplitude of gravitational waves from binary black holes, we usually express it in terms of Fourier modes of the spatial metric perturbations $h_{\mu
u}$. By imposing gauge conditions, which are most commonly the one studied in the Regge/string theory literature, we are left with physical observables (e.g. a complex number from a damped complex plane wave) which are independent of gauge choices. So, what is the origin of gauge invariance? And what exactly is gauge invariance in Regge/string theory?
I ask this because currently gauge invariance is usually taken for granted. E.g., in the Regge/string theory literature, gauge fields are introduced into the formalism from the outset.
A:
The answer is contained in your “currently”, in the sense that it is taken for granted that Regge/string theory is a gauge theory.
It is not, and it has always been not.
Gauge fields are a standard addition to most non-gravitational theories. They play a role as the tool for constructing the theory. Gauge fields, as it was understood, for example in QFT, play a role as “convenient ghosts” for handling the enormous number of quantum states that build up the theory.
As in QFT one may explain the excitations of gauge fields as “emergent” particle excitations, just as one can do for the gravitons, one has a theory of “emergent gravity”. The matter fields, just as in non-gravitational theories are non-emergent. That is they are not fundamental.
In any case it is irrelevant whether one starts with a gauge theory or with a “stringy theory”, as the same formalism (matter plus gauge fields plus strings) is always the result.
The fact that one may chose to define a gauge theory as having a vanishing string coupling constant and a vanishing dilaton, as done in the context of the weak coupling limit of string theory (i.e. perturbative string theory), does not make a difference.
In non-gravitational theory one may select the most convenient Lagrangian,
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