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Axel Maas has coauthored a series of papers applying one such formalism (the gauge-invariant perturbation theory of Fröhlich, Morchio, and Strocchi) to electroweak theory, here is a summary. A central issue is how to understand electroweak symmetry breaking, something normally understood as a result of the nonzero Higgs vev, given the fact that there are gauges in which the vev is zero. Maas says the FMS picture provides an answer to this problem.

Now he has moved on to applying the FMS framework to quantum gravity:

I can't vouch for this work's correctness, but here are some questions that suggest themselves.Axel Maas

(Submitted on 6 Aug 2019)

Taking manifest invariance under both gauge symmetry and diffeomorphisms as a guiding principle physical objects are constructed for Yang-Mills-Higgs theory coupled to quantum gravity. These objects are entirely classified by quantum numbers defined in the tangent space. Applying the Fröhlich-Morchio-Strocchi mechanism to these objects reveals that they coincide with ordinary correlation functions in quantum-field theory, if quantum fluctuations of gravity and curvature become small. Taking these descriptions literally exhibits how quantum gravity fields need to dress quantum fields to create physical objects, i.e. giving a graviton component to ordinary observed particles. The same mechanism provides access to the physical spectrum of pure gravitational degrees of freedom.

1. Despite the difficulties of quantum gravity, perturbative quantum gravity as an effective field theory at low energies has been known for some time. Has the gravitational dressing of states in that theory been addressed before?

2. Is there any relationship between FMS gravitational dressing, and the recent work of Strominger et al on BMS transformations which add "soft gravitons" to asymptotic particle states?

3. Can this be implemented in string theory? If so, how does it look? Reproducing known QFT phenomena within string theory is a way that both fields have progressed.