Curving the Space by Non-Hermiticity

> What are the implications on quantum gravity

On the one hand, none. The paper at the top may be a path towards a quantum theory of gravity corresponding to a classical theory with non-vanishing torsion.

Torsion is a gravitational field that causes vacuum observers to rotate in otherwise flat spacetime. The duality proposed in this paper might allow for a study of unusual quantum systems as a more-straightfoward quantum system on a flat spacetime with torsion.

Unfortunately nobody has written down a physically plausible theory with non-vanishing vacuum torsion. In most published torsion-containing gravity theories, the rotation depends on a coupling between the intrinsic spin of the elementary particles of (low-mass) observers and (high-mass) sources -- there is no torsion on low-mass objects from the vacuum, or sufficiently far from large masses, or if spin is "washed out" by large particle counts. When writing down a Hamiltonian formulation for a theory of gravitation with this form of torsion, one can find that the rotation breaks an important symmetry (see last paragraph below) for matter particles with intrinsic spin. The opposite direction of the duality in the paper might be helpful by making it easier to study a system with this type of (non-vacuum) torsion as a torsion-free system with additional non-Hermitian matter fields instead.

However, there is no torsion of this second type in gravitation that we know about in our universe. Put simply, the Universality of Free Fall in our cosmos means that electron neutrinos and W bosons fall in the same way. If there were torsion, they would fall differently, and we would see that in weak interactions like nuclear decays.

This scales upwards: we would expect dense objects like white dwarfs and neutron stars to have different spectra if they were sources of torsion on the subatomic particles they spit out.

We have done some limited direct testing of gravitational torsion in our solar system involving highly sensitive accelerometers, notably the gyroscopes of Gravity Probe B. There is no evidence for torsion from any of these tests, so we can discount torsion-containing theories wherein the Earth is a source. To go from discounting to ruling out, we would really want to test with a microstructured accelerometer: one incorporating well-understood radioactive material, and sensitive to it (put sloppily, it might show a small anomalous acceleration after each decay), or one without any nuclear spin at all.

Meanwhile (and perhaps even after such a test rules out torsion near Earth) one might still be able to develop a model in which there is torsion at the very largest scales in the universe, with the goal of replacing the Big Bang and its infinitely dense gravitational singularity with a cyclical bouncing cosmology. When we run what we know of our expanding universe backwards, we expect everything to collapse, heat, densify, and crunch together into a volume where the curvature scalars diverge. With a careful choice of torsion tensor and a framework in which the torsion tensor's nonzeros are useful (this usually takes one away from General Relativity to an alternative known as Einstein-Cartan-Sciama-Kibble), starting "today" and running backwards we end up collapsing, heating, and densifying, but at some point torsion becomes strong enough that it counteracts the influence of gravitation on sufficnetly dense matter and "bounces" the hot material away from ultimate collapse. The goal then would be to be to look for evidence of strong torsion in our cosmic history, and so far nobody's found any evidence for it (even in data that ought to reveal torsion if it's there).

It then becomes something of a chore to deal with torsion in extremely dense matter like in the heaviest neutron stars. Apart from spectral issues, there appear to be black holes in our universe, but those are hard to form if there is any torsion (to banish the big bang singularity, or to banish cosmic inflation) because torsion would push apart the densest pre-black-hole matter arrangements in ways that we would notice (but don't see when we look). One could alternatively deploy an escape hatch wherein dynamical torsion is strong in the very early universe, but fades away practically entirely very early, notably before the formation of photons, electrons/muons/taus, their respective neutrinos, and all of the above's antiparticles. This is in effect designing a species of cosmic inflation.

So, as the key generator of non-Hermiticity in theories of gravity is not needed classically, it is not expected to be needed in a quantum theory of gravity either.

On the other hand, maybe not strictly none.

A popular and reasonable approach to quantum gravity starts with a Hamiltonian formulation and then proceeds through to a canonical quantization. This is the approach followed by Perturbative Quantum Gravity, which treats gravitation as a background field with Hermitian operators such as the metric tensor and the field strength tensor. If one wanted to add a tensor field, for example while pursuing a nonsymmetric theory of gravity (usually represented as an additional massive antisymmetric tensor field), one might add it as a set of non-Hermitian operators, and there this paper might be mathematically interesting.

I don't think the theory is likely to be physically interesting, because our universe does not need torsion or an additional metric field to describe it. In the 90s and early 2000s, motivated by abolishing cosmic inflation and/or dark matter, such theories were explored and the problem has turned out to be that you can't hide the entirety of inflation or the entirety of dark matter using these types modifications to gravity alone, and as a consequence you have to hide even more fields (turning them off entirely early in the universe's history, before the formation of the cosmic microwave background). Asymmetrical and torsion-containing theories of gravity were also popular with high-dimension supersymmetry, which became less attractive after the Large Hadron Collider results for the Higgs boson in 2011-2013 or thereabouts killed off numerous families of supersymmetry models.