Your argument that gravitation is the square of electromagnetism has some resonance to it with my work. It is not difficult to quantize weak gravity. This is usually written as a bimetric theory g_{ab} = η_{ab} + h_{ab}, where η_{ab} is a flat spacetime (Minkowski) metric and h_{ab} is a perturbation on to of flat spacetime. Gravitons enter in if you write the perturbing metric term as h_{ab} = φ_aφ_b, or ω_a^c = φ_aφ^c. The Ricci curvature in this weak field approximation is
R_{ab} - (1/2)Tg_{ab} = □h^t_{ab},
with h^t_{ab} the traceless part of the metric, and □ the d'Alembertian operator. Which in a sourceless region this computes plane waves. The two polarization directions of the graviton may then be interpreted as a form of diphoton, or two photons in an entanglement or a "bunching" as in Hanbury Brown-Twiss quantum optical physics.
The gluon and graviton have some sort of curious relationship as well. From a stringy perspective the operator (a^†n)^α generates a string vibration mode of number = n. The index α refers to spacetime in some dimension D. This index is positive so there is a preferred direction around the string, which violates the Noether theorem. So physical states must be products (a^†n)^α(a^†-n)^β, where the total mode generation is zero, meaning there is no preferred direction around the string. The mode generated pertains to one transverse direction. For D = (1, d) there are d-1 possible directions of polarization, where the remaining direction is the longitudinal direction. The indices at the beginning of the Greek alphabet α, β are composed of an index μ for the uncompactified spacetime directions plus the Latin index j for compactified directions. In uncompactified spacetime there are then two directions of polarization, so we have raising operators (a^†n)^α and (b^†n)^α for the two directions, and a graviton in D dimensions is composed of these, such as
(a^†n)^α(a^†n)^β,
(a^†n)^α(b^†n)^β
and so forth, where each (a^†n)^α has a spin of s = 1, so the composite is spin s = 2. When restricted to spacetime we have various possible gravitons, using only the (a^†n)^α(a^†n)^β composition
(a^†n)^μ(a^†n)^ν
(a^†n)^μ(a^†n)^j
(a^†n)^j(a^†n)^k
The first of these is a graviton in spacetime corresponding to a g^{μν}. The second of these corresponds to a g^{μj}, which is a metric that extends into the compactified dimension. This from our perspective is a quantum field with one external index and an internal index corresponding to a gauge space. This is a gauge vector boson. The final term is a scaling factor and can be thought of as tracing over the spacetime dimension and corresponds to a scalar field. This in different forms is the dilaton and the axion.
Now suppose we have a gauge field (a^†n)^μ(a^†n)^j, which we will call a QCD gluon. Let us compose two gluons, so we think of a product
(a^†n)^μ(a^†n)^j(a^†n)^k(a^†n)^ν = (a^†n)^μ[(a^†n)^j(a^†n)^k](a^†n)^ν,
where we the portion in the square brackets are an axion or dilaton field. We factor that out and we see that this is a graviton times a scalar field. So the graviton sector can be seen as the "square" of the gluon sector. There is a further bit I have brushed over, which is that general graviton states involve circular polarized states with the (b^†n)^β operator in superpositions. The graviton can be seen as a quantum entanglement state of gluons. Whether the (a^†n)^j(a^†n)^k type of states are axions or dilatons depends on the entanglement --- a complication I will just refer to here.
My essay concerns this connection between gauge field or QCD and gravity.
Cheers LC
