Since Quantum Field Theory can't handle gravity, and gravity is mathematically equivalent to acceleration (equivalence principle), does this mean Quantum Field theory can't handle accelerated frames of reference?
No, quantum field theory is perfectly capable of handling accelerated frames of reference:
- Quantum Field Theory is based on special relativity. Contrary to some somewhat widespread belief, special relativity is perfectly capable of handling accelerated frames. As long as the spacetime is flat, special relativity is perfectly fine. And the curvature of spacetime doesn't depend on the frame of reference (it is a tensor, a geometric object). This is why you will often hear that the twin paradox can be solved with special relativity alone. The twin on the rocket is on an accelerated frame. But as long as they don't pass near a black hole or something like that, spacetime is flat and special relativity applies.
I understand your confusion, gravity is locally equivalent to acceleration. "Locally" is the important word. This means that in the presence of a gravitational field, for every point of spacetime, you can always choose a frame of reference where in the neighborhood of that point, you are in free fall (i.e. spacetime looks flat, the frame of reference is inertial). But there is no frame of reference such that spacetime looks flat everywhere. The crucial difference between being in an accelerating frame in flat spacetime versus being in a gravitational field is that in the former case there is a global inertial reference frame, where spacetime looks flat everywhere, not just in the neighborhood of some point.
So the difference between special and general relativity isn't that one treats acceleration and the other doesn't, but that one treats flat spacetime and the other treats curved spacetime.
- Quantum field theory is also capable of handling curved spacetime. Some things must be modified to make it work, but there aren't great issues as long as the spacetime is treated classically. A quantum field in a curved (static or time dependent) classical spacetime works well.
Problems arise when you try to quantize gravity, i.e. treat it as a quantum field. A quantum field theory of gravity. That's what we have not been able to do.
Gravity is not "mathematically equivalent to acceleration" - otherwise how could things be accelerated in theories without gravity? Physical principles like the different versions of the equivalence principle need to be stated carefully precisely to avoid issues like this that ultimately arise from sloppy phrasing.
When people say that quantum field theory cannot "handle gravity", they mean that a field theory with the spacetime metric as a dynamical field - as in the Einstein-Hilbert action of general relativity - is not renormalizable. See this question and its answers for more on the incompatibility between general relativity and quantum field theory.
Although it does not have renormalizable theories of gravity, quantum field theory does have specific predictions about what happens to accelerated observers, namely they will see radiation generated through the Unruh effect.