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Are all metric theories non-renormalizable?

As a hand-wavy argument, it seems like any theory where spacetime geometry itself is an "active" player would run into similar problems when trying to quantize the theory to give a quantum description of spacetime.

So my question is. QED is renormalizable, correct? Then what of Special Relativity (not GR) + Maxwell's equations included as a fifth dimension (using Kaluza-Klein theory). Now we have a metric theory of electromagnetism (with gravity ignored at the moment). Classically this should be equivalent to Maxwell's equations in regards to predictions for experiments, yes? But when trying to convert it to a quantum theory, is it suddenly no longer renormalizable?

If so, what exactly does that mean? Is there something important to learn here from this?

If not, what is special about the metric theory of electromagnetism that allows it to be renomalizable?

As a hand-wavy argument, it seems like any theory where spacetime geometry itself is an "active" player would run into similar problems when trying to quantize the theory to give a quantum description of spacetime.

So my question is. QED is renormalizable, correct? Then what of Special Relativity (not GR) + Maxwell's equations included as a fifth dimension (using Kaluza-Klein theory). Now we have a metric theory of electromagnetism (with gravity ignored at the moment). Classically this should be equivalent to Maxwell's equations in regards to predictions for experiments, yes? But when trying to convert it to a quantum theory, is it suddenly no longer renormalizable?

If so, what exactly does that mean? Is there something important to learn here from this?

If not, what is special about the metric theory of electromagnetism that allows it to be renomalizable?

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