Horizontal Fuzziness

The difference between classical and quantum physics is fuzziness. In classical mechanics all quantities can be measured to arbitrary precision, but observables in quantum mechanics are subject to quantum fluctuations. There is a fundamental limit to the precision with which certain pairs of physical properties, such as position and momentum, can be simultaneously known; this is Heisenberg's uncertainty principle. Hence measurements in quantum mechanics are fuzzy.

Our most accurate theories are theories of fields, i.e. systems which depend on space as well as time. Electromagnetism, weak and strong interactions are all described by Quantum Field Theory (QFT), i.e. quantum theory applied to fields. In QFT we have the same kind of fuzziness as in quantum mechanics. The result of an experiment is fuzzy, but the location of the experiment is not. This is unphysical in my opinion. The experiment takes place inside a detector, and in order to know where the detector is located, we must measure its position, e.g. with rulers or a GPS receiver. This measurement is a separate physical experiment and as such subject to quantum fluctuations. Hence we need to replace QFT with a more general theory which takes into account that the detector's location is fuzzy.

A fundamental theory must hence have both vertical fuzziness (the result of an experiment is fuzzy) and horizontal fuzziness (the location of the experiment is also fuzzy). We expect that QFT is recovered in the limit that horizontal fuzziness can be ignored. This is the case when the detector is a classical object whose position at all times can be known without fuzziness. This amounts to a hidden assumption that the detector's inert mass is infinite.

Now it is easy to see why QFT is incompatible with gravity. The equivalence principle asserts that inert mass equals heavy mass. However, an infinitely heavy detector will interact with the gravitational field and immediately collapse into a black hole, which sucks up the rest of the universe. This is not a very good description of most experiments. 

Quantum mechanics reached its final formulation a century ago, and people attempted to apply it to gravity soon thereafter. There has been essentially no progress on this problem over the past century, and the reason is now clear. By ignoring horizontal fuzziness, a hidden assumption about an infinitely massive detector is introduced, and gravity does not work well in the presence of infinitely massive objects. There have also been other well-advertised approaches to a quantum theory of gravity. It is not clear to me to what extent they are related to field theory, but it does not really matter. Absence of horizontal fuzziness amounts to the same hidden assumption about an infinitely massive detector, and hence such theories are also incompatible with gravity.

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