The time has come to wrap up this series of posts. I went to grad school in the early 1980s, and specialized in phase transitions and critical phenomena, often working on 2D models. A few years later conformal field theory (CFT) came along, and essentially finished my field of research. I was too young and isolated to notice at the time, but when I did notice a few years later I felt that I had missed the bandwagon. CFT explains everything worth knowing about 2D critical phenomena, but it says nothing about the much harder and physically more relevant case of critical phenomena in 3D. So I started to think that something similar might work in higher dimensions as well. It can not be conformal symmetry, because the conformal group is infinite-dimensional only in 2D. But the same group also described diffeomorphisms in 1D, and that has a natural generalization to higher dimensions.
This made me decide that a multi-dimensional Virasoro algebra had to exist and I went out to find it. That it really does exist and has natural realizations on p-jets was very satisfying, but completing this task took a long time and I ran out of funding in the process. Moreover, it is only a limited success, because important pieces of the puzzle are still missing.
- There are operators acting on a vector space, but I haven't found a nice way to identify this space with its dual. So there is no known inner product and we can not discuss unitarity.
- The vector space is too big because momenta and velocities are not identified, which they should be.
- This is a purely kinematical theory, like tensor calculus. To use it in physics we should introduce dynamics in some form, with a Lagrangian, a Hamiltonian, or equations of motion. Doing this for p-jets is surprisingly difficult, or at least it was for me.
Because of these shortcomings I haven't been able to apply this theory to physics. Nevertheless, the math exists which made the effort worthwhile in itself, and I hope that somebody will make progress on the above problems one day.
But even if much remains to be done, some physics insights can still be extracted. The Virasoro extension depends on the observer's trajectory, and therefore we need a theory with an explicit physical observer. Since the observer's position is an operator, we must measure it to know its value, and that measurements is subject to quantum fluctuations. This is really enough to rule out every approach to quantum gravity which does not take horizontal fuzziness into account, because the lack of horizontal fuzziness amounts to a hidden assumption about an infinite mass.
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