Laundry pile phenomenon and various dimensions
Sabine Hossenfelder talks about various notions of dimension in her posting What if the universe was like a pile of laundry?. SH introduces ordinary topological dimension, Hausdorff dimension and spectral dimension.
- Sabine claims that the Hausdorff dimension of 3-D pipe is lower than ordinary dimension. This is definitely not the case (see Wikipedia article). One needs very irregular sets, which are not smooth or not even continuous - typically fractals. Sabine seems to confuse spectral dimension of pipe with Haussdorff dimension. For the pipe one could think that spectral dimension could be lower than 3 since at the limit of very thin pipe one obtains line with dimension 1 and if the sound waves do not depend on the transversal coordinates of pipe they effectively live in 1-D line.
- Sabine states that one speak about spectral dimension in dimensional reduction. This makes sense. The intuitive understanding of this in case of field theories dimensionally reduced to 4-D space-time is that spectral dimension is just 4 if the field modes involved depend on the coordinates of base space only. Otherwise the spectral dimension is higher than 4. One could imagine defining spectral dimension as quantum/thermal average for the number of coordinates that the modes appearing in the quantum/thermal state depend on.
- Sabine also describes what might be called pile of laundry metaphor. Lower-dimensional surface in high-dimensional space effectively fills it. This is a very good metaphor but I am not sure whether it describes what would happen in loop quantum gravity, where one has just 4-D discretized spacetime degenerating locally to 2-D one - possibly by the failure of numerics or of the model itself.
Pile of laundry phenomenon would require 2-D clothes in 4-D space filling it: can one really have a situation in which 2-D orbits of loops effectively fill the space-time? To my best intuitive understanding "laundry" dimension is something different from Hausdorff and spectral dimensions.
In any case, the "laundry" dimension would be defined by what happens when one starts from point of cloth and is allowed to move in the higher-D space rather than only along cloth, where everything is just smooth and continuous. This typically leads to another fold and one is lost to the folds of the laundry pile. Perhaps one should speak just about "laundry" dimension. I am not sure.
- In TGD the twistorial reduction from 6-D twistor space to 4-D space-time defining its base the old-fashioned dimensional reduction could happen and spectral dimension would be 4 since no excitations are involved: twistorial description would be just an alternative description. It seems however that it brings in both Planck length via the radius of the sphere defining the fiber of twistor space and cosmological constant defined the coefficient of a volume term added to Kähler action in dimensional reduction.
Together with CP2 radius analogous to GUT unification scale one would have 3 fundamental scales. Cosmological constant would be dynamical and in recent cosmology it would correspond to neutrino Compton length, which is of order of cell size scale so that biology and cosmology would meet!
- Strong form of holography (SH) states that the information about space-time dynamics (and quantum dynamics in general) is coded by string world sheets and partonic 2-surfaces (or possibly by their metrically 2-D light-like orbits, I must be careful here). Although 4 goes to 2 this is very different to what is claimed to occur in loop gravity numerics. The reduction is at the level of information theory: 2-surfaces define "space-time genes".
- The reduction of metric dimensionality for light-like orbits of partonic 2-surfaces from 3 to 2 is however analogous to what is claimed to occur in loop gravity numerics. The point is that 4-D tangent space at points of the light-like 3-surface reduces locally to 3-D since the determinant of 4-D metric vanishes due the fact that the signature changes from Euclidian (-1,-1,-1,-1) to Minkowskian (1,-1,-1,-1) and is (0,-1,-1,-1) at the 3-surface.
- Laundry pile phenomenon applies to higher dimensional clothes drying in higher-D space-time. This could happen for the 3-space (4-D space-time) represented as surface in M4× CP2 in TGD - clothes would be space-time sheets drying in 8-D imedding space;-). In this case the "laundry" dimension would become higher than 3 (4) effectively.
One could imagine that the 3-D light-like orbits of 2-D partonic surfaces effectively fill the space-time surface. In similar manner, their 2-D partonic ends at the light-like boundaries of causal diamond (CD) could fill the 3-D ends of the space-time at the ends of CD. The 4-D regions with Euclidian signature of induced metric and analogous to lines of Feynman diagrams would almost totally fill the space-time surface leaving only very tiny volume of Minkowskian signature, where signals would progate. A very dense and also extremely complex and information rich phase would be in question.
This kind of net like Feynman diagrams were assigned to non-perturbative phase by t'Hooft in gauge theories and this led to the notion of holography.
- Could laundry pile phenomenon apply to the many-sheeted space-time? QFT limit of TGD is defined by replacing the sheets of with single slightly curve region of M4 and induced gauge potentials with the sum of induced gauge potentials from different space-time sheets to define standard model gauge potentials. Gravitational field as deviation of metric from flat M4 metric is defined in the same manner. Could it make sense to assign a "laundry dimension" larger than 4 to the resulting counterpart of GRT space-time and could one make the QFT-GRT limit free of divergences by using dimensional regularization using the "laundry dimension"?
- One can add to the list of dimensions also algebraic dimension, which is important in TGD: the dimension of algebraic extension of rationals becomes a measure for the level of number theoretic evolution and complexity.
For a summary of earlier postings see Latest progress in TGD.