What does one mean with space-time as covering space?
The central idea is that space-time corresponds to n-fold covering for heff=n× h0. It is not however quite clear what this statement does mean.
- How the many-sheeted space-time corresponds to the space-time of QFT and GRT? QFT-GRT limit of TGD is defined by identifying the gauge potentials as sums of induced gauge potentials over the space-time sheets. Magnetic field is sum over its values for different space-time sheets. For single sheet the field would be extremely small in the present case as will be found.
- A central notion associated with the hierarchy of effective Planck constants heff/h0=n giving as a special case ℏgr= GMm/v0 assigned to the flux tubes mediating gravitational interactions. The most general view is that the space-time itself can be regarded as n-sheeted covering space. A more restricted view is that space-time surface can be regarded as n-sheeted covering of M4. But why not n-sheeted covering of CP2? And why not having n=n1× n2 such that one has n1-sheeted covering of CP2 and n2-sheeted covering of M4 as I indeed proposed for more than decade ago but gave up this notion later and consider only coverings of M4? There is indeed nothing preventing the more general coverings.
- n=n1× n2 covering can be illustrated for an electric engineer by considering a coil in very thin 3 dimensional slab having thickness L. The small vertical direction would serve and as analog of CP2. The remaining 2 large dimensions would serve as analog for M4. One could try to construct a coil with n loops in the vertical direction direction but for very large n one would encounter problems since loops would overlap because the thickness of the wire would be larger than available room L/n. There would be some maximum value of n, call it nmax.
One could overcome this limit by using the decomposition n=n1× n2 existing if n is prime. In this case one could decompose the coil into n1 parallel coils in plane having n2≥ nmax loops in the vertical direction. This provided n2 is small enough to avoid problems due to finite thickness of the coil. For n prime this does not work but one can of also select n2 to be maximal and allow the last coil to have less than n2 loops.
An interesting possibility is that that preferred extremal property implies the decomposition ngr=n1× n2 with nearly maximal value of n2, which can vary in some limits. Of course, one of the n2-coverings of M4 could be in-complete in the case that ngr is prime or not divisible by nearly maximal value of n2. We do not live in ideal Universe, and one can even imagine that the copies of M4 covering are not exact copies but that n2 can vary.
- In the case of M4× CP2 space-time sheet would replace single loop of the coil, and the procedure would be very similar. A highly interesting question is whether preferred extremal property favours the option in which one has as analog of n1 coils n1 full copies of n2-fold coverings of M4 at different positions in M4 and thus defining an n1 covering of CP2 in M4 direction. These positions of copies need not be close to each other but one could still have quantum coherence and this would be essential in TGD inspired quantum biology.
Number theoretic vision suggests that the sheets could be related by discrete isometries of CP2 possibly representing the action of Galois group of the extension of rationals defining the adele and since the group is finite sub-group of CP2, the number of sheets would be finite.
The finite sub-groups of SU(3) are analogous to the finite sub-groups of SU(2) and if they action is genuinely 3-D they correspond to the symmetries of Platonic solids (tetrahedron,cube,octahedron, icosahedron, dodecahedron). Otherwise one obtains symmetries of polygons and the order of group can be arbitrary large. Similar phenomenon is expected now. In fact the values of n2 could be quantized in terms of dimensions of discrete coset spaces associated with discrete sub-groups of SU(3). This would give rise to a large variation of n2 and could perhaps explain the large variation of G identified as G= R2(CP2)/n2 suggested by the fountain effect of superfluidity.
- There are indeed two kinds of values of n: the small values n=hem/h0=nem assigned with flux tubes mediating em interaction and appearing already in condensed matter physics and large values n=hgr/h0=ngr associated with gravitational flux tubes. The small values of n would be naturally associated with coverings of CP2. The large values ngr=n1× n2 would correspond n1-fold coverings of CP2 consisting of complete n2-fold coverings of M4. Note that in this picture one can formally define constants ℏ(M4)= n1ℏ0 and ℏ(CP2)= n2ℏ0 as proposed for more than decade ago.
There is also a puzzle related to the identification of gravitational Planck constant. In TGD framework the only theoretically reasonable identification of Planck length is as CP2 length R(CP2), which is roughly 103.5 times longer than Planck length. Otherwise one must introduce the usual Planck length as separate fundamental length. The proposal was that gravitational constant would be defined as G =R2(CP2)/ℏgr, ℏgr≈ 107ℏ. The G indeed varies in un-expectedly wide limits and the fountain effect of superfluidity suggests that the variation can be surprisingly large.
There are however problems.
- Arbitrary small values of G=R2(CP2)/ℏgr are possible for the values of ℏgr appearing in the applications: the values of order ngr ∼ 1013 are encountered in the biological applications. The value range of G is however experimentally rather limited. Something clearly goes wrong with the proposed formula.
- Schwartschild radius rS= 2GM = 2R2(CP2)M/ℏgr would decrease with ℏgr. One would expect just the opposite since fundamental quantal length scales should scale like ℏgr.
- What about Nottale formula ℏgr= GMm/v0? Should one require self-consistency and substitute G= R2(CP2)/ℏgr to it to obtain ℏgr=(R2(CP2)Mm/v0)1/2. This formula leads to physically un-acceptable predictions, and I have used in all applications G=GN corresponding to ngr∼ 107 as the ratio of squares of CP2 length and ordinary Planck length.
If n2 corresponds to the order of finite subgroup G of SU(3) or number of elements in a coset space G/H of G (itself sub-group for normal sub-group H), one would have very limited number of values of n2, and it might be possible to understand the fountain effect of superfluidity from the symmetries of CP2, which would take a role similar to the symmetries associated with Platonic solids. In fact, the smaller value of G in fountain effect would suggest that n2 in this case is larger than for GN so that n2 for GN would not be maximal.
For a summary of earlier postings see Latest progress in TGD.