Comparison of AdS/CFT duality with the TGD based holographic duality

This posting is a continuation to an earlier posting Are wormholes really created in quantum computer? (see this).

In the experiment considered, the so-called SYK model (Sachev-Ye-Kitaev) was simulated using a quantum computer. The model is constructed to realize AdS₂/CFT correspondence and the quantum computer simulates the 1-D quantum system dual to wormhole in 2-D AdS₂.

The problem is that the AdS₂ is completely fictitious so that the physics at this side cannot be tested. However, TGD also predicts holographic duality between 3-D surfaces as boundaries of space-time surfaces and identifiable as outer boundaries of physical objects and the interior of the space-time takes the role of AdS. In particular, the system considered in the experiment should allow TGD based dual description.

The thesis Holographic quantum matter : toy models and physical platforms (see this) of Etienne Lantagne-Hurtubise gives a nice description of the SYK model and the following comments are based on the introduction of the thesis.

TGD should give a classical description of quantum dynamics coded by 3-D data holographically in terms of classical physics in the interior of space-time surface. Therefore the challenge is to also describe the reported findings.

How do AdS/CFT holography and TGD holography relate to each other?

There are obvious questions to be answered. How closely AdS/CFT and TGD holography could relate and how do they differ? Could there exist some kind of AdS/CFT-TGD dictionary?

1. AdS/CFT correspondence predicts 4→ 5 holography for AdS₅ interpreted as an emergent 5-D space-time. M⁴ would carry gauge fields and theory would be conformally invariant. The gravitational holography is 3→ 4. Skeptics could argue that there is total mess: for instance, what happens to the general relativistic description of gauge fields using 4-D space time? Should one have 3→ 4 gravitational holography followed by 4→ 5 for gauge fields?
2. TGD predicts 3→4 holography. Instead of AdS one has space-time but is realized as a 4-D surface. Light-like 3-surfaces with extended conformal symmetries due to their metric 2-dimensionality defined boundaries of 4-D space-time surfaces and contain holographic data defining the space-time surface and also the data defining the fermionic part of quantum state.

   4-D general coordinate invariance implies almost exact holography and classical deterministic dynamics becomes an exact part of quantum TGD: one has what one might call Bohr orbitology. This picture has a number theoretic counterpart at the level of M⁸: associativity assigns 4-D surface of M⁸_c to the roots of rational polynomials represented as 3-D mass shells in M⁴_c ⊂ M⁸_c.

Do the time loops of AdS has time-like loops have a TGD counterpart?

AdS time loops have indeed TGD countepart. The reason is that 4-D space-times are completely exceptional.

1. 4-D, and only 4-D, space-times allow exotic smooth structures (see this)! A continuum of exotic smooth structures are possible. Exotic smooth structure can be always regarded as ordinary smooth structure apart from a discrete set of points.

   Exotics break cosmic censorship so that global hyperbolicity fails and the initial-value problem becomes ill-defined because of time-like loops. Time-like loops are a heavy counter argument against AdS/CFT duality. They are however encountered also for the TGD variant of the holographic duality.

   Could it be that time-like loops are not a nuisance but something fundamental forcing the space-time dimension to be D=4.

2. In the TGD framework holography predicts the smooth structure of the space-time surface so that the non-uniqueness is not a problem.

   The discrete set of points spoiling the standard smooth structure is an analogue for a set of point-like defects. Outsides this set the standard smooth structure fails. The proposal (see this) is that this set of points is assignable to particle reaction vertices in TGD and have a topological interpretation. Two partonic 2-surfaces with opposite homology charges (monopole fluxes) touch at defect point and fuse together to a single particle 2-surface.

3. This makes possible time loops which are essential for understanding pair creation in TGD. It is essential that the interiors of the orbits of wormhole contacts have an Euclidian signature: this is obviously a completely new element when compared to AdS/CFT. The boundaries between these Euclidian regions and Minkowskian regions of the space-time surface are light-like and correspond to the orbits of wormhole throats at opposite Minkowskian sheets (see this). The creation of a fermion pair would correspond to a change of the time direction of the fermion at the defect point of the exotic smooth structure.
4. Could exotic smooth structures make possible quantum computations as evolution forth-and-back in space-time in the TGD framework? Could the time loops serve as microscopic classical correlates for this and could the defects give a topological realization for what happens. Note that wormhole throats can in principle have large sizes and scale like h_eff and can be very large for gravitational Planck constant h_gr.
5. Could wormholes correspond in the TGD framework to magnetic flux tubes? Or could they correspond to light-like orbits of wormhole throats/partonic 2-surfaces appearing analogous to lines of topological counterparts of Feynman diagrams? Orbits of wormhole contacts identified as orbits of pairs of wormholes give rise to light-like orbits of wormhole throats, which are always paired. Fermionic quantum numbers are associated with the light-like lines of the wormhole throat. They represent building bricks of elementary particle orbits. Could these structures be seen as analogues of wormholes?

What is the TGD counterpart of time reversal of the SYK model?

Time reversal is central in the SYK model.

1. Time reversed of time evolution as unitary time evolution with Hamiltonian having opposite sign is central in the model. This notion is somewhat questionable since usually one requires that the energy eigenvalues are positive. In TGD, this time evolution could correspond to a sequence of SSFRs in the reversed time direction following BSFR.
2. Shock wave in the wormhole appears as a negative energy signal. This could correspond to time reversed classical signals having effectively negative energy and propagating along the flux tube or the counterpart of the wormhole in TGD. Time reversal would be induced by BSFR.
3. One could also interpret reversed time evolution as a generation of Hawking radiation. Negative energy particles falling to the blackhole would correspond to the time reversed signal propagating from right to left after BSFR has occurred in the experiment considered.

TGD counterparts of scrambling time evolution and descrambling as its time reversal

Scrambling means generation of quantum chaos. Descrambling does the opposite and is in conflict with the second law unless the arrow of time changes. For a unitary time evolution descrambling can be considered if the negative of Hamiltonian makes sense.

Scrambling corresponds to a random unitary time evolution inducing mixing as dispersion of entanglement in the entire system. Actually a sequence of scramblings characterized by scrambling times described by random Hamiltonians is assumed to take place. Scrambling time is assumed to depend on blackhole entropy S= A/4Gℏ= 4π GM²/ℏ roughly as

T_s= r_s ×O(S^1/2 log(S)) ,

where r_s = 2GM is Schwarzschild time. Blackholes are assumed to be very fast scramblers.

1. A sequence of "small" state function reductions (SSFRs) as the TGD counterparts of "weak" measurements of quantum optics, generalizes Zeno effect to a subjective time evolution of self. The sequence of SSFRs as analog of a sequence of unitary time evolutions

   Scrambling could correspond to a sequence of SSFRs as an analogue for a sequence of random unitary evolutions in TGD. Since one has a sequence of SSFRs, scrambling might correspond to the emergence of thermodynamics chaos.

   An alternative interpretation of chaos is an increase of complexity. Mandelbrot fractal is complex but not chaotic in the thermodynamic sense. Could scrambling correspond to an effective increase of the extension of rationals during the sequence of SSFRs? More and more roots of polynomials defining light-cone propertime a=constant hyperboloids become visible at the increasing space-time surface inside the CD. This option does not look plausible.

2. De-scrambling time evolution is in conflict with intuition. In TGD, de-scrambling could correspond to scrambling with an opposite arrow of time emerging in "big" SFR (BSFR) and therefore dissipation with a reverse arrow of time looking like self-organization for an observer with an opposite arrow of time. This process is fundamental in biology and would correspond to processes like healing. BSFR corresponds to a "death" or falling asleep in TGD inspired theory of consciousness and self lives forth and back in geometric time.
3. It is interesting to look for the TGD counterpart of scrambling time T_s. For hbar →, where ℏ_gr = GM²/β₀ is the gravitational Planck constant and β₀≤1 is a velocity parameter, one obtains

   T_s= r_s ×4π β₀log(β₀) .

   Scrambling time would be negative for β₀< 1: could the interpretation be that scrambling takes place with opposite arrow of time? Blackhole entropy is equal to β₀ and smaller than 1 and practically zero. One must of course take this expression for the scrambling time with a big grain of salt and as found in the previous posting, TGD allows us to consider a more general picture in which the correspondence with blackholes is not so concrete.

See the earlier posting Are wormholes really created in quantum computer? and the article How do AdS/CFT- and TGD based holographic dualities relate?.

For a summary of earlier postings see Latest progress in TGD.

Articles related to TGD.