Sunday, May 02, 2021

The notion of holography in TGD framework

Thanks to Bob Tang for the link to Sabine Hossenfelder's article about holography. I will not comment about the content of the link but about TGD view of holography.

What "Universe as a hologram" does really mean must be first defined. In pop physics this notion has remained very loose. In the following I summarize the TGD based view about what holography means in the geometric sense.

In TGD 3-D surfaces are basic objects and replace 3-space. Holography is not a new principle but reduces to general coordinate invariance.

1. "Ordinary" holography

General coordinate invariance in 4-D sense requires that they correspond to single 4-D surface-space-time - at which general coordinate transformations act. Space-time surface is like Bohr orbit, preferred extremal for the action defining the space-time surface.

This is nothing but holography in standard sense and leads to zero energy ontology (ZEO) meaning that quantum states are superpositions of 3-surfaces or equivalently, of 4-D surfaces.

[The alternative to ZEO would be path integral approach, which is mathematically ill-defined and makes no sense in TGD framework due to horrible divergence difficulties.]

ZEO has profound implications for quantum theory itself and solves the measurement problem and also implies that the arrow of time changes in "big" (ordinary) state function reductions as opposed to "small" SFRs ("weak" measurements). Also the question at which length scale quantum behavior transforms to classical, becomes obsolete.

2. Strong form of holography (SH)

Besides space-time-like 3-surfaces at the boundaries of causal diamond CD serving as ends of space-time surface (initial value problem) there are light-like surfaces at which the signatures of the metric changes from Minkowskian to Euclidian (boundary value problem). Euclidian regions correspond to fundamental particles from which elementary particles are made of.

If either space-like or light-like 3-surfaces are assumed to be enough as data for holography (initial value problem is equivalent with boundary value problem), the conclusion is that their interactions as partonic 2-surfaces are enough as data. This would give rise to a strong form of holography, SH.

Intuitive arguments suggest several alternative mathematical realizations for the holography for space-time surfaces in H=M4×CP2. They should be equivalent.

  1. Space-time surfaces are extremals of both volume action (minimal surfaces) having interpretation in terms of length scale dependent cosmological constant and of Kähler action. This double extremal property reduces the conditions to purely algebraic ones with no dependence on coupling parameters. This corresponds to the universality of quantum critical dynamics. Space-time surfaces are analogs of complex sub-manifolds of complex imbedding space.
  2. Second realization is in terms of analogs of Kac-Moody and Virasoro gauge conditions for a sub-algebra of super-symplectic algebra (SSA) isomorphic with the entire SSA and acting as isometries of the "world of classical worlds" (WCW). SSA has non-negative conformal weights and generalizes Kac-Moody algebras in that there are two variables instead of single complex z coordinate: the complex coordinate z of sphere S2(to which light-one boundary reduces metrically) and light-like radial coordinate r of the light-cone boundary. Also the Kac-Moody type algebra assignable to isometries of H at light-like partonic orbits involve the counterparts of z and r. A huge generalization of symmetries of string theory is in question.

3. Number theoretic holography

M8-H duality leads to number theoretic holography, which is even more powerful than SH.

  1. In complexified M8 - complexified octonions - space-time surface would be "roots" of octonionic polynomials guaranteeing that the normal space of space-time surface is associative/quaternionic. Associativity in this sense would fix the dynamics. These surfaces would be algebraic whereas at the level of H surfaces satisfy partial differential equations reducing to algebraic equations due to the complex surface analogy.
  2. M8 would be analogous to momentum space and space-time surface in it analogous to Fermi ball. M8-H duality would generalize the q-p duality of wave mechanics having no generalization in quantum field theories and string models.
  3. M8-H duality would map the 4-surfaces in M8 to H=M4× CP2. Given region of space-time surface would be determined by the coefficients of a rational polynomial. Number theoretic holography would reduce the data to a finite number rational numbers - or n points of the space-time region (n is the degree of the polynomial). The polynomials would give rise to an evolutionary hierarchy with n as a measure for complexity and having interpretations in terms of effective Planck constant heff/h0=n.
For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD. 

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