It however turned out that it is not at all clear whether the boundary conditions stating that no isometry currents flow out of the boundary, can be satisfied. Therefore the cautious conclusion was that perhaps the boundaries are only apparent. For instance, the space-time regions correspond to maps M4 → CP2, which are many-valued and have as turning points, which have 3-D projections to M4. The boundary surfaces between regions with Minkowskian and Euclidean signatures of the induced metric seem to be unavoidable, at least those assignable to deformations of CP2 type extremals assignable to wormhole contacts.
There are good reasons to expect that the possible boundaries are light-like and possibly also satisfy the det(g4)=0 condition and I have considered the boundary conditions but have not been able to make definite conclusions about how they could be realized.
- The action principle defining space-times as 4-surfaces in H=M4× CP2 as preferred extremals contains a 4-D volume term and the Kähler action plus possible boundary term if boundaries are possible at all. This action would give rise to a boundary term representing a normal flow of isometry currents through the boundary. These currents should vanish.
- There could also be a 3-D boundary part in the action but if the boundary is light-like, it cannot depend on the induced metric. The Chern-Simons term for the Kähler action is the natural choice. Twistor lift suggests that it is present also in M4 degrees of freedom. Topological field theories utilizing Chern-Simons type actions are standard in condensed matter physics, in particular in the description of anyonic systems, so that the proposal is not so radical as one might think. One might even argue that in anyonic systems, the fundamental dynamics of the space-time surface is not masked by the information loss caused by the approximations leading to the field theory limit of TGD.
Boundary conditions would state that the normal components of the isometry currents are equal to the divergences of Chern-Simons currents and in this way guarantee conservation laws. In CP2 degrees of freedom the conditions would be for color currents and in M4 degrees of freedom for 4-momentum currents.
- This picture would conform with the general view of TGD. In zero energy ontology (ZEO) (see this and this) phase transitions would be induced by macroscopic quantum jumps at the level of the magnetic body (MB) of the system. In ZEO, they would have as geometric correlates classical deterministic time evolutions of space-time surface leading from the initial to the final state (see this). The findings of Minev et al provide (see this) lend support for this picture.
How the light-like 3- surfaces could be realized?
- A very general condition considered already earlier is the condition det(g4)=0 at the light-like 4-surface. This condition means that the tangent space of X4 becomes metrically 3-D and the tangent space of X3 becomes metrically 2-D. In the local light-like coordinates, (u,v,W,Wbar) g>uv= gvu) would vanish (guu and gvv vanish by definition.
Could det(g4)=0 and det(g3)=0 condition implied by it allow a universal solution of the boundary conditions? Could the vanishing of these dimensional quantities be enough for the extended conformal invariance?
- 3-surfaces with det(g4)=0 could represent boundaries between space-time regions with Minkowskian and Euclidean signatures or genuine boundaries of Minkowskian regions.
A highly attractive option is that what we identify the boundaries of physical objects are indeed genuine space-time boundaries so that we would directly see the space-time topology. This was the original vision. Later I became cautious with this interpretation since it seemed difficult to realize, or rather to understand, the boundary conditions.
The proposal that the outer boundaries of different phases and even molecules make sense and correspond to 3-D membrane like entities (see this) served as a partial inspiration for this article but this proposal is not equivalent with the proposal that light-like boundaries defining genuine space-time boundaries can carry isometry charges and fermions.
- How does this relate to M8-H duality (see this and this)? At the level of rational polynomials P determined 4-surfaces at the level of M8 as their "roots" and the roots are mass shells. The points of M4 have interpretation as momenta and would have values, which are algebraic integers in the extension of rationals defined by P.
Nothing prevents from posing the additional condition that the region of H3⊂ M4⊂ M8 is finite and has a boundary. For instance, fundamental regions of tessellations defining hyperbolic manifolds (one of them appears in the model of the genetic code (see this) could be considered. M8-H duality would give rise to holography associating to these 3-surfaces space-time surfaces in H as minimal surfaces with singularities as 4-D analogies to soap films with frames.
The generalization of the Fermi torus and its boundary (usually called Fermi sphere) as the counterpart of unit cell for a condensed matter cubic lattice to a fundamental region of a tessellation of hyperbolic space H3 acting is discussed is discussed in this. The number of tessellations is infinite and the properties of the hyperbolic manifolds of the "unit cells" are fascinating. For instance, their volumes define topological invariants and hyperbolic volumes for knot complements serve as knot invariants.
Euclidean space-time regions are not allowed in General Relativity. Can one allow them in TGD?
- CP2 type extremals with a Euclidean induced metric and serving as correlates of elementary particles are basic pieces of TGD vision. The quantum numbers of fundamental fermions would reside at the light-like orbit of 2-D wormhole throat forming a boundary between Minkowskian space-time sheet and Euclidean wormhole contact- parton as I have called it. More precisely, fermionic quantum numbers would flow at the 1-D ends of 2-D string world sheets connecting the orbits of partonic 2-surfaces. The signature of the 4-metric would change at it.
- It is difficult to invent any mathematical reason for excluding even macroscopic surfaces with Euclidean signature or even deformations of CP2 type extremals with a macroscopic size. The simplest deformation of Minkowski space is to a flat Euclidean space as a warping of the canonical embedding M4⊂ M4× S1 changing its signature.
- I have wondered whether space-time sheets with an Euclidean signature could give rise to black-hole like entities. One possibility is that the TGD variants of blackhole-like objects have a space-time sheet which has, besides the counterpart of the ordinary horizon, an additional inner horizon at which the signature changes to the Euclidean one. This could take place already at Schwarzschild radius if grr component of the metric does not change its sign.
Whether this scenario works depends on whether the normal components for the isometry currents are finite.
- det(g4)=0 condition gives boundaries of Euclidean and Minkowskian regions as 3-D light-like minimal surfaces. There would be no scales in accordance with generalized conformal invariance. guv in light-cone coordinates for M2 vanishes and implies the vanishing of det(g4) and light-likeness of the 3-surface.
What is important is that the formation of these regions would be unavoidable and they would be stable against perturbations.
- guv|det(g4)|1/2 is finite if det(g4)=0 condition is satisfied, otherwise it diverges. The terms gui∂ihk |det(g4)|1/2 must be finite. gui= cof(giu)/det(g4) is finite since guvgvu in the cofactor cancels it from the determinant in the expression of gui. The presence of |det(g4)|1/2|1/2 implies that the these contributions to the boundary conditions vanish. Therefore only the condition boundary condition for guv remains.
- If also Kähler action is present, the conditions are modified by replacing Tuk= guα∂αhk|det(g4)|1/2 with a more general expression containing also the contribution of Kähler action. I have discussed the details of the variational problem in this.
The Kähler contribution involves the analogy of Maxwell's energy momentum tensor, which comes from the variation of the induced metric and involves sum of terms proportional to Jα μJμbeta and gαβJμνJμν.
In the first term, the dangerous index raisings by guv appear 3 times. The most dangerous term is given by JuvJvv|det(g4>|1/2= guμgvνJαβ gvuJvu|det(g4>|1/2. The divergent part is guvgvuJuv gvuJvu|det(g4>|1/2. The diverging guv appears 3 times and Juv=0 condition eliminates two of these. gvu|det(g4>|1/2 is finite by |det(g4>|=0 condition. Juv=0 guarantees also the finiteness of the most dangerous part in gαβJμνJμν |det(g4>|1/2.
There is also an additional term coming from the variation of the induced Kähler form. This to the normal component of the isometry current is proportional to the quantity JnαJkl∂βhl|det(g4>|1/2. Also now, the most singular term in Juβ= guμgβνJμν corresponds to Juv giving guvgvuJuv|det(g4>|1/2. This term is finite by Juv=0 condition.
Therefore the boundary conditions are well-defined but only because det(g4)=0 condition is assumed.
- Twistor lift strongly suggests that the assignment of the analogy of Kähler action also to M4 and also this would contribute. All terms are finite if det(g4)=0 condition is satisfied.
- The isometry currents in the normal direction must be equal to the divergences of the corresponding currents assignable to the Chern-Simons action at the boundary so that the flow of isometry charges to the boundary would go to the Chern-Simons isometry charges at the boundary.
If the Chern-Simons term is absent, one expects that the boundary condition reduces to ∂vhk=0. This would make X3 2-dimensional so that Chern-Simons term is necessary. Note that light-likeness does not force the M4 projection to be light-like so that the expansion of X2 need not take with light-velocity. If CP2 complex coordinates are holomorphic functions of W depending also on U=v as a parameter, extended conformal invariance is obtained.
- Kähler Chern-Simons form, also identifiable as a boundary term to which the instanton density of Kähler form reduces, defines an analog of topological QFT.
- In the recent case the metric is however present via boundary conditions and in the dynamics in the interior of the space-time surface. However, the preferred extremal property essential for geometry-number theory duality transforms geometric invariants to topological invariants. Minimal surface property means that the dynamics of volume and Kähler action decouple outside the singularities, where minimal surface property fails. Coupling constants are present in the dynamics only at these lower-D singularities defining the analogs of frames of a 4-D soap film.
Singularities also include string worlds sheets and partonic 2-surfaces. Partonic two-surfaces play the role of topological vertices and string world sheets couple partonic 2-orbits to a network. It is indeed known that the volume of a minimal surface can be regarded as a homological invariant.
- If the 3-surfaces assignable to the mass shells H3 define unit cells of hyperbolic tessellations and therefore hyperbolic manifolds, they also define topological invariants. Whether also string world sheets could define topological invariants is an interesting question.
Quantum criticality is the basic dynamical principle of quantum TGD. What led to its discovery was the question "How to make TGD unique?". TGD has a single coupling constant, Kähler couplings strength, which is analogous to a critical temperature. The idea was obvious: require quantum criticality. This predicts a spectrum of critical values for the Kähler coupling strength. Quantum criticality would make the TGD Universe maximally complex. Concerning living matter, quantum critical dynamics is ideal since it makes the system maximally sensitive and maximallt reactive.
Concerning the realization of quantum criticality, it became gradually clear that the conformal invariance accompanying 2-D criticality, must be generalized. This led to the proposal that super symplectic symmetries, extended isometries and conformal symmetries of the metrically 2-D boundary of lightcone of M4, and the extension of the Kac-Moody symmetries associated with the light-like boundaries of deformed CP2 type extremals should act as symmetries of TGD extending the conformal symmetries of 2-D conformal symmetries. These huge infinite-D symmetries are also required by the existence of the Kähler geometry of WCW (see this and this).
However, the question whether light-like boundaries of 3-surfaces with scale larger than CP2 are possible, remained an open question. On the basis of preceding arguments, the answer seems to be affirmative and one can ask for the implications.
- At M8 level, the concrete realization of holography would involve two ingredients. The intersections of the space-time surface with the mass shells H3 with mass squared value determined as the roots of polynomials P and the tlight-like 3-surfaces as det(g4)=0 surfaces as boundaries (genuine or between Minkowskian and Euclidean regions) associated by M8-H duality to 4-surface of M8 having associative normal space, which contains commutative 2-D subspace at each point. This would make possible both holography and M8-H duality.
Note that the identification of the algebraic geometric characteristics of the counterpart of det(g4)=0 surface at the level of H remains still open.
Since holography determines the dynamics in the interior of the space-time surface from the boundary conditions, the classical dynamics can be said to be critical also in the interior.
- Quantum criticality means ability to self-organize. Number theoretical evolution allows us to identify evolution as an increase of the algebraic complexity. The increase of the degree n of polynomial P serves as a measure for this. n=heff/h0 also serves as a measure for the scale of quantum coherence, and dark matter as phases of matter would be characterized by the value of n.
- The 3-D boundaries would be places where quantum criticality prevails. Therefore they would be ideal seats for the development of life. The proposal that the phase boundaries between water and ice serve as seats for the evolution of prebiotic life, is discussed from the point of TGD based view of quantum gravitation involving huge value of gravitational Planck constant ℏeff= ℏgr= GMm/v 0 making possible quantum coherence in astrophysical scales (see this). Density fluctuations would play an essential role, and this would mean that the volume enclosed by the 2-D M4 projection of the space-time boundary would fluctuate. Note that these fluctuations are possible also at the level of the field body and magnetic body.
- It has been said that boundaries, where the nervous system is located, distinguishes living systems from inanimate ones. One might even say that holography based on det(g4)=0 condition realizes nervous systems in a universal manner.
- I have considered several variants for the holography in the TGD framework, in particular strong form of holography (SH). SH would mean that either the light-like 3-surfaces or the 3-surfaces at the ends of the causal diamond (CD) determine the space-time surface so that the 2-D intersections of the 3-D ends of the space-time surface with its light-like boundaries would determine the physics.
For a summary of earlier postings see Latest progress in TGD.