A strange behavior of hybrid matter-antimatter atoms in superfluid Helium

I received an interesting link to a popular article "ASACUSA sees surprising behaviour of hybrid matter antimatter atoms in superfluid helium" (this), which tells of a completely unexpected discovery related to the behavior of antiproton-⁴He⁺⁺ atoms in ⁴He superfluid. The research article by ASACUSA researchers Anna Soter et al is published in Nature (this).

The formation of anti-proton-⁴He⁺⁺ hybrid atoms containing also an electron in ⁴He was studied both above and below the critical temperature for the transition to Helium superfluid. The temperatures considered are in Kelvin range corresponding to a thermal energy of order 10^-4 eV.

Liquid Helium is much denser than Helium gas. As the temperature is reduced, a transition to liquid phase takes place and the Helium liquid gets denser with the decreasing temperature. One would expect that the perturbations of nearby atoms to the state should increase the width of both electron and antiproton spectral lines in the dense liquid phase.

This widening indeed occurs for the lines of electrons but something totally different occurs for the spectral lines of the antiproton. The width decreases and when the superfluidity sets on, an abrupt further narrowing of He⁺⁺ spectral lines takes place. The antiproton does not seem to interact with the neighboring ⁴He atoms.

Researchers think that the fact that the surprising behavior is linked to the radius of the hybrid atom's electronic orbital. In contrast to the situation for many ordinary atoms, the electronic orbital radius of the hybrid atom changes very little when laser light is shone on the atom and thus does not affect the spectral lines even when the atom is immersed in superfluid helium.

Consider now the TGD inspired model.

1. It seems that either antiprotons or the atoms of ⁴He superfluid effectively behave like dark matter. For the electrons, the widening however takes place so that it seems that the antiproton seems to be dark. In the TGD framework, where dark particles corresponds h_eff=nh₀ >h, h=n₀h₀ phases of ordinary matter, the first guess is that the antiprotons are dark and reside at the magnetic flux tube like structures.

   The dark proton would be similar to a valence electron of some rare earth atoms, which mysteriously disappear when heated (an effect known for decades, see this). Dark protons would indeed behave like a dark matter particle is expected to behave and would have no direct quantum interactions with ordinary matter. The electron of the hybrid atom would be ordinary.

2. Darkness might also relate to the formation mechanism of the hybrid atoms. Antiproton appears as a Rydberg orbital with a large principal quantum number N and large size proportional to N². N>41 implies that the antiproton orbital is outside the electron orbital but this leaves the interactions with other Helium atoms. For a smaller value of N the dark proton overlaps the electronic orbital. Note that for N=1, the radius of the orbital is 10^-3/8a₀, a₀∼ .53 × 10^-10 m, in the Bohr model.
3. The orbital radii are proportional to h_eff² ∝ (n/n₀)² so that dark orbitals with the same energy and radius as for ordinary orbitals but effective principal quantum number (n/n₀)N_d=N_eff, are possible. (n/n₀)N_d=N_eff condition would give the same radius and energy for the dark orbital characterized by N_d and ordinary orbital characterized by N.

One can consider both dark-to-dark and dark-to-ordinary transitions.

1. The minimal change of the effective principal quantum number N_eff in dark-to-dark transitions would be n/n₀ and be larger than one for n>n₀. There is evidence for n=n₀/6 found by Randel Mills (see this) discussed from the TGD view here. In this case one would have effectively fractional values of N_eff. One can also consider a stronger condition, h_eff/h=m , one has mN_d=N. The transitions would be effectively between ordinary orbitals for which Δ N_eff is a multiple of m. This could be tested if the observation of dark-to-dark transition is possible. The transformation of dark photons to ordinary photons would be needed.
2. Energy conserving dark-to-ordinary transitions producing an ordinary photon cannot be distinguished from ordinary transitions if the condition (n/n₀)N_d=N_eff is satisfied.

   The transitions (37,35)→ (38,34) and (39,35)→ (38,34) at the visible wavelengths λ =726 nm and 597 nm survive in the Helium environment. The interpretation could be that the transitions occur between dark and ordinary states such that the dark state satisfies the condition that (n/n₀)N_d=N_eff is integer, and that an ordinary photon with λ = h/Δ E is produced. This does not pose conditions on the value of h_eff/h.

   If the condition that (n/n₀)N_d=N_eff is an integer is dropped, effective principal quantum numbers N_eff coming as multiples of n/n₀ are possible and the photon energy has fractional spectrum.

If this picture makes sense, it could mean a new method to store antimatter without fear of annihilation by storing it as a dark matter in the magnetic flux tubes. They would be present in superfluids and superconductors.

See the article TGD and condensed matter or the chapter with the same title.

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

Articles related to TGD.

About the TGD based notions of mass, of twistors and hyperbolic counterpart of Fermi torus

The notion of mass in the TGD framework is discussed from the perspective of M⁸-H duality (see this, this, and this).

1. In TGD, space-time regions are characterized by polynomials P with rational coefficients (see this). Galois confinement defines a universal mechanism for the formation of bound states. Momenta for virtual fermions have components, which are algebraic integers in an extension of rationals defined by a polynomial P characterizing a space-time region. For the physical many fermion states, the total momentum as the sum of fermion momenta has components, which are integers using the unit defined by the size of the causal diamond (CD) (see this, this, and this).
2. This defines a universal number theoretical mechanism for the formation of bound states as Galois singlets. The condition is very strong but for rational coefficients it can be satisfied since the sum of all roots is always a rational number as the coefficient of the first order term.
3. Galois confinement implies that the sum of the mass squared values, which are in general complex algebraic numbers in E, is also an integer. Since the mass squared values correspond to conformal weights as also in string models, one has conformal confinement: states are conformal singlets. This condition replaces the masslessness condition of gauge theories (see this).

Also the TGD based notion of twistor space is considered at concrete geometric level.

1. Twistor lift of TGD means that space-time surfaces X⁴ is H=M⁴× CP₂ are replaced with 6-surfaces in the twistor space with induced twistor structure of T(H)= T(M⁴)× T(CP₂) identified as twistor space T(X⁴). This proposal requires that T(H) has Kähler structure and this selects M⁴× CP₂ as a unique candidate (see this) so that TGD is unique.
2. One ends up to a more precise understanding of the fiber of the twistor space of CP₂ as a space of "light-like" geodesics emanating from a given point. Also a more precise view of the induced twistor spaces for preferred extremals with varying dimensions of M⁴ and CP₂ projections emerges. Also the identification of the twistor space of the space-time surface as the space of light-like geodesics itself is considered.
3. Twistor lift leads to a concrete proposal for the construction of scattering amplitudes. Scattering can be seen as a mere re-organization of the physical many-fermion states as Galois singlets to new Galois singlets. There are no primary gauge fields and both fermions and bosons are bound states of fundamental fermions. 4-fermion vertices are not needed so that there are no divergences.
4. There is however a technical problem: fermion and antifermion numbers are separately conserved in the simplest picture, in which momenta in M⁴⊂ M⁸ are mapped to geodesics of M⁴⊂ H.The led to a proposal for the modification of M⁸-H duality (see this and this). The modification would map the 4-momenta to geodesics of X⁴. Since X⁴ allows both Minkowskian and Euclidean regions, one can have geodesics, whose M⁴ projection turns backwards in time. The emission of a boson as a fermion-antifermion pair would correspond to a fermion turning backwards in time. A more precise formulation of the modification shows that it indeed works

The third topic of this article is the hyperbolic generalization of the Fermi torus to hyperbolic 3-manifold H³/Γ. Here H³=SO(1,3)/SO(3) identifiable the mass shell M⁴\subset M⁸ or its M⁸-H dual in H=M⁴× CP₂. Γ denotes an infinite subgroup of SO(1,3) acting completely discontinuously in H³. For virtual fermions also complexified mass shells are required and the question is whether the generalization of H³/Γ, defining besides hyperbolic 3-manifold also tessellation of H³ analogous to a cubic lattice of E³.

See the article About the TGD based notions of mass, of twistors and hyperbolic counterpart of Fermi torus or the chapter Trying to fuse the basic mathematical ideas of quantum TGD to a single coherent whole.

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

Articles related to TGD.

Riemann zeta and the number theoretical vision of TGD

There are strong indications that Riemann zeta (see this) has a deep role in physics, in particular in the physics of critical systems. TGD Universe is quantum critical. For what quantum criticaly would mean at the space-time level (see this). This raises the question whether also Riemann zeta could have a deep role in TGD.

First some background relating to the number theoretic view of TGD.

1. In TGD, space-time regions are characterized by polynomials P with rational coefficients. Galois confinement defines a universal mechanism for the formation of bound states. Momenta for virtual fermions have components, which are algebraic integers in an extension of rationals defined by a polynomial P characterizing space-time region. For the physical many fermion states, the total momentum as the sum of fermion momenta has components, which are integers using the unit defined by the size of the causal diamond (CD).

   This defines a universal number theoretical mechanism for the formation of bound states. The condition is very strong but for rational coefficients it can be satisfied since the sum of all roots is always a rational number as the coefficient of the first order term.

2. Galois confinement implies that the sum of the mass squared values, which are in general complex algebraic numbers in E, is also an integer. Since the mass squared values correspond to conformal weights as also in string models, one has conformal confinement: states are conformal singlets. This condition replaces the masslessness condition of gauge theories (see this).

   Riemann zeta is not a polynomial but has infinite number of root. How could one end up with Riemann zeta in TGD? One can also consider the replacement of the rational polynomials with analytic functions with rational coefficients or even more general functions.

   1. For real analytic functions roots come as pairs but building many-fermion states for which the sum of roots would be a real integer, is very difficult and in general impossible.
   2. Riemann zeta and the hierarchy of its generalizations to extensions of rationals (Dedekind zeta functions) is however a complete exception! If the roots are at the critical line as the generalization of Riemann hypothesis assumes, the sum of the root and its conjugate is equal to 1 and it is easy to construct many fermion states as 2N fermion states, such that they have integer value conformal weight.

      One can wonder whether one could see Riemann zeta as an analog of a polynomial such that the roots as zeros are algebraic numbers. This is however not necessary. Could zeta and its analogies allow it to build a very large number of Galois singlets and they would form a hierarchy corresponding to extensions of rationals. Could they represent a kind of second abstraction level after rational polynomials?

   See the article About TGD counterparts of twistor amplitudes: part II or the chapter About TGD counterparts of twistor amplitudes.

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

   Articles related to TGD.

Invisible magnetic fields as a support for the notion of monopole flux tube

Physicists studying a system consisting of a layered structure consisting of alternate superconducting and spin liquid layers have found evidence for what they call invisible magnetic fields. The popular article is published in Scitechdaily (see this) and tells about research carried out by Prof. Beena Kalisky and doctoral student Eylon Persky in Bar-Ilan University. The research article is published in Nature (see this).

First some basic notions.

1. The notions of spin liquid and charge-spin separation are needed. Popular texts describe charge separation in a way completely incomprehensible for both layman and professional. Somehow the electron would split into two parts corresponding to its spin and charge. The non-popular definition is clear and understandable. Instead of a single electron, one considers a spin liquid as a many-electron system associated with a lattice-like structure formed by atoms. The neighboring electrons are paired. There are a very large number of possible pairings. In the ground state the spins of electrons of all pairs could be either opposite or parallel (magnetization). Pairing with a vanishing spin is favoured by Fermi statistics.

   If the opposite spins of a single pair become parallel and this state is delocalized, one can have a propagating spin wave without moving charge. If one electron pair is removed and this hole pair is delocalized ,one obtains a moving charge +2e without any motion of spin.

2. When a superconductor of type II is in an external magnetic field with a strength above critical value, the magnetic field penetrates to the superconductor as vortices. Inside these vortices the superconductivity is broken and electrons swirl around the magnetic field. This is how the magnetic flux quanta become visible.

In the layered structures formed by atomic layers of spin liquid and superconductor, magnetic vortices are created spontaneously in the superconducting layers. In the Maxwellian world, magnetic fields would be created either by rotating currents or by magnetization requiring a lattice-like structure of parallel electron spins. In the recent case spontaneous magnetization should serve as a signature for the presence of these magnetic fields.

Surprisingly, no magnetization was observed so that one can talk of "invisible" magnetic field.

In the bilayered structure 4Hb-TaS₂, the superconductivity is anomalous in the sense that the critical temperature is 2.7 K whereas in bulk superconductor 2H-TaS₂ it is .7 K. There is also a breaking of time reversal symmetry closely related t the presence of the magnetic flux quanta. The magnetic flux quanta survive above critical temperature 2.7 K up to 3.6 K and their life time is very long as compared to the electronic time scales (12 minute scale is mentioned). Therefore one can talk of magnetic memory.

The proposal is that a spin liquid state known as a chiral spin liquid is created and that the invisible magnetic field associated with the chiral spin liquid penetrates to the superconductor as flux quanta.

Could TGD explain the invisible magnetic fields?

1. TGD predicts what I called monopole flux tubes, which have closed, rather than disk-like , 2-D cross sections and carry monopole flux requiring no current nor magnetization to generate it.

   This is possible only in the TGD space-time, which corresponds to a 4-surface in 8-D space H=M⁴× CP₂, but not in Minkowski space or in general relativistic space-time in its standard form. The reason is that the topology of the space-time surface is non-trivial in all scales.

   The possibility of closed monopole flux tubes without magnetic monopoles, is one of the basic differences between TGD and Maxwell's theory and reflects the non-trivial homology of CP₂.

2. Monopole flux tubes solve the mystery of why there are magnetic fields in cosmic length scales and why the Earth's magnetic field B_E has not disappeared long ago by dissipation (see this)).
3. Electromagnetic fields at frequencies in the EEG range corresponding to cyclotron frequencies have quantal looking effects on brains of mammalians at the level of both physiology and behavior. The photon energies involved are extremely low.

   In the TGD based quantum biology they can be understood in terms of cyclotron transitions for "dark" ions with a very large effective Planck constant h_eff= nh₀ in a magnetic field of .2 Gauss, which is about 2/5 of the nominal value .5 Gauss of the Earth's magnetic field B_D. The proposal is that B_E involves a monopole flux contribution about 2B_E/5 (see this).

   The estimate for the invisible magnetic field was .1 Gauss so that the numbers fit nicely.

The findings suggest that the spin liquid phase atomic layer involves the monopole flux tubes assignable to the Earth's magnetic field and orthogonal to the layer. They would not be present in the superconducting layer but would penetrate from spin liquid to the superconductor.

See the chapter Magnetic Sensory Canvas Hypothesis.

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

Articles related to TGD.

Are space-time boundaries possible in the TGD Universe?

One of the key ideas of TGD from the very beginning was that the space-time surface has boundaries and we see them directly as boundaries of physical objects.

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.

1. 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.
2. 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.

3. 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.

Light-like 3-surfaces from det(g₄)=0 condition

How the light-like 3- surfaces could be realized?

1. 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= g_vu) would vanish (g_uu and g_vv 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?

2. 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.

3. 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.

Can one allow macroscopic Euclidean space-time regions

Euclidean space-time regions are not allowed in General Relativity. Can one allow them in TGD?

1. 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.
2. 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.
3. 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 g_rr component of the metric does not change its sign.

But are the normal components of isometry currents finite?

Whether this scenario works depends on whether the normal components for the isometry currents are finite.

1. 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. g_uv 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.

2. g^uv|det(g₄)|^1/2 is finite if det(g₄)=0 condition is satisfied, otherwise it diverges. The terms g^ui∂_ih^k |det(g₄)|^1/2 must be finite. g^ui= cof(g_iu)/det(g₄) is finite since g_uvg_vu in the cofactor cancels it from the determinant in the expression of g^ui. 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 g^uv remains.
3. If also Kähler action is present, the conditions are modified by replacing T^uk= g^uα∂_αh^k|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 J_{α μ}J_μ^{beta and gαβJμνJ_μν.}

   In the first term, the dangerous index raisings by g^uv appear 3 times. The most dangerous term is given by J^uvJ_v^v|det(g_4>|^1/2= g^uμg^vνJ_αβ g^vuJ_vu|det(g_4>|^1/2. The divergent part is g^uvg^vuJ_uv g^vuJ_vu|det(g_4>|^1/2. The diverging g^uv appears 3 times and J_uv=0 condition eliminates two of these. g^vu|det(g_4>|^1/2 is finite by |det(g_4>|=0 condition. J_uv=0 guarantees also the finiteness of the most dangerous part in g^αβJ^μνJ_μν |det(g_4>|^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 J^nαJ^k_l∂_βh^l|det(g_4>|^1/2. Also now, the most singular term in J^uβ= g^uμg^βνJ_μν corresponds to J^uv giving g^uvg^{vuJuv|det(g_4>|1/2. This term is finite by J_uv=0 condition.}

   Therefore the boundary conditions are well-defined but only because det(g₄)=0 condition is assumed.

4. 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.
5. 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 ∂_vh^k=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.

This picture resonates with an old guiding vision about TGD as an almost topological quantum field theory (QFT) (see this), which I have even regarded as a third strand in the 3-braid formed by the basic ideas of TGD based on geometry-number theory-topology trinity.

1. 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.
2. 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.

3. 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.

det(g₄)=0 condition as a realization of quantum criticality

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.

1. 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.

2. 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=h_eff/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.
3. 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 ℏ_eff= ℏ_gr= 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.
4. 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.
5. 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.

This condition is perhaps too strong but a fascinating, weaker, possibility is that the internal consistency requires that the intersections of the 3-surface with the mass shells H³ are identifiable as fundamental domains for the coset spaces SO(1,3)/Γ defining tessellations of H³ and hyperbolic manifolds. This would conform nicely with the TGD inspired model of genetic code (see this). See the article TGD inspired model for freezing in nano scales or the chapter with the same title.

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

Articles related to TGD.

Some comments of the notion of mass

In the sequel some comments related to the notion of mass.

M⁸-H duality and tachyonic momenta

Tachyonic momenta are mapped to space-like geodesics in H or possibly to the geodesics of X⁴ (see this, this and this). This description could allow to describe pair creation as turning of fermion backwards in time (see this). Tachyonic momenta correspond to points of de Sitter space and are therefore outside CD and would go outside the space-time surface, which is inside CD. Could one avoid this?

1. Since the points of the twistor spaces T(M⁴) and T(CP₂) are in 1-1 correspondence, one can use either T(M⁴) or T(CP₂) so that the projection to M⁴ or CP₂ would serve as the base space of T(X⁴). One could use CP₂ coordinates or M⁴ coordinates as space-time coordinates if the dimension of the projection is 4 to either of these spaces. In the generic case, both dimensions are 4 but one must be very cautious with genericity arguments which fail at the level of M⁸.
2. There are exceptional situations in which genericity fails at the level of H. String-like objects of the form X²× Y² ⊂ M⁴⊂ CP₂ is one example of this. In this case, X⁶ would not define 1-1 correspondence between T(M⁴) or T(CP₂).

   Could one use partial projections to M² and S² in this case? Could T(X⁴) be divided locally into a Cartesian product of 3-D M⁴ part projecting to M² ⊂ M⁴ and of 3-D CP₂ part projected to Y²⊂ CP₂.

3. One can also consider the possibility of defining the twistor space T(M²× S²). Its fiber at a given point would consist of light-like geodesics of M²× S². The fiber consists of direction vectors of light-like geodesics. S² projection would correspond to a geodesic circle S¹⊂ S² going through a given point of S² and its points are parametrized by azimuthal angle Φ. Hyperbolic tangent tanh(η) with range [-1,1] would characterize the direction of a time like geodesic in M². At the limit of η → +/- ∞ the S² contribution to the S² tangent vector to length squared of the tangent vector vanishes so that all angles in the range (0,2\pi) correspond to the same point. Therefore the fiber space has a topology of S².

   There are also other special situations such as M¹× S³, M³ × S¹ for which one must introduce specific twistor space and which can be treated in the same way.

During the writing of this article I realized that the twistor space of H defined geometrically as a bundle, which has as H as base space and fiber as the space of light-like geodesic starting from a given point of H need not be equal to T(M⁴)× T(CP₂), where T(CP₂) is identified as SU(3)/U(1)× U(1) characterizing the choices of color quantization axes.

1. The definition of T(CP₂) as the space of light-like geodesics from a given point of CP₂ is not possible. One could also define the fiber space of T(CP₂) geometrically as the space of geodesics emating from origin at r=0 in the Eguchi-Hanson coordinates (see this) and connecting it to the homologically non-trivial geodesic sphere S²_G r=∞. This relation is symmetric.

   In fact, all geodesics from r=0 end up to S². This is due to the compactness and symmetries of CP₂. In the same way, the geodesics from the North Pole of S² end up to the South Pole. If only the endpoint of the geodesic of CP₂ matters, one can always regard it as a point S²_G.

   The two homologically non-trivial geodesic spheres associated with distinct points of CP₂ always intersect at a single point, which means that their twistor fibers contain a common geodesic line of this kind. Also the twistor spheres of T(M⁴) associated with distinct points of M⁴ with a light-like distance intersect at a common point identifiable as a light-like geodesic connecting them.

2. Geometrically, a light-like geodesic of H is defined by a 3-D momentum vector in M⁴ and 3-D color momentum along CP₂ geodesic. The scale of the 8-D tangent vector does not matter and the 8-D light-likeness condition holds true. This leaves 4 parameters so that T(H) identified in this way is 12-dimensional.

   The M⁴ momenta correspond to a mass shell H³. Only the momentum direction matters so that also in the M⁴ sector the fiber reduces to S² . If this argument is correct, the space of light-like geodesics at point of H has the topology of S²× S² and T(H) would reduce to T(M⁴)× T(CP₂) as indeed looks natural.

Conformal confinement at the level of H

The proposal of this, inspired by p-adic thermodynamics, is that all states are massless in the sense that the sum of mass squared values vanishes. Conformal weight, as essentially mass squared value, is naturally additive and conformal confinement as a realization of conformal invariance would mean that the sum of mass squared values vanishes. Since complex mass squared values with a negative real part are allowed as roots of polynomials, the condition is highly non-trivial.

M⁸-H duality (see this and this) would make it natural to assign tachyonic masses with CP₂ type extremals and with the Euclidean regions of the space-time surface. Time-like masses would be assigned with time-like space-time regions. It was found that, contrary to the beliefs held hitherto, it is possible to satisfy boundary conditions for the action action consisting of the Kähler action, volume term and Chern-Simons term, at boundaries (genuine or between Minkowskian and Euclidean space-time regions) if they are light-like surfaces satisfying also det{g₄}=0. Masslessness, at least in the classical sense, would be naturally associated with light-like boundaries (genuine or between Minkowskian and Euclidean regions).

About the analogs of Fermi torus and Fermi surface in H³

Fermi torus (cube with opposite faces identified) emerges as a coset space of E³/T³, which defines a lattice in the group E³. Here T³ is a discrete translation group T³ corresponding to periodic boundary conditions in a lattice.

In a realistic situation, Fermi torus is replaced with a much more complex object having Fermi surface as boundary with non-trivial topology. Could one find an elegant description of the situation?

1. Hyperbolic manifolds as analogies for Fermi torus?

The hyperbolic manifold assignable to a tessellation of H³ defines a natural relativistic generalization of Fermi torus and Fermi surface as its boundary. To understand why this is the case, consider first the notion of cognitive representation.

1. Momenta for the cognitive representations (see this)define a unique discretization of 4-surface in M⁴ and, by M⁸-H duality, for the space-time surfaces in H and are realized at mass shells H³⊂ M⁴⊂ M⁸ defined as roots of polynomials P. Momentum components are assumed to be algebraic integers in the extension of rationals defined by P and are in general complex.

   If the Minkowskian norm instead of its continuation to a Hermitian norm is used, the mass squared is in general complex. One could also use Hermitian inner product but Minkowskian complex bilinear form is the only number-theoretically acceptable possibility. Tachyonicity would mean in this case that the real part of mass squared, invariant under SO(1,3) and even its complexification SO_c(1,3), is negative.

2. The active points of the cognitive representation contain fermion. Complexification of H³ occurs if one allows algebraic integers. Galois confinement(see this and this) states that physical states correspond to points of H³ with integer valued momentum components in the scale defined by CD.

   Cognitive representations are in general finite inside regions of 4-surface of M⁸ but at H³ they explode and involve all algebraic numbers consistent with H³ and belonging to the extension of rationals defined by P. If the components of momenta are algebraic integers, Galois confinement allows only states with momenta with integer components favored by periodic boundary conditions.

Could hyperbolic manifolds as coset spaces SO(1,3)/Γ, where Γ is an infinite discrete subgroup SO(1,3), which acts completely discontinuously from left or right, replace the Fermi torus? Discrete translations in E³ would thus be replaced with an infinite discrete subgroup Γ. For a given P, the matrix coefficients for the elements of the matrix belonging to Γ would belong to an extension of rationals defined by P.

1. The division of SO(1,3) by a discrete subgroup Γ gives rise to a hyperbolic manifold with a finite volume. Hyperbolic space is an infinite covering of the hyperbolic manifold as a fundamental region of tessellation. There is an infinite number of the counterparts of Fermi torus (see this). The invariance respect to Γ would define the counterpart for the periodic boundary conditions.

   Note that one can start from SO(1,3)/Γ and divide by SO(3) since Γ and SO(3) act from right and left and therefore commute so that hyperbolic manifold is SO(3)\setminus SO(1,3)/Γ.

2. There is a deep connection between the topology and geometry of the Fermi manifold as a hyperbolic manifold. Hyperbolic volume is a topological invariant, which would become a basic concept of relativistic topological physics (see this).

   The hyperbolic volume of the knot complement serves as a knot invariant for knots in S³. Could this have physical interpretation in the TGD framework, where knots and links, assignable to flux tubes and strings at the level of H, are central. Could one regard the effective hyperbolic manifold in H³ as a representation of a knot complement in S³?

   Could these fundamental regions be physically preferred 3-surfaces at H³ determining the holography and M⁸-H duality in terms of associativity (see this and this). Boundary conditions at the boundary of the unit cell of the tessellation should give rise to effective identifications just as in the case of Fermi torus obtained from the cube in this way.

2. De Sitter manifolds as tachyonic analogs of Fermi torus do not exist

Can one define the analogy of Fermi torus for the real 4-momenta having negative, tachyonic mass squared? Mass shells with negative mass squared correspond to De-Sitter space SO(1,3)/SO(1,2) having a Minkowskian signature. It does not have analogies of the tessellations of H³ defined by discrete subgroups of SO(1,3).

The reason is that there are no closed de-Sitter manifolds of finite size since no infinite group of isometries acts discontinuously on de Sitter space: therefore these is no group replacing the Γ in H³/Γ (see this).

3. Do complexified hyperbolic manifolds as analogs of Fermi torus exist?

The momenta for virtual fermions defined by the roots defining mass squared values can also be complex. Tachyon property and complexity of mass squared values are not of course not the same thing.

1. Complexification of H³ would be involved and it is not clear what this could mean. For instance, does the notion of complexified hyperbolic manifold with complex mass squared make sense.
2. SO(1,3) and its infinite discrete groups Γ act in the complexification. Do they also act discontinuously? p² remains invariant if SO(1,3) acts in the same way on the real and imaginary parts of the momentum leaves invariant both imaginary and complex mass squared as well as the inner product between the real and imaginary parts of the momenta. So that the orbit is 5-dimensional. Same is true for the infinite discrete subgroup Γ so that the construction of the coset space could make sense. If Γ remains the same, the additional 2 dimensions can make the volume of the coset space infinite. Indeed, the constancy of p₁• p₂ eliminates one of the two infinitely large dimensions and leaves one.

   Could one allow a complexification of SO(1,3), SO(3) and SO(1,3)_c/SO(3)_c? Complexified SO(1,3) and corresponding subgroups Γ satisfy OO^T=1. Γ_c would be much larger and contain the real Γ as a subgroup. Could this give rise to a complexified hyperbolic manifold H³_c with a finite volume?

3. A good guess is that the real part of the complexified bilinear form p• p determines what tachyonicity means. Since it is given by Re(p)²-Im(p)² and is invariant under SO_c(1,3) as also Re(p)• Im(p), one can define the notions of time-likeness, light-likeness, and space-likeness using the sign of Re(p)²-Im(p²) as a criterion. Note that Re(p)² and Im(p)² are separately invariant under SO(1,3).

   The physicist's naive guess is that the complexified analogs of infinite discrete and discontinuous groups and complexified hyperbolic manifolds as analogs of Fermi torus exist for Re(P²)-Im(p²)>0 but not for Re(P²)-Im(p²)<0 so that complexified dS manifolds do not exist.

4. The bilinear form in H³_c would be complex valued and would not define a real valued Riemannian metric. As a manifold, complexified hyperbolic manifold is the same as the complex hyperbolic manifold with a hermitian metric (see this) and this) but has different symmetries. The symmetry group of the complexified bilinear form of H³_c is SO_c(1,3) and the symmetry group of the Hermitian metric is U(1,3) containing SO(1,3) as a real subgroup. The infinite discrete subgroups Γ for U(1,3) contain those for SO(1,3). Since one has complex mass squared, one cannot replace the bilinear form with hermitian one. The complex H³ is not a constant curvature space with curvature -1 whereas H³_c could be such in a complexified sense.

See the article Some objections against p-adic thermodynamics and their resolution or the chapter About TGD counterparts of twistor amplitudes.

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

Articles related to TGD.

TGD inspired model for freezing in nano scales

Freezing is a phase transition, which challenges the existing view of condensed matter in nanoscales. In the TGD framework, quantum coherence is possible in all scales and gravitational quantum coherence should characterize hydrodynamics in astrophysical and even shorter scales. The hydrodynamics at the surface of the planet such as Earth the mass of the planet and even that of the Sun should characterize gravitational Planck constant h_gr assignable to gravitational flux tubes mediating gravitational interactions. In this framework, quantum criticality involving h_eff=nh₀>h phases of ordinary matter located at the magnetic body (MB) and possibly controlling ordinary matter, could be behind the criticality of also ordinary phase transitions.

In this article, a model inspired by the finding that the water-air boundary involves an ice-like layer. The proposal is that also at criticality for the freezing a similar layer exists and makes possible fluctuations of the size and shape of the ice blob. At criticality the change of the Gibbs free energy for water would be opposite that for ice and the Gibbs free energy liberated in the formation of ice layer would transform to the energy of surface tension at water-ice layer.

This leads to a geometric model for the freezing phase transition involving only the surface energy proportional to the area of the water-ice boundary and the constraint term fixing the volume of water. The partial differential equations for the boundary surface are derived and discussed.

If Δ P=0 at the critical for the two phases at the boundary layer, the boundary consists of portions, which are minimal surfaces analogous to soap films and conformal invariance characterizing 2-D critical systems is obtained. For Δ P≠ 0, conformal invariance is lost and analogs of soap bubbles are obtained.

In the TGD framework, the generalization of the model to describe freezing as a dynamical time evolution of the solid-liquid boundary is suggestive. An interesting question is whether this boundary could be a light-like 3-surface in H=M⁴× CP₂ and thus have a vanishing 3-volume. A huge extension of ordinary conformal symmetries would emerge.

See the article TGD inspired model for freezing in nano scales or the chapter with the same title.

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

Articles related to TGD.