Uncertainty Principle and M8-H duality

The detailed realization of M⁸-H duality (for what this duality means, see this) involves still uncertainties. The quaternionic normal spaces containing fixed 2-space M² (or an integrable distribution of M²) are parametrized by points of CP₂. One can map the normal space to a point of CP₂.

The tough problem has been the precise correspondence between M⁴ points in M⁴× E⁴ and M⁴× CP₂ and the identification of the sizes of causal diamonds (CDs) in M⁸ and H. The identification is naturally linear if M⁸ is analog of space-time but if M⁸ is interpreted as momentum space, the situation changes. The ealier proposal maps mass hyperboloids to light-cone proper time =constant hyperboloids and it has turned out that this correspondence does not correspond to the classical picture suggesting that a given momentum in M⁸ corresponds in H to a geodesic line emanating from the tip of CD.

M⁸-H duality in M⁴ degrees of freedom

The following proposal for M⁸-H duality in M⁴ degrees of freedom relies on the intuition provided by UP and to the idea that a particle with momentum p^k corresponds to a geodesic line with this direction emanating from the tip of CD.

1. The first constraint comes from the requirement that the identification of the point p^k∈ X⁴⊂ M⁸ should classically correspond to a geodesic line m^k= p^kτ/m (p²=m²) in M⁸ which in Big Bang analogy should go through the tip of the CD in H. This geodesic line intersects the opposite boundary of CD at a unique point.

   Therefore the mass hyperboloid H³ is mapped to the 3-D opposite boundary of cd⊂ M⁴⊂ H. This does not fix the size nor position of the CD (=cd× CP₂) in H. If CD does not depend on m, the opposite light-cone boundary of CD would be covered an infinite number of times.

2. The condition that the map is 1-to-1 requires that the size of the CD in H is determined by the mass hyperboloid M⁸. Uncertainty Principle (UP) suggests that one should choose the distance T between the tips of the CD associated with m to be T= ℏ_eff/m.

   The image point m^k of p^k at the boundary of CD(m,h_eff) is given as the intersection of the geodesic line m^k= p^kτ from the origin of CD(m,h_eff) with the opposite boundary of CD(m,h_eff):

   m^k=ℏ_effX× (p^k/m²),

   X= 1/(1+ p₃/p₀) .

   Here p₃ is the length of 3-momentum.

   The map is non-linear. At the non-relativistic limit (X\rightarrow 1), one obtains a linear map for a given mass and also a consistency with the naive view about UP. m^k is on the proper time constant mass shell so the analog of the Fermi ball in H³ ⊂ M⁸ is mapped to the light-like boundary of cd⊂ M⁴⊂ H.

3. What about massless particles? The duality map is well defined for an arbitrary size of CD. If one defines the size of the CD as the Compton length ℏ_eff/m of the massless particle, the size of the CD is infinite. How to identify the CD? UP suggests a CD with temporal distance T= 2ℏ_eff/p₀ between its tips so that the geometric definition gives p^k= ℏ_effp^k/p₀² as the point at the 2-sphere defining the corner of CD. p-Adic thermodynamics strongly suggests that also massless particles generate very small p-adic mass, which is however proportional to 1/p rather than 1/p^{1/2. The map is well defined also for massless states as a limit and takes massless momenta to the 3-ball at which upper and lower half-cones meet.}
4. What about the position of the CD associated with the mass hyperboloid? It should be possible to map all momenta to geodesic lines going through the 3-ball dividing the largest CD involved with T determined by the smallest mass involved to two half-cones. This is because this 3-ball defines the geometric "Now" in TGD inspired theory of consciousness. Therefore all CDs in H should have a common center and have the same geometric "Now".

   M⁸-H duality maps the slicing of momentum space with positive/negative energy to a Russian doll-like slicing of t≥0 by the boundaries of half-cones, where t has origin at the bottom of the double-cone. The height of the CD(m,h_eff) is given by the Compton length L(m,h_eff) = ℏ_eff/m of quark. Each value of h_eff corresponds its own scaled map and for h_gr=GMm/v₀, the size of CD(m,h_eff)=GM/v₀ does not depend on m and is macroscopic for macroscopic systems such as Sun.

5. The points of cognitive representation at quark level must have momenta with components, which are algebraic integers for the extension of rationals considered. A natural momentum unit is m_Pl=ℏ₀/R, h₀ is the minimal value of h_eff=h₀ and R is CP₂ radius. Only "active" points of X⁴⊂ M⁸ containing quark are included in the cognitive representation. Active points give rise to active CD:s CD(m,h_eff) with size L(m,h_eff).

   It is possible to assign CD(m,h_eff) also to the composites of quarks with given mass. Galois confinement suggest a general mechanism for their formation: bound states as Galois singlets must have a rational total momentum. This gives a hierarchy of bound states of bound states of ..... realized as a hierarchy of CDs containing several CDs.

6. This picture fits nicely with the general properties of the space-time surfaces as associative "roots" of the octonionic continuation of a real polynomial. A second nice feature is that the notion of CD at the level H is forced by this correspondence. "Why CDs?" at the level of H has indeed been a longstanding puzzle. A further nice feature is that the size of the largest CD would be determined by the smallest momentum involved.
7. Positive and negative energy parts of zero energy states would correspond to opposite boundaries of CDs and at the level of M⁸ they would correspond to mass hyperboloids with opposite energies.
8. What could be the meaning of the occupied points of M⁸ containing fermion (quark)? Could the image of the mass hyperboloid containing occupied points correspond to sub-CD at the level of H containing corresponding points at its light-like boundary? If so, M⁸-H correspondence would also fix the hierarchy of CDs at the level of H.

It is enough to realize the analogs of plane waves only for the actualized momenta corresponding to quarks of the zero energy state. One can assign to CD as total momentum and passive resp. active half-cones give total momenta P_tot,P resp. P_tot,A, which at the limit of infinite size for CD should have the same magnitude and opposite sign in ZEO.

The above description of M⁸-H duality maps quarks at points of X⁴ ⊂ M⁸ to states of induced spinor field localized at the 3-D boundaries of CD but necessarily delocalized into the interior of the space-time surface X⁴ ⊂ H. This is analogous to a dispersion of a wave packet. One would obtain a wave picture in the interior.

Does Uncertainty Principle require delocalization in H or in X⁴?

One can argue that Uncertainty Principle (UP) requires more than the naive condition T=ℏ_eff/m on the size of sub-CD. I have already mentioned two approaches to the problem: they could be called inertial and gravitational representations.

1. The inertial representations assigns to the particle as a space-time surface (holography) an analog of plane wave as a superposition of space-time surfaces: this is natural at the level of WCW. This requires delocalization space-time surfaces and CD in H.
2. The gravitational representation relies on the analog of the braid representation of isometries in terms of the projections of their flows to the space-time surface. This does not require delocalization in H since it occurs in X⁴.

Consider first the inertial representation. The intuitive idea that a single point in M⁸ corresponds to a discretized plane wave in H in a spatial resolution defined by the total mass at the passive boundary of CD. UP requires that this plane wave should be realized at the level of H and also WCW as a superposition of shifted space-time surfaces defined by the above correspondence.

1. The basic observation leading to TGD is that in the TGD framework a particle as a point is replaced with a particle as a 3-surface, which by holography corresponds to 4-surface.

   Momentum eigenstate corresponds to a plane wave. Now planewave could correspond to a delocalized state of 3-surface - and by holography that of 4-surface - associated with a particle.

   A generalized plane wave would be a quantum superposition of shifted space-time surfaces inside a larger CD with a phase factor determined by the 4-momentum. M⁸-H duality would map the point of M⁸ containing an object with momentum p to a generalized plane wave in H. Periodic boundary conditions are natural and would force the quantization of momenta as multiples of momentum defined by the larger CD. Number theoretic vision requires that the superposition is discrete such that the values of the phase factor are roots of unity belonging to the extension of rationals associated with the space-time sheet. If momentum is conserved, the time evolutions for massive particles are scalings of CD between SSFRs are integer scalings. Also iterated integer scalings, say by 2 are possible.

2. This would also provide WCW description. Recent physics relies on the assumption about single background space-time: WCW is effectively replaced with M⁴ since 3-surface is replaced with point and CP₂ is forgotten so that one must introduce gauge fields and metric as primary field variables.

As already discussed, the gravitational representation would rely on the lift/projection of the flows defined by the isometry generators to the space-time surface and could be regarded as a "subjective" representation of the symmetries. The gravitational representation would generalize braid group and quantum group representations.

The condition that the "projection" of the Dirac operator in H is equal to the modified Dirac operator, implies a hydrodynamic picture. In particular, the projections of isometry generators are conserved along the lifted flow lines of isometries and are proportional to the projections of Killing vectors. QCC suggests that only Cartan algebra isometries allow this lift so that each choice of quantization axis would also select a space-time surface and would be a higher level quantum measurement.

See the article TGD as it is towards the end of 2021 or the chapter with the same title.

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

Articles and other material related to TGD.