About TGD counterparts of twistor amplitudes

The twistor lift of TGD, in which H=M⁴ × CP₂ is replaced with the product of twistor spaces T(M⁴) and T(CP₂), and space-time surface X⁴⊂ H with its 6-D twistor space as 6-surface X⁶ ⊂ T(M⁴)× T(CP₂), is now a rather well-established notion and M⁸-H duality predicts it at the level of M⁸.

Number theoretical vision involves M⁸-H duality. At the level of ⊂H⊂, the quark mass spectrum is determined by the Dirac equation in ⊂H⊂. In M⁸ mass squared spectrum is determined by the roots of the polynomial P determining space-time surface and are in general complex. By Galois confinement the momenta are integer valued when p-adic mass is used as a unit and mass squared spectrum is also integer valued. This raises hope about a generalization of the twistorial construction of scattering amplitudes to TGD context.

It is always best to start from a problem and the basic problem of the twistor approach is that physical particles are not massless.  

1. The intuitive TGD based proposal has been that since quark spinors are massless in H, the masslessness in the 8-D sense could somehow solve the problems caused by the massivation in  the construction of twistor scattering amplitudes. However, no obvious mechanism has been identified. One step in this direction was the realization that in H quarks propagate with well-defined chiralities and only the square of Dirac equation is satisfied. For a quark of given helicity the  spinor can be identified as helicity spinor.
2.  M⁸ quark momenta are in general complex as algebraic integers. They are the counterparts of the area momenta x_i of momentum twistor space whereas H momenta are identified as ordinary momenta. Total momenta  of Galois confined states  have as components ordinary integers.
3.   The  M⁸ counterpart of  the 8-D massless condition in H is the restriction of momenta to mass shells m²= r_n determined as roots  of P. The M⁸ counterpart of Dirac equation in H is octonionic Dirac equation, which is algebraic as everything in M⁸ and analogous to massless Dirac equation. The solution is a helicity spinor λ associated with the massive momentum. 

The outcome is an extremely simple proposal for the scattering amplitudes.

1. Vertices correspond to trilinears of Galois confined many-quark states as states  of super symplectic algebra acting as isometries of the "world of classical worlds" (WCW). Quarks are on-shell with H  momentum p and M⁸ momenta x_i,x_i+1, p_i=x_i+1-x_i. Dirac operator x^k_iγ_k restricted to fixed helicity L,R appears as a vertex factor and has an  interpretation as a residue of a pole from an on-mass-shell propagator D so that a correspondence with twistorial construction becomes obvious. D  is expressible in terms of the helicity spinors of given chirality and gives two independent holomorphic factors as in case of massless theories.
2. MHV construction utilizing k=2 MHV amplitudes as building bricks does not seem to be needed at the level of a single space-time surface. One can of course ask, whether the M⁸ quark lines could be regarded as analogs of lines connecting different MHV diagrams replaced with Galois singlets. The scattering amplitudes would be rational functions in accordance with the number theoretic vision. The absence of logarithmic radiative corrections is not a problem: the coupling constant evolution would be discrete and defined by the hierarchy of extensions of rationals.
3. The scattering amplitudes for a single 4-surface X⁴ are determined by a polynomial. The integration over WCW is replaced with a summation of polynomials characterized by rational coefficients. Monic polynomials are highly suggestive. A connection with p-adicization emerges via the identification of the p-adic prime as one of the ramified primes of P. Only (monic) polynomials having a common p-adic prime are allowed in the sum. The counterpart of the vacuum functional exp(-K) is naturally identified as the discriminant D of the extension associated with P and p-adic coupling constant evolution emerges from the identification of exp(-K) with D.

See the article About TGD counterparts of twistor amplitudes 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. 


What would you choose: Alcubierre drive or superluminal quantum teleportation in zero energy ontology?

There was an interesting popular article (see this) about a theoretical article "Worldline numerics applied to custom Casimir geometry generates unanticipated intersection with Alcubierre warp metric" by Harold White et al published in European Physical Journal (see this).

The article claims that a calculation of the Casimir energy (see this) for a system of two parallel metal plates using what is called world-line numerics predicts that the region between capacitor plates has a torus-like region inside which the vacuum energy density is negative (note that the vacuum energy depends on the shape and size of the cavity for a quantum field theory restricted inside the capacitor by posing suitable boundary conditions).

The observation is that the vacuum energy density resembles that for the so-called Alcubierre drive (see this) claimed to make possible space-time travel with superluminal speeds with respect to the time coordinate of an asymptotically flat space-time region. The idea is that if the space contracts in front of the space-ship and expands behind, super-liminality becomes possible. Inside the space-ship the space-time would be in a good approximation flat. Alcubierre himself suggests that the Casimir effect might produce the needed negative energy density.

It is easy for a skeptic to invent objections. Consider first the calculation of the vacuum energy behind Casimir force, which is a real effect and has been experimentally detected.

1. The original calculation of Casimir was for van der Waals forces. It has been later show that Casimir effect could be interpreted as retarded van der Waals force (see this): the consideration of poorly defined vacuum energies was not needed in this approach.

   Later emerged the proposal that one can forget the interpretation as an interaction between molecules and that the calculation applies by considering the system as idealized conducting capacitor plates. There are objections against this interpretation.

2. Force is the negative gradient of energy. The predicted force is finite although the calculation of the vacuum energy gives an ultraviolet divergent infinite answer requiring a regularization. The regularization gives a finite result as an energy E per area A of the plates, which is negative and given by

   E/A= -ℏπ²/720a³.

   a is the distance between plates and force is proportional to 1/a⁴.

   Note that there is no dependence on the fine structure constant or any other fundamental coupling strengths. Casimir energy and force approach rapidly to zero when a increases so that practical applications to space-travel do not look feasible.

Also the notion of Alcubierre drive can be criticized.

1. The basic problem of general relativity (GRT) is that the notions of energy and momentum and corresponding conservation laws are lost: this was the starting point of TGD. In weak gravitational fields in which space-time is a small metric deformation of empty Minkowski space-time, one can expect that these notions make approximately sense. However, Alcubierre drive represents a situation in which the deviation from a flat Minkowski space is large. Does it make sense to speak about (conserved) energy anymore?

2. If one accepts GRT in this kind of situation one still has the problem that negative energy density violates the basic assumptions of GRT. Some kinds of exotic matter with negative energy would suggest itself if one believes that energy corresponds to some kind of particles

3. One can also argue that the proposed effect is a kind of Munchausen trick. The situation must allow an approximate GRT based description by regarding space-time ship as a single unit whose energy is determined by the sum of the energy of the space-time ship and Casimir energy and is positive so that the space-ship moves in a good approximation along time-like geodesic of the background space-time. The corrections to this picture taking into account the detailed structure of the space-ship should not change the description in an essential manner and only add small scale motion superposed to the center of mass motion.

What about the situation in TGD?

1. The notions of energy and momentum are well-defined and the classical conservation laws are not lost. The conserved classical energy assignable to space-time surface is actually analogous to Casimir energy although it is not assigned to vacuum fluctuations and consists of the contributions assignable to Kähler action and volume action. These contributions depend on Kähler coupling strength and cosmological constant which in the TGD framework is (p-adic) length scale dependent. Recall that for the parallel conductor plates at least, Casimir energy has no dependence on fundamental coupling strengths.

   If the energy is positive definite in TGD as there are excellent reasons to believe, the basic condition for the Alcubierre drive is not satisfied in TGD.

2. Here I must however counterargue myself. One can construct very simple space-time surfaces for which the metric is flat and Euclidean and they are extremals of the basic variational principle.

   1. Consider a surface representable as a graph of a map M⁴× CP₂ given by Φ= ω t, where Φ is angle coordinate of the geodesic circle of CP₂. The time component g_tt= 1-R²ω² of the induce flat metric is negative for ω >1/R.

      The energy density associated with the volume part of the action is non-vanishing and proportional to (g_tt^1/2 g^tt and negative. The coefficient is analogous to cosmological constant.

   2. Can these "tachyonic" surfaces correspond to preferred extremals of the action, which are physically analogous to Bohr orbits realizing holography. The 3-D intersections of this solution with two t= constant time slices are Euclidean 3-spaces E³ or identical pieces of E³. If the preferred extremal minimizes its volume action, then (g_tt^1/2=(1-R²ω²)^1/2 is maximum. This gives ω R=0 and a flat piece of M⁴.

      Interestingly, the original formulation for what is it to be a preferred extremal (as a condition for holography required by the realization of general coordinate invariance), was that space-time surfaces are absolute minima for the action which at that time was assumed to be mere Kähler action. The twistor lift of TGD forced the inclusion of the volume term. It seems that Alcubierre drive is not possible in TGD.

      It might be also possible to show this by demonstrating that the embedding of the Alcubierre metric as a 4-surface in M⁴× CP₂ is not possible.

   3. TGD also allows different kinds of Euclidean regions as preferred extremals. These correspond to what I call CP₂ type extremals. They have positive energy density and they have light-like geodesics as M⁴ projection and they serve as classical geometric models for fundamental particles.

Recall that the motivating problem of the article was how to avoid the restrictions posed by the finite light-velocity on space travel. The first thing that comes to mind is that it is not very clever to move a lot of steel and other metals to distant parts of the Universe. Quantum teleportation allows us to consider a more advanced form of space travel. One could send only the information needed to reconstruct the transferred system at the second end. Reconstruction of the space travelers is quite a challenge with existing technology but the less ambitious goal of sending just the information looks more promising and qubits have been already teleported.

Concerning superluminal teleportation, the problem is that in the standard quantum theory teleportation also requires sending of classical information. Maximal signal velocity makes superluminal teleportation impossible. This poses extremely stringent limits on the communications with distant civilizations.

In the zero energy ontology of TGD (see this and this), the situation changes.

1. In the so-called "big" state function reductions (BSFRs), which are the TGD counterparts for ordinary SFRs, the arrow of time changes.
2. For light-signal this means that the signal is reflected in time direction and returns back in time with a negative energy (this brings in mind the negative energy condition for Alcubierre drive). This is just like ordinary reflection but in time direction perhaps allowing seeing in time direction I have proposed conscious memory recall could correspond to this kind if seeing in time direction.
3. This might also make practically instantaneous classical communications over space-like distances possible. This in turn would also make possible superluminal quantum teleportation.

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

Articles and other material related to TGD. 


How bubbles in coffee cup, hyperbolic geometry, and magnetic body could relate?

I received in Facebook from Runcel D. Arcaya a beautiful photo of fractal bubble structure formed at the surface of liquid in coffee cup.

This picture is fascinating and inspires questions. What could be the physics behind this structure? Is standard hydrodynamics really enough?

The bubble structures from the point of view of standard physics

Let us list what looks like standard physics.

1. In the interior of the liquid bubbles of air are formed: this requires energy feed, say due to shaking.
2. The bubbles are lighter than liquid and rise to the surface and eventually develop holes and disappear.
3. The pressure inside the bubble is higher than in air outside and this gives rise to a spherical shape locally. Otherwise one would have a minimal surface with vanishing curvature (locally a saddle). The role of pressure means that thermodynamics is needed to understand them.

What kind of dynamics could give rise to the fractality and the beautiful structure with a big bubble in the center.

1. The structure brings in mind the illustrations of 2-D hyperbolic geometry and its model as Poincare disk.

   One can also imagine concentric rings consisting of bubbles with scaled down size as radius increases. Atomic structure or planetary system of bubbles also comes into mind!

2. The so- called (rhombitriheptagonal tiling) of the Poincare disk in the plane comes to mind. Could the bubble structure be associated with a tiling/tessellation of H² represented as a Poincare disk?

What about TGD description of the bubble structure?

Could one understand the bubble structures in classical TGD in which 4-D space-time surfaces in M⁴× CP₂ would be minimal surfaces with singularities?

1. Tessellations of 3-D hyperbolic space H³, which has H² as sub-manifold, are realized in mass shell of momentum space familiar to particle physicists and also in 4-D Minkowski space M⁴ \subset M⁴× CP₂ as the surface t²-r²= a²,where a is light-cone proper time and identified as Lorentz invariant cosmic time in cosmologies. H³ is central in TGD.
2. Tessellations of H³ induce tessellations of H² and they could in some sense induce 2-D tessellations at the space-time surface, magnetic body (MB) of any system is an excellent candidate in this respect.

   This would be a universal process. For instance, genetic code could be understood as a universal code associated with them and genes would be 1-D projections of the H³ tessellations known as icosa-tetrahedral tessellation.

   I have proposed that cell membranes could give rise to 2-D realizations of genetic code as an abstract representation of the 3-D tessellation at MB (see this).

3. Could these "bubble tessellations" somehow correspond to the 3-D tessellations of H³, not necessarily the icosa-tetrahedral one, since there is an infinite number of different tessellations of H³?

The basic problem is that the Poincare disk has a finite radius and H² consists of the entire Euclidean plane. Furthermore, the Euclidean plane has vanishing curvature while Poincare disk has a negative curvature. The simplistic attempts indeed fail.

1. There is no natural map from the hyperbolic plane H² to the Poincare disk. Projection is impossible since it would require an infinite compression of the Euclidean plane.
2. What about realizing Poincare plane as 2-surface in M⁴× CP₂ with induced metric equal to the metric of Poincare disk given by ds²_P =ds²_E/(1-ρ²): here ds²_E is the Euclidean metric of the plane and ρ its radial coordinate.

Simple realizations seem implausible. Presumably the negative curvature of H² in contrast to the positive curvature of CP₂ and vanishing curvature of the Euclidean plane is the problem.

The following represents a possible successful attempt.

1. Could the MB of the system, which can realize the tesselations, somehow induce the discrete Poincare disk based bubble structure as a discrete representation of its 2-D hyperbolic sub-geometry? The distances between the discrete points of the bubble representation, say the positions of bubbles, at MB would have hyperbolic distances and induced correspond to the distances at MB.
2. There is evidence that in a rather abstract statistical sense neurons of the brain obey a hyperbolic geometry. Neurons functionally near to each other are near to each other in an effective hyperbolic geometry. Hyperbolic geometry at the level of the MB of the brain could realize this concretely. Neurons functionally near each other could send their signals to points near each other at MB of the brain (see this.

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

Articles and other material related to TGD.

Spin Glasses, Complexity, and TGD

Spin glasses represent an exotic phenomenon, which remains poorly understood in the standard theoretical framework of condensed matter physics. Actually, spin glasses provide a prototype of complex systems and methods used for spin glasses can be applied in widely different complex systems.

A TGD inspired view about spin glasses is discussed.

1. TGD view about space-time leads to the notion of magnetic flux tubes and magnetic body. Besides spins also long closed magnetic flux tubes would contribute to magnetization. The basic support for this assumption is the observation that the sum of the NFC magnetization and the FC remanence is equal to the NFC magnetization. Magnetic field assignable to spin glass would correspond to a kind of flux tube spaghetti and the couplings J_ij between spins would relate to magnetic flux tubes connecting them.
2. Quantum TGD leads to the notion of "world of classical worlds" (WCW) and to the view about quantum theory as a "complex square root" of thermodynamics (of partition function). The probability distribution for {J_ij} would correspond to ground state functional in the space of space-time surfaces analogous to Bohr orbits.
3. Spin glass is a prototype of a complex system. In the TGD framework, the complexity reduces to adelic physics fusing real physics with various p-adic physics serving as correlates of cognition. Space-time surfaces in H=M⁴× CP₂ correspond to images of 4-surfaces X⁴⊂ M⁸_c mapped to H by M⁸-H duality. X⁴ is identified as 4-surface having as holographic boundaries 3-D mass shells for which the mass squared values are roots of an octonionic polynomial P obtained as an algebraic continuation of a real polynomial with rational coefficients. The higher the degree of P, the larger the dimension of the extension of rationals induced by its roots, and the higher the complexity: this gives rise to an evolutionary hierarchy. The dimension of the extension is identifiable as an effective Planck constant so that high complexity involves a long quantum coherence scale.

   TGD Universe can be quantum critical in all scales, and the assumption that the spin glass transition is quantum critical, explains the temperature dependence of NFC magnetization in terms of long range large h_eff quantum fluctuations and quantum coherence at critical temperature.

4. Zero energy ontology predicts that there are two kinds of state function reductions (SFRs). "Small" SFR would be preceded by a unitary time evolution which is scaling and generated by the scaling generator L₀. This conforms with the fact that relaxation rates for magnetization obey power law rather than exponent law. "Big" SFRs would correspond to ordinary SFRs and would change the arrow of time. This could explain aging, rejuvenation and memory effects.
5. Adelic physics leads to a proposal that makes it possible to get rid of the replica trick by replacing thermodynamics with p-adic thermodynamics for the scaling operator L₀ representing energy. What makes p-adic thermodynamics so powerful is the extremely rapid converges of Z in powers of p-adic prime p.

See the article Spin Glasses, Complexity, and TGD 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.