- The map pk→ mk= ℏeffpk/p2 defining M8-H duality is consistent with Uncertainty Principle but this is not quite enough. Momenta in M8 should correspond to plane waves in H.
Should one demand that the momentum eigenstate as a point of cognitive representation associated with X4⊂ M8 carrying quark number should correspond to a plane wave with momentum at the level of H=M4× CP2? This does not make sense since X4⊂ CD contains a large number of momentaassignable to fundamental fermions and one does not know which of them to select.
- One can however weaken the condition by assigning to CD a 4-momentum, call it P. Could one identify P as
- the total momentum assignable to either half-cone of CD
- or the sum of the total momenta assignable to the half-cones?
- Momentum conservation for a single CD is an ad hoc assumption in conflict with Uncertainty Principle, and does not follow from Poincare invariance. However, the sum of momenta vanishes for non-vanishing planewave when defined in the entire M4 as in QFT, not for planewaves inside finite CDs. Number theoretic discretization allows vanishing in finite volumes but this involves finite measurement resolution.
- Zero energy states represent scattering amplitudes and at the limit of infinite size for the large CD zero energy state is proportional to momentum conserving delta function just as S-matrix elements are in QFT. If the planewave is restricted within a large CD defining the measurement volume of observer, four-momentum is conserved in resolution defined by the large CD in accordance with Uncertainty Principle.
- Note that the momenta of fundamental fermions inside half-cones of CD in H should be determined at the level of H by the state of a super-symplectic representation as a sum of the momenta of fundamental fermions assignable to discrete images of momenta in X4⊂ H.
M8-H-duality as a generalized Fourier transform
This picture provides an interpretation for M8-H duality as a generalization of Fourier transform.
- The map would be essentially Fourier transform mapping momenta of zero energy as points of X4⊂ CD⊂ M8 to plane waves in H with position interpreted as position of CD in H. CD and the superposition of space-time surfaces inside it would generalize the ordinary Fourier transform . A wave function localized to a point would be replaced with a superposition of space-time surfaces inside the CDhaving interpretation as a perceptive field of a conscious entity.
- M8-H duality would realize momentum-position duality of wave mechanics. In QFT this duality is lost since space-time coordinates become parameters and quantum fields replace position and momentum as fundamental observables. Momentum-position duality would have much deeper content than believed since its realization in TGD would bring number theory to physics.
How to describe interactions of CDs?
Any quantum coherent system corresponds to a CD. How can one describe the interactions of CDs? The overlap of CDs is a natural candidate for the interaction region.
- CD represents the perceptive field of a conscious entity and CDs form a kind of conscious atlas for M8 and H. CDs can have CDs within CDs and CDs can also intersect. CDs can have shared sub-CDs identifiable as shared mental images.
- The intuitive guess is that the interactions occur only when the CDs intersect. A milder assumption is that interactions are observed only when CDs intersect.
- How to describe the interactions between overlapping CDs? The fact thequark fields are induced from second quantized spinor fields in in H resp. M8 solves this problem. At the level of H, the propagators between the points of space-time surfaces belonging to different CDs are well defined and the systems associated with overlapping CDs have well-defined quark interactions in the intersection region. At the level of M8 the momenta as discrete quark carrying points in the intersection of CDs can interact.
Zero energy states as scattering amplitudes and subjective time evolution as sequence of SSFRs
This is not yet the whole story. Zero energy states code for the ordinary time evolution in the QFT sense described by the S-matrix. What about subjective time evolution defined by a sequence of "small" state function reductions (SSFRs) as analogs of "weak" measurements followed now and then by BSFRs? How does the subjective time evolution fit with the QFT picture in which single particle zero energy states are planewaves associated with a fixed CD.
- The size of CD increases at least in statistical sense during the sequence of SSFRs. This increase cannot correspond to M4 time translation in the sense of QFTs. Single unitary step followed by SSFR can be identified asa scaling of CD leaving the passive boundary of the CD invariant. One can assume a formation of an intermediate state which is quantum superposition over different size scales of CD: SSFR means localization selecting single size for CD. The subjective time evolution would correspond to a sequence of scalings.
The crucial point is that scalings commute with Poincare symmetries. Subjective and Poincare time evolutions commute.
- The view about subjective time evolution conforms with the picture of string models in which the Lorentz invariant scaling generator L0 takes the role of Hamiltonian identifiable in terms of mass squared operator allowing to overcome the problems with Poincare invariance. This view about subjective time evolution also conforms with super-symplectic and Kac-Moody symmetries of TGD.
One could perhaps say that the Minkowski time T as distance between the tips of CDs corresponds to exponentiated scaling: T= exp(L0t). If t has constant ticks, the ticks of T increase exponentially.
For a summary of earlier postings see Latest progress in TGD.