### Too thin em waves in graphene

I got from Sebastian a link to a popular article about finding that light can be squeezed to much smaller volume than the wave length using single sheet of graphene. The original article by David Alcaraz Aronzo et al is published in Science. A naive application of Uncertainty Principle suggests that this is impossible since this would mean a very large expectation value of momentum squared in transverse momentum degrees of freedom.

The finding is interesting from the point of view of classical limit of TGD. So called massless extremals (MEs) or topological light rays are extremely general solutions of field equations (practically independently of the details of the action principle: it is enough that it is general coordinate invariant). The counterparts of MEs are not possible in Maxwellian electrodynamics but TGD allows them because of the extreme non-linearity of the underlying geometric variational principle for which the topological pair of Maxwell's equations involving no currents is identically satisfied.

**What MEs are?**

MEs are 4- surfaces describing the propagation of massless topological field quanta of induced classical fields characterized by light-like propagation direction and polarization orthogonal to it. Classical 4-momentum is light-like. The propagation occurs with maximal signal velocity, and there is no dispersion so that pulse shape is preserved. If there are several pulses they must propagate in the same direction. The analogy is propagation of laser beam in waveguide. MEs are be ideal for targeted communications and MEs associaged with magnetic flux tubes and carrying dark photons assignable with wormohole contacts play a key role in TGD inspired quantum biology. A possible interpretation is as space-time correlates Bose-Einstein condensate of photons. Photons themselves would correspond to wormhole contacts (actually pairs of them) connecting ME to another space-time sheet, which could be magnetic flux tube or even ME.

MEs have finite size scale in directions orthogonal to the direction of propagation and MEs can be arbitrary thin. I do not see any reason why they could not be thinner than wavelength. The graphene seems to provide a situation in which classical modelling by MEs makes sense.

The QFT limit is obtained from many-sheeted space-time by replacing many-sheeted structure with a region of Minkowski space made slightly curved. Gauge potentials and gravitational fields are sums of the corresponding induced fields at space-time sheets so that the effects of these fields on a test particle sum up although fields themselves are at different space-time sheets. The linear superposition of Maxwell's theory is replaced with a set theoretic union of the space-time sheets in M^{4}×CP_{2}. The effects of the fields of space-time sheets on the test particle sum up just like in superposition of fields in Maxwell's theory.

For instance, this allows a situation in which one has two MEs describing propagation of signals in opposite directions as far effects on test particles are considered. This gives rise to standing waves not possible in TGD as single sheeted extremals. Lorentz transforms give analogs of em signals propagating with arbitrary velocity smaller than light velocity. Even field patterns for which the QFT limit corresponds to vanishing fields because the effects on test particles are trivial are possible: both sheets however carry non-vanishing fields with non-vanishing energy-momentum density.

**Why the apparent breaking of Uncertainty Principle?**

Why the apparent breaking of Uncertainty Principle is then possible in TGD? The point is that in TGD particles do not correspond to wavefunctions in a fixed space-time - this is true only at quantum field theory limit of TGD. Instead, they correspond to wave functions in what I call "world of classical worlds" (WCW). 3-space as "world" is in TGD replaced with 3-D surface defining the "world". In zero energy ontology (ZEO) one can identify space-time surfaces as preferred extremals of an action, which in a well-defined sense generalizes the Maxwell action for a point like charged particle. Thanks to holography, the space-time surface is characterized by 2 3-surfaces at its opposite ends - loci for initial and final states - located at the opposite boundaries of causal diamond (CD), whose M^{4} projection is intersection of future and past directed light-cones and would look like diamond if it were 2-D.

The world as 3-D surface or equivalently 4-D surface is the quantum dynamical object and space-time ceases to be a passive arena of dynamics. Uncertainty Principle holds true for wave functions in WCW rather than for induced fields at space-time surfaces. Therefore the apparent breaking of Uncertainty Principle is possible.

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

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