Induction coils in many-sheeted space-time

I have been trying to concretize many-sheeted space-time by thinking what simple systems involving electric and magnetic fields would look like in many-sheeted space-time. The challenge is highly non-trivial since the basic difference between Maxwell's theory and TGD is that allows extremely limited repertoire of preferred extremals and there is no linear superposition.

1. By general coordinate invariance only 4 field like variables (say CP₂ coordinates) are possible meaning that all classical fields identified as induced fields are expressible in terms of only four field like variables at at given sheet. This has several implications.
   

   The classical field equations determining the space-time surface theory is extremely non-linear although they have simple interpretation as expression for local conservation laws of Poincare charges and color charges. Linear superposition of Maxwell's equations is lost.
   

   Only for so called topological light rays ("massless extremals"), MEs) the linear superposition holds true but in extremely limited sense: for the analogous of plane waves travelling in either direction along ME. One has pulses of arbitrary shape preserving their shape and propagating in single direction only with maximal signal velocity.
   
   

2. Strong form of holography (SH) implies that 2-dimensional data at string world sheets and partonic 2-surfaces fix the space-time surfaces. 2-D data include also the tangent spaces of partonic 2-surfaces so that the situation is only effectively 2-D and TGD does not reduce to any kind of string model.
   

   It is possible that the light-like 3-surfaces defining parton orbits as the boundaries of Minkowskian and Euclidian space-time regions possess dynamical degrees of freedom as conformal equivalence classes. Kac-Moody type transformations trivial at the ends of partonic orbit at boundaries of causal diamond (CD) would generate physically equivalent partonic orbits. There would be n conformal equivalence classes, where n would correspond to the value of Planck constant h_eff=n× h. At the ends of orbit all these n sheets of the singular covering would co-incide. Possible additional degrees of freedom making partonic 2-surfaces somewhat 3-D would be therefore discrete and make possible dark matter in TGD sense.
   
   

What is clear that single space-time sheet is very simple entity, and one can assign to it only extremely limited set of say solutions of Maxwell's equations. More complex solutions must correspond to many-sheeted space-time surfaces approximated as slightly curved pieces of Minkowski space at the GRT-QFT limit of TGD.

This limit is discovered by noticing that a test particle touches all sheets of the space-time surface in a given region of Minkowski space - they are extremely near to each other. Test particle experiences sums for the induced gauge potentials and gravitational fields defined is deviations of the induced metric from flat Minkowski metric. These sums corresponds naturally to the gauge potentials and gravitational fields assignable to the GRT-QFT limit. One obtains GRT plus standard model.

The challenge is to look whether one can indeed construct typical Maxwellian field configurations as sums of electromagnetic gauge potentials represented as induced gauge potentials at various sheets. The simplest configurations would be realizable using only two sheets.

I have already considered the realization of standing waves not possible as single sheeted structures as ≥ 2-sheeted structures carrying the analogs of sinusoidal waves (see this).

1. The proposal is that magnetic bodies (MBs) use this kind of standing wave patters to generate biological structures: charged biomolecules would end up the nodal surfaces of the standing wave and become stationary structures. Of course, also time varying nodes are possible.
   
   
2. MB could use the MEs parallel to flux tubes connected to a given node of tensor network to generate biological structures at the node. Note that the interference patter would be completely analogous to that of a hologram but allowing more than two waves. As a matter of fact, I considered a vision about living systems as conscious holograms for decades ago (see this) but was not able to invent a concrete model at that time. This Chladni mechanism - as one might call it - could be a general mechanism of morphogenesis and morphostasis.
   
   

Second challenge - raised by discussions with my friend Samppa - is provided by the field patterns of an inductance coil with AC current flowing around the boundary of cylinder.

1. The current is typically AC current. Oscillating magnetic field has direction parallel or opposite to the cylinder and electric field field lines rotate around the cylinder. That the geometry of field pattern is this is easy to understand by looking just the general form of the solutions of the Maxwell's equations in question.
   
   
2. What is essential is that one has standing wave type field pattern meaning that the fields at all points of the cylinder oscillates in the same phase. The temporal and spatial dependences of the magnetic field separate into product of sinusoidal function and spatial function, which in the simplest situation is constant. One might even regard the standing wave property as a signal of quantum coherence.
   
   

Could one use MEs as building bricks to construct the field pattern associated with the coil?

1. MEs define an extremely general set of (hopefully preferred) extremals of Kähler action. Basic type of ME corresponds to cylindrical regions inside which pulses propagate in the same direction along the cylinder. It is also possible to have waves propagating radially: either inwards or outwards. The ansatz for ME is following. Restrict the consideration to a geodecic sphere S² of CP₂, assume that S² coordinates (Θ,Φ) have dependence [sin(Θ)=sin(ω (t+/- ρ),Φ=nφ ], where [ρ,φ] are cylindrical radial and angle coordinates. The resulting magnetic field is in z-direction and the lines of the electric field rotate around the cylinder. The sum of these fields is a standing wave in radial direction representing just right kind of magnetic and electric fields.
   
   
2. What is amusing that the field experienced by the test particle would be expressible as sum of the two modes of TGD counterpart of radiation field. If the AC frequency is 50 Hz, the period of radial cylindrical wave characterized by wave vector k=ω (c=1) is of order wave length λ=2π/k, which is of order 10⁷ meters, order of magnitude of Earth radius! Hence the longitudinal magnetic field is essentially constant for the coils encountered in practical situation.
   
   
3. The boundary of the cylinder carries the AC current. The description of this current is a further challenge to TGD
   and will not be considered here.
   
   

One can of course have more general currents generating much more general waves, not necessary standing waves.

1. The general recipe would be simple. These fields can be expressed as a Fourier decomposition of simple sinusoidal field patters. Assign to each sinusoidal field pattern a space-time sheet in the proposed manner so that superposition for modes is replaced with union of space-time surfaces.
   
   
2. The more terms in the Fourier expansion, the larger the number of sheets for the many-sheeted space-time is. The number of space-time sheets gives a measure for the complexity of the system. For instance, a current with form of square pulse is an interesting challenge. Should one approximate the square pulse as a superposition of space-time sheets of its Fourier components?
   
   

See the chapter TGD Based View about Classical Fields in Relation to Consciousness Theory and Quantum Biology of "TGD and EEG".

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

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