On the Structure of Space-Time and Matter as Obtained from the Planck Scale by Period Doubling in Three and Four Dimensions
| Scientific Paper | |
|---|---|
| Title | On the Structure of Space-Time and Matter as Obtained from the Planck Scale by Period Doubling in Three and Four Dimensions |
| Read in full | Link to paper |
| Author(s) | Ari Lehto |
| Keywords | Structure, Space-Time, Matter, Planck Scale, Period Doubling |
| Published | 2006 |
| No. of pages | 16 |
Read the full paper here
Abstract
One of the most interesting questions in modern physics is the possible relation of the Planck scale to our perceived world. The Planck energy (1022 MeV) is extremely large as compared to the rest energies of the elementary particles and the Planck length (10-35 m) is too short to be directly connected to any real world distances. Why the Planck scale is interesting is that it is absolute, as it is determined by the natural constants h, c, G and eo. Another reason is that the Planck scale may represent the ultimate "graininess", i.e. the basic structure, of the space-time and matter.
It is well known that nonlinear systems show universal behavior in the form of period doubling, which is the same as frequency and energy halving. If period doubling is applied to the Planck energy Eo= h/to, an absolute and unadjustable set of sublevels is borne. Spatial period doubling will correspondingly yield a set of increasing lengths. The rest energy of the electron-positron pair is given directly by a Planck energy sublevel, whereas the nucleon rest energies originate from a sum energy of two adjacent sublevels.
It is also shown that the value of the elementary electric charge squared, which is proportional to energy, results from the Planck charge squared by the same period doubling process.
It is further shown that the planets in the Solar system occupy orbits, the radii of which can be calculated from the Planck length by spatial period doubling. A spectrum of velocities can be calculated from the speed of light by the same process. These velocities fit the consequent orbital velocities of the planets and the quantized redshifts of galaxies, if redshift is interpreted as velocity.
A hypothesis is made that the invariant properties and structures of matter are related to periodic structures obtained by a period doubling process in three and four dimensional nonlinear systems.
PACS numbers: 04.20.Gz, 05.45.-a, 05.45.Mt
Overview
Ari Lehto, of the Laboratory of Materials Science at Helsinki University of Technology, here sets out the mature form of a programme he had been developing since 1984: that the stable, invariant properties of matter — particle rest energies, magnetic moments, the elementary charge, and even planetary orbits and galaxy redshifts — are not independent empirical facts but successive halvings of a single set of absolute units fixed by h, c, G and ε0. The mechanism he borrows is period doubling, the route to chaos identified by Feigenbaum in nonlinear systems, which by the Planck relation E = h/t is the same thing as exact energy halving.
The paper's ambition is to close the gulf between the Planck scale and the observed world without introducing a single adjustable parameter. Nothing is fitted: the sublevels follow inexorably from the Planck units once the number of doublings N is chosen, and Lehto's whole claim is that the values of N picked out by nature are of the superstable form N = Σ2i. This departs from the mainstream account in three ways at once. It replaces the Standard Model's parameters with a counting scheme; it denies that spin is fundamental, treating magnetic moment as the primitive quantity instead; and it treats the observed quantization of galaxy redshifts — a phenomenon orthodox cosmology regards as a selection artefact — as a genuine structural effect on the same footing as the electron mass. The framework is developed alongside, and acknowledges a debt to, Suntola's Dynamic Universe, in which the fourth dimension is spatial and E = mc2 arises from expansion at the speed of light in that direction.
The model
Planck units and the doubling formulae
Lehto tabulates the units he needs: length lo = 4.0513 × 10−35 m, period to = 1.3513 × 10−43 s, single-level energy Eos = 3.0603 × 1022 MeV, Planck charge qo = 4.7012 × 10−18 As, temperature To = 3.55 × 1022 K, and two unit magnetic moments — an orbital type μoe = 1.5485 × 10−46 Am2 and a radial type μop = 1.9104 × 10−46 Am2. He notes candidly that G is the limiting uncertainty, at 150 ppm, giving 75 ppm on the Planck energy, length, moment and temperature — a figure that matters, because his agreements are quoted at the tens-of-ppm level.
The central postulate is that an observer perceives the geometric mean of the periods, so that the perceived number of doublings is n = N/3 for a three-dimensional structure and n = N/4 for a four-dimensional one, where N is an integer. Volume doubling in a three-dimensional box gives VN = 2i+j+kVo, and the triple (i, j, k) is used both as a label and as a description of the object's shape — cubical when the three are equal. This yields the working formulae En = 2−N/3Eo, ln = 2N/3lo, Tn = 2−N/3To, qn2 = 2−N/4qo2, and for velocities vn = 2−N/3c, the last derived directly from li/tj = 2i−jc.
Superstability
Since the number of sublevels is vast but stable objects are few, a selection rule is needed. Lehto obtains one from Feigenbaum's functional iteration. Taking volume V = x3 as an operator acting on the period, and using x3 = t2 (Kepler's law, which follows from the 1/x form of both the Coulomb and gravitational potentials), repeated operation squares the period: 21, 22, 24, 28, … so that t = 22ito and the admissible totals are sums of the form N = Σ2i. Structures satisfying this condition are called superstable. He also notes that adjacent 3-d levels differ by a factor 21/3, i.e. some 260,000 ppm — the scale against which his few-hundred-ppm agreements are to be judged.
Particle production and charge
Materialization is modelled on electron–positron pair production from a 1.022 MeV gamma. A single sublevel splits into a pair of energy levels which become particles when charges are attached; a double level splits first into two half-energies and then into four particles.
For the elementary charge, log(e2/qo2)/log(2) gives n = 9.75, i.e. N = 39 doublings in four dimensions, decomposing as the superstable (1, 2, 4, 32). Since α = e2/2ε0hc gives α/2π = e2/qo2, it follows that α−1 = 239/4/2π = 137.045, which Lehto reports as 65 ppm from the recommended value, and e = 1.60213 × 10−19 As, 30 ppm from the recommended value. That the fine structure constant should emerge as a pure power of two divided by 2π is the paper's most striking single result. The four-dimensionality of the charge, he suggests, may mean that electric interactions proceed partly via the fourth dimension.
Particles
The superstable level N = 224, or (32, 64, 128), has energy 1.0206 MeV; adding the Coulomb energy of the elementary charge (0.0012 MeV) gives 1.0218 MeV, and halving it gives 0.5109 MeV against the measured electron rest energy 0.5110 MeV — a difference of 190 ppm. Because magnetic moment is proportional to period while energy is inversely proportional to it, the split moments correspond to N = 227, giving 9.286 × 10−24 Am2 against a measured 9.285 × 10−24, a difference of 160 ppm. Since the model's moment already is the measured value to within that figure, Lehto's electron magnetic moment "anomaly" becomes ae = −0.00016 rather than the accepted +0.00116 — negative, far smaller, and in his reading not a real anomaly at all, since "the doubling process is exact."
Nucleons are treated as composite. The double level (64, 64, 64) at 3749.2 MeV splits into four, one quarter of it being the (66, 66, 66) energy 937.31 MeV. Adding the (32, 64, 128) electron–positron structure at 1.021 MeV gives 938.33 MeV for the proton (measured 938.27, difference 62 ppm), and adding also the n = 74.33 single sublevel at 1.29 MeV gives 939.62 MeV for the neutron (measured 939.57, difference 55 ppm). The proton moment is the (64, 64, 64) radial moment 3.5240 × 10−27 Am2 multiplied by four; the neutron's is modelled as two opposed concentric loops, μn = (264.33 − 266.00)μop, matching the observed sign and, Lehto stresses, requiring "no traditional gyromagnetic ratio". These moments agree to 720 ppm (proton) and 660 ppm (neutron).
Astronomical and cosmological applications
The velocity formula v = 2−N/3c is applied to the Solar system, with the planets falling on consecutive integers from N = 38 (Mercury) to N = 48 (Pluto), the asteroids at N = 42, and — a prediction of sorts — an empty orbit at N = 44 between Jupiter and Saturn "as if a planet were missing there". Radii follow from the corresponding length formula. Lehto's interpretation is that the periodic structures of massive bodies synchronize.
The same equation, modified by replacing the cube root with a ninth root (the "Lehto–Tifft rule") to allow transitions between states, is applied to Tifft's quantized galaxy redshifts, whose most prominent period is 73 km/s and its half. Because the doubling process is a function of time if the universe's volume is expanding, Lehto suggests it may also account for Tifft's reported variable redshift periods — and, since a galaxy 10,000–100,000 light years across cannot change state coherently in a few years by any light-speed signal, that the change must occur "in the fourth dimension".
Three further coincidences are offered: N = 336, superstable (16, 64, 256), gives 21.04 cm against the 21.12 cm hydrogen spin-flip wavelength; N = 320, superstable (64, 128, 128), gives T = 2.76 K against the 2.73 K cosmic background; and in the cosmic ray spectrum N = 128, superstable (32, 32, 64), gives 4.4 PeV for the "knee" while N = 96, superstable (32, 32, 32), gives 7.1 EeV for the "ankle". He predicts a further feature at N = 160 corresponding to 2.7 × 103 GeV. Finally, the difference between the perceived doublings of mass-squared (149.33) and charge-squared (9.75) is 139.58, giving a force ratio 2139.58 ≈ 1042, offered as the electric-to-gravitational strength ratio; and 2−128/3 = 1.4 × 10−13 is matched to the weak interaction strength.
Assessment
The genuinely attractive feature of this paper is its economy. There is no free parameter anywhere: once the Planck units are fixed by h, c, G and ε0, every predicted value is 2−N/3 or 2−N/4 times a unit, with N an integer restricted further by the superstability rule N = Σ2i. That is a far more constrained structure than numerological coincidence-hunting usually allows, and Lehto deserves credit for stating the constraint in advance rather than after the fact. The α−1 = 239/4/2π result is arithmetically correct and arresting. He is also unusually forthright about his own error budget, quoting the 150 ppm uncertainty in G and — crucially — printing the 3-d level separation of about 260,000 ppm as the yardstick against which his agreements should be read. Treating the electron–positron pair rather than the electron alone as the superstable object is a defensible and non-obvious choice, and the neutron's two-loop moment construction, which reproduces the correct sign without a fitted g-factor, is the model's most substantive physical result.
Against this, the central methodological difficulty is that the scheme's discriminating power is much weaker than the ppm figures suggest. Since n = N/3 or N/4, the available levels are spaced by factors of 21/3 ≈ 1.26 or 21/4 ≈ 1.19; any target quantity is therefore within about 13 per cent of some level before any theory is applied, and the choice between 3-d and 4-d, between single and double levels, and between adding or omitting the Coulomb term or an extra (32, 64, 128) structure supplies further freedom. The reported 62 ppm for the proton is not the accuracy of a prediction but the residual after a sum of three separately chosen terms; the raw (66, 66, 66) level, 937.31 MeV, misses the proton by about 1,000 ppm, and it is the added electron–positron structure — introduced precisely because the level "is a little short" — that closes the gap. The paper says this in as many words. The superstability rule is presented as a strong constraint, yet the decompositions actually used include (32, 64, 128), (64, 64, 64), (1, 2, 4, 32), (16, 64, 256), (64, 128, 128), (32, 32, 64) and (32, 32, 32); with sums of powers of two available in any combination, and i, j, k unordered, the number of admissible N below a few hundred is large enough that a hit is not by itself evidence.
Several steps are asserted rather than derived. That an observer "perceives the geometric mean of the periods", which is what licenses the division by 3 or 4 and hence the entire non-integer exponent structure, is referred to earlier work and not argued here; without it nothing follows. The identification of Feigenbaum's period doubling — a phenomenon of dissipative nonlinear maps under a varying control parameter — with the halving of Planck-scale energies is an analogy, not a dynamical result: no map, no control parameter and no attractor is exhibited, and the "operator" argument that yields N = 2i treats volume as acting on period by fiat. The claim that spin is unnecessary is likewise a claim about two numbers, not a derivation; the model reproduces magnitudes of magnetic moments but says nothing about angular momentum conservation, the Stern–Gerlach result, or Fermi–Dirac statistics, all of which require spin independently of any moment.
The most serious empirical difficulties lie in the treatment of the anomaly and of the redshift. Lehto's ae = −0.00016 does not merely disagree with the accepted 0.00116; the electron anomaly is the most precisely verified prediction in physics, agreeing with quantum electrodynamics to better than one part in 1012, whereas Lehto's residual is dominated by his own 75 ppm uncertainty in G — which is to say his model cannot even resolve the quantity it claims to explain away, and would need G known some seven orders of magnitude better before it could be tested at all. Similarly, taking Tifft's redshift quantization as real commits the model to a claim that later large redshift surveys have not sustained, and the invocation of the fourth dimension to explain galaxy-wide simultaneous transitions is unconstrained by anything else in the paper. The Solar system fit inherits the well-known weakness of Titius–Bode-type laws — an exponential ladder will accommodate almost any set of orbital radii, and the "missing planet" at N = 44 has no independent support.
Judged as a completed physical theory, then, the paper does not establish its case: too much depends on unstated selection among many available levels, and the one place where it makes a sharp quantitative departure from measurement — the magnetic moment anomaly — is a place where measurement is overwhelmingly strong. Judged as what it says it is, a hypothesis about the invariant properties of matter, it is a disciplined and unusually parameter-free proposal whose most testable content is the predicted cosmic-ray feature at 2.7 × 103 GeV.