4d sphere Welcome !!

"The mind is not a vessel to be filled but a fire to be kindled." Plutarch.

Franklin Potter is my name. Physics is my profession.

0000-0002-9440-2291 ORCID


And I thoroughly enjoy struggling with Nature's challenges to us humans who are trying to understand the most fundamental aspects of the natural world. The 4-D sphere in the image above plays a key role in my research into the nature of leptons and quarks, the fundamental particles of Nature. Below, I tell you about some of the physics research I am doing.

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Leptons and Quarks

I spent much of the last thirty years working on a project which I created for myself: to understand what underlying mechanisms are responsible for the leptons and the quarks that comprise all things around us.

I think that I have successfully answered the challenge by connecting the quarks to the symmetries of the regular polytopes in a real 4-D space and the leptons to the symmetries of the regular polyhedrons in the 3-D subspace. This approach stays within the realm of the Standard Model and enlists the finite rotational subgroups of its Lie group, i.e., I assign subgroups of SU(2) that represent discrete symmetries, a different subgroup for each lepton family and for each quark family. The papers listed in prediction 2 below discuss the amazing successes of this geometrical approach.

*** In 2016 the acid test of the ideas may be achievable because the LHC collider at CERN will be able to analyze many times the number of collisions per second than before, and I would expect dozens of FCNC b' quark decays into b + gamma and b + gluon to make their appearance. There are some indications of two jets being produced with a total energy around 120 - 150 GeV, which may be b' quarks, or it may be the Higgs particle. No final statements can be made yet. As of 2015, still no b' quark production, although there might be a remote possibility that a b' bound state exists at around 125 GeV to 135 GeV, hidden under the boson bump in that energy region. There is also the possibility of a t' bound state exists at around 5.2 TeV.

Some important predictions of this modification of the Standard Model(SM) are:

  1. Leptons and quarks are related through the finite rotational symmetry groups in the 3-D and 4-D real spaces that are part of the 2-D complex space called the unitary plane. The unitary plane is the only space mentioned explicitly in the SM. Its Lie Group SU(3)C x SU(2)L x U(1)Y may be operating on the finite rotational groups I have for the leptons and quarks, this discrete symmetry being contrary to expectations.

  2. 4 quark families and 3 lepton families are predicted, with the particles no longer considered to be point particles. As a result of 3-D states for the lepton families and 4-D states for the quark families, I have derived the neutrino PMNS mixing matrix AND the CKM (and CKM4) mixing matrix from first principles, available online at Progress in Physics: Geometrical Derivation of the Lepton PMNS Matrix Values (2013) AND CKM and PMNS Mixing Matrices from Discrete Subgroups of SU(2) (2014), and my talk at DISCRETE 2014 in London at CKM and PMNS Mixing Matrices from Discrete Subgroups of SU(2). Concern over triangle anomaly cancellation, which traditionally requires equal numbers of lepton and quark families, is no longer a problem because the derivations of the PMNS and CKM matrices dictate also that all 3 lepton families act as one family to cancel the anomaly contribution from all 4 quark families acting as one family, as I show in Weinberg Angle Derivation from Discrete Subgroups of SU(2) and All That (2015). In this same paper I derive the Weinberg angle of 30 degrees, previously thought to be determined by experiment only!

  3. A b' quark at 80 GeV and a t' quark at 2600 GeV should exist, both members of the 4th quark family. This b' quark at 80 GeV is the acid test for my modification, and I expect it to show up at the colliders soon. One problems is their longer lifetime than normal because of the FCNC type of decay, so most may escape the detector.

  4. Neutrinos and antineutrinos are distinct particle states. At least one neutrino has a mass > 0 eV in order to solve the solar neutrino problem. By taking linear superposition states, neutrinos can have a non-zero mass.

  5. No Higgs boson is required. The actual symmetry groups are finite rotational groups, so one does not need to introduce the artificial Higgs mechanism because the symmetry breaks to these discrete groups, i.e. also the source of electroweak symmetry breaking. I.e., the Standard Model gauge group is an excellent approximation to these finite rotational groups for the different families.

  6. Quark states are defined in a 4-D real space while lepton states are defined in the 3-D subspace. This spatial demarcation distinguishes leptons from quarks and may be the origin of baryon number. Quark confinement follows also because they are 4-D states that cannot exist in 3-D space.

  7. All mass ratios arise from the numerical invariants of the symmetry groups of the polytopes and polyhedrons. The invariants are related to the absolute invariant J of elliptic functions, so any Hamiltonian must incorporate the symmetries of the appropriate elliptic functions for each finite rotational group. Linear superposition of the two degenerate states in each finite rotational group leads to the two physical states in each family.

  8. Left-handedness and parity violation are dictated by the mathematical properties of the general rotation in the unitary plane whenever a W or Z boson acts. There is no choice here, and all left-right symmetric approaches are eliminated. Also, two particle states per family are required, eliminating approaches with one additional particle state.

  9. The three color charges for quarks represent the three pairs of rotation planes for a true 4-D rotation. If correct, lepton states must exist in the 3-D subspace as proposed, so they do not have color charge. One can also define all meson and baryon states in the 3-D subspace formed mathematically by intersecting 4-D entities.

  10. No more leptons or quarks will appear beyond the four quark families and three lepton families predicted here. No more finite rotational groups are available in the 3-D and 4-D subspaces of the unitary plane. In fact, leptons, quarks and the interaction bosons may be the only fundamental particles!

  11. These fundamental particles and all fundamental physics rules are dictated by the Fischer-Greiss Monster Group. The details are outlined in my 2011 paper titled "Our Mathematical Universe: I. How the Monster Group Dictates All of Physics" in the online journal Progress in Physics: Our Mathematical Universe: I. How the Monster Group Dictates All of Physics (2011)

  12. The particles of the Standard Model and their interactions are encoded by information theory There are 72 fundamental particle states - 30 lepton and quark states (6 leptons and 8 x 3 colors of quarks = 30) plus 30 antiparticle states plus 12 interaction bosons - for a grand total of 72 particles. There are exactly 72 available Golay-24 code words.

For more information, please read my original paper .
"Geometrical Basis for the Standard Model" .
International Journal of Theoretical Physics, Vol. 33 (1994), pp. 279-305,

which is available at Geometrical Basis for the Standard Model (1994).

Connection to Superstrings in a discrete spacetime:

A recent (2005) follow-up paper Leptons and Quarks in a Discrete Spacetime as well as a published article (2006) Unification of Interactions in Discrete Spacetime are now available (in PDF format) that show how to combine the finite subgroups of the Standard Model from the original paper above with finite subgroups of the Lorentz group SO(3,1) to form Weyl E8 x Weyl E8, a finite subgroup of SO(9,1) in discrete 10-D spacetime, unifying all the fundamental interactions. Its relationship to a discrete PSL(2,O) and to the continuous group E8 x E8 of superstrings and M-theory is discussed. Consequently, spacetime is discrete, and there can be only one universe and one set of physical constants because the result is unique. The discovery of the b' quark decaying to a b quark and an energetic photon at the LHC is the definitive experiment.

A short 2-minute Quicktime summary (2007) of several of the key concepts is available to whet your appetite at Let's Fix String Theory!. The video provides the "big picture" of how the unification of the fundamental interactions in discrete spacetime occurs in a unique way, leaving "no choice in the creation of the world".

Presentation slides in pdf format for my talk at the DISCRETE '08 Conference (2008) in Valencia, Spain with numerous images to help in conceptual understanding of the advantages to having finite rotation groups for the leptons and quarks, plus the unification in 4-D discrete space and 10-D discrete spacetime. Finite Rotation Group approach.

The details of connections to the Monster group and to coding theory are outlined in my 2011 paper titled "Our Mathematical Universe: I. How the Monster Group Dictates All of Physics" in the online journal Progress in Physics: Our Mathematical Universe: I. How the Monster Group Dictates All of Physics (2011).

Further progress has been achieved in 2013 toward verifying my geometrical approach because I have derived the neutrino PMNS mixing matrix from first principles, available online at Progress in Physics: Geometrical Derivation of the Lepton PMNS Matrix Values (2013).

In 2014 I derived the CKM matrix from first principles: CKM and PMNS Mixing Matrices from Discrete Subgroups of SU(2) (2014).

In 2015 I not only derived the Weinberg angle but also showed how 3 lepton families act together to be one equivalent lepton family cancel the triangle anomaly contribution exactly from 4 quark families acting together as one equivalent quark family: Weinberg Angle Derivation from Discrete Subgroups of SU(2) and ALL That (2015).

In 2019 I was able to use an engineering type approach, treating an electron and other fundamental particles as an antenna to emit and receive "feedback signals", to reveal a common origin for gravitation and quantum mechanics. My approach derives the path integral method. See GR=QM: Revealing the Common Origin of Gravitation and Quantum Mechanics ... (2019).

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Quantum Celestial Mechanics(QCM)

With general relativity physicist Howard Preston (deceased 2011), in 2003 we have posted a paper at the archive which has a new approach to understanding the states of gravitationally bound systems such as solar systems, galaxies, and the Universe. By knowing only the total mass and the total angular momentum of the system, all its quantization states are known. Predictions include angular momentum quantization states for orbiting bodies of mass M and angular momentum L obeying

L/M = m LT/MT

where m is an integer, MT is the total mass of the gravitationally bound system, and LT is the total angular momentum of the system.

Several of our recent papers are online:

Predicting Total Angular Momentum in TRAPPIST-1 and Many Other Multi-Planetary Systems Using Quantum Celestial Mechanics (2018) in which the TRAPPIST-1 system with its 7 planets in MMR are shown to obey QCM;

Update on Pluto and Its 5 Moons obeying the Angular Momentum Quantization per Unit Mass constraint Quantum Celestial Mechanics (2016) in which the Pluto system total angular momentum is now known well enough to be a possible definitive test of QCM;

Kepler-47 Circumbinary Planets obey Angular Momentum Quantization per Unit Mass predicted by Quantum Celestial Mechanics (QCM) (2014) in which the 3 known planets orbiting a binary star system are analyzed and shown to fit the QCM constraint;

Multi-planet Exosystems All Obey Orbital Angular Momentum Quantization per Unit Mass predicted by Quantum Celestial Mechanics (QCM) (2013) in which 15 multiple planet systems are analyzed and shown to fit the constraint;

Pluto Moons exhibit Orbital Angular Momentum Quantization per Mass (2012) shows that the known 5 moons of Pluto obey the QCM predictions;

Galaxy S-Stars exhibit Orbital Angular Momentum Quantization per Unit Mass (2012) shows that the innermost 27 well-measured S-stars of our Galaxy (Milky Way) obey QCM predictions even though they have seemingly random orbital planes.

Kepler-16 Circumbinary System Validates Quantum Celestial Mechanics (2012) shows how this angular momentum quantization condition produces m = 10 to within 1% for the only system found so far for which the pertinent values are known to within 1%. The Kepler-34 and Kepler-35 systems have 4% uncertainties in values and they produce m = 9 and m = 4, respectively. The acid test for QCM would appear to be finding another planet orbiting Kepler-16 and calculating its quantization integer.

QCM has a wave equation which, in the Schwarzschild metric, predicts quantization states that agree extremely well with the actual states of the Solar System, the Galaxy (without requiring dark matter!), and clusters of galaxies. With the interior metric, QCM predicts that the redshifts of supernovae SNe1a are actually gravitational redshifts and are not due to space expansion. In other words, we have discovered a repulsive gravitational effect that helps keep planets in quantized orbits, the stars revolving around the galactic center in quantization states, and the Universe in a static equilibrium. More details can be perused in these papers:

Exploring Large-scale Gravitational Quantization without h-bar in Planetary Systems, Galaxies, and the Universe (2003)

Gravitational Lensing by Galaxy Quantization States (2004)

Quantization State of Baryonic Mass in Clusters of Galaxies (2007)

Cosmological Redshift Interpreted as Gravitational Redshift (2007)

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Other physics research:


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What me worry?

If you would like to discuss any of these ideas with me and your age is between 8 and 80, my e-mail address is given below.

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Frank Potter (drpotter@lycos.com)
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