Archive for the 'Quantum Mechanics' Category

Nov 17 2022

The Coulomb Gauge    

There is another name for a free graviton, – it is “the identity isomorphism idEx, here denoted 1x, and the elements 1x, x ϵ M, act as unities for any multiplication in which they can take part” ([1], pg. 4). We see that unlike π (pi), idEx has some degree of circular polarization and/or skewed sine waves. In some writing instances π is the same as idEx and I am not trying to dictate how they should be used.

In the “Coulomb, radiation, or transverse gauge. This is the gauge in which ∇ · A = 0” ([2], pg. 241), we have a classical description. In the tensor sense, we have the forms Χij. The direction we choose for Χ is always transverse to the radial electric field at a chosen point, and the coordinate frame Ui is picked centered on the same point, creating a k-plane. We have that “The forms Χij are the transition forms for the Lie algebroid atlas {Ui, ψi, Θi}” ([1], pg. 206), and Θi varies with the density of the gamma ray field:

Considering the transition form TP/G [1], we may here call G the density of the gravitational field. It is seen that as the density goes up the transition angle Θi decreases for a given charge and distance from the charge.

In Jackson’s problem 6.19 (b), “the original and space-inverted vector potential differ by a gauge transformation” ([2], pg. 291). Though the earth catches some of the sun’s gravitons all the time, the sun’s gravitons during the day are greater at the face of the earth than at night, and inverted, changing the Coulomb gauge.

With the “Lorenz condition (1867), ∇ · A + (1/c2) ẟφ/dt = 0” ([2], pg. 240), it is mathematically shown that the system {Ui, ψi, Θi} acts fast compared to the gradient of A, and
           ιX (φ ˄ ψ) = ιX(φ) ˄ ψ + (-1)i φ ˄ ιX(ψ)              ([1], pg. 306)
Also, as small as gravitons are, we may as well call the k-planes “flat connections Θi“ ([1], pg. 206).

Since we have “t the fixed point set of θ” ([3], pg. 401), t is on the center line of a gamma ray, and “g0 = t0 + p0 is a Cartan decomposition of g0“ ([3], pg. 184). In certain situations the center can shift as well, in which case “c0 is the center of t0” ([3], pg. 452) as t0 moves back and forth.

With the polarization factor, it is interesting to call h the vector summation of two gamma ray electric fields. When a gravitational field is yet more compact, h is the summation of more than 2 electric fields, so that “f: MH be a smooth map” ([1], pg. 183), and “Let h be a proper subalgebra of g of maximum dimension” ([3], pg. 160).

Incidentally, the identity isomorphism reminds us of quantum 1:

[1] Mackenzie, Kirill C. H., “General Theory of Lie Groupoids and Lie Algebroids”, c. 2005 Kirill C. H. Mackenzie, London Mathematical Society
[2] Jackson, J. D., “Classical Electrodynamics, Third Edition”, c. 1999 John David Jackson, John Wiley & Sons, Inc
[3] Helgason, Sigurdur, “Differential Geometry, Lie Groups, and Symmetric Spaces”, American Mathematical Society, 2012

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Oct 05 2022

The Vector Potential

In electrodynamics we find that “A quantum-mechanical description of photons necessitates quantization of only the vector potential” ([1], pg. 242), as in the summation of all the manifolds of gravitational fields at a given location. In a more densely packed summation of manifolds, the action of an electric charge will have a lesser rotational effect on the electric fields of the gamma rays than on a less dense field. The power of the rotation is the same in either field however, as long as we are referring to a gravitational field that is not too sparse for electric fields to propagate.

“The definition of B = ∇ x A specifies the curl of A, but it doesn’t say anything about the divergence – we are at liberty to pick that as we see fit, and zero is ordinarily the simplest choice.” ([2], pg. 235) The reason we may pick the divergence as zero is that the manifolds “are frozen in time for phonon transmission”:

As far as group action, Mackenzie [3] calls these “groupoids”, such as an ellipsoid, a spheroid, or another 3-dimensional shape. The definition of a spheroid I find is that it is like a sphere, but not a perfect sphere, and in the present case we have “oscillations and accordion motion in multiple axes”:

On a side note, though related to manifolds of gravitational fields, the Nobel Prize in Physics is being given this year for essentially this:

[1] Jackson, J. D., “Classical Electrodynamics, Third Edition”, c. 1999 John David Jackson, John Wiley & Sons, Inc
[2] Griffiths, David J., “Introduction to Electrodynamics, Third Edition”, c. 1999, Prentice-Hall, Inc.
[3] Mackenzie, Kirill C. H., “General Theory of Lie Groupoids and Lie Algebroids”, c. 2005 Kirill C. H. Mackenzie, London Mathematical Society

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Aug 27 2022

Concentrated Group Action

There is another slightly different view in which the Coulomb field transmits when it comes to nuclei as parts of molecules, and that is by pulsating, concentrated Weyl groups or frames of a small conical angle, toward another nucleus. It is not wholly different from the last blog entry because spherical pulses could also be seen as groups, and for a given charge these concentrated groups are in addition to spherical groups.

When an O2 or N2 binuclear molecule forms, or let us say a benzene molecule, each nucleus senses the other nuclei closest. This is a strong repulsion, so the nuclei may start sending out groups concentrated toward the other nuclei for efficiency, while the electron cloud in between the nuclei offers attraction and keeps the molecule from flying apart. This also changes the Calabi-Yau structures within the nuclei.

For Coulomb attraction, a frame may wrap around another charge. For Coulomb repulsion, there may be partial contact and some backflush. As two close nuclei in a molecule sense each other, there may also be alternating, concentrated, group pulses between the two. This is likened to a synchronization between the nuclei, without the need for backflush. With phonon transmission this is a very fast process and transmits without intercepting electrons in orbitals. When we compare the size of nuclei and electrons to molecular size, there is a lot of empty space filled with gravitons, so this synchronization is reasonable.

In larger nuclei there are more compact spaces and more affine connections between them. For a nucleus we may call these irreducible representations, where the exception is fission, as a “reduced root system in V” ([1], pg. 461). The nuclear charge manufactures springboard groups repeatedly, with oscillations and accordion motion in multiple axes. A stable nucleus in a molecule is an isomorphism, though we must be careful here because as orientations change, there may be slight changes in structure. Particle colliders are excluded from this discussion.

A nucleus consists of involutive automorphisms, the summation adding to the entity’s spin, as it absorbs gravitons for the energy to send out groups or frames. Boothby calls these “inner automorphisms of G” ([2]. Pg. 237). A Weyl chamber is part of atomic mass, while a Weyl group transmits as a packet in the not so compact space of the gamma ray field.

We see that “π is a continuous and open mapping.” ([1], pg. 120]. In certain areas of deep space we may call this Riemannian globally symmetric space I, with perfect sine waves and no circular polarization. “Riemannian globally symmetric spaces of type II” ([1], pg. 516) are due to the bi-invariant structure of the Coulomb field. In both cases there is a “strong orthogonality” ([1], pg. 576) which produces a polarization factor of 2, as used in the blackbody radiation formula and in G = 4hf/3. π may be called a free graviton, since it is one half wavelength long.

As far as the Coulomb field produced by an electron in an atomic or a molecular orbital: “Let N0 be a bounded star-shaped open neighborhood of 0 ϵ g which exp maps diffeomorphically onto an open neighborhood Ne of e in G.” ([1], pg. 552) Let e be the electron, star-shaped be the lobes of orbitals, and exp be the growth of an orbital electron in size and charge. The increase in size of an electron in the orbital enables it to absorb more gravitons at a given time, thus increasing gravitational pull in the second half of the arc.

[1] Helgason, Sigurdur, “Differential Geometry, Lie Groups, and Symmetric Spaces”, American Mathematical Society, 2012

[2] Boothby, William M., “An Introduction to Differentiable Manifolds and Riemannian Geometry”, Academic Press, 2003

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Mar 29 2022

Transmission of the Coulomb Field

The gamma ray field we live in is extremely rich and dense.  For the forms we find in nuclei and assorted particles, there is all the energy needed to drive all physical processes.

A Calabi-Yau shape within a nucleus or particle needs an external energy supply to maintain it. Gravity provides the energy. Here we are talking about force and pressure within a nucleus or particle, with only indirect connection to the outside, or connection at a point, curve, or surface.  There may also be tears joining and reforming.

Occasionally we refer to neutrons, protons, electrons, and nuclei.  A proton can be a hydrogen nucleus, though we list it separately when we talk about free protons, such as in the solar wind, particle colliders, or elsewhere.  Let’s take an Oxygen nucleus for example with the makings of 8 protons and 8 neutrons. Inside the nucleus, at the top, parachutes with baskets attached through ropes, or strings, instead of a parachutist, may cause some gravitons to loop around the insides of the parachutes, or branes, and into the baskets with enough force to hold the parachutes against the highest flux density of gravitons. Then the gravitons would find ways to tunnel through the baskets, pushed from behind.  In the motions of O2 in air, the parachutes may slide around to stay opposite the maximum flux.

This may also help explain weak interaction parity violation, because as an electron forming within a nucleus tries to escape, out the bottom is easier, due to escape out the top involving going through the gaps in the parachutes.  More than 50% would come out downward.

The manifold of the sun’s gamma ray field, the manifold of the earth’s gamma ray field, and likewise with other celestial bodies, provides a combination of symmetric spaces. During the day, at noon let’s say, the vectors of the sun’s manifold are in the opposite direction as the vectors of the earth’s terrestrial manifold. The Coulomb field uses all vectors of all manifolds to propagate, because all vectors, within a distance of 10 meters at least, are frozen in time for phonon transmission.

Let’s say M1 is the earth’s manifold, and M2 is the combination of the earth’s and sun’s manifolds. “…a diffeomorphism F: M1 → M2 of manifolds oriented by Ω1, Ω2, is orientation-preserving if F*Ω2 = λΩ1, where λ > 0 is a C function on M.” ([1] pg. 209) In our example here, λ > 1, and we have neglected the earth’s moon for simplification.

We may call a negative charge a left coset space, and a positive charge a right coset space. Each creates its own homomorphism in the dense gamma ray field, by a diffeomorphism on the electric fields of the gamma rays.  For one thing, there is circular polarization. For another, perpendicular to the greatest flux density of gamma rays the electric fields of the gamma rays may have skewed sine wave lobes, somewhere between a normal sine wave and a sawtooth. The Coulomb field acts tangent to the R vector sphere, and “(∇XY)p depends not on the vector field X but only on its value Xp at p.” ([1] pg. 309]  The way that the Coulomb field transmits radially is by centrifugal force through the gamma ray field.

The inside of an atom may be called a geodesic.  An electron path in an atomic orbital may also be called a geodesic, and “a long geodesic may not be minimal.” ([2] pg. 62)  This is due to the Lorentz force:

Gravity is an integral manifold.  Each orbital arc is a line integral absorbing gravitons.  The Coulomb field, on the other hand, is a charge induced diffeomorphism in the gamma ray field. Substantially outside of neutral atoms there is a propensity for positive and negative charges to cancel, though in the near field we have van der Waals forces.

Phonons for the Coulomb interaction are generated inside a charge.  The field created, that acts on another charge, may act on the outside of another charge, possibly only 5% of the diameter deep.  The fields may also act in the interspace, producing backflush to the charges that generate the fields.  Phonons of opposite chirality attract, and of the same chirality repel.

As points meet for the Coulomb force, the acceleration would be periodic, and relates to the vector potential.  A Fourier Series can be applied to the vector potential, with the direction of force being the side of the ‘x’ axis where the sine or cosine function has larger lobes.  Often a geodesic is called piecewise smooth, due to gravitons being separate, though on a classical scale the motion is smooth.

Two electrons can occupy the same atomic orbital if they have opposite half-integer spin projections.  This is the Pauli exclusion principle.  In terms of tensor math, “the subspaces are mutually orthogonal and each is a nontrivial irreducible subspace.” ([1] pg. 242)

[1] Boothby, William M., An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, 2003

[2] J. Milnor, based on lecture notes by M. Spivak and R. Wells, Morse Theory, Princeton University Press, 1969

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Feb 10 2022


Inside a proton or electron, events may approach the Planck length.  The frequency of the waves would not change from that of a free space graviton, though wavelength and amplitude do change.

Waves inside a particle may make loops in certain circumstances, not necessarily around the perimeter, though internally, and required because of all the traffic.

Certainly, the ways these vibrations set up in a proton or electron determines whether we have a positive or negative charge.  If we did not have any loops and curves, the versatility needed would be hard to set up.  It is somewhat like a Hilbert space with wrapped up dimensions.

Put another way: “A string vibrating in one particular pattern might have the properties of an electron, while a string vibrating in a different pattern might have the properties of an up-quark, a down-quark, or any of the other particle species in Table 12.1.  It is not that an “electron string” makes up an electron …Instead the single species of string can account for a great variety of particles …” *

If you peruse this website, you will find other areas of unification.

* Greene, Brian, The Fabric of the Cosmos, c. 2004 Vintage Books, a division of Random House, Inc., p. 346-347

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Feb 09 2022

Core of an Electron or Proton

We can see from the calculation of the diameter of a free electron that as the density of the gravitational field goes down, the diameter increases.  This would be because of less gravitational pressure on the outside of the electron.

As gravitons enter a proton, electron, neutron, or nucleus, as conjugate waves or to take residence, the buildup takes on a fuzzy look that makes them look larger.  If we take a core diameter of 1.3335 x 10-15 m, the part that produces the fundamental charge, and add one graviton wavelength, we arrive at 5.30 x 10-15 m diameter, which is close to the classical diameter of the electron, 5.64 x 10-15 m *.  One graviton wavelength is used because one-half wavelength is on one side of the electron and one-half wavelength of a different graviton is on the other side.

We may call these outer layer gravitons tentacles or strings.  When nuclear fission occurs, the de Broglie wavelength of a neutron can come in at an angle where the strings on each entity hook and help pull the neutron into the nucleus. The cross section for this process is larger for slow neutrons vs fast neutrons in part because of the longer de Broglie wavelength.

* Jackson, J. D., Classical Electrodynamics, Third Edition, c. 1999 John David Jackson, John Wiley & Sons, Inc., p. 695

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Nov 14 2021

Greatest Lower Bound

One would guess that the particle physicists and quantum field theorists may like a 1.3335 x 10-15 m diameter of a free electron, because it is closer to a point particle than many estimates of the diameter.  It is possible that 1.3335 x 10-15 m is also the limit inferior of the sequence S137 to Sn in an atomic orbital.

The maximum diameter, on the other hand, will depend on the element and on the orbital.  At a spin flip, electrons in all orbitals may reduce to 1.3335 x 10-15 m, before taking off on a new trajectory and increasing in diameter again.  We cannot speak of a limit superior of the sequence of diameters of the electron in an atomic orbital nevertheless.  That will depend on the direction of electron travel, and on whether the atom is at the surface of the earth, or at some other planet.  For the latter, it depends on the density of the gravitational field.

We may also ask whether 1.3335 x 10-15 m is the greatest lower bound at all locations in the universe.  This raises the question of whether the fine structure constant is a universal constant, or whether or not the Coulomb gauge is the same everywhere.

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Sep 28 2021

Diameter of the Electron

Published by under Quantum Mechanics

It has been surmised that with conjugate wave gravitons, the number of gravitons in a free electron in the earth’s locale may be the reciprocal of the fine structure constant.  We may then use this number to determine a possible diameter of the electron.

First, start with the energy:

(5.011 x 10-11 J) x 137 = 6.8651 x 10-9 J

Then use the electron’s spin angular momentum:

ħ/2 = (½) mvr = (½) mr2ω


ω = ħ / (mr2)

Considering rotational kinetic energy, we have:

Ek = (½) [Iω2] = (½) [ (½) mr2 ] ω2 = 6.8651 x 10-9 J

Substituting ω from above:

Ek = (½) [ (½) mr2 ] [ħ / (mr2]2 = (½) (½) [ħ2 / (mr2)]

1/r2 = [(4) (6.8651 x 10-9 J) (9.1095 x 10-31 kg)] / ħ2

r = 6.6676 x 10-16 m

D = 2r = 1.3335 x 10-15 m

ω = ħ / (mr2) = 2.6040 x 1026 rad/sec

This angular velocity is fictitious, though it can be used in calculations due to Green’s Theorem (see April 2007 paper).

Of course the number of gravitons in an electron in an atomic orbital varies, and thus the electron diameter would vary also.

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Jan 25 2021

Final Mediator

Going back to the April 2007 paper once again: “… and the same Coulomb force being the final mediator of the gravitational force as proposed, the mass energy of the proton appears to be integral to gravitational field action.”, we can go further back in history to examine the path of how we arrived at this place.

In the scientific community, the cause of gravity is by many still considered as unknown.  As Hughes-Hallett et al. puts it in relation to the gravitational force: “How does acceleration come about?  How does the velocity change?  Through the action of forces.  Newton placed a new emphasis on the importance of forces.  Newton’s laws of motion do not say what a force is, they say how it acts.” 1

Newton himself says in The Principia: “…considering in this treatise not the species of forces and their physical qualities but their quantities and mathematical proportions, as I have explained in the definitions.” 2  In Definition 8 Newton says: “This concept is purely mathematical, for I am not now considering the physical causes and sites of forces.” 3

Roger Cotes, Plumian Professor of Astronomy and Experimental Philosophy at Cambridge University wrote in his Editor’s Preface to the Second Edition of The Principia: “But will gravity be called an occult cause and be cast out of natural philosophy on the grounds that the cause of gravity itself is occult and not yet found?” 4  Electromagnetic waves were not yet discovered in 1713 when Cotes wrote this, so it is difficult to imagine how anyone could have discovered the cause of gravity back then.

1  Hughes-Hallet, Gleason, McCallum et al., Calculus, John Wiley & Sons, Inc., 2005, p.304

2  Cohen, Whitman, Isaac Newton, The Principia, Mathematical Principles of Natural Philosophy, University of California Press, 1999, p. 588

3  Ibid., p. 407

4  Ibid,. p. 392

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Dec 04 2019

Electric and Magnetic Fields

Jon Rogawski has a cool diagram and calculation using Faraday’s Law on page 979 of his Multivariable Calculus book*. The magnetic field around the straight wire with an alternating current flowing in it produces a voltage in an adjacent looped wire with no conventional energy applied except that from the other wire.

Of course this magnetic field, and likewise for electric fields, must have a pervasive field in which to transmit. One may say it transmits through the air, however Faraday’s law also applies in space.

These E and B fields transmit by turning and bending the E and B fields of the dense gamma rays.

* Rogawski, Jon, “Multivariable Calculus”, W. H. Freeman and Company, c. 2008

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