The Geometry Corrections Of The Casimir Electron

[Dubbelosix]

On the Casimir electron and gravitational corrections.

 

Casimir wasn't daft [1] . He did believe the electron was a conducting sphere of radius and associated zero point energy as

 

[math]E = -C \frac{\hbar c}{2R}[/math]

 

Where [math]C[/math] is a dimensionless constant that if positive implied an inward force - the constant is simply the fine structure constant. That inward force balances the outward Couloumb force (as Poincare stress) when the magnitudes of the corresponding energies are equal

 

[math]\frac{e^2}{2r} = C \frac{\hbar c}{2r}[/math]

 

A detailed calculation of C by Boyer, however, shows that Casimir’s intuitive approach was off the mark: The constant is negative, equal to about −0.09; that is, the stress on a conducting sphere tends to make it expand. Boyer’s result remains of interest because it highlights the geometry dependence of the Casimir force, a subject that has received considerable attention.

 

Point particles are a problem for physics and definitely problematic to our understanding of them. I stated that electrons where best seen in a phase space as a Planck cell, by which terminology was given by Von Neumann who stated that points in phase space do not make any physical sense. When I stated this earlier in the year on a different physics forum I was banned for ''not fitting in'' because my views where openly regarded ''as pseudoscientific.''

 

On what basis this was determined was never made clear, only that academics are more like parrots than true physicists since no one seemed to be concerned with the unavoidable divergence problems found when dealing with point like dynamics, such as an electron possessing an infinite curvature in relativity or even the self energy of an electron blowing up to infinity simply because of they way we model these particles.

 

Moreover I said that rotational bands where actually evidence that things like atoms and even molecules possess a real rotation period which was met with largely skepticism. Anything with spatial distribution doesn't require a notion of ''intrinsic spin'' which was a fancy set of words just to decribe, in every mathematical sense of the word, an identical phenomenon to a real spin or rotary feature to system. Why did we do this to particle physics? The reason is simple, particles of all kinds interact as though they where point like particles - this is not a surprise, classical physics predicts that small objects should behave point like.

 

The real question we need to rethink is whether they actually are. Why does a rest electron in the Casimir model have an energy equal to the charge of the electron interaction with the zero point energy off-shell photon?

 

The intuitive answer is one of those ‘of course(!)’’ moments. It is because the electron is confined only approximately to a position and so contains a motion that interacts with one virtual particle every periodic motion of its spin around a very small differential area. This means its’ interactions with virtual particles interaction with the electron in its most ground state causes a small ‘’jiggling’’ motion of an electron due to the uncertainty principle in regards to position and momentum.

 

What we may learn then, is that the uncertainty principle truly is only a measure of [our] limitations of extracting enough information to know deterministically how it will unravel. Underneath the observable dynamics we can see, extra computations and adjustable parameters from vacuum energy could explain the motion of the electron due to uncertainty as really being a phenomenon which had origin in dynamics revealiing itself in the vacuum (just not in an observable way) > off shell particles, or virtual particles, are not observable. Their interactions are so short, that perhaps only a negligable amount of pressure from Casimir forces contribute on the global scale to a very small cosmological constant!

 

I was surprised that no one has thought about that restriction since I got my idea from Puthoff that virtual particles only interact for a short time and interact in such a way that their couplings to the ‘’observable energy of a vacuum’’ could be minimal. He made clear this was his own hypothesis but I came to realize it makes sense when you realize virtual particles are not described by Hermitian matrices and therefore cannot contribute in the normal laws as we understand them to ‘’observable energy'' in the vacuum.

 

The Casimir Electron Self Energy

 

The Casimir self rest energy of an electron is

 

[math]E = \frac{\hbar c}{2R}[/math]

 

Note, this equation is very simplistic but there is some truth behind it. Keep in mind, Casimir made an assumption of a dimensionless coefficient which is in all regards probably an adjustable parameter of the theory. Regardless, Casimir’s electron as a sphere makes it manifest from geometry intrinsically. There may in fact be support from this, since attempts to measure the ‘’shape of an electron’’ had found that it appears remarkably spherical. It seems we may need to abandon the idea virtual particles are constantly interacting with the electron in ‘’virtual particle shielding’’ an early quantum field interpretation to ‘’do-away’’ with the divergence problems of point particles.

 

The spectral energy density of the modes is

 

[math]\rho = \hbar c \int k^3\ dk \times C(1) + C(2) \ne 0[/math]

 

(Here the C is a multiplicative constant usually to denote two photon processes C = 2 and the final constant C(2) is a renormalizing constant set to zero for flat space.

 

The forth power over the momentum of an off-shell particle was once believed to be zero. In the case above we have adjusted it so that there are non-negligible effects. I will present an argument for this even though previous references other than found in other blog posts have elucidated to the non-vanishing over the forth power of the momenta; such as the work shown by Sakharov.

 

Whatever conditions others have applied, we will imply in this work a dependence to the Sakharov zero point fluctuation equation. As noted, the Casimir electron is in fact geometry-dependent. Moreover, sphere's of that size corresponding to the electron classical radius must invoke a corresponding large curvature on the boundary of the sphere due to relativity. That curvature could be large enough to overcome electrostatic interactions, meaning, the Poincare stress could be gravitational in nature. This idea though is similar to Motz who believed there was a discontinuity of the gravitational field ‘’past the boundary’’ in which its order of magnitude would match the corresponding electric forces wishing to tear the electron apart.

 

In the Sakharov model, he has found that the forth power over virtual particles momenta in the presence of a curved space may not be zero in contrast. Still his model is a little different, including the physics when it comes to the Casimir electron but there are clear similarities as well. One difference is that the the curved space of the sphere is not producing the fluctuations, but instead its a fact of the sphere being conductive in contrast to a ‘’Faraday Cage’’ which is probably totally classical which is why it is incompatible.

 

The total cross section [math]\sigma[/math] involved in the interaction between the charge inside of the electron with an electromagnetic scattering is

 

[math]U = \frac{\hbar c}{4 \pi R^2} \int \sigma\ dk[/math]

 

and the corresponding density

 

[math]\rho = \frac{\hbar c}{4 \pi R^2} \int \sigma k^3\ dk[/math]

 

From here, you really should chase the references as they then form the equations with bounds and speculate on the UV divergence for zero point energy - the divergence will tell us at what scales the virtual photon is allowed to interact with the conducting electron sphere.

 

The Basic Model

 

Conducting plates are generic, you can apply the interaction to a conducting sphere. When spheres are involved, relativity naturally assumes curvature associated to the system - the smaller the curvature, the larger the contribution. Let's take a look at the standard equation Casimir suggested;

[math]\frac{F}{A} = -\frac{d}{da} \frac{<\mathbf{E}>}{A} = - \frac{\hbar c \pi^2}{240 a^4}[/math]

With [math]a[/math] being the radius of an electron, (instead) of it being the distance between two plates.

The energy associated to the system is

[math]E = \frac{\pi^2}{720} \frac{\hbar c}{V}A[/math]

 

Let's keep in mind for now, that there is a geometric dependence to the Casimir electron and the Casimir force it experiences when coupled with vacuum fluctuations.

 

The Sakharov Correction

 

The Sakharov fluctuation term, which is the modes of the zero point fluctuations was presented as a Langrangian with a geometric correction that deviates from flat spacetime using a power series

 

[math]\mathcal{L} = \hbar c\ R \int k dk... + \hbar c\ R^n \int \frac{dk}{k^{n-1}} + C[/math]

 

Where [math]R[/math] is the curvature tensor.

 

This equation can reinterpret the Casimir electron which seems to be a non-trivial application when taking into respect the geometry-dependence we spoke about. So let's get this rounded up.

 

Unifying the Sakharov correction with the Casimir electron shell model

 

To do what I have described, requires that we replace the geometric properties into the language of the wave numbers associated to the fluctuations:

 

[math]\frac{F}{A}  = - \frac{\hbar c \pi^2}{240}\ R\ \int k\ dk ... +\ \frac{\hbar c \pi^2}{240}\ R^n \int \frac{dk}{k^{n-1}} + C[/math]

 

This equation, for the Casimir electron has now taken into consideration the gravitational correction to the system. There is an extra interesting addition to understand the left hand side of the equation, for instance, in general relativity, the action of spacetime depends on the curvature as

[math]S = -\frac{c^4}{16 \pi G} \int\ \sqrt{|-g|}\ R\ d^4x[/math]

 

Contained in this, is an area term:

[math]A = \sqrt{det|-g|}\ dxdy[/math]

 

and when dealing with four space derivatives, we use the notation

[math]d^4x = dx\ dy\ dz\ dt[/math]

So concentrating on the area term in [math]\frac{F}{A}[/math] can be rewritten as

 

[math] F\ \frac{1}{\sqrt{det|-g|}\ dx\ dy}  = - \frac{\hbar c \pi^2}{240}\ R\ \int k\ dk ... +\ \frac{\hbar c \pi^2}{240}\ R^n \int \frac{dk}{k^{n-1}} + C[/math]

 

Now what about the ''force'' term?

 

The Christoffel symbol can be 'loosely' though of as being analogous to a force in Newtons equations (where mass has been set to 1 to denote that it is a constant in this formulation):

 

[math]\Gamma = \frac{1}{2} \frac{\partial g_{00}}{\partial x}[/math]

 

Newtonian formulation of this acceleration is

 

[math]F = -\frac{\partial \phi}{\partial x}[/math]

 

However as mentioned MANY times by myself, the gravitational force is not actually a true definition of a force as we come to expect say, in the proposed fundamental fields of nature, which are inherently complex (when quantum gravity is not in the WDW equation) and that require the quantization of field particles acting as mediators of the force (something which gravity is expected to use to form the unification theory in the opinions of many scientists). It's actually a crucial component of many theories, most notably string theory, which seems to be dying out.

 

Gravity is a pseudo force and can be understood in the following (neat) and (concise and short) way:

 

[math]\frac{d^2x^{\mu}}{d\tau^2} + \Gamma^{\mu}_{\nu \lambda} \frac{dx^{\nu}}{d \tau} \frac{dx^{\lambda}}{d\tau} = 0[/math]

 

where

 

[math]\Gamma^{\mu}_{\nu \lambda} = \frac{\partial x^{\mu}}{\partial \eta^{a}}\frac{\partial^2 \eta^a}{\partial x^{\nu}\partial x^{\lambda}}[/math]

 

or more compactly

 

[math]\Gamma^{\mu}_{\nu \lambda} = J^{\mu}_{a} \partial_{\nu} J^{a}_{\lambda} = J^{\mu}_{a} \partial_{\lambda} J^{a}_{\nu} \equiv J^{\nu}_{a} J^{a}_{\nu \lambda}[/math]

 

Which represents a pseudo force for gravity which makes it in the same league as the Coriolis and the Centrifugal forces.

 

Now... going back to the force issue, we simply take general relativity's correction of the force from the Newton force and apply it to the equation:

 

The simplest way to represent the force would be with mass now included;

 

[math]\frac{1}{\sqrt{det|\ -g\ |}}\ \frac{m \Gamma}{\ dx\ dy}  = - \frac{\pi^2}{240}\ \hbar c\ R\ \int k\ dk ... +\ \frac{\hbar c \pi^2}{240}\ R^n \int \frac{dk}{k^{n-1}} + C[/math]

 

And here endeth the investigation.

 

REFERENCES:

 

http://www.casimir-network.org/IMG/pdf/Lamoreaux.pdf

 

https://arxiv.org/ftp/hep-th/papers/0606/0606227.pdf

 

[1] - In fact Casmir is one of several physicists that have openly admitted considering an electron not truly pointlike since there is in fact a geometric dependance to the Casimir effect. One particular physicist who was also off the mark in his own construction was Motz who believed the gravitational field took the role of a Poincare stress. Though he attributed the mass in normal Planck standards and that just doesn’t make any sense right now within the standard model as he desperately tried to attribute that large mass to the top quark.

https://arxiv.org/ftp/hep-th/papers/0606/0606227.pdf

 

http://earthtech.org/publications/puthoff_casimir_electron.pdf

Edited April 19, 2019 by Dubbelosix

[Dubbelosix]

To be fair, it could have been a bit superfluous to try and modify [math]\frac{F}{A}[/math] in terms of general relativity, it is just as acceptable to remain the simpler form of

 

[math]\frac{F}{A}  = - \frac{\pi^2}{240}\ \hbar c\ R\ \int k\ dk ... +\ \frac{\hbar c \pi^2}{240}\ R^n \int \frac{dk}{k^{n-1}} + C[/math]

 

The energy equation too can be interpreted through the Sakharov method in at least two different ways, the first being replacing the area term for a general relativity interpretation, which was again for clarity

 

[math]A = \sqrt{det|-g|}\ dxdy[/math]

 

In which case it enters like;

 

 

[math]E = \frac{\pi^2}{720} \frac{\hbar c}{V}\ A= \frac{e \pi^2}{720} \rho_q \ \sqrt{det|-g|}\ dx\ dy[/math]

 

(with [math]\frac{\hbar c}{V} = \rho_q e[/math] as a charge density where the charge though is a squared component, note though, there could be another additional correction in which a system is not just measured by a charge [math]\hbar c[/math] but also as [math]Gm^2[/math] in which I showed that a relativity of charges for systems involves both terms)

 

Or in terms of the wave number:

 

[math]E = \frac{\pi^2}{720} \hbar c\ \int dk[/math]

 

This energy can be interpreted as a flat space solution, so as usual, we introduce the curvature corrections, identified as the Langrangian density

 

[math]\mathcal{L} = - \frac{\pi^2}{240}\ \hbar c\ R\ \int k\ dk ... +\ \frac{\hbar c \pi^2}{240}\ R^n \int \frac{dk}{k^{n-1}} + C[/math]

Edited April 19, 2019 by Dubbelosix

[Dubbelosix]

Here is a reference to the work about the ''relativity of charges'' and explains historically why it should hold.

 

https://physics.stackexchange.com/questions/472438/electromagnetic-mass