# Hypothesis about the formation of particles from fields

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4 hours ago, swansont said:

Fusion, though fission is used to create the high temperature and pressure.

Apologies, you are of course correct, “fusion” was what I meant to say (I typed that in a hurry).

10 hours ago, Kartazion said:

But the interpretation of the energy E is scalar through the stress-energy-momentum tensor.

I’m not sure what you mean here - could you elaborate?

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58 minutes ago, Markus Hanke said:

I’m not sure what you mean here - could you elaborate?

It's something about Klein–Gordon equation. I thought it was used.

Energy and flux is a scalar quantity. That's why I thought that.

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On 8/4/2022 at 10:49 PM, SergUpstart said:

The particle has energy E=mc^2

In terms of field, rather m = E / c^2 (or u / c^2 in the notation I used). But the difference is only what value is primary. Of course, there is a deep connection between mass and energy.

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9 hours ago, Kartazion said:

It's something about Klein–Gordon equation. I thought it was used.

Energy and flux is a scalar quantity. That's why I thought that.

Ah I see. The Klein-Gordon equation is the quantum version of the energy-momentum relation (not the tensor though), and is generally used to describe relativistic fields/particles without spin - scalar particles.

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• 1 month later...

Klein–Gordon equation is very controversial.

In one place of Wikipedia is writen: "related to the Schrödinger equation",

in another place: "The wave function cannot therefore be interpreted as a probability amplitude. The conserved quantity is instead interpreted as electric charge, and the norm squared of the wave function is interpreted as a charge density. The equation describes all spinless particles with positive, negative, and zero charge."

Completely arbitrary assumption.

Other thing, there is no first time derivative, only second, so this expression is incomplete and requires intermediate values. Also can not be simulated on a computer.

And its stationary state has the unique spherically symmetrical solution, like sin(k • r) / r, with some constant k and distance from center r. But all-space intregral of this function, of its square, of its first and second derivatives, and of derivatives squares is infinite, and can not be represented as finite charge or probability.

Edited by computer
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Nice analysis by Markus on the previous page.

On this off topic discussion ( I recommend splitting if possible ), energy, even rest energy, can be described as due to the configuration of the system, but not internal configuration ( as there is no internal aspect of elementary particles; if they had internal structure, they could be divided, and would not be fundamental ), rather the external configuration with respect to other fields, such as the Higgs, EM, color, and even space-time ( which gives rise to variable energies and masses in differing frames ).

But back to the OP.
I would appreciate a brief summary of your conjecture, Computer, so I don't have to slog through your lengthy post.
Qed is best described as a perturbative theory of the quantized electrodynamic vacuum fields.
And it does an excellent job of describing all phenomena involving charged particles.

What does your theory add, or modify, to QED that is currently lacking, or in need of modification ?
What new predictions does it make, and are any experimentally verified ( as is the case with current theory ) ?

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11 hours ago, MigL said:

1. if they had internal structure, they could be divided, and would not be fundamental

2. What does your theory add, or modify, to QED that is currently lacking, or in need of modification ?
What new predictions does it make, and are any experimentally verified ( as is the case with current theory ) ?

1. In my understanding, the structure, for example, of electron is detailed description of the fields that make up the central part with high energy density. An electron cannot be divided, it does not have many point-centers of energy compaction, although it can annihilate under certain conditions. The structure of it is described rather by the scaling constant R that I used in equations.

2. My calculations have nothing to do with quantum physics, this is an exit to the level of "pure fields" on continuum, where we can dig as deep as we want and there will be always something even smaller. Theories are designed mostly for computer simulation of processes at distances comparable to the dimensions of elemental particles. Using raster matrices, finite differences or finite elements methods. What will happen, for example, in the collision of particles. And  such simulation is much cheaper than natural experiments.

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14 hours ago, computer said:

in another place: "The wave function cannot therefore be interpreted as a probability amplitude. The conserved quantity is instead interpreted as electric charge, and the norm squared of the wave function is interpreted as a charge density. The equation describes all spinless particles with positive, negative, and zero charge."

Please clarify this, preferably with a prope reference/quotation.

Which wave function are you referring to?  The the dependent variable in KG or QM ?

In neither case is the probability directly dependent on the  wave function.  It is the square of the wave function that determines probability.

Wave functions have inappropriate physical dimensions to be directly associated with probabilities.

In the case of KG the wave' is a soliton.

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5 hours ago, computer said:

In my understanding, the structure, for example, of electron is detailed description of the fields that make up the central part with high energy density.

How are these 'fields' compactified, or localized to this 'central part' ?
The QED fields, which manifest quantum particles, are not.

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On 7/31/2022 at 9:55 PM, computer said:

[...]

The main differences from the more classical approach are:

1) Velocity vector V had introduced as an "independent" and essential part of field,

as physical reality along with E and H

2) The expression for the energy flux W was changed, with an additional member ε0 · (E · V) · E

and I hope someone will conduct experiments to confirm or reverse that assumption

3) There are non-trivial suggestions, what "forces" may affect the velocity derivative over time

I applaud your presentation. Does your .htm file have more background or detailed information? Could you provide a .pdf or https:// link for it?

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23 hours ago, studiot said:

Please clarify this, preferably with a prope reference/quotation

23 hours ago, studiot said:

It is the square of the wave function that determines probability

All-space integral of square of the function sin(k · r) / r (spherically symmetric stationary solution) also is infinite. k is some constant, r is the distance from the center of coordinate system.

19 hours ago, MigL said:

The QED fields, which manifest quantum particles, are not

My speculations have nothing to do with QED. The most similar analogy is like how stars and planets form from clouds of cosmic dust and gas, so I imagine the formation of elementary particles from fields. But this process requires nonlinear equations. In my calculations they are still linear.

17 hours ago, NTuft said:

Does your .htm file have more background or detailed information?

Some useful thoughts can be taken from here:

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On 10/2/2022 at 5:28 AM, computer said:

Klein–Gordon equation is very controversial.

No it’s not. It’s the simplest possible wave equation for a relativistic scalar field without spin. There’s nothing controversial about it.

On 10/2/2022 at 5:28 AM, computer said:

Completely arbitrary assumption.

It’s not an arbitrary assumption - the Klein-Gordon equation is simply the Euler-Lagrange equation corresponding to the simplest possible Lagrangian for a scalar field.

On 10/2/2022 at 5:28 AM, computer said:

Other thing, there is no first time derivative, only second

Of course - that’s because the equation is Lorentz invariant, so space and time need to be treated on equal footing. This follows directly from the Lagrangian.

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Let us consider what field " blobs" can be, moving at the speed of light in a certain direction while maintaining their shape. That is, compact formations capable of traveling long distances compared to their size without significant changes in structure. Unlike dipole radiation, which propagates spherically in all directions. Perhaps such structure have emissions of atoms during the transitions of electron clouds to less energetic levels. Discussion of how justified use of the term "photon" in relation to such objects is beyond the scope of this article.

Let us take as basis the equations, existence of which in the real world is justified in the topic on dipole radiation: ?

The following symbols are used:

Scalar potential = a

Vector potential = A

Electrical field = E

Speed of light in vacuum = c

Time derivatives are denoted by singlequote '

a' = - c2 · div A

A' = - E - grad a

E' = c2 · rot rot A

The formulas are given in cylindrical coordinate system (ρ,φ,z),

associated with the point of space where the geometric center of field blob is located at the time of observation.

Let us put r2 = ρ2 + z2

Motion occurs along z-axis at the speed of light and structure of field object remains unchanged,

that is, ∂/∂t = - c · ∂/∂z for all physical quantities.

Also, integral of internal energy throughout all the space must be finite, density of which is expressed by the law:

u = ε0/2 · E2 + μ0/2 · H2

where E 2 = Eρ2 + Eφ2 + Ez2, H2 = Hρ2 + Hφ2 + Hz2

H = 1/μ0 · rot A, B = rot A = μ0 · H

Let us put J = rot B = rot rot A

Let us start with the mathematically simplest descriptions possible from the point of view of field laws mentioned above. In cylindrically symmetric case, when ∂/∂φ = 0 for all physical quantities.

Basic equations are divided into two independent systems:

1. With circular electric field.

Aφ' = - c · Aφ/z = - Eφ

→ Eφ = c · Aφ/z

→  Eφ/z = c · 2Aφ/z2

Eφ' = - c · Eφ/z = c2 · Jφ

= c2 · (- 2Aφ/z2 - 2Aφ/∂ρ2 - Aφ/∂ρ / ρ + Aφ / ρ2)

Eφ/z = c · (2Aφ/z2 + 2Aφ/∂ρ2 + Aφ/∂ρ / ρ - Aφ / ρ2)

Equating ∂Eφ/z from two equations, we get

2Aφ/∂ρ2 + Aφ/∂ρ / ρ - Aφ / ρ2 = 0

/∂ρ (Aφ/∂ρ + Aφ / ρ) = 0

If Aφ is not zero in all the space,

so ∂Aφ/∂ρ + Aφ / ρ = 0, and Aφ is proportional to 1 / ρ, that gives infinite energy integral. Hence, such non-zero components of compact radiations can not exist. After artificial creation or computer modeling such structures will diverge in waves in all directions, instead of moving in one direction at the speed of light.

2. With circular magnetic field.

a' = - c · ∂a/∂z = - c2 · (∂Aρ/∂ρ + Aρ / ρ + ∂Az/∂z)

∂a/∂z = c · (∂Aρ/∂ρ + Aρ / ρ + ∂Az/∂z)

Aρ' = - c · ∂Aρ/∂z  = - Eρ - ∂a/∂ρ

Eρ = c · ∂Aρ/∂z - ∂a/∂ρ

∂Eρ/∂z = c · 2Aρ/∂z2 - 2a/∂ρ/∂z

Az' = - c · ∂Az/∂z = - Ez - ∂a/∂z

Ez = c · ∂Az/∂z - ∂a/∂z

∂Ez/∂z = c · 2Az/∂z2 - 2a/∂z2

Eρ' = - c · ∂Eρ/∂z = c2 · Jρ

∂Eρ/∂z = c · (2Aρ/∂z2 - ∂2Az/∂ρ/∂z)

Ez' = - c · ∂Ez/∂z = c2 · Jz

∂Ez/∂z = c · (2Az/∂ρ2 - ∂2Aρ/∂ρ/∂z - ∂Aρ/∂z / ρ + ∂Az/∂ρ / ρ)

Equating ∂Eρ/∂z from the equations for Aρ' и Eρ', we get

c · 2Aρ/∂z2 - 2a/∂ρ/∂z = c · (2Aρ/∂z2 - ∂2Az/∂ρ/∂z)

and conclude that a = c · Az if we are talking about quantities decreasing to zero with distance from the center goes to infinity.

From the equation for a' then follows ∂Aρ/∂ρ + Aρ / ρ = 0,

which means Aρ = 0 if Aρ is not proportional to 1 / ρ with infinite energy integral.

From the equation for Az' follows Ez = 0 at a = c · Az

The following equations remain valid:

Eρ = - a/∂ρ = - c · Az/∂ρ

whereas from ∂Ez/z = c · (2Az/∂ρ2 + Az/∂ρ / ρ) = 0

it follows that with non-zero Az must be Az proportional to ln(ρ) and energy integral is infinite.

Thus, no valid expressions for field formations were found. The situation changes if we assume that div E ≠ 0 (non-zero charge density) and introduce additional terms into formulas for E' using the velocity field:

E = c2 · J - grad (E · V) - V · div E

where div E = ∂Eρ/∂ρ + Eρ / ρ + ∂Ez/∂z

in case of circular magnetic field, whereas case of circular electric field remains within previous calculations, since there div E = 0

Assuming that Vz = c is in the entire space around isolated field object, whereas Vρ = 0 and Vφ = 0,

and since E · V = Ez · c, we get

Eρ' = - c · ∂Eρ/∂z = c2 · Jρ - c · ∂Ez/∂ρ - 0 · div E

∂Eρ/∂z = ∂Ez/∂ρ - c · Jρ

∂Eρ/∂z = ∂Ez/∂ρ - c · (2Az/∂ρ/∂z - ∂2Aρ/∂z2)

Ez' = - c · ∂Ez/∂z = c2 · Jz - c · ∂Ez/∂z - c · div E

∂Ez/∂z = - c · Jz + ∂Ez/∂z + div E

c · Jz = div E

c · (2Aρ/∂ρ/∂z - 2Az/∂ρ2 + ∂Aρ/∂z / ρ - ∂Az/∂ρ / ρ) = div E

The following equations remain true

∂a/∂z = c · (∂Aρ/∂ρ + Aρ / ρ + ∂Az/∂z)

Eρ = c · ∂Aρ/∂z - ∂a/∂ρ

Ez = c · ∂Az/∂z - ∂a/∂z

From the expression for Ez' after substitutions it follows:

c · (2Aρ/∂ρ/∂z - 2Az/∂ρ2 + ∂Aρ/∂z / ρ - ∂Az/∂ρ / ρ)

= ∂Eρ/∂ρ + Eρ / ρ + ∂Ez/∂z = c · 2Aρ/∂ρ/∂z - 2a/∂ρ2

+ c · ∂Aρ/∂z / ρ - ∂a/∂ρ / ρ + c · 2Az/∂z2 - 2a/∂z2

2a/∂ρ2 + ∂a/∂ρ / ρ + 2a/∂z2 = c · (2Az/∂ρ2 + ∂Az/∂ρ / ρ + 2Az/∂z2)

Which leads to the conclusion a = c · Az

Then Ez = 0, also ∂Aρ/∂ρ + Aρ / ρ = 0, hence Aρ = 0 to avoid infinity of energy integral.

As result we get:

a = c · Az, Aρ = 0, Ez = 0

Eρ = - ∂a/∂ρ = - c · ∂Az/∂ρ

Which corresponds to the equation derived earlier from Eρ'

∂Eρ/∂z = ∂Ez/∂ρ - c · (2Az/∂ρ/∂z - ∂2Aρ/∂z2)

Herewith Bφ = - ∂Az/∂ρ = Eρ/c

Charge, spin and polarization

If one looks in the direction of movement of field object, it is easy to notice that in the above version with annular magnetic field it is possible to orient this field clockwise or counterclockwise. Accordingly, radial electric field will be directed from z-axis outward or inward to this axis. To one type of field formations can be attributed conditional positive "spin", to the second negative.

Let us try to find out how intensity of fields can decrease at distance from the geometric center of object.

Let a = A0 / s, где A0 = amplitude constant,

and s2  = R2 + ρ2 + z2, where R = object's scaling constant, possibly having an indirect relation to conditional "wavelength" in experiments. Note that ∂s/∂ρ = ρ / s, ∂s/∂z = z / s

Then Az = A0 / c / s, Aρ = 0, Eρ = A0 · ρ / s3, Ez = 0

div E = ∂Eρ/∂ρ + Eρ / ρ = A0 · (2 / s3 - 3 · ρ2 / s5)

The integral of charge density (divided by dielectric constant) over the entire space will be equal to

-∞+∞02·π0 (2 / s3 - 3 · ρ2 / s5) · ρ ∂ρ ∂φ ∂z = 0

That is, although charge density is not locally zero, the object as a whole is charged neutrally. This is natural, for example, for radiation arising from atoms and molecules, taking into account laws of conservation, since the particles located there will not give up part of their charge.

In general, when E = Eρ = - ∂a/∂ρ, the subintegral expression

ρ · div E = ρ · (∂Eρ/∂ρ + Eρ / ρ) = ρ · (- ∂2a/∂ρ2 - ∂a/∂ρ / ρ)

= - ρ · 2a/∂ρ2 - ∂a/∂ρ = ∂/∂ρ (- ρ · ∂a/∂ρ)

Computing the integral 0 ρ · div E ∂ρ we get

for ρ = 0 the function - ρ · ∂a/∂ρ = 0,

for ρ = ∞ the function - ρ · ∂a/∂ρ = 0

if ∂a/∂ρ decreases by absolute value with a distance faster than 1 / s

Further computation of integrals by φ and z will not change zero result. The author of this article tested using MathCAD zero equality of the triple integral for a = A0 · ρ2 / s3 with Eρ = A0 · (2 · ρ / s3 - 3 · ρ3 / s5), also for a = A0 · ρ4 / s5 with Eρ = A0 · (4 · ρ3 / s5 - 5 · ρ5 / s7), for a = A0 · ρ / s2, a = A0 · z / s2, a = A0 / s2

Very wide range of such objects is neutrally charged in general, although it is likely that field formations are statistically inclined to take simplest geometric shapes, with minimum number of spatial extrema.

It should be noted that when a = A0 / s2 or s appears with even higher degrees, field formation receives significantly greater ability to penetrate matter than with a = A0 / s or a = A0 · ρ2 / s3

Accordingly, the probability of registration of field object by measuring instruments is reduced. Which may be similar to the behavior of neutrinos in experiments.

Polarized field object can be described as follows:

s2 = R2 + X · x2 + Y · y2 + Z · z2

where R, X, Y, Z are scaling constants

∂s/∂x = X · x / s, ∂s/∂y = Y · y / s, ∂s/∂z = Z · z / s

If a = A0 / s, where A0 is amplitude

Az = A0 / c / s, Ax = 0, Ay = 0

Ex = A0 · X · x / s3, Ey = A0 · Y · y / s3, Ez =  0

Bx = - A0 / c · Y · y / s3, By = A0 / c · X · x / s3, Bz = 0

div E = ∂Ex/∂x + ∂Ey/∂y + ∂Ez/∂z

= A0 · (X / s3 - 3 · X · x2 / s5 + Y / s3 - 3 · Y · y2 / s5)

At the same time, all the above formulas for case of circular magnetic field remain true,

E = c2 · J - grad (E · V) - V · div E

Ex' = c2 · (∂Bz/∂y - ∂By/∂z) - 0 - 0 = 3 · A0 · c · X · Z · x · z / s5

Ey' = c2 · (∂Bx/∂z - ∂Bz/∂x) - 0 - 0 = 3 · A0 · c · Y · Z · y · z / s5

Ez' = c2 · (∂By/∂x - ∂Bx/∂y) - 0 - c · div E = 0

That is, there may be no cylindrical symmetry, with different X and Y, the field object will be stretched or extended along x- axis or y-axis. Compression or extension along z-axis is determined by multiplier Z. With significant differences between coordinate multipliers, structures arise with predominant orientation of fields in one direction (and the opposite also) in areas with high field energy density.

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As Markus says (+1), the KG equation is not controversial at all today, because we understand it in terms of field operators, not in terms of the probability amplitude of just one relativistic particle. When the kinematics enters the relativistic regime, you no longer are dealing with one particle, and enter the realm of particle-antiparticle pairs, so your field variables are not interpreted in terms of localisation amplitudes for one particle.

The KG equation was in fact hypothesized by Schrödinger, but he originally ruled it out on account of producing "negative probabilities."

In quantum field theory, the φ(x) field variable does not represent a probability amplitude at spacetime point x, but annihilating a particle --or producing an antiparticle-- at x.

Edited by joigus
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1 hour ago, joigus said:

but annihilating a particle --or producing an antiparticle-- at x.

I mean "quantum amplitude for either annihilating a particle or producing an antiparticle at x=$$\left( t, \boldsymbol{x} \right)$$.

I prefer the rather less theology-laden terms appearing and disappearing, rather than "creating" and "annihilating." But that's me. I find God-fearing Pauli suspect of having introduced these semi-religious terms, but who knows. I'm a stickler for clean, minimally-assuming language that really tells you what's going on, and nothing more.

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4 hours ago, computer said:

Very wide range of such objects is neutrally charged in general, although it is likely that field formations are statistically inclined to take simplest geometric shapes, with minimum number of spatial extrema.

Take a long hard look at this sentence you wrote. Why would anyone believe any of that? What do you mean "very wide range"? What do you mean "in general"? What do you mean "likely"? How do you know "statistically inclined"? What do you mean "simplest geometric shapes"? What do you mean "minimum number"?

Does any of your reasoning depend on this statement?

Also, you seem to be looking for topological solutions of Maxwell equations. Do you know there's an extensive field of work on that already?

Also, you're implying "classical" all the time, but you're saying "spin." Are you aware that spin is fundamentally non-classical?

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!

Moderator Note

Similar topics merged

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For a recent work on knotted topologic solutions of Maxwell equations:

The biblio will take you back to the origins of this extensive body of work.

My --recently deceased-- and dearest professor Antonio Fernández Rañada* was one of the pioneers, along with José L. Trueba. Both of them I knew personally, and I can attest to the fact that they have done very interesting work in the field.

* I will never forget Rañada. I got my paper on quantum theory of measurement peer-review-published thanks to him, in the face of staunch opposition of other members of the Faculty.

Edited by joigus
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11 hours ago, Markus Hanke said:

It’s the simplest possible wave equation for a relativistic scalar field without spin.

It is too simple to describe something like particle. But maybe useful for some specific cases, like 1-dimensional or 2-dimensional current in frozen conductor. Anyway, stationary point-like solution gives infinite integrals over all the space.

Relativistic theories arose on the principle of long-range action. Of classical and quantum physics, only small fraction of equations work on the locality principle, so that first time derivative depends on the spatial state of some fields. Maxwell's equation for independent field without sources, Schrödinger's equation. The rest usually implies presence of material points, hamiltonians and long-range action.

7 hours ago, joigus said:

KG equation is not controversial at all today, because we understand it in terms of field operators, not in terms of the probability amplitude of just one relativistic particle. When the kinematics enters the relativistic regime, you no longer are dealing with one particle, and enter the realm of particle-antiparticle pairs, so your field variables are not interpreted in terms of localisation amplitudes for one particle.

The KG equation was in fact hypothesized by Schrödinger, but he originally ruled it out on account of producing "negative probabilities."

In quantum field theory, the φ(x) field variable does not represent a probability amplitude at spacetime point x, but annihilating a particle --or producing an antiparticle-- at x.

In no way KG equation is "controversial", but it came from long-action area. The most prominent feature of short-action (locality) is explicit first time derivative for any field (or at least for fundamental ones). Schrödinger's statistical equation is built so, and Maxwell's (for pure field without charges). It seems relativity is completely incompatible with motion of particles and even "big" charged bodies, because of problems with field internal energy conservation laws. Relativity is useful for such objects as rockets with clocks and stations with observers, nothing more.

Schrödinger's equation works only in the presence of powerful centers of attraction, like atomic nuclei. Each member is important, and only the nuclear attraction collects an electron cloud, other members dissipate.

6 hours ago, joigus said:

Take a long hard look at this sentence you wrote. Why would anyone believe any of that? What do you mean "very wide range"? What do you mean "in general"? What do you mean "likely"? How do you know "statistically inclined"? What do you mean "simplest geometric shapes"? What do you mean "minimum number"?

Does any of your reasoning depend on this statement?

Also, you seem to be looking for topological solutions of Maxwell equations. Do you know there's an extensive field of work on that already?

Also, you're implying "classical" all the time, but you're saying "spin." Are you aware that spin is fundamentally non-classical?

My work is not designed to believe, but to think and calculate.

"Very wide range" does mean if field intensity decreases with distance as 1 / r, 1 / r2, 1 / r3 and so on.

"Simplest geometric shapes" does mean namely with minimal number of local extrema. If a photon emitted by electron cloud with statistical background, there is no cause to be very complex in shape. For example, visualization of function 1 / s looks much simpler than of r2 / s3 having minimum at r = 0 and other at f'(r) = 0 with other root of quadratic equation 2 · r / s3 - 3 r3 / s5

6 hours ago, joigus said:

Also, you're implying "classical" all the time, but you're saying "spin." Are you aware that spin is fundamentally non-classical

I have not found any word "classic" or "classical" in my last article about objects moving at the speed of light.

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33 minutes ago, computer said:

It is too simple to describe something like particle. But maybe useful for some specific cases, like 1-dimensional or 2-dimensional current in frozen conductor. Anyway, stationary point-like solution gives infinite integrals over all the space.

Relativistic theories arose on the principle of long-range action. Of classical and quantum physics, only small fraction of equations work on the locality principle, so that first time derivative depends on the spatial state of some fields. Maxwell's equation for independent field without sources, Schrödinger's equation. The rest usually implies presence of material points, hamiltonians and long-range action.

In no way KG equation is "controversial", but it came from long-action area. The most prominent feature of short-action (locality) is explicit first time derivative for any field (or at least for fundamental ones). Schrödinger's statistical equation is built so, and Maxwell's (for pure field without charges). It seems relativity is completely incompatible with motion of particles and even "big" charged bodies, because of problems with field internal energy conservation laws. Relativity is useful for such objects as rockets with clocks and stations with observers, nothing more.

Schrödinger's equation works only in the presence of powerful centers of attraction, like atomic nuclei. Each member is important, and only the nuclear attraction collects an electron cloud, other members dissipate.

No. The KG equation is local. Relativistic theories arose from the demand of complying with Lorentz symmetries, and there's nothing long-range about that. AAMOF, relativistic theories are far-better locality-compliant than Newtonian ones, and you can see that in the fact that exact dynamic solutions are expressed in terms of potentials that take account of the delay. Eg, the Liénard-Wiechert potential.

In fact, the standard solutions of the Schrödinger equation for hydrogen-like atoms assumes an instantaneous action, not because it's non-local, but because the proton is considered as having infinite mass, so the CoM coincides with the position of the proton, which plays the role of a classical object for all intents and purposes. As you know the reduced mass of a pair of objects, when one of them is enormous in comparison, reduces to the smaller one, while the position of the smaller one with respect to the big one reduces to the position of the smaller one with respect to the CoM.

Relativity is useful for muons, rockets, and most everything else. It's the real deal.

Schrödinger's equation works for an infinite chain of paramagnetic atoms, for just one atom, or for an electron moving in a constant electric field.

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Thanks for reference to Arxiv article. Anyway Maxwell equations for "pure field" are insufficient to describe even a photon. They are prone to deal only with E (D) and H (B) vectors, but at least four fundamental fields required: a, A, E, V.

Maxwell's equations were often written before (specially in 19 century) with a term as product of velocity by charge density instead of current. But it seems that this term has come to be seen only as being associated with point charges forming some kind of cloud. Not as a fundamental field existing in the zero state everywhere on the continuum.

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The Schrödinger equation with Coulomb potential also assumes that the motions of the electron are much slower than the speed of light.

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7 minutes ago, joigus said:

Schrödinger's equation works for an infinite chain of paramagnetic atoms, for just one atom, or for an electron moving in a constant electric field.

Of course. But without external source of attraction or directing field electron cloud dissipates. So I wonder how KG equation can be used by itself and extremely simplified. In Schrödinger's equation also often present potential from other electron clouds, specific member of additional repulsion of clouds with similar spin. One can add magnetic interactions. In KG are nothing similar.

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4 minutes ago, computer said:

Thanks for reference to Arxiv article. Anyway Maxwell equations for "pure field" are insufficient to describe even a photon. They are prone to deal only with E (D) and H (B) vectors, but at least four fundamental fields required: a, A, E, V.

Equations are not prone to anything.

Microscopic version of Maxwell's equations can be expressed either in terms of $$\boldsymbol{B}$$, $$\boldsymbol{E}$$, or in terms of $$\varphi$$ and $$\boldsymbol{A}$$ --the scalar and vector potentials.

Macroscopic version of Maxwell's equations can be expressed in terms of $$\boldsymbol{E}$$, $$\boldsymbol{B}$$, $$\boldsymbol{D}$$, $$\boldsymbol{H}$$, which in turn can be expressed in terms of $$\boldsymbol{E}$$, $$\boldsymbol{B}$$, $$\mu$$, $$\epsilon$$, which in turn can be expressed in terms of $$\varphi$$ and $$\boldsymbol{A}$$, $$\mu$$, $$\epsilon$$. Mu and epsilon carry the properties of materials.

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8 minutes ago, joigus said:

The Schrödinger equation with Coulomb potential also assumes that the motions of the electron are much slower than the speed of light.

And an electron in an atom has no reason to move at speeds close to of light.

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