Is matter infinitely divisible?

No, because the charged particles taken as fundamental (electrons, quarks, etc.) have intrinsic magnetic moments which cannot be classically reconciled with their angular momentum. Therefore fundamental electrodynamics has to deal with a unified electromagnetic field.

There is no separate "electric field" -- only an frame-dependent electric aspect of an electromagnetic field.

 

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Apparently, it is impossible to find the quantum of electric field if there is no electric field. I am still looking
for it though.

My situation resembles the problem of those working with quantization of gravity: is there time or is there only spacetime?
In 1908 did the German mathematician Hermann Minkowski announce his “radical” idea that space and time were a single entity: “Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”

What is time? Nothing but a fourth dimension, after length, breadth, and thickness. There is only a four-dimensional continuum, spacetime, where all events are baked together.
The theoretical physicist Lee Smolin is aiming to convince us that time is real after all. He is frankly recanting the accepted doctrine—an apostate:"I used to believe in the essential unreality of time. Indeed, I went into physics because as an adolescent I yearned to exchange the time-bound, human world, which I saw as ugly and inhospitable, for a world of pure, timeless truth….I no longer believe that time is unreal. In fact I have swung to the opposite view: Not only is time real, but nothing we know or experience gets closer to the heart of nature than the reality of time."

http://www.nybooks.com/articles/archives/2013/jun/06/time-regained/

I guess we cannot quantize time and space if they don't exist.
Maybe only spacetime can be quantized in that case.

 

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I think rpenner is saying the same thing as I was trying to, which is that the free-standing electric field has to be seen as a part of the joint EM field and thus is mediated by the same force carrier, the photon. So you do not need to search for something else.

I admit to being out of my depth on the issue of whether spacetime itself may be quantised. I would have thought not, as they would appear to be metrics rather than physical things, ad I can't envisage the constraint on them that would determine the size of the quantisation, but I'm willing to be told otherwise by a knowledgeable person.

 

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It means division of a quantum of a field. For example, the photon is not divisible, there is no half
of a photon.

Maxwell's equations describe what is an electromagnetic field. A field can be static, but there
is no static wave, there are standing (stationary) waves, though. Wavelet is a wavepacket ,a pulse.

Charged particles, electrons, protons produce electric fields. Matter is the source of fields. Coulomb's law describes a force between charged particles. The force is "transmitted" with the help of a field: the electric field. The mechanism of this transmission of force is explained in QED as an interaction between the charges. QED introduces a force transmitting particle, a virtual particle, a virtual photon. The interacting charged particles exchange a virtual photon between them.In this way we understand the mechanism by which the charged particles exert their characteristic forces. The magnetic field is also introduced in the interaction picture, because the photon is the quantum of EM-field. This happens although Coulomb's law does not contain magnetic field. In other words, the interaction picture introduces magnetic field into QED when it explains the mechanism by which the charged particles exert their forces.

Difficulties with infinities.

Yes, Coulomb's law says that as the distance to the particle approaches zero, the electric field strength becomes infinite.

I don't know. Spacetime is not yet quantized. It is not known if space can be quantized, or time.
Space and time are unified, so they may not exist as separate entities. On the other hand, if
what Lee Smolin says is true, that time is real, it may be possible to quantize time and perhaps
also space.

If space can be quantized we will understand what is the meaning of proximity at the quantum scale.We must also take into account the Heisenberg uncertainty principle.

Electron is both a wave and a particle.

Mathematical equation, for example Coulomb's law, seems to allow a continuum of values for the strength
of the field, it allows infinitely strong values of electric field at very short distances to the charged
particle. It is, however, likely the nature does not follow the absolute mathematical law and the
value of electric field cannot increase without bound.Instead of acquiring an infinite value, the electric
field stays instead finite and becomes quantized at very short distances to the source of the field, the particle.

 

A photon is not a field. A photon is a wavelet. A wavelet is a windowed segment of a continuous wave.

The standing wave is the cause of discrete energy levels of the electron. There can only be integer harmonics in a standing wave. This is the case of normal matter, that the electron is bound to the orbital it occupies, in the form of the standing wave allowed for that orbital, as defined by its principal quantum number.

A static field is an idealization that applies to the hypothetical existence of a stationary charge. Coulomb's law defines the the force arising from the field intensity according to that idealized condition. The real static fields are the ones for which all charges within an ensemble are superimposed, which may or may not resemble the idealized case.

The photon is a wavelet. A wavelet is the time-domain product of (a) a continuous sine wave and (b) a window function. But a light ray is an ensemble of these wavelets, which, when superimposed, reconstruct the continuous sine wave.

All charged particles are attended by electric fields. By the same token, all electric fields attending point charges combine under the principle of superposition. The superposition of fields imposes a requirement that the charges mutually interact, and the form of that interaction is a mutual force. The magnitude of the force follows Coulomb's Law.

A photon impinging on normal matter delivers energy E = hν, which, when absorbed by an electron, raises that electron's energy level. The photon is annihilated. Conversely, when an electron falls in energy level, a photon is created with energy E = hν. This applies to the interaction of normal matter with light (EMR). The interaction of charge with fields may or may not involve such an exchange. If a cathode ray is immersed in an electric field, the then force that arises acts on the particles to accelerate them. Their frequency remains constant but their kinetic energy increases.

The photon is a quantum of energy which exists as an electromagnetic wavelet. There is no magnetic field per se. There is a single wave, having two orthogonal components: an electric field component and a magnetic field component. All interactions with particles are at the level of the photoelectric effect, namely, it's an energy exchange, not a field interaction.

Coulomb's law applies to point charge interactions with electrostatic fields.

There is an interaction per Coulomb's law which applies to pairs of charged particles and there is an interaction with light and matter which applied to photons interacting with bound electrons.

The singularity can't occur under Coulomb's law since two particles can not coexist at the same point in space. Conversely, Coulomb's law tells you it would take infinite energy to superimpose two particles.

Maybe so.

Regardless of that fact, two particles can't be superimposed.

Ok.

Yes, it takes infinite energy to superimpose particles into infinitely dense matter.

The laws of electromagnetics are physical, not mathematical, laws. They are represented mathematically as Maxwell's equations. Coulomb's law contains a singularity at r=0, but it does not mean the field is infinite.

The thing you are calling an infinite field is not physically an infinite field. It is a particle.

 

The self-energy problem of the electron arises because if the radius of the electron were zero, the electric field
becomes infinite, \(E = e/r^2\), r is the distance to the electron. This is the reason why classically it was assumed that the electron has a finite radius, and thus isn’t a point particle.

http://arxiv.org/pdf/physics/0608108v1.pdf

Einstein wrote about the elementary particles of matter in his article Physics and Reality from 1936. He wrote that "the elementary particles of matter carry unalterable electric charges and, on this
account are subject on the one hand to the actions of ponderomotive forces and on the other hand possess
the property of generating a field. The elementary particles obey Newton's law of motion for the material point.......The unsatisfactory part of the theory showed up externally by the necessity of assuming finite dimensions for the particles in order to prevent the electromagnetic field existing at their surfaces from becoming infinitely great."

http://www.kostic.niu.edu/physics_and_reality-albert_einstein.pdf

Physically there cannot be an infinitely strong field, although mathematical model leads to it. Therefore we need to assume finite dimensions for the particles to avoid this. We need to assume that mathematics
does not model physical world exactly.

 

That's meaningless. You are conflating the field with the particle.

That doesn't matter. The electron is "there", not the field.

So? That does nothing to support your claim that there is a field at r=0.

Your "model" is wrong. You are confusing math and physics. There is a particle "there" at r=0. Hence there "is" no field. The field exists only in the vacuum.

No. you simply need to understand that there is a particle "there" not a field.

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Simply stated: You are ignoring the boundary conditions which limit the range of r for which the laws of electromagnetics apply. Wherever you find a singularity, or a value converging to a singularity which exceeds the maximum value allowed, the math is telling you that you made a mistake. In fact you've probably made a chain of them. You have to go back and figure out what the boundary conditions are for the physical world you are trying to model. The laws of electromagnetics are not sufficient to do that. You need more information.

As a general rule, you can consider that the electron is known by its wavelength, therefore it makes no sense to say you are "evaluating the field" at less than one wavelength of the particle in question. At that "proximity" you've forced into the model an uncertainty about whether the electron is even there.

Also, you should reconsider all of the assumptions you are making. When you say "point charge" you probably aren't even close to the conventional meaning. For one thing, you have to remember that whenever you try to resolve the world to around 1 Plank length, just about all of physics breaks down, beginning with those qualities and quantities which can only exist as valid integer amounts per the principal quantum number. Trying to resolve space to this scale is problematic in itself. Once you grasp what I'm saying, you'll realize it makes no sense to say "the field at the center of the point occupied by the electron" any more than it does to talk about the field "at one Planck length from the center of the point charge." Back off by the wavelength of the electron, and you may be able to say you are "as close as possible" to the elusive particle.

Finally, you are approaching this the wrong way. You shouldn't be asking "what is wrong with nature" and/or "Coulomb's Law is obviously broken", but tather, what did you omit? What assumptions did you make? How do you test whether your assumptions are valid? . . . etc.