# Do point particles have volume?

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If not, is it possible, for instance, to have an infinite number of photons in a given space. Also how can such particle(f.e quarks), then make something that has both mass and volume.
Can mass exist without volume?

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50 minutes ago, Sunzap said:

If not, is it possible, for instance, to have an infinite number of photons in a given space. Also how can such particle(f.e quarks), then make something that has both mass and volume.
Can mass exist without volume?

Point particles do not have volume, but there may be other factors involved.

It is possible to have an infinite number of photons in a given space because they obey something called Bose-Einstein statistics (particles like this are classed as bosons).

However, particles like electrons (and quarks) follow Fermi-Dirac statistics. This means that you cannot have two particles with exactly the same quantum state. So electrons, for example, have to take up different positions (energy levels) around the nucleus of an atom. (Actually, because they have half-integer spin, you get two electrons at each level. Simplifying, somewhat.) This is the Pauli exclusion principle.

And when the electrons in the atoms of your hand meet the electrons in the atom of the table they push against each other. Partly because they all have negative charge, but also because they have to maintain their separate quantum states.

There are other ways in which even point particle appear to have a finite size. They will interact (collide or annihilate or whatever) of they can get within a certain distance of one another. This is known as the interaction cross section (because this cross section area is very small, scientists invented a new unit for it: the "barn").

There are various other ways of defining the "size" of particles, even though they are treated as points.

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A confounding factor is that the particles interact. So electrons can "act bigger" than a point particle because of that. Also, photons can't exist in an optical cavity or a waveguide if the wavelength is too big, but that ties back to the fact that the electric and magnetic fields have to satisfy certain boundary conditions at the surface of the waveguide (one of the electric field components has to be zero at the edge, forcing it to be a node of the field) so that, to me, muddies the water a bit. It's hard to separate out the interaction from the particle, but there are experiments and analyses which do so.

When we say e.g. an electron is a point particle, the implication is that there is no "ball of material" that contains the charge. This is perfectly consistent, I think, with the idea that the charge is a property, and not stuff unto itself, which might require a volume. (A lot of this confusion, I think, is a residual effect of thinking about things in terms of classical physics)

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1 hour ago, Sunzap said:

Can mass exist without volume?

Both mass and volume are extensive properties.
The issue of the point value of extensive properties has been investigated for centuries and several ways to dealing with the problem have been used.

Considering a previous answer it is interesting that mass is said to be imparted by a boson.

If so here is a discussion of the size of the Higgs.

I will be interested to see answers by Mordred and Joigus in due course.

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5 minutes ago, studiot said:

Both mass and volume are extensive properties.

Interesting statement. In what sense is mass "extensive" (not a word I am familiar with in this context).

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1 hour ago, Strange said:

Interesting statement. In what sense is mass "extensive" (not a word I am familiar with in this context).

Sure, but it was part of the whole response.

An intensive property (of a system or body etc) is one that has a single value to represent the whole body.

For example the temperature of a poker has meaning when the poker is standing quietly unused by the side of the fireplace and all parts of it are at the same temperature.
But the 'temperature of the poker' has no meaning when one end is red in the fire and the other is cold in the hand.

Set against that an extensive property is additive in that every small or large part of the body contributes to the property.

So the volume or mass of a gallon of beer is twice that of half a gallon and so on.

This summation leads directly to subsidiary properties such as density which is summed over the length to obtain the mass or volume or other extensive property/area/volume.

Density, of course, needs a limit since it is the ratio of properties one or more of which approach zero at a point.

So bingo you need the Calculus.

So this brings in Continuity.

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3 minutes ago, studiot said:

An intensive property (of a system or body etc) is one that has a single value to represent the whole body.

Intersting. I have never come across these terms before.

4 minutes ago, studiot said:

Set against that an extensive property is additive in that every small or large part of the body contributes to the property.

Does that apply to fundamental particles (whether zero sized or not) when arguably there is no smaller part? And, if they are considered point particles, then there is definitely no smaller part.

So mass might be an extensive property of macroscopic objects, but the concept doesn't seem to apply here. (A bit like spin, when applied to particles.)

6 minutes ago, studiot said:

Sure, but it was part of the whole response.

Do you sometimes forget to write down some of the stuff you were thinking of? (I often do.) Because none of this was in your original answer.

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Posted (edited)
19 minutes ago, Strange said:

Do you sometimes forget to write down some of the stuff you were thinking of? (I often do.) Because none of this was in your original answer.

Not this time, I was waiting to gauge the interest.

I am not sure who introduced the terms, I met them in Thermodynamics, but they have wider application.

The Mathematics and Physics started developing with the probing of distributed forces (pressure, stress...) leading to the introduction of the Dirac Delta function, and the unit impulse function.
These are not true functions because they are 'multi valued'.
This was resolved by the introduction of a wider class of 'generalised functions' called distributions.

This all links into QM and SM

So once again the theory started in the macroscopic domain which we can more easily grasp, before being transplanted into the microscopic.

But it is a big subject.

So the OP asked a good question, albeit a little flippantly, perhaps deeper than he realised.

Edited by studiot

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4 minutes ago, studiot said:

So the OP asked a good question, albeit a little flippantly, perhaps deeper than he realised

Often the case.

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22 minutes ago, Strange said:

Intersting. I have never come across these terms before.

Yes, I know what he means. In thermodynamics and statistical mechanics they are very common terms. Statistical mechanics is only rigorously defined when you make all the extensive parameters go to infinity. It's called the thermodynamic limit. Although I must say that talking about that wouldn't have been my choice, because the question of mass is difficult enough as it is.

Just now, joigus said:

Yes, I know what he means. In thermodynamics and statistical mechanics they are very common terms. Statistical mechanics is only rigorously defined when you make all the extensive parameters go to infinity. It's called the thermodynamic limit. Although I must say that talking about that wouldn't have been my choice, because the question of mass is difficult enough as it is.

As I've been mentioned, I'm struggling to add something significant to what has been said. But it's not easy...

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Posted (edited)

While a multitude of bosons are allowed in the same state, according to B-E statistics, and F-E statistics puts limits on the number of fermions in any one state. Keep in mind that 'state' is not necessarily equivalent to 'place'. IOW, take a look at electron orbitals.

Another factor to keep in mind...
To establish a 'size' for a quantum particle, you are, in effect, placing it in a 'box' and seeing how small you can make that box.
Unfortunately the HUP has a large effect when that constraining 'box' becomes too small, and the particle's momentum  becomes so indeterminate that it may possibly exceed the speed of light. At this point the situation becomes non-sensical, and the best you can do is establish a lower bound for the size of the particle.

Edited by MigL

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2 hours ago, studiot said:

I will be interested to see answers by Mordred and Joigus in due course.

The issue of what mass is and what it's to do with space is very deep indeed, and I think it can be discussed at several levels, has acquired successive layers of sophistication throughout history, and I don't think I will be able to answer to the full satisfaction of anybody.

I've tried to compile a brief excursion through it that I will separate, so that nobody needs to read it. But it could serve as reference.

</start of digression>

Brief history of mass

There are several concepts of mass, which should really be carefully distinguished.

-Inertia: The opposition to acceleration as defined by Newton in the Principia. But this definition is tautological (force, inertia, what comes first?), and Newton could get away with it only because he was incredibly clever and instinctively new how to get out of the tautology by adding auxiliary hypothesis in the force law.

-Gravitational mass (Newton again!): The source of the gravitational field. Some people even distinguish between active gravitational mass (source) and passive (test particle.) But we can immediately identify both if we want momentum to be conserved, as otherwise F_12 =/= F_21.

-Then comes Newton, with hypothesis that amounts to equivalent pple. (EP) ==> inertia = gravit. charge (define all as 'mass'.)

-Ernst Mach: Mass must be a property of the filling-in of the surrounding space in relation to the particle. ==> mass appears to be a 'non-local' property in the sense that it codifies some properties of the whole universe as related to the particle. Any possible local anisotropy for inertia has always tested negative.

-Special Relativity: Einstein finds out that what we call mass is really rest energy reasoning in terms of inertial systems.

-GR: Einstein gives up Mach's principle as a constructive principle for a theory of gravity and opts for EP, although he remains impressed by its logical clarity to the end of his life.

Then comes QFT, which sees mass as a scale-dependent parameter. Safely defined as a constant for the infrared as the concept we all know and love, but blowing up in our faces when we try to go to higher and higher energies (UV limit.)

Then comes a symmetry-related problem with the electroweak force (EW) in the Standard Model (SM). In order to model short range interactions people need to give mass to the gauge bosons, because gauge bosons must be massless if gauge symmetry is to be preserved. Higgs et al. realize that a quantum field with 2 complex (4 real degrees of freedom) does the trick. Three of them are swallowed up by the bosons, giving them mass, and the other one must have an ephemeral life as the Higgs. But the thing becomes much more far-reaching when people realize in the laboratory that parity in its different versions (P, CP) is violated maximally all over the place. Dirac's equation connects the left-handed e- with the right-handed e- precisely through the mass term. But what's the right-handed electron doing there? All fermions need some kind of weird scalar companion to give them mass and account for parity violation. So every fermion must be coupled to a Higgs 4-plet.

</end of digression>

As you see, it's all a mess that English physicists have got us into!!

3 hours ago, Strange said:

Point particles do not have volume, but there may be other factors involved.

Not in any ordinary sense, no. I agree. Not like a gas, nor like a piece of crystal.

3 hours ago, swansont said:

A confounding factor is that the particles interact. So electrons can "act bigger" than a point particle because of that.

I like the way you put it. Yes, that tends to confound things a lot. I actually was quite satisfied when I read the first two answers.

17 minutes ago, MigL said:

To establish a 'size for a quantum particle, you are, in effect, placing it in a 'box' and seeing how small you can make that box.

Yes! Good point. The Klein paradox: When you consider a particle confined to a place of the size of (half) its Compton wavelength (=twice its mass,) it not longer makes sense to even consider it as a particle, as you are in the particle-antiparticle pair creation regime. I almost forgot. But I think that's an extremely important point.

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5 minutes ago, joigus said:

As you see, it's all a mess that English physicists have got us into!!

Hey, stop blaming the Brits
Neither A Einstein or E Mach were British

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Posted (edited)
4 minutes ago, MigL said:

Hey, stop blaming the Brits
Neither A Einstein or E Mach were British

I'm just trying to engage Studiot.

I tried to +1 you at your reminding me of the Klein paradox, but I messed up. Well deserved anyway.

Edited by joigus

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Wasn't aware of the Klein paradox.

Wiki mentions it was a problem with the Rutherford atom's neutral nuclear particles ( electron-proton confinement before the discovery of the neutron ).
I got the electron confinement, to a diminishing size box, from S Chandrashekar's treatment of maximum size of white dwarf stars, and collapse to neutron stars, which he wrote on the ship voyage to England, to do his graduate study under A Eddington.

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There are some illuminating examples, I think.

When you look at the structure of Hydrogen, you don't get quite the right answer if you assume the nucleus is a point particle, because it isn't. The proton has an extent, and therefore using a point mass and point charge in the theory lead to subtle differences in the spectra as compared to experiment. The fact that the electron spends time near r=0, for example, means there is a difference between assuming a point and an extended particle. The  result is that the hydrogen spectrum is more complex than it might be, if all of our first-order approximations actually represented the situation.

In a similar vein, you can do scattering experiments with an electron, where you can model the interaction coming from a point particle, or a particle of finite size. Lo and behold, there is no size you can assign to the electron. Any nonzero result is indistinguishable from the experimental error. It is interacting as if it were a point particle.

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It isn't only the nucleus that can define the volume of a particle.

Using the reciprocal or k space of Brillouin zones for a crystal lattice you can calculate the number of electrons that will 'fit' into a given space (volume).
Since the electron can roam the entire space does this mean that dividing the volume by the number of electrons gives an estimate of the volume of an electron ?

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3 hours ago, joigus said:

The issue of what mass is and what it's to do with space is very deep indeed, and I think it can be discussed at several levels, has acquired successive layers of sophistication throughout history, and I don't think I will be able to answer to the full satisfaction of anybody.

I've tried to compile a brief excursion through it that I will separate, so that nobody needs to read it. But it could serve as reference.

</start of digression>

Brief history of mass

There are several concepts of mass, which should really be carefully distinguished.

-Inertia: The opposition to acceleration as defined by Newton in the Principia. But this definition is tautological (force, inertia, what comes first?), and Newton could get away with it only because he was incredibly clever and instinctively new how to get out of the tautology by adding auxiliary hypothesis in the force law.

-Gravitational mass (Newton again!): The source of the gravitational field. Some people even distinguish between active gravitational mass (source) and passive (test particle.) But we can immediately identify both if we want momentum to be conserved, as otherwise F_12 =/= F_21.

-Then comes Newton, with hypothesis that amounts to equivalent pple. (EP) ==> inertia = gravit. charge (define all as 'mass'.)

-Ernst Mach: Mass must be a property of the filling-in of the surrounding space in relation to the particle. ==> mass appears to be a 'non-local' property in the sense that it codifies some properties of the whole universe as related to the particle. Any possible local anisotropy for inertia has always tested negative.

-Special Relativity: Einstein finds out that what we call mass is really rest energy reasoning in terms of inertial systems.

-GR: Einstein gives up Mach's principle as a constructive principle for a theory of gravity and opts for EP, although he remains impressed by its logical clarity to the end of his life.

Then comes QFT, which sees mass as a scale-dependent parameter. Safely defined as a constant for the infrared as the concept we all know and love, but blowing up in our faces when we try to go to higher and higher energies (UV limit.)

Then comes a symmetry-related problem with the electroweak force (EW) in the Standard Model (SM). In order to model short range interactions people need to give mass to the gauge bosons, because gauge bosons must be massless if gauge symmetry is to be preserved. Higgs et al. realize that a quantum field with 2 complex (4 real degrees of freedom) does the trick. Three of them are swallowed up by the bosons, giving them mass, and the other one must have an ephemeral life as the Higgs. But the thing becomes much more far-reaching when people realize in the laboratory that parity in its different versions (P, CP) is violated maximally all over the place. Dirac's equation connects the left-handed e- with the right-handed e- precisely through the mass term. But what's the right-handed electron doing there? All fermions need some kind of weird scalar companion to give them mass and account for parity violation. So every fermion must be coupled to a Higgs 4-plet.

</end of digression>

I will be adding further details to this later on as it's a good discussion that deserves extra attention.

The key points I would like to add is that mass is best treated as resistance to inertia change. How a particle couples to the fields they interact with contributes to the mass term. With the Higgs field a good tool to use is the CKMS and PMNS matrix. This matrix is what is described by the Covariant derivatives in the Link Studiot posted earlier.

(Though I would like to examine the equation in more detail later on. It's not a form I recognize though looks accurate).

Anyways I will add more later on after work.

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