Are Photons Actually Infinitely Small Particles?

[sophiecentaur]

Of course. The above post is ironic. I think you mean 'facetious'. [wink]

I have really enjoyed this thread. I'd been meaning to start one like this for ages.

That's a really really, really.
It's almost a 'model' thread.

 

[Cthugha]

Phase always refers to some time origin so, unless you could be more specific then the concept of the phase of a single photon would have no meaning - unless you could say something about when it was 'created'.

Yes, of course. What I was trying to say was that even for an ensemble of measurements performed on equally prepared single photons, the phase is ill defined. The phase of a single photon indeed has no meaning.

The only thing we can say fairly definitely is that a photon is a defined amount of energy that can be transferred when em power interacts with a system.

In fact, not even the exact amount of energy is well defined as most experimentally realized single photon sources are pretty broad spectrally. However, we can say that the energy transfer is quantized. While this is more or less indeed the only thing we can say fairly definitely, I think it is already quite a remarkable and non-trivial finding.

Nevertheless the "classical concept" of a photon, as you call it, should indeed be dead and buried.

 

[sophiecentaur]

Nevertheless the "classical concept" of a photon, as you call it, should indeed be dead and buried.

That's ma boy!
If only the unthinking public could say as much.

 

[Q-reeus]

Polarisation seems a red herring to me - taken care of by the wave model, entirely (afaics). A classical wave with circular polarisation carries angular momentum so the statistics of the photons 'in' that wave allows the photons to have spin. Linear polarisation can be regarded as consisting of suitably paired spinning photons.

Last bit is fine, but of course for a classical EM plane wave that works both ways - circular polarization can be decomposed into orthogonal linearly polarized waves. The matter of associating spin = intrinsic angular momentum, with CP (circular polarization) is a bit tricky. While it's easy to show that say crossed dipole antennas as a source of CP waves react on each other to give a net time-averaged torque, there is a seemingly paradoxical lack of any EM reaction torque when a normal incident CP wave is absorbed by a resistive sheet say. Which makes it very hard to reconcile photon spin with field CP. One is forced to find the field angular momentum as due to a net non-radial component in the Poynting vector of the combined radiation field of the CP source emitter.

Consequently just how a 'point' particle photon can carry an intrinsic angular momentum, apart from mathematical postulate, is hard if not impossible to visualize. In the case of an electron say, there is this concept of the 'dressed' charge and spin angular momentum might be considered as residing in some finite effective volume of virtual particles surrounding the 'bare' charge. But for a photon - is there a feasible 'point particle corkscrew motion' model applicable, or must one accept sheer mathematical postulate only? But then I suppose we are meant to accept the lesson is 'stop trying to visualize - there are no physical models that work, period!'

...Again, using wavelengths that are not optical, can open the view of what goes on; does a particle that would have to extend to most of a country make sense in a capacitor that is 2cm long?

Not sure of the point here. At 50Hz the capacitor field is local = near-field = virtual photons right out to such distances it becomes negligible. There is some finite radiation, but incredibly weak.

 

[sophiecentaur]

Last bit is fine, but of course for a classical EM plane wave that works both ways - circular polarization can be decomposed into orthogonal linearly polarized waves. The matter of associating spin = intrinsic angular momentum, with CP (circular polarization) is a bit tricky. While it's easy to show that say crossed dipole antennas as a source of CP waves react on each other to give a net time-averaged torque, there is a seemingly paradoxical lack of any EM reaction torque when a normal incident CP wave is absorbed by a resistive sheet say. Which makes it very hard to reconcile photon spin with field CP. One is forced to find the field angular momentum as due to a net non-radial component in the Poynting vector of the combined radiation field of the CP source emitter.

Consequently just how a 'point' particle photon can carry an intrinsic angular momentum, apart from mathematical postulate, is hard if not impossible to visualize. In the case of an electron say, there is this concept of the 'dressed' charge and spin angular momentum might be considered as residing in some finite effective volume of virtual particles surrounding the 'bare' charge. But for a photon - is there a feasible 'point particle corkscrew motion' model applicable, or must one accept sheer mathematical postulate only? But then I suppose we are meant to accept the lesson is 'stop trying to visualize - there are no physical models that work, period!'

So - no classical torque resulting from CP absorption? Awkward. Would there be no circular induced currents, to account for it?

Not sure of the point here. At 50Hz the capacitor field is local = near-field = virtual photons right out to such distances it becomes negligible. There is some finite radiation, but incredibly weak.

My point was that one has to 'bend' the geometrical characteristic of the photon in order to 'fit' the practical situation. You have introduced the notion of virtual photons to take care of what you call near field and that could be ok, I suppose. If you look at the fields due to a transmitting dipole, the fields change from E & H in quadrature in the near field and E & H in phase in the far field. The virtual photons presumably relate to this local quadrature fields?

There still seems, to me, to be a wierdness with wanting photons to, somehow, be different from individual to individual. A photon that is sourced in a distant star must, surely, be identical to one sourced locally if the two of them can interact in the same way with the same receiver.

 

[Q-reeus]

So - no classical torque resulting from CP absorption? Awkward. Would there be no circular induced currents, to account for it?

Well there is a net circular acting sheet current, but the magnetic interaction, making the usual assumption purely transverse motion of charges applies, yields no net torque. This is most easily seen by considering the CP wave as two spatially and temporally orthogonal linearly polarized plane waves. For each such component, E and B are mutually orthogonal and transverse to the propagation vector k. Consequently the B component of one wave is parallel to the current (which is in the direction of E) induced by the other wave. And as we know from Lorentz force law, when J and B are parallel, there is no magnetic force. Hence no mutual interaction, regardless of relative phase.

When field strengths get very high and frequency is not too great, as for EM propagation through a weakly ionized plasma, there is an induced longitudinal motion, but it tends to be very small wrt to transverse motion.
It may be of comfort to know some researchers can show a CP wave will induce angular momentum in media, in some circumstances at least: http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-14-5315 (unlocked file, so can be downloaded)

My point was that one has to 'bend' the geometrical characteristic of the photon in order to 'fit' the practical situation. You have introduced the notion of virtual photons to take care of what you call near field and that could be ok, I suppose.

Well vp's are not my original idea - and I'm aware the notion is hotly disputed here at PF/QM. Was merely equating 'accepted' terminology. Personally I'm agnostic as to reality of vp's - out of sheer ignorance of all the subtle arguments if nothing else.

If you look at the fields due to a transmitting dipole, the fields change from E & H in quadrature in the near field and E & H in phase in the far field. The virtual photons presumably relate to this local quadrature fields?

That more or less fits many peoples view - quadrature = zero time-averaged Poynting vector = reactive field(s) = 'virtual photons'. We left out static fields but let's not go there!

There still seems, to me, to be a wierdness with wanting photons to, somehow, be different from individual to individual. A photon that is sourced in a distant star must, surely, be identical to one sourced locally if the two of them can interact in the same way with the same receiver.

I agree. Yet while the search for a physical model that is universally applicable may be futile, surely along the way we can gain insights, if nothing else by eliminating models that just do *not* work other than on an ad hoc basis.

 

[Dickfore]

Photons do not even have a position, because, unlike non-relativistic QM, in relativistic QFT there is no probabilistic meaning attached to the wave function considered as a function of space-time coordinates.

Instead, its meaning is revealed in Second Quantization. Namely, the wave function gets promoted to a field operator that creates (annihilates) at a particular point in space, at a particular instant in time.

In QM, as well as QFT, elementary particles (those with which we associate a wave function or a field) are truly point particles. So, even if you had asked what is the size of the electron, you would have gotten an answer that it is a point particle.

 

[MikeGomez]

Re my question in post #38.

With a large number of photons, we get a diffraction pattern just like the one we get with light when we use a much smaller hole:

http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/cirapp.html

Single photons arrive at the screen or detector randomly according to a probability distribution which is just the classical intensity distribution:

http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/cirapp2.html#c2

As the wavelength increases, for the same diameter hole, the width of the central maximum increases. With radio waves (e.g. UHF with a wavelength of around 0.2 m) going through a hole 0.02 m in diameter, the central peak of the diffraction pattern more than fills the entire forward hemisphere beyond the aperture, so the aperture behaves almost as a "point" source of radio waves.

O.k. so with the radio waves I have simply described circular aperture diffraction at a different scale (than visible light). But what I was really wondering about is what you called the classical intensity distribution. The number of photons received, not the diffraction pattern.

If the wavelength is 100 times the hole diameter, and this causes a severe reduction in the probability of receiving a given photon, then that does seem to imply that a photon is somehow extended perpendicular to its direction of motion. The photon was directed at the center of the hole, let’s say in a direction we call the Z-axis. Now the photon’s ability to pass through in the XY plane depends on its wavelength!

I can only think of two ways this can happen.

1: The photon possesses a virtual field which extends perpendicular to its direction of motion, which travels faster than the speed of light in order to detect the edge of the hole and react with it. No, I don’t really believe this, so please don’t think I’m a crackpot.

2: The space that the hole occupies possesses virtual EM fields (ZPE). The virtual fields already know how big the hole is long before the real photon arrives. By this I mean that the virtual fields have natural preferred resonance frequencies which correspond with the size of the hole, due to interactions within the inside edges.

The real photon is some manifestation of the EM field, and it therefore seems to me quite reasonable for it to react with the virtual EM fields in such a way as to create diffraction patterns, probability distributions, etc.

Or are there other explanations?

 

[Q-reeus]

Photons do not even have a position, because, unlike non-relativistic QM, in relativistic QFT there is no probabilistic meaning attached to the wave function considered as a function of space-time coordinates.

Instead, its meaning is revealed in Second Quantization. Namely, the wave function gets promoted to a field operator that creates (annihilates) at a particular point in space, at a particular instant in time.

Which in order to square with Lorentz invariance, would seem to demand truly point particles, yes?
I'm wondering if QFT creation/annihilation is somehow the answer to the problem I posed here https://www.physicsforums.com/showthread.php?t=457544 but still without takers second time around (shameless plug :shy:)?

In QM, as well as QFT, elementary particles (those with which we associate a wave function or a field) are truly point particles. So, even if you had asked what is the size of the electron, you would have gotten an answer that it is a point particle.

Implying that for electron, virtual particle dressing cannot be invoked as seat of intrinsic angular momentum? If so this is a little disturbing because it signifies a total departure from any classical notion of what angular momentum entails at minimum - a finite moment arm! Showing my ignorance of QM/QFT here.

 

[Dickfore]

angular momentum in quantum mechanics is the generator of rotations and is not defined as in classical mechanics. But, you are derailing this thread.

 

[Q-reeus]

angular momentum in quantum mechanics is the generator of rotations and is not defined as in classical mechanics.

This is an example to my mind of what was said earlier - mathematical postulate completely divorced from any intuitive physical connection. I'm sure it 'works', but there must be some explanation of how say magnetization reversal does yield classical angular momentum change.

But, you are derailing this thread.

In what way?

 

[Q-reeus]

The real photon is some manifestation of the EM field, and it therefore seems to me quite reasonable for it to react with the virtual EM fields in such a way as to create diffraction patterns, probability distributions, etc.

You are suggesting one EM field reacts upon another *directly*. But that is contrary to the (classical) superposition principle.

Or are there other explanations?

Gave you one in #41. Incident field (bunch of photons) excites sheet currents in plate surface, which flow also within the hole boundary and out the other side of plate. In turn acting as re-radiators (secondary source of photons). A small number go on to the detector. That was for plate as a good conductor. If on the other hand it was a good absorber, transmission would still occur but radically altered in magnitude, and I think the estimate in #40 of direct proportionality to hole area may then be closer to the mark.

 

[Dickfore]

This is an example to my mind of what was said earlier - mathematical postulate completely divorced from any intuitive physical connection.

Define "physical connection" and then we can accept your criticism as justifiable. Also, rotations form a symmetry group. Specifying the angular momentum quantum number tells in what representation of the symmetry group does the object belong to. Continuous symmetries and conservation laws and selection rules play a fundamental role in Quantum Mechanics.

I'm sure it 'works', but there must be some explanation of how say magnetization reversal does yield classical angular momentum change.

I am not denying that angulatr momentum exists. I am simply saying that your logic of reasoning that defining angular momentum as \vec{L} = \sum_{a}{\vec{r}_a \times \vec{p}_a} and saying that a point particle has a zero arm means that it cannot have an intrinsic angular momentum (spin) is flawed because the definition does not apply in QM. You can see this if you make the classical limit \hbar \rightarrow 0, while keeping the spin quantum number fixed (an intrinsic property of the particle). We see that the spin angular momentum tends to zero, as we would reach by your suggested line of reasoning.

 

[Q-reeus]

...Also, rotations form a symmetry group. Specifying the angular momentum quantum number tells in what representation of the symmetry group does the object belong to. Continuous symmetries and conservation laws and selection rules play a fundamental role in Quantum Mechanics.

No doubt correct, but like a million others have bemoaned, it seems to be just one layer of abstraction upon another until there is no 'connection' with anything that can be sensibly visualized. For instance certain states that are mathematically distinct but identical re observables. Having a very classical background doesn't help warm one to that, but if theory and experiment agree, well that's that I guess.

I am not denying that angulatr momentum exists. I am simply saying that your logic of reasoning that defining angular momentum as \vec{L} = \sum_{a}{\vec{r}_a \times \vec{p}_a} and saying that a point particle has a zero arm means that it cannot have an intrinsic angular momentum (spin) is flawed because the definition does not apply in QM.

Fine but I was not arguing that a point particle cannot have angular momentum - merely that it cannot be reconciled with fundamental classical criteria.

You can see this if you make the classical limit \hbar \rightarrow 0, while keeping the spin quantum number fixed (an intrinsic property of the particle). We see that the spin angular momentum tends to zero, as we would reach by your suggested line of reasoning.

As per above, would just like to know how the quantum point particle regime transitions to the classical, spatially extended regime - if possible. No doubt that means reading a good book.

 

[Dickfore]

No doubt correct, but like a million others have bemoaned, it seems to be just one layer of abstraction upon another until there is no 'connection' with anything that can be sensibly visualized. For instance certain states that are mathematically distinct but identical re observables. Having a very classical background doesn't help warm one to that, but if theory and experiment agree, well that's that I guess.

First of all, about the "million others" part. There are of the order of 100 good universities in the U.S., each giving of the order of 10 physics graduates each year. That amounts to the order of 1000 U.S. graduates per year. Take one more order of magnitude and you get 10⁴ graduates in the World per year. At that rate, it would take of the order of 100 years to reach million physicists. Quantum Mechanics had been known for about that time. So your claim is highly improbable, because it would mean that every physicist that graduated doubts Quantum Mechanics.

Second of all, the word "visualize" means that you try to create an alternative model based on outdated concept involving macroscopic objects such as "billiard balls" traveling at very low speeds compared to the speed of light and pulled by strings or elastic bands. You cannot "visualize" properties of microscopic particles, particles from which the above physical constructs are created in the first place, in that manner. See this video:
https://www.youtube.com/watch?v=wMFPe-DwULM

Fine but I was not arguing that a point particle cannot have angular momentum - merely that it cannot be reconciled with fundamental classical criteria.

Which is fine, since the classical criteria do not extrapolate to Quantum Mechanics. This is why I pointed out to the different point of view on physical quantities through symmetry in Quantum Mechanics.

As per above, would just like to know how the quantum point particle regime transitions to the classical, spatially extended regime - if possible. No doubt that means reading a good book.

Okay, you mentioned yourself that elementary particles must be point particles due to the finite speed of propagation of interactions (second postulate of Relativity in a modified form). This has nothing "quantum" about it. The word "classical" means two things in Physics:
Classical as in non-quantum, and Classical as non-relativisitc.

The requirement for "zero size" of elementary particles is more of a requirement of Relativity than it is of Quantum Mechanics.

 

[edguy99]

...I started this thread in an attempt to put to bed the very classical picture of a photon that most people seem to want (going so far as to construct animations, even). It seems to be a very diehard concept; almost like angels of a pinhead.

I am not sure if you think of the animator as the angel or the pinhead, but I suspect given the context that I would be better off if I don't ask.

I would like to say that the animator needs to understand physics as best he can, and hopes that the physicist will remain somewhat open. Properly animating light is difficult but is done a lot. Every animated movie these days contains extensive "simulation" of light. Mostly done with rays, but if you are tracking light through mirrors, layers of glass, crystals and around corners (diffraction), and if you wish to stay accurate with reality, the assistance of the physicist if very important.

Back to the size of the photon. Consider a collection of 630nm photons in a small cavity.

The particles are clearly demonstating a "wavelike" property. Their size in the direction of motion is up to 1/2 wavelength in size but if viewed head on, their size is less then 3 fm.

If I suddenly open one side of my laser and point it at a double slit experiment, these photons (one at a time) will display the proper interference pattern as they will diffract around edges the exact way a wave would since the calculations are exactly the same. The particle is acting as a representative of the wave, with a particular direction, energy and phase at a particular time.

 

[Q-reeus]

First of all, about the "million others" part. There are of the order of 100 good universities in the U.S., each giving of the order of 10 physics graduates each year. That amounts to the order of 1000 U.S. graduates per year. Take one more order of magnitude and you get 104 graduates in the World per year. At that rate, it would take of the order of 100 years to reach million physicists. Quantum Mechanics had been known for about that time. So your claim is highly improbable, because it would mean that every physicist that graduated doubts Quantum Mechanics.

Nice math - except I never referred to graduates. Had in mind more the millions of ordinary folk who pick up a pop-sci best seller and get that feeling of sinking into a dark place.

Second of all, the word "visualize" means that you try to create an alternative model based on outdated concept involving macroscopic objects such as "billiard balls" traveling at very low speeds compared to the speed of light and pulled by strings or elastic bands. You cannot "visualize" properties of microscopic particles, particles from which the above physical constructs are created in the first place, in that manner. See this video:
http://www.youtube.com/v/wMFPe-DwULM

Interesting interview. Could take this a number of ways, but I'm trying not to take it as a personal put-down implying I'm 'intellectually challenged' like Feynman (having a real bad day - never seen him so combative) seems to have regarded the clueless interviewer. I will rather try and just get the eventual point there that classical concepts are pretty useless when it comes to QM/QFT.

Okay, you mentioned yourself that elementary particles must be point particles due to the finite speed of propagation of interactions (second postulate of Relativity in a modified form). This has nothing "quantum" about it. The word "classical" means two things in Physics:
Classical as in non-quantum, and Classical as non-relativisitc.
The requirement for "zero size" of elementary particles is more of a requirement of Relativity than it is of Quantum Mechanics.

But isn't the relativistic requirement in turn only because QFT casts it all in terms of those creation/annihilation operators that act instantly at a point - or so I gathered from #57:
"Instead, its meaning is revealed in Second Quantization. Namely, the wave function gets promoted to a field operator that creates (annihilates) at a particular point in space, at a particular instant in time.".
So the two go hand-in-hand surely? Not trying to be argumentative here - like crabby Feynman above.

 

[Dickfore]

Nice math - except I never referred to graduates. Had in mind more the millions of ordinary folk who pick up a pop-sci best seller and get that feeling of sinking into a dark place.

Even the Physics graduates are considered incompetent to do relevant research in the topics mentioned, let alone laymen. Their opinion is irrelevant for the discussion at hand.

Interesting interview. Could take this a number of ways, but I'm trying not to take it as a personal put-down implying I'm 'intellectually challenged' like Feynman (having a real bad day - never seen him so combative) seems to have regarded the clueless interviewer. I will rather try and just get the eventual point there that classical concepts are pretty useless when it comes to QM/QFT.

DId you watch through the whole thing. At one point he discusses (electro)magnetic forces and the inability to "visualize" them in terms of elastic bands. This is a similar situation where quantum phenomena (such as intrinsic spin of a particle) are visualized in terms of spinning tops.

But isn't the relativistic requirement in turn only because QFT casts it all in terms of those creation/annihilation operators that act instantly at a point - or so I gathered from #57:
"Instead, its meaning is revealed in Second Quantization. Namely, the wave function gets promoted to a field operator that creates (annihilates) at a particular point in space, at a particular instant in time.".
So the two go hand-in-hand surely? Not trying to be argumentative here - like crabby Feynman above.

QFT stands for Quantum Field Theory. There is also Classical (meaning non-quantum in the sense of the above mentioned disambiguation of the term "classical") Field Theory.

In fact, electrodynamics is a classical field theory. It fails miserably when predicting the behavior of a charged point particle. The easiest way to see this is to calculate the energy of the electrostatic field due to a charged point particle. The electric field is inversely proportional to the distance squared and the energy density is proportional to the intensity of the field square, thus inversely proportional to the fourth power of the distance. The volume of a spherical shell is proportional to its surface, which in turn is proportional to the distance squared. Thus, the volume integral is an integral over the radial coordinate from zero to infinity of a function that is \frac{1}{r^4} \times {r^2} = \frac{1}{r^2}. The integral of this function diverges as \frac{1}{r} at r = 0.

If we use the relation between energy and mass, it would mean that the charged point particle should have infinite electromagnetic mass. If we assign a finite mass to it, it means that we had subtracted off an infinite mass of non-electromagnetic origin to get a finite result. Since this procedure of subtracting two infinities is mathematically ambiguous, we get all sorts of inconsistencies. For example, Classical Electrodynamics predicts that an accelerated charged point particle should emit electromagnetic waves. These waves carry energy and momentum, so the particle should "feel" some deceleration force (Radiation reaction force). This force is proportional to the time derivative of the acceleration of the particle. Solving the equation. It means that if the particle had a non-zero acceleration, it would accelerate arbitrarily close to the speed of light, which is absurd.

Quantum Field Theory tries to tackle such divergences and many more in a procedure known as "renormalization", where the "subtraction of infinities" is done in a controlled manner at every order of a small parameter (in the Quantum theory of Electrodynamics, the small parameter is the fine-structure constant). This elaborate procedure has become known as Renormalization Group.

So, to recapitulate:
The "pointness" (in a lack of a better term) is a necessary requirement of Relativity. When coupled to a fundamental interaction of Nature - electromagnetism - it leads to absurd results in the classical (non-quantum) regime. Quantum Theory (which in the relativistic regime is necessarily a Field Theory) does a good job of eliminating many of the absurdities.

Asking for the size of an elementary particles is meaningless within QFT, because an extended object cannot be characterized by 4 space-time coordinates. There must be additional degrees of freedom that describe its internal structure - thus making it not elementary. The photon, at our present level of understanding, is an elementary quantum of the electromagnetic field. Thus, it does not have any size associated with it.

 

[A. Neumaier]

Asking for the size of an elementary particles is meaningless within QFT, because an extended object cannot be characterized by 4 space-time coordinates.

Elementary particles are not characterized by 4 space-time coordinates, and it is meaningful to ask for their size. For example, in the book
S. Weinberg,The quantum theory of fields, Vol. I,Cambridge University Press, 1995,
Weinberg defines and explicitly computes in (11.3.33) a formula for the charge radius of a physical electron.

'QED, or relativistic quantum field theory in general, is not based on the notion of ''point particles'', as one sees stated so often and yet so erroneously.

(emphasis as in the original; see p.2 of the book
O. Steinmann, Perturbative quantum electrodynamics and axiomatic field theory, Springer, Berlin 2000)

It is often said that the electron is a point particle without structure in contrast to the proton, for example. We will see in this section that this is not true. The electromagnetic structure of the electron is contained in the form factors

(beginning of Section 3.9 of
G. Scharf, Finite Quantum Electrodynamics: The Causal Approach, 2nd ed., Springer, New York 1995. )

 

[Q-reeus]

In fact, electrodynamics is a classical field theory. It fails miserably when predicting the behavior of a charged point particle. The easiest way to see this is to calculate the energy of the electrostatic field due to a charged point particle. The electric field is inversely proportional to the distance squared and the energy density is proportional to the intensity of the field square, thus inversely proportional to the fourth power of the distance. The volume of a spherical shell is proportional to its surface, which in turn is proportional to the distance squared. Thus, the volume integral is an integral over the radial coordinate from zero to infinity of a function that is 1/r⁴×r²=1/r². The integral of this function diverges as 1/r at r=0.

No argument here - familiar enough with the classical electron radius/'Poincare-stresses' issues.

If we use the relation between energy and mass, it would mean that the charged point particle should have infinite electromagnetic mass. If we assign a finite mass to it, it means that we had subtracted off an infinite mass of non-electromagnetic origin to get a finite result. Since this procedure of subtracting two infinities is mathematically ambiguous, we get all sorts of inconsistencies. For example, Classical Electrodynamics predicts that an accelerated charged point particle should emit electromagnetic waves. These waves carry energy and momentum, so the particle should "feel" some deceleration force (Radiation reaction force). This force is proportional to the time derivative of the acceleration of the particle. Solving the equation. It means that if the particle had a non-zero acceleration, it would accelerate arbitrarily close to the speed of light, which is absurd.

Yes looked at some of the Abraham-Lorentz self-force issues in Jackson etc and no classical way out. But never studied the quantum resolution. Leaving that to you folks.

Quantum Field Theory tries to tackle such divergences and many more in a procedure known as "renormalization", where the "subtraction of infinities" is done in a controlled manner at every order of a small parameter (in the Quantum theory of Electrodynamics, the small parameter is the fine-structure constant). This elaborate procedure has become known as Renormalization Group.

Elaborate is the word, and I have no intentions of challenging such an edifice.

So, to recapitulate:
The "pointness" (in a lack of a better term) is a necessary requirement of Relativity. When coupled to a fundamental interaction of Nature - electromagnetism - it leads to absurd results in the classical (non-quantum) regime. Quantum Theory (which in the relativistic regime is necessarily a Field Theory) does a good job of eliminating many of the absurdities.

Asking for the size of an elementary particles is meaningless withing QFT, because an extended object cannot be characterized by 4 space-time coordinates. There must be additional degrees of freedom that describe its internal structure - thus making it not elementary. The photon, at our present level of understanding, is an elementary quantum of the electromagnetic field. Thus, it does not have any size associated with it.

Thanks for that helpful summary - no further questions; what's more badly overdue for the cot. :zzz:

 

[edguy99]

...In fact, electrodynamics is a classical field theory. It fails miserably when predicting the behavior of a charged point particle. The easiest way to see this is to calculate the energy of the electrostatic field due to a charged point particle. The electric field is inversely proportional to the distance squared and the energy density is proportional to the intensity of the field square, thus inversely proportional to the fourth power of the distance. The volume of a spherical shell is proportional to its surface, which in turn is proportional to the distance squared. Thus, the volume integral is an integral over the radial coordinate from zero to infinity of a function that is 1r4×r2=1r2. The integral of this function diverges as 1r at r=0. ...

A very solid proof that these things are not "charged point particles".

 

[Dickfore]

Elementary particles are not characterized by 4 space-time coordinates, and it is meaningful to ask for their size. For example, in the book
S. Weinberg,The quantum theory of fields, Vol. I,Cambridge University Press, 1995,
Weinberg defines and explicitly computes in (11.3.33) a formula for the charge radius of a physical electron.


(emphasis as in the original; see p.2 of the book
O. Steinmann, Perturbative quantum electrodynamics and axiomatic field theory, Springer, Berlin 2000)


(beginning of Section 3.9 of
G. Scharf, Finite Quantum Electrodynamics: The Causal Approach, 2nd ed., Springer, New York 1995. )

From http://en.wikipedia.org/wiki/Form_factor_%28QFT%29:

In effective field theory, a form factor is a function which gives the properties of a certain particle interaction without including all of the underlying physics. It is measured experimentally when a theoretical calculation is unavailable or too difficult.

I find it very hard to believe that the absence of knowledge of the underlying physics can give information about the internal structure of an object. Also, please define a covariant definition of "size", particularly for a massless photon.

 

[edguy99]

...So, to recapitulate:
The "pointness" (in a lack of a better term) is a necessary requirement of Relativity. When coupled to a fundamental interaction of Nature - electromagnetism - it leads to absurd results in the classical (non-quantum) regime. Quantum Theory (which in the relativistic regime is necessarily a Field Theory) does a good job of eliminating many of the absurdities...

I would appreciate a better understanding of what you mean by "pointness" as it relates to "necessary requirement of Relativity"

 

[Dickfore]

I would appreciate a better understanding of what you mean by "pointness" as it relates to "necessary requirement of Relativity"

If the particle has finite linear dimensions, then the simultaneous positions of different parts of the particle in some reference frame are separated by space-like intervals. Due to the finite speed of propagation of interactions in Nature, it means that these parts are causally disconnected. This would mean that the particle is not "held together" while it exhibits complicated motions as a whole.

Put in more simpler terms, suppose a ball rotates around a circle with a center outside of the ball. The nearer side moves along a circle of smaller radius and the further side along a circle of a bigger radius. The question is how do these two parts of the ball "know" how to move with different speed so as to keep the shape of the ball intact? If there is some binding force keeping them together, there surely has to be some retardation effect that will cause "elastic waves" to propagate within the ball.

 

[A. Neumaier]

From http://en.wikipedia.org/wiki/Form_factor_%28QFT%29:

I find it very hard to believe that the absence of knowledge of the underlying physics can give information about the internal structure of an object. Also, please define a covariant definition of "size", particularly for a massless photon.

The form factors can be derived from the underlying physics if the latter is understood well enough, but they can also be measured experimentally (if the object is not too small). If one does the latter, one has information even in the absence of the underlying physics. For example, the proton form factor is quite well known, although it currently cannot be derived from the underlying theory, QCD, as the latter is not developped enough to give good predictions of bound state properties.

For the size of a photon, see, e.g.,
http://arnold-neumaier.at/ms/lightslides.pdf
http://arnold-neumaier.at/ms/optslides.pdf

 

[Dickfore]

Could you quote where exactly is the size of the photon evaluated? I don't feel like going through some mess of slides full of incoherent text.

 

[A. Neumaier]

Could you quote where exactly is the size of the photon evaluated?

The size of the photon is not a number - a photon is an extended but fuzzy object, as becomes clear from reading the slides.

I don't feel like going through some mess of slides full of incoherent text.

What is messy and incoherent in these slides?

 

[YummyFur]

Wasn't Feynman asked at his oral PhD exam what was the size of a photon? Which was a trick question. Or perhaps I'm imagining this.

 

[Dickfore]

The size of the photon is not a number - a photon is an extended but fuzzy object

Lol, what does that even mean?! How much extended?

What is messy and incoherent in these slides?

I'm not here to do editing of your work. Please don't derail the thread.

 

[edguy99]

If the particle has finite linear dimensions, then the simultaneous positions of different parts of the particle in some reference frame are separated by space-like intervals. Due to the finite speed of propagation of interactions in Nature, it means that these parts are causally disconnected. This would mean that the particle is not "held together" while it exhibits complicated motions as a whole.

Put in more simpler terms, suppose a ball rotates around a circle with a center outside of the ball. The nearer side moves along a circle of smaller radius and the further side along a circle of a bigger radius. The question is how do these two parts of the ball "know" how to move with different speed so as to keep the shape of the ball intact? If there is some binding force keeping them together, there surely has to be some retardation effect that will cause "elastic waves" to propagate within the ball.

Maybe my question was worded poorly, I will try again.

You seem to imply that relativity implies that all particles (electrons, protons and photons) must be points. Does it, or am I misunderstanding you?

 

[The Lobster]

I think this is an acurate description. "The photon, like the electron, is a standard building block of the
universe. This suggests that it is a tiny, invisibly-small sort of thing. (As
if you could see the very thing that we see with!) But if you could catch
a photon you would find that it weighed nothing at all. So why should
something that weighs nothing be small? Why couldn't it be naturally
any size at all? In the wave idea of light, colour is determined by the rate
at which the wave wiggles; a rapid, energetic wiggle for blue and a more
leisurely wiggle for red. You can measure the distance that the light
travels for each wiggle, and this is the 'wavelength' - it appears to be less
than a thousandth of a millimetre for light. A photon is sometimes
thought of as a 'wave packet', a short snippet of wave which zips along
like a cosmic caterpillar. This picture fits in well with the smallness that
we desire of photons. However, light has many close cousins which
share its same properties but wiggle at different rates. One such is a radio
wave. Long-wave radioähas a wavelength as long as one mile- quite
big for a caterpillar! The image of such colossal energy-vehicles
zooming around the Earth from radio transmitters and crashing into our
homes suggests that there may be a flaw in this picture description of a
photon. " -http://www.oxygraphics.co.uk/photons.htm

 

[Dickfore]

Maybe my question was worded poorly, I will try again.

You seem to imply that relativity implies that all particles (electrons, protons and photons) must be points. Does it, or am I misunderstanding you?

Relativity implies that if a particle has finite (non-zero) linear dimensions, then it must have an internal structure, and, therefore is not elementary.

By contra-position, this is equivalent to:

Elementary particles are point particles.

Protons are not elementary particles according to our current understanding. Rather, they are a complicated bound state of three quarks and a sea of gluons.

A scattering cross-section is not a measure of the size of the particles. It is a measure of their mutual interaction. For example, the Rutherford scattering formula gives a differential cross-section:
<br /> \frac{d \sigma}{d \Omega} = \left( \frac{\alpha}{4 E} \right)^2 \csc^4 \left( \frac{\theta}{2} \right)<br />
which, if we integrate over all the angles, gives:
<br /> \sigma = 2 \, \pi \, \left( \frac{\alpha}{4 E} \right)^2 \, \int_{0}^{\pi}{\sin \theta \, \csc^4 \left( \frac{\theta}{2} \right) \, d \theta}<br />
The integtral over the angles reduces to:
<br /> \int_{0}^{\pi}{\sin \theta \, \csc^4 \left( \frac{\theta}{2} \right) \, d\theta} = 2 \, \int_{0}^{\pi}{\frac{\cos \left( \frac{\theta}{2} \right)}{\sin^3 \left( \frac{\theta}{2} \right)} \, d\theta} \stackrel{x = \sin \left( \frac{\theta}{2} \right)}{ = } 4 \, \int_{0}^{1}{\frac{dx}{x^3}} \rightarrow \infty<br />
which diverges at the lower bound (small angle scattering - large impact parameters). This would imply that charge particles have an infinite radius! But, if you go through the (classical) derivation of this formula, you will see that the assumption of point particles had been used.

 

[edguy99]

Relativity implies that if a particle has finite (non-zero) linear dimensions, then it must have an internal structure...

Thanks, I understand your comments now.

 

[Cthugha]

Lol, what does that even mean?! How much extended?

Relativity implies that if a particle has finite (non-zero) linear dimensions, then it must have an internal structure, and, therefore is not elementary.

By contra-position, this is equivalent to:

Elementary particles are point particles.

Somehow I fail to grasp your point. Of course it is well known textbook knowledge that elementary particles are point particles. But it is also basic textbook knowledge that - despite the name - you cannot localize a point particle down to a single point in space in the conventional sense. The bible of quantum optics, the Mandel/Wolf, devotes a whole chapter to this topic and also some books with focus on relativity like "Principles of quantum general relativity" by Eduard Prugovecki discuss this topic, although I am sure that I do not have to tell you and you are much more knowledgeable on books on relativity than I am.

So let me try to get your point: Do you think it is a bad idea to associate non-localilzability with size or is your point something entirely different?

 

[Dickfore]

Somehow I fail to grasp your point. Of course it is well known textbook knowledge that elementary particles are point particles. But it is also basic textbook knowledge that - despite the name - you cannot localize a point particle down to a single point in space in the conventional sense. The bible of quantum optics, the Mandel/Wolf, devotes a whole chapter to this topic and also some books with focus on relativity like "Principles of quantum general relativity" by Eduard Prugovecki discuss this topic, although I am sure that I do not have to tell you and you are much more knowledgeable on books on relativity than I am.

So let me try to get your point: Do you think it is a bad idea to associate non-localilzability with size or is your point something entirely different?

Why does a region where the particle is localized have to be its size? Does it mean that the size of the electron is one Bohr radius in the hydrogen atom?

 

[A. Neumaier]

Lol, what does that even mean?! How much extended?

Extended like a cloud. One cannot tell the exact size of the latter, either.

In principle extended as far as the e/m field reaches. In practice, there is a limit beyond which one cannot
tell.

I'm not here to do editing of your work. Please don't derail the thread.

You made unjustified accusations. Please apologize or justify your accusations, or I'll report you.

 

[A. Neumaier]

Elementary particles are point particles.

You claim this again, against the testimony of three authoritative books that I cited.

Only the bare, noninteracting, unrenormalized, unobservable, and hence unphysical particles are point particles.

 

[YummyFur]

Wouldn't a point particle by definition be something the size of a Planck length?

 

[Dickfore]

Extended like a cloud. One cannot tell the exact size of the latter, either.

In principle extended as far as the e/m field reaches. In practice, there is a limit beyond which one cannot
tell.

But one can certainly give an order of magnitude estimate. Please quote the part of your previous two links that does so.

As for your second paragraph, I really think you should stick to a scientific terminology. This way, one gets an impression that you simply have the need to intrude your personal psychological constructs to the general public about the subject, whithout there really being any need for it.

You made unjustified accusations. Please apologize or justify your accusations, or I'll report you.

I really don't feel obliged to justify anything in front of you. If you feel a report should be made, then please do, but do not take it up to yourself to moderate the discussion.

 

[sheaf]

I have a question on the slides from A. Neumaier:

A photon particle is a particular field mode whose energy is
approximately spatially localized.
Thus the QED photon is a global state of the whole space,
a time-dependent solution of the Maxwell equation.
It acts as a carrier of photon particles,
which are extended but localized lumps of energy
moving with the speed of light along the beam
dened by a QED photon state.

According to my understanding, in QED electromagnetism is quantized by first identifying a bunch of modes of the classical field - these are solutions of Maxwell's equations, then quantizing by treating the mode expansion coefficients as raising and lowering operators. This way the one-photon state has certain non classical properties etc etc.

If I understood correctly, in your proposal this one-photon state can carry "photon particles" which are localized lumps of energy. Does this mean that the energy in the one-photon state (which has total energy \hbar\omega) can be delivered to a detector in lumps smaller than \hbar\omega ? Or does it simply mean that the modes you originally choose to excite to define the one-photon state have these localization properties ? Have you written more on this issue ?

 

[Cthugha]

Why does a region where the particle is localized have to be its size? Does it mean that the size of the electron is one Bohr radius in the hydrogen atom?

That is a clever way of turning around the question. I am not saying that the region where a particle is localized HAS TO BE its size. I am saying that it can be (and in some subfields it also is) defined as such.

My position is rather: Why does the internal structure of a particle have to determine the size? It is defined that way in relativity where the internal structure is of most interest. It is usually defined and used differently in e.g. quantum chemistry (see e.g. PNAS 106, 1001-1005 (2009) by Su et al.), chemical physics, some branches of semiconductor physics and other areas where the internal structure is not of interest.

Localizability is also more heavily studied in quantum optics where e.g. the energy density and the detection probability are nonlocally connected for polychromatic photons (see Mandel/Wolf, chapter 12.11.5).

I just think it is pointless to argue about semantics here. The "natural" meaning of size differs from discipline to discipline and I do not think that it is disputed that many quantities of interest associated with photons/electrons have some spatial extent. If this was the relativity forum I would agree that one should stick to the internal structure meaning of size. In the QM section, however, in my opinion the situation is quite different.

 

[A. Neumaier]

I have a question on the slides from A. Neumaier:

According to my understanding, in QED electromagnetism is quantized by first identifying a bunch of modes of the classical field - these are solutions of Maxwell's equations, then quantizing by treating the mode expansion coefficients as raising and lowering operators. This way the one-photon state has certain non classical properties etc etc.

This is not quite accurate. The modes simply form a basis of all solutions of the Maxwell equations - arbitrary superpositions of these modes represent arbitrary solutions. All these superpositions are quantized as well. Quantization is therefore independent of how one chooses the modes - different choices give equivalent quantizations.

If I understood correctly, in your proposal this one-photon state can carry "photon particles" which are localized lumps of energy. Does this mean that the energy in the one-photon state (which has total energy \hbar\omega) can be delivered to a detector in lumps smaller than \hbar\omega ?

No. In the situation where a 1-photon state describes a particle, one has only a single lump, and the whole energy is delivered in one piece.

A general 1-photon state can be an arbitrary solution of the Maxwell equation. It deserves to be regarded as a particle if and only if this solution is essentially localized in a single lump (or wave packet). Such 1-photon states are called ''photons on demand''. The lump then moves with the speed of light along the beam. Essentially the full energy of the state is then localized in the lump, and therefore moves to the detector, where it causes a detection event (with a certain probability).

Or does it simply mean that the modes you originally choose to excite to define the one-photon state have these localization properties ?

A photon on demand is prepared in such a lumpy mode.

Have you written more on this issue ?

No. But the slides contain references to the literature on photons on demands.

 

[A. Neumaier]

Wouldn't a point particle by definition be something the size of a Planck length?

No. By definition, a point particle has size 0, which is infinitely many orders of magnitude smaller than the Planck length. A particle of the size of the Planck length is obviously extended.

 

[A. Neumaier]

if you could catch
a photon you would find that it weighed nothing at all.

How do you know?

A photon with energy E has an inertial mass of m=E/c^2, hence would weight something if it could be reliably weighted...

 

[YummyFur]

Hang about, even I know that weight is not mass. Tighten it up a bit fellas.

 

[YummyFur]

No. By definition, a point particle has size 0, which is infinitely many orders of magnitude smaller than the Planck length. A particle of the size of the Planck length is obviously extended.

Yes that makes sense but wouldn't something the size of the Planck length fall underneath the radar of the uncertainty principle? Is it allowable to suggest something smaller than the Planck length.

Also if string theory suggests strings about the Planck size, which are said to be many orders of magnitude smaller than the size of currently referred to point particles like an electron there would appear to be some confusion, at least to the lay public, about what do physicists mean by 'point particle'.

It annoyed me when years ago I would marvel at particles that had mass but no size, naively thinking that physicists would be telling the truth, only to find later that these we're not point particles at all.

Even the aforementioned strings are referred to as point particles while at the same time presenting as objects with length and breadth, I mean by the very definition a string cannot be a point.

 

[Fredrik]

Yes that makes sense but wouldn't something the size of the Planck length fall underneath the radar of the uncertainty principle? Is it allowable to suggest something smaller than the Planck length.

The Planck length is not special in any of the established theories (QM, GR, etc.). These theories all describe things that are smaller, in the same way as the describe things that are much larger. They are however expected to be very wrong about things at small enough distance scales, and a simple order-of-magnitude estimate suggests that the Planck length is "small enough" in this sense.

What do I mean by "a simple order-of-magnitude estimate"? I mean something like estimating the volume of a sphere to be r³ where r is the radius, because the volume clearly depends on the radius, and r³ has the right units. This estimate is wrong by a factor of about 4. This is of course to be expected since the method ignores almost all the details. But experience tells us that these crude estimates are rarely wrong by many orders of magnitude.

Also if string theory suggests strings about the Planck size, which are said to be many orders of magnitude smaller than the size of currently referred to point particles like an electron there would appear to be some confusion, at least to the lay public, about what do physicists mean by 'point particle'.

In a classical theory with spacetime M, a point particle can be defined as a pair (x,m) where x:(a,b)→M is a function that satisfies an equation of the form mx''(t)=F(x'(t),x(t),t). The number m is of course called the "mass" of the particle. Point particles in quantum theories are much harder to define. Non-interacting particles can be defined in terms of irreducible representations of groups, but I don't know if there even is a good definition of interacting point particles in 3+1-dimensional spacetime.

 

[technician]

The best and only answer I give to my students is : photons are emitted in less than 10^-10secs.perhaps it is 10^-9 secs.
They travel at the speed of light... 3 x 10^8 m/s
therefore photons have a length of the order 0.3m...
go and join in the many discussions about the size of photons

 

[A. Neumaier]

It annoyed me when years ago I would marvel at particles that had mass but no size, naively thinking that physicists would be telling the truth, only to find later that these we're not point particles at all.

Well, there are levels of rigor in talk. For the laymen, one has to take all concepts with a large grain of salt, in order to be able to communicate at least a bit. With more specialized education, one can be more and more precise about what things really mean. If one would be allowed to say only things that are rigorously true, almost nobody would understand it...

 

[A. Neumaier]

Hang about, even I know that weight is not mass. Tighten it up a bit fellas.

An object that has a nonzero mass will have a nonzero weight when put on a scale in a nonzero gravitational field: F=mg. How much it weighs depends on the strength g of the graviitation. Thus weight is not mass, but from the existence of a nonzero mass one can conclude the existence of a nonzero weight.

Moreover, by general relativity, it is not the rest mass that counts here (which for photons is zero) but the mass equivalent of the total energy (which for photons in nonzero, E = hbar*omega).

So if one had a scale with a resolution high enough to detect the difference of the presence and the absence of a photon, it could be weighted.