Are Photons Actually Infinitely Small Particles?

[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:
$$\frac{d \sigma}{d \Omega} = \left( \frac{\alpha}{4 E} \right)^2 \csc^4 \left( \frac{\theta}{2} \right)$$
which, if we integrate over all the angles, gives:
$$\sigma = 2 \, \pi \, \left( \frac{\alpha}{4 E} \right)^2 \, \int_{0}^{\pi}{\sin \theta \, \csc^4 \left( \frac{\theta}{2} \right) \, d \theta}$$
The integtral over the angles reduces to:
$$\int_{0}^{\pi}{\sin \theta \, \csc^4 \left( \frac{\theta}{2} \right) \, d\theta} = 2 \, \int_{0}^{\pi}{\frac{\cos \left( \frac{\theta}{2} \right)}{\sin^3 \left( \frac{\theta}{2} \right)} \, d\theta} \stackrel{x = \sin \left( \frac{\theta}{2} \right)}{ = } 4 \, \int_{0}^{1}{\frac{dx}{x^3}} \rightarrow \infty$$
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.

 

[fizzle]

BUT, what about when we are dealing with low Radio Frequency em? Consider a photon with an 'extent' of just one wavelength. For a 200kHz transmission, that represents a wavelength of 1500m. Now take a very simple transmitter with, say, the collector of a transistor connected to a short wire. Take an equally simple receiver, with a short wire connected to the base of transistor. Separate them by 10m. The receiver will receive photons that the transmitter is sending it. These photons, if they were to have the proposed extent would have to extend from the transmitter to a region that is 100 times as far away as the receiver input or, they would somehow need to extend ('coiled up?' somehow) from within the transmitter to somewhere within the nearby receiver. This just has to be a nonsense model. In fact you just can't allow a photon to have any extent al all or there will be some circumstance like the above that spoils the model.

If I understand the whole "photon" business correctly, a radio wave would be composed of waves of billions of photons (one for each electron excited in the antenna), not a single "photon" at 200 kHz. The number of photons would be proportional to the magnitude of the radio wave and that number would vary at 200 kHz.

A photon is definitely not a localized em wave or "packet". For ordinary optical frequencies the required electric field strength is 10+ orders of magnitude too high. In addition, em waves don't "stick" together, so any sort of localized packet would disperse fairly quickly.

 

[sophiecentaur]

. The number of photons would be proportional to the magnitude of the radio wave and that number would vary at 200 kHz.

That is definitely not the model of a photon that is generally accepted.

 

[Dickfore]

Actually, radio waves emitted from an antenna are best modeled by a coherent state of the electromagnetic field in which the number of photons is not specified, but obeys a Poisson distribution.

 

[ThomasT]

Hence the spatial and temporal extent of a photon must be considered as zero.

Can't a photon be thought of as a wave front (or sequence thereof, ie., a wave train) whose size is constrained by the channel via which it's transmitted?

 

[YummyFur]

It seems to me that until an agreed definition of what is meant by 'size' when referring to a photon, then the debate is as meaningless as 'does god exist'.

Being a quantum object if we are going to apply a classical concept like size to a photon we really should agree on what we mean by size before saying what this size is.

Can't a photon be thought of as a wave front

that's up to your definition of size. That's what I'm getting at. You have to first contend that a photon is a wave front for your purposes of defining what size is. Then if someone else has an equally valid definition of what size means to them but their definition is different to yours then the two of you would be talking over each other while both being right by your own definitions.