How Big is a Photon and What are its Dimensions?

[marlon]

No-one said it's a point particle.

Well, a photon IS a point particle...BUT in energy space. A photon is defined as a chunk of energy, nothing more. Ofcourse it is NOT defined as a point particle that has finite spatial boundaries, so asking about the "magnitude" of a photons is useless because this concept is defined (in most cases ) based upon spatial coordinates.

Finally, the only photon quantity we know that is defined using spatial coordinates is the photon's wavelength. BUT, we are now talking about the wavelike properties of a photons while in paragraph one, we were using the particle like properties. Ofcourse, no problem with that thanks to the particle wave duality. We should however not forget that because we use a wavelike concept to describe a physical object (ie the photon) that we, intuitively, like to imagine as being a particle (ie a little ball ).

So, one can use this concept to talk about "a photon's magnitude" as long as we keep in our minds what the special nature of the "photon versus magnitude" discussion is.

regards
marlon

 

[teal4two]

uh.. where'd you get this idea from?

Do you know anything about x-ray crystallography?
Crystals are regular geometric arrays of atoms (molecules, ions). The electrons of the atoms scatter x-rays and by measuring the position and intensity of the scattered rays, information about the structure can be obtained.
Often the structure is not homogeneous. If there is local disruption, the scattering pattern shows the average local structure. If the differences are being larger domains, each containing the same local structure, a superposition of the two patterns is observed. The difference between local disorder and larger scale differences seems like it has to be based on photon size;i.e., how much of the crystal scatters the x-ray.
1) If the entire crystal scattered (large photons), all we would see is the scattering pattern of the averaged structure.
2) If the photon was tiny compared to atoms, we would not see the interference pattern of spots.

 

[CarlB]

Well, a photon IS a point particle...BUT in energy space. A photon is defined as a chunk of energy, nothing more. Ofcourse it is NOT defined as a point particle that has finite spatial boundaries, so asking about the "magnitude" of a photons is useless because this concept is defined (in most cases ) based upon spatial coordinates.

Back in grad school, I was taught the same, but recently, there has been work on this by Margaret Hawton which says otherwise. The original was published in Phys Rev A, 59, 954 (1999):
http://prola.aps.org/abstract/PRA/v59/i2/p954_1

The latest arXiv preprint from the same author extends this:

In the standard formulation of quantum mechanics the real space wave function is the projection of the state vector onto an orthonormal basis of eigenvectors of a Hermitian position operator. However, it has been claimed since the early days of quantum mechanics that there is no position operator that defines such a basis for the photon. Here we will briefly review our recent work on the construction of a photon position operator and obtain a photon wave function by projecting onto its eigenvectors.
http://www.arxiv.org/abs/quant-ph/0609086

My explorations of QM have been in applications of geometric / Clifford algebra to density operator states, and it is very important to me that a position representation exist for the photon. Another Hawton article, that reads more closely to what I am doing because it discusses the gauge problem is:

Angular momentum and the geometrical gauge of localized photon states
Margaret Hawton, William E. Baylis, 2004
Localized photon states have non-zero angular momentum that varies with the non-unique choice of a transverse basis and is changed by gauge transformations of the geometric vector potential "a". The position operator must depend on the choice of gauge, but a complete gauge transformation of a physically distinct state has no observable effects. The potential "a" has a Dirac string singularity that is related to an optical vortex of the electric field.
http://www.arxiv.org/abs/quant-ph/0408017

Another article is:
Photon Position Operators and Localized Bases
Phys. Rev. A64, 012101 (2001)
Margaret Hawton, William E. Baylis
http://www.arxiv.org/abs/quant-ph/0101011

These articles have received very little attention. As an aside, Baylis seems to have his fingers in a remarkable number of interesting pies that intersect what I am working on:

Comments on "Dirac theory in spacetime algebra"
J. Phys. A: Math. Gen.35, 4791-4796
William E. Baylis, 2002
In contrast to formulations of the Dirac theory by Hestenes and by the current author, the formulation recently presented by W. P. Joyce [J. Phys. A: Math. Gen. 34 (2001) 1991--2005] is equivalent to the usual Dirac equation only in the case of vanishing mass. For nonzero mass, solutions to Joyce's equation can be solutions either of the Dirac equation in the Hestenes form or of the same equation with the sign of the mass reversed, and in general they are mixtures of the two possibilities. Because of this relationship, Joyce obtains twice as many linearly independent plane-wave solutions for a given momentum eigenvalue as exist in the conventional theory. A misconception about the symmetry of the Hestenes equation and the geometric significance of the algebraic spinors is also briefly discussed.
http://www.arxiv.org/abs/quant-ph/0202060

Hestenes geometrized spinors, I've been applying his techniques to the density matrix formalism. Here's a somewhat similar idea couauthored by Baylis, sort of half spinor and half "idempotent" theory:

A Geometric Basis for the Standard-Model Gauge Group
J. Phys. A. 34, 3309-3324 (2001).
Greg Trayling, William E. Baylis, 2001
http://www.arxiv.org/abs/hep-th/0103137

An introduction to the Clifford algebra / geometric algebra ideas we are using is here:
Relativity in Introductory Physics
William E. Baylis
http://www.arxiv.org/abs/physics/0406158

Carl

 

[ZapperZ]

But to what extent is the "position" of a photon has anything to do with the "size" of a photon?

Note that anytime we detect a photon, be it on a CCD or via a PMT, we have narrowed down its position. So a photon's location in space is not meaningless concept simply based on experimental observation. However, this doesn't mean that its "size" is a meaningful concept.

Zz.

 

[inha]

Do you know anything about x-ray crystallography?
Crystals are regular geometric arrays of atoms (molecules, ions). The electrons of the atoms scatter x-rays and by measuring the position and intensity of the scattered rays, information about the structure can be obtained.
Often the structure is not homogeneous. If there is local disruption, the scattering pattern shows the average local structure. If the differences are being larger domains, each containing the same local structure, a superposition of the two patterns is observed. The difference between local disorder and larger scale differences seems like it has to be based on photon size;i.e., how much of the crystal scatters the x-ray.
1) If the entire crystal scattered (large photons), all we would see is the scattering pattern of the averaged structure.
2) If the photon was tiny compared to atoms, we would not see the interference pattern of spots.

I've performed numerous XRD and SAXS experiments while working in an x-ray laboratory and taken many courses in solid state and x-ray physics so yes, I do know something about crystallography. When one derives the elastic scattering cross section, Bragg's law etc. the photons "size" does not come up in the calculation. Wavelength does but it doesn't equal size. As many others have pointed out it's meaningless to talk about a photons size. And photon-solid interactions are many body processes in the end so it's also pointless to talk about only a part of the solid being involved in the process. Unless of course one has a beam of a certain size and a sample of a certain size, but that's not what you were after.

 

[teal4two]

However, this doesn't mean that its "size" is a meaningful concept.

Let me rephrase an earlier question:
Light photons can pass through the metal mesh in a microwave oven; microwaves can't.
1) If this isn't a measure of something similar to "size" for a macroscopic object, then what is it a measure of?
2) If there are different techniques to measure this quantity, do they give similar (or at least relatively similar) values?
[If no explanation is possible in simple terms, please don't waste your time responding. The physicist Joyce referenced earlier seems as impenetrable as the literary one. I'm spinoring about eigenvectors on a string trying to gauge what any of that means.]

 

[teal4two]

Thanks for the response Inha. It sounds like you have the background to answer my question. [The x-ray question I asked is not as intuitive to me as the microwave oven one.] If size doesn't matter (in the x-ray experiment), then what is this author (found via Google) referring to?
Dauter, Z. (2003). Acta Cryst. D59, 2004-2016.
"Introduction. ...
The size of the twin domains is large in comparison with the crystal cell dimensions, so that X-rays diffracted by the separate parts do not interfere and the total diffraction corresponds to the sum of intensities originating from individual domains. This is in contrast to disorder in a crystal, where there are differences in the location of the content of the neighbouring unit cells. In the case of disorder (either static or dynamic), diffracted X-rays represent the averaged (spatially or temporally) content of all unit cells."

 

[CarlB]

Let me rephrase an earlier question:
Light photons can pass through the metal mesh in a microwave oven; microwaves can't. 1) If this isn't a measure of something similar to "size" for a macroscopic object, then what is it a measure of?

Wavelength. Electrons act similarly, but are still considered point particles with no size.

 

[ZapperZ]

Let me rephrase an earlier question:
Light photons can pass through the metal mesh in a microwave oven; microwaves can't.
1) If this isn't a measure of something similar to "size" for a macroscopic object, then what is it a measure of?

It is a measure of the wavelength of the microwave. We have already had a discussion on the vagueness of associating this with "size". If you think that this is the "size" of a microwave photon, try making a slit of this size or smaller and tell me if you see any microwave passing through that slit.

2) If there are different techniques to measure this quantity, do they give similar (or at least relatively similar) values?

Different techniques? I'd settle for ONE technique that would claim to measure the size of a photon AND the physics that defines such a size.

Zz.

 

[jnorman]

inre:
"In the standard formulation of quantum mechanics the real space wave function is the projection of the state vector onto an orthonormal basis of eigenvectors of a Hermitian position operator. "

about all i can say to that is HA HA HA - is that actually supposed to mean something?? oh, i guess i am just not smart enough to understand it...

 

[Amr Morsi]

Hi,

Just to comment, it is the rest mass of photon which is zero, not its mass. Its mass is h*nu / c^2, where nu is the frequency of the "wave".

As believed today, according to the standard model, photons are point particles, i.e. they don't have size. I just also want to comment as previously mentioned in the thread, that there is a difference between real size and width of distribution of the probability function. No one has ever dealt with photon through Shrodinger Eq. (or even Dirac's.). QFT, especially QED, are concerning the quantized field due to the probability of the source (charges). Epsi Involved in the QED's equation are still for the particles (charges), not photons, which are the mediators.

Amr Morsi.

 

[jnorman]

again, photons are NOT point particles. the concept of point particle implies localization. photons do not exhibit localization. yes, they have no size, but not because they are similar in some way to, for example, an electron. we simply cannot say anything meaningful about a photon in between the time it is emitted and the time it is absorbed.

 

[Amr Morsi]

O.K. Jnorman,
Then, we agree that it has no size.

But, this (photon is not a particle as a consequence) will imply that Shrodinger's (or, Dirac's or, Klien-Gordon's) can't be applied upon the photon itself. But, they, however can be applied on particles, and can account for quantization (spreading) of photons (fields) only through QFT's.

As far as I can understand the word "localization" from your words, photon is an effect. Or, do you mean by localization that it cannot be said to be at a certain position (even if "probably")? If so, then this is the case of elementary particles.

Am I true?

 

[Fredrik]

It is a measure of the wavelength of the microwave. We have already had a discussion on the vagueness of associating this with "size". If you think that this is the "size" of a microwave photon, try making a slit of this size or smaller and tell me if you see any microwave passing through that slit.

I don't think it makes any more sense to say that this is a measure of the wavelength of the photon. It's just as acceptable, or just as unacceptable if you prefer, to say that it's a measure of the size of the photon.

A photon state is in general a superposition of many wavelengths. The same state can also be described as a superposition of many different positions. If the position space wavefunction has a peak, it makes some sense to say that the width of this peak is "the size of the photon". It also makes some sense to say that the location of the peak of the momentum space wave function is "the momentum of the photon". Then we can also use the standard formulas to calculate a wavelength that corresponds to this "momentum", and call it "the wavelength of the photon".

None of the concepts "size", "position" and "wavelength" are well defined here, but with a little effort we can find things in the quantum mechanical formalism that correspond to all of these classical concepts. This is of course not the same as saying that the things we find are the size, position and momentum of a photon. Those concepts aren't well defined in quantum mechanics.

 

[Fredrik]

again, photons are NOT point particles.

I would say that they are. I would also say that "point particle" means something different in quantum mechanics than in classical mechanics.

 

[CarlB]

again, photons are NOT point particles. the concept of point particle implies localization. photons do not exhibit localization.

Photons are used to provide images on photographic plates. The images consist of a collection of spots that have absorbed photons.

yes, they have no size, but not because they are similar in some way to, for example, an electron. we simply cannot say anything meaningful about a photon in between the time it is emitted and the time it is absorbed.

In the quantum theory, one cannot say anything meaningful about an electron in between the time it is emitted and the time it is absorbed. This is the basis for the wave particle duality interpretation of the electron slit and interference experiments. The behavior of electrons and photons in QM is not easily distinguished.

On the other hand, the phrase "point" and "size" do not need to have any meaning in the quantum theory. They're just collections of words, the quantum theory is a bunch of rules on how to make calculations. Arguing about words may not be the best way to learn anything new.

I looked up "point particle" in wikipedia, and they said it means that the particle is assumed to be elementary, with no constituent parts. This definition would apply to photons as well as electrons. I think we've exhausted this particular discussion.

 

[teal4two]

Wavelength. Electrons act similarly, but are still considered point particles with no size.

Thanks. Do you have some resource(s) (website, book) to recommend for a more detailed, yet simple, explanation? [The websites listed at the start of this thread are dead.] To give you an idea of my current level of comprehension, I think of wavelength as being a property in the direction of propagation and thus it is hard to see how it is related to hole size - e.g., a pencil can fit through a hole much smaller than a pencil length.

 

[ZapperZ]

A photon state is in general a superposition of many wavelengths. The same state can also be described as a superposition of many different positions. If the position space wavefunction has a peak, it makes some sense to say that the width of this peak is "the size of the photon". It also makes some sense to say that the location of the peak of the momentum space wave function is "the momentum of the photon". Then we can also use the standard formulas to calculate a wavelength that corresponds to this "momentum", and call it "the wavelength of the photon".

I disagree.

The "peak" in the position tells you the most probable location that the photon will be found. The "width" of that peak is in no way correspond to the "width" of that particle. It is simply the statistical width of where one would find that particle.

If what you say is true, then all those "resonance peaks" that we observe in high energy physics experiments would somehow imply the "size" of that particle. No such inference has ever been made in all those experiments. So your connection between the two is neither obvious, nor standard.

Zz.

 

[marlon]

Photons are used to provide images on photographic plates. The images consist of a collection of spots that have absorbed photons.

Hey CarlB,

I know you wrote the above post as a response to someone who wrote "point particle implies locality". Ofcourse this person is wrong because he mixes locality in coordinate space with locality in an energy basis. I looked through your references but i still do not get how my original post contains something incorrect. A photon is a point particle in energy space because it is a chunk of energy. Again, a point particle is defined as a particle with finite boundaries in some chosen base (spatial, energy, etc etc).


Concerning, the photons you "see" in a detector. What we see there is the result of an interaction between a photon and the detector. The spots we see do not, however, show us actual photons. So, IMO, saying that photons are seen through spots on a detector is NOT a reason to say that photons are not point particles (in a spatial base).

I looked up "point particle" in wikipedia, and they said it means that the particle is assumed to be elementary, with no constituent parts. This definition would apply to photons as well as electrons.

Well Wikipedia, i would not use this website (though it is a good one) for the definition of basic physical concepts.

I am sure you agree with me that if 99.9 % of the people talk about "point particle", they are referring to a particle with finite spatial boundaries. If we reread most of the posts in this very thread, we can only conclude that the same is happening here. I believe that we clearly need to make the distcintion between the spatial base and the energy base : THAT IS THE MAIN ASPECT HERE.

greets
marlon

 

[CarlB]

Marlon,

Part of my problem here is that so many different arguments are very convincing. I don't see what's incorrect about your first post. But the belief that photons can only be point particles in the energy basis is so common that I wanted to bring up the Hawton result, which is not well known.

If one takes to heart the concept that the intermediate states cannot be described by the theory, (i.e. so as to avoid issues with the two slit interference problem), then the situation of a photon being emitted by a radioactive atom, and absorbed by a grain of photographic emulsion, seems to me to be about as close an example as one can have of a point emission and a point absorption.

In particular, the object that emits the photon can have dimensions far smaller than the wavelength of the photon, and it can be localized in space (before the emission) to dimensions far smaller than the emitted photon wavelength. (I.e. a heavy nucleus emitting a low energy photon.)

So it's really not surprising that there is a position representation for the photon. What is surprising is that it was denied for so long. It really makes me wonder how many other odd facts about QM are wrong.

Carl

 

[jnorman]

carl - if i read cohen, et al, correctly, it sounds to me like they are implying that localization effect of photons in energy space may be an artifact of RMT strategy. http://physics.bgu.ac.il/~dcohen/ARCHIVE/wbr_PRL.pdf

still, energy space is hardly a concrete or measureable entity, and to indicate that photons demonstrate "point particle" characteristics similar to that of an electron (which can actually be identified as having a scpecific location, HUP notwithstanding), surely seems a misconception to me. i do not doubt that my understanding of this issue is incomplete, but i am just not following your logic here. i stand by my comment that you cannot say anything meaningful about a photon during its "travels" - it simply cannot be defined as having any location. i do agree, of course, that photons are always detected as particles, but this results from collapse of the probablilty function at the time of absorption, and not from any macro-perspective measureable characteristic of photons. feel free to elucidate further for my edification. danke.

 

[teal4two]

Light scattering.
A visible light photon comes from the sun and enters the atmosphere. An atom in the atmosphere has electrons, whose positions can be described as probabilities (orbitals). The electric field of the photon can cause an electron in the atom to vibrate and in turn the electron radiates another photon.
What determines whether a given photon and a given electron will interact? Is this an all-or-nothing interaction (photon is either scattered or it isn't)? If the photon is on one side of the Earth and the atom on the other, presumably there will be no interaction. How close do they have to be?

 

[lightarrow]

Light scattering.
A visible light photon comes from the sun and enters the atmosphere. An atom in the atmosphere has electrons, whose positions can be described as probabilities (orbitals). The electric field of the photon can cause an electron in the atom to vibrate and in turn the electron radiates another photon.
What determines whether a given photon and a given electron will interact? Is this an all-or-nothing interaction (photon is either scattered or it isn't)? If the photon is on one side of the Earth and the atom on the other, presumably there will be no interaction. How close do they have to be?

The photon is the quantized interaction between light and matter. Just my opinion, of course.

 

[reilly]

No Virginia, there ain't no point particles, except on the pages of physics books and papers. Any particle with charge necessarily has an internal structure due to a cloud of (virtual) photons, e-p pairs, quarks, ... it then follows that any particle, massless or massive cannot be a point particle, but necessarily has an internal structure.

The problem is: we don't know how to compute, even approximately, such structures, form factors, if you will. But the structure of QFT interactions says those structures must exist. Further, we can measure nucleon form factors with electron scattering, for example. But whatever structure electrons or photons have is beyond our capability to measure or compute -- other than basic QED phenomena like the perturbative results for the Lamb Shift or the anomalous magnetic moment of the electron.

Nonetheless, QFT almost always deals with point particles, for mathematical convenience. For the most part, the physics community accepts this approximation as appropriate for comon practice.

If one, for the moment, accepts point particles, then position operators for massive spinning particles, at least, are a bit peculiar. As is noted in the work referenced by CarlB, the Lorentz transformation of a standard position operator, +i d/dp, in 1-D with the momentum=p, gets tangled up with spin, and, more generally, with angular momentum. this happens because, in the Poincare Group, the generators of Lorentz transformations do not commute with the generators of rotation, which in turn causes the product of two Lorentz transformations to result in a rotation followed by a Lorentz xform, or vica versa.

So, the net effect is that under a Lorentz transformation, the transformed position of a particle depends on the spin state -- that is the transformed position of an spin-up electron at (T,X) will not be the same as the position of a spin-down similarly at (T,X). While I realize the dangers of ascribing classical ideas to the arcane phenomena of the quantum world, I think of this as suggesting an extended object -- and there is some classical justification for this, as discussed by Moller in his Relativity text -- a classical object with an internal angular momentum, the Earth or a baseball thrown as a curveball,..., has a radius greater than |L|/(c M0), where L is the angular momentum, M0 is the rest mass, and c is the speed of light.(See section 64 in Moller's Theory of Relativity.)

It seems to be the case that these fine points of position, spin, and Lorentz transformations are, at present, more curiousities that anything else. but certainly, these ideas make the notions of positions of point particles with spin rather a dicey proposition.

Regards,
Reilly Atkinson

 

[Careful]

If one, for the moment, accepts point particles, then position operators for massive spinning particles, at least, are a bit peculiar. As is noted in the work referenced by CarlB, the Lorentz transformation of a standard position operator, +i d/dp, in 1-D with the momentum=p, gets tangled up with spin, and, more generally, with angular momentum. this happens because, in the Poincare Group, the generators of Lorentz transformations do not commute with the generators of rotation, which in turn causes the product of two Lorentz transformations to result in a rotation followed by a Lorentz xform, or vica versa.

So, the net effect is that under a Lorentz transformation, the transformed position of a particle depends on the spin state -- that is the transformed position of an spin-up electron at (T,X) will not be the same as the position of a spin-down similarly at (T,X). While I realize the dangers of ascribing classical ideas to the arcane phenomena of the quantum world, I think of this as suggesting an extended object -- and there is some classical justification for this, as discussed by Moller in his Relativity text -- a classical object with an internal angular momentum, the Earth or a baseball thrown as a curveball,..., has a radius greater than |L|/(c M0), where L is the angular momentum, M0 is the rest mass, and c is the speed of light.(See section 64 in Moller's Theory of Relativity.)

It seems to be the case that these fine points of position, spin, and Lorentz transformations are, at present, more curiousities that anything else. but certainly, these ideas make the notions of positions of point particles with spin rather a dicey proposition.

Regards,
Reilly Atkinson

I agree; let me first add that spin has to enter the position operator for massless particles if you want it to commute with the helicity operator.
On the other hand it wouldn't be unreasonable at all that the spin operator is included even for massive spinorial particles (apart from the obvious reason that you want the limit for m -> 0 to be well defined) : Zitterbewegung (due to spin) makes it hard to ``localize'' -say- an electron, so it is not unreasonable to think that position and spin operators are noncommuting. Indeed, in the book ``Relativistic quantum mechanics and introduction to field theory'', F.J. Ynduarain gives at page 191 a ``Newton - Wigner'' operator for spin 1/2 particles containing the spin operators.

 

[epv]

http://www.pbs.org/wgbh/nova/programs/ht/tm/3501.html?site=10&pl=wmp&rate=hi&ch=10
Talks about slowing down light and length of photon.

 

[ZapperZ]

http://www.pbs.org/wgbh/nova/programs/ht/tm/3501.html?site=10&pl=wmp&rate=hi&ch=10
Talks about slowing down light and length of photon.

I saw this when it first aired. Where exactly did it talk about "length of photon"? Maybe length of a light pulse, but not length of a photon.

You do know that this thread is more than a year old, don't you?

Zz.

 

[CarlB]

I'd forgotten about this thread. Margaret Hawton's position operator was recently discussed briefly on sci.physics.research in the topic of the title
"EM Field of Photon":
http://groups.google.com/group/sci.physics.research/topics?hl=en

Hawton herself chimed in. I put together a blog post showing the relationship between her photon operator and the consistent histories interpretation of quantum mechanics and she linked it into the discussion. The blog post is here:
http://carlbrannen.wordpress.com/2008/01/14/consistent-histories-and-density-operator-formalism/

As you can see from the sci.physics.research discussion, the local expert is not convinced that a photon position operator makes sense, but he also makes it quite clear that he doesn't bother to read the articles that are linked in, LOL. This was from a few days ago, and I suspect that Hawton is busily working on another paper extending these ideas.

 

[jambaugh]

According to Quantum Mechanics, a photon, like any other point particle, has no size at all. It is a point, if treated as a particle. If it is treated as a wave, it has wavelength and amplitude instead, amplitude being the "size".

Not quite. True it "has no size at all" but that means it nether has zero size nor some other value. It isn't a classical object with a size (even zero) while a "point particle" is such an animal. A photon is a process of quantized electro-magnetic energy transmission. This can happen in the very small volume of the nucleus of an atom or equally spread over a planet sized antenna array. "It has no size" in that size of any numerical value isn't a property of the photon.

 

[agkyriak]

Well, a photon IS a point particle...BUT in energy space. A photon is defined as a chunk of energy, nothing more. Ofcourse it is NOT defined as a point particle that has finite spatial boundaries, so asking about the "magnitude" of a photons is useless because this concept is defined (in most cases ) based upon spatial coordinates.

Finally, the only photon quantity we know that is defined using spatial coordinates is the photon's wavelength.

It is absolutely right. From a theoretical point of view the proof was given long years ago in the work of Landau and Peierls and confirmed recently in the works of other scientists Cook, Inagaki and others. Let us consider briefly this proof, using the book of (Akhiezer and Berestetskii, 1969):

The wave function of photon is here introduced as follows. The vectors of the EM field \vec {{\rm E}} and \vec {{\rm H}}, as the solutions of the wave equation of the second order, which follow from the Maxwell equations, are considered as the classical wave functions \vec {\varepsilon }\left({\vec {r},t} \right) and \vec {H}\left( {\vec {r},t} \right).

Representing the wave equation as multiplication of two equations for the advanced and retarded waves, we obtain two linear equations, which correspond to the wave vector \vec {f}_k and is a certain generalization of vectors of EM field. The equation for this function is equivalent to the system of the Maxwell equations. For this reason it is possible to consider the Maxwell's equation as the equation of one photon (Gersten, 2001). The quantization of classical wave function is produced by means of the quantization of energy of this wave by the introduction of the relationship \varepsilon =\hbar \omega. It turned out that in this case the function \vec {f}_k could be interpreted as the quantum wave function of photon in the momentum space.

But with the attempt to introduce the function of photon in the coordinate representation was revealed the insurmountable difficulty According to the analysis of Landau and Peierls (Landau and Peierls, 1930), and later of Cook (Cook, 1982a; 1982b) and Inagaki (Inagaki, 1994), the wave function of photon by its nature is nonlocal (see also the review (Bialynicki-Birula, 1994)). .

Actually, after completing the inverse Fourier transformation of above function \vec {f}_k we obtain:
\frac{1}{\left( {2\pi } \right)^3}\int {\vec {f}_k e^{i\vec {k}\vec {r}}d^3k=\vec {f}\left( {\vec {r},t} \right)}.

It seems that it is possible to determine \vec {f}\left( {\vec {r},t} \right) as the wave function of photon in the coordinate representation. Actually, because of normalization condition for \vec {f}_k the function \vec {f}\left( {\vec {r},t} \right) will be also normalized by the usual method: \int {\left|{\vec {f}\left( {\vec {r},t} \right)} \right|} ^2d^3x=1

However, the value \left| {f(\vec {r},t)} \right|^2 will not have the sense of the probability density distribution to find the photon at the given point of space. Actually, the presence of photon can be established only by its interaction with the charges.

This interaction is determined by the values of the EM field vectors \vec {{\rm E}} and \vec {{\rm H}} at the given point, but these fields are not determined by the value of the wave function \vec {f}\left( {\vec {r},t} \right) at the same point, and they are defined by its values in entire space.

In fact, the component of the Fourier field vectors, expressed byf_k, contain the factor \sqrt k. Formally this can be written down in the form
\[\vec {\varepsilon }\left( {\vec {r},t} \right)=\sqrt[4]{-\Delta }\vec {f}\left( {\vec {r},t} \right)\]
where \Delta is the Laplace operator. But \sqrt[4]{-\Delta } is integral operator, and therefore the relationship between \vec {\varepsilon }\left( {\vec {r},t} \right) and \vec {f}\left( {\vec {r},t} \right) is not local, but integral. In other words, \vec {f}(\vec {r},t) is not determined by field value \vec {{\rm E}}(\vec {r},t) at the same point, but it depends on field distribution IN A CERTAIN REGION, WHOSE SIZE IS THE ORDER OF WAVELENGTH.

This means that localization of photon in the smaller region is impossible and, therefore, the concept of the probability density distribution to find the photon at the fixed point of space does not have a sense.

This conclusion of theory is confirmed by experiment, since all measurements with the use of EM waves or photons (interference, diffraction and so forth) can be carried out to the region, not smaller as wavelength.

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