The strange world of Phil the photon

[Q-reeus]

First, a short story to illustrate a philosophical issue I have not seen discussed let alone resolved:
Mac was about to press the 'On' button of his super-duper special microwave oven with perfectly reflecting interior walls. Inside, a random jumble of thermal photons, comprising a 'photon gas', criss-crosses the interior. One of them, let's call it 'Phil', reflects on the fact that apart from an absurdly small coupling from gravitational 'charge' computed on the basis of their individual energies E = hf, they are non-interacting massless Bosons. Blissfully indifferent to each others presence, they pass through each other without in any way effecting the others energy, momentum, phase, polarization or spin - 'ghosts through ghosts'. Not only total energy but also net gravitational influence is just the sum of their individual contributions. All is sensible thinks Phil. But then Mac gives that 'On' button a short burst (quickly realizing there was nothing inside to absorb the energy), and suddenly Phil is living in a strangely different environment. By chance Phil has the correct frequency, phase, orientation and location to be in lock-step with billions of newly added photons, all in lock-step with each other (microwave oven as an energized cavity resonator). Phil, along with each lock-step neighbor, becomes aware by calculation that now each has amazingly acquired billions of times their individual energies E = hf and momentum p = E/c, since by superposition the net cavity field amplitude is just the linear sum of their individual contributions, whereas the cavity field energy density goes as the square of that amplitude. (Phil realizes there is no net system energy anomaly - the oven had to work harder and harder in pumping in all those new photons). What's more, Phil and co now have to each churn out billions times more virtual gravitons in keeping with their mutually enhanced energies (optionally, replace 'virtual gravitons' with 'spacetime curvature'). Noticing that the thermal photons continue on exactly as before, Phil frowns "this collectivist madness is insane; how can we be 'non-interacting' in every other way except for this incredible blow-up in mutual energy/momentum/gravity?" End of story.

Might just as well have used a different scenario comparing the mutual energy/momentum/gravity of say light beams intersecting at right angles, one being a phase-coherent laser beam, the other an incoherent beam from say a flash-light. Same dilemma arises. Sure, system behavior works out OK in accordance with the formal mathematical framework, but is that good enough? Is it all covered satisfactorily in QFT - 'overlapping wave-functions' or some such? Input please.

 

[ghwellsjr]

Phil does an awful lot of thinking in his brief lifetime, which in a microwave oven is about a nanosecond or two.

All the photons in a given situation may not know of each other's existence but they all know the entire environment which creates and absorbs them and that causes them to behave in a particular aggregate way that is exactly like what is determined by classical physics. It is only when the number of photons gets down to a countable few that anybody bothers to analyze the situation from a quantum mechanical viewpoint, at least in situations like a microwave oven.

Why is this post on the relativity forum?

 

[Mentz114]

I don't see a problem. If the microwave generator puts energy E into the cavity, then the gravitating energy is E.

 

[Q-reeus]

Phil does an awful lot of thinking in his brief lifetime, which in a microwave oven is about a nanosecond or two.

Worse, in Phil's own frame he has precisely zero time - but then it is only an illustrative story.

All the photons in a given situation may not know of each other's existence but they all know the entire environment which creates and absorbs them and that causes them to behave in a particular aggregate way that is exactly like what is determined by classical physics.

Can't argue with that as a formal statement, but can't find any real explanation there either. Point is, HOW do they 'know' to enormously multiply energy and gravitation? If the seat of energy and gravitation is in the fields themselves, to me that's a big question mark. Hinting here that maybe the long discarded alternate formulation of EM in terms of charge/potentials being the real seat of energy (and thus gravity) might make more sense.

It is only when the number of photons gets down to a countable few that anybody bothers to analyze the situation from a quantum mechanical viewpoint, at least in situations like a microwave oven.

Whether or not one bothers to analyze from a QED perspective, it's still there.

Why is this post on the relativity forum?

That gravity is inextricably part of the picture is not sufficient? If the moderator wishes to remove to some other section then so be it.

 

[sheaf]

If I have N photons of a given frequency, the total energy doesn't increase if they happen to be coherent, (eg as in a laser beam) compared to the non coherent case. Or was that not what you were saying ?

 

[bcrowell]

Phil, along with each lock-step neighbor, becomes aware by calculation that now each has amazingly acquired billions of times their individual energies E = hf and momentum p = E/c, since by superposition the net cavity field amplitude is just the linear sum of their individual contributions, whereas the cavity field energy density goes as the square of that amplitude.

The frequency f is fixed, so Phil's energy can't go up.

As far as I can tell, there is nothing really quantum-mechanical about your paradox. The same paradox exists, and can be resolved, in classical E&M. You have an apparent nonconservation of energy, because fields add linearly, but energy goes like the square of the field. But if you try working it out in some simple cases, you'll see that energy really is conserved. For example, try two plane-polarized plane waves colliding head-on. If their polarizations are in the same plane, then at a moment when their E fields are in phase, the total energy in the E field is quadrupled. But at this moment the B fields are exactly out of phase, so they cancel. The total energy is the same as when the waves were separate.

 

[Q-reeus]

If I have N photons of a given frequency, the total energy doesn't increase if they happen to be coherent, (eg as in a laser beam) compared to the non coherent case. Or was that not what you were saying ?

It is my understanding. If what you say is true then problem solved in one sense (there is no strange collective phenomena). On the other hand, how is the superposition principle avoided here. Field amplitude would then go as the square root of coherent number density, which seems odd also. Can you please point to some article(s) explaining that?

 

[Q-reeus]

The frequency f is fixed, so Phil's energy can't go up.

As far as I can tell, there is nothing really quantum-mechanical about your paradox. The same paradox exists, and can be resolved, in classical E&M. You have an apparent nonconservation of energy, because fields add linearly, but energy goes like the square of the field. But if you try working it out in some simple cases, you'll see that energy really is conserved. For example, try two plane-polarized plane waves colliding head-on. If their polarizations are in the same plane, then at a moment when their E fields are in phase, the total energy in the E field is quadrupled. But at this moment the B fields are exactly out of phase, so they cancel. The total energy is the same as when the waves were separate.

Violation of conservation of energy was nowhere implied or stated, quite the opposite. People seem to have a hard time understanding the point, but the last poster seemed to get what I was on about. Your example of the counter-propagating waves is interesting but not relevant here, as the scenario involved co-propagating and phase-coherent photons.

 

[Mentz114]

Violation of conservation of energy was nowhere implied or stated, quite the opposite. People seem to have a hard time understanding the point, but the last poster seemed to get what I was on about. Your example of the counter-propagating waves is interesting but not relevant here, as the scenario involved co-propagating and phase-coherent photons.

If there's conservation of energy, as you took pains to point out, how did this happen ?

Phil, along with each lock-step neighbor, becomes aware by calculation that now each has amazingly acquired billions of times their individual energies E = hf and momentum p = E/c,

Maybe that's the bit I'm not getting.

I think BCrowell got it spot on.

 

[PeterDonis]

Field amplitude would then go as the square root of coherent number density, which seems odd also. Can you please point to some article(s) explaining that?

This is a whole discussion, not a short article, but along the way it does talk about the issue you raise:

http://math.ucr.edu/home/baez/photon/schmoton.htm

 

[Q-reeus]

Originally Posted by Q-reeus View Post

Violation of conservation of energy was nowhere implied or stated, quite the opposite. People seem to have a hard time understanding the point, but the last poster seemed to get what I was on about. Your example of the counter-propagating waves is interesting but not relevant here, as the scenario involved co-propagating and phase-coherent photons.

If there's conservation of energy, as you took pains to point out, how did this happen?
Maybe that's the bit I'm not getting.
I think BCrowell got it spot on.

What's overlooked is the parenthetical statement in the first posting:
"(Phil realizes there is no net system energy anomaly - the oven had to work harder and harder in pumping in all those new photons)." and later "Sure, system behavior works out OK in accordance with the formal mathematical framework, but is that good enough?" I tried to anticipate and head off the very comments you are making, but obviously to no avail.
Brief summary of my point: Overall energy is conserved - the oven magnetron does all the work as needed. BUT within that mix we have - example - nominally non-interacting photons somehow strangely pumping out billions of times 'more gravity' EACH than when phase incoherent to meet those formal requirements. The numbers are required all right, but is that not mutual interaction? But how? How do they somehow signal and stimulate each other in a precise numerical way if non-interacting? This is all based on that linear superposition applies. If post #5 by sheaf is true (I have strong doubts) then superposition somehow doesn't apply to photons and each photon puts out eg. the same virtual graviton count independent of other's phase and number density etc. That would be a solution, but is it correct?

 

[bcrowell]

Brief summary of my point: Overall energy is conserved - the oven magnetron does all the work as needed. BUT within that mix we have - example - nominally non-interacting photons somehow strangely pumping out billions of times 'more gravity' EACH than when phase incoherent to meet those formal requirements.

The gravitational field made by the EM waves only depends on the total energy. The total energy is conserved, so it can't depend on the phase relationships between the superposing waves. I still don't see anything quantum-mechanical in this, and I still don't see anything paradoxical.

 

[Q-reeus]

This is a whole discussion, not a short article, but along the way it does talk about the issue you raise:

http://math.ucr.edu/home/baez/photon/schmoton.htm

Thanks for the link Peter. Will plow through it as can. Meanwhile, way past bed time here in Oz.

 

[pervect]

I'm still not sure what the issue is here. But if you are thinking that photons don't interact gravitationally, that's not quite true. It's fairly well known that light beams (and hence, photons) traveling in the same direction don't interact, while light beams traveling in opposite directions do attract. In fact, in theory you can even find (unstable) configurations where the gravitational attraction of light beams moving in opposite directions binds them into an entity which has been given the name of a "geon".

While GR doesn't have anything to say about photons, one can handle quantum gravity pertubatively, and in "Quantum Field Theory in a Nutshell", Zee goes into this, describing how the gravitational interaction of photons (which is present) is handled in perturbative quantum gravity. You might want to have a look, they even derive the above behavior about same/opposite directions. IIRC you can find it by the name Tolman-Ehrenfest-Podolsky paradox. While it used to be online, I think the sample pages with the needed info have been changed so that they aren't online anymore.

Or perhaps you aren't asking a question about GR at all, but about QFT. If photons each come in fixed energy packets, then the total energy should be proportional to the number of packets. But if the fields add, and the photons are coherent, then the energy should go up as the square of the number of packets. Obviously only one of these choices can be right for predicting the number of photons (photons in the microwave cavity in your example).

I can't personally explain the discrepancy with confidence. The best place to ask this would be the QFT forum, if this is the question you're interested in. It doesn't seem like gravity is particularly relevant to the quesiton.

 

[bcrowell]

If photons each come in fixed energy packets, then the total energy should be proportional to the number of packets. But if the fields add, and the photons are coherent, then the energy should go up as the square of the number of packets.

This is a purely classical issue, not a quantum-mechanical one. If the waves are initially separate and then move into the same space, then the total energy stays constant; see #6. If you have a source like an antenna radiating waves coherently, then the energy quadruples with a doubling of amplitude, and the extra energy comes from the work that the RF amplifier has to do in order to push the current up and down the antenna.

Note that it's never possible for EM plane waves that are initially spatially separated to move into the same space and add coherently. If their wave-vectors aren't parallel, then you can go into the center of mass frame, and in that frame you get two waves colliding head-on, as in #6.

 

[Dale]

One thing about coherent states is that the number of photons is not well defined. Because of that it is rarely helpful to think of it in terms of photons rather than in terms of classical fields. Particularly in situations like a microwave where the photon energy is relatively small and they are hard to detect.

 

[bcrowell]

One thing about coherent states is that the number of photons is not well defined.

Why do you say that? I can see how a pulse of finite length would have a spread of frequencies, leading to an uncertainty in the photon number. But I don't see why coherence would relate to an uncertainty in photon number.

 

[Dale]

A quantum mechanical state with a definite number of photons is called a Fock state. The definition of a coherent state is that it is an eigenstate of the anhillation operator, meaning that it is unchanged by the detection of a single photon. The eigenstates of the anhillation operator, in turn, are not Fock states but instead follow a Poisson distribution in terms of Fock states and therefore have a Poisson distribution of number of photons.

 

[bcrowell]

A quantum mechanical state with a definite number of photons is called a Fock state. The definition of a coherent state is that it is an eigenstate of the anhillation operator, meaning that it is unchanged by the detection of a single photon. The eigenstates of the anhillation operator, in turn, are not Fock states but instead follow a Poisson distribution in terms of Fock states and therefore have a Poisson distribution of number of photons.

I'm sure you're right, but can you explain it with crayons for me?

 

[sheaf]

I'm sure you're right, but can you explain it with crayons for me?

I think DaleSpam has hit the nail on the head. I'd forgotten (If I ever knew!) that coherent states weren't eigenstates of the number operator. It's treated in chapter 3 of Gerry and Knight's "Introductory Quantum Optics". Hopefully this Google books link isn't too long...

http://books.google.co.uk/books?id=... number operator&pg=PA43#v=onepage&q&f=false"

 

[Dale]

I'm sure you're right, but can you explain it with crayons for me?

Unfortunately my knowledge of QM is very superficial (even more superficial than my knowledge of GR). However, recall e.g. that a photon does not have a definite state of polarization unless it is in an eigenstate. When it is not in an eigenstate you cannot say what polarization it has, all you can say is that if you measure the polarization you will have a certain likelihood of getting a given answer.

Similarly with coherent states. Coherent states are not eigenstates of the number of photons. So you cannot say how many photons are in a coherent state. All you can say is that if you measure the number of photons you will get a certain likelihood (Poisson distributed) of getting a given answer.

 

[bcrowell]

Thanks, sheaf and DaleSpam, for the additional info. Unfortunately the google books link doesn't work for me. (I think they must randomly decide which pages to make available to which users.)

Is this lack of definite particle number for a state with a definite frequency and a spread in total energy, or one with a spread in frequency but a definite energy?

Anyway, if the particle number is Poisson-distributed, then it seems like this will have no effect on the current discussion. If you want to measure the gravitational field of a box full of photons, using futuristic technology, the number of photons n will have to be very large. The relative fluctuations will then be [itex]1/\sqrt{n}[/itex], which should be negligible compared to measurement uncertainties with any foreseeable technology. (If it's a state of definite energy with a spread in frequency, then this spread in n doesn't even affect the gravitational field at all.)

Basically I still haven't seen any aspect of this question that is quantum-mechanical in any important way.

 

[Dale]

Basically I still haven't seen any aspect of this question that is quantum-mechanical in any important way.

Yes, that is basically my point. The number of photons in a coherent state is fundamentally a tricky concept and it doesn't add anything to the problem except confusion and difficulties. So, IMO the QM treatment is not recommended and this problem should be approached classically.

 

[sheaf]

Is this lack of definite particle number for a state with a definite frequency and a spread in total energy, or one with a spread in frequency but a definite energy?

Anyway, if the particle number is Poisson-distributed, then it seems like this will have no effect on the current discussion. If you want to measure the gravitational field of a box full of photons, using futuristic technology, the number of photons n will have to be very large. The relative fluctuations will then be [itex]1/\sqrt{n}[/itex], which should be negligible compared to measurement uncertainties with any foreseeable technology. (If it's a state of definite energy with a spread in frequency, then this spread in n doesn't even affect the gravitational field at all.)

Basically I still haven't seen any aspect of this question that is quantum-mechanical in any important way.

I think that the coherent states do not have definite energy, but do have definite frequency (but not definite phase). However as you say, for a reasonably energetic field the energy spread will be mind bogglingly small.

 

[PeterDonis]

I think that the coherent states do not have definite energy, but do have definite frequency (but not definite phase). However as you say, for a reasonably energetic field the energy spread will be mind bogglingly small.

The website I linked to in post #10 (link posted again below) talks quite a bit about coherent states. As I understand what is said there, a coherent state is not an eigenstate of the energy operator (Hamiltonian), and therefore would not have a definite energy.

http://math.ucr.edu/home/baez/photon/

Classically, however, I think bcrowell and DaleSpam are correct: the mean of the Poisson distribution for energy will determine the classical stress-energy tensor, as long as we're working on time scales that are long compared to the time scale of fluctuations around that mean.

 

[Q-reeus]

I'm still not sure what the issue is here. But if you are thinking that photons don't interact gravitationally, that's not quite true. It's fairly well known that light beams (and hence, photons) traveling in the same direction don't interact, while light beams traveling in opposite directions do attract. In fact, in theory you can even find (unstable) configurations where the gravitational attraction of light beams moving in opposite directions binds them into an entity which has been given the name of a "geon".

In the opening 'problem statement', I took the standard position that photons gravitationally interact but that it was in context negligible. Later in entry #4 there was a hint that a more coherent picture might be to treat the (conduction) charges as the real seat of energy-gravity. That would then be problematic when applied to the light beam scenario, no question. Problems either way if one believes in superposition applying (or not!). Found an 'interesting' link to geon at http://www.angmalta.net/clients/alan/existence/geon.pdf. Main title raised my eyebrows.

While GR doesn't have anything to say about photons, one can handle quantum gravity pertubatively, and in "Quantum Field Theory in a Nutshell", Zee goes into this, describing how the gravitational interaction of photons (which is present) is handled in perturbative quantum gravity. You might want to have a look, they even derive the above behavior about same/opposite directions. IIRC you can find it by the name Tolman-Ehrenfest-Podolsky paradox. While it used to be online, I think the sample pages with the needed info have been changed so that they aren't online anymore.

Thanks for reference but could find no link to TEPP.

Or perhaps you aren't asking a question about GR at all, but about QFT. If photons each come in fixed energy packets, then the total energy should be proportional to the number of packets. But if the fields add, and the photons are coherent, then the energy should go up as the square of the number of packets. Obviously only one of these choices can be right for predicting the number of photons (photons in the microwave cavity in your example).
I can't personally explain the discrepancy with confidence. The best place to ask this would be the QFT forum, if this is the question you're interested in. It doesn't seem like gravity is particularly relevant to the quesiton.

Perhaps QFT forum was more appropriate - it was thoughts about gravitational interactions that got my attention in the first place. But agreed about the expected square relation - that and the role of phase is the nub of the issue it now appears.

 

[Q-reeus]

Definitely first and last attempt at a Short Story - bad idea.
Following a link given in entry #10 led to http://math.ucr.edu/home/baez/photon/odd-ques.htm" , where John Baez states:
"First let's think of light as being classical. Then the energy density is (E² + B²)/2,...So the energy density is something like A²...
On the other hand, let's think of light as being made of photons. The energy of a photon is E = hbar w where hbar is Planck's constant and w is the angular frequency, not the number of full wiggles per second but that times 2 pi (which is why hbar = h/2pi). So the energy density should be hbar w d, where d is the average number of photons per cubic meter. So we should have A² = hbar w d
Does that make sense? The lower the frequency, the more photons we need to get a particular energy, so that part makes sense. It's sort of odd how the number of photons is proportional to the square of the amplitude; you mighta thunk it would just be proportional." (agreed; odd.)

Baez's remarks are in keeping with entry #5 comments and probably the views of several contributors to this thread. What is the picture here? For a random photon gas OR an EM wave, energy and gravitational mass is just the sum of all the individual photons. So phase apparently has no effect. Or is it that quantum randomness as suggested elsewhere causes just the right spread in phase to maintain energy/photon unchanged. Big problem as I see it with that proposition is the need for a continual readjustment of phase distribution depending on field strength. Which seems totally incompatible with the notion of massless Boson non-interaction. If I have the gist of recent entries right, phase spread is insignificant in this scenario, so mute argument.
To recap:
Experience and classical EM theory say if a probe maintained at a fixed rms current feeds a high Q cavity resonator:
* Cavity field rms amplitude rises linearly with time.
* Completely balanced rms feed energy input and interior field rms energy output is directly proportional to rms amplitude squared.
* There is a corresponding fixed rate of photon creation by the probe current - photon number rises linear with time.
Simple maths - divide cavity energy (and 'gravity') by photon count, it's not constant and tends to a huge value relative to E = hf.
This may seem stubborn but what seems the standard QFT view applied here makes no sense to me.

 

[bcrowell]

Q-reeus, since there's nothing inherently quantum-mechanical about your question, could you please try rewriting #27 without any references to photons, Planck's constant, etc.? The Baez link is not relevant, because it's about photons.

 

[Phrak]

Is this about what you had mind, Q-reeus?:

"I have about 10 coherent photons in phase. The amplitude is 10 times as much as a single photon. The energy is the amplitude squared, so I expect 10 photons to share 100 units of energy, or about 10 units of energy per photon rather than 1 unit that a single photon would have."

It may help to know that, as theory has it, photons can be coherent but have no comparable phase.

 

[Q-reeus]

Q-reeus, since there's nothing inherently quantum-mechanical about your question, could you please try rewriting #27 without any references to photons, Planck's constant, etc.? The Baez link is not relevant, because it's about photons.

bcrowell, consider the following please. We're in a classroom. The teacher refers to an apple in his/her hand and, as a pedagogical exercise, points out that adding one atom at a time to the apple's mass merely increases the apple's gravitational mass (wrt the Earth as exterior gravitational source) by that atom's mass value alone. 'Teach' explains it's essentially a linear thing because in that context the apple's constituent atoms can to a very good approximation be considered independent 'non-interacting' entities. Then 'teach' gives a second example - add one electron at a time to the negatively charged plate of a charged capacitor, and lo and behold, net energy/mass has not gone up by one electron unit m_e = 9.11 × 10⁻³¹ kg, but 'amazingly' by a factor m_e(effective)/m_e = 1+ (0.5*CV²/N_e)/(m_ec²), = 1+fN_e, where f is a geometrical proportionality factor, C is capacitance, N_e ~ applied voltage V being the number of electrons already on each plate. For most any specific configuration, m_e(effective) works out to be considerably larger than m_e. 'Teach' begins to explain this 'anomaly' in terms of work done in transporting the electron against the combined field of fellow electrons to that negative plate, and how that relates to change in total stored field energy/mass. The electrons are here very much mutually interacting particles. Suddenly, one of the students, a childhood genius well versed in QM, shouts "NO, this is all wrong! You have no right in principle mixing classical concepts of 'apple mass' or 'capacitor energy/mass' with notions of 'atom' or 'electron'. One regime is purely classical, the other purely QM. It is VERBOTEN to compare these in any way." Who is out of order here, 'teach' or student? Don't get me wrong - been to your website at http://www.lightandmatter.com, clearly you are very knowledgeable in a wide range of physics topics, and in reality I'm just an unlearned member of the general public. That's not the inference here - it's just that taking your criticisms as purely objective I fail to find any logic or detail other than prior misdirected references to conservation of energy issues.

 

[Q-reeus]

Is this about what you had mind, Q-reeus?:

"I have about 10 coherent photons in phase. The amplitude is 10 times as much as a single photon. The energy is the amplitude squared, so I expect 10 photons to share 100 units of energy, or about 10 units of energy per photon rather than 1 unit that a single photon would have."

Phrak - thanks for precisely condensing down my point of view! I believe both sheaf (#5) and pervect (#14) (and I think PeterDonis (#10)) also 'got it' but this is the first precise reproduction of my argument (combined with the assumption of 'non-interaction' of photons).

It may help to know that, as theory has it, photons can be coherent but have no comparable phase.

This is where I absolutely don't get it. I've been picturing a photon as a wave packet with an SR frame dependent but otherwise well defined Fourier spread of frequencies. In a given inertial frame, combine a bunch of such photons via eg. microwave oven, and there should be some at least averaged phase relation. This averaged phase relation should not as I see it 'mysteriously' change in a precise way related to photon number density such that energy per photon is invariant, especially as numbers grow large. Would much appreciate an explanation in layman's terms what's wrong with this picture. If it's something that just comes out of the math and can't be visualized in any classical way ('non-locality/entanglement' comes to mind), well OK that's that I guess.
Edit: Let's suppose energy per photon is somehow maintained via phase-spread shifting. This surely still conflicts with points 1 & 3 in the recap of entry #27:
"Experience and classical EM theory say if a probe maintained at a fixed rms current feeds a high Q cavity resonator:
1: * Cavity field rms amplitude rises linearly with time.
3: * There is a corresponding fixed rate of photon creation by the probe current - photon number rises linearly with time."
Only possibility here is that 3: is wrong - rms photon production rate is decoupled from rms current amplitude. Makes sense?

 

[Phrak]

Q-reeus,

I don't mean to mislead you, and Dale's posts have both informed me, after some research, and left me some doubts about my understanding of accepted theory. So I will try to do my best.

A photon can be represented as a wave with complex amplitude. To keep things simple, the photon can be a sinusoidal planar wave traveling in the x direction.

[tex]\psi = A e^{i (kx-\omega t)}[/tex]

Photons are independent in phase and amplitude so a bunch of them, all planar waves traveling in the same direction and having the same energy don't add like this

[tex]\Psi_{sum} = \psi_{1} + \psi_{2} + \psi_{3} + ... + \psi_{n}[/tex]

but like perpendicular vectors

[tex]\Psi_{sum} = \psi_{1}\hat{i} + \psi_{2}\hat{j} + \psi_{3}\hat{k} + ... + \psi_{n}\hat{n}[/tex]

 

[Q-reeus]

Phrak - thanks again for trying to get it down to something understandable to a layman (myself).

Q-reeus,
...A photon can be represented as a wave with complex amplitude. To keep things simple, the photon can be a sinusoidal planar wave traveling in the x direction.

[tex]\psi = A e^{i (kx-\omega t)}[/tex]

'Complex amplitude' I'm thinking relates to the notion of 'spin 1' - meaning in classical terms a photon is circularly polarized and therefore that both 'E' and 'B' parts each have spatially and temporarily orthogonal components. About right?

Photons are independent in phase and amplitude so a bunch of them, all planar waves traveling in the same direction and having the same energy don't add like this

[tex]\Psi_{sum} = \psi_{1} + \psi_{2} + \psi_{3} + ... + \psi_{n}[/tex]

'Independent in phase' is the tricky part as I see it. Implied is the sum here is always 'scalar' and phase has no meaning/relevance in the vector/phasor addition sense for photons?

but like perpendicular vectors

[tex]\Psi_{sum} = \psi_{1}\hat{i} + \psi_{2}\hat{j} + \psi_{3}\hat{k} + ... + \psi_{n}\hat{n}[/tex]

This is where implied 'quantum weirdness' really bites. Just cannot picture how phase becomes an irrelevance here but is vital in the classical situation - eg. interference patterns in optics. Still thinking that a classical wave is sensibly a sum (in the phasor-vector sense) of a bunch of photons that each retain their independent existence - ie. that 'superposition of photons' has some sensible meaning.

Returning this discussion back to something relevant to SR/GR, let's consider the following:
A spatially narrow laser beam comprised of two pure monochromatic frequencies f₁ and f₂ of equal amplitude will exhibit beats - nodes and anti nodes spaced apart according to a 'beat frequency' f_b = |f₁-f₂| and 'beat wavelength' c/f_b. Place a gravitating object close to the beam at a node. If 'Photons are independent in phase and amplitude' applies then I expect mutual gravitational attraction is just that given by the average energy density/mass for each beam frequency - that is, indifferent to node/anti-node location. By contrast, classically phase rules and here mutual attraction ~ local energy density is at a minimum, and vice versa a maximum at the anti-node location. So what will actually apply?

Still coming back to the consequences as per recap in #27 (expanded a little here):
"Experience and classical EM theory say if a probe maintained at a fixed rms current feeds a high Q cavity resonator:
(in most circumstances net rms current is not constant - rather voltage. That is because a fixed driving voltage is increasingly opposed by a rising cavity field voltage. But this can be compensated or allowed for and when done, the following points 1 and 2 hold FOR SURE)
*1: Cavity field rms amplitude rises linearly with time.
*2: Completely balanced rms feed energy input and interior field rms energy output is directly proportional to rms cavity field amplitude squared.
*3: There is a corresponding fixed rate of photon creation by the probe current - photon number rises linearly with time. (logically necessary surely - an antenna at fixed rms current must be outputting a fixed rms photon rate)
Simple maths - divide cavity energy (and 'gravity') by photon count, it's not constant and tends to a huge value relative to E = hf."

So how to reconcile what I see as the inescapable logic of 1, 2, 3 above with standard QFT view? I haven't a clue!

 

[jfy4]

Worse, in Phil's own frame he has precisely zero time - but then it is only an illustrative story.

I know I am really late, but this isn't true. Remember we derived the length contraction and time dilation under the assumption that were were not massless! taking the limit as your mass goes to zero or your speed goes to c, logically does not imply that you turn into light. it important to remember how these amazing results came about...from original assumptions, hold onto them.

 

[Phrak]

Place a gravitating object close to the beam at a node.

This is really the thrust of your query, isn't it?: If we have a bunch of photons, arranged in a particular way, where is their gravity (or stress-energy tensor) located?

The short answer is that nobody knows.

I don't have a good long answer. Quantum mechanics is a castle in the air looking for a foundation. The uncertainty principle is enough; if you don't know where something is, you don't know how the very space, upon which you build your quantum mechanics is shaped. If you can't define the shape of the space upon which you base your theory, your theory is suspect. So theorists get by with presuming spacetime is flat and well behaved and call it Lorentz invariant as if this is enough.