The strange world of Phil the photon

[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.

\psi = A e^{i (kx-\omega t)}

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

\Psi_{sum} = \psi_{1} + \psi_{2} + \psi_{3} + ... + \psi_{n}

but like perpendicular vectors

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

 

[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.

\psi = A e^{i (kx-\omega t)}

'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

\Psi_{sum} = \psi_{1} + \psi_{2} + \psi_{3} + ... + \psi_{n}

'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

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

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.

 

[Q-reeus]

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.

A little amazed you picked up on that point! It's standard position that one can never be in the frame of a photon. Think of the absurdities - photon wavelength is infinite, while the rest of the universe has zero length and has aged infinitely.

 

[Q-reeus]

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?

Not exactly. There are now two related issues:
1: Photon count vs classical field amplitude.
My money is on the 1-2-3 'semi-classical accounting procedure' given prior - direct proportionality applies. It fundamentally conflicts with the relation implied by QFT - field amplitude going as the square root of photon count. Can't see any way to reconcile.
EDIT: Just realized that for a free-space dipole antenna, the radiation field drops off such that QFT relation holds. Dropping this point for now - a puzzle.

2: As per two frequency laser beam setup there is this question of what constitutes beam effective mass. Making the usual assumption object mass >> laser beam 'mass', clearly the net angle of beam deflection by nearby massive object is purely a spacetime curvature 'diffraction' effect indifferent to beam energy density. By contrast gravitational action of beam on mass should be directly proportional to the beam energy density and thus a function of beat phase. Bare in mind that modern laser frequency stability can be around 1 part in 10¹⁵ (http://congrex.nl/ICSO/Papers/TPosters/FCXNL-10A02-1987002-1-ACEF_ICSO_PAPER.pdf" ). Meaning at optical frequencies one could readily achieve beat wavelengths many thousands of kilometers long - many orders of magnitude larger than the size of a typical setup. So quantum uncertainty surely plays no significant role here. Action vs reaction holds here if the beat envelope is much larger than the significant gravitational range of the object, or at such times it is symmetrically disposed wrt object. Otherwise there may well be a temporal imbalance, although averaging out to zero over a full beat cycle.

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.

Thanks for that frank assessment!

 

[Q-reeus]

Issue 1 in thread #37 has become deeply problematic, but the conflict now is between what should be equivalent analysis - time and space evolution of EM fields within: an energized cavity resonator, and that of fields propagating away from an antenna source. The assumption of linear amplitude gain with time in a resonant cavity is well established; eg.
http://www.jpier.org/PIER/pier78/15.07090605.Wen.pdf", sections 5.2, 5.3, fig's. 6, 9, 10. It is definitely not a 'square root' relation. Nevertheless if the free-space antenna picture was hit upon first, this thread would likely never have started.

Issue 2, originally proposed to show that phase matters despite a quirky indifference for photon addition, is now also in limbo. Even the fluctuations in net force balance are not inherently gravitational in nature. The same thing will show up for propagation through a bent waveguide, or even a straight waveguide. It simply shows the vital influence of phase, and just how 'phase insensitive' photons become 'phase sensitive' in an aggregate situation (EM wave) remains for me a mystery without any simple explanation.

 

[Dale]

You may want to start a new thread in the QM forum. This doesn't seem to be very related to SR or GR now that your 2 key issues are not about gravitation. You probably will get better responses if the right people see it.

 

[Q-reeus]

You may want to start a new thread in the QM forum. This doesn't seem to be very related to SR or GR now that your 2 key issues are not about gravitation. You probably will get better responses if the right people see it.

Agreed - will give that some thought. Just wasn't evident at the start of what, apart from a few helpful inputs about the strangeness of photon addition, has mainly been a self discovery exercise.

 

[Dale]

Yes, and at the beginning with the discussion of gravitational interactions the fit here was not bad at all, I don't think that was a mistake. I just think a new set of eyes will be better for the remainder of the question in the way that it has since evolved.