QFT and the Art Hobson Paper

[KipIngram]

A couple of weeks back Mordred posted a link to this paper in some thread or another:

 

https://arxiv.org/pdf/1204.4616.pdf

 

I read it and have been mulling it over - I think it's causing my brain to "click" to some things that have previously been fuzzy for me. I'm going to summarize the new insights briefly in hopes that some kind folks can validate / correct me.

 

So, what I'm getting from the paper is as follows:

 

1. There are no "particles" per se - what we call particles are just certain excitations of this or that field.
2. There's not just one field - there are quite a few (I believe someone said twenty-ish in some other thread).
3. Like all wave phenomena, we can decompose excitations of these fields into modes of various frequencies.
4. The amount of energy associated with any given mode is quantized, E = Nhf for frequency f, where N is an integer.
5. Each mode of any field has infinite spatial extent.
6. Changes in energy for any mode are instantaneous events.

I bolded item 5 because that one was a key part of my change in thinking from reading the paper. I just hadn't been clear on that before.

 

So, based on this a lot of things become more clear to me. It's easy to see why we get interference in the double slit experiment even when we run it one photon at a time - the quantum is spatially extended and interferes with itself in the usual way we think of interference happening in such a situation. The Bell-type "spooky" entanglement stuff becomes clear - the quanta are spread out all over the place, and when Alice makes a measurement the field mode that is involved changes everywhere instantly, including where Bob is doing his measurements. Superluminal signaling is prevented by the uncertainty principle. And so forth - things that seem paradoxical with the particle perspective just don't with the field perspective.

 

Where I begin to get a little fuzzier is on explaining, for instance, why we wind up getting localized responses on the photographic film or whatever used to detect the interference pattern in the double slit experiment. The arriving quantum is spread out. It's stronger where the interference is constructive and weaker where it's destructive, but it's still extended over a substantial area of the film. But when the interaction happens it happens in one place. Hobson talks about this but he doesn't go into great depth.

 

So, I've tried to ponder through it myself. I recognize that the matter comprising the film is also composed of matter field excitations, and that those excitations (which we recognize as electrons, atoms, molecules, etc.) also have modes that are space-filling. Hobson mentioned that the spot on the film involved a substantial number of atoms (500, for example). I assume that has to do with the size of the emulsion crystals and so on.

 

The only way I can think of to "get at" a localized spot on the film is to regard a collection of modes of the matter field that correspond to the Fourier transform of such a localized spot. It does seem that if I started with the spot (basically an impulse function) and took the FFT that I could pinpoint a specific set of modes. But then it seems like I'd have to have a quantum of energy for each mode, and we may only be sending one quantum of light at a time. I realize the energy quanta might not be the same size, but I'm still not sure that's the right track to be on.

 

As an alternative to that line of reasoning, I recognized that the slits are very small too. For one photon, which strikes the barrier in a spatially extended way, the whole quantum of energy either goes through the slits or does not go through the slits, and that's just a probabilistic thing. So is it that way too for the film? If the energy of the photon is spread out over the whole film, one "quantum unit of response" is just chosen at random (i.e., according to Born's rule) to register the event?

 

I guess I'll stop there. This is nice stuff - I think I understand more than I did. But it feels like step one (or maybe ten or so - I've been at it for a while) of a lengthy journey.

 

Oh - one other question. Each of these modes can have a variable number of quanta associated with it; Hobson presented it that way for the photon field. Is this where the exclusion principle comes in? Is it the case that a single mode can have any number of quanta of energy associated with it for bosons, but only one quantum of energy (or zero, leaving just the vacuum energy) for fermions?

hobson.pdf

Edited May 22, 2017 by KipIngram

[Mordred]

If the destructive interference leads to less than a quanta of action. No detector can detect these fluctuations. Not even an idealized perfect detector. The photgraphic paper (idealized) will only ever record the amplitudes of the excitations in excess of a quanta.

 

Side note photographic paper is insensitive to wavelengths greater than 600 nm. This allows you to work with it under red light. (recall the photon is a packet) a tightly bundled excitation of the field.

 

Now there is with work function associated with any detector. (including your eyes This is called the photelectric work function.

 

https://en.m.wikipedia.org/wiki/Work_function

 

Changes in frequency will only affect the kinetic energy not the intensity of light (increasing intensity only increases the number of photons in your light beam.

Edited May 22, 2017 by Mordred

[KipIngram]

Ok, that makes sense - that's just where the probability of an interaction goes all the way to zero. But everywhere it's not zero, the possible interactions just occur per Born's rule based on the appropriate field amplitude? And there is no predictability of that whatsoever beyond Born's rule?

Which photons go through the slit arise in the same way, right? The whole panel with the slits in it gets exposed to the light quanta, and it's Born's rule that determines when a photon passes through vs. getting absorbed by the barrier?

[Mordred]

Yeah in its simplest form it states that the probability density of finding the particle at a given point is proportional to the square of the magnitude of the particle's wavefunction at that point. However its actually a complex conjugate.

 

A wavefunction is a complex conjugate probability amplitude.

 

https://en.m.wikipedia.org/wiki/Wave_function

 

Here is a good paper covering the two slit experiment with the Born rule applied in terms of degrees of freedom.

 

https://www.google.ca/url?sa=t&source=web&rct=j&url=http://file.scirp.org/pdf/JMP20110100004_79863016.pdf&ved=0ahUKEwi5yO2p0ILUAhWIr1QKHXHGDOcQFggeMAE&usg=AFQjCNFaMYu6gR4z1qkC_Lkp3KF0aw5Urw&sig2=JgDkvaMiaNUluyGGAxAVcw

[KipIngram]

Oh, I did remember one other question I had. I often read that wave functions are defined "up to an arbitrary phase factor." In the context of what I've discussed so far, is that just the phase of these modes? Based on what I know about Fourier techniques those phases matter when you add up a bunch of components to get a total, but I don't really see another phase floating around in this discussion to apply that bit too.

[Mordred]

Yes phases matter when the waves are in phase you have constructive interference.

Edited May 22, 2017 by Mordred

[KipIngram]

So what is being referred to when I see these references to "arbitrary phase factor"? The double slit paper you linked above mentions this, when it says that "phase is arbitrary, and this can't be manifested physically" or something like that.

Does that refer to the same phase, and do they mean that even though the phase matters when adding amplitudes to compute probabilities, once we have a probability the phase no longer affects the measurement outcomes?

I.e., the phase doesn't affect the measured results, only the probability of obtaining them?

[Mordred]

lol yeah I was digging up a reference to cover that.

 

https://en.wikipedia.org/wiki/Photon_polarization also the Jones calculus is handy. https://en.wikipedia.org/wiki/Jones_calculus

 

"in which the arbitrary (scalar or vector) function τ ↦ f ( τ ) {\displaystyle \tau \mapsto f(\tau )} {\displaystyle \tau \mapsto f(\tau )} gives the variation of the wave's amplitude, and the fixed unit"

 

https://en.wikipedia.org/wiki/Plane_wave

 

here is an example of the arbitrary vector or scalar. in the last link. In essence the arbitrary waves function is just some function that gives rise to the variations in amplitude. In this example however you can also have arbitrary direction as per

https://en.wikipedia.org/wiki/Plane_wave#Arbitrary_directione

 

one the better definition of arbitrary is

 

An arbitrary function simply means that it is a function that you are free to define in any way you want. The simple constraint is that it should be a function

 

when its arbitrary you are free to define it any way you want.

Edited May 22, 2017 by Mordred