Double slit experiment violates triangle inequality?

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Main Question or Discussion Point

Imagine a light source, double-slit, and a curved screen in vacuum, shaped so that all parts of the interference pattern are created simultaneously. Define distance as proportional to the time light requires to reach a point. Detectors at each slit can be operating or not. Call the source S, the slits A and B, a point of constructive interference at the screen C, and a point of destructive interference at the screen D. With the detectors at slits on, one can say that the photon traveled (for example) from S to A to D, destroying the destructive interference there. The distance from S to D is given then by d(S,A) + d(A, D). However, with the slit detectors off, no light arrives at D. This implies that d(S,D) is infinity (light does not arrive in any finite time.)

The above shows a violation of the triangle inequality. d(S,A) + d(A,D) is finite, thus less than d(S,D).
Therefore I conclude that spacetime is not metrizable with this metric. However, there seems to be no reason why this metric should be any less valid than many equivalent ones in use.

The idea of spacetime not having a metric is understandably uncomfortable. Obviously, something that acts much like a metric must arise on the large scale of classical objects. We do have a perception of distance. However, it cannot be a true metric. I think this idea is usually rejected on the suspicion that all of physics would collapse if it were true. There are two closely linked ideas here which I consider important to separate.

1. “We must be able to ignore the price of tea in China while measuring the mass of an electron.” The separability of the universe into distinct non-interacting parts is essential to the functioning of physics. We cannot consider everything in the universe, we have to be able to ignore most data in order to perform any effective analysis.

2. “Spacetime locality must not be violated.” This is I suspect justified using #1, but is actually a separate assumption. Spatiotemporal proximity is not the only possible means of separation of the universe into analyzable parts.

I think we have a challenge in geometry and topology, not philosophy. I'm in the minority of people who see no reason to flirt with abandonment of the notion of reality, nor that of logic. Therefore, as John Bell proved, spatiotemporal locality must go. I have attempted to construct various models of “distance” that do not satisfy the requirements of a metric. I regret my skills are not equal to those of a typical Ph.D. in physics. I offer my idea for better minds to follow if interested.

But I must ask--is this widely known? Is it part of the “oral culture” of the physics community which never gets written down? Or have I made an original contribution? I am a physics grad school dropout, because I never really understood how the physics community operates nor what I had to do to be part of it. Anyway, I have been unable to find any material that seems to follow or refute this reasoning. Please let me know what I have missed. Thank you.
 

Answers and Replies

  • #2
Cthugha
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However, with the slit detectors off, no light arrives at D. This implies that d(S,D) is infinity (light does not arrive in any finite time.)
I do not see how one could draw that conclusion. First, you will get a different interference pattern for every possible wavelength of light you could possibly use giving you a wavelength-dependent distance which is odd. Second, the mere fact that no intensity is arriving at some point in space does not mean that no light field arrives there. The electromagnetic field will arrive at that point - destructive interference is already a proof for that - which invalidates your reasoning.
 
  • #3
Drakkith
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What does the interference of waves have to do with a triangle inequality about the distance between the source, slit, and screen?
 
  • #4
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Actually, Cthugha, I would say a zero field means no field. No light, no field at that point. Energy density zero. That's the point.

You are correct that every wavelength would have different places it cannot go. I don't see how that is any weirder than diffferent particles passing through the same point at different velocities end up in different places. You are quite correct that it says something important about how spacetime is stitched together.

Drakkith, it is an operational definition of distance: proportional to the time required to get directly from point S to point D without observation in the meantime. Since it never gets there, the time is infinity, or if you don't like that, even one second is plenty of time to show the triangle inequality is violated. By observing in the middle, you are *shortening* the distance, which metrics do not allow.
 
  • #5
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If an airplane crossing the Atlantic falls into the ocean, does that mean that the distance between Paris and NYC is infinity(airplane does not arrive in any finite time in NYC)?


"Optics: Destructive interference - Where does the light go?"

 
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  • #6
Cthugha
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Actually, Cthugha, I would say a zero field means no field. No light, no field at that point. Energy density zero. That's the point.
No, that is exactly not the point. Zero field does not mean no field. If you modify the light field - say by introducing a phase shift somewhere, modifying the intensity ratio going to the slits or whatever - any change will still arrive at the point in question with some speed. This is the important quantity.

According to your reasoning you could just put some screen somewhere and a light source in front of it. As no light reaches the back side of the screen, the distance will be infinite according to your definition. This is an external perturbation and not a matter of the metric.
 
  • #7
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Cthugha,
If you introduce a relative phase shift between the slits then the point of destructive interference will move. If you introduce a phase shift in both, no light continues to be no light, and a non-event propagates at the speed of light, is that what you are claiming?

And yes, Cthugha, that is exactly my point--if you have a light on one side of a screen, light does not get to the other side. The *direct* distance from the source to the back of the screen is infinity. You can get light back there, but only by using a mirror, which use can be detected--and then you're not going directly there any more, you are measuring *two* distances, not one. My point is that they don't add up in accordance with the triangle inequality.

As for "external perturbation"--that's interaction. That's changing the measurement from direct distance to the sum of two different distances. It is all about the metric. The metric does not behave in the way people assume. That is my point.

Maui: The *direct* distance between Paris and NYC is infinite, yes. Because you can't go straight there, you have to go around the earth.
 
  • #8
Drakkith
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The triangle equality doesn't take into account effects like interference. But so what? It's about geometry, not wave mechanics, right?
 
  • #9
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Drakkith, I am saying that we are using the wrong math to describe distance as covered by actual particles. Quantum mechanics is weird because there is something weird about spacetime.

If you watch a play, and the first actor to come out on stage slips and falls, you think they are clumsy. If every other actor in the play also slips and falls when they come out on stage, you begin to realize that the stage is too slippery. Likewise, if one quantum particle showed interference, that would be a weird particle. But since *all* of them do it, it seems to me that that means it isn't the particles, it's the spacetime they are trying to move through. Interference phenomena of single particles I believe are consequences of the small scale structure of spacetime.

It has been obvious for a long time that some sort of "spooky action at a distance" is going on, and I think it is because distant points of space are closer than we think. Now if you try to work with a metric and don't use the triangle inequality, you slide down a slippery slope where all distances become zero. I have not found a way around that yet, because I'm not a good enough mathematician. I have toy models, so I can see there are possibilities.

Is this really a new idea?
 
  • #10
Drakkith
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Could you elaborate on why you think interference of single particles is "weird" and why you believe it requires different math?
 
  • #11
Cthugha
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Cthugha,
If you introduce a relative phase shift between the slits then the point of destructive interference will move. If you introduce a phase shift in both, no light continues to be no light, and a non-event propagates at the speed of light, is that what you are claiming?
I am claiming that already the first part about moving the point of destructive interference shows that the change in the light field will arrive at the point in question and your definition of distance is not a sensible measure of physical distance.

And yes, Cthugha, that is exactly my point--if you have a light on one side of a screen, light does not get to the other side. The *direct* distance from the source to the back of the screen is infinity. You can get light back there, but only by using a mirror, which use can be detected--and then you're not going directly there any more, you are measuring *two* distances, not one. My point is that they don't add up in accordance with the triangle inequality.
So by using a screen which only transmits some wavelengths, you get a wavelength dependent metric again? If you get a time-dependent shutter you get a time-dependent distance? What about gravitational influences which travel at the speed of light, too? Do these have a different distance? This is just not a sensible concept of physical distance. Physical distance is regarded as a property of space. Making it a property of the whole setting is not really tenable. You can come up with effective distance measures as much as you like, but how does that contain any progress besides shifting the physics happening into a strange, complicated and wavelength-, time- and interaction-dependent metric?

There are semimetrics and premetrics which do not fulfill the triangle equality. However, I am not aware of any sensible attempt to attribute these to physical distances. They are instead used in rather abstract notions of distance like in complex networks (I have seen them used in neural networks or social network analysis).
 
  • #12
Funny you should mention that, I was just partying with two chicks east of Vegas the other night :)
 
  • #13
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Drakkith, last I checked everybody thinks interference of particles is weird--it's just that most people have given up on trying to explain or understand it. That's why we had the long debate between Bohr and Einstein which is now carried on by others. Many like to pretend there is no problem, or handwave and call it "the measurement problem" and follow that up with deafening silence. But there is always going to be a strong minority of physicists working on the problem because solving things like that is the whole *point* of physics for us.
Look into Bell's Theorem if you're unfamiliar with any of this.

Cthugha, I accept that I have not sufficiently made my case; I guess I will have to try to get more results on my own before I try again to present this. The answers to most of your case questions are "yes", btw. I'll have to work on my explanations; honestly I was mostly expecting to hear either "well, duh, everybody knows that" or "Oh, why didn't I see that?"

I hadn't realized that I had created YAPIOQM--Yet Another Personal Interpretation of Quantum Mechanics--as I call the many worlds, the transactional, many histories, etc.
As far as I can see, having the "fabric" of spacetime be woven with a complicated "stitch" seems far superior conceptually to the mystical gibberish and arbitrary restrictions on "allowable" questions I find in all the textbooks.

And spacetime already interacts with matter--that's general relativity. Distance is already defined with geodesics followed by light. I honestly didn't think I had much new here.

As far as I am concerned, distant entanglement, to say nothing of spin 1/2, means that our concepts of spacetime are fundamentally broken, and the need to fix that is the elephant in the room that most of the physics community seems to be ignoring--or to be fair, many of them probably made a few attempts and saw it was really hard and decided to tackle something else. My only beef with them is the way they pretend the problem isn't there and actively pressure people not to investigate it.

At root, the motivation is understanding, and yes that means a picture or a model. The claim that "no such model is possible" is not only an unprovable assumption but a pessimistic one to boot. And the reason I for one insist on searching for that model is because I don't feel I can trust *any* of the physics I have learned until this gets settled. Why should I study QFT if tomorrow it's going to be shown to be as useless as phlogiston? Can I even trust conservation of energy? Which parts of physics are still going to be standing? How can I ever build an intuition, a feel for problems, when I can't trust any of it except classical physics in the classical regime, and there only approximately?

I have taken six courses in quantum mechanics at various levels and I don't feel that I understood a thing. I can solve an equation if presented but I have *no idea* how to look at a physical situation and set up the equations in the first place. I don't have this problem in any other area. I can work GR and classical physics just fine, start to finish. I have no intuition for QM and I see no way to build any without a model which does not yet exist.

Sorry if that was overexplaining but this is the first time in many years that I have been able to vent that rant to anyone who had a chance of understanding more than one word in three. I am actually very grateful for the feedback and I am not attacking you, just the damned orthodoxy.
 
  • #14
Drakkith
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Drakkith, last I checked everybody thinks interference of particles is weird--it's just that most people have given up on trying to explain or understand it. That's why we had the long debate between Bohr and Einstein which is now carried on by others. Many like to pretend there is no problem, or handwave and call it "the measurement problem" and follow that up with deafening silence. But there is always going to be a strong minority of physicists working on the problem because solving things like that is the whole *point* of physics for us.
Look into Bell's Theorem if you're unfamiliar with any of this.
I didn't ask what most people think, I asked why YOU think it's weird. Do you think that the universe should work in a simpler way or more intuitive way at the quantum level?

And spacetime already interacts with matter--that's general relativity. Distance is already defined with geodesics followed by light. I honestly didn't think I had much new here.
I don't see how your example deals with this issue. The EM wave still travels to the spot on the screen, it just interferes so we don't see any photons, correct?

As far as I am concerned, distant entanglement, to say nothing of spin 1/2, means that our concepts of spacetime are fundamentally broken, and the need to fix that is the elephant in the room that most of the physics community seems to be ignoring--or to be fair, many of them probably made a few attempts and saw it was really hard and decided to tackle something else. My only beef with them is the way they pretend the problem isn't there and actively pressure people not to investigate it.
First you need to explain why this is even a problem. What is wrong with entanglement and spin 1/2 particles? Why is this a problem at all? Perhaps this is genuinely what happens.

At root, the motivation is understanding, and yes that means a picture or a model. The claim that "no such model is possible" is not only an unprovable assumption but a pessimistic one to boot. And the reason I for one insist on searching for that model is because I don't feel I can trust *any* of the physics I have learned until this gets settled. Why should I study QFT if tomorrow it's going to be shown to be as useless as phlogiston? Can I even trust conservation of energy? Which parts of physics are still going to be standing? How can I ever build an intuition, a feel for problems, when I can't trust any of it except classical physics in the classical regime, and there only approximately?
Because it isn't always about right and wrong, it is about accuracy and ease of use. Classical physics is obviously wrong, but is still used all over the world for almost everything because it is accurate enough for practically all purposes and is MUCH simpler than more accurate theories and models. Science strives to build more accurate theories and models but unfortunately we run into the problem that what we find does NOT look like what we THINK it should look like, what we are used to dealing with. This doesn't mean that we are wrong, it may mean that we are right! But we don't know. That's why we keep looking.

I have taken six courses in quantum mechanics at various levels and I don't feel that I understood a thing. I can solve an equation if presented but I have *no idea* how to look at a physical situation and set up the equations in the first place. I don't have this problem in any other area. I can work GR and classical physics just fine, start to finish. I have no intuition for QM and I see no way to build any without a model which does not yet exist.
Without taking a course in QM I cannot say much on the matter, but it APPEARS to me that you simply never learned how. This is a problem for many people in many different areas in life. I don't see this as being a problem with QM itself or any other theory.
 
  • #15
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Drakkith, you raise a lot of points. Now that I know you have never studied QM, I'll see if I can rephrase and explain.

Start with the double slit experiment with light. We get an interference pattern, as Young showed in 1811. We know that waves do that, so we interpret light as a wave. Then we developed the ability to register individual photons with CCD counters and such. So light acts like a particle. In short, it travels like a wave, but interacts with the world like a particle. So de Broglie tried in 1927 to describe the situation as a particle having a location, though unknown, and motion guided by the "pilot wave." But this doesn't work because if you turn the light intensity down so far that only one photon per second comes through, the interference pattern still shows up. So it seems that the photon is spread out over the entire wave, right? It has to be in both places at once to be affected by both slits. But now we have a problem when the light wave arrives, because if it hits all parts of the screen simultaneously (we can curve the screen slightly to guarantee this) then if the photon "decides" to arrive at a particular spot on the screen, somehow every other point on the screen *instantaneously* knows *not* to register a photon. So information is going faster than light in what seems to be a violation of relativity, but which always seems to weasel out on a technicality when you try to use it for FTL signalling. It's as if Nature is cheating to keep her books straight, violating her own laws.

Now, one can ignore this, and "shut up and calculate", as the proponents of the so-called "Copenhagen Interpretation" command, but some of us want to understand what is going on. Is something going faster than light or not? Many say, "well, we can't expect classical thinking to work in the quantum realm," to which I reply, Yes, but we can expect *some* kind of thinking to work, instead of just assuming the answer and refusing to think about it.

To my mind, the particle is actually spread out over a volume of space in some way, and when it is detected by interacting with the world, somehow all the distant scattered parts of the particle have to instantaneously teleport all into the same spot to get absorbed by the electron or what have you. Suddenly, instantly, there isn't any wave any more, anywhere, because a particle got detected in one spot. I find it a reasonable question to ask "how does it do that? How is spacetime woven such that a particle can spread out all over space and then suddenly all get sucked into one tiny region?" I think it is a geometry problem, and a very hard one, but most people seem to go off into mystical semi-philosophical handwaving and insist that some questions are "improper" but don't provide any way of identifying in advance which questions are "forbidden", which smells suspicious. It is a way to cheat and win every argument--if you don't know the answer, declare the question forbidden. I don't like that one bit.

It would be one thing if they said, "that kind of question is ill-defined because of this specific feature, and this is what is well-defined in replacement," but instead they say, "well, *sometimes* that question is forbidden and sometimes it isn't," and always seem to forbid it when things get interesting. I don't like theories that claim that some knowledge is impossible when they keep moving the criteria. I find it philosophically offensive, not to mention an unscientific and pessimistic assumption.

I don't disagree that these things *happen*--the calculations of QM are verified to some huge number of decimal places. The theory works, it gets valid results. But it is a cookbook recipe, not a model, not a way of understanding the world. It is blind magic. And all of my professors and all of my 30+ textbooks on quantum mechanics are incoherent when it comes to giving anything like rules as to how and when to work the magic--hence I can't set up problems because I can't trust any of it. I can solve equations all day long but it's useless if I don't know what they mean.

QM is a theory written by human beings for human beings, and as such it should be comprehensible to human beings. I'm willing to admit I have blind spots, but I studied physics at MIT, I'm not an idiot, I've been a physics instructor for 19 years and a damned good one by all accounts, and it's not just me. Yes, I didn't learn how, I missed something that all the other students got. But I know pedagogy. I construct many different explanations every day to help students understand. Either every single instructor and author of quantum mechanics is atrociously incompetent, or there is something wrong with the theory. I vote the latter.
 
  • #16
Drakkith
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Ok. Everyone has their opinion. I don't see what so "incomprehensible" about QM, but perhaps your use of the word is different than mine.
 
  • #17
Cthugha
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I hadn't realized that I had created YAPIOQM--Yet Another Personal Interpretation of Quantum Mechanics--as I call the many worlds, the transactional, many histories, etc.
As far as I can see, having the "fabric" of spacetime be woven with a complicated "stitch" seems far superior conceptually to the mystical gibberish and arbitrary restrictions on "allowable" questions I find in all the textbooks.
I still do not see your point of putting the physics happening into the metric, but feel free to follow your ideas - though I do not find mystical gibberish in reasonable textbooks.

And spacetime already interacts with matter--that's general relativity. Distance is already defined with geodesics followed by light. I honestly didn't think I had much new here.
The geodesics are those along which changes in the light field travel, not those where intensity arrives. This is a huge conceptual difference.

So de Broglie tried in 1927 to describe the situation as a particle having a location, though unknown, and motion guided by the "pilot wave." But this doesn't work because if you turn the light intensity down so far that only one photon per second comes through, the interference pattern still shows up. So it seems that the photon is spread out over the entire wave, right? It has to be in both places at once to be affected by both slits.
I am not a friend of pilot wave theory, but this is incorrect. The guiding wave will go through both slits in pilot wave theory. The photon will not. Those two things are not the same. If it is the ontology you are after, Bohm's approach will at least give you a tenable way of approaching the problem. Other approaches work fine, too.

Anyway, you somewhat seem to still imagine photons as tiny bullets. This point of view is outdated by at least 40 years. At least with the advent of quantum optics that simple picture became obsolete. Unfortunately, some textbooks from "older times" are still widely used and influential, most prominently the Feynman lectures. If one takes this naive view, one indeed runs into problems with particle-wave duality. However, I do not see why one should even follow that route as duality is just a consequence of taking this wrong approach. Btw, the reality of wave-particle duality is such a common topic around here, that it even got its own FAQ entry: https://www.physicsforums.com/showthread.php?t=511178
 
  • #18
kith
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[...] somehow every other point on the screen *instantaneously* knows *not* to register a photon.
Ignoring the problem with photon positions: The point didn't register a photon before you started the experiment and it didn't register one during the experiment. So nothings changes for it. Why do you think this is remarkable?

Now, one can ignore this, and "shut up and calculate", as the proponents of the so-called "Copenhagen Interpretation" command, but some of us want to understand what is going on.
"Shut up and calculate" is not really the Copenhagen approach. But I agree that Copenhagen seems more like "shut up and calculate" when compared to other interpretations. In the beginning, I didn't like the Copenhagen interpretation and read a lot about different interpretations. I still don't have a favourite, but Copenhagen is special in one way: it is very close to the scientific method by acknowledging that we get our knowledge by doing measurements. Keeping this in mind, it seems much more rational that measurements may play a special role in a fundamental physical theory.

To my mind, the particle is actually spread out over a volume of space in some way, and when it is detected by interacting with the world, somehow all the distant scattered parts of the particle have to instantaneously teleport all into the same spot to get absorbed by the electron or what have you. Suddenly, instantly, there isn't any wave any more, anywhere, because a particle got detected in one spot. I find it a reasonable question to ask "how does it do that?"
The idea that particles are actually spread out over finite volumes is very uncommon even between people who are working on fundamental interpretational questions. It was first advocated by Schrödinger and I don't know much about. Maybe someone else can comment on it. I just wanted to say that starting with a uncommon idea and asking "how does this work" is maybe not the best way to get a grip on the difficult question of interpretation.

QM is a theory written by human beings for human beings, and as such it should be comprehensible to human beings.
The main problem I see is that comprehension comes from humans interacting with the world. These interactions are described by classical physics, so comprehension really means understanding something in classical terms. This is often possible, but I don't see why we should assume that it always has to be so.
 
  • #19
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Okay Drakkith, thank you for the conversation.
 
  • #20
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Ok. Everyone has their opinion. I don't see what so "incomprehensible" about QM, but perhaps your use of the word is different than mine.


Feynman may of help here:

"If you think you understand quantum mechanics, you don't understand quantum mechanics"
 
  • #21
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Cthugha, I will think a while on this distinction you are drawing between "light" and "changes in the light field" and see if I can make sense of it. I've studied EM simple and advanced and optics (that kind link about interference given above told me nothing I don't know already) but I'll go over it all again, I guess. Ah, yes, you are correct about the pilot wave, my apologies.
So perhaps I am old fashioned in wanting things to have locations, extended or otherwise. I just have never understood how one can throw away ideas of space and time at one moment and then use them later in the same calculation. Temperature can be understood as an emergent property of kinetic energy of molecules. Space and time don't appear to be emergent from anything--they are just there when the QM worker says they are and they go away when the QM worker says they are inconvenient. Now, I can learn to calculate textbook problems this way, and have, but it adds nothing to my comprehension of the world and it certainly bestows no ability to take a physical description and turn it consistently into a math problem.
 
  • #22
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Kith, thanks for your reply. The remarkable thing is that we never get two photons out the far end--all the different parts of the detector, despite all being possible points of arrival all the way up to the end, somehow magically confer with each other faster than light speed and agree on which one is going to go off so that one and only one photon is detected. And this seems to not be useable for FTL signaling because...a different technical argument each time. It feels more like a good lawyer than a law of nature.

I totally agree that measurements are important--this whole issue is called the measurement problem, and I was stipulating that measurement was what determined the end points of a distance in my model. My view of Copenhagen is that it tells us not to interpret at all--calling it an interpretation is like calling bald a hair color, or an atheist religious. But I respect that you have looked into this too, and perhaps "agnostic" is a better analogy.

Umm..."spread out all over" is called a "wave." I guess I did not say that very well. It acts like a wave when we don't look at it--spreads out all over--and then when we look at it, it all instantly condenses in a single spot. I think a lot of people just shrug and say, "well, that's the way it is," and move on, but I think there's something to be learned there.
Of course there is no reason to assume the world is comprehensible. But there is no reason to assume it is not, and assuming it is incomprehensible is defeatist. No science would ever be invented, nothing would ever be understood without someone trying to understand.

Cthugha, thanks for the link, I'll go read that FAQ through and see if there is anything in it I haven't already heard a bunch of times.
 
  • #23
kith
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Umm..."spread out all over" is called a "wave." I guess I did not say that very well. It acts like a wave when we don't look at it--spreads out all over--and then when we look at it, it all instantly condenses in a single spot.
But the question is if it is really the matter which is spread out or something else. If you don't assume that it is the matter, you don't have problems with an instantaneous contraction of the wave (but you have other problems probably ;-)). For example, if the wavefunction encodes a probability, an instantaneous update corresponds to ordinary rolling the dice.

Another problem for such a simple form of realistic matterwave interpretation seems to be decoherence. The modern picture of measurements is this: a measurement apparatus constitutes an environment for our measured system turning it into an open system. Interactions with this environment cause the system to undergo a transition from a coherent superposition (single wavefunction) to a incoherent mixture (statistical mixture of wavefunctions). This mixture is what eventually undergoes collapse. You can read about this in "Decoherence, the measurement problem, and interpretations of quantum mechanics" from Schlosshauer.

Of course there is no reason to assume the world is comprehensible. But there is no reason to assume it is not, and assuming it is incomprehensible is defeatist. No science would ever be invented, nothing would ever be understood without someone trying to understand.
Yeah, this was poorly expressed. Let me try again: Our interaction with the world leads us to the assumption that the world behaves logically. I agree that if we abandon this, we cannot do science. But our interaction also leads us to think that the world behaves classically. Now when we approached smaller and smaller time scales, it seems that we cannot express the encountered phenomena in classical terms. But why should we expect this in the first place? Can science really provide more than a theory which makes accurate predictions?

Most controversies of quantum mechanics are not about science, but about ontology (the relation between the theory and reality). Classically, the physical theories suggest an ontology which is accepted by most people. But why should we demand this from every theory?
 
  • #24
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Imagine a light source, double-slit, and a curved screen in vacuum, shaped so that all parts of the interference pattern are created simultaneously.

Photons/electrons don't arrive at the slit simultaneously but one after another in very short series and accumulate to form an inteference pattern or 2 fringes, depending on the setup.



Define distance as proportional to the time light requires to reach a point. Detectors at each slit can be operating or not. Call the source S, the slits A and B, a point of constructive interference at the screen C, and a point of destructive interference at the screen D. With the detectors at slits on, one can say that the photon traveled (for example) from S to A to D, destroying the destructive interference there. The distance from S to D is given then by d(S,A) + d(A, D). However, with the slit detectors off, no light arrives at D. This implies that d(S,D) is infinity (light does not arrive in any finite time.)


There is always light arriving at D, though at a different point on the curved screen that you call C(this where "light" has gone).

The above shows a violation of the triangle inequality. d(S,A) + d(A,D) is finite, thus less than d(S,D).

No, there is no violation, d(S,D) is not infinite.


Therefore I conclude that spacetime is not metrizable with this metric. However, there seems to be no reason why this metric should be any less valid than many equivalent ones in use.

The idea of spacetime not having a metric is understandably uncomfortable. Obviously, something that acts much like a metric must arise on the large scale of classical objects. We do have a perception of distance. However, it cannot be a true metric. I think this idea is usually rejected on the suspicion that all of physics would collapse if it were true. There are two closely linked ideas here which I consider important to separate.

1. “We must be able to ignore the price of tea in China while measuring the mass of an electron.” The separability of the universe into distinct non-interacting parts is essential to the functioning of physics. We cannot consider everything in the universe, we have to be able to ignore most data in order to perform any effective analysis.

2. “Spacetime locality must not be violated.” This is I suspect justified using #1, but is actually a separate assumption. Spatiotemporal proximity is not the only possible means of separation of the universe into analyzable parts.

.


I agree that spacetime metrics are heavily observer dependent(contrary to human intuition), but in this partuicular case i don't see anything strange with the chosen metric.
 
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Most controversies of quantum mechanics are not about science, but about ontology (the relation between the theory and reality). Classically, the physical theories suggest an ontology which is accepted by most people. But why should we demand this from every theory?


Maybe it's because all that we know from all fields of science very strongly suggests that the classical newtonian universe can only exist as we observe it in a naive realistic form. And this naive realistic universe cannot be supported by experimental evidence.
 

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