# Can a point of space be identified?

Consider a single particle in empty space; by what argument can we say that it always occupies the same position or place?

Space itself has no identifying mark or label being everywhere the same.

Suppose, otherwise; and every point of space has a label; and as a concrete illustration say along the three axes they are labelled by the three primary colours - red, green and blue; so any point of space was identified by a colour that is unique.

Then were we in motion in space - we could see it - by the change of colour as we move; or not - if we do not.

But given space has no such structure then it seems we cannot justifiably say that a particle is at some identifiable position.

• That is true in infinite simply connected space, but if space is finite or non-simply connected or both it doesn't have to be. Commented Mar 29, 2015 at 0:58

If there were some obvious universal coordinate frame ("center of the universe") with some sort of obvious choice of axes ("um...???"), then it would of course be no problem.

But the universe isn't like that. It's not even entirely clear what the geometry of the universe is. (Worse still: when you're embedded into a manifold of some higher geometry, you can't tell what shape the manifold "really" is, though you can for instance notice of local geometric relationships that would be true in Euclidean space.)

Happily, this isn't what we mean when we say a particle is somewhere. It's enough to know the relationship between the particle and some other reference point that you can identify (e.g. the measurement device in your physics lab). And, better yet, because of relativity that's usually enough; maybe you're moving at 0.99c compared to some other far-away thing, and maybe in some sense that thing is a better choice of universal reference frame, but it's irrelevant because your local physics isn't affected by your relative velocity compared to that far-away thing.

So when one says a particle is somewhere, one needs, at least implicitly, to have some reference frame in mind. But identification of the reference frame usually isn't a major challenge in practice (when it is, one must take on that challenge). So we can speak freely of particles having a position.

(Even worse: they're delocalized due to quantum mechanics. We can still say that the center of the probability distribution has "a position" if we like the idea of exactness, or we can just give up and accept a pretty strongly peaked distribution function in place of "a position". But this is a separate issue from not being able to find a universal reference frame.)

Finally, what would happen if we have only one particle and nothing else (setting aside the concern about what "we have" means in such a case)? Well, then there is no need for the concept of space at all. Saying "it is always at the same place" is meaningless. Rather, we say that we do not need to consider embedding into a geometry or manifold in order to understand the properties of this particle.

• I'm not sure that one needs a reference frame to identify a point - we do physically of course; but mathematically speaking can't one can choose a fixed point p of a manifold? Commented Mar 29, 2015 at 18:14
• @MoziburUllah - If there exists a unique fixed point for some homeomorphism, and if the (effect of the) homeomorphism is observable, sure. But that's not different from a "obvious universal coordinate frame"; it's just a way to get one. Commented Mar 29, 2015 at 22:01
• I can't see what 'fixed points of a homeomorphism' have got to do with 'naming points'? General covariancy, in the sense of GR, simply says that any system of naming points is as good as any other - which seems like a perfectly reasonable idea - a physical space doesn't change its behaviour because we choose what we call its points; and a coordinate frame is a device to 'name points'; one could argue that all this is a fiction, in that there are no actual points to name... Commented Mar 30, 2015 at 14:57
• But one could suggest that post-GR that space is substantive. Commented Mar 30, 2015 at 14:58

This is an answer of a mathematician describing his understanding of what a point in space means for a physicist.

Although obviously philosophy does not reduce neither to physics nor to mathematics I think it is still useful to take into account these disciplines when speaking about points in space.

In physics, spatial coordinates are merely some of the numeric characteristics of an object, like speed, electrical charge or colour. They may be measured within certain accuracy and they may undergo change.

Even in quantum physics it does not contradict the prevalent theory that one may measure some of these quantities (including the collection of all spatial coordinates) to arbitrary precision (at the expense of making the resulting perturbation of values of some of the other quantities unknown).

Having that, you may view a point in space just as an intuitively convenient way to speak about a triple of numbers (calling them, say, latitude, longitude and altitude).

As for why this particular abstract convention is most convenient for us - well, maybe it has to do with particularities of physiological/kinesthetic/optical mechanisms of our perception.

Of course you may not say that this point is precisely that particular triple of numbers since in other coordinate systems the same point will be described by other triples of numbers; but the transformation rules from one system to other are also well understood. So finally a point in space is still a rigorously definable combination of numerical quantities.

The short answer is "because we said so," and the long answer is an interesting journey into ontology and epistemology with a slight jaunt into Quantum Mechanics just for "fun."

I am going to focus on one of your wordings, and use it to pry apart the question you are looking to ask:

... it always occupies the same position or place?

This is a phrase written in the English language, so much be interpreted according to the rules of language. To make an argument saying "it always occupies the same position or place," requires us to have some interpretation of what "occupies" means. This definition is going to require a quick cursory overview of ontology vs epistemology.

Ontology is the study of what things "are." Phrases like "this knife is sharp" show up in ontology, where "sharp" is a true trait of the knife. Epistemology is a study of how things "behave." Epistemology would change that phrase to "this knife behaves sharply," where "sharply" describes how the knife will behave.

When we get into talking about things like particles in space, we are almost always coming from an epistemological perspective. We don't actually know anything about the object. However, we do find that objects like it behave "predictably," and those predictions can be made based on points in space. A more technically correct wording for the original wording would be "it always behaves as though it occupies the same position or place." The argument for that particular phrasing can easily be founded in statistics and is the basis of modern western science.

However, in the case of some small objects where we've done many many studies on them (like electrons), it starts to get a bit pedantic to continuously talk about "behavior this" and "behavior that." After we can declare a 99.999999999% certainty that an electron behaves as though it occupies a the same position or place, it gets really tiring. At some point we make an axiomatic and unprovable claim that its epistemological behavior is its ontological essence. When we do so, we get away from ugly phrases like "we are 99.999999% certain that an electron occupies the same position or place" and get to shortcut it to the simpler "an electron occupies the same position or place." As stated in this paragraph, this change in wording has no logical argument beyond "we find the convenience of pretending the statement is ontological outweighs the risks from any inaccuracies in such a pretense."

So, to sum up the argument so far, the phrasing "it always occupies the same position or place" is a slightly inaccurate phrasing, but it's often close enough because we come across no particular reason to believe it is wrong.

This process has burned us before, so the counterargument against making such a statement is simply "we have counterexamples that prove such arguments can be false."

Take our lovely electron. We're going to put it in a section of doped silicon known as a diode. Diodes are really neat: you can mathematically show that any electron trying to approach the junction in the middle of the diode can never possibly cross it, because the greater the voltage (pushing the electron harder) the greater the force required to push the electron across teh gap. You can then go into experimentation mode, and prove that you are hopelessly wrong. Somewhere between 3 and 100V the diode suddenly lets electrons jump the gap in a way you mathematically proved was impossible (the actual voltage it occurs at is a feature of the geometry, and we actually use tuned "avalance breakdown" voltages in modern limiter circuits). Somehow, our diode did the impossible.

What we find is something known as "quantum tunneling." As the voltage across the diode piled up, the gap between the two sides got harder to cross, but it also got narrower. Eventually we find that an electron approaching this impossible to traverse barrier simply ceases to exist on one side of the barrier and starts to exist on the other side. This is terribly unintuitive, but experimental evidence has shown that the actually happens in real life in very predictable ways. In diodes, this is very evident because that first electron to skip over the gap begins neutralizing the forces within the gap, making it easier for the next electron to jump, causing something called "avalanche breakdown."

Do people wish this wasn't the case. Yes. But this is the physics we get. If the experiment says the world is round, it really doesn't matter how badly we wish the world was flat.

Quantum mechanics provides an explanation for this. It models the electron not as a point-mass with a position, but as a wave packet centered on that position in space. QM makes the claim that you can't talk about the "position of an electron," but rather you must talk about "the expectation of the position of an electron." You can run the QM numbers to show that each time this wave packet gets near the thin-but-steep barrier, part of the wave packet extends across the gap, indicating that the electron "might actually be on the other side." The math works. It describes the behaviors we see in diodes and other circuits perfectly.

So QM's argument would be that no particle has a position, just an expectation of its probabilistic distribution of positions. It would argue you cannot make an argument like "it always occupies the same position or place" because QM actually argues that it doesn't.

And so we come full circle. QM does such a good job of describing the behaviors we see, that we start getting lazy and talking in ontological terms. I see talk phrase like "the quantum numbers of this electron are (1, -1, 1, +1)," which is an ontological phrase. Even in this answer I probably made a mistake of using such wording regarding QM, and I'm actively writing an answer to raise awareness of what happens when people do it!

This has shown up in quantum gravity. It turns out that the QM models don't play well with the models that come from relativity. Both of them were actually epistemological models based on empirical evidence which have seen such good track records that people now talk about them using ontological phrasings. However, they are decidedly inconsistent when it comes to modeling gravity. The real physicists exploring this problem recognize that, ontologically, neither relativity nor QM is actually the "true" nature of a particle. They understand that both are epistemological models of behaviors, and we're going to collect new data to improve both models. However, the rest of us laypeople often forget this key detail, and start trying to phrase things like "QM must be wrong because it doesn't work with relativity." The better phrasing would be "The behaviors predicted by QM and relativity are not consistent with each other, so we should collect more data to try to understand how the real world differs from each model."

• +1: another possibility, which is what I was mulling over, is simply to deny particles; and in fact, if one considers electrons then one can identify them with their fields, which is continuous with space; and this field establishes place by potential; but also Commented Mar 29, 2015 at 15:04
• I was thinking about the gauge principle in physics - though I'm unsure quite what this means - other than it represents equivalent but not identical states. Commented Mar 29, 2015 at 15:05
• I am no physicist, but my best layman's understanding of gauge theory is that QM claims that the world is actually a continuous waveform in a Hilbert space. However, for most practical applications (such as electron emission of a photon), the waveform can be well modeled as a set of discrete numbers which basically end up being "numbers of harmonics." It's not possible to vibrate a ring at a frequency whose wavelength is not a multiple of the lenght of the ring because there would be a discontinuity (and thus we wouldn't have thought of it as a ring in the first place) Commented Mar 29, 2015 at 16:32