What has this to do with Newtonian mechanics and where Mars is?Demystifier said:
What then, according to you, is real? You think there lies nothing behind the observations? The math of virtuality is not real either. I can't observe the integral over the four-momentum of the Green function of the field (a rough description of a virtual propagator, a quantum bubble). I can imagine what the bubble describes though. A weird particle, going forward and backwards in time, with all energies and momenta, all over the place. Which is an ideal stuff for charged particles to couple to when interacting with other particles. Or to realize real particles out of, moving in one time direction only. I think they even made up the pre-big bang stuff, fluctuating in time, waiting to inflate into reality. Waiting for the right condition.PeterDonis said:
It's the B-field. As demonstrated above for the idealized case of an infinite solenoid, the phase, describing the shift of the interference pattern when switching on this B-field, is given by the gauge-invariant magnetic flux through the solenoid. Outside the solenoid the B-field is vanishing and ##\vec{A}## is a gradient, but it's a "potential vortex" so that the closed line integral along a path encircling the solenoid does not vanish but gives the said magnetic flux. The important point here is that the region outside the solenoid is not simply connected. In this sense the AB effect is a topological effect.JandeWandelaar said:
Thanks. Indeed. Ryder gives a nice exposition of the effect being topological because of the non-simply-connectedness (is this good English?). How is the electron informed about the thin solenoid? You would expect the thin flux of magnetic field in the solenoid not to reach outside it. But it does have effect. Is it the very topology that causes the shift? Or virtual photons?vanhees71 said:
Consistency, one should use the same standards of rational reasoning in all of physics. You use one set of standards in classical physics, another in Bohm's theory, and yet another in interpretation of the AB effect.martinbn said:
Don't you mean "invariant" instead of "not invariant"?Demystifier said:
No.JandeWandelaar said:
I mean it corresponds to something real, just as particle position ##x## corresponds to something real. For example, if ##x=5##, it's not the number ##5## that's real, but it's the particle at a definite position (which we label with number 5) that's real. With a change of origin we can turn 5 into any other number we wish, but it doesn't change the reality of the particle at a definite position. Reality of the potential is completely analogous.JandeWandelaar said:
Demystifier said:
I see what you mean. The potential of a particle in the Earth's potential springs up in my mind. You can change, gauge, the global potential field without changing the differences. Doesn't that mean the potential is invariant under a global transformation? Which leads to an assignment of it being non-real?. Though real in the sense it has effect? Performing a global gauge keeps everything the same. Which doesn't mean it's unreal because the differences have effect. In the AB case, the A-field is not globally changed nor locally. A local change would imply changes independent of position. So an independent gauge at every position of the position between slits and screen. A global gauge means the same gauge at every position. Can't we say the solenoid induces a gauge which is a function of position?Demystifier said:
I am not sure what you are referring to!Demystifier said:
One thing that people don't say is that not only the A-field is not uniquely defined, but it is also not defined at all if the region is not simply connected. On every simplyconnected open subset a field A exists, but you cannot choose them so that they patch to a global field. This doesn't seem to bother people that want to think of A as real!vanhees71 said:
Yes. The fact that you don't get it explains why we can't reach much agreement on quantum interpretations.martinbn said:
The electrons are not localized in this setup, so that you get an interference pattern to begin with. So you cannot say that the electron is never within the region where a magnetic field is present.JandeWandelaar said:
It resonates well with my particle position analogy. On a sphere we can't define a global system of coordinates, but it doesn't bother people who want to think that position of a classical particle on a sphere is real.martinbn said:
Let me get this straight. According to you in classical mechanics the position observable doesn't have a specific value at a given time if an observation is not made at that time?Demystifier said:
This shows that you don't understand it. It is not analogous to coordinates. A function on a sphere is defined everywhere on the sphere although there are no global coordinates and the function cannot be expressed in coordinates globaly. On the other hand on a nonsimplyconnected regen an one-form ##\omega##, which is closed (i.e. ##d\omega=0##) need not be exact (i.e. there may be no function ##f## such that ##df=\omega##) even though locally that is true.Demystifier said:
That's not what I'm saying. I'm saying that if classical mechanics is interpreted strictly operationally (in a sense in which quantum mechanics is often interpreted strictly operationally), then classical mechanics is agnostic on the question whether position has a value when not measured. I am also saying that such a strictly operational interpretation does not contradict any observed fact, that almost nobody uses such a strict operational interpretation of classical mechanics (and certainly not me), and that many people use such a strict operational interpretation of quantum mechanics. I am also saying that anybody who uses strictly operational interpretation in quantum physics, but not in classical physics, uses double standards.martinbn said:
I think it's a good analogy. In both cases you can do something locally, but due to topological reasons you cannot do it globally. The two things are not mathematically equivalent, but there is an analogy.martinbn said:
I think you just refuse to admit it although you very well know that you were wrong.Demystifier said:
No it is not. There is a difference between a mathematical object exists globaly, but cannot be represented globaly in coordinates (in case there are no global coordinates) and a mathematical object does not exist. If you don't understand that, I cannot help you. I can see why you are confused and how you can believe what you believe without realizing that it makes no sense.Demystifier said:
Ah yes. The electron "scans" the whole space, so to speak. Will it's presence inside the solenoid influence it's state outside it? Will it couple to the virtual photon condensate inside it, or are there virtual photons in the whole region (be they real or a math aid)?vanhees71 said:
If something (call it ##A##) exists globally, but cannot be represented globally in coordinates, then coordinate representation ##R(A)## does not exist globally. A coordinate representation ##R(A)## is something too, right?martinbn said:
Wrong about what?martinbn said:
It can be represented in any coordinate system. If there are no coordinates systems of some kind, that has nathing to do with ##A##.Demystifier said:
Not sure how this is relevent, but it is not true. There are no global coordinates on any compact manifold, but there are global nonvanishing vector fields on many compact manifolds. For example on odd dimensional spheres or on the usual two dimensional torus.Demystifier said:
I have wondered about that too. The space in the experiment is still simply connected. The insertion of a finite piece of mass doesn't introduce non-simply-connectedness. Or is the non-simple-connectedness a local phenomenon?Davephaelon said:
You are mistaken. The 2-dimensional torus is a compact manifold that does have global coordinates.martinbn said:
That's quite common. For a differentiable manifold you may necessarily need atlasses with more than one map to cover the entire manifold. The same geometrical/topological properties you have in the case of the solenoid concerning the gauge field. It's a very well known property. See, e.g.,Demystifier said:
That's not true. If there were global coordinates for ##\mathbb T^2##, then there would have to be a homeomorphism ##f:\mathbb T^2\rightarrow U##, where ##U## is an non-empty open subset of ##\mathbb R^2##. Since homeomorphisms preserve compactness, ##U## would have to be compact. In ##\mathbb R^2##, a set is compact iff it is both closed and bounded. The only open sets ##U\subseteq\mathbb R^2## that are also closed are ##\varnothing## and ##\mathbb R^2##. ##\varnothing## is not non-empty and ##\mathbb R^2## is not bounded. So there is no subset ##U\subseteq\mathbb R^2## that qualifies as a global coordinate chart for ##\mathbb T^2##.Demystifier said:
Things that appear in all of our models. For example, electrons are "real" because every model we have of matter includes them. But that does not mean real electrons are exactly the same as a theoretical "electron" in some particular model. Models are not reality.JandeWandelaar said:
Of course not. See above.JandeWandelaar said:
It seems that we have different notions of "global coordinates" in mind. Take, for example, circle. I would say that the usual angle variable ##\varphi## is a global coordinate for circle, but obviously you would say it isn't.Nullstein said:
This way you could get global coordinates on anything.Demystifier said:
It does not meet the requirements for a global coordinate chart, since an open interval of ##\varphi## either does not cover the entire circle, or covers at least one point more than once. To cover every point on the circle exactly once, you need a half-closed interval of ##\varphi##.Demystifier said:
The usual angle variable ##\varphi## is a coordinate, but you need two charts to cover the circle. That's the standard definition of a coordinate system in differential geometry and there is a good reason to allow only open sets: If you allow non-open sets, the notion of continuity isn't preserved by coordinate changes. If a function on the circle is given (in coordinate representation) by ##f:[0,2\pi]\rightarrow[0,2\pi], f(\varphi)=\frac\varphi 2##, then it seems to be continuous, but it is really discontinuous in a different coordinate system. That would also result in the impossibility to define differentiability. Also, some points may be represented by several coordinates. If you use a non-standard definition, you should mention that, because it may have unnoticed consequences and others generally assume the standard definition.Demystifier said:
No, it isn't. There are compact manifolds that do have everywhere non-vanishing vector fields on them, but still cannot be covered by a single global coordinate chart. The circle, ##S^1##, which I discussed in post #242, is an example.Demystifier said:
I guess my implicit "definition" was that coordinates are "good" everywhere if there are no coordinate singularities in the metric tensor. But of course, a manifold does not need to have a metric tensor at all, so I see my error now. Thank you for explaining why only open sets are allowed, that's much more instructive than quoting a definition which says that it must be so.Nullstein said:
