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AharonovBohm topological explanation 
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#91
Dec512, 05:06 PM

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I have written down (explicitly) an Afield valid for r>0. So this Afield certainly does exist on R³/R. And of course you can (and should) calculate rot A which again is perfectly valid for r>0. What else do you need to be convinced that A does exist ??? 


#92
Dec612, 05:08 AM

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15.9 onwards,However it is not written here that one can eliminate magnetic field,but it is possible to do so.In very simple cases,whenever there is a motion of charge producing magnetic field,one can transform to an inertial frame in which this field is zero and only electric field remains.Still there is a possibilty of vector potential,but since the gauge invariant integral only depends on curl of A,it is zero because of absence of magnetic field.The law given in the reference is directly a consequence of lagrangian of charged particle coupled to electromagnetic field,as you are already aware of it. 


#93
Dec612, 05:14 AM

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#94
Dec612, 08:26 AM

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Sorry, I don't get it. It's not the wrong subforum but perhaps the wrong way asking questions



#95
Dec612, 10:50 AM

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I believe this is the typical case where "you can't have your cake and eat it too".
We have here a potential Afield that we've agreed that is not globally defined in the space of the AB effect, we have to cut out the origin where the solenoid is, leaving a singularity. The only problem being that by removing the origin we eliminate the justification for having the Afield at all, that is the EM field that is switched on in the experiment to obtain the shift in wave function pattern. 


#96
Dec612, 10:54 AM

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#97
Dec612, 11:05 AM

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#98
Dec612, 11:37 AM

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I mean that the EMfield is zero!



#99
Dec612, 02:18 PM

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What has to happen in order to observe the phase shift in the interference pattern of the electrons? Hint: It's got to do with switching something on. So we have a situation for the AB experimental setup in which something is switched off, where the EMfield is zero for the electrons and there's no Afield, and a situation in which something is switched on where the EM field is still zero for the electrons but the Afield is nonzero and produces a phase shift. See? We can relate that nonzero Afield with the switching on of something, I'll let you call that something however you want, but I think it's called an EMfield. The take away point is that the Afield is related with switching it on, fine so far? What I want to underline is that I'm not doubting the effect, I'm only concerned about the usual explanation of it because it seems to mix two incompatible scenarios in a contradictory way, the R^3 scenario and the R^3/R scenario. The situation before the switching on is compatible with both spaces, but the situation after is compatible with the R^3/R space only, however an effect that is only causally compatible with the activation of something that doesn't exist for the R^3/R space is produced. If that is ok for you guys , then fine. I won't bother about it anymore 


#100
Dec612, 05:24 PM

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I wonder why in the quantum physics forum nobody has mentioned the nonlocality of the effect. Probably that's all my quibble amounts to.



#101
Dec612, 05:34 PM

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There are two ways you can look at the experiment: 1) you constructed an apparatus with the solenoid and you can switch the current on and off. In that case you know what you are doing. You can solve the Maxwell equations for the current; you find the nonvanishing EMfield inside the solenoid and the vanishing EMfield but nonvanishing A outside; you can calculate the phase shift of the wave function and you'll find that it agrees with experiment. 2) you haven't constructed the apparatus and you can't switch anything on and off. All there is is an interference pattern. You observe that this pattern deviates from the usual expectation, so it's not symmetric w.r.t. the symmetry axis of the experimental setup. OK, now you may guess that the apparatus has been constructed as described above (1) and that there's a current inside which produces the Afield. Fine. But it could be as well that nothing is inside, except for a singularity, a onedim. line removed from R³  and that due to some unknown reason there is an Afield which is puregauge, locally flat, w/o any EMfield, w/o any energy stored in the Afield etc. w/o looking into the solenoid you can't distinguish between 1) the solenoid with a current and 2) the solenoid wrapping vacuum w/o any current, a onedim. singularity, and a sourceless Afield. The interference patterns are identical. Physically (2) seems to be unacceptable, but as I said: w/o looking into the solenoid and inspecting the apparatus in detail there is no way to distinguish between (1) and (2). This is fine for me: mathematical I can do it either way, and physically I know what the clever guys in the lab have constructed. No problem for me. EDIT: In #22 I wrote "The Afield ... is puregauge locally, but not globally; that's what's measured by the loop integral"; in #77: "in other words A is pure gauge locally i.e. A ~ A' = 0 but not globally"; in #82 you wrote "... this topological nontriviality, which can be expressed as a number , say, is a global topological invariant and so is not expressible by a local formula"; #83: "... F=dA is granted locally but not globally, so F and A may required patching, cutting out singularities etc. In the case of the Afield as described above one has to remove r=0. On this R³ / R the relation F=dA=0 is valid, so #not globally' means 'not on R³ but on R³ / R'". 


#102
Dec612, 06:26 PM

P: 3,060

Anyway in the case 1), when you switch the current on, is the nonzero Afield in R^3/R or in R^3? If the former you have to recur to the magic sourceless Afield that somehow knows when you switch it, if the latter you have the Afield but don't have the vanishing Bfield so no experiment. You seem content with the former, but I don't, I guess that's all. 


#103
Dec612, 07:23 PM

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Refer to #77. 


#104
Dec712, 02:42 AM

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Further if you declare R^3 indistinguishable from R^3/R in the AB experiment, the logic of the explanation of the experiment based on the latter topology is invalidated. As I said you can't have your cake and eat it too, ;) 


#105
Dec712, 03:26 AM

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I still think you understand nothing :(
The solenoid does not "create a singularity"; it's a physical solenoid with a current, nothing special. You don't even use deltafunctions or something like that. Math is perfectly OK, physics is standard, don't worry. The electrons are shielded by some mechanism from the current, so they don't penetrate the solenoid and feel only the Afield (no EMfield, and not directly the current). That's perfectly valid as a "physical" explanation w/o strange math, singularities R³/R etc. And of course I can "declare R^3 indistinguishable from R^3/R in the AB experiment" b/c as I said: a) the solenoid is impenetrable, therefore the electrons can't distinguish between the scenario 1) with the current and the scenario 2) with "vacuum + singularity" b) the Afield the electrons feel is identical in case 1) and 2) The Afield the electrons feel in both cases (it can be calculated exactly !!) which creates the phase shift is the same (!) for 1) and 2) so the math to solve the Schrödinger equation, to calculate the phase shift etc. is identical. In that sense the scenarios cannot be distinguished, neither experimentally nor mathematically (the guy preparing the solenoid can tell you the difference). So a minor correction to what I wrote above is ion order: this is not really a "declaration" but a consequence of the math. Remark: the cake was fine ;) 


#106
Dec712, 03:57 AM

P: 3,060

I don't really want to enter in your "you understand nothing" dynamic. I believe I made my point sufficiently clear so I don't need to make that kind of remarks.
It was clear enough I was referring to the ideal case, which is the one used in the topological explanation of the effect. In that case it is clear the infinite solenoide creates a stringlike singularity. I know the physical case of the experiment is not this ideal case but as long as the solenoid is shieldd for the electrons, the consequence is the same, or else the ideal explanation of the different topology is not valid, doesn't explain anything. With respect to the "declaration of indistinguishability": the math is an instrument of physics, not an end, I know the math of QM doesn't care, that's because of its nonlocality, many physicists feel uneasy about nonlocality, but it is accepted by the community as one of the "weird" things of QM one should not question but accept. But remember you started you participation in this thread saying something like the AB effect is really classical, that just its experimental realization is quantum mechanical. Now this is wrong as long as one considers classical mechanics doesn't include nonlocality, unlike QM. So I guess I accept the indistinguishable argument and the topological explanation within the QM nonlocal frame of mind, but not within the classical frame. No more need to argue about this. Thanks. 


#107
Dec712, 04:18 AM

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[tex]\oint_C A[/tex] is a classical (nonquantized), nonlocal, gauge invariant observable already present in Maxwell's theory of electromagnetism? This entity exists classically, but you don't have any classical measuring device; that's where the quantum mechanical description of the electron enteres the stage. btw.: text books about (algebraic) topology in physics are full of classical nonlocal entities like AharonovBohm phase, Dirac strings, 't Hooft monopoles, solitons, instantons, ... many of them considered in a QM/QFT context but formulated w/o quantizing the gauge field! So it's always the same: nonlocal entities do exist classically, whereas experimentally a qm device is required. And this nonlocality we are discussing here has NOTHING to do with the nonlocality a la EPR! 


#108
Dec712, 04:22 AM

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About the "spooky action at a distance" that shows up in the AB effect it might be relevant to cite here the words of Newton, though he was referring to gravity:"that one body may act upon another at a distance thro' a Vacuum, without the Mediation of any thing else, by and through which their Action and Force may be conveyed from one to another, is to me so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it."
The absurdity was apparently solved in the case of gravity by Einstein's relativity. But I guess in the case of EM after being for a moment substituted by the "field" concept, it returned with a vengeance with QM. Will that be the last word? 


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