Aharonov-Bohm effect

[Anonymous No.10056808]

>To put it differently, global symmetry is a property of a system, but gauge symmetry in general is a property of a description of a system. What we really learn from the centrality of gauge symmetry in modern physics is that physics is described by subtle laws that are “geometrical.” This concept is hard to define, but what it means in practice is that the laws of Nature are subtle in a way that defies efforts to make them explicit without making choices. The difficulty of making these laws explicit in a natural and non-redundant way is the reason for “gauge symmetry.”....This suggests that in a model of Nature complete enough to include both quantum mechanics and gravity, the only true symmetries are gauge symmetries.
[https://arxiv.org/abs/1710.01791]

does anybody find the Aharanov-Bohm effect kind of bizarre? see pic related -- the particles (electrons i guess) don't pass through the solenoid's magnetic field, yet the magnetic field changes the interference pattern. and this applies no matter how huge or small you make your experiment, so it's not that the electrons have some "tail" of their wavefunction in the physical B field. nope, it's that the vector potential is interacting with them. but A can be written in a bunch of different gauges -- it's not uniquely defined

the vector potential and gauge symmetry are pretty crazy bros

[Anonymous No.10056839]

>>10056808
No one can explain this in brainlet terms can they? Not sure I can even wrap my mind around the concept of symmetry itself

[Anonymous No.10056889]

>>10056839
explain what in brainlet terms? symmetry? the aharanov-bohm effect? or what witten is talking about?

the aharanov-bohm effect is pretty much shown by the diagram in OP. symmetry is a pretty elaborate concept but as far as what witten is talking about is basically that global symmetries are akin to how things look the same after transforming them somehow (like e.g. you can rotate a hexagon by 60 degrees and it looks the same as before you rotated it) but gauge symmetries are mathematical changes in the description which don't alter the physics (like e.g. in the picture you can describe the B field by many different but equivalent vector potentials). witten is trying to make sense of why symmetries like parity swaps and charge inversion and isospin rotations are not exact (or "true") symmetries but OTOH gauge symmetries are the shiznit

[Anonymous No.10056920]

>>10056808
quality post

[Anonymous No.10056986]

>>10056839
Basically what the picture is showing is a key difference between qm and cm. If you have a charged particle traveling in a region with zero magnetic field there should be no effect, right? Well that's a classical picture of how things work but in qm things are a bit more subtle, when you introduce the solenoid the magnetic field is only non-zero within the solenoid, so if you have two beams of charged particles being split and then being recombined like in the diagram you expect the usual interference pattern, but weirdly enough you find that it's shifted, why is that? It turns out because the quantum description of nature requires potentials, not fields, so even if the field strength is zero in the region the particle lies, the introduction of some non-zero potential does have an effect on the phase of the particle and thus changes the interference pattern. But one also remembers that potentials introduce the concept of gauge symmetry, so it ends up being completely necessary in quantum mechanics. Seiberg talks a bit more about the same things as Witten in a more accessible manner
https://www.youtube.com/watch?v=eBAXZ2yfJdA

[Anonymous No.10057220]

>>10056808
The AB effect is nothing but the effect of the first cohomology group on the prequantum bundle. Let [math]L \rightarrow M[/math] be a prequantum bundle on the punctured plane [math]M = \mathbb{R}^2\setminus\{0\}[/math] such that [math]\omega = -\frac{1}{r}(dy \wedge dx)[/math] is the symplectic form. The prequantum bundle then acquires the Hermitian conenction 1-form [math]\theta = -\frac{i\hbar}{2\pi r} (xdy- ydx)[/math] which generates the first de Rham cohomology group [math]\frac{2\pi}{\hbar}H^1(M) \cong \mathbb{Z}[/math]. This means that the prequantum bundles [math](L,A)[/math] on [math]M[/math], with [math]A = \theta - i\frac{\hbar}{2\pi}\lambda[/math], are mutually inequivalent for [math]\lambda \in [0,1)[/math]. Each prequantum bundle are then labeled by an integer for which the local sections [math]\psi\in\Gamma(M,L)[/math] satisfy [math]\exp\left(i\frac{2\pi}{\hbar}\otimes_\gamma \theta_m\right) \psi = \exp\left(i2\pi m\right)\psi[/math] by Weil integrality condition.

[Anonymous No.10057243]

>>10057220
sure, buddy, whatever floats your boat

just out of curiosity, where does the solenoid's field enter into your gibberish?

[Anonymous No.10057248]

>>10056808
Everything points to natural laws not being symmetrical at this point. This idea of perfect symmetry has held us back from discovering the true reality, both now and in the past. Before it was particles it was niggers who couldnt come to grip with the fact that celestial orbits are elliptical and not perfectly round.

[Anonymous No.10057272]

>>10057243
The magnetic flux threading through the plane manifests as the puncture since no particle can access its core due to its Lorentz force. In addition, the symplectic potential I gave is exactly the Landau gauge for the magnetic field. You would've known this if you did even the tiniest bit of research; all of this very basic stuff is covered in Wen and Auerbach.
You should've use the 3 brain cells left in your head first before posting, dear.

[Anonymous No.10057298]

>>10057220
>>10057272
pls give me some books to study this.

[Anonymous No.10057313]

>>10057298
I've already mentioned Wen and Auerbach, which should give you decent physical description. Then read Woodhouse for the geometric quantization.

[Anonymous No.10057318]

>>10057313
ok, tnx fren

[Anonymous No.10058946]

bump