Timeline for Why not regard all large gauge transformations as genuine ones?
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25 events
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| Apr 11, 2019 at 15:35 | comment | added | gj255 | @knzhou Not sure I have enough expertise to concretely say that it is OK to treat large gauge transformations as genuine ones... I don't understand the answers of ACuriousMind or David Bar Moshe, for instance... What I will say is that I know this picture must change when massless fermions are coupled. The axial anomaly now ensures that different $|n\rangle$ vacua are physically different. Thus the anomaly has the effect of changing the configuration space to some covering space thereof. | |
| Apr 11, 2019 at 10:42 | comment | added | knzhou | @gj255 I would appreciate if you made this an answer! | |
| Apr 9, 2019 at 17:28 | comment | added | gj255 | Just for reference, Rubakov also mentions in his book (p277) that identifying the different $|n\rangle$ vacua is an equally valid approach to the problem of instantons and $\theta$-vacua. He credits this idea to Manton (1983), who personally told me one should identify these $|n\rangle$ vacua in pure Yang-Mills. This is also the approach found in Shifman's book (p177), who espouses the idea that the potential through which to tunnel is really defined on a circle, not periodically on a line. | |
| Dec 21, 2018 at 17:09 | history | edited | knzhou | CC BY-SA 4.0 |
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| Jul 31, 2018 at 14:42 | comment | added | David Bar Moshe | Take a trivial configuration $A=0$ (zero instanton number) act on it with a large gauge transformation $U$, you obtain $A = U^{-1} dU$ with a different instanton number. | |
| Jul 31, 2018 at 14:24 | comment | added | knzhou | @DavidBarMoshe I still think we're just using different nomenclature. The way I learned it (from E. Weinberg), large gauge transformations do not change the instanton number; you cannot gauge away an instanton. On the other hand they do change the winding number of vacua (the spatial slices the instantons tunnel between), and I'm proposing to identify all the vacua as the same state. | |
| Jul 31, 2018 at 14:12 | comment | added | David Bar Moshe | I don't think that I will be able to convince you but for the sake of precision the above gauge transformation is indeed a fiber preserving bundle automorphism: It acts by a $U(1)$ transformation on the $U(1)$ fibers and doesn't mix fibers. It is indeed discontinuous but why is this different from the singular (large) gauge transformations in the instanton theory. In my understanding both are large gauge transformations. | |
| Jul 31, 2018 at 13:42 | comment | added | knzhou | Another argument: the Aharahov-Bohm effect relates a phase to a magnetic flux. Gauge transformations, large or small, can never change the magnetic flux, so they can't change the Aharahov-Bohm phase. | |
| Jul 31, 2018 at 13:41 | comment | added | knzhou | @DavidBarMoshe I define a gauge transformation to be a fiber-preserving bundle automorphism $P \to P$, and a large gauge transformation to be one not connected smoothly to the identity. Your proposed gauge transformation is not a map to begin with, since it's not single valued. | |
| Jul 31, 2018 at 12:46 | comment | added | David Bar Moshe | A large gauge transformation is an element of the gauge group that cannot be connected smoothly to the identity. The above transformation is such. It has the form of a gauge transformation, but you propose not calling it a large gauge transformation, then what is it ? It cannot be a small gauge transformation because it changes the physics. | |
| Jul 31, 2018 at 11:04 | comment | added | knzhou | @DavidBarMoshe I wouldn't call what you propose a large gauge transformation. My impression is that large gauge transformations would have $\phi(2 \pi) = \phi(0)$ but with the map $\phi(\theta)$ not homotopic to the identity, as it can nontrivial winding. Such maps only change the Aharanov-Bohm phase by multiples of $2\pi$, so there's no problem. | |
| Jul 31, 2018 at 10:59 | comment | added | David Bar Moshe | On $S^1$ outside the solenoid is $A = \frac{\Phi}{2 \pi r} d\theta$. If large gauge transformations were allowed what can prevent us from gauging out this field $A \rightarrow A + d\psi = 0$ with $\psi = -\frac{\phi}{2 \pi r} \theta$. This is a large gauge transformation because the gauge function $e^{i\psi}$ is discontinuous on $S^1$, thus cannot obtained by smooth deformation of the trivial gauge transformation. On the other hand there is no constraint on applying small gauge transformations for which $\psi(2 \pi) = \psi(0)$. In this case no change in the interference pattern exists. | |
| Jul 31, 2018 at 10:15 | comment | added | knzhou | @DavidBarMoshe I don't see why that would be. In the bundle picture ($U(1)$ bundle over, say, $S^1$) I thought the Aharanov-Bohm phase was built into the bundle's transition functions, and hence cannot be changed by any gauge transformation, large or small. | |
| Jul 31, 2018 at 8:19 | comment | added | David Bar Moshe | If you treat large gauge transformations as redundencies you gauge out the whole Aharonov-Bohm effect. | |
| Jul 30, 2018 at 22:19 | comment | added | Ryan Thorngren | Thanks I think I understand now and I agree with you. I guess there is a physical consequence of these theta vacua it must be very subtle, along the lines of the electromagnetic memory effect. | |
| Jul 30, 2018 at 22:12 | comment | added | knzhou | @RyanThorngren I think we're having a mixup between vacua (on $\mathbb{R}^3$, possibly compactified to $S^3$) and instantons (on $\mathbb{R}^4$, possibly compactified to $S^4$). In the standard picture, it is impossible for any gauge transformation, large or small, to change the instanton number, but large gauge transformations can go between the vacua. In Weinberg's picture, the vacua are all identified as one state, but instantons are still distinguished by their instanton number. (In either case, the vacua are always trivial fiber bundles, as you said.) | |
| Jul 30, 2018 at 21:42 | comment | added | Ryan Thorngren | Either you study transformations on $R^4$ which wind at infinity or ones with a singularity at the one-point-compactification. Either way, there is no gauge transformation that takes you between instanton sectors. And now I'm confused by what you say because there are no nontrivial $SU(n)$ bundles on $S^3$. | |
| Jul 30, 2018 at 21:01 | history | tweeted | twitter.com/StackPhysics/status/1024037267990560770 | ||
| Jul 30, 2018 at 17:30 | answer | added | ACuriousMind♦ | timeline score: 2 | |
| Jul 30, 2018 at 17:30 | history | edited | AccidentalFourierTransform | CC BY-SA 4.0 |
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| Jul 30, 2018 at 17:23 | history | edited | Qmechanic♦ |
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| Jul 30, 2018 at 17:22 | history | edited | knzhou | CC BY-SA 4.0 |
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| Jul 30, 2018 at 17:21 | comment | added | knzhou | @RyanThorngren I think that's an orthogonal issue; the large gauge transformations I'm considering here are constant at infinity, they're just not continuously connected to the identity. Everything I said was really implicitly on compactified space, where you still get many vacua if you don't mod out by large gauge transformations, and a unique vacuum if you do. | |
| Jul 30, 2018 at 17:15 | comment | added | Ryan Thorngren | I don't know the history, but here is my perspective. One can only change the instanton number by a local transformation which winds around $G$ at infinity. When quantizing theories on noncompact manifolds, we require that gauge transformation parameters tend to constant (or flat in the higher symmetry case) parameters. Invariance under these gauge transformations ensures that the charge coming in equals the charge coming out. If you enforce gauge invariance under local transformations at infinity, there won't be any way to compute the $S$-matrix between charged states. | |
| Jul 30, 2018 at 16:51 | history | asked | knzhou | CC BY-SA 4.0 |