Timeline for Why gauge-invariant Berry curvature commutator looks like torsion?
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8 events
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| Nov 21, 2017 at 20:33 | comment | added | Gareth Meredith | (by the way, I am not a mathematician, just someone who is very keen) about physics too. | |
| Nov 21, 2017 at 20:31 | comment | added | Gareth Meredith | I wrote them in commutator language, I picked it up from a tutorial. Never mind, it really isn't important. The last term (would be) completely analogous to a torsion, the complete symmetry of the Berry curvature and then the final commutator matches the full Riemann tensor with non-zero torsion. The similarities are striking. | |
| Nov 18, 2017 at 17:30 | comment | added | ACuriousMind♦ | But the object in the definition of the curvature is not a commutator! The curvature of a connection form $A$ is $F = \mathrm{d}A + A \wedge A$ in component-free notation, which yields $F_{ij} = \partial_i A_j - \partial_j A_i + [A_i,A_j]$ - the last term is a genuine commutator, but the first two are not. Where did you get the impression they are commutators, and why do you think the last term is "torsion"? | |
| Nov 17, 2017 at 19:14 | comment | added | Gareth Meredith | The Berry geometric phase is related to the Berry curvature - I use $$[\partial_i,A_j]$$ because its standard notation for me to write things as commutators - I am asking about the last commutator $$[A_i,A_j]$$ which appears identical to $$[\Gamma_i,\Gamma_j]$$ - where the latter here is the torsion related to the gravitational field equations. If there is a deep connection (like suggested to me in my previous thread) between the Berry curvature and the curvature tensor, then I would like to know why the last object would not have similar meaning, hence a ''Berry Torsion.'' | |
| Nov 17, 2017 at 18:58 | comment | added | Qmechanic♦ | Related post by OP: physics.stackexchange.com/q/369233/2451 | |
| Nov 17, 2017 at 18:56 | history | edited | Qmechanic♦ | CC BY-SA 3.0 |
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| Nov 17, 2017 at 18:53 | comment | added | ACuriousMind♦ | 1. Your notation is idiosyncratic - why do you write $[\partial_i, A_j]$ instead of just $\partial_i A_j$? 2. Are you familiar with the more general concept of (Yang-Mills) gauge theory? 3. The notion of torsion needs soldered bundles, since the Berry curvature is not on a tangent bundle, it has no solder form. What do you mean by "geometric Berry torsion"? | |
| Nov 17, 2017 at 18:49 | history | asked | Gareth Meredith | CC BY-SA 3.0 |