Timeline for Mathematical explanation for connections on gauge bundles in curved spacetime for spinors
Current License: CC BY-SA 4.0
17 events
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| Jul 17 at 21:04 | comment | added | Malkoun | And of course, when I say that $\omega$ is a connection, I am making an abuse of language. Strictrly speaking it is $\partial_\mu + \omega_{\mu a b} \sigma^{ab}$ which is a connection. A connection is a way to differentiate fields along curves. The difference between two connections on the same bundle is a $1$-form with values in the adjoint bundle. | |
| Jul 17 at 21:00 | comment | added | Malkoun | The Levi-Civita connection is a kind of canonical connection associated to a metric. It is defined to be metric-compatible and torsion-free. The so-called fundamental theorem of Riemannian geometry says that there is a unique such connection associated to any given metric. Regarding the reference you are using, I am guessing that the $\omega$ is either the spin connection of the metric $g$ or the Levi-Civita connection of $g$ with respect to a tetrad. I will have to read the reference to understand things better. I think $\omega$ is probably a connection which is canonically attached to $g$. | |
| Jul 17 at 20:25 | vote | accept | trying | ||
| Jul 17 at 20:24 | comment | added | trying | Thanks. Notation is much more clear to me now. The exact factors I can work out by myself. Just one last thing i didnt ask in the original post, is this connection Levi-civita? | |
| Jul 17 at 18:58 | history | edited | Malkoun | CC BY-SA 4.0 |
added 654 characters in body
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| Jul 17 at 18:48 | comment | added | Malkoun | Regarding the $\sigma^{ab} = [\gamma^a, \gamma^b]$, for $1 \leq a < b \leq 4$, those form a basis for the Lie algebra of $O(1, 3)$, but note for each $(a, b)$ with $a < b$, $\sigma^{ab}$ is a $4$ by $4$ matrix, so it has two additional spacetime indices, which are hidden from the notation. | |
| Jul 17 at 18:24 | comment | added | Malkoun | Regarding the Dirac operator vs covariant derivative, well, the Dirac operator is $i$ times the Gamma matrices contracted with the covariant derivative. It does not have any remaining free index, unlike the covariant derivative, and is usually denoted by $D$ with a slash through it. | |
| Jul 17 at 16:49 | comment | added | trying | third, is the spin connection $\omega_{\mu a b}$ Levi-Civita, in the sense that it is metric compatible (check, that is how we get tetrad postulate) and torsion free, it is antisymmetric in a,b index. So i don't know. Also, thank you so much for taking time to clarify my doubts. | |
| Jul 17 at 16:34 | comment | added | trying | second, is the last term $iA^{j}_{\mu k} \phi^{\mu A k}$ correct? It does not match the index of LHS. If $A_{\mu}$ is the gauge connection, from my understanding it can only have two indices one for spacetime, one for bundle index. What is this 3 index object supposed to represent? unless it is supposed to represent the gauge group commutator but It is represented by $f^{a}_{bc}$ | |
| Jul 17 at 16:23 | comment | added | trying | I am more confused now, first, isnt the total Dirac operator($D_{\mu}$) in curved manifold is$\partial_{\mu} + \omega_{\mu ab} \sigma^{ab}+ iA_{\mu}$ where $\sigma^{ab} = [\gamma^{a}, \gamma^{b}]$ upto some constant? Why is $\sigma^{ab}$ missing in your example? | |
| Jul 15 at 18:17 | history | edited | Malkoun | CC BY-SA 4.0 |
added an example for how to differentiate composite objects
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| Jul 15 at 17:59 | comment | added | Malkoun | The way it works is as follows. Let us say you have an object $\phi$ with many different types of indices. To differentiate it, you use the usual partial derivatives first, then use the corresponding algebraic part for each index. I will edit my post and try to provide an example. | |
| Jul 15 at 17:57 | comment | added | Malkoun | The $\gamma^\mu \partial_\mu$ is (up to a factor of $i$) the Dirac operator. Please distinguish between all of these. | |
| Jul 15 at 17:54 | comment | added | Malkoun | Hmm, there are really many connections appearing in your setting. We often use the same letter, such as $D$, to denote various different connections. Please distinguish between them. To differentiate a spinor, you use the spin connection, to differentiate a tensor, you use the Levi-Civita connection, to differentiate a section of a vector bundle, associated to a gauge group $G$, you use a gauge connection. We may use the same letter $D$ in all 3 cases, but they are really different connections/objects. | |
| Jul 15 at 15:15 | vote | accept | trying | ||
| Jul 17 at 20:25 | |||||
| Jul 15 at 15:15 | comment | added | trying | thanks, I understand. So $D A_{\nu}$ for gauge field will have $D =\gamma^{\mu}( \partial_{\mu} + \omega_{\mu a b} \sigma^{ab} + i A_{\mu})$. | |
| Jul 9 at 23:32 | history | answered | Malkoun | CC BY-SA 4.0 |