Timeline for What's the space of eigenvalues/field configurations for a fermion?
Current License: CC BY-SA 4.0
34 events
| when toggle format | what | by | license | comment | |
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| S Jul 15, 2022 at 9:30 | history | bounty ended | Quantumwhisp | ||
| S Jul 15, 2022 at 9:30 | history | notice removed | Quantumwhisp | ||
| Jul 9, 2022 at 9:30 | comment | added | Blind Miner | Have you read Appendix A.2 of Polchinski's String Theory, Volume 1? It addresses fermionic field configurations and path integrals, but I don't know if it meets your standard of rigor. Maybe give it a try. | |
| Jul 8, 2022 at 13:05 | answer | added | Qmechanic♦ | timeline score: 3 | |
| Jul 8, 2022 at 8:25 | comment | added | Qmechanic♦ | Links to abstract pages for Jackiw 89, Symanzik 81 and Hatfield 92? Which pages? | |
| S Jul 8, 2022 at 8:13 | history | bounty started | Quantumwhisp | ||
| S Jul 8, 2022 at 8:13 | history | notice added | Quantumwhisp | Draw attention | |
| Aug 24, 2019 at 5:25 | history | edited | alexchandel | CC BY-SA 4.0 |
sign error, lowercase fields
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| Aug 21, 2019 at 5:02 | history | edited | alexchandel | CC BY-SA 4.0 |
add ref
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| Aug 17, 2019 at 3:00 | history | tweeted | twitter.com/StackPhysics/status/1162559765958090752 | ||
| Aug 14, 2019 at 6:21 | history | edited | alexchandel | CC BY-SA 4.0 |
cite Jackiw for fermionic wave-functional
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| Aug 1, 2019 at 15:37 | history | edited | alexchandel | CC BY-SA 4.0 |
more examples
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| Aug 1, 2019 at 8:44 | history | edited | Qmechanic♦ |
edited tags; edited tags
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| Aug 1, 2019 at 8:43 | comment | added | alexchandel | WP, PlanetMath, Wolfram, & nLab and their sources all disagree with you. Jackiw literally defines a fermionic wave-functional (as does Symanzik). And saying "$\theta_{x,0}$ takes no values" is as vacuous as saying "the Euclidean basis vector $\mathbf{e}_1$ takes no values;" no one ever suggested otherwise. | |
| Aug 1, 2019 at 8:14 | history | edited | alexchandel | CC BY-SA 4.0 |
organize
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| Aug 1, 2019 at 7:57 | history | edited | alexchandel | CC BY-SA 4.0 |
cite Jackiw for fermionic wave-functional
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| Aug 1, 2019 at 6:31 | history | edited | alexchandel | CC BY-SA 4.0 |
minor
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| Aug 1, 2019 at 6:19 | history | edited | alexchandel | CC BY-SA 4.0 |
reword
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| Aug 1, 2019 at 0:37 | comment | added | octonion | I said Grassmann 'numbers' have no values. Something like $\theta_{x,0}$ appears in integrals much like a bosonic $\phi_x$ would. The difference is $\phi_x$ will take different values depending on the state so a functional makes sense. $\theta_{x,0}$ is already a member of the Grassmann algebra, it does not take new values. This is my last reply, good luck with your painstaking rigor | |
| Jul 31, 2019 at 9:02 | comment | added | alexchandel | Is $\chi(\mathbf x)$ a field configuration, as in $\mathrm X(\mathbf x)$? I'd need my question answered to say. | |
| Jul 31, 2019 at 9:01 | comment | added | alexchandel | $\chi(\mathbf x)=e^{i \mathbf{k}\cdot\mathbf{x}} \theta_{\mathbf x}$ isn't in $\mathbb{R}^3\to\mathbb C$; it's in $\mathbb{R}^3\to\Lambda(\mathbb{C}^{\mathbb R^3})$, specifically $\mathbb R^3\to\Lambda_1(\mathbb{C}^{\mathbb R^3})$. There are many other possibilities, e.g. $\chi:\mathbf{R}^3\to\Lambda(\mathbb{C}^{2\mathbb R^3})$ where $\chi(\mathbf x)=e^{i \mathbf{k}\cdot\mathbf{x}} \theta_{\mathbf x,0} + e^{-i \mathbf{k}\cdot\mathbf{x}} \theta_{\mathbf x,1}$. Painstaking rigor is needed in part because of all these superstitions about Grassmanns, like "Grassmann algebras have no values." | |
| Jul 31, 2019 at 8:15 | history | edited | alexchandel | CC BY-SA 4.0 |
extend notation for the struggling
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| Jul 31, 2019 at 7:35 | history | edited | alexchandel | CC BY-SA 4.0 |
note emphasizing question
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| Jul 30, 2019 at 22:57 | comment | added | octonion | Is your $\chi$ ever a field configuration in the sense that it is like an eigenvalue associated with a state? You essentially turned the field configurations into a map from $R^3\rightarrow C$ by multiplying each $\theta_x$ by a complex number. But by using only $\theta_x$ at each $x$ I think you're starting to see what I'm getting at. Please think about it a little, sorry if I was a little harsh in my comment. | |
| Jul 30, 2019 at 22:30 | comment | added | alexchandel | Using the above notation, a simple counterexample to the notion there are no distinct functions yielding Grassmanns is $\chi:\mathbf{R}^3\to\Lambda(\mathbb{C}^{\mathbb R^3})$ where $\chi(\mathbf x)=e^{i \mathbf{k}\cdot\mathbf{x}} \theta_{\mathbf x}$ with $\mathbf{k}\in\mathbb R^3$, but there are infinitely more. | |
| Jul 30, 2019 at 22:28 | comment | added | alexchandel | I might be, I mirror the definitions that Wolfram, PlanetMath, and WP (and its sources) use to define the Grassmann algebra, with values $z\in\Lambda(V)$. Are they all wrong? | |
| Jul 30, 2019 at 22:04 | history | edited | alexchandel | CC BY-SA 4.0 |
cleanup
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| Jul 30, 2019 at 21:58 | comment | added | octonion | I suspect you are the one confusing something. Back up a little. Take a single component of a fermion field at a given point of spacetime and define its eigenvalue as Grassmann number. You can form an exterior algebra with other components and other points of spacetime, but that eigenvalue itself doesn't take distinct values. This is not like the bosonic case. How are you going to define a wave functional like this? | |
| Jul 30, 2019 at 21:54 | comment | added | alexchandel | Nope. Grassmann "numbers" are elements of a complex exterior algebra, and take values. You might be confusing them with their generators. | |
| Jul 30, 2019 at 21:48 | history | edited | alexchandel | CC BY-SA 4.0 |
cleanup
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| Jul 30, 2019 at 21:47 | comment | added | octonion | I think you're on the wrong track. Grassmann 'numbers' don't take values, so there is no way to define the distinct field configurations that would appear in a wave functional. | |
| Jul 30, 2019 at 21:44 | history | edited | alexchandel | CC BY-SA 4.0 |
shorten title, cleanup
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| Jul 30, 2019 at 21:28 | history | edited | alexchandel | CC BY-SA 4.0 |
shorten title
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| Jul 30, 2019 at 19:41 | history | asked | alexchandel | CC BY-SA 4.0 |