advanced_tools:spinors
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advanced_tools:spinors [2018/03/30 14:49] jakobadmin [Researcher] |
advanced_tools:spinors [2019/02/09 11:15] 129.13.36.189 [Why is it interesting?] |
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| ====== Spinors ====== | ====== Spinors ====== | ||
| - | <tabbox Why is it interesting?> | ||
| - | Spinors are the appropriate mathematical objects to describe particles with [[basic_notions:spin|spin]] 1/2, like, for example, electrons. | ||
| - | |||
| - | <blockquote> | ||
| - | "One could say that a spinor is the most basic sort of mathematical object that can be Lorentz-transformed." | ||
| - | |||
| - | <cite>[[https://arxiv.org/abs/1312.3824|An introduction to spinors]] by Andrew M. Steane</cite> | ||
| - | </blockquote> | ||
| - | <tabbox Layman> | + | <tabbox Intuitive> |
| - | A spinor is a mathematical object similar to a [[basic_tools:vector_calculus|vector]]. However, while a vector points in some spatial direction, like, for example, in the direction of the north pole, a spinor points in a direction in an [[advanced_tools:internal_symmetries|internal space]]. | + | A spinor is a mathematical object similar to a [[basic_tools:vector_calculus|vector]]. However, while a vector points in some spatial direction, like, for example, in the direction of the north pole, a spinor points in a direction in an [[advanced_tools:internal_symmetry|internal space]]. |
| A curious property of a spinor is that if you rotate it by 360° it isn't the same but get's a minus sign. Only after a rotation by 720° a spinor is again the same. In contrast a vector is completely unchanged if you rotate it by 360°. | A curious property of a spinor is that if you rotate it by 360° it isn't the same but get's a minus sign. Only after a rotation by 720° a spinor is again the same. In contrast a vector is completely unchanged if you rotate it by 360°. | ||
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| * https://en.wikipedia.org/wiki/Plate_trick | * https://en.wikipedia.org/wiki/Plate_trick | ||
| | | ||
| - | <tabbox Student> | + | <tabbox Concrete> |
| A Dirac spinor field $\Psi$ and its conjugate $\overline\Psi$ | A Dirac spinor field $\Psi$ and its conjugate $\overline\Psi$ | ||
| are equivalent to two left-handed Weyl spinors $\chi$ and $\tilde\chi$ | are equivalent to two left-handed Weyl spinors $\chi$ and $\tilde\chi$ | ||
| and their right-handed conjugates $\chi^\dagger$ and $\tilde\chi^\dagger$; | and their right-handed conjugates $\chi^\dagger$ and $\tilde\chi^\dagger$; | ||
| $\chi$ and $\chi^\dagger$ describe the left-[[advanced_notions:chirality|chiral]] fermion and the right-chiral | $\chi$ and $\chi^\dagger$ describe the left-[[advanced_notions:chirality|chiral]] fermion and the right-chiral | ||
| - | antifermion (\eg\ $e^-_L$ and $e^+_R$), | + | antifermion (e.g. \ $e^-_L$ and $e^+_R$), |
| while $\tilde\chi$ and $\tilde\chi^\dagger$ describe | while $\tilde\chi$ and $\tilde\chi^\dagger$ describe | ||
| - | the left-chiral antifermion and the right-chiral fermion (\eg\ $e^+_L$ and $e^-_R$). | + | the left-chiral antifermion and the right-chiral fermion (e.g. $e^+_L$ and $e^-_R$). |
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| * A nice introduction is [[https://arxiv.org/abs/1312.3824|An introduction to spinors]] by Andrew M. Steane | * A nice introduction is [[https://arxiv.org/abs/1312.3824|An introduction to spinors]] by Andrew M. Steane | ||
| + | * [[http://www.weylmann.com/spinor.pdf|A Child’s Guide to Spinors]] by William O. Straub | ||
| * See also http://www-personal.umich.edu/~williams/notes/spinor.pdf | * See also http://www-personal.umich.edu/~williams/notes/spinor.pdf | ||
| * https://users.physics.ox.ac.uk/~Steane/teaching/rel_C_spinors.pdf | * https://users.physics.ox.ac.uk/~Steane/teaching/rel_C_spinors.pdf | ||
| * https://physics.stackexchange.com/questions/74682/introduction-to-spinors-in-physics-and-their-relation-to-representations/112041#112041 | * https://physics.stackexchange.com/questions/74682/introduction-to-spinors-in-physics-and-their-relation-to-representations/112041#112041 | ||
| + | ---- | ||
| - | **Things to take note of:** | + | **Things to take care of:** |
| <blockquote>Representing the u, as vectors is a heuristic oversimplification though, and in fact is not really correct, as operations like spinor addition work a little differently than vector addition. (See Winter 3.) However, temporarily visualizing them as such can aid in our understanding of how they and spin behave, relative to the at-rest coordinate system, for varying particle velocities.<cite>page 99 in [[ftp://srdconsulting.com/USB_BackUp/Data/Articles/QFT/StudentFriendlyQFT/SF_QFT_Chap04.pdf|Student Friendly Quantum Field Theory]], by R. Klauber</cite></blockquote> | <blockquote>Representing the u, as vectors is a heuristic oversimplification though, and in fact is not really correct, as operations like spinor addition work a little differently than vector addition. (See Winter 3.) However, temporarily visualizing them as such can aid in our understanding of how they and spin behave, relative to the at-rest coordinate system, for varying particle velocities.<cite>page 99 in [[ftp://srdconsulting.com/USB_BackUp/Data/Articles/QFT/StudentFriendlyQFT/SF_QFT_Chap04.pdf|Student Friendly Quantum Field Theory]], by R. Klauber</cite></blockquote> | ||
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| - | <tabbox Researcher> | + | <tabbox Abstract> |
| Spinors arise as mathematical objects when we study the [[advanced_tools:group_theory:representation_theory|representations]] of the Lorentz group. | Spinors arise as mathematical objects when we study the [[advanced_tools:group_theory:representation_theory|representations]] of the Lorentz group. | ||
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| + | <tabbox Why is it interesting?> | ||
| + | Spinors are the appropriate mathematical objects to describe particles with [[basic_notions:spin|spin]] 1/2, like, for example, electrons. | ||
| + | <blockquote> | ||
| + | "One could say that a spinor is the most basic sort of mathematical object that can be Lorentz-transformed." | ||
| - | | + | <cite>[[https://arxiv.org/abs/1312.3824|An introduction to spinors]] by Andrew M. Steane</cite> |
| - | <tabbox Examples> | + | </blockquote> |
| - | --> Example1# | + | <blockquote>No one fully understands spinors. Their algebra is formally understood, but their geometrical significance is mysterious. |
| + | In some sense they describe the ‘‘square root’’ of geometry and, just as understanding the concept of $\sqrt{-1}$ took centuries, | ||
| + | the same might be true of spinors. <cite>Sir Michael Atiyah</cite></blockquote> | ||
| - | |||
| - | <-- | ||
| - | --> Example2:# | + | |
| - | + | ||
| - | + | ||
| - | <-- | + | |
| <tabbox FAQ> | <tabbox FAQ> | ||
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| <-- | <-- | ||
| | | ||
| - | <tabbox History> | ||
| </tabbox> | </tabbox> | ||
advanced_tools/spinors.txt · Last modified: 2022/09/07 21:52 by 147.92.69.196
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