Spinors [The Physics Travel Guide]

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--- advanced_tools:spinors [2017/12/17 12:44]
jakobadmin [Layman]
+++ advanced_tools:spinors [2022/09/07 21:52] (current)
147.92.69.196 [FAQ]
@@ Line 1: / Line 1: @@
 ====== Spinors ======
-<tabbox Why is it interesting?>
-Spinors are the appropriate mathematical objects to describe particles with spin 1/2, like, for example, electrons.
-<blockquote>
+<tabbox Intuitive>
-"One could say that a spinor is the most basic sort of mathematical object that can be Lorentz-transformed."
+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]].
-<cite>[[https://arxiv.org/abs/1312.3824|An introduction to spinors]] by Andrew M. Steane</cite>
-</blockquote>
-<tabbox Layman>
-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 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°.
@@ Line 24: / Line 16: @@
   * https://en.wikipedia.org/wiki/Plate_trick
-<tabbox Student>
+<tabbox Concrete>
-Spinors arise as mathematical objects when one studies the [[advanced_tools:group_theory:representation_theory|representations]] of the Lorentz group.
+A Dirac spinor field $\Psi$ and its conjugate $\overline\Psi$
+are equivalent to two left-handed Weyl spinors $\chi$ and $\tilde\chi$
+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
+antifermion (e.g. \ $e^-_L$ and $e^+_R$),
+while $\tilde\chi$ and $\tilde\chi^\dagger$ describe
+the left-chiral antifermion and the right-chiral fermion (e.g.  $e^+_L$ and $e^-_R$).
-The objects that transform under the $(\frac{1}{2},0)$ or $(0,\frac{1}{2})$ representation of the Lorentz group are called **Weyl spinors**, objects transform under the (reducible) $(\frac{1}{2},0) \oplus (0,\frac{1}{2})$ representation are called **Dirac spinors**.
 ----
   * 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
   * 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
+----
-**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>
 Reference 3 is  Winter, Rolf G., Quantum Physics, Wadsworth (1979), Chap. 9.
-<tabbox Researcher>
+<tabbox Abstract>
+Spinors arise as mathematical objects when we study the [[advanced_tools:group_theory:representation_theory|representations]] of the Lorentz group.
+The objects that transform under the $(\frac{1}{2},0)$ or $(0,\frac{1}{2})$ representation of the Lorentz group are called **Weyl spinors**, objects transform under the (reducible) $(\frac{1}{2},0) \oplus (0,\frac{1}{2})$ representation are called **Dirac spinors**.
+<blockquote>"spinor representations are the square root of a principle [[advanced_tools:fiber_bundles|fiber bundle]]”
+https://particlephd.wordpress.com/2008/11/28/why-spinors/</blockquote>
+----
   * [[https://www.youtube.com/watch?v=SBdW978Ii_E|Sir Michael Atiyah, What is a Spinor ?]]
@@ Line 51: / Line 66: @@
+<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>
@@ Line 74: / Line 90: @@
 <cite>Student Friendly Quantum Field Theory by Klauber</cite></blockquote>
+ * [[https://geocalc.clas.asu.edu/pdf/SPINORPM.pdf| Spinor Particle Mechanics]] by David Hestenes
 <--
-<tabbox History>
 </tabbox>

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