Spinors

Prerequisites: Lorentz Group, SU(2). Full path (11 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?
• FAQ

Intuitive

A spinor is a mathematical object similar to a 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 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°.

This crazy property can be illustrated as shown, for example, here:

• Dirac's belt trick for spin 1/2 particle
• Dirac | Greg Egan
• Dirac String Trick
• Plate trick - Wikipedia

Concrete

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-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 $e^-_R$).

• A nice introduction is An introduction to spinors by Andrew M. Steane
• 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 care of:

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.page 99 in Student Friendly Quantum Field Theory, by R. Klauber

Reference 3 is Winter, Rolf G., Quantum Physics, Wadsworth (1979), Chap. 9.

• No-Nonsense Quantum Field Theory by Jakob Schwichtenberg, section "Scalars, Vectors, Tensors, and Spinors"

Abstract

Spinors arise as mathematical objects when we study the 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.

"spinor representations are the square root of a principle fiber bundle”

https://particlephd.wordpress.com/2008/11/28/why-spinors/

• Sir Michael Atiyah, What is a Spinor ?
• The standard reference for spinors is Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry

Why is it interesting?

Spinors are the appropriate mathematical objects to describe particles with spin 1/2, like, for example, electrons.

"One could say that a spinor is the most basic sort of mathematical object that can be Lorentz-transformed."

An introduction to spinors by Andrew M. Steane

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. Sir Michael Atiyah

FAQ

Path to this page

1. Calculus
2. Linear Algebra
3. Group Theory
4. Newton's Second Law
5. Vector Calculus
6. Lie Groups
7. Newtonian Mechanics
8. SO(3)
9. Special Relativity
10. Lorentz Group
11. SU(2)

Leads to: Chirality, Dirac Equation, Spinor-Helicity Formalism.