Understanding Spinor Rotations

In summary, the conversation discusses a confusion about the rotation of a spinor and how one component changes sign while the other remains unchanged. The conclusion is that this is correct, as shown by the equation using the rotation matrix and the fact that the exponential of iπ is equal to -1.
  • #1
96
1
Hi,

I am confused on a very basic fact. I can write [itex]\xi = (\xi_{1}, \xi_{2}) [/itex] and a spin rotation matrix as

[tex]
U =
\left( \begin{array}{ccc}
e^{-\frac{i}{2}\phi} & 0 \\
0 & e^{\frac{i}{2}\phi}
\end{array} \right)
[/tex]

A spinor rotates under a [itex]2\pi[/itex] rotation as

[tex]
\xi ' =
\left( \begin{array}{ccc}
e^{-i\pi} & 0 \\
0 & e^{i\pi}
\end{array} \right)
\left( \begin{array}{c}
\xi_{1} \\
\xi_{2}
\end{array} \right)
=
\left( \begin{array}{ccc}
-\xi_{1} \\
\xi_{2}
\end{array} \right)
[/tex]

which is [itex](-\xi_{1}, \xi_{2})[/itex], and not [itex]-\xi[/itex], so only one component changes sign. Is this correct?
 
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  • #2
How do you come to this conclusion? Since ##\exp(\mathrm{i} \pi)=\exp(-\mathrm{i} \pi)=-1## your rotation by ##2 \pi## leads to ##\hat{U} \xi=-\xi## as it should be.
 
  • #3
Ahah, ok, I should not try to do physics today.
 

What is a spinor rotation?

A spinor rotation is a mathematical concept used to describe the transformation of a spinor, which is a mathematical object representing the inherent quantum mechanical properties of particles. It is a type of rotation in a complex space that includes both rotation and reflection, and is commonly used in quantum mechanics and particle physics.

Why are spinor rotations important?

Spinor rotations are important because they allow us to mathematically describe the behavior of particles with spin in quantum mechanics. They also play a crucial role in understanding the symmetries and transformations of physical systems, as well as in the development of new theories and models.

How do spinor rotations differ from regular rotations?

Unlike regular rotations, which only involve rotation around an axis, spinor rotations involve both rotation and reflection. This is due to the complex nature of spinors, which have both real and imaginary components. Additionally, spinor rotations are non-commutative, meaning that the order in which rotations are performed can affect the result.

What are some real-world applications of spinor rotations?

Spinor rotations have various applications in both theoretical and experimental physics. They are used in quantum computing, where spin qubits are manipulated using spinor rotations. They are also used in particle physics to describe the behavior of subatomic particles, and in condensed matter physics to study properties of materials with spin.

How are spinor rotations represented mathematically?

Spinor rotations are represented using mathematical objects known as spinors, which are typically represented by matrices or arrays of complex numbers. These matrices can be multiplied together to combine multiple spinor rotations, and the resulting matrix represents the overall spinor rotation. The specific form of these matrices depends on the type of spinor being rotated and the type of rotation being performed.

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