- #1

- 96

- 1

## Main Question or Discussion Point

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?

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?