#### Hans de Vries

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A spin 1/2 particle has total angular momentum √3/2. At most 1/2 points in any given direction, so its spin is most assuredlynotin some definite direction.

Indeed, the socalled "spin-direction" is actually the axis of precession of the total spin, where the total spin is given by ##s=s^x+s^y+s^z## as shown in the figure below for a single spinor ##\xi##

The three spin vectors ##s^x##, ##s^y## and ##s^z## can be calculated with:

$$ \begin{matrix}

s^z~~=&\{&\xi^\dagger\sigma^x\xi~&\xi^\dagger\sigma^y\xi~&\xi^\dagger\sigma^z\xi&\} \\

s^x~~=&\{&\xi^\dagger \sigma^2\sigma^x\xi^*&\xi^\dagger \sigma^2\sigma^y\xi^*&\xi^\dagger \sigma^2\sigma^z\xi^*&\}\\

s^y~~=&\{&\xi^\dagger i\sigma^2\sigma^x\xi^*&\xi^\dagger i\sigma^2\sigma^y\xi^*&\xi^\dagger i\sigma^2\sigma^z\xi^*&\}

\end{matrix}$$

The three spin vectors ##s^x##, ##s^y## and ##s^z## can also be considered as defining the

*local reference frame*of the spinor which can be oriented in any arbitrary direction. The expressions above are quite horrible and nontransparent. They can be written in a very elegant way using the Euler Parameter representation:

$$ \begin{matrix}

s^x~~=&\{&\xi^\intercal\sigma^x_x\,\xi~&\xi^\intercal\sigma^y_x\,\xi~&\xi^\intercal\sigma^z_x\,\xi&\} \\

s^y~~=&\{&\xi^\intercal\sigma^x_y\,\xi~&\xi^\intercal\sigma^y_y\,\xi~&\xi^\intercal\sigma^z_y\,\xi&\} \\

s^z~~=&\{&\xi^\intercal\sigma^x_z\,\xi~&\xi^\intercal\sigma^y_z\,\xi~&\xi^\intercal\sigma^z_z\,\xi&\}

\end{matrix}$$

Spinors are much better and much easier understood in the older, real valued Euler Parameter representation. Many essential spinor properties are hopelessly lost and obscured in the complex Cayley-Klein representation that Pauli adopted and which is generally the only representation that physicist are familiar with. As a result (quantum) physicist have an incomplete understanding of spinors.

The (real) Euler parameter representation of a

**SPINOR**is obtained as below in such a way that 3 of the 4 elements can be directly associated with ##x##, ##y## and ##z##. This already makes interpretation much simpler.

$$\left(\begin{array}{r} a +ib \\ c+id \end{array}\right) ~\longrightarrow~ \left(\begin{array}{r} a \\ b \\ c \\ d \end{array}\right) ~\longrightarrow~ \left(\begin{array}{r} u \\ x \\ y \\ z \end{array}\right) $$

The Euler parameter representation of a

**PAULI MATRIX**and any other 2x2 complex matrix is obtained by replacing each complex element by a real valued 2x2 matrix.

$$a+ib ~\longrightarrow~ \left(\begin{array}{rr} a & -b \\ b & a \end{array}\right) ~~~~~~~~~~ * ~\longrightarrow~ \left(\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right)$$

Note that the complex conjugate operator becomes simply another real valued matrix further simplifying interpretations. Group theoretically: SU(2) combined with ##*## gives us SO(4):

The three ##\mathsf{j}## and the three ##\mathsf{i}## form the matrix representation of the left-acting quaternions and the right-acting quaternions respectively.

Note that the colored rectangles correspond with the three SO(3) generators for x, y and z. Simplifying things again because the ##\mathsf{j_x,j_y,j_z}## are in fact just the three spinor rotation generators!

The standard Pauli matrices are ##\sigma^1##, ##\sigma^2## and ##\sigma^3##. The indices x, y and z in the ##(\mathsf{j_x,j_y,j_z})##, the ##(\mathsf{i_x,i_y,i_z})## and ##(\sigma^x,\sigma^y,\sigma^z)## are chosen so that they correspond with the SO(3) rotation matrices.

With this we now get a very elegant definition of the spinor vectors combining the above matrices into 9 different Pauli matrices using ##\sigma^a_b=\mathsf{i_b\,j^a}## where ##\xi^\intercal## is a row-vector, the transpose of ##\xi##.

$$ \begin{matrix}

s^x~~=&\{&\xi^\intercal\sigma^x_x\,\xi~&\xi^\intercal\sigma^y_x\,\xi~&\xi^\intercal\sigma^z_x\,\xi&\} \\

s^y~~=&\{&\xi^\intercal\sigma^x_y\,\xi~&\xi^\intercal\sigma^y_y\,\xi~&\xi^\intercal\sigma^z_y\,\xi&\} \\

s^z~~=&\{&\xi^\intercal\sigma^x_z\,\xi~&\xi^\intercal\sigma^y_z\,\xi~&\xi^\intercal\sigma^z_z\,\xi&\}

\end{matrix}$$

But it gets even better. All nine expressions above can be calculated with a single(!) matrix multiplication. The following very important formula to calculate the spinor's local coordinate system is completely absent from the (quantum) physics textbooks.

For an arbitrary spinor (u,x,y,z) we can calculate its local coordinate system with the following matrix multiplication:

$$\check{\xi}\,\hat{\xi} ~=~ (\xi\cdot\mathsf{i})(\mathsf{j}\cdot\xi)~=~\big|\xi \big|^2\!\!\left(\begin{smallmatrix}1~&0~&0~&0~\\0~&\mathsf{X}^{^x}&\mathsf{X}^{^y}&\mathsf{X}^{^z} \\0~&\mathsf{Y}^{^x}&\mathsf{Y}^{^y}&\mathsf{Y}^{^z} \\0~&\mathsf{Z}^{^x}&\mathsf{Z}^{^y}&\mathsf{Z}^{^z}\end{smallmatrix}\right)$$

Here ##\mathsf{X,Y,Z}## are the normalized versions of the three spin-vectors ##s^x,s^y,s^z##. Together they also form a rotation matrix that defines the orientation of the spinor. The spinor operator matrices ##\hat{\xi}## and ##\check{\xi}## are written out as:

$$\begin{array}{c}\hat{\xi} ~~=~~ (\xi\cdot\mathsf{j})~~=~~(u\mathsf{j}_o+x\mathsf{j}_x+y\mathsf{j}_y+z\mathsf{j}_z) ~~=~~\left(\begin{smallmatrix} ~u & ~x & ~y & ~z \\\!\!-x & ~u &\!\!-z & ~y \\\!\!-y & ~z & ~u &\!\!-x \\\!\!-z &\!\!-y & ~x & ~u\end{smallmatrix}\right)\\ \\\check{\xi} ~~=~~(\xi\cdot\mathsf{i})~~=~~(u\mathsf{i}_o+x\mathsf{i}_x+y\mathsf{i}_y+z\mathsf{i}_z) ~~=~~\left(\begin{smallmatrix} ~u &\!\!-x &\!\!-y &\!\!-z \\ ~x & ~u &\!\!-z & ~y \\ ~y & ~z & ~u &\!\!-x \\ ~z &\!\!-y & ~x & ~u\end{smallmatrix}\right)\end{array}$$

Although the matrix multiplication is mine, the result is exactly identical to the Euler-Rodrigues formula

Now ##(\xi^\intercal\sigma^y_x\,\xi)## is the y-component of ##s^x##. There is a very simple geometric explanation for this: The Pauli matrix ##\sigma^y_x## acting on ##\xi## performs a 180##^o## rotation of the spinor around the y-axis and a -180##^o## rotation of the spinor around its own ##\mathsf{X}##-axis. We will explain this further on.

- If the ##\mathsf{X}##-axis of the spinor is parallel to the y-axis then ##~~~~~~~\xi^\intercal\sigma^y_x\,\xi~=~~~\xi^\intercal\xi~=~~~|\xi|^2##

- If the ##\mathsf{X}##-axis of the spinor is anti-parallel to the y-axis then ##\xi^\intercal\sigma^y_x\,\xi~=-\xi^\intercal\xi~=-|\xi|^2##

- If the ##\mathsf{X}##-axis of the spinor is orthogonal to the y-axis then ##~\xi^\intercal\sigma^y_x\,\xi~=~~0##

Now we come to the remarkable fact that the ##(\mathsf{j_x,j_y,j_z})## rotate the spinor around the coordinate axis while the ##(\mathsf{i_x,i_y,i_z})## rotate the spinor in its own local reference frame. There is a relation with the fact that one is left-acting and the other is right-acting. Consider the matrix shown below which corresponds with the (transposed) rotation matrix.

$$\left(\begin{matrix}1~&0~&0~&0~\\0~&\mathsf{X}^{^x}&\mathsf{X}^{^y}&\mathsf{X}^{^z} \\0~&\mathsf{Y}^{^x}&\mathsf{Y}^{^y}&\mathsf{Y}^{^z} \\0~&\mathsf{Z}^{^x}&\mathsf{Z}^{^y}&\mathsf{Z}^{^z}\end{matrix}\right)$$

Acting from one way rotates the columns of the matrix which corresponds with a rotation in the (world-) coordinate system ##x,y,z## while operating from the other direction rotates the rows which corresponds with a rotation in the spinors own reference frame based on the unit-vectors ##\mathsf{X,Y,Z}##.

See the sections 1.2, 1.3 and 1.4 of chapter 1 and the whole of chapter 2 here:

To see how QED (1 boson field, 1 fermion generation) can be perfectly extended to the full Electroweak Theory (extra boson triplet and 3 fermion generations) using the Euler parameter representation see here: https://thephysicsquest.blogspot.com/

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