Total Angular Momentum
 Particle Type Definition a spin-up particle $N^{ \mathsf{U}} > N^{ \mathsf{D}}$ a non-rotating particle $N^{ \mathsf{U}} = N^{ \mathsf{D}}$ a spin-down particle $N^{ \mathsf{U}} < N^{ \mathsf{D}}$

Consider a particle P, that is characterized by the coefficients of its rotating seeds $\; \it{N}^{\, \mathsf{U}}$ and $\it{N}^{\, \mathsf{D}}$. We say that P has a spin that depends on these coefficients, as noted in the accompanying table. We also define a number $\delta_{z}$, called the helicity, by

$\delta _{z} \equiv \begin{cases} +1 \; \; &\sf{\text{if}} \; \; &N^{\mathsf{U}} \; > \; N^{\mathsf{D}} \\ \; \; 0 \; \; &\sf{\text{if}} \; \; &N^{\mathsf{U}} \; = \, N^{\mathsf{D}} \\ -1 \; \; &\sf{\text{if}} \; \; &N^{\mathsf{U}} \; < \; N^{\mathsf{D}} \end{cases}$

which is used for making quantitative descriptions of P's And later, if P is also being used as a then $\; \it{N}^{\, \mathsf{U}}$ and $\it{N}^{\, \mathsf{D}}$ may be used to establish the phase of other particles. So rotating seeds have primary role in describing motion. This task is expanded by considering the coefficients of $\, \it{N}^{\, \mathsf{A}}$, $\, \it{N}^{\, \mathsf{M}}$, $\, \it{N}^{\, \mathsf{G}}$ and $\, \it{N}^{\, \mathsf{E}}$, to define

\begin{align} {\rm{J}}_{m} \equiv \delta _{\hat{m}} \frac{h}{\mathrm{16} \pi} \sqrt{ \left( N^{\mathsf{A}}-N^{\mathsf{M}} \right)^{2} + 8 \, \left| N^{\mathsf{A}}-N^{\mathsf{M}} \right| \; \vphantom{{\left(N^{\mathsf{D}} \right)^{2}}^{3} }} \end{align}

\begin{align} {\rm{J}}_{e} \equiv \delta _{\hat{e}} \frac{h}{\mathrm{16} \pi} \sqrt{ \left( N^{\mathsf{G}}-N^{\mathsf{E}} \right)^{2} + 8 \, \left| N^{\mathsf{G}}-N^{\mathsf{E}} \right| \; \vphantom{{\left(N^{\mathsf{D}} \right)^{2}}^{3} }} \end{align}

\begin{align} {\rm{J}}_{z} \equiv \delta_{z} \frac{h}{\mathrm{16} \pi} \sqrt{ \left( N^{\mathsf{U}}-N^{\mathsf{D}} \right)^{2} + 8 \, \left| N^{\mathsf{U}}-N^{\mathsf{D}} \right| \; \vphantom{{\left(N^{\mathsf{D}} \right)^{2}}^{3} }} \end{align}

where $\delta _{\hat{e}}$ is the electric polarity and $\delta _{\hat{m}}$ is the magnetic polarity. Then we specify the total angular momentum vector as \begin{align} {\rm{\overline{J}}} \equiv \left( {\rm{J}}_{m} , \ {\rm{J}}_{e} , \ {\rm{J}}_{z} \right) \end{align}. Exchanging quarks for anti-quarks does not alter seed counts, so ${\rm{\overline{J}}} ( \sf{P} ) = {\rm{\overline{J}}} ( \sf{\overline{P}} )$. In general, the components ${\rm{J}}_{m}$, ${\rm{J}}_{e}$ and ${\rm{J}}_{z}$ have non-zero values, and a P's motion is complicated. But for a solitary, undivided particle that is not electrically or magnetically polarized we may construct a framework where P is centered on the electric and magnetic axes. Then it is easy to assess the norm of ${\rm{\overline{J}}}$ because $N^{\mathsf{A}} = N^{\mathsf{M}}$ and $N^{\mathsf{G}} = N^{\mathsf{E}}$. The vector is aligned with the polar-axis, $\, {\rm{\overline{J}}} = \left( 0, \ 0, \ {\rm{J}}_{z} \right)$ and

\begin{align} {\large{\parallel}} \, {\rm{\overline{J}}} \, {\large{\parallel}} = \frac{h}{\mathrm{16} \pi} \sqrt{ \left( N^{\mathsf{U}}-N^{\mathsf{D}} \right)^{2} + 8 \, \left| N^{\mathsf{U}}-N^{\mathsf{D}} \right| \; \vphantom{{\left(N^{\mathsf{D}} \right)^{2}}^{3} }} \end{align}

This expression is simplified by defining the total angular momentum quantum number as

 \begin{align} 𝘑 \equiv \frac{ \, \left| \, N^{\mathsf{U}} - N^{\mathsf{D}} \, \right| \, }{8} \end{align} so that \begin{align} {\large{\parallel}} \, {\rm{\overline{J}}} \, {\large{\parallel}} = \frac{h}{\rm{2} \pi} \sqrt{ \, 𝘑 \, \left( 𝘑 + \rm{ 1} \right) \; \vphantom{1^{2}} } \end{align}

## Conservation of Angular Momentum

If $𝘑 \ne 0$ then the $z$-component of the angular momentum vector can be expressed in terms of $𝘑$ as

\begin{align} {\rm{J}}_{z} = \delta_{z} \frac{𝘑 h}{\rm{2} \pi} \sqrt{ \, 1 + \frac{1}{ 𝘑 \, } \vphantom{{\frac{1}{ 𝘑 \, }}^9} \; } \end{align}

And if $𝘑 \gg 1$ then the radical is approximately one, and

\begin{align} {\rm{J}}_{z} \simeq \, \delta_{z} \frac{𝘑 h}{\rm{2} \pi} = \frac{h}{16 \pi} \delta_{z} \left| \, N^{\mathsf{U}} - N^{\mathsf{D}} \, \right| = \frac{h }{16 \pi} \left( N^{\mathsf{U}} - N^{\mathsf{D}} \right) \end{align}

Similar results obtain for the other axes so that

\begin{align} \overline{{\rm{J}}} \, \simeq \frac{h }{16 \pi} \left( N^{\mathsf{A}} - N^{\mathsf{M}}, \ \ N^{\mathsf{G}} - N^{\mathsf{E}}, \ \ N^{\mathsf{U}} - N^{\mathsf{D}} \vphantom{W^{W^{W}}} \right) \end{align}

But seeds are indestructible. When particles are formed or decomposed, the quantity and character of the seeds in a description cannot change. All conservation laws are logically based on this principle. And as just shown, the total angular momentum vector is approximately given by differences in seed counts. So by the of subtraction, the angular momentum must be approximately conserved too. For macroscopic particles, $𝘑$ is huge because $h$ is so small, and the approximation is excellent.
Sensory interpretation: Rotating seeds are objectified from achromatic visual sensations, so for spin-up particles, white sensations outnumber black sensations. Collectively they are bright. For spin-down particles, black sensations are more numerous than white sensations, they look dark. Non-rotating particles are objectified from colourless sensations. So $\delta_{z}$ indicates if a complex achromatic visual sensation is brighter or darker than some medium grey visual experience. And $𝘑$ notes the size of the difference.

 Summary
 Adjective Definition Total Angular Momentum Quantum Number \begin{align} 𝘑 \equiv \frac{ \, {\large{\mid}} \, N^{\mathsf{U}} \! -N^{\mathsf{D}} {\large{\mid}} \, }{8} \end{align} 4-11
 Noun Definition Spin-Up Particles $N^{ \mathsf{U}} > N^{ \mathsf{D}}$ 5-30
 Noun Definition Spin-Down Particles $N^{ \mathsf{U}} < N^{ \mathsf{D}}$ 5-32
 Noun Definition Non-Rotating Particles $N^{ \mathsf{U}} = N^{ \mathsf{D}}$ 5-31
 Noun Definition Total Angular Momentum Vector \begin{align} &\ \ \ \ \ \ \ \ \ \ \ \ {\rm{\overline{J}}} \equiv \left( {\rm{J}}_{m} , \ {\rm{J}}_{e} , \ {\rm{J}}_{z} \right) \ \ \ \ \ \ \sf{\text{where}} \\ \ \\ \ \\ {\rm{J}}_{m} &\equiv \delta _{\hat{m}} \frac{h}{\mathrm{16} \pi} \sqrt{ \left( N^{\mathsf{A}}-N^{\mathsf{M}} \right)^{2} + 8 \, {\large{\mid}} N^{\mathsf{A}}-N^{\mathsf{M}} {\large{\mid}} \; } \\ \ \\ \ \\ {\rm{J}}_{e} &\equiv \delta _{\hat{e}} \frac{h}{\mathrm{16} \pi} \sqrt{ \left( N^{\mathsf{G}}-N^{\mathsf{E}} \right)^{2} + 8 \, {\large{\mid}} N^{\mathsf{G}}-N^{\mathsf{E}} {\large{\mid}} \; } \\ \ \\ \ \\ {\rm{J}}_{z} &\equiv \delta _{z} \frac{h}{\mathrm{16} \pi} \sqrt{ \left( N^{\mathsf{U}}-N^{\mathsf{D}} \right)^{2} + 8 \, {\large{\mid}} N^{\mathsf{U}}-N^{\mathsf{D}} {\large{\mid}} \; } \end{align} 5-40
page revision: 453, last edited: 01 Aug 2022 21:18