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Angular Momentum

SpinHelicitySeeds
spin-up\delta_{z}=+1 N^{\mathsf{U}} > N^{\mathsf{D}}
non-rotating\delta_{z}=0 N^{\mathsf{U}} = N^{\mathsf{D}}
spin-down\delta_{z}=-1 N^{\mathsf{U}} < N^{\mathsf{D}}

Consider a particle P, that is described by the coefficients of its rotating seeds N^{\mathsf{U}} and N^{\mathsf{D}}. We say that P has a spin that is defined by these coefficients, as noted in the accompanying table. We also use the helicity, written as

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

to make quantitative descriptions of P’s spatial orientation. And later, if P is also being used as a frame of reference, then N^{\mathsf{U}} and N^{\mathsf{D}} may be used to establish the phase of other particles. So rotating seeds have an important role in describing motion. This task is expanded by considering the coefficients of leptonic seeds, N^{\mathsf{A}}, N^{\mathsf{M}}, N^{\mathsf{E}} and N^{\mathsf{G}}, to define

{\mathrm{J}}_{m} \equiv \delta_{\hat{m}} \dfrac{h}{16\pi}  \sqrt{ \left( N^{\mathsf{A}}-N^{\mathsf{M}} \right)^{2} + 8 \left| N^{\mathsf{A}}-N^{\mathsf{M}}   \right|  \; \vphantom{{\left(X^{\mathsf{X}} \right)^{X}}^{X} }}

{\mathrm{J}}_{e} \equiv \delta_{\hat{e}} \dfrac{h}{16\pi}  \sqrt{ \left( N^{\mathsf{G}}-N^{\mathsf{E}} \right)^{2} + 8 \left| N^{\mathsf{G}}-N^{\mathsf{E}}   \right|  \; \vphantom{{\left(X^{\mathsf{X}} \right)^{X}}^{X} }}

{\mathrm{J}}_{z} \equiv \delta_{z} \dfrac{h}{16\pi}  \sqrt{ \left( N^{\mathsf{U}}-N^{\mathsf{D}} \right)^{2} + 8 \left| N^{\mathsf{U}}-N^{\mathsf{D}}   \right|  \; \vphantom{{\left(X^{\mathsf{X}} \right)^{X}}^{X} }}

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  {\mathrm{\overline{J}}} \equiv \left( {\mathrm{J}}_{m} , \ {\mathrm{J}}_{e} , \ {\mathrm{J}}_{z} \right). Exchanging quarks for anti-quarks does not alter seed counts, so  {\mathrm{\overline{J}}} ( \mathsf{P} ) = {\mathrm{\overline{J}}} ( \mathsf{\overline{P}} ). In general, the components  {\mathrm{J}}_{m},  {\mathrm{J}}_{e} and  {\mathrm{J}}_{z} have non-zero values, and 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  \mathrm{\overline{J}} because N^{\mathsf{A}} = N^{\mathsf{M}} and N^{\mathsf{G}} = N^{\mathsf{E}}. The vector is aligned with the polar-axis, {\mathrm{\overline{J}}} = \left( 0, \ 0, \ {\mathrm{J}}_{z} \right) and so

\left\| \, {\mathrm{\overline{J}}} \, \right\| =  \dfrac{h}{16 \pi}  \sqrt{ \left( N^{\mathsf{U}}-N^{\mathsf{D}} \right)^{2} + 8 \, \left|    N^{\mathsf{U}}-N^{\mathsf{D}}   \right|  \; \vphantom{{\left(X^{{X}} \right)^{X}}^{X} }}

This expression is simplified by defining the total angular momentum quantum number {\textsl{\textsf{J}}} as

{\textsl{\textsf{J}}}  \equiv   \dfrac{ \, \left| \,  N^{\mathsf{U}}   - N^{\mathsf{D}} \, \right| \, }{8}

so that

\left\| \, {\mathrm{\overline{J}}} \, \right\| = \dfrac{h}{\rm{2} \pi} \sqrt{ \, {\textsl{\textsf{J}}} \, \left( {\textsl{\textsf{J}}} + 1 \right) \; \vphantom{X^{X}} }

Total Angular Momentum is Conserved

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

{\mathrm{J}}_{z}   =  \delta_{z}  \dfrac{{\textsl{\textsf{J}}} h}{2\pi} \sqrt{  \,  1 + \dfrac{1}{ {\textsl{\textsf{J}}} \, }    \vphantom{{\dfrac{1}{ X }}^{X}}        }

And if {\textsl{\textsf{J}}} \gg 1 then the radical is approximately one, and

{\mathrm{J}}_{z}    \simeq \,  \delta_{z}   \dfrac{ {\textsl{\textsf{J}}} h}{2\pi}    =  \dfrac{h}{16 \pi}   \delta_{z}    \left| \,  N^{\mathsf{U}}   - N^{\mathsf{D}} \, \right|       =   \dfrac{h }{16 \pi}     \left( N^{\mathsf{U}} - N^{\mathsf{D}} \right)

Similar results obtain for the other axes so that

\overline{{\mathrm{J}}}  \,   \simeq      \dfrac{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)

But seeds are conserved, so the quantity and character of the seeds in a description cannot change. Whenever some generic compound quarks  \mathbb{X},  \mathbb{Y} and  \mathbb{Z} interact, if \mathbb{X} + \mathbb{Y} \leftrightarrow \mathbb{Z} then the coefficients for any sort of seed Z are related as

N^{\mathsf{Z}} \left( \mathbb{X} \right) + N^{\mathsf{Z}} \left( \mathbb{Y} \right) = N^{\mathsf{Z}} \left( \mathbb{Z} \right)

Then by the associative properties of addition, the angular momentum must be approximately conserved too. For macroscopic particles, {\textsl{\textsf{J}}} is huge because h is so small, and the approximation is excellent.

Sensory Interpretation

The internal energy of down quarks is represented by the black background in this icon for achromatic visual sensation.

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 in between, they are greyish. So \delta_{z} indicates if a complex achromatic visual sensation is brighter or darker than some medium grey visual experience. And {\textsl{\textsf{J}}} notes the size of the difference.

The angular momentum is used to describe spatial relationships, somewhat like this ajat basket from Borneo.
Ajat basket, Penan people. Borneo 20th century, 20 (cm) diameter by 35 (cm) height. Photograph by D Dunlop.