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The Norm

Consider an ordered-set of three numbers  \rho_{m},  \rho_{e} and  \rho_{z} that constitute an algebraic vector written as \overline{\rho} \equiv \left( \, \rho_{m}, \rho_{e}, \rho_{z} \right). These numbers can be compressed into a single number called the norm of   \overline{\rho}. Evaluating the norm depends on several more numbers that are called components of a metric. The numerical values of these metric components are established by the context of a calculation, they are often implicit.

A complete mathematical discussion of metrics and norms can be quite extended, so for EthnoPhysics we focus on the specific case where  \rho_{m}, \rho_{e} and  \rho_{z} are the three radii used to describe the shape of a particle. These radii are determined by counting quarks, so to assess their norm we use components of the quark metric, written as  k_{\alpha \beta}. First we define a directed surface area by

    \begin{equation*} \begin{split}  \widehat{A} & \equiv  k_{mm} \rho_{m}^{2} +  k_{ee} \rho_{e}^{2} +  k_{zz} \rho_{z}^{2}  \\  & \hspace{20px} + 2 k_{em}  \rho_{m}  \rho_{e} + 2k_{mz}\rho_{m}  \rho_{z} \\ & \hspace{40px} + 2 k_{ez}\rho_{e}  \rho_{z} \end{split} \end{equation*}

In general, radii may be positive or negative, so  \widehat{A} may be positive or negative too. If \widehat{A}>0 we say that the surface of P is outside facing. And if \widehat{A}<0, the surface is facing inside. The norm of   \overline{\rho} is defined as

\left\| \,   \overline{\rho} \,  \right\|  \equiv \sqrt{\widehat{A} \; \rule[-3px]{0px}{15px} }

This number may be imaginary1The term imaginary is used here with its mathematical meaning . if P’s surface is facing inward. Note that particles and anti-particles have opposing radius vectors, \overline{\rho} ( \mathsf{P} ) = - \overline{\rho} ( \mathsf{\overline{P}} ). And all radii appear as paired factors in the expression for  \widehat{A}. So both vectors have the same norm, and we write

\left\|  \rule[-3px]{0px}{12px} \,   \overline{\rho} \,  \right\|^{\mathsf{P}} =  \left\| \,  \rule[-3px]{0px}{12px} \overline{\rho} \,  \right\|^{\mathsf{\overline{P}}}

The norm is used to organize spatial relationships, somewhat like this ajat basket from Borneo.
Ajat basket, Penan people. Borneo 20th century, 23 (cm) diameter by 36 (cm) height. Hornbill motif. Photograph by D Dunlop.

The Inner Product

Here is another useful way to distill two radius vectors into a single number. Let us call the vectors \overline{a} = \left( \, a_{m}, a_{e}, a_{z} \right) and \overline{b} = \left( \, b_{m}, b_{e}, b_{z} \right). Then the inner product is defined by

    \begin{equation*} \begin{split}  \overline{a} \cdot \overline{b}  & =  k_{mm} a_{m} b_{m} + k_{ee} a_{e} b_{e} + k_{zz} a_{z} b_{z}  \\  & \hspace{20px} + 2 k_{em} a_{m} b_{e} + 2k_{mz} a_{m} b_{z} \\ & \hspace{40px} + 2 k_{ez} a_{e} b_{z} \end{split} \end{equation*}

We say that \overline{a} and \overline{b} are orthogonal if \overline{a} \cdot \overline{b} = 0.

References
1The term imaginary is used here with its mathematical meaning .