The Euclidean Metric

This article extends an earlier, more general discussion of metrics that considered a particle's shape as described by its radius vector $\, \overline{\rho}$. Initially, the radius was established from a particle's quark content, and a vector space $\mathbb{S}$ was defined from sets of particles and their radii. $\mathbb{S}$ was characterised using statistical averages $\tilde{\rho}$, standard deviations $\delta \rho$ and correlation coefficients noted by $\chi$. These quantities have already been assessed for generic compound quarks.

Now let us consider that $\mathbb{S}$ might be filled with a more restricted set of particles that have special attributes. Specifically we examine collections of atoms where a radius vector can be expressed in Cartesian coordinates. Characterize atomic aggregates using their orbital radius $R$, a wavelength $\lambda$ and $\theta$ the phase angle. Definition: we say that a metric is Euclidean if almost every particle in $\mathbb{S}$ satisfies the following conditions.

1. Atoms and molecules must be fully three dimensional like cylinders or rods so that $R \ne 0$ and $\lambda \ne 0$. Wavelengths and radii are all presumably positive, not nil.
2. Particles must exhibit some variation in their shape so that $\delta \rho \ne 0$. The standard deviation for any component of the radius is presumably positive.
3. $\mathbb{S}$ must be well-stirred. We require spatial homogeneity at an atomic level so that $\delta \rho _{x} = \delta \rho_{y} = \delta \rho_{z}$.
4. The particles in $\mathbb{S}$ cannot all be extremely low temperature, or all resonating like the atoms in a laser. We assume that $\delta \theta \ne 0$ so there must be some variation among atomic phase angles.
5. Finally, atoms must interact incoherently so that the variation among phase angles is random. Then we can presume that large sums over odd powers of circular functions of $\theta$ add-up to zero.
For a specific example of a Euclidean space consider a big collection of atoms, and larger particles that are composite atoms, noted by $\mathbf{P}$1, $\mathbf{P}$2, $\mathbf{P}$3$\, \mathbf{P}$N. Let these particles be described using the rotini model where atomic shapes are like a corkscrew spiral. They approximate a geometric curve called a so that atomic radii can be mathematically written as

$\rho_{x} = R \cos{\! 2 \theta}$and$\rho_{y} = R \sin{\! 2 \theta }$and\begin{align} \rho_{z} = \frac{ \lambda \theta}{2 \pi} \end{align}

The radius vector is expressed in Cartesian coordinates as $\, \overline{\rho} = \rho_{x} \hat{x} + \rho_{y} \hat{y} + \rho_{ z} \hat{z}$ so a space like $\mathbb{S}$ can be generically represented as

$\mathbb{S} = \left\{ \overline{\rho}^{ 1 }, \ \overline{\rho}^{2}, \ \overline{\rho}^{3} \ldots \ \overline{\rho}^{\, i} \ldots \ \overline{\rho}^{\, N} \right\}$

The average radii of the particles in $\mathbb{S}$ are given by

\begin{align} \tilde{\rho}_{x} = \frac{R}{ N} \sum_{i=1}^{N} \cos{ 2 \theta^{i} } \end{align} and \begin{align} \tilde{\rho}_{y} = \frac{R}{ N} \sum_{i=1}^{N} \sin{ 2 \theta^{i} } \end{align} and \begin{align} \tilde{\rho}_{z} = \frac{ \lambda \tilde{\theta} }{2 \pi } \end{align}

As noted above, we presume that the atoms in $\mathbb{S}$ have phase angles that are completely incoherent so that sums over the circular functions add-up to zero. Then

\begin{align} \tilde{\rho}_{x} \approx 0 \end{align} and \begin{align} \tilde{\rho}_{y} \approx 0 \end{align}

and the variations in particle radii are

\delta \rho_{x} = \sqrt{ \frac{1}{N} \sum_{i=1}^{N} \left( \rho_{x}^{i} \right)^{2} \; } = R \sqrt{ \frac{1}{N} \sum_{i=1}^{N} \left( \cos{2 \theta^{i} } \right)^{2} \; }\begin{align} \; \approx \frac{R}{2} \end{align} \delta \rho_{y} = \sqrt{ \frac{1}{N} \sum_{i=1}^{N} \left( \rho_{y}^{i} \right)^{2} \; } = R \sqrt{ \frac{1}{N} \sum_{i=1}^{N} \left( \sin{2 \theta^{i} } \right)^{2} \; }\begin{align} \; \approx \frac{R}{2} \end{align}

$\delta \rho_{z} \equiv \sqrt{ \frac{1}{N} \sum_{i=1}^{N} \left( \rho_{z}^{i} - \tilde{\rho}_{z} \right)^{2} \; } = \frac{ \lambda }{2 \pi } \sqrt{ \frac{1}{N} \sum_{i=1}^{N} \left( \theta^{i} - \tilde{\theta} \right)^{2} \; }$\begin{align} \; = \delta\theta \frac{ \lambda }{2 \pi } \end{align}

We presume that all the particles in $\mathbb{S}$ are three-dimensional so that $R \ne 0$ and $\lambda \ne 0$. And as noted above, we also assume that phase angles vary. So $\delta \theta \ne 0$ and the standard deviations in particle radii must be positive

$\delta \rho_{x} \ne 0$and$\delta \rho_{y} \ne 0$and$\delta \rho_{z} \ne 0$

We also presume that $\mathbb{S}$ has spatial homogeneity at the atomic-level. Then any variations cannot depend on direction, and so we require that

$\delta \rho _{x} = \delta \rho_{y} = \delta \rho_{z}$

But by definition, correlation coefficients and standard deviations are related as $\; \chi _{\alpha \alpha} = \delta \rho _{\alpha}$ so that

$\chi _{xx} = \chi_{yy} = \chi _{zz} > 0$

The other correlation coefficients are given by

$\chi _{x y} = \sqrt{ \frac{1}{N} \sum_{i=1}^{N} \rho_{x}^{i} \rho_{y}^{i} \; } = R \sqrt{ \frac{1}{N} \sum_{i=1}^{N} \left( \cos{ 2 \theta^{i} } \right) \left( \sin{ 2 \theta^{i} } \right) \ } \approx 0$

$\chi _{x z} = \sqrt{ \frac{1}{N} \sum_{i=1}^{N} \rho_{x}^{i} \left( \rho_{z}^{i} - \tilde{\rho}_{z} \right) \ } =\sqrt{\frac{R}{N} \sum_{i=1}^{N}\cos{2\theta^{i}} \left( \rho_{z}^{i} - \tilde{\rho}_{z} \right) \ } \approx 0$

$\chi _{y z} = \sqrt{ \frac{1}{N} \sum_{i=1}^{N} \rho_{y}^{i} \left( \rho_{z}^{i} - \tilde{\rho}_{z} \right) \ } =\sqrt{\frac{R}{N} \sum_{i=1}^{N}\sin{\,2\theta^{i}} \left( \rho_{z}^{i} - \tilde{\rho}_{z} \right) \ } \approx 0$

 The Euclidean Metric
 $k_{zz} \equiv 1$ $k_{xz} = 0$ $k_{xx} = 1$ $k_{xy} = 0$ $k_{yy} = 1$ $k_{yz} = 0$

But these coefficients must all be nil because, as noted above, we presume that large sums over odd powers of circular functions of $\theta$ add-up to zero. Then recall that the metric of $\mathbb{S}$ is given by

\begin{align} k_{\alpha \beta} \equiv \frac{ \chi_{\alpha \beta} ^{ \mathbb{S} } }{ \chi_{zz} ^{ \mathbb{S} } } \end{align}

where $\alpha$ and $\beta \in \{ x, y, z \}$. So overall

$k_{xx} = k_{yy}= k_{zz} = 1$ and $k_{xy} = k_{xz} = k_{yz} = 0$

and we say that the metric of $\mathbb{S}$ is Euclidean.