A Note on the Measure of Vector and Pythagorean Theorem

Abstract

Why the square? We present a geometry-axiom-free derivation of the Pythagorean theorem and the square at its core, establishing their algebraic origin from within the bare vector-space framework. Such concepts as the (right) angle, rotation, inner product, orthogonality etc also emerge as a logical construct rather than taken as given. They are necessitated by the square, and the ensuing theory, in turn, $\text{canonically}$ stems from a $\text{single}$ definitional primitive $-$ the ($\mathbb R^{\vcenter{\hbox{$\scriptscriptstyle+$}}}\!$-quantitative) invariant $\mathcal{Q}$-measure of a vector. This provides the core of an algebraic justification for Euclidean geometry. Equally important, these findings account (also canonically) for the complex modulus-squared $p=|\mathfrak{a}{|}^2$ $-$ the quantum Born rule $-$ and point out what is even admissible for being quantitatively interpreted. The linear structure and its endomorphisms are rigid in the sense that the $\text{well-defined}$ interpretable turns out to be, up to gauge $\mathcal Q {\,\to\,} \mathrm{const} {\,\vcenter{\hbox{$\scriptstyle\times$}}\,} \mathcal Q$, the unique gauge-invariant measure $\mathcal{Q}=||\cdot \cdot \cdot |{|}^2$; independently of the field $\mathbb{R}$ or $\mathbb{C}$.

LaTeX 6 pages, no figures (v2: minor improvements of text)