Positive chiral current and exponential test-coordinate model — Examples

Assume Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) as defined in The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). Let $u$ be a length coordinate measured in metres, let $\kappa >0$ have units of inverse metres, and let $b>0$ be dimensionless; set $u^{\mathrm{\prime }}=u\mathrm{/}b$ and ${\kappa }^{\mathrm{\prime }}=b\kappa$. Write $\beta =2\pi \mathrm{/}\kappa$ and ${\beta }^{\mathrm{\prime }}=2\pi \mathrm{/}{\kappa }^{\mathrm{\prime }}$ for the corresponding strip widths, measured in metres. Here “generator frequency” means only the angular Fourier-dual parameter: for the convention $\widehat{g}(\omega )=\int g(u)e^{-i\omega u}du$, $\omega$ has units of inverse metres and is not the eigenvalue of a separately asserted operator generator. To express it using an energy parameter, take $E>0$ in joules and a fixed formal $\sigma >0$ in joule-metres, and define $\omega =E\mathrm{/}\sigma$; no physical energy or clock identification is assumed. For a primed parameter $E^{\mathrm{\prime }}>0$ in joules define ${\omega }^{\mathrm{\prime }}=E^{\mathrm{\prime }}\mathrm{/}\sigma$. Matching the phase means ${\omega }^{\mathrm{\prime }}{u}^{\mathrm{\prime }}=\omega u$ for every $u\in \mathbb{R}$, and holds exactly when $E^{\mathrm{\prime }}=bE$, equivalently ${\omega }^{\mathrm{\prime }}=b\omega$. For $g^{\mathrm{\prime }}\in C_c^{\mathrm{\infty }}(\mathbb{R};\mathbb{C})$ define the transformed current by $j^{\mathrm{\prime }}(g^{\mathrm{\prime }})=j_{\kappa }(g^{\mathrm{\prime }}(\cdot \mathrm{/}b))$. If $j_{\kappa }$ is represented by a regular density, changing variables $u=b{u}^{\mathrm{\prime }}$ in this test pairing gives $j^{\mathrm{\prime }}(u^{\mathrm{\prime }})=b{j}_{\kappa }(b{u}^{\mathrm{\prime }})$; the same formula defines the density transformation distributionally. For ${\phi }^{\mathrm{\prime }}\in C_c^{\mathrm{\infty }}(\mathbb{R};\mathbb{C})$, the corresponding two-point test pairing transforms as $⟨C^{\mathrm{\prime }},{\phi }^{\mathrm{\prime }}⟩=⟨C_{\kappa },b{\phi }^{\mathrm{\prime }}(\cdot \mathrm{/}b)⟩$, equivalently $C^{\mathrm{\prime }}=b^2{C}_{\kappa }(b\cdot )$ as a distribution. Thus the covariance retains its chiral form, the spectrum acquires the factor $b$, and ${\beta }^{\mathrm{\prime }}{\omega }^{\mathrm{\prime }}=\beta \omega$, so the detailed-balance exponent is unchanged. This is a coordinate and parameter identity within the mathematical model; no physical clock normalization or temperature assignment is included.

This is a coordinate and parameter identity within the specified one-dimensional model. The inverse-length generator frequency and the strip width are reference-normalized quantities; no invariant physical energy or black-hole temperature is inferred without a separate physical clock and unit assignment.