Dimensions, SI unit conventions and quantity types of the framework

(a) Dimensions. Write $\mathrm{\ell }$, $\tau$, $\mu$, $\vartheta$ for the four independent dimension symbols of length, duration, mass and electric charge. A dimension is an element of ${\mathbb{Q}}^4$, written ${\mathrm{\ell }}^a{\tau }^b{\mu }^c{\vartheta }^d$ with $a,b,c,d\in \mathbb{Q}$; products and inverses of dimensions are componentwise addition and negation, and the dimensionless dimension ${\mathrm{\ell }}^0{\tau }^0{\mu }^0{\vartheta }^0$ is abbreviated $1$. The conventions assign to each quantity of the framework a dimension $[X]$, subject to $[XY]=[X][Y]$, $[X\mathrm{/}Y]=[X][Y{]}^{-1}$ and $[dX\mathrm{/}du]=[X][u{]}^{-1}$, with sums meaningful only between quantities of equal dimension, in which case the sum carries that dimension. For a rational power $q$, dimension powers mean $[X{]}^q={\mathrm{\ell }}^{qa}{\tau }^{qb}{\mu }^{qc}{\vartheta }^{qd}$ when $[X]={\mathrm{\ell }}^a{\tau }^b{\mu }^c{\vartheta }^d$. Rational exponents include the half-integer wave-function dimensions in clause (d); the integer-exponent dimensions form the subgroup ${\mathbb{Z}}^4$. These are consistency requirements on the conventions, not laws of nature.

For extensions using temperature, adjoin an independent dimension symbol $\mathrm{\Theta }$: the enlarged group is ${\mathbb{Q}}^4\times \mathbb{Q}$, written ${\mathrm{\ell }}^a{\tau }^b{\mu }^c{\vartheta }^d{\mathrm{\Theta }}^e$, with the same componentwise rules. The preceding group embeds by $e=0$, so all previously assigned dimensions are preserved. Temperature has dimension $\mathrm{\Theta }$, the Boltzmann constant has dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}{\mathrm{\Theta }}^{-1}$, and entropy in joules per kelvin has that same dimension; a dimensionless entropy remains dimensionless. This convention does not assign a temperature or an entropy to any system.

(b) Reference units. The metre (m), second (s), kilogram (kg) and coulomb (C) are the reference units of length, duration, mass and charge; a quantity of dimension ${\mathrm{\ell }}^a{\tau }^b{\mu }^c{\vartheta }^d$ is expressed in the unit ${\mathrm{m}}^a{\mathrm{s}}^b\mathrm{k}{g}^c{\mathrm{C}}^d$. Coordinates of an inertial frame are read with the time coordinate $t$ in seconds, $[t]=\tau$, and the spatial coordinates $x_i$ in metres, $[x_i]=\mathrm{\ell }$; slice distances of the arena are read in metres. In the temperature extension the reference unit of $\mathrm{\Theta }$ is the kelvin (K), so the general reference unit gains a factor ${\mathrm{K}}^e$.

(c) Defining constants of the current SI. The framework keeps $\mathrm{\hslash }$ explicit. In the current SI the constants $h$, $c$ and $e$ are exact defining constants, with the values $h=\mathrm{6,626}07015\times {10}^{-34}$ J s, $c=\mathrm{2,99792458}\times {10}^8$ m s${}^{-1}$ and $e=\mathrm{1,602}176634\times {10}^{-19}$ C, and $\mathrm{\hslash }:=h\mathrm{/}(2\pi )$. These are real scalar constants with dimensions and units $[h]=[\mathrm{\hslash }]=\mu {\mathrm{\ell }}^2{\tau }^{-1}$ (J s), $[c]=\mathrm{\ell }{\tau }^{-1}$ (m s${}^{-1}$), and $[e]=\vartheta$ (C). The vacuum permeability ${\mu }_0$ is a positive real scalar that is experimentally determined in the current SI, with $[{\mu }_0]=\mu \mathrm{\ell }{\vartheta }^{-2}$ (N A${}^{-2}$ = H m${}^{-1}$); ${\epsilon }_0:=({\mu }_0{c}^2{)}^{-1}$ is a positive real scalar with $[{\epsilon }_0]={\mu }^{-1}{\mathrm{\ell }}^{-3}{\tau }^2{\vartheta }^2$ (F m${}^{-1}$). These are unit-convention statements of the SI, not measurements reported by this framework; $c$ enters the framework only through the separate radiation assumptions adopted on their own pages, not in the nonrelativistic spacetime of The Newtonian spacetime arena of the framework.

(d) Dimensions and types of the framework's quantities. The dimension map is fixed on the following quantities.

1. State quantities: a position wave function of a system with $d$ spatial dimensions and $r$ internal components is an almost-everywhere equivalence class in $L^2({\mathbb{R}}^d;{\mathbb{C}}^r)$ with dimension ${\mathrm{\ell }}^{-d\mathrm{/}2}$, its norm $\mathrm{\parallel }\psi \mathrm{\parallel }$ is dimensionless, and an abstract normalized state and the density operators of the state postulate are dimensionless.
2. Probability: the outcome probability $\mu (B)$ of the observable postulate is a dimensionless number; a position density $\mathrm{\mid }\psi {\mathrm{\mid }}^2$ has dimension ${\mathrm{\ell }}^{-d}$; a momentum wave function normalized against ${\mathrm{d}}^{d}p$ has dimension $(\mu \mathrm{\ell }{\tau }^{-1}{)}^{-d\mathrm{/}2}$; an amplitude for a finite outcome is a dimensionless complex number, while an amplitude read against a detector coordinate carries the square-root of the inverse of the detector measure.
3. Frame parameters (The Galilean transformations of the coordinate space): $[b]=\tau$, $[a_i]=\mathrm{\ell }$, $[v_i]=\mathrm{\ell }{\tau }^{-1}$ and $[R_{ij}]=1$.
4. Mechanical quantities: action $[\mathrm{\hslash }]=\mu {\mathrm{\ell }}^2{\tau }^{-1}$; energy and the Hamiltonian eigenvalues $\mu {\mathrm{\ell }}^2{\tau }^{-2}$; momentum $\mu \mathrm{\ell }{\tau }^{-1}$; position components $\mathrm{\ell }$; angular momentum and spin $\mu {\mathrm{\ell }}^2{\tau }^{-1}$; mass $\mu$; velocity of a free spinless model $\mathrm{\ell }{\tau }^{-1}$; force $\mu \mathrm{\ell }{\tau }^{-2}$; a real potential energy $\mu {\mathrm{\ell }}^2{\tau }^{-2}$.
5. Probability current: in the ordinary continuous-field regime, fix an integer $d\ge 1$, an open spatial region $\mathrm{\Omega }\subseteq {\mathbb{R}}^d$ and an open time interval $I\subseteq \mathbb{R}$. A probability current density is a declared continuous map $j:\mathrm{\Omega }\times I\to {\mathbb{R}}^d$, whose spatial components have dimension ${\mathrm{\ell }}^{1-d}{\tau }^{-1}$ and reference unit ${\mathrm{m}}^{1-d}{\mathrm{s}}^{-1}$. At fixed time it is a spatial vector field; its signed normal component is read as probability transported per unit spatial $(d-1)$-area and time. Explicitly, for a flat face $S=\{x:x_k=a,a_i\le x_i\le b_i(i\mathrm{\ne }k)\}\subset \mathrm{\Omega }$, with $a_i<b_i$ and chosen normal $n=ϵe_k$, $ϵ\in \{-1,1\}$, define its signed probability flux at $t\in I$ as ${\mathrm{\Phi }}_S(t):=ϵ{\int }_{{\prod }_{i\mathrm{\ne }k}[a_i,b_i]}{j}_k(x_1,\dots ,x_{k-1},a,x_{k+1},\dots ,x_d,t){\prod }_{i\mathrm{\ne }k}d{x}_i$. Here $e_k$ is the $k$th coordinate unit vector and the integral uses $(d-1)$-dimensional Lebesgue measure; for $d=1$ this means ${\mathrm{\Phi }}_{\{a\}}(t)=ϵj(a,t)$. The flux has dimension ${\tau }^{-1}$ and unit ${\mathrm{s}}^{-1}$; positive sign means transport in the chosen normal direction. Fluxes through curved surfaces, or currents with weaker regularity, require separately specified surface measures or traces. This fixes the quantity type and flux convention; it supplies neither a current associated to an arbitrary $L^2$ state nor a conservation or detector law. Each dynamical model supplies its current construction and balance hypotheses.
6. Electric quantities of the model parameters: charge has dimension $\vartheta$; a magnetic moment has dimension ${\mathrm{\ell }}^2\vartheta {\tau }^{-1}$, equivalently $\mathrm{A}{\mathrm{m}}^2=\mathrm{J}{\mathrm{T}}^{-1}$, and a gyromagnetic parameter $\gamma$ in the relation $\mathit{\mu }=\gamma \mathbf{S}$ has dimension ${\mu }^{-1}\vartheta$, equivalently ${\mathrm{s}}^{-1}{\mathrm{T}}^{-1}$; these quantities appear only in the model assumptions that introduce them.
7. Statistical quantities: a mean has the dimension of its outcome coordinate, a variance the square of it and a standard deviation it; a mean, variance or higher moment exists only when the corresponding moment of the outcome probability is finite, so a symbolic average is not a number unless that integrability is established. Angles and phases are dimensionless under the radian convention.
8. Quantities of later extension assumptions - radiation frequencies and energies, electric dipole moments, cross sections, temperatures and entropies - are declared with their dimensions and units by the items that introduce those assumptions, under the rules fixed here; they are not part of the present list.

(e) Conventions only. No numerical value of any model parameter is fixed by this item: no mass, charge, spin label, potential or measured constant is asserted, and natural-unit shorthand (for example setting $\mathrm{\hslash }=1$ or $c=1$) is read as an abbreviation for the corresponding SI-dimensioned statement with the constant restored.

• The dimension rules fix bookkeeping only. They neither predict a value nor restrict which models are admissible; each later item that introduces an operation outside the product, quotient and derivative rules declares the dimensions of its outputs.
• The framework's exact constants of (c) are conventions of the current SI: they assert nothing about the validity of the model, and the measured constants of the model (masses, charges, magnetic parameters) are qualified empirical inputs wherever they enter.