Physlib.Units.ISQBridge

This function maps a physical dimension $d$ from the LTMCT basis (Length $L$, Time $T$, Mass $M$, Charge $Q$, and Temperature $\Theta$) to the ISQ basis (Length $L$, Mass $M$, Time $T$, Electric Current $I$, Thermodynamic Temperature $\Theta$, Amount of Substance $N$, and Luminous Intensity $J$). The mapping preserves the exponents for length, mass, and temperature, but converts the LTMCT charge generator into the derived ISQ charge $Q = I \cdot T$. Specifically, if $d$ has exponents $(l, t, m, q, \theta)$ in the LTMCT system, the resulting ISQ dimension has the following exponents: - $\text{Length} = l$ - $\text{Mass} = m$ - $\text{Time} = t + q$ - $\text{Electric Current} = q$ - $\text{Thermodynamic Temperature} = \theta$ - $\text{Amount of Substance} = 0$ - $\text{Luminous Intensity} = 0$

The group homomorphism $\text{toISQHom}$ maps a physical dimension from the LTMCT basis (Length $L$, Time $T$, Mass $M$, Charge $Q$, and Temperature $\Theta$) to the ISQ basis (consisting of Length $L$, Mass $M$, Time $T$, Electric Current $I$, Thermodynamic Temperature $\Theta$, Amount of Substance $N$, and Luminous Intensity $J$). This homomorphism is the lifted version of the mapping function $\text{toISQFun}$, ensuring that the product of dimensions is preserved. It treats the LTMCT charge generator as the derived ISQ charge $Q = I \cdot T$. For a dimension $d$ with exponents $(l, t, m, q, \theta)$ in the LTMCT system, the corresponding ISQ dimension has exponents: - $\text{Length} = l$ - $\text{Mass} = m$ - $\text{Time} = t + q$ - $\text{Electric Current} = q$ - $\text{Thermodynamic Temperature} = \theta$ - $\text{Amount of Substance} = 0$ - $\text{Luminous Intensity} = 0$

For any physical dimension $d$ in the LTMCT basis (consisting of Length, Time, Mass, Charge, and Temperature), the result of applying the group homomorphism $\text{toISQHom}$ to $d$ is equal to the value of the mapping function $\text{toISQFun}(d)$.

Given a dimension $d$ in the ISQ basis (comprising length $L$, mass $M$, time $T$, electric current $I$, temperature $\Theta$, amount of substance $N$, and luminous intensity $J$), this function maps it to a dimension in the LTMCT basis (length $L$, time $T$, mass $M$, charge $Q$, and temperature $\Theta$). The transformation is defined by: - The exponents for length, mass, and temperature are preserved. - The exponent for charge $Q$ in the LTMCT basis is assigned the value of the exponent for electric current $I$ in the ISQ basis. - The exponent for time $T$ in the LTMCT basis is calculated as the ISQ time exponent minus the ISQ electric current exponent (reflecting the relation $I = Q/T$). - The exponents for amount of substance $N$ and luminous intensity $J$ are discarded.

The group homomorphism $\text{fromISQHom}$ maps a physical dimension from the ISQ basis (length $L$, mass $M$, time $T$, electric current $I$, thermodynamic temperature $\Theta$, amount of substance $N$, and luminous intensity $J$) to the LTMCT basis (length $L$, time $T$, mass $M$, electric charge $Q$, and temperature $\Theta$). For a dimension $d$ represented by exponents $(n_L, n_M, n_T, n_I, n_\Theta, n_N, n_J)$ in the ISQ system, the resulting dimension in the LTMCT system is $L^{n_L} T^{n_T - n_I} M^{n_M} Q^{n_I} \Theta^{n_\Theta}$. This mapping reflects the physical relation $I = Q/T$ and discards the information regarding the amount of substance and luminous intensity.

For any physical dimension $d$ in the ISQ basis (comprising length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity), the result of applying the group homomorphism $\text{fromISQHom}$ to $d$ is equal to the result of the projection function $\text{fromISQFun}$ applied to $d$. Both operations map the ISQ dimension to the LTMCT basis (length, time, mass, charge, and temperature) by preserving $L, M, \Theta$, mapping current $I$ to charge $Q$, and adjusting time $T$ according to the relation $I = Q/T$.

Let $\text{Dimension}(\text{LTMCT})$ be the commutative group of physical dimensions based on length ($L$), time ($T$), mass ($M$), charge ($Q$), and temperature ($\Theta$). Let $\text{Dimension}(\text{ISQ})$ be the commutative group of physical dimensions based on the seven ISQ base quantities: length ($L$), mass ($M$), time ($T$), electric current ($I$), thermodynamic temperature ($\Theta$), amount of substance ($N$), and luminous intensity ($J$). We define two group homomorphisms: 1. $\text{toISQHom} : \text{Dimension}(\text{LTMCT}) \to \text{Dimension}(\text{ISQ})$, which embeds LTMCT dimensions into the ISQ system by identifying the charge $Q$ as the derived quantity $I \cdot T$. 2. $\text{fromISQHom} : \text{Dimension}(\text{ISQ}) \to \text{Dimension}(\text{LTMCT})$, which projects ISQ dimensions back onto the LTMCT system by identifying electric current $I$ as $Q/T$ and discarding the dimensions for amount of substance and luminous intensity. The composition of these homomorphisms satisfies: $$\text{fromISQHom} \circ \text{toISQHom} = \text{id}_{\text{Dimension}(\text{LTMCT})}$$ where $\text{id}_{\text{Dimension}(\text{LTMCT})}$ is the identity homomorphism on $\text{Dimension}(\text{LTMCT})$.

The group homomorphism $\text{toISQHom} : \text{Dimension}_{\text{LTMCT}} \to \text{Dimension}_{\text{ISQ}}$, which maps physical dimensions from the LTMCT basis (Length, Time, Mass, Charge, Temperature) to the ISQ basis (the seven ISO/IEC 80000-1 base quantities) by identifying the charge generator $Q$ as $I \cdot T$, is injective. This means that the LTMCT system of dimensions is faithfully embedded into the ISQ system.

The group homomorphism $\text{fromISQHom} : \text{Dimension}(\text{ISQ}) \to \text{Dimension}(\text{LTMCT})$, which maps physical dimensions from the International System of Quantities (ISQ) basis to the length-time-mass-charge-temperature (LTMCT) basis, is surjective. That is, every physical dimension in the LTMCT system can be represented as the reduction of at least one dimension from the ISQ system.

The embedding $\text{ltmctToISQ}$ defines a dimension-preserving inclusion of the LTMCT basis (consisting of Length $L$, Time $T$, Mass $M$, Charge $Q$, and Thermodynamic Temperature $\Theta$) into the ISQ basis (consisting of Length $L$, Mass $M$, Time $T$, Electric Current $I$, Thermodynamic Temperature $\Theta$, Amount of Substance $N$, and Luminous Intensity $J$). This embedding is characterized by the injective group homomorphism $\text{toISQHom}$, which maps the LTMCT charge generator $Q$ to the ISQ product $I \cdot T$, ensuring that the LTMCT system is faithfully represented as a subset of the ISQ system.

The projection `isqToLTMCT` is a dimension-preserving mapping from the ISQ basis (Length $L$, Mass $M$, Time $T$, Electric current $I$, Thermodynamic temperature $\Theta$, Amount of substance $N$, and Luminous intensity $J$) to the LTMCT basis (Length $L$, Time $T$, Mass $M$, Electric charge $Q$, and Temperature $\Theta$). This projection is defined by a surjective homomorphism that interprets the ISQ base current $I$ as the derived quantity $Q/T$ (charge per unit time) and discards the dimensions of amount of substance $N$ and luminous intensity $J$ by mapping them to the dimensionless unit.

The composition of the dimension-preserving embedding from the LTMCT basis to the ISQ basis ($\text{ltmctToISQ}$) and the projection from the ISQ basis back to the LTMCT basis ($\text{isqToLTMCT}$) is equal to the identity homomorphism on the group of LTMCT dimensions. That is, $\text{isqToLTMCT} \circ \text{ltmctToISQ} = \text{id}$. This identifies the projection as a retraction of the embedding, meaning that any physical dimension expressed in the LTMCT basis (Length, Time, Mass, Charge, Temperature) is recovered exactly after being mapped into the ISQ system and then projected back.

Under the dimension group homomorphism $\text{toISQHom}$ that maps physical dimensions from the LTMCT basis to the ISQ basis, the LTMCT charge generator $\mathsf{C}_d$ is equal to the ISQ derived charge dimension $Q$, which is defined as the product of the base dimensions of electric current $I$ and time $T$.