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However, this is only one perspective, and changing $\alpha$ will change other realms of physics differently. In deeply relativistic quantum regimes, for example, it makes much more sense to make $\hbar$ and $c$ the benchholders of your physics, and you switch from atomic units to natural units; here $\alpha$ is more meaningful as a measure of the Coulomb constant $e^2/4\pi/\varepsilon_0$, which determines the strength of the electromagnetic interaction between two elementary charges.

This is a different perspective, one among many others, and they are all equally valid, and for each a change in $\alpha$ will affect physics in different ways. For all of them you need to choose some dimensionless ratio as a hallmark that physics has changed - but the reference quantity in that ratio will be different depending on what physics you're exploring, so you always need to say what it is you're keeping constant.

However, this is only one perspective, and changing $\alpha$ will change other realms of physics differently. In deeply relativistic quantum regimes, for example, it makes much more sense to make $\hbar$ and $c$ the benchholders of your physics, and you switch from atomic units to natural units; here $\alpha$ is more meaningful as a measure of the Coulomb constant $e^2/4\pi\varepsilon_0$, which determines the strength of the electromagnetic interaction between two elementary charges.

This is a different perspective, one among many others; they are all equally valid, and for each a change in $\alpha$ will affect physics in different ways. For all of them you need to choose some dimensionless ratio as a hallmark that physics has changed - but the reference quantity in that ratio will be different, depending on what physics you're exploring, so you always need to say what it is you're keeping constant.

where it is crystal clear that the only observable which can be reliably used as a gauge for changes in $\alpha$. In this specific case the two quantities being compared are the frequencies of optical radiation from a single trapped ion (because they can be measured to exquisite precision, and because there is strong theoretical evidence that those frequencies have a different dependence on $\alpha$), but the principle can be applied to the ratio of any two lengths, times, velocities - you name it.

where it is crystal clear that the only observable which can be reliably used as a gauge for changes in $\alpha$ is the ratio between two commensurate quantities. In this specific case the two quantities being compared are the frequencies of optical radiation from a single trapped ion (because they can be measured to exquisite precision, and because there is strong theoretical evidence that those frequencies have a different dependence on $\alpha$), but the principle can be applied to the ratio of any two lengths, times, velocities - you name it.

However, this is only one perspective, and changing $\alpha$ will change other realms of physics differently. In deeply relativistic quantum regimes, for example, it makes much more sense to make $\hbar$ and $c$ the benchholders of your physics, and you switch from atomic units to natural units; here $\alpha$ is more meaningful as a measure of the Coulomb constant $e^2/4\pi/\varepsilon_0$, which determines the strength of the electromagnetic interaction between two elementary charges.

This is a different perspective, one among many others, and they are all equally valid, and for each a change in $\alpha$ will affect physics in different ways. For all of them you need to choose some dimensionless ratio as a hallmark that physics has changed - but the reference quantity in that ratio will be different depending on what physics you're exploring, so you always need to say what it is you're keeping constant.