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( Probably a stupid question. But the thought crossed my mind. I'm not a physicist; I'm a mathematician)

Is there any way that the fundamental quantities (like length, time ) be "unified" in some way? Or have some deep equivalent analogy or similarities between them?

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  • $\begingroup$ This is probably not quite the answer you were expecting, but there is the notion of planck length, planck time, and planck mass. They are merely conjectured to exist, and I wouldn't call this an all-out unification. But it may satisfy your curiosity, or at least be a semi-answer to your question. $\endgroup$
    – the_photon
    Apr 6, 2020 at 23:00
  • $\begingroup$ Could you be more specific as to which ones you want to unify. Space and time are already unified in special relativity. Mass seems interesting as a real challenge. If you mean the other SI units, they seem pretty independent. $\endgroup$
    – user257090
    Apr 7, 2020 at 0:15

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Distances may be defined by the radar method, which reduces measurement of distance to measurement of time. This is a fundamental principle in special relativity. It has been said that all measurements can be reduced to measurement of time and position, and I am not aware of a counter example. Certainly most measurements can reduced to a set of measurements of position, including measurements of the position of a pointer. For example, a classical measurement of velocity may be reduced to a time trial over a measured distance. In quantum mechanics momentum is a conjugate variable to position, meaning that in natural units (setting Planck's constant) equal to 1, in has units of $1/\mathrm{time}$. Then mass and energy have the same units.

There are a number of systems of natural units which set fundamental constants to $1$, explicitly reducing other measurements to measurements of time. Perhaps the most "natural" are Planck units.

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