What does it mean for two quantities to be on equal footing? I often see this said for time/space or energy/momentum, but I do not understand what this means beyond a mathematical convenience.
I have taken a look at What does it mean to treat space and time on equal footing? but the answers in that thread are unsatisfactory. Is there some physical interpretation of this statement?
I use this phrase myself (most recently in this answer on Astronomy SE). The key equation to know is
$$ds^2 = -dt^2 + dx^2 + dy^2 + dz^2$$
This equation comes from special relativity and describes flat spacetime (so-called Minkowski space). You don't need to know what it represents; it's good enough to know that $t$ is time, and $x, y, z$ are the three spatial dimensions.
Note you can add time and space. This is a sign they are on equal footing.
To illustrate two physics concepts that are not on equal footing, consider energy $E$ and electric current $I$. The former is measured in $J$ (joules), while the latter is measured in $A$ (amperes). These two units are different, which makes any expression like $E + I$ meaningless. This shows they are not on equal footing.
You might point out that time and space are measured in different units, seconds ($s$) and meters ($m$) respectively, but in relativity, both these units measure the same thing, and you can convert from one to the other using the speed of light. Hence relativity puts time and space on equal footing. Time and space are indeed not on equal footing in Newtonian mechanics.
To my understanding, for two quantities to be on equal footing, both quantities must be the same type of mathematical object. For example, in textbook Introductory Quantum Mechanics, space coordinates are operators (as well as parameters) and the time coordinate is only a parameter (it is not an operator). This mismatch in object type means that space and time are not on the same footing.
I think the direct consequence of two quantities not being on equal footing is that you cannot manipulate said quantities in the same way. For example, suppose one theory of physics puts space and time on the same footing and thus allows one to manipulate space and time in the same way. Suppose another theory of physics does not put space and time on equal footing so that one cannot manipulate space and time in the same way. Then, you have an inconsistency to solve if you want to combine the two theories of physics.
