Feb 7, 2018

## Why time is different in quantum mechanics and its relation to Heisenberg’s uncertainty relations

“Canonically conjugate” is maybe a bit of a hard-core jargon word to use, but it’s not hard to understand the effect that it has in quantum mechanics. In our everyday, classical world, we can always measure two things at the same time, for example (as in the article above), location and velocity of a particle. Another way of saying this is, that if we measure first the location and then the velocity of something gives us the same result as first the velocity and then the location. If I denote the measurement of location with X and velocity with P, that means that XP=PX.

In quantum mechanics, exactly that is not true. One says that “*X and P don’t commute with each other*”, which just means that XP is not PX, but rather

XP=constant+XP, where the constant is a number related to Planck’s constant. Pairs of measurements, which “do not commute” are also called “canonically conjugate”. And the equality I wrote above with the constant is nothing but Heisenberg’s uncertainty relation.

Now, energy and time fulfill a similar relation and it actually looks exactly the same. But the origin of this relation is very different, because, as I said, time is not an observable, i.e. you can not “measure” time the same way as you can measure space or velocity. It is simply a parameter, not a property that particles have. You can say “this particle has the property of being at this location at this time” but you can not say “the property of this particle is that it has that time”. This unequal footing of space and time in quantum mechanics simply shows that it is a theory which does not fulfill the basic requirements of Einstein’s theory of relativity: time and space are one and the same thing in relativity. This is why quantum mechanics was extended to fulfill special relativity in the framework which is called quantum field theory, which is also the language in which the standard model of particle physics is formulated. In it, space and time have exactly the same nature: They both are simply parameters of space-time, but not strictly speaking something which we can observe. They simply lay the stage for all magic to happen.