Simple question (I think).
Is time infinitely divisible?
I heard that it was, although not from a particularly explanatory source. If so, are we sure that it is, without a shadow of a doubt?
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In most of physics we treat it like it's continuous, but treating it as quantized might have important implications for quantum gravity, etc. etc. etc.
(Even though the Planck time is the smallest meaningful amount of time that can exist between two events, that doesn't mean time is quantized into Planck times.)
TLDR; no one knows... yet.
You are asking if time is a continuous variable. This is on par with asking if space is a continuous variable.
Time we measure with clocks is as quantized as space we measure with meter sticks, the accuracy of our measurements.
In the past hundred years we have explored the microcosm and have found that it is quantized, from the existence of countable atoms to the existence of quarks we have found matter to be quantized and have been studying it with elegant theories which assume continuous variables for both space and time.
At present the whole construct of particle physics is well described in this manner, and theories with quantized space and time violate Lorenz invariance which is a well validated part of the theory, so quantization of either space or time is contradicted by data. What the case is for regions of space and time not explorable with our instruments yet is an open research question for the future.