I am having difficulty understanding time at the most fundamental level, especially I am wondering whether there exists an indivisible unit of time (i.e. whether time at some fundamental level is discrete). If the question is non-nonsensical, please help me with how to think about time.
Research for and against this notion are very welcome.
At this point, nobody knows.Certainly there is no experimental evidence in favor of such an unit. On the other hand, there is no evidence against it, except that we have been unable to find it, yet.
However, by putting together $G$ (Newton's constant of gravity), $h$ (Planck's constant) and $c$ (the velocity of light), we can compute the smallest meaningful time coming at about $10^{-44}$ second. At this scale, quantum effects should be dominating gravity and hence, because Einstein's theory links gravity and time, dominating the ordinary notion of time. Simply put, any 'time' smaller than this would hold no meaning according to our notion of time.
The question can be answered in two ways -
Is time discrete for us? Yes, it is. How discrete? will depend upon the smallest event we can use to measure the time. If we can not measure time with higher precision than that, then it does not make sense to talk about smaller units of time than that as there is no way we can measure it. Just like how high the temperature can be - as high as we can measure.
Is time discrete in nature? No! that can not be true. Simply because - suppose there is a smallest unit of time in nature. That smallest unit has to belong to the smallest possible event in nature. That means all such events in the whole universe have to be synchronized, otherwise, you can come up with a smaller event by intersecting boundaries of two such events.
Also, suppose there was a smallest unit of time in nature, then what would wake up the nature when that time is elapsed? That means, there has to be a more precise nature underlying the nature, eventually making it a continuous nature.
