There are two distinctly separate perceptions of time travel. Firstly, time travel by the human mind, which, as a reader of history well knows, is possible by means of our imagination. Secondly, the notion of physical transport' from one time to another, by means of some sort of unimaginable time machine. Well, a Quantum Physicist, I am not. However, will you allow me to apply logical deduction to this question ?
The latter question of physical transport' from one time to another has two main difficulties, one being whether or not the speed of light can be exceeded and the other being whether or not the same matter can exist in two places at once. Lets start with this matter difficulty.
The frequently stated requirement, that the same particulate matter cannot exist in two places at one time provides a starting point. This postulation seems sensible and stems from the idea that once matter has been created, it is guided between the limits set by time, as the fourth dimension. There are logically derived difficulties for matter in two places at the same time. If it came into contact with itself what would happen?
Suppose a man travels back in time and meets himself, who then proceeds to handcuff the time traveling self and keep him in a cage until the time when he is supposed to start his later' journey back through time. Quite rightly, such a thing cannot be. Time travel to a time when the same matter exists would involve the destruction of one or other or both of the subjects of travel.
For time travel to be permitted by the laws of physics, therefore, matter can only travel to a time when it did not or will not exist.
The vast majority of matter exists continually and is merely mixed, sorted or re-combined with other matter during its existence. You might think of a steel shell for a time machine, for instance and perceive it to have been created', but of course the iron ore from where it was derived existed long before man's ascent, let alone before the fabrication of the steel shell.
Living matter is different because it reproduces itself. Nevertheless, after death, living matter merely decomposes rather than being destroyed, so a man could not travel forward from his natural' time, even if it was to a time after he had died. However, theoretically, a man could travel in time, without his clothes, spectacles and credit cards, provided that the destination was a time before he was born and his return journey was back forwards to a time after he had departed in the first place. This latter forward travel is only possible because it does not start in his natural' time.
All this logical deduction is all very well, but how would the naked man without his gold tooth actually achieve it ? A living time machine would be necessary, devoid of fabricated steel, devoid of plastic and devoid of glass covers to digital display screens. This living thing would have to evolve in such a way so that, in distant hindsight, it would be seen to have been selected.
It appears, then, that time travel would only be possible in a distant future where, after many millions of mutations, the highest species has evolved by having a meaningful advantage through its existence as a living, breathing, heart thumping time machine.
Such a living time machine would require certain circumstances to evolve. For instance, in an infinite Universe where time is infinite and it is otherwise physically possible to travel faster than the speed of light, there would be no likelihood of naturally selecting such a living time machine because there would be no need for it to be. There would be a comfortable expectation of survival without the need to travel in time and thus it would not happen.
However, in a finite Universe, which may be the case for the one in which we find ourselves, time is also finite. Hence, it is conceivable that in a distant future, when time itself is under threat, there may be a need for a species to be selected which is capable of time travel in order to survive. Such a scenario is the likeliest route by which time travel could be achieved, but even then, the confining effects of the Universal constant, identified by Einstein as the speed of light, would have to be overcome in some currently inconceivable way.
Having written all this, if there were time travelers in our midst then they would have to be keeping themselves to themselves!
Maybe the current lack of time travelers from our future is proof that we don't have a future, because the global ecosystem fails just as the oil resources are used up and a massive environmental deterioration leads to the slow death of the human species.