my puzzling grandfather

A short story in the New Yorker, The Region of Unlikeness by Rivka Galchen, led me to a time traveling problem called the Grandfather Paradox. The paradox states that time travel must be impossible because it would lead to paradox. For instance I might use a time travel machine to go back and kill my grandfather (if I hated him for some reason). But if my grandfather didn't give birth to my father then I wouldn't be born - so then I couldn't go back in time and kill him, so he lived and I was born and did go back and kill him, and so on impossibly.

People who wish to defend time travel resolve the paradox by theorizing (and this is the premise of the New Yorker story) that if I did go back and meet my grandfather that something would happen that would prevent the murder - even something that was otherwise highly unlikely. And thus the one solid state line of history would be preserved, even as I moved through out of sequence.

I think there's a better answer, though. The problem with time travel as it's always imagined in science fiction, is that a person will get into a machine, now, and the machine will transport them quickly to a past or future, by somehow moving outside of the regular sequence of events, as though a separate time traveling path could be created and followed beside the regular path of time. But when you look at the Feynman line drawings that make time travel theoretically possible at the quantum level they don't show a second (quicker) backward or forward path, all they show is that the action of a particle illustrated by the line can be read in either direction. A particle goes back in time (if that's the correct interpretation) on the same line.

So if I went back in time attempting to murder my grandfather I would have to follow the same path that brought me where I am now. That is, I would get younger, re-experiencing my life in reverse, until I was back in the womb and then cease to exist. That's as far as I could go. I could never go back to meet my grandfather as a young man because I didn't exist then.