This type of drive explicitly requires exotic matter (you need negative mass/negative energy density). There is no evidence that such exotic matter exists, and many of our best physical theories explicitly predict that it doesn't exist. So what you have here is basically something that works within some of our rules of physics (General Relativity), but violates other areas of physics (the Standard Model). That is likely an insurmountable problem (at least until if and when we have something beyond the Standard Model).
You can imagine the solution to some problems by waiving your hands and saying "meh, good engineering". But when you have something that requires a form of matter that is likely to not exist, well, thats a rather fundamental problem.
I'm not saying that this shouldn't be worked on by physicists, I'm just saying that I wouldn't exactly get your hopes up.