You are just expected to produce some research which makes some papers for your advisor. This is all that is required to get a PhD. The goal is to do so while developing your own field which is entirely your own work.
From my experience, you will not get a good problem from an advisor unless you get lucky with advisor, so you must make your own luck by doing your own research. You have a better nose than your advisor or the professors, so just learn what is necessary and try to figure out some aspect of the world to the point where you are sure of the answer. Scientific publishing is undergoing a revolution which is especially pronounced in physics, and it is opening up, so whatever you discover cannot be suppressed and cannot be taken from you. So don't worry about discussing your work, or keeping it secret. This hurts you more than it protects you. If someone can steal your result, it is not original enough to be a good result.
False open problems
The biggest problem for grad students is that there are many problems people will tell you are open, because they don't understand them, that are actually completely well understood and closed for a long time. After you figure out the answer to a question you think might be open, look to see if it is already solved. Don't trust people's statements about it.
In school, I at one time or another heard the following were open (they are obviously not):
- Equivalence of Polyakov and Nambu actions
- How to do Path integrals in p-q space, where the coordinates don't commute.
- electromagnetic arrow of time
- getting the beta function of strings from worldsheet actions
- The measurement problem in standard QM
There were many more I don't remember. These I remember, because I got the answer, and then I tended to get pissed off that the answer was well-known and it was not presented to me. People in the US tend to hoard actual open problems, and work on them, and they present fake open problems to students to get you to think "gosh, people don't understand anything". The effect is to steer you toward useless thinking.
Basically, any problem that you see in a textbook isn't open. Just solve it as an exercize, and if you can't, read related literature until you can. There are no real open problems in book subjects, or else they wouldn't be book subjects.
Political fields
Further, there are some fields where certain things are known, but for political reasons people say it isn't:
- The pairing mechanism of HighTc superconductors.
The pairing is purely electronic, but it is politically impossible to say this, because there are stupid people who say otherwise. The mechanism is just BCS theory (although in unusual circumstances), but again, nobody can say it. I personally think I know the detailed mechanism, but when I presented it to an expert he said "even if you are right, this is not the thing that anyone cares about in HighTc anymore". I didn't listen about the idea not being important, but I decided the experts in the field are political idiots, and there is no point in trying to penetrate it.
Although this field is political, one can do interesting things if one has access to experimental data. The thing to know is the pairing mechanism. I'll write it up here.
- The rigorous formulation of quantum field theory
Here again, Wilson and Kadanoff made the path clear, and it is only politics and the political structure of mathematics that prevents the work from getting incorporated. Avoid rigorous field theory, it is not productive.
- Large extra dimensions (in any form)
This is junk, and when I was in graduate school, it was expected of people to write about it assuming it is possible. It is better to be homeless and starving than to promote junk science. I was supposed to write a paper on experimental methods to detect large extra dimension, but at some point I said "no, this I cannot do", and I left grad school and started working on biology. I do not regret this decision, and neither will you.
Do not work on junk science, even if it gets you a PhD. Find non-junk things to work on, there are plenty. You might not get a job, you might wander the wilderness like Kraichnan or Einstein or Onsager, but you will discover new things about nature, and those that do junk science cannot and will not.
Left vs right
One recurring issue with research is that the major breakthroughs are almost always made by people on the political left, and these people hardly ever get credit for these, because by the time the work is well understood, it is picked up by the right, and these people are easier for society to reward.
This leads to the marginalization of many great physicists: Ernst Stueckelberg, Geoffrey Chew, the Italian school (Regge, Veneziano, etc). If you are working in the US, bend over backwards to read people from the former USSR and Europe, they had excellent work, and they were not compelled by market forces to write junk for publicity.
This doesn't mean "right wing science" can be ignored--- the development of quantum field theory in the 1970s and 1980s was essentially right-wing science, since it was reviving 1950s work and suppressing more radical 1960s work. But everything radical is eventually tamed, and string theory was at first a radical experiment in nuclear democracy, then taken over by traditional liberals in the 1980s, and is now a conservative's science (although still great).
Try to avoid politics, but be aware of it, since it will allow you to identify work that others cannot because of the political biases.
Problems
Just ask your own questions here, and quickly you will get to research questions that nobody has answered. I brought up a few here that would be nice to solve, like the degeneracy of even and odd trajectories, the emissions of near-extremal black holes, and so on.
You will not find such open discussions in academic writing anymore, since all the discussion of active questions has moved online. I have some answers for open problems here: What is currently incomplete in M-theory?
