Is it possible to pass a PHD without full understanding of ones own thesis?

Question

This is likely to be a rather strange question.

I have been working in physics in an attempt to obtain a PHD. My initial plan was entirely experimental, however I have not obtained enough data by itself. This is due to a combination of difficult experiments, and temperamental equipment and its repeated failures; I was and am simply not capable of repairing the systems without any replacement components by myself.

This is ok I had thought, I can attempt to slightly extend a pre-existing theoretical framework to explain the few results that I do have. However this is not my supervisor, or even departments expertise, so I have for the last year and half attempted this myself. After much effort I have been able to replicate the prior theoretical calculations of the model.

However despite having the best grasp of the model out of anyone I have spoken to in my department, there are significant parts where I just do not have a clue.

I have reached a point where I have 2 months left to submit, and I do not yet have a sufficient understanding of the theory to be able to tell if it is actually valid for what I am trying to use it for.

Is it acceptable to just shut up and apply it anyway, and be deliberately vague and sparse in describing it and the necessary mathematics so that I do not include things I do not understand?

Or do I just throw in the towel and save myself another 2 months of absolute torment?

Many thanks.

Edit in response to comments: it's rather that I do not understand some of the mathematics necessary.

There is a set of 6 eigenfunction equations, where they all have the same eigenvalues. All the authors in the literature only care about the eigenvalues, where I need the eigenfunctions. I had assumed that they were all the same, like the eigenvalues, but upon attempting to write a thesis chapter on them I now realise that they are not. This is a problem as my calculations are based upon the eigenfunctions.

There is only one of the eigenfunction equations in the literature. I am able to, with initial difficulty, solve it and obtain the same eigenvalues as the literature. I then do some stuff with the eigenfunction that was not done, and obtain another different eigenvalue equation based upon an operator corresponding to the measuremrnt data I do have. But I cannot derive the original equation in the literature, and I cannot derive the other, not stated in the literature, 5 equations.

Noone has worked on it for decades, and the original authors, and those who initially extended their work, are all dead.

Hence why I state that I do not understand it. I can state what I have done, that I cannot derive the equations, and that I know it is wrong without the other 5 solutions. I am 90% certain I can solve the other 5 equations if only I could understand how they are obtained.

Unfortunately I just get bogged down in attempting to follow the derivation, 1.5 years of attempting to follow it and I still just do not follow it.

Answer 1

Of course, it would be good if someone around you can help you see what needs to be done to legitimize your use of "the other 5 equations". If you post to MathStackExchange, someone might be able to help, for example, if no one physically nearby can.

In any case, honesty and at least a nominal "dispassion" about things is the best approach, I think. Say what you assume, say what you can verify, say what previous sources say, etc. It's ok, I think, if your results are "relative" (=relative to assuming that certain sources are ok, plus assuming that certain extensions, which you've tried but failed (so far) to validate yourself first-hand, are ok...), as long as you are very clear on what your results are relative to.

Don't bluff... :)

Answer 2

Or do I just throw in the towel and save myself another 2 months of absolute torment?

Nobody except you would be able to decide. But, given what you write, failing could be an option. Academia is a hard sport, and your health might be more important than some theory. I see bad examples around me.

In case you decide to proceed, here are some thoughts:

• Ask an online service (such as Google scholar) for which papers cite the original framework. Some of the authors referencing the framework might be living. Write them e-mails.

• Apply Feynman's problem-solving algorithm. (It's not a joke, I'm very serious.) In less extravagant terms, improve your working style.

• Apply for an extension.

• ...

Bear in mind that, after all, the problem might be unsolvable with the methods at your disposal.

E.g., the mature Ostrogradsky once asked Paris mathematicians for help for solving a particular problem. They took quite some time, and finally responded "This task can be solved only by one person: the Russian professor Ostrogradsky. He lives in Petersburg. You should contact him.".

But one proper piece of advice: no torment; go home at 6pm whatever you do.

Answer 3

For a PhD you should demonstrate two things: that you have become an expert in your (very very small) chosen topic, and that you have learned how to do credible research in that topic. From your description you don't seem to have demonstrated either at this point, so I would find it concerning if you did get a PhD.

Kicking in the towel isn't the only option though.

• You say no-one in your department understands the material. But it is extremely unlikely that no-one in the world does. There are many theoretical physicists, and many mathematicians. Even if they are not familiar with the exact problem, there will be someone you can talk to who knows far more about eigenfunctions than you seem to.
• Get an extension. If you've been allowed to go substantially beyond the expertise of those supervising you where you don't understand and no-one else is supporting you, then someone has not really been taking their responsibilities seriously enough.
• Try and find a summer school on the mathematics you need.
• Explore the options for graduating with a lower degree.