In this blog entry I will try to explain my most recent paper. The theme of the paper is rather simply put: You should not compare apple with oranges. The subtlety comes from knowing whether you have an apple or an orange in your hand. This is far less simple than it sounds.
The origin of the problem are once more gauge theories. In gauge theories, we have introduced additional degrees of freedom. And, in fact, we have a choice of how we do this. Of course, our final results will not depend on the choice. However, getting to the final result is not always easy. Thus, ensuring that the intermediate steps are right would be good. But they depend on the choice. But then they are only comparable between two different calculations, if in both calculations the same choice is made.
Now it seems simple at first to make the same choice. Ultimately, it is our choice, right? But this is actually not that easy in such theories, due to their mathematical complexity. Thus, rather than making the choice explicit, the choice is made implicitly. The way how this is done is, again for technical reasons, different for methods. And because of all of these technicalities and the fact that we need to do approximations, figuring out whether the implicit conditions yield the same explicit choice is difficult. This is especially important as the choice modifies the equations describing our auxiliary quantities.
In the paper I test this. If everything is consistent between two particular methods, then the solutions obtained in one method should be a solution to the equations obtained in the other method. Seems a simple enough idea. There had been various arguments in the past which suggested that this should be he case. But there had been more and more pieces of evidence over the last couple of years that led me to think that there was something amiss. So I made this test, and did not rely on the arguments.
And indeed, what I find in the article is that the solution of one method does not solve the equation from the other method. The way how this happens strongly suggests that the implicit choices made are not equivalent. Hence, the intermediate results are different. This does not mean that they are wrong. They are just not comparable. Either method can still yield in itself consistent results. But since neither of the methods are exact, the comparison between both would help reassure that the approximations made make sense. And this is now hindered.
So, what to do now? We would very much like to have the possibility to compare between different methods at the level of the auxiliary quantities. So this needs to be fixed. This can only be achieved if the same choice is made in all the methods. The though question is, in which method we should work on the choice. Should we try to make the same choice as in some fixed of the methods? Should we try to find a new choice in all methods? This is though, because everything is so implicit, and affected by approximations.
At the moment, I think the best way is to get one of the existing choices to work in all methods. Creating an entirely different one for all methods appears to me far too much additional work. And I, admittedly, have no idea what a better starting point would be than the existing ones. But in which method should we start trying to alter the choice? In neither method this seems to be simple. In both cases, fundamental obstructions are there, which need to be resolved. I therefore would currently like to start poking around in both methods. Hoping that there maybe a point in between where the choices of the methods could meet, which is easier than to push all all the way. I have a few ideas, but they will take time. Probably also a lot more than just me.
This investigation also amazes me as the theory where this happens is nothing new. Far from it, it is more than half a century old, older than I am. And it is not something obscure, but rather part of the standard model of particle physics. So a very essential element in our description of nature. It never ceases to baffle me, how little we still know about it. And how unbelievable complex it is at a technical level.