There’s a very interesting new paper on the arXiv by Joe Polchinski, a survey article for Studies in History and Philosophy of Modern Physics, entitled just Dualities. It’s an unusually lucid summary of the story of dualities in quantum field theory and string theory. This is a very complex subject which has been a central one in theoretical physics for the last few decades, but most expository writing on the subject has tended to be either superficial promotional material or mired in technical detail obscuring fundamental issues.

One reason for this is that, as Polchinski does an admirable job of making clear, in a very real sense we still do not understand at all the fundamental issues raised by these dualities. He notes that “we are still missing some big idea”, and points to the same comments from Nati Seiberg last month that I blogged about here. For most of the dualities at issue, our current standard technology for dealing with QFTs (the Lagrangian and the path integral over classical fields) is capable of capturing the two QFTs that are in some sense “dual”, but we lack a viable larger framework that would give the two QFTs in two different limits and explain the duality relationship.

For an example of the problem, probably the oldest and most well-studied case where we are missing something is Montonen-Olive duality, a non-abelian duality between electric and magnetic charges and fields. A currently popular idea is to find the explanation of this in “Theory X”, a 6d superconformal QFT, with duality coming from compactifying the theory on a torus (for more about this, see talks last week in Berkeley). The problem with this is that we don’t have a definition of the “Theory X”.

Polchinski places this problem in the context of a conjectural “M-theory” with various string theory limits. This has been the dominant idea in the subject for nearly 20 years now, but we seem no closer now to finding an actual realization of this conjectural picture than we were back in the mid-90s. Twenty years and thousands of papers have just given better understanding that various possible ideas about this don’t work.

One place where I think Polchinski’s survey is weak is in the treatment of this conjecture, where at times he takes as solid result something highly conjectural. For instance he starts off at one point with:

String-string dualities imply that there is a unique string/M-theory.

and moves on to the conjecture that

In this sense it may be that every QFT can be understood as a vacuum state of string/M-theory.

The problem here is that he’s built a speculative view of the unification of physics, constructed on an assumption about a “unique” theory, when we don’t know at all that such a thing exists. One basic lesson of mathematical research is that you need to keep very clear the distinction between what you really understand and what is speculation, because your speculation is often wrong and if so will lead you in the wrong direction. I think particle theory of recent decades likely suffers from people forgetting that some ideas are speculative, not firmly grounded, and may be pointing in the wrong direction.

One wrong direction this takes Polchinski is to the non-predictive, pseudo-scientific landscape of supposed string theory solutions and the multiverse, which he blithely invokes as our best fundamental explanation of physics. Tellingly, unlike the clear explanations of other topics, here he makes no attempt to describe these ideas other than to note that

they rest on multiple approximations and no exact theory.

In a final section, Polchinski addresses the question of what all this tells us about what is “fundamental” and what is the role of symmetries. This is the crucial question, and I’d argue that our lack of understanding of where these dualities come from likely is due to our missing some understanding of how symmetries are realized in QFT or string theory. This has been the lesson of history, with the Standard Model only coming into being when people better understood how symmetries, especially gauge symmetries, could act in QFT. Polchinski largely takes the opposite point of view, arguing that the fundamental theory maybe has no symmetries, local or global. He quotes Susskind as suggesting that symmetries have nothing to do with fundamental equations, are just calculational tools for finding solutions. I think this is completely misguided, that a strong case can be made (and I do it here) that “symmetry” (in the sense of the mathematics of groups and their representations) lies at the very foundation of quantum mechanics, and thus any quantum mechanical theory, even string/M-theory, whatever it might be.

Wondering whether there will be an arXiv trackback to this, and whether Polchinski has something to say about it…