Over at Backreaction, Sabine has given a nice overview of Garrett Lisi's new theory of everything. I, for one, am very impressed, and have been spreading the news among my friends. Of course, it's pretty far beyond my capabilities to understand the math involved, but as far as I can tell it has several very nice features. Mainly, the fact that things fit so well into the structure of E₈, with most of everything else falling out of this, is very nice.

However, as Sabine mentions, there are a few missing things. And this got me thinking: what would I really want to have, from a theory of "everything"? Perhaps this is moving beyond the conventional definition as "something that unifies gravity and the Standard Model under one set of laws," but ultimately, we do want to have an explanation for everything in one theory. What do I mean, exactly? Well, I think instead of rambling on about general ideas, I'd better start giving some examples.

Cosmological Mysteries

Some of the most mysterious data in modern physics are present in cosmology. And, given the fact that accelerator technology is starting to have increasingly low return-on-investment (imagine if all the LHC finds is the Higgs? Or worse, if it doesn't even find that?), this makes sense. So what kind of cosmological mysteries would we like a theory of everything to explain?

• Dark Matter. Although I've been convinced that dark matter is, in fact, matter—and not a modification of our theories of gravity—but this still doesn't explain exactly what dark matter is. There are certainly many possibilities, but it would be nice if our TOE had some candidate particle (or some such) that we could point to and say "ahah! That perfectly fits all of our dark matter data!"
• Dark Energy. Even worse than dark matter is dark energy, as it quite plausible could be a modification of our theories. That is, while we can try to fit the observed rate of the expansion of the universe into existing theories, via a cosmological constant or some kind of quintessence field, there isn't a compelling reason to say that it should fit into such a framework. Any natural explanation for dark energy—that is, a non-fine–tuned one—would be a welcome feature of any TOE, whether it comes in the form of a conventional (but not fine-tuned) cosmological constant, or in another form entirely.
• Large-Scale Structure. Most graphically, where do we get that huge hole in the universe? And similarly, what's with the axis of evil? There is increasing evidence that our universe's large-scale structure has a number of strange properties that are difficult (impossible?) to explain with conventional models. Of course, such an explanation would be closely tied to our next cosmological mystery…
• Creation and Evolution. The Big Bang hypothesis, along with its counterpart of inflation, are both slightly unsatisfactory. While they certainly have some explanatory power (challenged by the above results, perhaps, but in the end remarkably successful), at the same time they raise a lot of questions. The Big Bang itself is a singularity, which we cannot describe with our current physics: perhaps a TOE can? And inflation is simply missing a mechanism—hopefully we can get one of those out of our hypothetical TOE. Alternately, scrapping inflation and replacing it with something different is a possibility; the Big Bang is probably here to stay, however.

Black Holes

Black holes, of course, are the ultimate frontier for quantum gravity. Predicted by general relativity, they cannot comfortably be accommodated by quantum field theory, although we can kinda-sorta mash them together to get neat stuff like Hawking radiation. Here are some black hole-related things that a good TOE should resolve:

• No Singularity. The center of a black hole contains (according to general relativity) all of the object's mass, but within zero volume. Thus infinite density and spacetime curvature. And everything breaks when you throw infinities at it. So, if we want a theory to describe the universe, it can't break down at the center of black holes!
• Zero Volume, Really? If spacetime is quantized, in any meaningful sense, you just can't have a zero-volume object. So what happens when these things just keep gravitating toward each other? Where does it end, and why, and how?
• Hawking Radiation. This process is unobserved and on somewhat shaky ground, in that it arises from merging quantum field theory and general relativity. While it's quite conceivable that such a process is explicable in terms of interactions between gravitons and virtual particle pairs, it's also possible that our TOE will have something else to say about the matter.
• Information Loss. Closely tied to the issue of the no hair theorem and Hawking radiation, there is the possibility of black holes being able to destroy information, something which none of our current physical theories allow (the quantum measurement problem notwithstanding). A TOE should either exclude this possibility, or explain why black holes are special in this respect. Hawking claims to have solved this by letting the information be slowly emitted via Hawking radiation, but apparently not everyone is satisfied with this solution.
• Thermodynamics and the Holographic Principle. The issues regarding black holes, entropy, temperature, and the holographic principle are all mixed up as of now. It's a mess of classical statistical mechanics grafted onto classical general relativity with a bit of quantum-mechanical trickery thrown in to get thermal radiation (and thus "temperature"). And once we start characterizing a black hole by its entropy, we get interesting connections to the holographic principle, but it's not entirely clear what role the holographic principle plays in our hypothetical TOE anyway. A good TOE should allow us to get a complete explanation for these connections entirely within itself.

Some Foundational Issues

There are some pretty important issues that it seems possible a TOE could entirely skip, in the same way current approaches do. These affect the very foundations of what our theory means, but of course a "shut up and calculate" approach is usually available to bypass them. I suppose, due to the availability of such an approach, these issues won't be on everyone's TOE wish list; however, me and some others do insist on resolving them. One good paper that sums up most of these is Isham's "Prima Facie Questions in Quantum Gravity."

• The Problem of Time. One of the more fascinating papers I've been reading recently spells out what is called "the problem of time" in quantum gravity. Mainly, it boils down to the fact that in general relativity time is completely relational, arising only from the relations that events have to each other. To put this more graphically, if everything in the universe stopped moving right now—if events stopped happening—then there would be no time. However, in quantum theory, time is a parameter that we need to use to evolve our dynamical system, according to our Schrödinger/Klein-Gordon/Dirac equation. We need time as a fundamental background in which to draw our Feynman diagrams. Time isn't an "observable"—in fact, you can show that any real clock will always have a nonzero probability of being observed as running backward. These are two completely different views of time, both ontologically and mathematically. A good TOE needs to resolve them. (Googling around found me this paper, which I haven't read but seems like it would be a good introduction to the idea.)
• The Measurement Problem. The mother of all foundational problems is the quantum measurement problem. I'm sure I'll talk more about this in a future blog post, but let's just say that you'd better resolve this or else people will be bickering about your theory for the next 80 years or so, as has happened with quantum theory.
• Relational vs. Absolute. Best posed as a vivid series of questions. For example: if there were only one object in the universe, could you measure it's motion, or size? If there were only two objects in the universe, could you measure how far away from each other they were? If all objects (and interactions) in the universe "slowed down" by the same, constant factor, would this affect anything? General relativity answers all of these with a "no," a property which we often call diffeomorphism invariance or background-independence. Quantum field theory, however, relies on an absolute background spacetime in which to set the stage of our grand saga, and let the participants evolve. Can we describe the universe accurately without diffeomorphism invariance? It's very pretty, but then again, so is Lorentz invariance, and we can "fake" that in Bohmian hidden variable theories to the extent of matching observation without integrating it into our theory's fundamental structure. What will our TOE choose? Is there a "right" choice?
• What's Really Going on Down There? Ultimately, we'd like to be able to package a nice picture of our universe into some great graphics that can go on TV. You know, have some nice researcher go on PBS and broadcast a multi-part special explaining what we think is happening. But this time it should be, well, supported by experiment. Anyway, the point remains: can we talk about what's happening in our universe, or do we just get to talk about group representations and symmetry transformations? A principle like "everything is made out of vibrating strings, and each way of vibrating gives you a new particle" is pretty good; I can, in some sense, "picture" that universe. On a different level, a Bohmian hidden-variables model tells me exactly what's happening, without any of the messy quantum randomness and mysterious measurement processes. It would be nice if our TOE gave us such a picture.

Meta-Universe and Meta-Theory Questions

As discussed previously, it would certainly be nice if our ultimate TOE emerged "naturally," allowing no other possibilities than the one we observe. This can be taken to various levels of severity, so we'll start from the very basic and end with the most fundamental.

• Constants of Nature. It almost goes without saying that we would like to avoid any "input parameters" to our theory, avoiding the Standard Model's 26 fundamental constants. This is, perhaps, one of the most attractive features of Garrett's theory; as far as I can tell, it all arises from the structure of E₈.
• 4 Forces. Why four? Why are they unified in certain ways, but not others? Of course we have the famous hierarchy problem to contend with here, but I think more fundamentally is the question that we might lose track of in the midst of all this unifying: why are we starting with these particular forces to unify, anyway? I suppose this might not be too difficult to answer, in the end: say we have this perfectly-described unified force at high energies; then it simply becomes a matter of predicting how that force breaks apart at lower energies.
• 3 Generations. Why are there three generations of particles, and not some other number? Again, Garrett's theory answers this nicely, tying it to the structure of E₈. One might then ask, "why E₈?" But I'll leave that for a few bullet points down.
• 3+1 Spacetime. While there are plenty of anthropic arguments for 3+1 space/time dimensions, this isn't very satisfying for a number of reasons. One of the most obvious is that an arbitrary number of dimensions can be fit in, "string theory style," by requiring them by "curled up" to the extent of having no visible effects. (Does this work for time too?) More generally, there's no reason to believe that physics itself could not exist in any other dimensionality (the existence of human beings aside); some sort of selection principle as to why 3+1 is "best" would be a great feature for a TOE.
• Where Do We Get Those Equations From? One of Sabine's most prominent complaints about Garrett's theory was that he needed to make many assumptions to pick out the right action that would give the desired equations of motion. And in general, many equations in physics are necessarily derived from empirical data, and not from first principles. Ideally, our TOE should leave us no choice in the equations it contains, just as it (ideally) leaves us no choice in the constants of nature.
• Why These Fundamental Principles? Whether the fundamental principles turn out to be gauge invariance, diffeomorphism invariance, certain decompositions into irreducible representations over a given group, or even just the plain-old action principle, it would be nice if there were a way to say "it must be this way!" instead of "our observed universe maps very well onto a theory satisfying these fundamental principles." As you can tell, we've reached the end of the list, where we start getting more demanding than usual.

To Conclude

Of course, it's pretty easy to sit back and say "hah, all you working theorists, give me a theory that satisfies all these!" But I didn't mean this post as a list of demands; more as a solidification of my thoughts on the question, "After we have a TOE, what is there left to do?" And the answer is, "investigate all these issues."

I think it's a pretty interesting list. If you have any additions, be sure to leave a comment with them; I might update it if I agree with you, or we could get into an interesting debate as to whether it's really important.

P.S.: Does anyone pronounce "TOE" as "toe"? Like, the thing you have ten of, on your feet? Because that would be really weird.