Hidden Variables

When Media Coverage Does Matter

Domenic — Mon, 26 Nov 2007 14:21:00 -0400

I was hoping to get an interesting post on applying Gödel's incompleteness theorem to physics up sometime last week, but it turned out to be quite difficult to make my arguments rigorous, and so that little essay has been placed on the back burner. In the meantime, I'd like to comment on Sabine's post regarding science reporting and media coverage. While the post spans a great deal of interesting material, the latter half deals mainly with the thesis that public discussion of physics should not be thought of as relevant by physicists themselves. I agree with this in almost all instances, but there is one point where I think the public discussion and media coverage does matter. Since it's a worthwhile point, and also ties in to another upcoming post, I thought I'd spend some time on it here.

A Story

I think it'd be best to start off with a little story from my past, regarding how I became interested in physics research. Until rather recently, say three years ago, I was mostly committed to becoming a mathematician. I'd always thought—and mostly, still do—that there wouldn't be much point in doing physics research unless my research program was part of an overall drive toward a theory of everything. And unfortunately, the only path toward a theory of everything that I knew about was string theory.

You see, at the time I was beginning to contemplate career paths, string theory was at one of its heights of popularity. Brian Greene's The Elegant Universe had been on my shelf since seventh grade, and I re-read it periodically. My parents had rented the PBS television adaptation, which we all watched together as a family. A lot of media coverage was spawned from these, and so whenever I picked up a copy of some popular science magazine in the school library, the physics content would almost always mention some connection to string theory (baby universes… from string theory! the big bang… from colliding D-branes! etc.). So it was pretty clear that if I was going to be doing physics research, it would end up taking me toward string theory.

Now, even then, with what little mathematical knowledge I had, I could see that string theory was pretty… messy. Far from the simple diagrams and thought experiments in Greene's book, once I started looking through the way that they formulated some of the problems, or even just checking out the consequences of supersymmetry, I could see that this wouldn't be very much fun to work in. My prejudices also caused me to be displeased by the fact that general relativity was supposed to emerge as a low-energy limit, instead of being incorporated from the beginning. So at that point, I had decided that pure math would be a lot more fun.

Then, in my senior year of high school (or perhaps a little before), I found Lee Smolin's book Three Roads to Quantum Gravity. This bypassed all the media hype about string theory as the only approach to quantum gravity, and showed me that there were actually alternatives to be found: loop quantum gravity being the foremost competitor. Now that I knew where to look, I discovered the physics was not nearly as homogenous a research field as I had thought: people were working on all kinds of interesting paths toward a theory of everything. And thus, to cut a long story of reading and discovery short, my enthusiasm for physics was re-kindled, and I find myself where I am today.

Think of the Children

The point of our story, if you hadn't guessed, is to show exactly why working physicists might want to care about the media coverage of physics: because it effects the up-and-coming generation of physicists. If string theory had continued to be unfailingly popular, how many others like me might have abandoned physics? You might think that I'm a special case, but I've got a lot of anecdotal evidence that says otherwise (and I'll be writing a post about that shortly).

But more generally, if the media portrays physics as stagnant, you might not attract many young physicists looking to make some exciting progress. If the media portrays quantum physics as inextricability connected to mysticism, then we might scare off those who are in to serious science, or even attract those who are more interesting in consciousness than in wavefunction collapse. And if the LHC is going to settle everything in a few years, then why bother getting into fundamental physics at all? Or even worse, if Garrett Lisi has a theory of everything already, what is there left to do?

And I'm not talking about those "young physicists" who are knowledgeable enough to see the big picture. Yes, if you're in graduate school, you obviously can figure out that the LHC isn't going to settle anything, and that if string theory doesn't seem like your cup of tea then you'll be able to find something else. (I personally know two graduate students who have taken the latter path.) Those people will be fine; they know enough to realize that the media coverage of their field isn't worth basing decisions on.

I'm talking about the kids for whom media coverage and popular science books are all they know of physics. The gifted high school student with nobody to guide him toward the right books, or show him the arXiv's broad spectrum of research areas, or point out the flaws in popular science articles. The middle schooler whose mom buys him The Dancing Wu Li Masters for his birthday, instead of A Brief History of Time, since she's heard that the former is a seminal work in the field. Or the college freshmen who oscillates between courses at registration, not sure whether she should go into physics based on what she's heard so far, or stick with something like business that she knows will guarantee her a job.

If we don't manage to steer the popularization of physics toward the kind of balanced sanity that Sabine discusses, then these people will be affected negatively, and in turn, so will our field. So I think that in this aspect, at least, media coverage does matter.

Pathological Monsters! Cried the Terrified Mathematician

Domenic — Wed, 14 Nov 2007 03:48:00 -0400

What's this now? Doing my weekly browsing through seminar calendars in search of something interesting, I come across an event titled "Curved Space, Monsters and Black Hole Entropy" in the Caltech High Energy Physics Seminar series. Check this out:

We use curved space to construct objects of extremely high entropy -- potentially higher than that of a black hole of equal mass. Due to their pathological properties, we refer to these objects as monsters. However they seem to be legitimate physical configurations and should be part of the Hilbert space of gravity. Our results suggest that the relation between black hole entropy and the number of microstates of the hole is more subtle than perhaps previously appreciated.

A Calculation Excursion

Now, the way I'd always heard it, black holes were objects with maximum entropy for their given volume, not mass, so I was a little confused here. I started working out the equations, which led to me installing a TeX(-like) equation renderer on this blog, so now both you and I can use <math /> tags to have equations in posts and comments! Oh, we lead such an exciting life.

Right then, let's start working through things. This is basic algebra, but what the variables represent might be unfamiliar to you, so I suggest poking around Wikipedia if that's the case. Also, we're going to use Planck units, since they make life easy. And finally, note that the geometric quantities here—viz. radius, area, and volume—refer to the event horizon of the black hole, and not the singularity itself (for which all of these are zero, at least until quantum gravity shows up to save the day).

For black holes, the entropy is given by , and (for non-rotating ones) the radius is . , so . We also get , and hence . This latter "density" is necessarily the mean density of matter within the black hole's event horizon, for (as mentioned above) all of this mass is actually concentrated at the singularity.

So the way that I've heard it, is maximal. That is, given any other object of mass and volume , that object has .

The article is claiming that there is some "monster" object such that is higher than the same value would be for a black hole, which is . Our question is, under what conditions does this give ? Well,

This will imply iff . Simplifying, this means . But this latter expression is exactly equal to !

So! These "monsters" only violate our usual maximum entropy principle if they have a greater density than black holes do. Can such a configuration exist? We'll have to read on to find out...

The Story

All right, now that we're done having fun with our new equation renderer, let's try to figure out what's really going on. Well, it appears that the arXiv has a paper of almost the same title, with one of the coauthors being the person who will be giving the upcoming talk. And if we check the trackbacks, we find that this coauthor has his own blog, where he has a post on the paper! So now we have several sources of summarization: the Caltech talk summary; the arXiv abstract; and the blog post. But then again, the paper's only five pages—four, if you don't count references—so you might just want to go read it. I'll do that now.

Well huh, that was interesting. The abstract really doesn't tell the whole story here. Here are some highlights, keeping in mind their definition the crucial quantity , where is the "energy within radius ."

As demonstrated, curved space configurations can have greater entropy than their flat space counterparts of the same mass or size. This is because of their small : the configurations have proper surface area , but have internal proper volume much larger than . Equivalently, they have very large proper mass relative to mass . It is easy to see that the ratio can be made as small as desired if approaches zero for large . The large negative gravitational binding energy allows us to pack substantially more proper mass into the region than suggested by a flat space analysis.

And the arbitrarily large entropy we were promised:

Without a constraint on how close can get to zero, can be made arbitrarily large. Invoking quantum effects, one might require that a Planck length uncertainty in the proper radial distance not cause horizon formation [...]. This implies [...]. This is still potentially problematic for the area entropy of black holes. A limit of would require that . This would be the consequence of the previous logic if one assumed a Planck length uncertainty in the radial coordinate rather than the proper radial distance (or equivalently an uncertainty in proper radial distance which grows as ). This seems unphysical, but nevertheless cannot be excluded as a consequence of quantum gravity.

And the real shocker:

To obtain entropy scaling faster than , we must consider configurations in which is close to zero in regions containing significant entropy and energy density. We now show that such configurations have the following pathological properties.

They inevitably evolve into black holes, even in the absence of any outside perturbation.

Even their time-reversed evolution leads to black hole formation.

They are therefore neither ordinary black holes nor ordinary matter configurations. We refer to them as monsters.

Finally, from the comments in the linked blog post, Steve adds:

Note these configurations have so much entropy (number of possible states) that they can't be accommodated in any kind of holographic dual description.

I'll say! So if these things exist—which apparently could be possible as a result of quantum tunneling from an ordinary matter configuration—then it looks like the holographic principle is doomed. At least, as far as I can tell...

Tying Things Together

How does this fit with our above derivation? Well, the authors do not mention an entropy/volume ratio at all, which at this point makes me think that all the cool physicists must be thinking about entropy/mass ratios instead and only us amateurs who are constantly referring to Wikipedia end up thinking about volume. But really, this is pretty immaterial: the main points don't relate to the entropy/volume ratio, but instead to the entropy/area ratio—because this is what's really of interest when we're trying to overthrow the holographic principle, for instance. Nevertheless, I'll try to extract some information on these monsters' densities, to tie things back to where we started.

They give two specific examples of such monster configurations, both spherically symmetric. The first, a "blob of matter," has scaling with , but interestingly enough has a density less than . We can see this from the fact that at the very densest region, the uniformly-dense core (outside of which the density tapers off as ), the density is chosen to relate to via . So this object has entropy greater than any ordinary matter (which is bounded by , as discussed early in the paper), and indeed entropy on the order of a black hole, but has density less than a black hole, and so by our opening reasoning does not violate the (supposed) maximality of black-hole . Interesting!

The second configuration consists of a thin shell of mass, with inner radius and outer radius . The authors give two instances of this: one which obeys the ordinary-matter bound , and one which can take on arbitrarily high entropy, independent of the area, in the region for some . And this time, it appears to happen via arbitrarily high density. Actually, the details on this—which are too technical to really do here; ask me in the comments if you insist—are pretty interesting. They seem to imply that, by choosing differing values of and for these shells, you would be dealing with a density that was either higher or lower than black hole density depending on the radius of your object. For example, take , : then if (approximately), the overall density of the shell is greater than the density of a black hole of the same radius, whereas if then it is less.

This becomes really fascinating once you realize that we're still working in Planck units. So, that 1.5? That's 1.5 Planck lengths. A popular conjecture (i.e., something that people heuristically argue will follow from quantum gravity) is that all length must be quantized in integer multiples of the Planck length, and that the "minimum length" of anything is one Planck length. So what we have here, for this denser-than-a-black-hole shell, is , , . That is, an empty sphere of radius 1, surrounded by a width-1 shell of bounded entropy, surrounded by another width-1 shell of arbitrarily high entropy. This is more or less a concrete picture of what such a monster configuration would look like, if I'm not mistaken.

Now What?

That was pretty fun. I'm going to have to write another blog post about the process of what I just did—find a thing, play with it, find a paper on the thing, work through it, come up with interesting stuff—because it's pretty much representative of the sort of free-to-dabble "research" that undergraduates get to indulge in. (Other groups in academia might do this too, but I would imagine after a certain point you start becoming locked in to your specialty?)

But seriously. Where do I go from here? Well, I'll certainly attend that seminar next Thursday—I'll have to make sure to wake up before 16:00, but that's usually doable. I'll play around with these equations some more, and see if I can get any more conceptual insight into both "monsters" and black holes in general. (I encourage you to do the same!) I'll follow some references in that article: the sentence "see [18] for a discussion of highly entropic objects and their effect on black hole thermodynamics" points to a particularly intriguing-looking paper.

And, well, I'm almost embarrassed to admit this, but despite having just discoursed extensively on black holes, I still don't really know general relativity. So I'd better continue moving through my (excellent) textbook on the subject. I'm about a third of the way through... two more chapters until we start talking about curved spacetime. I think I'd better get on that.

P.S.: Kudos to anyone who got the (probably very obscure) reference that the title was making.

P.P.S.: there's a video too!! Be sure to listen to the last ten seconds. Most excellent.

Criteria for a Theory of Everything

Domenic — Thu, 08 Nov 2007 22:55:00 -0400

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.

The Interface of Mathematics and Physics

Domenic — Thu, 01 Nov 2007 06:06:00 -0400

I could make excuses about why I haven't posted for two weeks, but that would be boring, so I won't. (Besides saying that John Dies at the End is a really fun novel that you can read online as a way to reduce your available free time.) Let's just say that I don't want this blog to die, and with any luck will have fun posts on an engaging schedule.

"The Unreasonable Effectiveness of Mathematics in the Natural Sciences"

One of the things I've recently tried to nail down is what the ideal interface is between mathematics and physics. Of course, plenty of smart people have wondered about this before; this section's title is taken from a very famous paper by the same name. And I don't mean to reiterate those points, but rather see where they take us.

While talking with a friend, we agreed that neither of us could really conceive of a world in which you could not mathematically describe "how it works." I simply cannot believe that it would even be theoretically possible for the world to not be governed by laws. How would things "know what to do"? There is always a guiding principle, a set of equations or constraints, that explain why the system behaved this way.

People might like to invoke free will here, but to me this kind of inconceivability is exactly what makes free will such a problematic concept. Why did you choose to do this, and not that? You can say that it was the result of signals traveling throughout a biological neural network, according to the laws of physics and the initial conditions of the system. Sure, your actions might not be predetermined, but if so only because of some fundamental randomness in nature (stemming from quantum effects, or whatever is behind them). Even Penrose's speculative theories about consciousness are ultimately grounded in physical laws. Whereas, if you invoke some kind of mystical nonphysical ability to "choose," in a way that is not governed by physics, I have to ask "where did that come from? Why did you end up choosing this over that?" And in the end, you'll be reduced to something silly, like "just because" something circular like "because I wanted to." But why? Why isn't that behavior describable? In other words, even if there were some mystical choice-force at work, I should be able to describe it via a set of laws. Essentially we are talking about causality here: and free will (and gods, etc.) is not just some kind of interesting causality violation that we can theorize about, but something that stands entirely outside of the causal structure and the ability to theorize at all. So, that's why I cannot conceive of free will as more than an illusion.

That was a bit of a tangent, but if we're generous we could instead think of it as an example of the fact that our way of thinking is so entirely dependent upon the idea of mathematical laws governing the world. Why we have such a conviction is more of a philosophical question, and why it works so well is metaphysical, but it's important to note that we do. And given this, I ask, what is the right way to describe the universe mathematically?

Some Examples

Ultimately, we seem to derive our mathematical formalisms from how we view the world. For example, it may seem obvious that "of course the world should be invariant under general coordinate transforms!" or "of course things of unequal mass should fall at the same speed!" But ultimately, even though these are beautiful principles with even deeper reasons behind them, it's possible to imagine a world in which this were not the case. And the whole point of my section above was that we could still describe such a world by mathematical laws—we would have to be able to—even though they would be different ones entirely.

To further drive home this point, we note that such "of course!" moments break down at other levels. Most conspicuously, I don't see anyone going "of course nature should evolve complex probability amplitudes over time, instead of actual probabilities!" or "of course a point particle should have intrinsic angular momentum!" Even the ones that people often think of as an "of course!" are really not very good, in the end: "of course everything should ultimately be a point particle!" has, in the last few decades, come under much greater scrutiny, and now we're hoping for something more like loops in spacetime or vibrating strings.

So really, saying that one formalism is "better" or "makes more sense" for describing the universe isn't really possible. I often fall into the trap of thinking that general relativity, with its nice tensor equations based in differential geometry and mapping directly to events and gravity-as-spacetime-geometry, has a "better" mathematical framework than quantum theory. I mean, who really likes having concepts such as "operators on a Hilbert space" or "probability amplitudes" be a fundamental part of your theory? But in the end there really is no difference: each is a way of mapping the real world into certain quantities that obey observed laws. In one case, these laws fit naturally into the framework of differential geometry, and the quantities we work with and think of as "real" end up being tensors and their ilk. In the other case, they fit into the linear algebra formalism, and we work with operators on these spaces and the associated eigenstuffs.

That said, in the end we do want to have a single theory, one that fits into a single mathematical framework. There are lots of people who think they have the right framework, and I'm not really able to judge or even point you in the right direction, since I'm still attempting to master our two main theories separately before worrying about the right way to combine them. But the point is that we hope that what we come up with is grounded in some beautiful mathematics, something that isn't just a hodgepodge of what we already have—"start with our linear algebra, add some differential geometry to the right places, and poke it with a stick until they all fit together."

The Basis of All Mathematics

I seem to have neglected to talk about one of the areas of mathematics that plays a fundamental role in modern physics, namely group theory/group representation theory. The fact that we find this all over the place, from quantum mechanics's operator representations to the standard model's gauge groups to general relativity's diffeomorphism groups, indicates that it's going to end up as a crucial part of our ultimate mathematical framework. This is probably a rather obvious thing to say, but it has some interesting implications.

Because ultimately, groups and their representations and the transformations among them are just a small facet of the much more all-encompassing category theory. Category theory can essentially be used to describe all of mathematics, as it encompasses in the most general sense the idea of "mathematical object," "relationship," and "transformation." Indeed, a subset of category theory, called topos theory, can replace axiomatic set theory as the foundations of mathematics. And, even though the conventional foundations of mathematics are some of my favorite areas to study, in the end this is really a good thing. For when you ultimately trace your physical theory back to its very deepest roots, I personally would rather not have the epiphany be "Oh! The entire time, we were simply talking about sets, building them all up from this list of axioms!" One more along the lines of "oh! we were talking about relations between generalized mathematical objects!" seems much better.

Certainly, categories have come up in the work of lots of mathematical physicists. John Baez has a great page about them. Chris Isham is known for applying them to quantum theory (you have to love the title of one of his relevant papers, "The Representation of Physical Quantities With Arrows"). Indeed, a quick search through the arXiv finds 26 papers on topos theory (in the physics sections). So, wouldn't it be cool if this ended up being physics? If to describe our universe in the most general, fundamental level, wouldn't it be neat if we had to go to the most general, fundamental level of mathematics? It would be an adventure!

No Anthropic Reasoning Necessary?

One of the most frustrating counterpoints to my above "I must have laws!" viewpoint is the emergence of arbitrary constants throughout the standard model. (And elsewhere, of course—the gravitational constant G is an obvious example, but also things like the number of space dimensions, the number of time dimensions, and so forth.) Where do these things come from? A popular solution these days is to invoke anthropic reasoning, which is a pretty feeble attempt at giving us an explanation. The problems with it have been detailed in numerous places, so I won't go into it, but I think that most people will agree with me that it would be nice if we could get all those constants to emerge naturally, without invoking the anthropic principle.

Well, if one looks through category theory, one finds many instances in which something emerges "naturally." For example, I was shown the "natural" definition of a general product in category theory. I'm afraid I'm stepping outside of my knowledge-zone here, but I am assured by both Wikipedia and people who have taken the appropriate courses that many things emerge naturally in category theory, in that there is only one way to reasonably define them.

Wouldn't it be nice if we could do this with the universe? I'm not sure if this even makes sense to say, but wouldn't it be great if there was only one "natural" definition of the universe as a category-theoretic structure? How much of this would be based on what we observe, I wonder—would there be room for speculation as to "if we lived in a universe that were different in these ways, then we'd expect a different natural structure"—or would we simply come to the conclusion that nothing else would make sense? That the math just "doesn't let" anything else have the possibility of existing? I think this is every physicist's dream, in some sense (although I know a lot of them these days seem to be unfortunately smitten by anthropic reasoning).

It's an interesting idea. In the end, it'll be a fun adventure, however it turns out. That is, no matter what the ultimate interface between mathematics and physics ends up being, I will really enjoy spending my life exploring that interface, and trying to pin it down into a simple set of principles and structures.

The Dirac Sea and Quantum Field Theory

Domenic — Thu, 18 Oct 2007 19:33:00 -0400

So, I was planning on writing a blog post much earlier than this. But then, I came down with a cold, and as a secondary effect I got behind on my homework. But I'm almost healthy by now—if not entirely caught up on my homework—and a blog post popped up in my RSS feeds that I just can't ignore. Over at Tommaso Dorigo's blog, Rick "Island" Ryals has been given the floor for a very interesting guest post. As far as I can make out, the claim is that taking into account the gravitational effects of pair production solves many open problems in physics, including the inflationary universe, fine-tuning, and even the Higgs mechanism. The clearest statement of how this should be done seems to be the following sentence:

…an increasing anti gravitational *effect* is offset by the local increase in positive gravitational curvature that accompanies the created massive particle pair.

While I'm dubious that this truly has as many interesting consequences as Ryals claims, it does seem likely that he's on to something. I do wonder what the "conventional" solution to this problem is, however: that is, what would most physicists answer when asked Ryals's motivating question, "Does particle creation from vacuum energy change the gravity of the universe?"

Connection to the Dirac Sea?

This may be a flaw in my understanding (or in Ryals's writing, as opposed to his theory), but the post mentioned above makes a pretty abrupt jump into the Dirac sea theory, and as far as I can tell doesn't actually make any kind of even semi-mathematical connection between the two concepts. That is, I'm not seeing anything discussing the implications of gravitational pair-production effects in the Dirac sea paradigm, but instead just an attempt to use the Dirac sea idea to point out that quantum field theory has problems with the vacuum state.

(There's also the puzzling matter of this quote: "Dirac’s theory was flawed though, in-spite its success at predicting the existence of the 'positron', because it can’t fully account for particles of negative energy, since it is restricted to positive energy particle." Um, Dirac's theory works pretty much in the opposite way of what he's describing: it does fully account for particles of negative energy, using them to fill the vacuum state. It does not contain conventional quantum field theory's restriction to positive-energy particles only, but instead allows both to exist and relegates one to filling the vacuum state. Or is he claiming that there are actual negative-energy particles—as opposed to positive-energy antiparticles—and then saying that they shouldn't be confined to filling the vacuum state?)

However, what really caught my eye in this section was his link to one of Dan Solomon's papers. During the summer, while browsing idly through the arXiv, I too ran across Dan Solomon and the many papers he has written on the vacuum state in quantum field theory, with emphasis on how it differs from that of the Dirac sea theory. Curiously, nobody seems to have cited him or picked up on his results. I emailed Dan about his ideas, and received the following summary of them:

As you evidently know quantum field theory (QFT) is suppose to be gauge invariant. However when calculations are done non-gauge invariant terms can appear. These terms must be eliminated in order to achieve a physicaly correct result. There are a number of mathematical techniques that are used to eliminate the problem. The question is why does the problem exist in the first place? As you know from my papers I claim to show that the problem occurs due to the fact that when the vacuum state is defined in the standard way it is the state of minimum energy.

Now this has an interesting implication. It implies that if the vacuum state were defined correctly it would not be the state of minimum energy. This possibilty is a frightening thought to most physicists. If you look in any book on QFT the vacuum state is always assumed to be the state of minimum energy. This is where my work on Dirac Hole theory comes in. Hole theory is not of much interest to current physics because it has been replaced by Quantum field theory. However at a "simple" level Hole theory and QFT should be equivalent. Therefore results from hole theory should carry over to QFT. Now consider the hole theory vacuum, or the dirac sea, as it is sometimes called. This consists of electrons that occupy the negative energy solutions to the Dirac equation. These electrons are like "normal" electrons in that they interact with an electromagnetic field. So the question I asked myself is whether or not the Hole theory vacuum is a state of minimum energy. That is, can you extract energy from the vacuum through interaction with an electric field? In my paper "Some new results concerning the vacuum in Dirac's hole theory" I show that the anwer to this question is yes. You can extract energy from the hole theory vacuum therefore it is not the state of minimum energy. The result of all this is that perhaps some of the assumptions we have made about the vacuum should be re-examined.

He also confirmed that he has received almost no feedback on his ideas, and admits that one of the possible reasons might be that "there is something fundamentally wrong with my work and everbody knows it except me." And that might be a possibility, just as it might be a possibility for Ryals's ideas as well. But for an interested, open-minded amaeuter like me, without any research grants to waste or supervisors to displease, I'm willing to give them the benefit of the doubt.

Is There Something Wrong with QFT?

Back during the summer, shortly after exchanging emails with Dan, I showed his papers to another undergraduate friend of mine with a passion for physics. While he generally knows more than me, neither of us have the background to check over the papers in all their mathematical detail, so I wasn't able to get any conclusive answers. But we both agree that this kind of problem ultimately seems to stem from the shakiness of the mathematical foundations of perturbative quantum field theory. For a funny introduction to these problems, and also an idea of how much I understand of them, check out the section "Life Cycle of a Theoretical Physicist" in these quantum field theory lectures.

In essence, as far as I understand, we've been trying ever since QFT came out to apply quantum mechanical methods and logic to our new theory; this often doesn't work too well. For some reason we're able to get a lot of predictions out of QFT, but we do so by through renormalization and cutoffs and all of these messy, often ad-hoc procedures that, at least to me, say that we're missing the right combination of conceptual and mathematical framework. I mean, virtual particles? What are those supposed to be? Mathematical artifacts, or actual particles that travel from here to the moon and back—faster than the speed of light, no less—before contributing to our path integral? The work of people like Dan Solomon and Rick Ryals, assuming it's at least partially valid, is bringing this kind of thing to light. But it would wbe nice if we were just able to start from some simple axioms (like, say, "the laws of physics are the same for all comoving observers" or "coordinates don't matter") and come up with a way of reproducing the predictive power of quantum field theory, while naturally being able to handle things like gauge invariance, Lorentz covariance (or, better yet, general covariance), etc. without making a big perturbative mess and fixing our gauge and so on.

Going Forward

As we move forward, it seems like there might be a lot to be gained from attempting to reshape quantum field theory into something more elegant, instead of hoping that quantum gravity will solve all of our problems. (Our current major contenders seem to have calculational difficulties of their own that remind me vaguely of QFT's, but here I'm getting way out of my league so I should stop speculating.) If we manage to axiomatize it in a nice way, with general relativity-style axioms (as above) instead of quantum mechanics-style axioms (states are vectors in an abstract space that contains an inner product; upon "measurement" these collapse to eigenstates; etc.), then maybe we'll solve some problems in quantum foundations too! And if we're lucky, once this hypothetical "beautiful-QFT" is worked out, quantizing gravity might become easy in its framework.

Unfortunately, there's another alternative that also seems likely. Namely, that we won't be able to solve QFT's problems without incorporating some features of gravity. This would certainly make things more complicated, and explain why quantum field theory is such a mess right now. And one the one hand, it'd be pretty exciting: everything is connected to such a degree that we can't even separately account for these aspects of nature! On the other hand, it might just reduce to our existing quantum gravity programs (which, while not boring, aren't yielding any "eureka!" moments currently). I don't know, though; has anyone looked at quantum field theory, and said "what do we need to fix, and can gravity help?" It seems more as if people are asking "how can we quantize gravity?" Ryals's work seems to be driven at least somewhat from the former direction, which is part of why it interests me.

Clearly, the only thing for me to do is learn quantum field theory myself so I can start working these questions out to my satisfaction! Curse my silly wave mechanics homework for getting in the way…

Let's Get This Blog Started

Domenic — Sat, 13 Oct 2007 14:18:00 -0400

All right, first blog post. How exciting!

It's Finally Up!

Yeah, after maybe two years of existence, domenicdenicola.com is finally going to have some content. I've removed all the broken links that mapped to sections of the site I was hoping to, at one point, fill in with interesting content; now it's just the blog and the résumé. Hopefully I'll get around to putting up the "About" section (about me, about my philosophies, about my interests, about my future) sometime in the near future, but you never know. A blog should be simple enough to maintain though, right? Once it's started?

Speaking of getting started, that wasn't exactly easy. Sure, I could have started something over at wordpress.com and called it a day. But where's the fun in that? It's much nicer if the blog is well-integrated into one's personal website, preferably with a minimum of code duplication. There's also the issue of getting blog software that works with my preferred web framework, ASP.NET: over the last year or two, I've been searching for something that wasn't absurdly complicated or filled with legacy code, and finally I found BlogEngine.NET. Of course, it wasn't perfect, so I spent about three weeks tweaking the settings, source code, themes, etc. But now, it's ready to go. Yay me!

What Am I Going to Write About?

I'm hoping to make this into an actually-interesting blog, in the tradition of those that appear on my blogroll (see sidebar, bottom right). That is, not a diary of my observations of the world or day-to-day life, and certainly not the quintessential "what I had for breakfast this morning" blog, but instead something with interesting and informative posts about specific subjects.

These subjects will probably be mostly focused around physics. And by "physics," I mean interesting physics: quantum mechanics, quantum field theory, general relativity, quantum gravity, and such. None of this condensed-matter or particle-physics stuff for me; I am definitely not an experimentalist, and in fact am majoring in mathematics so that I don't have to do any physics labs. I recently immersed myself in some quantum foundations research at the Perimeter Institute over the summer, and so have many thoughts swimming around in my head about quantum foundations and related subjects. I thought this would be a nice outlet for me to share my thoughts on topics such as issues in the foundations of quantum mechanics, the physics community, and current events in physics. Of course, there should occasionally be diversions from this diet of physics posts, but the content should still be of general interest and provoke some thought on the subject. Some topics that come to mind might be mathematics, undergraduate life, or something neat I found online and would like to discuss in some depth.

As a sample, here are some things I'm hoping to blog about in the near future:

Why I like the de Broglie–Bohm interpretation of quantum mechanics, and what problems remain even after adopting it.
The essential division of interpretations of quantum mechanics into four categories.
The string theory debate from an undergraduate's perspective.
Career ideas, for my future.
How I became interested in physics over time.

Doesn't that seem interesting?

A Word of Caution

I am not an expert on pretty much any of the subjects I blog about. I mean, this should be fairly obvious from the fact that I'm a 19-year-old second-year undergraduate student, who doesn't even get any formal education in quantum mechanics until second term this year (which is currently eight weeks away). But, I've done my reading; I have the enthusiasm; and when I don't know something, I'll try to make that clear. For example, I would conjecture that I know more about de Broglie–Bohm theory than any professor at my school, as I've spent a comparatively large amount of time on it (and nobody here even has "quantum foundations" listed as an interest, much less hidden variable theories). But there are tons of people who know more about string theory than me; my knowledge is only slightly above that of someone who's read (and understood) Brian Greene's The Elegant Universe.

So, with that parting note, farewell! Until next time…