Sunday, March 23, 2008

Fields and their Discontents

How does science make progress? In school we all learned about the "scientific method": data, hypothesis, experiment, new hypothesis...resulting in incremental improvement to our models of the world. When it comes to fundamental physics, however, this paradigm is rather inadequate, because it gives the impression that hypotheses are arbitrary, unconstrained, and concocted as needed to fit new data. Nothing could be farther from the truth.

Hypotheses in fundamental physics take the form of mathematical theories of the underlying structure of the universe, and mathematics is neither arbitrary nor unconstrained. Only certain mathematical structures exist and our theories must be built using these. New structures can be discovered, of course, but the latitude for constructing them is tightly limited by the requirement of logical consistency. For this reason, very sweeping hypotheses may often be put forth on the basis of little, or even no new data, but simply by investigating the mathematical consequences of our existing theories and fixing purely mathematical flaws.

In this case theorizing proceeds, not inductively, by gathering more data and seeking models to fit it, but rather deductively, by seeking some new mathematical structures which can resolve problems in the existing framework. Often enough there is only one mathematical structure which can achieve this. Of course it must be validated by experiments before we believe it, but if we find a complete, mathematically coherent hypothesis, chances are good that it is correct, because such hypotheses are not common. One or two key experiments may be all it takes to convince the community of a new theory when no mathematically compelling rival has been found.

It is not too much of an exaggeration to say that all of modern physics was born in this fashion. In the early to mid 19th century, Michael Faraday and James Clerk Maxwell had introduced a major new mathematical concept to the world, the "field", and had argued convincingly that the phenomena of light, electricity, and magnetism could all be unified in a new theory based on this new concept. The new theory was called Electromagnetism, and was the first major advance in physics since Newton's laws. The new theory scored success after success, but after several decades some clear thinkers began to notice that the field concept contained certain inherent difficulties. These were Lord Kelvin's famous "two small clouds" on the horizon of physics, and they would grow, respectively, into the revolutionary storms of Relativity and Quantum Mechanics.

To describe these problems let's back up and review physics as it was before the advent of the field. Before fields, there were particles. Particles were discrete bundles of matter, not subject to further analysis, and had a definite location at each moment in time. They exerted forces on each other by instantaneous action-at-a-distance (Newton's law of gravity, and the similar laws of electric charge attraction and repulsion).

The phenomenon of light, however, is very difficult to understand with a particle model. Its diffraction, refraction, and interference behaviors can only be explained by assuming light is a wave. But a wave of what? Something has to be "waving", just like the water whose up-and-down movement constitutes water waves; and that something is the newly invented concept of the field.

A field, unlike a particle, exists everywhere. In every nook and cranny of space, at all times, within and without any other matter, the field is there. It is somewhat analogous to temperature and pressure in the Earth's atmosphere; for every point in the space above the Earth, there is a temperature number and a pressure number. Likewise, a field is described by a certain set of numeric values at every point in space and time (for electromagnetism, there are six values). The larger the values, and the more rapidly they are changing, the more energy the field contains at a particular location. A disturbance at one location, like a pebble dropped in water, spreads by waves into the surrounding space.

We can't go further into the physics of fields and waves here, but the important point to grasp is that the field is a new kind of mathematical beast. Particles are defined by a location; fields are defined by a value at every possible location.

Every possible location is a lot of locations, and therein lies the first, and most crucial problem with fields: they have too much energy-storage capacity. You can always pack more energy into a given little region just by making the field fluctuate more rapidly in that region. This, unfortunately, makes it impossible to cook food! An oven works by heating up the surroundings of the food, so that heat is transferred to the food. The surroundings of the food include, of course, the electromagnetic field, so this must be heated up. But no matter how much energy you pump into the field in the oven, there is always room for more - there are always higher frequency modes of fluctuation which are not yet filled. The field, therefore, can never be heated to any temparature; both the oven, and the food in it, would see all of their energy sucked away by the field, making them colder than they started (indeed, taking them to absolute zero).

This paradox of fields was known for technical reasons as the "ultraviolet catastrophe", and it shows quite starkly that a classical field theory such as electromagnetism cannot be a fundamental theory of nature. No matter how well it seems to match many experiments, it is not mathematically possible for it to truly represent a universe in which any structure, e.g. life, could exist.

Max Planck was obsessed with this problem and, in perhaps the most remarkable bout of theorizing in the history of physics, he concocted a mathematical formula to resolve the oven problem, and a profoundly non-intuitive mechanism to underly it. Planck's formula was ad-hoc and just the tip of the iceberg - the first glimpse of a new, consistent mathematical structure which contains the old field theory, mostly, and resolves its problem of "too muchness". The new structure, called Quantum Field Theory (QFT), was created in the 1930's and its profound mathematical depths are being plumbed to this day.

Between Planck's discovery, in 1900, and the advent of QFT in the 1930's, physicists were engaged in working out a preliminary stage of this theory, namely Quantum Mechanics. Quantum Mechanics is a theory of particles, not fields, and this has obscured the fact that it came into existence to resolve a problem with fields. The universe could be made of classical, Newtonian particles; or, it could be made of Quantum particles; but it cannot be made of classical, Faraday/Maxwell fields.
Classical field theories cannot underly a real universe because of the oven problem, and so far no way has been found to resolve this outside of the Quantum. It appears that the "purpose" of the Quantum is to make field theories mathematically possible.

Thus Quantum physics arose out of the mathematical difficulties of fields. The Theory of Relativity also arose from the mathematics of fields, not as a problem but rather a very unexpected mathematical consequence.

Recall that energy propagates through a field by waves, just like the water waves when a pebble drops. So what? Well, the funny thing about waves is that they have a predetermined speed. You can't push on water waves to make them go any faster; any kind of splashing or pushing you do just makes more waves, but the new waves move at the same, predetermined speed as the old ones. This is completely different from particles, which move faster if you push them harder.

Now let's imagine that everything in the universe is described by a field of one kind or another (which, in fact, is believed to be the case). Imagine an object, for example a wristwatch, which consists of various different parts. These parts have to communicate with each other in order for the watch to work. The communication happens by waves of the fields, and these waves move at a certain speed. Now here's the kicker: what if the watch itself is moving at a speed close to the wave speed? Then the waves emitted from the parts behind are going to have an awfully hard time "catching up" to the parts ahead. This moving watch is very unlikely to tick at the same rate as a stationary watch; indeed, when we look at it this way it seems surprising that it can keep working at all.

Einstein thought very hard about this problem, albeit from a somewhat different angle, and the result is his famous Theory of Relativity, in which moving clocks run slow, moving objects shrink, and matter equates to energy. I will fill in more of the logical steps here in a later blog, but the point to take away is that things built from fields act funny when they move, because waves travel with a fixed speed. Depending what the fields are like exactly, moving things can act funny in a simple way or in arbitrarily complex ways. Einstein's hypothesis is that they act funny in the simplest possible way. His theory is often regarded as a theory "about space and time", but I think it is more correct to regard it as a theory about the behavior of moving matter; however, this discussion must wait for a later blog.

In closing let me note that the problems and mathematical developments brought about by the field concept are far from finished. It turns out that most Quantum Field Theories still suffer from the problem of "too muchness". In Quantum Physics, particles (i.e., local field fluctuations) can pop into existence temporarily from nothing, and if there are too many possible modes of fluctuation (roughly speaking) the theory doesn't make mathematical sense. This appears to be the case for any possible Quantum theory of gravity, so that gravity cannot coexist with the theories we have now for other types of matter. Something beyond a QFT is needed - and so far the only compelling candidate is String Theory.

Therefore, with some exaggeration, we can say that all of modern fundamental physics, from Relativity to Quantum Physics to String Theory, was implicit in the purely mathematical difficulties which arise from the field hypothesis. Had all scientific experimentation stopped in 1850, it is quite possible that all of modern physics would still have been discovered by mathematicians, and that they would have become convinced of its truth based on consistency alone, and lack of any other discoverable alternatives.

Saturday, March 15, 2008

Relativity I : Newtonian Spacetime

This is the first post in a series which will cover one aspect or another of Einstein's Special Theory of Relativity. I probably won't put these posts all in a row...that would be too simple :-)In this post I will focus on the concept of spacetime.

If you're reading this you probably have heard at some point in the past that one of the consequences of Relativity is that space and time have to be merged together into a new, exotic-sounding beast known as "spacetime". Perhaps you have gained the impression that space and time have somehow been shown to be equivalent, or to be different aspects of the same thing, in spite of the apparently enormous differences between them in everyday life.

I will argue here for a considerably more prosaic interpretation, one which I hope will demystify spacetime to some extent. Relativity is a theory of motion and it is motion, i.e. changing position with time, which leads to mixing up position and time and necessitates the concept of spacetime. Motion is nothing new and indeed the concept of spacetime is just as relevant to old-fashioned Newtonian physics as it is in Einstein's theories. It has always been part of our commonsense understanding of movement, but it didn't receive a name until the more bizarre predictions of Einstein's Relativity drew renewed attention to it.

To understand the actual content of the spacetime concept, in either classical or modern physics, we ask a very simple question: are you sitting still right now? Before answering, consider that you are sitting on the Earth, which revolves around the sun, which revolves around the galaxy center, which moves within its supercluster, etc., etc. By what criterion can we distinguish any of these as being "sitting still" while the others are "moving"? It is hard to think of any, and indeed nobody has managed to think of any so far.

So we have no absolute criterion to determine whether an object is moving or not. But this means that we also have no absolute way to specify its spatial location! For if we could define locations absolutely, then we could also define motion simply by saying that something is moving if and only if its location changes with time.

To put this a different way, when you move you take your concept of "here" along with you. If you eat dinner and then watch a movie on an airplane, you feel that these two things happen in the same place - seat 27D, or wherever you are sitting. Of course in this case you might think that it's more "correct" to say that you ate dinner over Colorado and watched a movie over Kansas, but this just means defining the Earth itself as "sitting still", which we have just seen to be equally unjustified; and if we extend the example from airplane to interstellar spaceship moving through space at some random location in the universe, it becomes pretty clear that anyone's concept of "here" is as good as anyone else's.

And more than just "here", observers in different states of motion have different, equally valid perceptions of the locations of all objects and events in the universe. If someone asks, for example, how far you had to walk to get to the restroom at the front of the airplane, you would probably say about 100 feet, even though the distance you covered over the Earth during that walk was probably closer to a mile. In your "reference frame" aboard the plane, you walked 100 feet, while in the reference frame of the Earth below, you covered a mile; and there's nothing to prefer either reference frame over the other.

We can describe this situation by saying that motion causes space to get "mixed with" time. I see myself as sitting still, but if someone else sees me as moving, then they see my position as changing with time, i.e., "mixed with" time. Mathematically, I see my position as a constant x = x0, while the other person sees my position as x = x0 + vt, where v is the speed he sees for me. Space gets mixed with time by using a factor of speed. Note that we are certainly not suggesting that space and time have some equivalence or even similarity; the "mixing" doesn't need to go any deeper than what was just described.

Now, in Newtonian physics there is an asymmetry, because spatial reckoning varies depending on motion, but time measurements do not vary at all. Time, in Newtonian physics, is absolute, and every observer measures the same time for a given event, regardless of motion. This is the big difference between Newtonian Relativity and Einsteinian Relativity, for in the latter theory the assignment of times to events does vary among observers. The addition of time, however, doesn't add greatly to the rationale for "spacetime", it just makes things more complicated. The main point remains the same - moving observers see things differently, and the different viewpoints are equally valid.

So we see that the spacetime concept arises from very trivial observations. We simply note that different states of motion give rise to different conceptions of the locations of things. The different viewpoints are related to one another in a simple way, so we can easily convert between them, but we can't refer just to "the position" of a thing, without specifying which viewpoint we are using.

Since all of this is just common sense, we see that "spacetime" is just a fancy word for our ordinary understanding of space and time. We use spacetime concepts every day when we discuss our experiences aboard jetliners or any other form of conveyance. It doesn't mean that space is somehow "equivalent" to time, and it has nothing to do with the bizarre time-slowing and ruler-shrinking effects for which Einsteinian Relativity is famous.

All of this is not to say that space isn't merged to time in some way. It may be; nobody knows at this point. But if it proves to be the case, it won't be because of Special Relativity. General Relativity - Einstein's theory of gravity - is a stronger candidate, but here too the "merging" of space and time is less deep than it appears. If it were otherwise, we would call it Einstein's theory of spacetime rather than Einstein's theory of gravity, and theorists around the world would not be racing each other to figure out the actual underlying structure of space and time.

Wednesday, March 12, 2008

Surface Tension and the "Floating Needle"

At some point in the early years of my science education I was shown how to float a needle in a cup of water by placing it ever so gingerly on the surface, and it was explained to me that this was the result of "surface tension", a mysterious phenomenon which occurs at the surface of a liquid, where it meets the air. Exactly how this tension developed and why it could support a needle (or pond-skating bugs, the other canonical example) was never made very clear to me.

Years later it is still not very clear to me. After considerable thought, it seems to me that surface tension actually has nothing to do with the flotation of needles and bugs. The following is my best understanding this phenomenon; should any liquid physics experts happen by, I hope they will comment.

First, let us recognize that the molecules in a liquid are held together by mutual attraction. Where the attraction comes from is another subject, but without it one would have a gas and not a liquid.

Within the liquid, the molecules are packed as densely as possible, since otherwise the attraction would draw them yet closer. At the surface, however, the situation is more complicated. One might think that the surface molecules would have the same density as in the interior - but in fact this simple situation is not stable.

The surface molecules feel the attraction from their next-nearest neighbors on the surface, and also from molecules even farther away, and they would like to pop out of the surface so they can get around their near neighbor and draw closer to their further neighbors. In other words, the surface would like to crumple into a ball (i.e., a droplet) just as it would if it were an isolated sheet of water floating in the air.

Of course the surface layer can't crumple because it is attached to the rest of the water, but what happens is that it partially crumples, and some molecules do pop out, with the result that the layers near the surface have a decreasing density compared to the liquid bulk. In particular, the molecules on the surface are farther apart than those in the bulk. Since they are farther apart, yet still experiencing mutual attraction, they are in a state of "tension" compared to their colleagues in the bulk of the liquid.

We can picture the liquid as a collection of balls connected together by springs. Inside the liquid, the balls are packed so close together that the springs are mostly slack - they have lost their tension. At the surface, however, the balls are farther apart and the springs are stretched and tense. This tension tends to pull the liquid into the shape having the smallest surface area possible - a spherical droplet, for example.

So surface tension arises because the surface wants to crumple but it can't. Now we ask, does surface tension have any relevance for the "floating" of needles and bugs? This would mean that the tension makes it more difficult to penetrate the surface of the liquid, i.e., makes it more difficult to break the bonds between the liquid molecules at the surface. But as far as I can see, surface tension doesn't make these bonds any harder to break. If anything, since the surface molecules are already farther apart, they are easier to separate than those in the bulk.

Therefore it seems to me that the "floating" of needles and bugs is a separate phenomenon from surface tension. Needles and bugs are held up more by virtue of their shape, and by being placed very flat against the water, than by any special condition of surface tension.

When an object is placed with its broad side flat on a surface, then it can only penetrate the surface by breaking a bunch of molecular bonds at once; if the object is light enough, it can then rest on the surface. Conversely, if the narrow end is against the surface, only a few bonds need to be broken for the object to slide in.

Once the object is already submerged, regardless of how it is oriented, it will sink steadily because molecular bonds broken beneath it are replaced by bonds re-forming above it, so there is no overall retarding effect from the bonds. (More precisely, there is a smaller effect, because molecules still have to rearrange and their bonds give rise to viscosity).