Background for Bell's theorem

Quantum Mechanics is known for having a fundamental element of randomness to it. If you try to measure the position of a particle it is impossible to predict exactly where you will find it. That's because it was in several places all at once, a probability wave smeared through space. When we observe the particle, it "collapses" to a single location.

But why can't we just say that the particle was there all along? Maybe we couldn't know exactly where it was. But that doesn't mean it didn't have an exact location prior to measurement. What gives? Why do we need all this metaphysical junk about a particle existing in several places at once?

It turns out that there's a fairly good scientific reason for this. It was most famously demonstrated in what is known as Bell's theorem. Bell's theorem looks at the correlations between the spins of two entangled particles. I think it will help to cover some background first, so we understand the setup.

Particle Spin

In classical physics, the spin of an object tells you how quickly it is rotating around its own axis. The Earth, for instance, spins around once per day. We can describe the spin as a vector, which has a magnitude and direction. The magnitude tells you how quickly it is spinning, and the direction tells you where the axis of rotation is pointing.

Spin in quantum mechanics is analogous, but there are a few wrinkles. We can measure the magnitude of the spin without much problem. Electrons always have the same magnitude of spin, no uncertainty about it. However, we cannot fully know the direction of the spin, because the uncertainty principle prevents us. We can only measure one component of the spin at a time. For example, if I measure the vertical component of an electron's spin, I will always find that it is "spin up" or "spin down". If I measure the horizontal component of the electron's spin, I will always find it "spin left" or "spin right". But an electron cannot simultaneously be spin up and spin right, no more than it can simultaneously have a definite momentum and location.

This is relatively easy to demonstrate with the Stern-Gerlach experiment. If you shoot a bunch of electrons through a magnetic field, their path will be deflected by an amount which is proportional to the vertical component of their spin. This experiment showed that electrons are always either spin up or spin down, and never in-between. A few modifications show that the vertical and horizontal components of spin cannot be simultaneously known.

Entangled particles

In quantum mechanics, we describe particles as having quantum states. For example, we might say that an electron is in a state of 100% spin up, meaning that if we measured the vertical component of its spin, we would always find it to be spin up. We could also have an electron which is 50% spin up, and 50% spin down, meaning that we have a 50-50 chance of measuring either outcome.

When we have two particles, we do not describe them as each having a distinct quantum state. We instead talk about the quantum state of the whole system. For example, we could have a state where there is a 50% chance both are spin up, a 50% chance that both are spin down. If we measured the first electron's spin, there would be a 50% chance that it was spin up. Whatever we measure in the first particle is guaranteed to also be the result of the second particle. Unlike independent coin flips, the two particles' states depend on each other. This is called quantum entanglement.

This works even if the two entangled electrons are very far apart! In principle, we could send two entangled electrons to opposite sides of the galaxy. I could measure the first electron on this side of the galaxy while you measure the second electron on the opposite side of the galaxy. As soon as I find it to be spin up or spin down, I know what result you would get too. This is strange because if you were to send me a message from across the galaxy, it would take 100,000 years to reach me at the speed of light.

This "action at a distance" is known as the EPR Paradox, but in fact there is no paradox. Einstein's relativity requires that no information travels faster than light, because that would imply that you can send a message into the past. However, quantum entanglement does not allow you to send any information. The result of my measurement is random, and so is the result of your measurement, on the other side of the galaxy.

The set-up

First, we need a source of entangled pairs of electrons (or photons, if we like). I'm not really sure what people use for an entangled electron source these days, but I'm fairly sure that it's nothing too exotic. Each of the electrons goes into a separate detector. If both detectors are measuring the vertical component of the spin, then they will always get the same results. Either both will be spin up, or both will be spin down. Therefore, we would say that the results are identical 100% of the time.

But what happens if one detector is measuring the vertical component, while the other is measuring the horizontal component? How much correlation is there with the two electrons?

Next page

My doubts

Let's talk about emotions for a moment. My emotions. My doubts. Mostly, this is just an exercise in selfish self-expression, but I suppose I should also preface it with some sort of Point, just for all you readers.

Many religious people are rather quick to relate the fact that they've had "doubts" in the past. I believe there are basically two kinds of doubts which get seriously confused in the process. The first kind of doubt is simply the uncertainty in one's beliefs. The second kind of doubt is that emotional angst which seems to exist in all our lives. If religious people are using "doubt" to refer to the angst they've experienced, then I guess that's okay. But if religious people are using "doubt" to refer to their uncertainty in their beliefs, then there is something very wrong with their attitude towards doubt. They talk about doubt as if it were their shame, something which can only be considered good in the same sense that being "only human" is good. But what is so bad about uncertainty? Uncertainty drives free inquiry, which drives personal progress.

The thing is, I can sort of understand why these two types of doubt get so mixed up. Emotion leads to uncertainty. One kind of doubt leads to the other. However, it may be a useful exercise to separate out the two kinds of doubt. Perhaps we can pick out the good from the bad? We can improve ourselves through adversity? I don't know.

Anyways, onward with the selfish self-expression.

Sometimes, I just get that feeling of, "What am I doing in life?" Which is ridiculous, because I know exactly what I'm doing in life. I've gotten straight A's all through college, and I'll eventually go to grad school, perhaps become a professor. And I still have enough free time that I can maintain a blog. I guess I have it pretty good. So now I just feel bad about feeling bad.

I worry that one day someone will discover I'm a fraud. The Journal of Geophysical Research will discover that my research is all done wrong because it turns out I had no clue what I was doing. The band will discover that I'm actually the worst flautist in the group (and I believe that one too). Readers of my blog will discover that I'm actually just this college student who likes to write, and that my opinions are complete BS. Even when I write about physics, something I know, I worry that someone will discover that I was merely spreading misinformation.

I'm worried that people will find out that I'm a fraud as a humanitarian and an activist. I don't have the disposition for it. I cannot be "for" something without having serious doubts about it first. I won't commit. I have crippling buyer's regret. I will not spend my money on anything. Not for me, not for anyone else. And I won't donate blood either. I just--I just don't want to.

Sometimes, I feel like I just can't get enthusiastic about anything. This makes me feel like I'm letting people down. I can't root for the home team. I will never have any sort of school spirit. I will never feel proud as an American. I can't really get into the messages of "hope" and "change" in Obama's campaign, and not because I don't like him. I will never feel proud of my family heritage. Even when it comes to the subject matter of this blog, at some level, I don't think I will ever be excited about atheism, skepticism, or science. I could never inspire people about them, because I don't really feel inspired myself.

What's worse is I feel like I'm falling into some sort of stupid rationalist stereotype. Like we're all repressed emotionally, incapable of feeling. I hate that stereotype... but I fit it. And what am I supposed to do about that? Stop repressing my feelings? There are no feelings to be repressed--just the feeling that everyone else expects me to be excited when there is nothing worth being excited about. Every time I see a fictional story about how some character turns his life around by "opening up", I think I hate myself a little more for being who I am.

But I suppose I take comfort in the fact that none of the above paragraphs really make a whole lot of sense, that it's all just a bunch of rambling that I wrote in a cynical state of mind. I mean, what is this about feeling angst because I'm unable to feel? Ridiculous!

Why I'm not an agnostic

I consider myself to be an atheist, but I don't consider myself to be an agnostic. What exactly does that tell you about me? Does it imply that I have dogmatic certainty in my beliefs? Does it imply that I'm a hardcore new atheist who is obnoxiously anti-religious? No, silly.

So it's time for a bit of a story. I went to a Catholic Jesuit high school. I stopped believing right around my graduation. Depending on how you look at it, you might say I actually stopped believing the year before, or the month after graduation. I don't know. My thoughts were fluid. I didn't think or worry about it much. In particular, I didn't think at all about "What should I call myself?" Labels are not important. The ideas behind labels are important, and I was still sorting those ideas out. It was not as if I was trying to explain my journey to anyone else, so there was just no reason to try to contain it within words.

But I don't mean to demonize the very concept of a label. Some months later, when I entered college, I did start to think about what to call myself. See, I had started to interact with people online, and I needed to organize my thoughts into something easily expressed. I also needed a way to explain to myself what had happened. Labels are useful for both of these purposes. So I began to scout out for what sort of labels people use, and what they mean by them. After a while, I settled on "atheist".

The main point of this story is that deciding between "agnostic" and "atheist" was not a matter of sorting out my beliefs, but rather, a matter of sorting out descriptions of those beliefs. It was not a matter of self-searching, but a matter of internet-searching. The purpose of a label is primarily communication, so it's important to know what other people think about them, not just what I think about them.

So what did I discover on the internet that made me think "atheist" is so much better? Here are my impressions and conclusions:

1. Ask nearly any atheist, and they will tell you that atheists are not certain about their belief that they do not take it on faith. Some will tell you that they are only "de facto" atheists, in that they go through their lives in such a way that they might as well not believe in god. Some will tell you all about the difference between "belief in no god" and "no belief in a god".
   
   
2. Ask most agnostics, and they will tell you that agnostics do not think that God is a 50-50 bet. Usually, instead they'll tell you about how it is impossible to prove or disprove God.
   
   
3. In general, people will draw lots of fine distinctions between agnosticism and atheism. But the more I looked around, the more I found that these fine lines criss-crossed all over each other, blurring the distinctions more than ever.
   
   
4. Ask atheists about agnostics, and they might tell you that agnostics are wishy-washy fence-sitters who can't be bothered to make up their minds. Ask agnostics about atheists, and they might tell you that atheists are just as dogmatic as fundamentalists, only in the other direction.
   
   On the spectrum between "know-nothing" and "know-everything", it's rather clear that both atheists and agnostics overlap somewhere in the middle. And yet, some in each group will tell you that the other group is being extreme. I don't believe it, and I think it's kind of insulting.
   
   
5. Many atheists also consider themselves to be agnostics. Even if they never talk about themselves as agnostic, many will respond in affirmative if asked if they are agnostic. Why, if you asked me whether I'm agnostic, I'd probably say yes too. Or not. It depends on the context. I realize the title of this post is "Why I'm not an agnostic", but that's because I have an entire essay to explain my meaning. If asked in a different context, I might just say yes, and that would give people approximately the right idea.
   
   
6. Atheists are not necessarily mean or overly confrontational. Agnostics are not necessarily nice or underly confrontational. Even towards religion. This much should have been obvious.

So far, I've been pretty neutral. But there are two additional things that really bother me about agnosticism.

First of all, who is to say that agnostics are the only people who are unsure of themselves? Agnosticism seems premised on the idea that everyone else is completely sure of themselves, and no one else appreciates how difficult or impossible it is to prove or disprove the existence of a god. Atheists can appreciate the uncertainty of the situation too, as can religious people. Other people have doubts too. Why would humble doubt and uncertainty make me so special that I need a label to express it?

Second of all, while agnosticism prides itself on being epistemologically accurate, I feel that it is not. It's just not a big deal to say that a claim can be neither proven nor disproven. Most claims can't. That's just not how reasoning works. We usually only talk about supporting or detracting evidence. I'm not going to say that there is much evidence which is relevant to the existence of a god, but there is some (disregarding unimportant gods like the deist god). It's just a matter of flipping around the absence of evidence, and you've got something. Not much, but then I never said I was certain in my unbelief, I just said I was an atheist.

If anyone here is really wondering what to call themselves, this is my advice: Just pick whatever you like, whatever serves you best. Don't worry about the fine distinctions between the definitions, because different people draw different distinctions, and all details get completely muddled in the mess. Just try not to insult anyone in the process.

Extraordinary vs impressive

When people put forth extraordinary claims, this evokes in me a very specific feeling. It's hard to put this feeling into words, but I might express it as "What do you take me for?" or "Is this supposed to impress me?" It is a feeling of disappointment. I am disappointed if there is not enough evidence for an extraordinary claim. I am even more disappointed when people seem to think there doesn't need to be any evidence. The claims become ever more fantastic, as if this would make up for the decreasing likelihood.

But I am rambling. Perhaps this feeling is best conveyed not through words, but through math. Math is the highest art form, or at least the highest art form accessible through Mathematica. The following graph can express my feelings.

The important thing to notice is that the more extraordinary a claim is, the more impressed I am, but only up to a certain point! After that point, further extraordinariness merely strains credulity. The more extraordinary a claim, the less impressed I am. Because then it's just less likely to be true. In fact, I would say that it's exponentially less likely to be true.

Why is this so hard for people to understand?

But I think my graph is missing something. Evidence! Evidence is pretty important. After all, extraordinary claims can be impressive if they have extraordinary evidence. In fact, you might say that I'm more impressed by an extraordinary claim than an ordinary claim if it has extraordinary evidence supporting it.

This can only be expressed in a 3-d graph!

The height of this curve represents the how highly impressed I am, while the two horizontal axes represent the extraordinariness of the claim and the extraordinariness of the evidence.

These graphs, by the way, are based on the function I(x,E) = x/(e^(x-E)+1), where I is the impressiveness, x is the extraordinariness, and E is the evidence. One interesting feature of this function, is that even if the evidence is zero, there exists a particular nonzero value of extraordinariness which gives you maximum impressiveness. This particular value would be a fundamental constant of skepticism! But then again, my function is just a model, and I could have picked any number of other curves to serve the same purpose. So I guess we shouldn't read too much into the details.

What's that you're telling me? You're saying that normal people don't spend their free time modeling skeptical expressions with math equations?

Two measuring problems

A classic puzzle:
You have two unmarked, asymmetrical containers. One of them holds exactly 3 liters when full, while the other holds exactly 5 liters. You also have a sink faucet and a drain. You need to measure exactly four liters of water. How can you measure exactly 4 liters? You cannot simply fill the larger container four fifths of the way, because you have no way of knowing when it reaches that point.

And since I realize that I have at least a few readers who are just too good for the classics, here's a challenge problem, custom-made by me.

You have three timers. After you start each timer, it stays silent for a period of time, and then it dings. The first timer stays silent for exactly 10 seconds, the second timer for 40 seconds, and the third timer for 45 seconds. Until the timer dings, you have no way of knowing how much time has past. Using these timers, can you measure an unbroken time interval of exactly 20 seconds? The catch is that after each timer dings, you must rewind it before starting it again. You cannot rewind a timer instantaneously; however, you may assume it takes less than 5 seconds to rewind.

Dots and boxes solution

Remember the Dots and Boxes puzzle? Of course you do.

Before I give away the whole solution, I will state the targets. For a 5x5 grid, the target is 15 lines. For a 6x6 grid, the target is 23 lines. For a 7x7 grid, the target is 31 lines. If you show me a solution that is better than any of these targets, I will be amazed, because I just don't think it's possible to do any better.

Solution to the 5x5 grid

You should know, by the way, that the reason I reveal these one at a time is in case anyone is inspired to go back and try to solve the larger grids. Eduard Baumann was the first solver of the 5x5 grid, and Secret Squïrrel was the first solver of the 6x6 grid. Nobody reached the target for the 7x7, but you still have a chance to try it yourself.

Solution to the 6x6 grid

There is another distinct solution to the 6x6 grid using the same number of lines.

Solution to the 7x7 grid

When is Theistic Evolution acceptable?

If you thought I would go another Darwin Day without any mention of the Evo/Creo wars, you were wrong! This has little to do with Darwin himself, but I hope you enjoy it anyway.

When you first join a social networking site (ie Facebook or Myspace), one thing that takes a little getting used to is how friendship is treated in such a black and white manner. You're either a friend with a person or you're not. There is no "we say hi every time you run into each other", "we like to chat, but we've never actually met", or any other in-between. There are workarounds, but I'm already used to the black and white, and I'm just not willing to invest the effort to change it.

When it comes to respect or acceptability, I am not used to black and white. I don't like to say that I respect or don't respect a person, or that I find a certain view acceptable or unacceptable. There are many different levels of respect and acceptability, and I'm perfectly willing to invest the effort to yammer on about their distinctions on my blog.

When it comes to Theistic Evolution (which is merely any system of belief which includes evolution and God), there are many levels on which I consider it acceptable or unacceptable. In one sense, it is unacceptable, because theistic belief is unacceptable to me. In another sense, it is acceptable to the extent that it politically opposes the Intelligent Design movement and other Creationist movements. On this same level, I would also find it acceptable if a Young Earth Creationist opposed Creationist movements on the grounds of separation of church and state. The enemy of my enemy--that's another thing that doesn't exist on Facebook.

I want to outline another in-between level on which Theistic Evolution is acceptable, but Intelligent Design is not. The distinction: A belief is acceptable if and only if it would not hamper a person's efforts to academically study any specific branch of science. Let's apply it to various positions on evolution!

• Young Earth Creationism - Obviously not acceptable, because there are lots of things, from evolution to geology to astronomy, which you just can't study while believing the Earth is six thousand years old.
• Subcategory: Omphalism (aka Last Thursdayism) - Omphalism is the belief that, even though the Earth and everything was created six thousand years ago, it was created to look exactly as if it would look if it were much older. Omphalists believe that sciences which go beyond six thousand years ago are incorrect, but still have predictive power because God made it that way. Omphalism is acceptable to the extent that they accept the predictive power of evolution (and other sciences).
• Gap Age Creationism - This is the type of Creationism which states that life was created on a previously existing old earth. Technically, this is slightly better than Young Earth Creationism, in that it might not hamper earth science as much--but that's just not good enough!
• Intelligent Design - Usually by "Intelligent Design", we refer to the political movement which wants to promote "alternatives" to evolution in schools. But for the moment, I am referring to general attitude of evolution denial which underlies the Intelligent Design movement. This is bad. You can't be a respectable biologist and simultaneously support Intelligent Design. And most Intelligent Design arguments just don't pass muster. Examples:
  
• Irreducible Complexity - The explanation for this has been around for a long time.
  
• Only microevolution, no macroevolution - Not good enough! If you believed in Electromagnetism but not Special Relativity, that wouldn't cut it either.
  
• Theistic Evolution - This is acceptable, because it accepts evolution. As far as I can tell, theistic evolutionists do not make any worse scientists than anyone else. Argue all you like about how hard it is to reconcile science and religion, but that's only in theory. In practice, theistic evolutionists don't have those problems, so that's good with me.
• Exception: God made altruism - If you believe that altruism can't evolve, and that divine intervention was needed, you got problems. You may not be barred from evolutionary biology, but you are barred from a particular branch of biology, the evolution of cooperative behavior. Similarly, if you have a problem with junk DNA, then you have a problem.
  
• God made the soul - If you believe that it required divine intervention to create the human soul at one point of evolutionary history, then that's actually okay. Unless you think the insertion of the soul makes some prediction about the natural world, I just don't see how this would hurt your scientific study. However, this may change with developments with psychology, if the common conception of the soul becomes noticeably at odds with science.
  

I leave it as an exercise to the reader to apply this principle to other beliefs.

Darwin's Flatfish Flounder

And now for something completely different. Today is Darwin Day, the bicentennial of Darwin's birth, and this year is the sesquicentennial of the publication of On the Origin of Species. I don't know about you, but I think an ideal topic for today is to talk about one of the ways in which Darwin was wrong.

He was wrong about flatfish.

Flatfish, by the way, are those weird fish which have both their eyes on the same side of their head. They look sort of like they were designed by Picasso. They spend their adult lives swimming sideways, flat.

A few interesting facts: While they're usually camouflaged on top, their underside is usually just a pale white. Unlike most fish which flex side to side to swim, flatfish flex up and down (because they're sideways, of course). However, they do not look like this their entire lives. They begin their lives looking like normal fish, with one eye on each side, and then one eye migrates over to the other side. Most species are exclusively right-eyed, or exclusively left-eyed, but the more primitive forms often have a mix of left-eyed and right-eyed fish within each species.

In Darwin's day, the flatfish was used as an argument against the theory of evolution. Darwin's theory of evolution (keeping in mind that he did not know much about the mechanisms of heredity and development) required that evolution occur in small, gradual steps. Each step of the way must be evolutionarily advantageous, or it will never catch on. But then, how could flatfish evolution be possible? What possible advantage could be gained by having an eye which migrates only part way to the other side of one's head? Until the eye reaches the other side of the head, it seems like there is no advantage at all, or even a disadvantage.

Darwin, of course, had an explanation. On the Origin of Species has a chapter called Difficulties on Theory in which Darwin answers many of the contemporary objections to his theory. He devoted several paragraphs to flatfish. Let us ponder his wisdom:

The Pleuronectidae [flatfish], while very young and still symmetrical, with their eyes standing on opposite sides of the head, cannot long retain a vertical position, owing to the excessive depth of their bodies, the small size of their lateral fins, and to their being destitute of a swim-bladder. Hence, soon growing tired, they fall to the bottom on one side. While thus at rest they often twist, as Malm observed, the lower eye upward, to see above them; and they do this so vigorously that the eye is pressed hard against the upper part of the orbit. The forehead between the eyes consequently becomes, as could be plainly seen, temporarily contracted in breadth. On one occasion Malm saw a young fish raise and depress the lower eye through an angular distance of about seventy degrees.

I realize Darwin's writing is pretty dense, so allow me to parse it for you. Flatfish, when they are young, vigorously try to move their eye. Because of the extended stress, their eye eventually moves over to the other side. This behavior and condition are inherited and acted upon through natural selection.

The problem is that this explanation is rather Lamarckian (though there's a bit of natural selection in it too). Lamarckianism was a sort of alternative to Darwin's natural selection, which stated that a creature's behavior would give it new characteristics over time, and that these acquired characteristics would be inherited by later generations. For example, the Lamarckian explanation for giraffes might be that they kept on reaching for the tops of trees, and eventually, over generations, they grew long necks. Lamarckianism is currently unaccepted. Though creatures may acquire new characteristics in their lives, these acquired characteristics are not heritable unless it some how affects their genes or gene expression.

To be clear, Darwin did not reject Lamarckianism (as the example with flatfish demonstrates). In fact, he was sympathetic to it. What Darwin said was that in addition to the Lamarckian mechanism, there is also the mechanism of natural selection, which he dare says is the dominant mechanism. Darwin was too cautious a scientist to go so far as to say that Lamarckian evolution was impossible. After all, Darwin didn't know anything about genetics, so he couldn't have ruled it out.

So the question is, if Darwin couldn't explain it, have we come up with an explanation in the 150 years since that time? The answer, I found, is no. We are not sure of the explanation. However, there's a recent article in Nature which sheds some light into the question!*

The evolutionary origin of flatfish asymmetry

This paper discusses two extinct genus of flatfish, called Amphistium and Heteronectes. These are the most primitive forms of flatfish that we know of. By examining the fossil evidence, the author concluded that there was no torsion-induced damage. This suggests that the eye migration is not caused by the fish's vigorous effort, as Darwin suggested. So we know that whatever the answer, Darwin's is not the correct one.

The other interesting thing about Amphistium and Heteronectes is that their eyes do not completely migrate to the other side. Their eyes are asymmetrical, but they still remain on opposite sides, even in the adult forms. These genus' were evolutionarily stable, and existed for at least two geological stages. This shows that there was a gradual transition from having eyes on the opposite sides of the head, to having both eyes on the same side of the head. This was not the result of a sudden mutation. No punctuated equilibrium here.

Perhaps the transitional forms were not as maladaptive as we previously thought. The author suggests that primitive flatfish were still able to use the part-way migrated eye by propping up their body (as certain modern flatfish sometimes do). But we can't say for sure! I think the flatfish are mocking us.

*Some years ago, I wrote a research paper on this topic for a class, and I was pretty disappointed to find that there was no known explanation for Darwin's flatfish problem. When this Nature paper came out, I was pretty excited. If only the Nature paper had appeared before I took that class.

You may be interested to know that other people are also blogging for Darwin Day. Click these for some blogging carnivals!

A ghost dream

I had a dream! I don't remember details, but it was like I was a character in some TV drama or something. The character I played was dead, a ghost, but I could still speak and interact with the living. In particular, I spoke to a lot of relatives and friends. We were plotting for me to come back alive somehow. I forgot how I actually accomplished all that, but it involved traveling all across town, and it was possibly very dramatic. But somehow I did it.

But there was a twist ending! A voice over said something like "And I could not have done it, if it weren't for the faith of my friends and family, who, without ever seeing me, trusted that I was there the whole time." Then there was a flashback montage, cutting to several scenes where I had been speaking to people, only this time, I'm not actually there! There's just this woman on the subway, talking to herself the whole time. Despite what the previous scenes in my dream led me to believe, I could not in fact interact with the living when I was dead. No one had any evidence that I was ever there, except that they "felt" my presence.

After that, I recall thinking about how stupid the ending was. Feeling a "presence" is just not sufficient evidence of, well, anything! I think my subconscious is just making this stuff up as it goes along.

Dimensions in String Theory

See the previous post in this two-part series: Dimensions in the Universe

One of the obstacles in String Theory is that it predicts 11 dimensions (10 of space, 1 of time). Clearly, our universe has 4 dimensions (3 of space, 1 of time). So how could String Theory possibly be a good description of reality?

To reconcile this discrepancy, it is good to know why String Theory predicts so many dimensions. The basic idea of String Theory is that elementary particles are not point-like objects, but are tiny string-like objects. How tiny? If particles are point-like particles, then their size is precisely zero. But according to String Theory, they are strings with length of roughly 10^-35 meters. Let's just say that if you shrunk down to the size of an atomic nuclei, then you'd be less than halfway there. One of the problems with String Theory is that, due to the uncertainty principle, it is practically impossible to directly observe the size of the tiny particles.

In order for this idea to be consistent with the kinds of particles we see, the strings need to be able to vibrate in 10 different directions. To the string, the seven extra dimensions are just as real as the first three. But you have to remember that these strings are tiny! If the strings can see the extra dimensions, but we cannot, perhaps the extra dimensions are just really small.

What does that even mean, for a dimension to be small? There are essentially two different ways this can happen. Both of these will be illustrated using an analogy with a hose. (I'm stealing this analogy from Lisa Randall. But the graphics are mine.)

Let's say you're a giant. If you see a hose, you might think it was a one-dimensional object. You can only go up the hose or down the hose. If there is any multi-dimensional structure in the hose, it's just too small to see. The hose, as far as you're concerned is one-dimensional, the same way that a piece of paper is two-dimensional. Of course, paper has a small thickness, but for most practical purposes, you can ignore it.

However, if you were that little ladybug on the surface of the hose, then it would be a different story. Then you'd see two dimensions. You can go up and down the hose, but you can also go clockwise or counter-clockwise around the hose. Of course, if you go far enough clockwise, you'll end up back where you started. Nevertheless, it still counts as a second dimension. Strings, because they are so small, might be like that ladybug, with access to extra dimensions. But that's only one extra dimension. How do we get seven?
This is another way of drawing the hose, in more abstract form. The blue line represents the large dimension, that even the giant can see. The green loops represent the extra dimension that only the ladybug can see. But what if, instead of loops, we have more complicated shapes? For instance, if each loop were replaced by the surface of a sphere, we would get two extra dimensions rather than one. In String Theory, we replace the loop with a weird seven-dimensional manifold called the Calabi-Yau manifold. I have no idea why it needs to look that way, but that's what String theorists have found.

As I said before, there are two different ways to explain the extra dimensions. One is if strings are like that ladybug. The second is if strings are like the water inside the hose.

A hose is a three-dimensional object. If you are a giant, it appears as if the water can only go upstream or downstream, but in fact there are two extra directions in which the water can go. The water can also go up/down, and left/right. However, the water analogy is distinct from the ladybug analogy in an important way. The ladybug could go clockwise indefinitely, eventually reaching its starting point. The water cannot go left indefinitely, because there is a wall in the way. The walls of the hose confine the water to a small space, and that's why it appears to be one-dimensional to the giant.

In String Theory, there is something analogous to the walls of the hose. There are objects called membranes, or just "branes" for short. Branes can be of any dimension, and might be very large. These branes do not act quite like walls. Supposedly, each of the two loose ends of a string can be attached to a brane, unable to move away. This has prompted speculation among scientists that the entire universe as we know it is stuck on a very large three-dimensional brane. The three-dimensional brane would exist in a larger 11-dimensional braneworld.

To further complicate things, the graviton, which is the hypothetical particle which carries the gravitational force, is theorized to be a closed loop. It has no loose ends! Thus, gravity is unlikely to be affected by branes the same way that other forces are. If we're lucky, this fact can be used to explain why gravity is such a weak force.* Perhaps gravitons are escaping from our brane!

*Gravity may seem like the strongest, but think of it this way: the entire Earth is pulling you down, but it can be stopped in its tracks the moment your feet touch the ground. The electrons in your feet repel the electrons in the ground. The reason gravity seems dominant is because most electric charges cancel each other, while there is no "negative mass" to cancel gravity.

So you get the basic idea. There are a lot of exotic ideas that have been proposed to solve the dimension discrepancy in String Theory, but most of them are built on one of the two basic concepts above. Either the extra dimensions are made of tiny little loops, or there is a wall or brane or something which is preventing us from moving in the extra dimensions. The rest, as far as we non-String-theorists are concerned, are details.

[Most of this, I learned from Warped Passages by Lisa Randall and The Fabric of the Cosmos by Brian Greene]

Some blogging stuff

I think I may need to slightly slow down my posting rate on this blog. Not for want of topics, but for want of time to write it all. But expect something cool for Darwin Day, Darwin's bicentennial.

This is a good time to spread the word about RSS feed. RSS is a tool to greatly assist your browsing experience. If you read a lot of blogs, comics, TV shows, or any other regularly updated webpages, you should definitely be using RSS. All you need is to get a "reader", and then start subscribing to different websites. I use Google Reader myself. If you have a reader, then you don't need to waste your time going to each and every website to see if they've updated. If you subscribe to a website, then all updates will appear in your reader.

Before I started using Google Reader, I found that I was discouraged from looking at infrequently updated sites, because most of the time I checked for updates, I found none.

If you want to subscribe to this blog, there's a link on the sidebar underneath "About this blog", or you can click here. Most of the time, you'll just see the RSS symbol, and you click on it to subscribe to a website.

If I slow down the posting rate, that also means that I'll slow down the appearance of puzzles. But there's still the puzzle archives. A lot of them were never really solved by any readers, because I often make them too difficult. Speaking of which, the recent Dots and Boxes puzzle has not yet fully been solved. The optimal solution for a 7x7 grid has not yet been found (the target is 31 lines).

Dimensions in the universe

A reader asked, "How many dimensions are there?" But first, it is worth answering, "What is a dimension?"

Without defining it too rigorously, I would say a dimension is essentially a direction. If we consider a drawing on a sheet of paper, it has two dimensions. This is because there are two directions in which you can move your pencil. You can move left and right, or you can move up and down.* If you want to move your pencil from point A to point B, you have move left/right a certain distance, and then move up/down a certain distance. That is, it takes two numbers to specify the coordinates of any point. Two numbers, two dimensions.

*We don't count "up" and "down" as distinct dimensions, because if you move up by a negative distance, it's just as if you had moved down.

Of course, our universe is not a mere drawing on a piece of paper. In our universe, we have three dimensions. You need three numbers to specify the location of any object. Since we're all Earth-dwellers here, the most convenient set of numbers to use are called latitude, longitude, and altitude. That is, the directions are north/south, east/west, and up/down.

But I omitted a fourth dimension: time. I think you'll agree that time is rather different than the other three dimensions. But it is a dimension nonetheless. You may argue that it is impossible to move backwards in time, but that is not important. What's important is that if we compare event A and event B, we need three numbers to specify the relative location, and one number to specify how much later/earlier event B is. The difference between time and space is a big one (don't let any amount of physics talk convince you otherwise). For instance, it is easy to get the directions north, south, east, and west all confused, because there's hardly any difference when you rotate yourself around. But no matter what you do, you will never confuse time and space with each other. (That said, there is a bit of space-time "rotation" that occurs when you approach the speed of light.)

Therefore, four dimensions is my final answer.

Wait, you wanted to hear about String Theory? Okay. It used to be that various versions of String Theory posited that the universe has 10 dimensions, or 26 dimensions. But in the 1990s, the science progressed, and the current most accepted version, called M-theory, now posits that the universe has 11 dimensions. This often prompts questions like, "WTF are those physicists thinking?" If I understand correctly, they are thinking that String theory is the most promising way to solve the problem of quantum gravity, because it naturally predicts the existence of a spin-two boson which behaves like the graviton. But that answer's much too arcane for my purposes here, so don't worry about it.

But it is a good question to ask, "How could the universe possibly have 11 dimensions when we so clearly see 4?" It is not merely a philosophical question. There are some things in physics that absolutely rely on having only three space dimensions. In particular, there is the so-called inverse-square law. If you have a light bulb in the middle of the room, then its light spreads out in all directions. The further you are away from the light bulb, the more the light spreads out before it gets to you. Similarly, the further away you are from the Earth, the more its gravity spreads out. The more spread out it is, the weaker its strength. More precisely, the strength of gravity is proportional to the inverse-square of distance. This is directly related to the fact that we have three dimensions of space. For instance, if we had four dimensions of space, we would instead have the "inverse-cube" law. It's not just humans who can't access the extra dimensions implied by String theory--it seems that gravity and light can't access them either!

This post got split into two. See part two: Dimensions in String Theory

BASS meeting minutes

I should probably mention that I've been writing up weekly meeting minutes for BASS, UCLA's skeptics and secularists club, and putting them online. They're on the BASS website, and here are the latest minutes.