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Friday, July 10, 2009
I like this one
So, my friend and for fun start flipping coins. We keep track of the string of results, which could, for example, look like:
HHTHTHTTHTHTTTTHTHHHTHTH
I then suggest to my friend that we make a wager. We begin a new sequence of coin flips. I say that if the sequence "HTT" comes up first, he owes me $10, but if the sequence "HTH" comes up before mine, then I owe him $10. My friend says, "Sure, that seems fair. Since they're just arbitrary triplets of heads and tails, and each one comes up with a 50/50 probability, there is nothing to distinguish the two, so the probability of one chain coming up first should be the same as the other." A reasonable assumption!
We play this game many times and it becomes clear that I'm by far getting the best of it, and my friend is quickly going broke of his life's savings. What's the catch?
// <-------------------------------------------------------->
Oh, and here's my solution to the 4-d cube. There's probably a simpler way to say the solution, but I like being wordy and clear.
What is a cube? In one dimension, a cube is a line. We can choose that line to be the one connecting the points (0) and (1). In two dimensions, it is a square. We can choose the four corners of the square to be (0,0), (1,0), (0,1), and of course (1,1). Simple so far. Each edge of the square connects two corners. So, how many edges are there. Well, we simply count the ways that we can connect two corners, right? No, not quite, because we don't want to count the diagonal. The line between (0,0) and (1,1) is not an edge So, let's take a step back.
Each edge of the square connects two corners. But the better way to think about it is by saying that each corner has two edges touching it. Since there are 4 corners, we must have 4*2 = 8 edges But of course by doing it this way we are exactly double counting, so we include a factor of 1/2. Thus, we arrive at our formula:
# of edges = # of corners * edges touching a corner * 1/2
So, we have to figure out how many edges touch each corner and the number of corners for a given dimension of cube. It's pretty easy to see that the number of edges touching a corner must be N. In 2-d, each corner touches two edges in 3-d it touches 3, etc. This is obvious by taking the origin as an example. The origin is defined as the point (0,0,.....0,0) with N zeros, where N is the dimension. This only connects to the corners (1,0,.....0), (0,1,0,.....0), ....., (0,.....,0,1). Clearly there are N of those. So, this one particular corner touches N edges. Thus, by the symmetry of the cube we can generalize this property of the origin to all corners and we learn that there are N edges touch at a corner.
This may seem like a stupid way of thinking about what a side of the square is, but it becomes easy to generalize to 3, 4, and N dimensions. Often times, the best way to find a solution is to find a really complicated way to solve the easy version.
Finally, we just need the number of corners for an N dimensional square. Well, what are the corners? For the cube, the corners are (0,0,0), (0,0,1), (0,1,0), (1,0,0), (1,1,0), (1,0,1), (0,1,1), (1,1,1) (we got all 8 of them). In general, we can list all the corners by listing the N-tuples (meaning the string of n digits) that contain only 0's or 1's. Each corners is of the type, and this spans all the corners. For each digit, we have two choices, and there are N digits, so the total number is just 2^N.
Thus, number of toopicks we need, meaning the number of edges on an N-d cube, are:
2^N * N * 1/2
For a 4-d cube, this is 32. A 4-d cube has 32 edges, and you'd need 32 toothpicks to build it.
HHTHTHTTHTHTTTTHTHHHTHTH
I then suggest to my friend that we make a wager. We begin a new sequence of coin flips. I say that if the sequence "HTT" comes up first, he owes me $10, but if the sequence "HTH" comes up before mine, then I owe him $10. My friend says, "Sure, that seems fair. Since they're just arbitrary triplets of heads and tails, and each one comes up with a 50/50 probability, there is nothing to distinguish the two, so the probability of one chain coming up first should be the same as the other." A reasonable assumption!
We play this game many times and it becomes clear that I'm by far getting the best of it, and my friend is quickly going broke of his life's savings. What's the catch?
// <-------------------------------------------------------->
Oh, and here's my solution to the 4-d cube. There's probably a simpler way to say the solution, but I like being wordy and clear.
What is a cube? In one dimension, a cube is a line. We can choose that line to be the one connecting the points (0) and (1). In two dimensions, it is a square. We can choose the four corners of the square to be (0,0), (1,0), (0,1), and of course (1,1). Simple so far. Each edge of the square connects two corners. So, how many edges are there. Well, we simply count the ways that we can connect two corners, right? No, not quite, because we don't want to count the diagonal. The line between (0,0) and (1,1) is not an edge So, let's take a step back.
Each edge of the square connects two corners. But the better way to think about it is by saying that each corner has two edges touching it. Since there are 4 corners, we must have 4*2 = 8 edges But of course by doing it this way we are exactly double counting, so we include a factor of 1/2. Thus, we arrive at our formula:
# of edges = # of corners * edges touching a corner * 1/2
So, we have to figure out how many edges touch each corner and the number of corners for a given dimension of cube. It's pretty easy to see that the number of edges touching a corner must be N. In 2-d, each corner touches two edges in 3-d it touches 3, etc. This is obvious by taking the origin as an example. The origin is defined as the point (0,0,.....0,0) with N zeros, where N is the dimension. This only connects to the corners (1,0,.....0), (0,1,0,.....0), ....., (0,.....,0,1). Clearly there are N of those. So, this one particular corner touches N edges. Thus, by the symmetry of the cube we can generalize this property of the origin to all corners and we learn that there are N edges touch at a corner.
This may seem like a stupid way of thinking about what a side of the square is, but it becomes easy to generalize to 3, 4, and N dimensions. Often times, the best way to find a solution is to find a really complicated way to solve the easy version.
Finally, we just need the number of corners for an N dimensional square. Well, what are the corners? For the cube, the corners are (0,0,0), (0,0,1), (0,1,0), (1,0,0), (1,1,0), (1,0,1), (0,1,1), (1,1,1) (we got all 8 of them). In general, we can list all the corners by listing the N-tuples (meaning the string of n digits) that contain only 0's or 1's. Each corners is of the type, and this spans all the corners. For each digit, we have two choices, and there are N digits, so the total number is just 2^N.
Thus, number of toopicks we need, meaning the number of edges on an N-d cube, are:
2^N * N * 1/2
For a 4-d cube, this is 32. A 4-d cube has 32 edges, and you'd need 32 toothpicks to build it.
Tuesday, July 7, 2009
4-d Cube
Here's a quick puzzle:
Imagine that you live in 4 dimensions and are playing with toothpicks and sticky tack. Someone asks you to construct a 4-d cube out of the toothpicks (with the tack to hold them together at the corners).
In 4-d, how many toothpicks do you need?
How about in n-d?
Imagine that you live in 4 dimensions and are playing with toothpicks and sticky tack. Someone asks you to construct a 4-d cube out of the toothpicks (with the tack to hold them together at the corners).
In 4-d, how many toothpicks do you need?
How about in n-d?
Friday, June 26, 2009
The Standard Model, Part 2: QCD
The other particle that many of us are familiar with from school is the proton. Protons have positive charge. Since electrons, as you will of course recall, have negative charge, they are attracted to protons. They want to bind together, and this is how atoms are formed. The electromagnetic force holds protons to electrons (the electromagnetic force is described by QED). But protons are fundamentally different than electrons. It turns out that, unlike electrons, they are composite particles. They are made up of smaller particles. These particles are called “quarks” after a line from James Joyce’s Finnegan’s Wake (“Three Quarks for Muster Mark”). Like protons, neutrons are also made of quarks. The term Hadron refers to particles that are made of quarks (there are more than just protons and neutrons, but those are by far the best known). So let’s talk about quarks.
There are 6 types of quarks that we have created and measured in a lab: the up quark, the down quark, the charm quark, the strange quark, the top quark, and the bottom quark. A proton is two up quarks and a down quark (sometimes written uud), and a neutron is an up dark and two down quarks (udd). The up quark has charge +2/3 and the down quark has charge -1/3, so we recover the well known properties that a proton has charge +1 and a neutron has charge 0.
Since quarks make up protons and neutrons (among many other particles), we know that there must be something that makes them want to stick together to form these composite particles. After all, we don't see free quarks flying around, we only see groups of quarks (quarks come in groups of two or three for group theoretical reasons). The things that bind quarks together, that make them stick to each other, are appropriately known as gluons. Gluons are analogous to photons. In QED, particles have positive and negative charge feel a force provided by photons. Therefore, one would guess that there was an analogous “charge” that quarks have which is related to gluons (just so we don’t get confused, quarks DO have ELECTRIC charge, and so they do interact with photons. But since they interact with gluons, they must have another type of charge).
It turns out that quarks and gluons are much more complicated than electrons and photons. In QED, there is only one “charge” (electric charge, of course). But with quarks and gluons, there are three types of charge. For lack of any better ideas, these three qualities that a quark can have were labeled “colors.” Most people call the three charges “red,” “green,” and “blue.” So, there are 6 types of quarks, but each quark can also be red, green, or blue. So, I can have a red up quark, or a green down quark, or a blue strange quark, or a red top quark, etc etc. And since it involves colors, the theory of quarks and gluons is called “Quantum Chromodynamics,” or QCD for short.
If you ask a physicist, they will tell you that QED is a very nice and clean theory. QCD is very, very messy. The main difference between QCD and QED is the difference between photons and gluons. Photons cause interactions between charged particles (like electrons). But they are not charged themselves. This means that photons don’t “talk” to one another. Photons only interact with charged particles, and since photons themselves aren’t charged, they don’t directly interact with photons. But gluons are different. Gluons interact with colored particles. But gluons themselves, it turns out, have color. Therefore, gluons are able to interact with other gluons. Two gluons can come together to form another gluons. One gluon can split apart into two gluons, or even three gluons. So, if you have one gluon, you have many. And if you have a quark, you have gluons. And if you have gluons, you have quarks. So, QCD is a mess. A quark flying along will emit gluons, which will make more gluons, which can make quarks, which make more gluons, etc. In the end, you end up with a bunch of particles that are all flying along. This big blob of particles that comes about from QCD is collectively called a “jet.”
So, to recap, QCD holds quarks together to form protons and neutrons. They can also become more exotic particles, like pions, kaons, and others that are named after Greek letters. QCD holds protons and neutrons together to form the nucleus of an atom. So, initially, QCD was called the “Strong Nuclear Force.” It is called strong because it indeed is strong. It is able to hold two protons together even though they should be repelled by the electric force (like charges repel each other). The study of nuclear physics is the study of QCD (usually just called the strong force in the context of nuclear physics). The reason that nuclear bombs are so powerful is that the strong force is so powerful. Nuclear explosions unleash the power of the strong force, of QCD. A gluon turns one quark into another, and releases a lot of energy in the process. QCD can indeed by quite messy.
There are 6 types of quarks that we have created and measured in a lab: the up quark, the down quark, the charm quark, the strange quark, the top quark, and the bottom quark. A proton is two up quarks and a down quark (sometimes written uud), and a neutron is an up dark and two down quarks (udd). The up quark has charge +2/3 and the down quark has charge -1/3, so we recover the well known properties that a proton has charge +1 and a neutron has charge 0.
Since quarks make up protons and neutrons (among many other particles), we know that there must be something that makes them want to stick together to form these composite particles. After all, we don't see free quarks flying around, we only see groups of quarks (quarks come in groups of two or three for group theoretical reasons). The things that bind quarks together, that make them stick to each other, are appropriately known as gluons. Gluons are analogous to photons. In QED, particles have positive and negative charge feel a force provided by photons. Therefore, one would guess that there was an analogous “charge” that quarks have which is related to gluons (just so we don’t get confused, quarks DO have ELECTRIC charge, and so they do interact with photons. But since they interact with gluons, they must have another type of charge).
It turns out that quarks and gluons are much more complicated than electrons and photons. In QED, there is only one “charge” (electric charge, of course). But with quarks and gluons, there are three types of charge. For lack of any better ideas, these three qualities that a quark can have were labeled “colors.” Most people call the three charges “red,” “green,” and “blue.” So, there are 6 types of quarks, but each quark can also be red, green, or blue. So, I can have a red up quark, or a green down quark, or a blue strange quark, or a red top quark, etc etc. And since it involves colors, the theory of quarks and gluons is called “Quantum Chromodynamics,” or QCD for short.
If you ask a physicist, they will tell you that QED is a very nice and clean theory. QCD is very, very messy. The main difference between QCD and QED is the difference between photons and gluons. Photons cause interactions between charged particles (like electrons). But they are not charged themselves. This means that photons don’t “talk” to one another. Photons only interact with charged particles, and since photons themselves aren’t charged, they don’t directly interact with photons. But gluons are different. Gluons interact with colored particles. But gluons themselves, it turns out, have color. Therefore, gluons are able to interact with other gluons. Two gluons can come together to form another gluons. One gluon can split apart into two gluons, or even three gluons. So, if you have one gluon, you have many. And if you have a quark, you have gluons. And if you have gluons, you have quarks. So, QCD is a mess. A quark flying along will emit gluons, which will make more gluons, which can make quarks, which make more gluons, etc. In the end, you end up with a bunch of particles that are all flying along. This big blob of particles that comes about from QCD is collectively called a “jet.”
So, to recap, QCD holds quarks together to form protons and neutrons. They can also become more exotic particles, like pions, kaons, and others that are named after Greek letters. QCD holds protons and neutrons together to form the nucleus of an atom. So, initially, QCD was called the “Strong Nuclear Force.” It is called strong because it indeed is strong. It is able to hold two protons together even though they should be repelled by the electric force (like charges repel each other). The study of nuclear physics is the study of QCD (usually just called the strong force in the context of nuclear physics). The reason that nuclear bombs are so powerful is that the strong force is so powerful. Nuclear explosions unleash the power of the strong force, of QCD. A gluon turns one quark into another, and releases a lot of energy in the process. QCD can indeed by quite messy.
Monday, June 8, 2009
The Standard Model, part 1: QED
The generally accepted picture of particle physics is known as the Standard Model. This collection of theories is the most successful physical model ever known and is able to make predictions that agree with experiment to unprecedented accuracies. For example, a quantity known as the anomalous magnetic moment of the electron can be calculated using Quantum Electrodynamics, which is one part of the Standard Model, and it matches with experimental measurements up to 14 decimal places. More or less, this means that one can zoom in by a factor of a trillion and the theoretical answer is still correct.
The standard model is an example of a Quantum Field Theory, meaning that it describes several "fields" and lists the ways that these fields can interact. Each field describes a different type of fundamental particle, so really the Standard Model is a list of the fundamental particles and the ways that they interact with each other. One doesn't need to really know anything about field theory in order to understand the particle content of the standard model, so let's dive into that.
Most of us are familiar with the concept of atoms and understand that they are made up of protons, neutrons, and electrons. However, if we zoom in closer, we learn that of these well known particles, only the electron is "fundamental," meaning that is isn't composed of smaller particles (as far as we know, of course). We should know a few things about the electron from high school. It has a property called "charge." The charge of the electron is -1, but that's just an arbitrary defintion of some units. The important point is that electrons are attracted to things with positive charge and repelled from things with negative charge (opposites attractive, likes repell, we see this among people as well as elementary particles). But what is charge really? Is there a more fundamental way to describe it beyond the ad hoc description above? Indeed there is. One a fundamental level, one can think of charge as the ability to interact with light. This seems odd, so let's qualify it a bit. Electrons interact with other charged particles via the exchange of light. Really, they interact with the smallest possible bunches of light known as "photons."
It turns out that all forces come about as the exchange of a particle. For every force, there is one or more particle that is exchanged. When an electron comes near another electron, they exchange a photon which carries the message that they should repell each other, and so they do. So, charge is the ability to communicate with photons, and therefore anything that is charged can talk to anything else that is charged by sending out or receiving a photon messenger.
The light that we see with our eyes is really just photon messengers talking to our rods and cones. There is an explosion in the sun. Two protons fuse together, and this process creates many photon messengers that go forth into the universe in all directions. Some of these messengers, after 8 minutes of traveling, hit the chemicals in our rods and cones and say, "There was a big explosion in the sun. Part of that explosion made red come this way in the form of me, the red photon." That's how we are able to know what happened 93 million miles away without feeling it with our hands. The photons told us!!
Let's go a bit deeper. In addition to electrons, there are electrons' evil cousins: the positron. The positron is just an electron but it has positive charge. It weights exactly the same. Since they have the opposite charge, they attract each other: they exchange a photon and tell each other to come closer. How romantic. The positron is known as anti-matter because it has the opposite charge as the electron, which is matter. Of course, the distinction is arbitrary: if we were made of positrons, we would call the electron the anti-matter. When electrons and positrons touch, they can destroy each other and release a lot of photons in the process. It's a cataclysmic event and they need to send out lots of messengers to tell the world. We can produce positrons in particle colliders, and we readily do, but they are quickly destroyed since the world is full of electrons to cancel them out.
The theory of electrons, positrons, and photons is known as Quantum Electrodynamics, or QED for short. It is a major component of the Standard Model. But really, it's quite simple. QED can be summarized by the fact that an electron and a positron interact via the exchange of a photon, two electrons interact via the exchange of the photon, or two positrons interact via the exchange of a photon. It's even simpler than that sentence if one understands Feynman diagrams (there's only one vertex between an electron, positron, and photon, and all possible interactions are built out of this one vertex like legos).
As you can see in the following diagram, each interaction vertex (where two lines meet) contains an electron and a positron (the two straight lines) and a photon (the wiggly line). Any such diagram that can be drawn that is made up of straight and wiggly lines where each wiggly line meets with two straight lines is a valid "Feynamn Diagram" in QED. If you aren't familiar with these sort of diagrams, don't worry, maybe I'll describe them in more detail in another blog.
Richard Feynman, the man see above playing the bongos, was a central figure in the creation of QED in the 1970's.
The other parts of the standard model are Quantum Chromodynamics (QCD) and Weak physics (which will merge with QED to form ElectroWeak physics). So, stay tuned.
The standard model is an example of a Quantum Field Theory, meaning that it describes several "fields" and lists the ways that these fields can interact. Each field describes a different type of fundamental particle, so really the Standard Model is a list of the fundamental particles and the ways that they interact with each other. One doesn't need to really know anything about field theory in order to understand the particle content of the standard model, so let's dive into that.
Most of us are familiar with the concept of atoms and understand that they are made up of protons, neutrons, and electrons. However, if we zoom in closer, we learn that of these well known particles, only the electron is "fundamental," meaning that is isn't composed of smaller particles (as far as we know, of course). We should know a few things about the electron from high school. It has a property called "charge." The charge of the electron is -1, but that's just an arbitrary defintion of some units. The important point is that electrons are attracted to things with positive charge and repelled from things with negative charge (opposites attractive, likes repell, we see this among people as well as elementary particles). But what is charge really? Is there a more fundamental way to describe it beyond the ad hoc description above? Indeed there is. One a fundamental level, one can think of charge as the ability to interact with light. This seems odd, so let's qualify it a bit. Electrons interact with other charged particles via the exchange of light. Really, they interact with the smallest possible bunches of light known as "photons."
It turns out that all forces come about as the exchange of a particle. For every force, there is one or more particle that is exchanged. When an electron comes near another electron, they exchange a photon which carries the message that they should repell each other, and so they do. So, charge is the ability to communicate with photons, and therefore anything that is charged can talk to anything else that is charged by sending out or receiving a photon messenger.
The light that we see with our eyes is really just photon messengers talking to our rods and cones. There is an explosion in the sun. Two protons fuse together, and this process creates many photon messengers that go forth into the universe in all directions. Some of these messengers, after 8 minutes of traveling, hit the chemicals in our rods and cones and say, "There was a big explosion in the sun. Part of that explosion made red come this way in the form of me, the red photon." That's how we are able to know what happened 93 million miles away without feeling it with our hands. The photons told us!!
Let's go a bit deeper. In addition to electrons, there are electrons' evil cousins: the positron. The positron is just an electron but it has positive charge. It weights exactly the same. Since they have the opposite charge, they attract each other: they exchange a photon and tell each other to come closer. How romantic. The positron is known as anti-matter because it has the opposite charge as the electron, which is matter. Of course, the distinction is arbitrary: if we were made of positrons, we would call the electron the anti-matter. When electrons and positrons touch, they can destroy each other and release a lot of photons in the process. It's a cataclysmic event and they need to send out lots of messengers to tell the world. We can produce positrons in particle colliders, and we readily do, but they are quickly destroyed since the world is full of electrons to cancel them out.
The theory of electrons, positrons, and photons is known as Quantum Electrodynamics, or QED for short. It is a major component of the Standard Model. But really, it's quite simple. QED can be summarized by the fact that an electron and a positron interact via the exchange of a photon, two electrons interact via the exchange of the photon, or two positrons interact via the exchange of a photon. It's even simpler than that sentence if one understands Feynman diagrams (there's only one vertex between an electron, positron, and photon, and all possible interactions are built out of this one vertex like legos).
As you can see in the following diagram, each interaction vertex (where two lines meet) contains an electron and a positron (the two straight lines) and a photon (the wiggly line). Any such diagram that can be drawn that is made up of straight and wiggly lines where each wiggly line meets with two straight lines is a valid "Feynamn Diagram" in QED. If you aren't familiar with these sort of diagrams, don't worry, maybe I'll describe them in more detail in another blog.
Richard Feynman, the man see above playing the bongos, was a central figure in the creation of QED in the 1970's.
The other parts of the standard model are Quantum Chromodynamics (QCD) and Weak physics (which will merge with QED to form ElectroWeak physics). So, stay tuned.
Monday, May 25, 2009
Symmetry
Oh, right, so why the title "Spontaneous Symmetry"? At that rate, what exactly do I mean by symmetry?
The concept of symmetry is one of the most important in physics. It can dramatically simplify problems, and often it is the only road toward an answer. The symmetries that are used in modern physics are often very abstract and esoteric, and are described by the beautiful field of mathematics known as group theory. The goal of group theory is to find general rules and patterns for systems that frequently appear in nature, math, physics, computer science, and nearly every field. Group theory describes things as ranging as a Rubik's cube, the energy levels of the hydrogen atom, cryptography, and countless other applications. But let's start off slowly.
We are all familiar with things that are "symmetric" in the layman sense. Something is symmetric if I can draw a line through it and it is the same on both sides. People are nearly symmetric if you draw a line through their head, torso, and between their legs, splitting their body in twain. For the sake of description, let's imagine a 2-d drawing of a human that indeed is perfectly symmetric. I can cut the drawing in half, flip one side over, and it will perfectly match the other side. A mathematician would describe this sort of mirror symmetry in the following way: I can take every part of the person, each arm, its hands, its fingers, its feet, etc, and I can reflect them to the opposite of the person and I will end up with the exact same image of the person.
One can imagine that if the person were holding a cane in his left hand and not in his right hand, he would no longer be symmetric. If I were to preform the above operation and switch the left and right sides, the cane would now be in the person's right hand. So, by swapping left and right, I DON'T get back the same image. The cane in this example breaks the symmetry.
Mathematically, this symmetry is described by taking things at position x and moving them to position -x (if we assume that the origin is aligned with the center of the person. In physics, this is called parity. The person that we imagined above (without the cane) is "symmetric under parity" or is "symmetric under mirror symmetry" or is "symmetric under reflections" or whatever term you want to use to describe it.
The point is that the symmetry that is most common to us is only a very simple example of symmetries. In general, a symmetry is defined as an operation that I can preform on an object that will leave the object the same. Consider a perfect circle. Certainly this circle also possesses the mirror symmetry that we described above. I can draw a line through the circle and flip each half over this line and the circle will be unchanged. But for the person, there was only one such line that I could draw. For the circle, there are infinitely many. The symmetries of the circle, we are starting to see, are much more rich and interesting than those of the person.
Aside from mirror symmetry, we are also free to rotate the circle. If I place my finger in the middle of the circle and spin the circle, it will appear unchanged (assuming the circle is indeed perfect). No matter how much I rotate it, I won't do anything. The circle, we say, is symmetric under rotations. Notice how this group of symmetries is different than the reflection symmetry from before. The act of reflecting is in a sense "discrete." You either do it or you don't. It's like being pregnant, you either are or you aren't. But rotations are "continuous." You can rotate by any amount, including none at all or any arbitrarily small or large amount.
Each of the sets of actions that we can do that leave an object the same form a "group." The set of rotations that we can do to a circle that leave it the same form what is known as the "rotation group" (specifically, for the mathematicians in the audience, it is called U(1), or equivalently SU(1) or O(2). This is also equivalent to the real numbers modulo the integers, and a fun exercise would be to find the isomorphism). The defining feature that makes these "symmetries" a "group" is the fact that any two symmetries done in succession make another symmetry. For example, if I rotate by 30 degrees and then by 80 degrees, it is the same as rotating by 110 degrees. It's as simple as that. I can also rotate backwards (ie rotating by -90 degrees is the same as rotating by a right angle in the opposite direction as 90 degrees), or I could just not rotate at all (identity).
Groups can be as simple as reflecting things and rotating squares and circles, but they can also be complicated. The more abstract ones involve matrices and complex numbers, but it's not necessary to bring them up in order to understand the main concepts.
So, how does any of this apply to physics? It turns out that for each continuous symmetry of a system, there is a corresponding "conserved" quantity. For those unfamiliar with this term, a conserved quantity is something that remains the same no matter what we do. For example, charge is conserved in the sense that you can't create charge, you can only move it. Energy is conserved, and you can only change its form, meaning heat is created when I use my laptop's battery. Momentum is conserved, meaning if I throw something in space in one direction, I will go flying in the opposite direction.
All of these conserved quantities can be derived from a symmetry of nature. The mechanism for doing this is one of the most beautiful theorems in all of physics. It's called Noether's theorem, which was first discovered by Emmy Noether (seen below). Einstein apparently called her the most important woman in the field of mathematics. It says what I just said, that each continuous symmetry leads to a conserved quantity (it just says it in fancy math language).
Conservation of energy comes from the fact that the laws of physics are constant in time (meaning if I preform an experiment at 3:00 in the morning, it should have the same results as if I did it at 5:00 in the afternoon, ignoring external differences associated with the different times of day such as daylight, weather, etc). Conservation of momentum comes from the fact that the laws of physics don't care where they take place (meaning they are the same here as well as in the Andromeda galaxy, or they are "symmetric under translations").
Conservation of charge comes from the fact that a rotation of electron fields, similar to the circle rotation above, leaves the energy of the system unchanged. This is of course mostly nonsense to those who aren't familiar with Quantum Field Theory, but I assure you that it's no more difficult than spinning a circle about its center.
The concept of symmetry is one of the most important in physics. It can dramatically simplify problems, and often it is the only road toward an answer. The symmetries that are used in modern physics are often very abstract and esoteric, and are described by the beautiful field of mathematics known as group theory. The goal of group theory is to find general rules and patterns for systems that frequently appear in nature, math, physics, computer science, and nearly every field. Group theory describes things as ranging as a Rubik's cube, the energy levels of the hydrogen atom, cryptography, and countless other applications. But let's start off slowly.
We are all familiar with things that are "symmetric" in the layman sense. Something is symmetric if I can draw a line through it and it is the same on both sides. People are nearly symmetric if you draw a line through their head, torso, and between their legs, splitting their body in twain. For the sake of description, let's imagine a 2-d drawing of a human that indeed is perfectly symmetric. I can cut the drawing in half, flip one side over, and it will perfectly match the other side. A mathematician would describe this sort of mirror symmetry in the following way: I can take every part of the person, each arm, its hands, its fingers, its feet, etc, and I can reflect them to the opposite of the person and I will end up with the exact same image of the person.
One can imagine that if the person were holding a cane in his left hand and not in his right hand, he would no longer be symmetric. If I were to preform the above operation and switch the left and right sides, the cane would now be in the person's right hand. So, by swapping left and right, I DON'T get back the same image. The cane in this example breaks the symmetry.
Mathematically, this symmetry is described by taking things at position x and moving them to position -x (if we assume that the origin is aligned with the center of the person. In physics, this is called parity. The person that we imagined above (without the cane) is "symmetric under parity" or is "symmetric under mirror symmetry" or is "symmetric under reflections" or whatever term you want to use to describe it.
The point is that the symmetry that is most common to us is only a very simple example of symmetries. In general, a symmetry is defined as an operation that I can preform on an object that will leave the object the same. Consider a perfect circle. Certainly this circle also possesses the mirror symmetry that we described above. I can draw a line through the circle and flip each half over this line and the circle will be unchanged. But for the person, there was only one such line that I could draw. For the circle, there are infinitely many. The symmetries of the circle, we are starting to see, are much more rich and interesting than those of the person.
Aside from mirror symmetry, we are also free to rotate the circle. If I place my finger in the middle of the circle and spin the circle, it will appear unchanged (assuming the circle is indeed perfect). No matter how much I rotate it, I won't do anything. The circle, we say, is symmetric under rotations. Notice how this group of symmetries is different than the reflection symmetry from before. The act of reflecting is in a sense "discrete." You either do it or you don't. It's like being pregnant, you either are or you aren't. But rotations are "continuous." You can rotate by any amount, including none at all or any arbitrarily small or large amount.
Each of the sets of actions that we can do that leave an object the same form a "group." The set of rotations that we can do to a circle that leave it the same form what is known as the "rotation group" (specifically, for the mathematicians in the audience, it is called U(1), or equivalently SU(1) or O(2). This is also equivalent to the real numbers modulo the integers, and a fun exercise would be to find the isomorphism). The defining feature that makes these "symmetries" a "group" is the fact that any two symmetries done in succession make another symmetry. For example, if I rotate by 30 degrees and then by 80 degrees, it is the same as rotating by 110 degrees. It's as simple as that. I can also rotate backwards (ie rotating by -90 degrees is the same as rotating by a right angle in the opposite direction as 90 degrees), or I could just not rotate at all (identity).
Groups can be as simple as reflecting things and rotating squares and circles, but they can also be complicated. The more abstract ones involve matrices and complex numbers, but it's not necessary to bring them up in order to understand the main concepts.
So, how does any of this apply to physics? It turns out that for each continuous symmetry of a system, there is a corresponding "conserved" quantity. For those unfamiliar with this term, a conserved quantity is something that remains the same no matter what we do. For example, charge is conserved in the sense that you can't create charge, you can only move it. Energy is conserved, and you can only change its form, meaning heat is created when I use my laptop's battery. Momentum is conserved, meaning if I throw something in space in one direction, I will go flying in the opposite direction.
All of these conserved quantities can be derived from a symmetry of nature. The mechanism for doing this is one of the most beautiful theorems in all of physics. It's called Noether's theorem, which was first discovered by Emmy Noether (seen below). Einstein apparently called her the most important woman in the field of mathematics. It says what I just said, that each continuous symmetry leads to a conserved quantity (it just says it in fancy math language).
Conservation of energy comes from the fact that the laws of physics are constant in time (meaning if I preform an experiment at 3:00 in the morning, it should have the same results as if I did it at 5:00 in the afternoon, ignoring external differences associated with the different times of day such as daylight, weather, etc). Conservation of momentum comes from the fact that the laws of physics don't care where they take place (meaning they are the same here as well as in the Andromeda galaxy, or they are "symmetric under translations").
Conservation of charge comes from the fact that a rotation of electron fields, similar to the circle rotation above, leaves the energy of the system unchanged. This is of course mostly nonsense to those who aren't familiar with Quantum Field Theory, but I assure you that it's no more difficult than spinning a circle about its center.
Tuesday, May 19, 2009
Two Envelopes: Answer
I'll start off by addressing the comments.
Just to reiterate, the paradox is the idea that for all possible values that we find in the envelope, switching is advantageous. To Erik, this clearly makes no sense because there is nothing that distinguishes the envelopes. This is different from a situation where we have a particular value, say $100, and are given the option of switching for either $200 or $50 with equal probability. Clearly in that case switching is advantageous.
To Mr. Zrake's argument, you have shown that having a strategy of always switching will not be advantageous. This is different than the idea that once we've opened the envelope and seen a particular value, we always want to switch. Let me explain with an example. Imagine that I tell you in advance that the envelopes contain exactly $50 and $100 and you get one envelope with a 50/50 chance. Clearly, in this case, a strategy of always switching is clearly going to have the same expectation value as a strategy of always staying, which is just the mean of the two envelopes. However, if you open the envelope and see $50, you should always switch and if you see $100 you should always stay.
The last example I think is a big hint toward the end of the paradox. See, if we know in advance how the envelopes are filled, then our strategy should be clear. Let's go back to the beginning. We haven't discussed how the person decides how much money to put in every envelope. The only constraint thus far is that one envelope should have twice the money that the other has. When we open a particular envelope, we know that the other has either twice the money or half the money. The key, however, is that there isn't a 50/50 chance of it either being half or double as one would naively expect. The relative probability between half and double depends on how the envelopes are filled, ie it depends on the probability distribution that the filler uses to pick what amount of moeny goes into the envelopes.
And here's the real key: It is impossible to have a probability distribution where, for every x that we see in the envelope, there is a 50% of the other having half and a 50% of it having double.
To be more concrete, let's say that the filler chooses to fill the envelope in the following way: he has some probability distribution p(x) that he uses to randomly fill one envelope. He then fills the other envelope with double that amount, and flips a coin which determines whether he gives us the larger or the smaller envelope.
So, if we open our envelope and see X dollars, it means that one of two things happened. Either the "seeded" number was X and we got the smaller envelope, or the seeded number was X/2 and we got the higher number. The expectation value of switching when we see X in our envelope is:
EV(switching) = N * { p(X)*2X + p(X/2) * X/2}, where N is a normalization constant such that the probability adds up to 1.
Clearly, this expecation value depends intimately on the probability distribution function p(X). If we were to know explicitely this function going in, then we could determine the proper strategy by comparing the above expectation value to X. If we DON'T know this function going in, which is the case presented in the original problem, then we CAN'T come up with the optimal strategy. However, that doesn't mean that ALWAYS IS correct to switch, it just means that we can't determine the correct answer because we don't have enough information.
Just to reiterate, the paradox is the idea that for all possible values that we find in the envelope, switching is advantageous. To Erik, this clearly makes no sense because there is nothing that distinguishes the envelopes. This is different from a situation where we have a particular value, say $100, and are given the option of switching for either $200 or $50 with equal probability. Clearly in that case switching is advantageous.
To Mr. Zrake's argument, you have shown that having a strategy of always switching will not be advantageous. This is different than the idea that once we've opened the envelope and seen a particular value, we always want to switch. Let me explain with an example. Imagine that I tell you in advance that the envelopes contain exactly $50 and $100 and you get one envelope with a 50/50 chance. Clearly, in this case, a strategy of always switching is clearly going to have the same expectation value as a strategy of always staying, which is just the mean of the two envelopes. However, if you open the envelope and see $50, you should always switch and if you see $100 you should always stay.
The last example I think is a big hint toward the end of the paradox. See, if we know in advance how the envelopes are filled, then our strategy should be clear. Let's go back to the beginning. We haven't discussed how the person decides how much money to put in every envelope. The only constraint thus far is that one envelope should have twice the money that the other has. When we open a particular envelope, we know that the other has either twice the money or half the money. The key, however, is that there isn't a 50/50 chance of it either being half or double as one would naively expect. The relative probability between half and double depends on how the envelopes are filled, ie it depends on the probability distribution that the filler uses to pick what amount of moeny goes into the envelopes.
And here's the real key: It is impossible to have a probability distribution where, for every x that we see in the envelope, there is a 50% of the other having half and a 50% of it having double.
To be more concrete, let's say that the filler chooses to fill the envelope in the following way: he has some probability distribution p(x) that he uses to randomly fill one envelope. He then fills the other envelope with double that amount, and flips a coin which determines whether he gives us the larger or the smaller envelope.
So, if we open our envelope and see X dollars, it means that one of two things happened. Either the "seeded" number was X and we got the smaller envelope, or the seeded number was X/2 and we got the higher number. The expectation value of switching when we see X in our envelope is:
EV(switching) = N * { p(X)*2X + p(X/2) * X/2}, where N is a normalization constant such that the probability adds up to 1.
Clearly, this expecation value depends intimately on the probability distribution function p(X). If we were to know explicitely this function going in, then we could determine the proper strategy by comparing the above expectation value to X. If we DON'T know this function going in, which is the case presented in the original problem, then we CAN'T come up with the optimal strategy. However, that doesn't mean that ALWAYS IS correct to switch, it just means that we can't determine the correct answer because we don't have enough information.
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