FROM PRINCIPLES TO PARADOXES AND BACK AGAIN

Welcome to the first lecture of our course on the physics of impossible things. I'm Benyamin Schumacher, a professor of Physics Kenyon College, and I will be your instructor for this course. Now, I must tell you from the outset that course is a paradoxe. the one hand, it is a course on physics. It is a serious explor of the fundamental laws of nature and we will be thinking about the way really. Yet, on other hand, our central focus, our basic theme, and our indispensable tool, will be the impossible. So we also thinking about the way the word really is not. In this course, we will be talking about crazy, We will be talking about going back in time, going faster the speed light, and from inside a black hole. We will discussion machines limitless energy, We will discuss of size. We also consider a few impossibility, like exactly duplicat a quantum or producing an electromagnetic miracel. Now, many of this idea are familiar science fiction stories. We will use of those stories for  inspiration. in fact and this is an interesting physicis think about impossible  quite a lot. is it a useful  to study impossible thing. As I see it are three  reasons for this.

1.     The first is that the boundary between the possible and the impossible is an important line. good thing to know what is and is not possible in our universe. The laws of physics determine the line the possible and the impossible should be drawn. So it a good thing to think about just where that line

2.       Second, thinking about the impossible is a tool for understanding the laws of physics. This is the main idea of this course. You see, by the impossible we inside into the meaning of physical laws, and the connections between different of physics. It turn out that when ever we find that something is really and  impossible, there is always a great principle of physics at work.

3.       Third, our own understanding of the laws of nature is imperfect; it's provisional. History tells us that can lead to revolutionary changes in our knowledge of physics. Whenever our theories advance, we have to redraw our  the possible and the impossible. Sometime big changes are made, Sometimes things that were once thought possible, turn out to be impossible. For example, motion machines were as possible for a long time, but now we know they impossible. It also we once thought were impossible, turn out to be possible after all. For example, finding out are made , was once impossible; now we do it all the time.

Now, I said there were three, but of course there are really . You see, thinking about impossible things is. It is a really fun game. So, what makes the game of the impossible so fun to play Well, on the one hand, it challeng our. It inspires our creativity. It force us to think out side of the box. On the other hand, to impossible well, we have to use careful and iron . A game that uses both of these things, both imagination and logic, and brings them the laws of nature, that a game worth playing.So, first of all, what do word impossible When is something is impossible, In this  different of impossibility and each of is useful to think about. So, three types of impossibility.The first is absolute impossibility. Something is an absolute if it involves a mathematic contradiction. This is a of impossibility that does not realy on any special. The thing is impossible. What I mean is that we kind of impossible is actually true. So an absolut impossibility. The second type of impossibility is what we call a  impossibility. This is that's not impossible in itself, but it impossible because it  some about the world. This is most what we mean by a physical impossibility. We mean the laws of physics. We can such an impossibility, but such a world It is a world that follows different laws. Now, of course, his type of impossibility. If we change the assumptions that we use, for example, if we are laws of physics that we must this type of impossibility.

Finally, the third type of impossibility is what we call statistical impossibility. Now, a  impossibility is not impossible in the strictest sense, but it's something that is so overwhelmingly improbable that we can regard it as effectively impossible. 

So, for example, lets imagine that we take a fair coin. It equally likely to land hands or tails each time. The answer is, yes, this is possible. Yet, it is so unlikely that it might as well be impossible. So, what are the odds of flipping this coin.Well, the odds are about 2 followed by  zeroes, to 1 against. We could imagine this happening; it wouldn't contradict any laws of physics for this to happen. It could happen even in our own world, but it is for it to happen, and so we can regard it as impossible. It's a statistical impossibility.

Now, I'd like to give you some illustrations of these 3 types of impossibilities, the absolute impossibilities, the derived physical impossibilities, and the statistical impossibilities.  also like to nominate for each type. These patron saints will be significant figures in the history of science, about the impossible, and who used the impossible to advance our understanding of the world. So, we will start out with our patron saint of absolute impossibility. Euclid, of course, was a mathematician. He flourished around BCE. His great book, The Elements, is a treatise on geometry and the theory of numbers. The Elements is without doubt the math textbook in history. It's used as a textbook right up to our present day. Then, he used logic to prove more advanced propositions. One of Euclid favorite methods was something called, which is also , reduction to an absurdity.  So, how does a proof by Suppose you want to prove . What you do is you pretend that x is not true. This will turn out to be a mathematical impossibility. We're going to prove that x is true. Yet, you pretend that x is not true. Then, you logically show that this hypothesis leads to a logical contradiction; therefore, x must be true after all, because assuming that it's false, leads to a contradiction.I'm going to give you a specific example. This is actually from Euclid Elements. It's in Book 1 and it's Proposition 6 of Book 1. This is actually the very first proof by contradiction in Euclid Elements. It a proposition about triangles, like the triangle ABC. 

Euclid has just shown in Proposition five that if ABC has two sides of equal length, lets say AB and AC are of equal length that is to say, if it's what's called an triangle then the angles at the bottom, at B and C, must be equal to each other. 

Now Euclid wants to prove the reverse. If you have a triangle ABC and the angles at point B and point C are equal to each other, then the sides AB and AC have to have equal length. You have to have an isosceles triangle. So, what Euclid does is he says suppose that not true. Euclid imagines an impossible triangle at the bottom are equal, the angles at B and C are equal, but the length of AB is longer than the length of AC. Given such a triangle, what do you do? What Euclid says is, you can pick a point D, which is on the AB side, so that the distance between D and B is the same as the distance between A and C. So DB and AC are equal to each other. So we can take our diagram and we can divide it into two pieces. There an angle DBC and an angle ACB. Those 2 angles are equal, our original assumption, and those 2 pieces of the diagram share a common side, the base, BC, which CB. That segment has the same length as itself. The other two sides, DB on the one side and AC on the other side, have sides that are equal by our hypothesis. That is how we pixed the point D. Therefore, the two triangles that we formed, DBC and ACB, must be the same; they are congruent triangles. Actually, they must be exact mirror images of each other in the plane. If they're congruent triangles, then they must have exactly the same area. 

Yet, one of those triangles is actually a part of the other.. One area must be greater than the other one, and so we've arrived at a contradiction; we arrived at an absurdity. Therefore, our imagined triangle must really be impossible. That means that the proposition is proved.Now, notice what Euclid has done, he has helped to establish what is really true about actual triangles. He's used the impossibility of the triangle as a tool. That's what makes Euclid our patron saint of absolute impossibility.

So, who's our patron saint of derived or physical impossibility. For that we are going to choose none other than Isaac Newton. He was, of course, an English physicist of the late 17th and early 18th century. He was one of the greatest scientific minds in all of history. His great work, Principia Mathematica Philosophiae Naturalis, published in 1687, was a fantastic foundation for the science of mechanics. 

Mechanics, of course, is the science of force and motion. Newton founded this science based on laws of motion and gravitation. His mechanics was able to describe the motion of everything from projectiles to planets based on a single set of laws. So, I want to focus on something in Book 1 of the Principia, just after he introduces the laws of motion. He's discussing the law of action and reaction, one of those laws of motion. He imagines two objects, Number one and Number two, two balls, perhaps. If Number one exerts a force on Number two, the force of Number two on Number one must be exactly equal in strength and opposite in direction. That's a general law, in fact; it applies to all forces between all objects. So, Newton asks, if the law of action and reaction had an exception, Newton then considered what we might call Newton's dumbbell. You have two spheres and they're connected to each other by a stiff rod.. One of the spheres experiences a greater force. What happens to the dumbbell. Well, on the dumbbell, there's a net force to one side. Yet this violates another of the laws of motion, the law of inertia, which states that in the absence of an outside force, the dumbbell should remain at rest or move with constant velocity. Yet this dumbbell accelerates without an outside force.  So, by thinking about a physically impossible, Newton shows how the laws of motion fit together. That's a significant insight into mechanics. That's why we nominate Newton as the patron saint of this type of impossibility. So, who shall we nominate for our patron saint of statistical impossibility. Now, almost everyone has heard of Euclid and Newton, but I think that Maxwell deserves that kind of fame, among other things. He's one of my heroes in physics and he will come up in this course several times.
Our present concern is a passage from his book Theory of Heat, published in 1871. Maxwell invites us to consider a container with a gas inside. Initially, the gas only fills half of the container and the rest is empty, and now we ask what will happen Well, what happens to the gas of course is that the gas rapidly expands and fills the whole container.
Now, the reverse process, where the never seen. In the mid century, they understood that this fact illustrates something called the second law of thermodynamics, the science of heat and energy transformations. We're going to discuss this a lot in lectures five and six.

now, Maxwell pointed out that the gas is made up of trillions upon trillions of tiny molecules. The motion of these molecules is immensely complex. We must regard it in fact as chaotic and random, although it actually is following So, Maxwell asks, His answer is rather startling. His answer is nothing prevents it. It could happen, but it is exceedingly improbable.OK, what are the odds. So what are the odds that all of the gas molecules will gather on one side of the container? Well, that's a 1 followed by 150 billion trillion zeroes, to 1 against. Yet this is statistically impossible.  What Maxwell realizes, is that the laws of thermodynamics, at least the second law. This is what makes Maxwell the patron saint of statistical impossibility.Now Maxwell actually took this a further step to illustrate his point. He devised a famous. This is a tiny being that manipulates the atoms and can radically change the odds of what happens. We will discuss Maxwell's demon at some length in lecture six. OK, we've seen three different kinds of impossibility—absolute or mathematical impossibility, and for changing our views about those laws in the light of new discoveries. Now, since our course is about the impossible, we're going to range all over the place in topic. We are going to talk about many, many branches of physics. We are going to talk about phenomena large and small, theories old and very new. This will give us, I hope, not at all like an ordinary physics course.Yet the danger is that this might seem like a disconncted series of entertaining stories, a mere catalog of curiosities. Here's what we will find, that quite different impossible things turn out to be impossible for very similar reasons, and that these facts are markers of fundamental principles of nature. Now, our future lectures are going to fall into some natural groupings. We will discuss why we cannot reach absolute zero, the absolute minimum temperature, the limit of cold. We'll also see why we cannot exactly predict the future. After this, we're going to do several more lectures where we get into issues about space and time, cause and effect. We will talk about why it is impossible to escape from inside that black hole. We'll talk about why time travel is nevertheless a real topic of fundamental research today.
We will spend a few lectures talking about the notion of symmetry and geometry. Now, the idea of symmetry is a crucially important idea for physics, but how is it connected to the impossible. Well, we say that a shape has a symmetry lets say it's left right symmetric if it is impossible to tell whether the shape has been reflected. The reflected version and the original are exactly the same. So, symmetry is a principle of impossibility. In the same way, We can ask, would it be possible to distinguish those mirror worlds from our own. We can also talk about magnified worlds, worlds where everything has just been enlarged. After this, we  going to turn to the quantum world. We are going to talk about how the quantum revolution changed physics, and why it is impossible to know precisely where a particle is and where it is going. At the end of our course, We're going to pose the question, what does all of this impossible stuff tell us about the laws of nature. We're going to ask, how does the universe actually enforce its laws. We will actually be able to give a partial answer to that question. We'll ask, what kinds of impossibility are there. What kinds of impossibility are possible? As we learn more about nature, what is the future of the impossible. Now, this is inspired, in part, by my own work as a physicist, my own thinking about impossible things. My research specialty is in something called quantum information theory, which is how quantum physics governs the way we store and retrieve and transmit and process information. It's a relatively new branch of physics; it's still full of surprises. We're still mapping out the border between the possible and the impossible. So, I find myself using the impossible as a tool for figuring out what the laws of physics mean and how they work. I keep running into astonishing connections between apparently disparate thermodynamics and relativity, and quantum physics and information theory, and more. Sometimes I'm as often surprised by what turns out to be possible, as I am by what turns out to be impossible. That experience has been the inspiration for this course. I know of no way to get more directly at more fundamental about the physical world, than to use the impossible. If our goal, therefore.then there is nothing more practical than the impossible. Yet of course. They said it couldn't be done, and then somebody did it. There have also been natural and scientific impossibilities. They said that it could happen, they said that it was scientifically impossible for something to happen, and then someone did happen. So, in our discussion of the impossible, we'd like to avoid such failures of the imagination. Next time, we are going to take a look at a few of these not-quite impossible things.