Physics and Mathematics Guidance

[lisab]

This site will give you some guidance, perhaps.

http://www.phys.uu.nl/~thooft/theorist.html

This is a website (still under construction) for young students - and anyone else - who are (like me) thrilled by the challenges posed by real science, and who are - like me - determined to use their brains to discover new things about the physical world that we are living in. In short, it is for all those who decided to study theoretical physics, in their own time.

 

[Matthewkind]

The site seems to be a bit... hard-to-follow. Is it saying I should search these topics, or is it going to provide me with texts, or...?
Sorry. I told you I'm a tad bit slow. At any rate, I'll keep on studying linear algebra and classical mechanics is parallel. After that, I'll work on Euclidean geometry, trigonometry, single-variable calculus, multivariable calculus, and then I'm not really sure. Perhaps those methods of approximations and analysis classes.

 

[hadsed]

It's recommending some good books to learn those certain topics, which I think is its primary function. I don't believe he posts his own lecture notes or literature or anything like that.

 

[twofish-quant]

But to Matthewkind: I would like to give you a sincere warning. Physics and theoretical physics is a really fun subject if you enjoy it. But academia is a very harsh world. I've seen a lot of brilliant people who didn't make it because they were broken by the system.

On the other hand, even though its harsh, I made it through.

One other thing that you need to remember is there is a huge amount of "dumb luck" in research. You could be mathematically brilliant but just be born in the wrong time and meet the wrong people. Or you could be not so talented, but just get lucky. I think most physicists I know have roughly the mathematical abilities of Einstein, and I'm sure that everyone one of them could get a Nobel prize if you just had a time machine back to 1905.

One other thing to remember is that GR requires a bit of mathematical knowledge, but EInstein got his Nobel for his explanation of the photoelectric effect, and to understand that requires no more than basic Algebra. His explanation of Brownian motion requires no more than first year undergraduate math. It's not solely mathematical brilliance that will get you somewhere, you just need to see the key thing that no one else did, and its likely that the key thing that people see in 2031 is something that we couldn't possibly have detected in 2011.

The other thing about Einstein is that "unifying" electro-magnetic and gravity in a classical field theory is pretty easy. Kaluza showed how to do it in 1925. If you take general relativity in 4 dimensions and then add a fifth dimension, you get electromagnetism. This is a very significant fact about the universe that means something really important, except that no one knows what.

The hard part is that we've observed all sorts of messy things since 1925, and there is no way that Einstein in 1925 or even 1945 could have gotten a unified field theory just because there are some experimental things that we know about that he didn't. He didn't know about quarks. So after getting lucky a few times, he got unlucky.

It may also be that physics is not what you expect it to be.

Or it could be pretty much what you expected.

Believing in yourself is the first step towards a bright future. But just don't be blind in your beliefs...

One thing that makes physics tough is that it's not all about you. The universe has something to say. You could be the most brilliant physicist in the world, but you just might be following a totally wrong path. One reason I try to get people into physics, is that the odds of one person accidentally hitting the right answer is rather low, but if you have a lot of people working on different things, it's likely that someone will trip over the right answer. If you are lucky, it could be you. It probably won't be, but you'll figure out some useful things along the way.

 

[twofish-quant]

But I don't think you need to be a genius in order to uncover the secret of Nature

One bit of good news is that the universe has a lot of secrets. Figuring out one thing that no one knew before is not that hard. About tensors. In my job, we figured out that if you calculate tensors equation using this neat trick, you can do it a lot faster than if you don't use the math trick. I don't think that anyone knew this five years ago, because it involves some computer hardware that didn't exist five years ago.

I just think if you don't give up, no matter what has happened, and no matter what will happen, you'll definitely find what you're searching for.

You need to think differently. You almost certainly will not find what you were looking for. The good news is that you'll likely find something different, and what you find might be more interesting.

Just to give an example of this. One way that you can get closer to a unified field theory is to say that for every "matter-like" particle there is a "energy-like" particle. This makes the equations balance, but it has the problem of predicting a lot of particles that no one has observed. This is called "supersymmetry." Now someone asked what the universe would look like if it was filled with supersymmetric particles. When they did this they figured out that the important thing about how a particle would affect the universe is how "hot" it is, and that you'd get a universe that looks pretty much a lot like what we see if we assume that the universe is filled with cold dark matter.

Now its possible that dark matter are these "supersymmetric particles" but it turns out that even if supersymmetry is totally wrong (which most people think it is), that you haven't wasted your time, because it lead you onto thinking about "dark matter."

And even if I don't make some ground-breaking discovery

Making a ground breaking discovery is a requirement for getting your Ph.D. (seriously) They won't hand you your doctorate until you've discovered something new. Setting things up so that you will discover something new over the course of five years is what you learn to do in graduate school.

simply being able to marvel at the beautiful solutions hitherto made and standing before the sheer elegance of the universe...

They again, maybe not. One reason that people are amazed when you come up with something elegant is that most of the time, the universe is messy and incomprehensible. The reason that physics is hard and interesting is that the universe *isn't* very elegant. So when you find a deep connection somewhere, it's really cool, because it really doesn't happen very often.

But when you find some connection, its often quite interesting. Who knew that the mathematics of stock prices are very similar to the mathematics of particle flow in supernova? You find a "magic connection" and then you stare at the equations very closely to see if you understand *why* there is a magic connection.

I should point out this is why banks hire physicists. You can find the formula for stock prices in any basic finance textbook (google for Black-Scholes model). If you want to just run the formula, you can put it into an excel spreadsheet. However, that's not good enough for me. I want to really understand *why* the formula works, which is useful because there are a lot of situations when it suddenly stops working.

 

[Matthewkind]

How can you use tensors and vectors in banking and computation to begin with? Don't you need some sort of direction for the values to be considered "vectors"? I realize I probably sound very unintelligent, but I would like to know how you think of purely abstract numbers in terms of direction. My apologies for being slow.
Also, what exactly do tensors do? I realize that if you input a vector quantity into a "magic tensor box" the output is a different vector.. But that doesn't satisfy my understanding. Please be just a little more specific. Again, my apologies for my slowness.

Random Sidenote (Side-question?): If the presence of mass and energy distort the fabric of space-time and that is the very description of gravity, then why is this concept irreconciable with quantum mechanics? We've already measured the masses of particles, and mass distorts space-time. Plus, the particles have energy. I don't quite understand why then the subatomic particles don't adhere to general relativity. I realize I don't know enough about either fields, and this is precisely why I've asked such a question.

Another Side-Note While I'm Here: Do particles really jump into and out of existence, or is the fabric of space-time just topologically distorted to such a degree that they're hitherto unnoticeable?

 

[Ryker]

How can you use tensors and vectors in banking and computation to begin with? Don't you need some sort of direction for the values to be considered "vectors"? I realize I probably sound very unintelligent, but I would like to know how you think of purely abstract numbers in terms of direction.

I don't know exactly how twofish-quant does it, since at this time stuff like that is far beyond me, but as my linear algebra professor said, vectors are things that act like vectors. This should give you a good idea on where to start, and help you realize that those abstract numbers need not be pure numbers, but can be thought of as coordinates. 3 and 2 are just two numbers, as well, but knowing where they appear they could just as easily represent coordinates of the (3, 2) vector. So don't worry, you're not slow, just try and start thinking of things more abstractly, so that you can extract the structures that pervade different mathematical constructs.

 

[Matthewkind]

You made me smile, friend.