Physics and Mathematics Guidance

[twofish-quant]

I thought I should start with linear algebra, then move onto Euclidean geometry. As for the physics, I'm studying classical mechanics and am actually currently working with angular motion in a plane. Should I be doing something else, or am I on the right track?

It's pretty decent.

Start with calculus and partial differential equations.

 

[twofish-quant]

Thanks! Einstein actually mentioned that tensors were a big part of his mathematics. Aren't they a step-up from vectors? But if scalars are quantities with magnitude, and vectors are quantities with both direction and magnitude, what then are tensors?

Great question! A tensor is a rule for turning one thing (like a vector) into another thing (like a vector). For example, suppose you have a sailboat. The wind is a vector. The force that the wind pushes against the sailboat is another vector. A tensor is a rule for calculating the force on the sailboat if someone gives you the direction of the wind.

In other words

vector 1 -> (magic tensor box) -> vector 2

Also that's one type of tensor. There is another type that turns two vectors into a scalar. For example, I draw two vectors on a piece of graph paper. Those two vectors form a parallelogram. So I get

vector 1 and vector 2 -> (magic tensor box) -> scalar (area)

 

[twofish-quant]

Actually, therein lies the problem. I don't have any money whatsoever - my mom works at McDonalds. Unfortunately, owing to the slowness of public high schools and my own aforementioned lethargy, getting into college doesn't seem very likely.

Community colleges are cheap and you can transfer to a state university.

 

[twofish-quant]

You remind me of a younger me. I wish someone had told me this earlier. You dream about theoretical physics and yet have a weak grasp of mathematics. There is more to physics then string theory and Quantum field theory. It is a siqn of your ignorance that you want to work in them!

Curious because he reminds me of a younger me. :-) :-) :-)

I ended up getting a Ph.D. in astrophysics, and that was a lot of fun. I work in an investment bank where I use a lot of my physics skills. For example the magic tensor box. You can write a list of stocks in Tokyo as one vector, a list of stocks in NYC as another vector, and then the magic tensor box will let you see how closely one set of stocks match another set.

 

[twofish-quant]

Kid. Please. You are not going to be einstein.

I think he could be. It's just a matter of being in the right place at the right time.

You are not going to revolutionize physics.

Not obvious why not. The cool think about physics is that it's not that hard to come up with something new and original. Pretty much everyone in physics comes up with *something* interesting.

 

[twofish-quant]

Well, I never really thought down on anyone. I merely wish to assist Einstein. I don't believe that anyone is more intelligent than anyone else. I just refuse to do anything but theoretical physics.

Bad idea. In order to be a good theoretical physicist, you have to get yourself very dirty with the data. Something that might be useful for you is to read what Einstein was doing between 1900 and 1905. He was spending most of his time getting familiar with the data, understanding the experiments and thinking about it.

If you just lock yourself in a room and think, you aren't going to be a very good theoretician.

Also you need to understand the experiments because, it's the experiments that make coming up with a unified field theory hard. Let me give you an example. It's overly simplified, but it gives you just one of the problems that you come up with.

I take a vector. Then I have a rule for stretching and shrinking it. The normal mathematical rules of stretching and shrinking don't care if the vector points to the right or to the left. They are the same for any vectors pointing in any direction.

Now it so happening that if you have a neutrino, the direction of spin points more heavily in one direction than another. Now what this means that I can't get this by just using math operations that stretch or shrink things, I have to come up with some other math operations that favor one direction over the other.

People have done this. One way of doing this is to assume that if the laws of the universe are such so that if you favor one particles in one direction, then you you have an lot of other particles that point in the other direction so things balance. Cool idea. Cool math. We haven't detected those other particles. Uh oh...

If you go into your room, you can come up with a nice beautiful theory about how the universe works. It's when you true to compare this to messy reality that it gets hard.

 

[twofish-quant]

I think people are being more discouraging than they should be.

If you keep learning math and keep thinking about the universe, you may not discovering the unified field theory, but you will find *something* interesting. In my case, it turns out that I wasn't in the right place at the right time, so I wasn't able to discover the unified field theory. But it also turns out that I can use my math skills to find all sorts of interesting things.

Let's go back to the magic tensor box.

(vector 1) -> (magic tensor box) -> (vector 2)

One thing that you can do is

(vector 1 - the price of all of the stocks on the NYSE right now) -> (magic tensor box) -> (vector 2 - the most likely price of all of the stocks on the NYSE six months from now)

Personally, I've ended up thinking that this problem is more interesting than unified field theory because ...

1) I can figure out if I'm right or not.

2) it's also interesting mathematically. In talking about space and time, you have to deal with vectors that are 4 dimensions. Some people think that the universe has 11 dimensions. Now if you think of the NYSE stock prices as a vector, the "universe" of stock prices is a *2000* dimensional space. If you think of all of the securities that are traded in the world, you are talking about space that has tens of thousands of dimensions.

One of the things that I get paid to do is to figure out ways of quickly calculating things in 2000 dimensional spaces.

 

[twofish-quant]

One other thing. It's actually quite easy to think that you can figure out the unified field theory when you don't know that much about physics.

If you keep learning math and physics, then at some point a light bulb will go off in your head, and you'll think, oh! so *that's* why the some of the smartest people in the planet have been spending decades and can't figure it out... I can give you a sketch of some of the big problems (the renormalization problem), and give you things to think about, but it's going to take a lot of work to get yourself to the point that you understand why it's hard.

So it's funny. If you don't know any math or physics, then solving the mysteries of the universe is easy. It's after you've spent years of your life thinking about something, that you realize that it's quite hard. So the more you know, the dumber you'll feel.

 

[Fizex]

I think twofish has found his protégé.

 

[Matthewkind]

Thanks. I will take all of that into consideration. So you're saying that tensors are just a mathematical way to transform vectors into vectors? I may have to do a little more studying before I understand that. Right now I'm going to focus on mathematical reasoning and physical principles. I believe that's a good first step in the right direction. And as for becoming the next Einstein - that's not my intention. I just want to help him achieve his dream. I don't mean to sound so pretentious - it's just, I'm really excited about physics. Also, I suppose I was being a bit too...overzealous. I'll accept any job that I can get - but I won't stop studying vehemently to become a theoretical physicist capable of assisting humanity in the overall understanding of our universe. Sure it's going to be a long and arduous road, but that's what makes it so fun. Without a challenge, life's boring. I'm not going to be scared off of this road. After all, if I only stick to the familiar, I'm limiting my own humanity.

 

[Shackleford]

I think twofish has found his protégé.

her.

 

[hadsed]

Also, I suppose I was being a bit too...overzealous. I'll accept any job that I can get - but I won't stop studying vehemently to become a theoretical physicist capable of assisting humanity in the overall understanding of our universe. Sure it's going to be a long and arduous road, but that's what makes it so fun.

Great. This is what I was trying to say, that you should just have a backup plan. Of course it's awesome to think about these things, and I don't mean to say that you shouldn't.

Also, you'll find out the more you read that there is a lot of stuff that let's you study both things and make you employable just in case. If you're interested in particle physics, you can check out lattice gauge theories and other computational approaches to particle physics problems. Even in quantum gravity there are approaches trying to discretize space, but that's a little too complicated for me to properly explain (it's still a bit above my understanding right now). If you're cynical, yeah it's like a compromise. But be positive about it, because you're still helping people and if it doesn't work out and if you're not lucky enough to be an academic, you'll still be able to do things that helps society and let's you live comfortably.

 

[Fizex]

her.

I wish english had gender-neutral pronouns then, but I heard twofish talking about a wife...

 

[MaxwellsDemon]

You and I should correspond more frequently. I don't have any physics friends - perhaps you could teach me? Do you have an MSN?

None of my friends are really into physics either, that's why I like visiting this forum occasionally. Its not a bad place to discuss interesting things with like minded people. I do have MSN, but I'm not on there often. We could always correspond here on the forum though. There certainly are a lot of knowledgeable people here who can help you learn. I'll do my best to help with any questions you might have.

As far as tensors go, mathematicians have an abstract definition of vector that includes tensors. So tensors are a type of vector in this abstract mathematical sense. I've always thought of tensors as sort of a higher dimensional generalization of the idea of a matrix. I think of them as a collection of indexed numbers that follow certain addition and multiplication rules.

 

[micromass]

Wow, there are a lot of harsh reactions to this kids question I'm really disappointed. Matthewkind simply has a dream of studying physics, there's no need to be rude for that...

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. And a lot of "fake people" who did make it because they could manipulate people. So the odds are big that you will meet some of the "academia ogres" now and then, and that's what I would like you to warn you for.

It may also be that physics is not what you expect it to be. It really happens a lot. And for these reasons it's best to get a backup plan. Try to minor or double major in some engineering field. If you end up in physics: no problem, you got what you wanted. And otherwise, you still have engineering to fall back on.

And don't be discouraged by other people. Especially people on this forum who don't know what you are capable off. Believing in yourself is the first step towards a bright future. But just don't be blind in your beliefs...

 

[lawsofform]

The reason a lot of people are (unjustly, imo) putting you down is that you're putting the horse before the carriage. It's good to be ambitious, but a lot of people have dreams of discovering a grand unifying theory only to find calculus too hard and end up graduating as C students in engineering. It's not impossible to become a reasearch professor and do what you want to do, but what you should focus on now is the immediate steps to get you there.

So, here are some things I wish I'd known when entering college:

It doesn't matter where you start. A lot of people who begin in community college end up in top universities. No matter where you go, you'll always have the chance to work hard and distinguish yourself, because that's something that's entirely under your control. So if you can't afford to go to a university, start in community college, work part-time or even full-time if you have to. If you do well, your options (academic and financial) will open up, and you should be able to graduate from a university without too much debt.

Spend a lot of time on your homework. Do all the problems in your books, even ones that aren't assigned. If most of the problems are computational rather than conceptual, ask your professor if they have any suggestions on studying something more advanced. Really successful students, the ones who go to graduate school, are the ones who take their education into their own hands.

Related to that, make sure you're taking the right courses. You should be taking the most rigorous ones available. There are lots of ways to a physics degree, and in my experience, advisors often don't know one class from another. Ask your professors, or the people around here, which classes you should be taking.

This is all assuming you want to go the traditional route, i.e. undergraduate-->PhD-->professor. A lot of people will tell you that this path is unrealistic, and they're right, but if it's what you want to do then those are the steps you have to take. Keep in mind that if you're smart enough to get a PhD, you'll probably able to find work somewhere to earn a comfortable living. Of course, it's not entirely necessary to follow this route. While you probably won't find anyone to pay you to do physics if you're self-taught, it's still something you can do. There's no shame in working an unskilled job and doing physics in your free time. The only thing is that, if physics does not occupy a dominant position in your day-to-day life, it's easy to slip up and never accomplish anything, so you've got to have a lot of discipline.

But in fact, sometimes I think that by the time a person has earned a PhD they are so indoctrinated in the existing structure of science that they lose the ability to create anything truly revolutionary. Many of history's geniuses in math and physics were outsiders to acadamia. Now odds are you are not a genius, but there's still something to be said for making a living some other way and persuing physics as a hobby.

My advice is to go to school while you're young, because if you don't do it now there's a good chance that you never will. It's a good place to learn the fundamentals. Once you get your hands dirty with some physics you'll be in a better place to make goals and decisions about it. Don't spend too much time worrying about string theory or what some jerks have to say about it, because none of that really has any bearing on your life right now.

 

[Shackleford]

The point is it's a lot harder than you think. It's also a lot more work than you think.

 

[qspeechc]

It's not that Matthewkind has no chance of becoming a physicist or professor but that the odds are extremely low. Indeed, you are competing against 10 year old wizz kids, not to mention thousands of other highly motivated, brilliant physics students; not just those graduating today, but of the decades before too; and not just from your country, but from around the world; all competing for a very few positions.

Study physics because you love thinking about it, you spend most of your time thinking about it. There's nothing wrong with that. Even someone with an undergrad degree in physics has employable skills. If all else fails, it's relatively easy to switch from physics to engineering, or computer science etc.

 

[Matthewkind]

Thanks! You've all been of immense help and I value everything you've said thus far! Of course I will continue to study physics, and I'll take any job I can get. I'll even get a part-tmie (or full-time) job at a community college if necessary in the beginning. In fact, owing to my lethargy in the slowness of high school, that might be my only way into a college at this rate. I really am a slow person. But I don't think you need to be a genius in order to uncover the secret of Nature; 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. And even if I don't make some ground-breaking discovery, simply being able to marvel at the beautiful solutions hitherto made and standing before the sheer elegance of the universe... that just seems like an amicable thing to me.

 

[micromass]

That's the spirit! Try to study what you enjoy. And even if you didn't accomplish what you wanted, you did enjoy yourself in the meanwhile...

 

[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.