Math in Physics: What the heck are they talking about?

[BluberryPi]

Although I am 12 years old, I do enjoy learning physics. However, sometimes their is a common issue when I look up something like " Introduction to Gauge Field Theory": whenever I read the document, their is some complicated math stuff that I don't understand. Here's an example:

I really need help with math like this. I can't understand physics without it.

So my question is: What type of math is this? Where can I learn how to do it and understand it? Is their any specific thing I should study? Any website or book?

 

[fresh_42]

The by far most difficult part of your question is how to answer it without discouraging you because this is the last thing I want to happen.
I once asked a teacher at my school something similar. His answer has been: "Oh, you have to study math for that, at least two years." Many years later as I knew the answer from my study by myself it turned out that my question at school could easily had been answered and would had been only a little apart from school math. Probably my teacher hadn't had the answer in mind and just tried to get rid of me.
You see my difficulty? I don't want to get rid of you.

This brings me to the crucial point underlying your question. To answer the above it would be enough to say it is something about group theory. Groups are nothing mysterious, it is simply a multiplication on a set of things, e.g. here the set are mappings. You know groups from multiplying numbers, or from the addition of numbers which also form a group.
However, this answer might not really help you as it leads to many more questions. Me or someone else could try and explain them in simple words like I did above, neglecting a lot of more details necessary to understand the lines you quoted. But almost certain there will be a point, where someone simply answers: you have to study some college math first. Then you might feel as I did when I asked my teacher.
Nevertheless, it is true. Mainly, because of the way math is taught at school and not necessarily because of the level of difficulty.
Unfortunately math at school is very much doing calculations of all kind. Math at college is more abstract. And, e.g. calculations by a physicist might look as the ones you quoted: $U(g)TU(g)^*$ which means "I know there is a transform $T$ and a mapping $U(g)$ for an element $g$ of $G$, such that the equation holds. Which $T,g,U$ are exactly meant isn't important. Their properties are. (And to even raise confusion, the $G$ in the quote is taken from a strange alphabet.)

It is not really difficult to understand all this. As in school, it's a matter of practise. But at school, everybody seems to be glad, if the students can buy some bread and vegetables without accidents. The concept of a group is not really needed in everyday life. On the other hand it is well needed in many scientific applications such as above.
So I could leave you saying: It's group theory you need. But this would be only a rather small part of the answer. To explain everything around, it would lead us to what hopefully will lead you to: a study of math and physics at a college. With every answer there appear new questions. Which is part of the fun learning math or physics. If you are really interested you could read or watch some introductions on, e.g. group theory or linear algebra. At least the latter could serve you as well at school. But don't be disappointed, please, if there are no fast answers. Unless you're a genius it will certainly take you some time and patience to go forward step by step..

 

[BluberryPi]

Okay, So I need to learn college math.
Is their anyway to do that without waiting 6 years?
Plus, I need to learn about Group Theory. Is their any books I can read about it?

 

[fresh_42]

Okay, So I need to learn college math.
Is their anyway to do that without waiting 6 years?
Plus, I need to learn about Group Theory. Is their any books I can read about it?

There are probably plenty. But I don't know an English match of a book's content and your level of knowledge. I found the most different part between math at school and at college the fact, that all of a sudden everything has become abstract and not numerical anymore. As I've mentioned above: properties are more important than actual values. The most important numbers are probably $0,1$ and $2$. To start with and see where your interest will lead you to, you could start and learn some terms using Wikipedia. E.g. What is a group? Examples? More important would certainly be to learn something about vector spaces and linear mappings. It is the most important concept! And your quote is essentially about linear functions. This will be useful anyway. Even if you'll decide to study something scientifically beside math or physics. The links on Wikipedia will lead you deeper and deeper. From there you can search the internet if you'll have more concrete questions. Moreover you may always return to PF and post questions. (But keep in mind, that PF expects you to show some effort to find answers on pages like Wikipedia or alike by yourself. And you should label your posts here as B (beginner's level) or I (undergraduate's level).)

 

[Dale]

Although I am 12 years old, I do enjoy learning physics. However, sometimes their is a common issue when I look up something like " Introduction to Gauge Field Theory":

I would strongly recommend starting with Newton's laws rather than gauge field theory. It will be much more useful and intuitive, but will still be challenging and can serve as the motivation for learning new math too.

 

[PeroK]

Okay, So I need to learn college math.
Is their anyway to do that without waiting 6 years?
Plus, I need to learn about Group Theory. Is their any books I can read about it?

The most important mathematics will be vectors, calculus, vector calculus, differential equations, complex numbers and linear algebra. Most of this should be good, practical mathematics that directly relates to physics. Start with vectors and calculus. You might be surprised how far that can take you.

 

[Delta2]

How did you find out about gauge field theory in your age of just 12? You might be able to read about and understand advanced for your age material but still you ll have to start from math like those suggested by PeroK. With those math you ll be able to understand better classical mechanics, classical electromagnetism and classical field theory. Once done with those maybe you can move to gauge field theory...

Jumping straight from the start into the hard core of modern theoretical physics, gauge field theory that is , is just over the top..Its like one trying to build a spaceship while all he knows is how to make woods get fire...

You might be a genius and be able to learn and understand stuff really fast but all I am saying is that you have to start from basic and work all the way up to the most advanced things like gauge field theory.

 

[Nugatory]

Is their anyway to do that without waiting 6 years?

You aren't "waiting" for most of that six years, you'll be spending most of that six years learning enough math to be ready for the college-level math courses. Between now and your first year of college you'll go through two years of algebra, a year of geometry, a year of trig and precalculus, and then AP-level calculus. That will fill most of the six years before you get to college and will put you in a good position to take on the college-level math you'll meet when you get there. You can knock a few years off that if you're motivated, but it's still a fair amount of ground to cover and there's no substitute for covering it all.

I understand that when you're twelve this all sounds very discouraging - You've been in school for maybe six years, that's half your life, and you're only halfway to having a foundation you can build on? But you have to remember that what you're learning was developed over almost five centuries by some of the smartest people who ever lived. It's really quite amazing that you can start from zero and be completely caught up by the time you're twenty. It's a lot of work, but it's also a lot of fun... the people who have been through it are generally glad that they did.

 

[BluberryPi]

Technically, All I need is Algebra and Pre - calculus, but isn't their anyway I can learn those two from books, or online? And after that, what should I study?

 

[Nugatory]

Technically, All I need is Algebra and Pre - calculus, but isn't their anyway I can learn those two from books, or online? And after that, what should I study?

Both can be learned on your own if you can get hold of a standard textbook (whatever the high schools you'll be attending use might be a good choice) and are willing to spend a few hours a day working through it. It will help a lot if you can find a sympathetic teacher or other tutor who can help you through the hard spots.

But don't kid yourself with this "technically all I need..." thinking. You're still building a foundation, and skipping steps in the foundation is what makes buildings fall down. You absolutely need geometry and trig before you go anywhere near any college-level math classes. As for intro calculus... Well, it was invented by Isaac Newton because without it he couldn't even get started on basic physics, and that was 350 years ago... without it, everything that physicists have done since then will be inaccessible to you.

 

[fresh_42]

I've found a few links which might (or not) be appropriate to start with:

https://www.math.ucdavis.edu/~anne/linear_algebra/mat67_course_notes.pdf
http://www.astro.caltech.edu/~golwala/ph106ab/ph106ab_notes.pdf
http://www.mecmath.net/calc3book.pdf
[ http://www.damtp.cam.ac.uk/user/tong/dynamics/clas.pdf ]

However, it is tough and especially the last one is most likely more than challenging. I have the feeling you underestimate the amount of calculus needed, but in the end it depends on your effort, patience and willingness to solve questions which will certainly arise during the lecture. You should be prepared to eventually make a step backwards.
(I still think that Wikipedia is a better way to start with and develop the areas you like to examine in greater detail. The links above might be well at the wrong end. At least the page numbers will tell you that you are on a way to climb a really high mountain.)

 

[Fervent Freyja]

Okay, So I need to learn college math.
Is their anyway to do that without waiting 6 years?
Plus, I need to learn about Group Theory. Is their any books I can read about it?

Find some buddies with the same interests- that will certainly expedite the process over a few years. Ask your parents for private tutoring when needed. Make sure that you know about the school and community resources in your area that are related to mathematics and science. If you are in the US public school system then you will probably have to take initiative in this area, there is too much focus on the humanities. Not all school systems in the US are equal, so it wouldn't hurt to determine where your school falls among national standards.

Check the universities around you to see if you have access to their libraries. If it is a state university then you can probably already get access to what you need. You may even get away with spending the summer there in the library.

 

[votingmachine]

Technically, All I need is Algebra and Pre - calculus, but isn't their anyway I can learn those two from books, or online? And after that, what should I study?

At 12, you should just have finished the 6th grade. Possibly the 7th. A lot of High Schools have better math than gets mentioned. You will have more than those two things in high school, as they also have to cover trigonometry, geometry, and some statistics. Most HS have full calculus for advanced students. I know my daughter's High School will send you up to the University of Utah if you have a math class need that they can't teach. I've certainly been impressed with how some kids are at incredibly high levels of math very early, and end up taking University math in High School.

A large part of math is understanding the derivation and proof of math "formulas" and tools. And learning the language and symbology. I would caution you not to shortcut past those proofs. To me, those are the really fun and beautiful part of math. I have never had any interest in memorizing the formula's, but I love understanding and knowing the derivation and proofs. I also have found that nothing beats sitting and listening to a good teacher writing the proof on a blackboard. The same thing in a book is not as good.

You will most likely get to much better math instruction in high school. It is contrary to the spirit of youth, but it is important to remember that classes and subjects are not a list to get checked off. They actually are things that can be enjoyed. Some impatience is good, too much is bad.

 

[Khashishi]

It's hard to answer the question because we don't know how much math you already know. Math builds on simpler math. A good explanation will put it in terms of math you already know.

In math and physics, there are several ways of representing or thinking about the same object. A polar vector is an object with a direction and magnitude, but it can also be represented as a 1 by 3 matrix. Actually, almost all objects can be represented as matrices (sometimes infinitely large matrices). Yes, f is a function, but it can also be written as a giant matrix (with one cell for each point in space). So we can use the math of linear algebra. The equations here are very abstract. They don't specify much about the transformations or the states, but we know they can be written as matrices, so they can be multiplied.

g is a transformation. Again, it's very abstract and could stand for any transformation, but for example, it could represent the action of turning an object around by 90 degrees about the z axis. U(g) refers to the matrix form of this transformation. So if you multiply U(g)f, where f is the matrix form of a state, then you get a matrix form of a rotated state.
U(g)* means the inverse transformation of U(g). So, rotating 90 degrees in the opposite direction in the example. The asterisk means complex conjugate transpose in math, but for a unitary transformation U, this is the same as the inverse.

T is an observable, which is a matrix (or function... you can use matrices or functions as two representations for the same object) which is used to extract the value of some measurement of the state f. It's abstract. For example, it could extract the angular momentum of state f. But you get a different angular momentum if you measure along a different axis. That's where $T^g$ comes in. It is the observable in some transformed view of the state. In our example, it would get the angular momentum of the state rotated by 90 degrees in some z axis.

 

[ogg]

I'm not sure if I have anything useful to contribute here. OP writes that s/he enjoys learning physics and is trying to learn "something like" gauge theory. At age 12, that is enormously impressive. Unfortunately, we have only two data points as far as determining where OP is in learning both the physics and the maths on the path(s) to having the requisite skills to enable understanding of the desired subject. Point 1 is OP's age, point two is the material OP sites as being subject of interest. The gap between those is enormous. The problem is that unless we know what you have already learned (that a 'typical' 12 or 13 year old hasn't) we can't tell you what (else) you need - we can't tell you how to get there without knowing where you are.The other problem, which I've not noticed anyone else addressing, is that without a lot of practice and application of any subject, the learning will be superficial. Like "learning" that Henry 8 had 6 wives, and executed 2 of them without understanding why (really) he had them executed while not executing the others - the politics, the economics, the law, the Church, his faith all contributed significantly to why...So, I diverge from the crowd in forking the discussion into two threads: superficial facility and deep, intuitive learning. (of course, this is a simplification!). For deep learning, you need to spend hours and hours practicing the stuff you've learned. This is simply how the brain needs to be wired. So, even if you're genius level, you'll need to exert a stupendous effort to acquire and practice the various things you'll need. OTOH, learning to become fluent in the language is much easier. I strongly recommend Susskind's (Stanford) video lectures on physics. They are very math lite. But they do have maths that you have probably not learned. I'd suggest you follow them in their logical order and perhaps buy the book (or books?) he co-wrote as supplements and when you encounter a concept you don't understand, dig into other resources to get enough background. You'll see that math is the LANGUAGE you must use to understand physics. And just as someone fluent in a language THINKS in that language (rather than their 'native' one), you'll see that thinking in math is required to fully grasp what he's doing on his whiteboard. Its been a few years since I watched his courses (they're free on-line) but iirc he assumes students have some knowledge of vectors, matrices, and calculus - both differential and integral, perhaps with a bit of trigonometry and series as well. A LOT of what he does is based on the infinitesimal, taking the limit, which is the basis of calculus (among other things). You'll also need to be familiar with the idea that space is a concept that can be generalized to be any number of dimensions (into vectors of any rank, in other words). But this is gently shown to you in his first course on classical dynamics (Mechanics). Ahhh, this post is long enough and I've just started. What isn't mentioned and rarely explicitly taught is that much of what you'll need to learn involves the abstraction of concepts and formal (logical-mathematical) systems. Physics uses both heavily, and little of either has been used by the time a student has finished 7th or 8th grade. Also, just as symmetry is a feature of the natural world, mathematics uses symmetry a lot, but Physics even more. Without the mathematics of symmetry, group theory, gauge theory, is meaningless. I just reread my post, and I'd quibble with myself - I shouldn't have used the word "fluent", you can't become fluent imho without extensive practice, but you might become 'adequate' enough to vaguely understand with just a little...which is probably the best you should hope for without mastering (practicing) the multitude of concepts.

 

[Sherwood Botsford]

At a guess you need some progression like this:

High school algebra, including trig.
Geometry is nice, but not necessary, but it gives you some concepts about similarity and symmetry and the nature of proof. Also a different way of looking at math. I prefer the outcomes of the more traditional Euclidean approach.

Calculus and Differential equations. Usually 2 years at college. This goes well with intro physics courses as you can put the calculus to work right away.

Linear algebra -- simultaneous equations, matrices. 1 semester course.

Abstract Algebra -- this is where you will be first exposed to groups. Although I've heard that mathematicians and physicists use a different notation. This is a whole specialty in Math, but try an intro course.

Good luck. Keep reading. Where you don't understand, skim across and try something else. It will all come together.

 

[Jon Richfield]

Although I am 12 years old, I do enjoy learning physics. However, sometimes their is a common issue when I look up something like " Introduction to Gauge Field Theory": whenever I read the document, their is some complicated math stuff that I don't understand. Here's an example:
View attachment 102075
I really need help with math like this. I can't understand physics without it.

So my question is: What type of math is this? Where can I learn how to do it and understand it? Is their any specific thing I should study? Any website or book?

Congratulations on your attitude and your insights so far. Also on the range of good answers you have received so far.
And good luck.
Now, I don't know whether I have missed the following suggestion in the foregoing, and I don't know whether it will be remarkably good advice or remarkably bad, so just take it for what it is worth. Check out the book by Roger Penrose "The Road to Reality". If local bookshops don't have it, Amazon should. It is designed to take you from a very elementary level of maths to a very advanced one.
Also, read a lot of popular books on assorted scientific subjects. They tend not to be too mathematical, but help with insight. Martin Gardner's or Isaac Asimov's collected essays would be good starters. If you are interested in a different line of maths, books like: Concrete Mathematics Ronald L. Graham and Donald E. Knuth do some brilliant lines of development. Books on problem solving also help one's mental development. Your mental development will be all the sounder and smoother if you don't just concentrate on maths. You might like to try Paul Zeitz The Art and Craft of Problem Solving anf How to solve it by Polya.
Surf Amazon if you can't find the stuff directly.
Good luck,and a great future to you.

 

[eri]

As for textbooks, the free OpenStax collection has everything from pre-algebra to calc III, and more are coming. They also have college (and soon university) physics, which are a good place to start after you've learned some pre-calc and trig.

https://openstax.org/subjects