Math for Physics: Learn Beyond Basics Books

[Wyman91]

I was wondering what math I need to teach myself in order to progress in learning physics after the point of the basic books such as Stephen Hawking's The Universe in a Nutshell and Michio Kaku's Hyperspace. I have learned some from those but I want to know more advanced things now so I need to know some books to read in order to understand the physics behind things. I do realize I will need to know calc to understand most physics but what I won't need that for I can learn.

Thanks,
Wyman91


P.S. I am going to be a freshmen this fall so take that into consideration.

 

[rick1138]

You will need calculus, and a lot more - it takes about 10 years or more from the point where you are to get a full understanding of Superstring Theory, but you will able to get a partial understanding long before that. Here is a list, hopefully complete:

Pre-Calculus
Calculus
Vector Calculus
Linear Algebra
Tensor Calculus
Ordinary Differential Equations
Partial Differential Equations
Differential Forms
Probability
Number Theory
Homology
Cohomolgy
Homotopy
Discrete Group Theory
Lie Theory
Lie Supertheory
Algebraic Geometry
Differential Geometry
Surgery Theory
Local Topology
Global Topology
Twistor Theory
Spin Networks
Knot Theory
K-Theory
Non-Commutative Geometry
Category Theory
Complex Analysis


That should about cover it.

 

[Wyman91]

Thank you very much that helps a lot but I have one more question, what on that list should I start with. And I am going to need to know some books to read to learn some of these things myself...

 

[Janitor]

Some of us have never heard of 'Lie Supertheory.' Can you give a quick description of that?

 

[rick1138]

Lie supergroups and superalgebras are extensions of Lie groups and algebras wherein the multiplication law is symmetric in even dimensions and antisymmetric in odd dimensions, if the two objects commute. Lie groups are groups that have a continous, rather than a discrete, parameter. An example is the rotation group in three dimensions, known as SO(3), which is most often represented by a set of 3 matrices. If you have ever played a 3D video game, the equations that allow objects to be rotated are some form of SO(3). One of the heterotic string theories is named after the Lie group SO(32), which is the rotation group of 32 dimensional spaces, and has 496 generators, wheras SO(3) had 3. I can go into a more formal definition of Lie groups, algebras, supergroups, and superalgebras when I have more time, hopefully in a way that they can be accessibe. As far as order of study is concerned, I have grouped everything roughly in order, when there is a group of subjects, that means the first members of a group need to be studied before the later. I've also remarked on the difficulty of each subject.


Probability (Somewhat easy - can be studied at anytime, but advanced probability does depend on first year or so calculus.)

Pre-Calculus(Not easy)
Calculus(Difficult at first)
Vector Calculus(Ditto)
Ordinary Differential Equations(Ditto)
Partial Differential Equations(Extremely difficult)
Tensor Calculus(Extremely difficult)
Complex Analysis(Calculus in complex spaces - difficult)

Linear Algebra(Moderately hard)
Differential Forms(Abstract, but easy and beautiful)

Hyperdimensional Geometry(Very Easy)

Discrete Group Theory(Moderately hard)
Lie Theory(Difficult, but the most of the difficulty lies in the lack of decent beginning materials)
Lie Supertheory(Easy if you have mastered Lie theory)
Category Theory(Considered difficult, but there is a great introduction to the subject that makes it fairly easy to master the basics)
Twistor Theory(Not too hard, but obscure)

Local Topology(Not Too hard)
Global Topology(Very Easy)
Surgery Theory(Easy if you have mastered the previous)
Homotopy(Moderately hard)
Homology(Moderately hard)
Cohomolgy(Difficult)
K-Theory(Extremely difficult)

Algebraic Geometry(Hard to extremely difficult)

Differential Geometry(Hard)

Spin Networks(Know almost nothing about)

Knot Theory(Easy to hard - not used much)

Non-Commutative Geometry(Very difficult)

 

[rick1138]

I forgot to mention that the motivation for supergroups and superalgebras is to construct superspace, which contains both symmetric and antisymmetric wavefunctions. Bosons (particles that transmit forces) have symmetric wavefunctions, and fermions (particles that form matter) have antisymmetric wavefunctions. This is where the "super" in superstring theory comes from.

 

[Janitor]

Thank you for the explanation.

 

[Alem2000]

Are these math classes you stated for all physicits or just theoretical ones. I know for undergrad physics you have to take differential calculus, integral calculus, vector calculus, differential equations, and linear algebra. All that other stuff I am assuming is for grad school physics. Do experitmental physicists have to take those classes also?

 

[rick1138]

Right, most of it is grad school math. Experimental physics revolves around the standard model, and the math required is less abstract, and includes a lot more material on methods of solving equations by approximation, such as perturbation theory. As far as places to start are concerned, a good website is http://www.sosmath.com .


Also of high value are these two sites, one on calculus, the other on linear algebra:
http://www.hverrill.net/courses/math006/
http://www.hverrill.net/courses/linalg/

Remember, these are for college level courses, so don't be intimidated if you don't understand all of the material, or even most of it, but there is a lot of value there.

As for a basic introduction to group theory, I highly recommend the book in the Teach Yourself series titled "Teach Yourself Mathematical Groups". It explains the basics of discrete group theory in a very simple format.