A request for advice on studying theoretical physics

[rstaylor]

I am a 25 year old high school teacher who wishes to further his understanding of physics to a point where some recent papers on theoretical physics can be read and understood. I graduated with a mathematics degree around 4 years ago and regards myself as fairly talented and able with mathematics, but have unfortunately not spent much time on it in the past few years so am very rusty. I have recently read many popular qualitative accounts of theoretical physics (Lee Smolin, Hawking etc) and wish to properly understand and study some topics, in particular to be able to understand recent attempts at unification (string theory, quantum gravity etc).

I have a reasonable knowledge on the following areas of mathematics that could be brushed up fairly quickly: Linear Algebra (pretty comprehensive), Group Thoery (up to Sylow's thoerems), Multi-variable calculus (Green/Stokes theorems), lots of real and complex analysis (spent much of my degree on this), some algebra (simple theorems on fields, rings etc), a little knowledge on differentiable manifolds, early undergrad physics (special relativity, Newton's laws, basic quantum mechanics). I skipped a lot of learning about differential equations, perhaps I shall regret that now...

I would really like to have a good working knowledge of:

1. General relativity
2. Quantum mechanics
3. The standard model.

I would really appreciate advice on the main areas of maths I should learn/brush up on and also which areas other areas of physics I sure explore to help with the above three topics.

How would experienced physicists here approach this if they were in my position? Rush to the frontier of the subject and hopefully fill in the blanks later or a slower more careful approach? My aim is to spend around 15-20 hours per week on average, and to be around the stage of a beginning postgrad physics student in a couple of years.

 

[confinement]

The first subject should be the calculus of variations, the Euler-Lagrange equation. This is something you could learn in one day, and it is fun. In basic terms you have an integral over dt where the integrand L(y(t),y'(t),t) depends on an unknown function y(t) and its derivative y'(t) , and you want to find the function y which extremizes (max or min) this integral.

In physics the integrand is called the Lagrangian, and all the forces of the standard model are based on finding the Lagrangian that describes the theory (where, for example, the unknown function y(t) becomes a vector field of electromagnetism that depends on position in spacetime). In classical mechanics the path of a particle is the one for which the integral of the Lagrangian is extremized; in quantum physics we integrate over all possible paths, which is generally impossible to do analytically, and so we do a perturbative expansion using the mnemonic of Feynman diagrams to make a term by term approximation to the infinite-dimensional path integral that no one can do (that isn't even defined by mathematicians). The point is that the rules for manipulating Feynman diagrams come directly from the Lagrangian itself.

The question then becomes, how do we generate Lagrangians? The answer is that we use gauge symmetry, group theory. This is why you see the standard model comes from SU(3) x SU(2) x U(1), where the elements of these groups generate Lagrangians in a mechanical way.

The important thing about SU(3) , SU(2) , U(1) is that they are lie groups i.e. continuous groups with infinitely many elements. Every Lie group is also a differentiable manifold, and the tangent space at the identity is called the Lie Algebra of that group. Unfortunately most of the theorems about finite groups do not apply, no Sylow theorems here.

The subjects you need to study are:

1) differential geometry: tangent space, vector bundles, tensors, differential forms
2) Lie Groups and Lie Algebras
3) Relativistic Quantum Mechanics: dirac's equation, path integrals

For GR I recommend Gravitation by Misner, Thorne, and Wheeler.

 

[confinement]

I almost forgot a very nice guide page from the man who put Gauge theories back on the map in the 1970s:

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

 

[rstaylor]

I did do a course on mechanics involving the lagrangian and the calculus of variations but this was about 6 years ago now - I can remember so little. All I can really remember is that it was a different way of solving classical mechanics problems but equivelent to Newtons laws - I had no idea it was important in quantum mechanics. I shall add it to my list of things to check out - thank you.

I almost forgot a very nice guide page from the man who put Gauge theories back on the map in the 1970s:

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

This looks like an excellent guide! Thank you again...

 

[reasonableman]

I'm an experimental physicist so not exactly from the same field. Still I think you've asked a very difficult question.

First I think the best thing would be to know where your going. I'm no expert in any of those fields but I'd guess that it's really very difficult to read a new paper from any of those fields and understand it from a general point of view. New papers are usually quite specialised. What might be useful in general QM might be quite irrelevant to some new exotic frontier QM...

If you have a preference then I suggest leaning towards that - gaining enough knowledge to read new papers in 3 fields is really rather formidable.

Then as well as working forwards work backwards. For this you'll need journal access. Get a new paper you like the sound of, briefly look up anything you don't know and try to identify any recent seminal papers in the field.

As mentioned in Hooft's guide, getting a teacher is very handy and will speed things up. Not sure how you can do that...some sort of correspondence may be possible with a professor at a local Uni.

 

[cristo]

For GR I recommend Gravitation by Misner, Thorne, and Wheeler.

Noone really learns GR from MTW, do they?

 

[rstaylor]

If you have a preference then I suggest leaning towards that - gaining enough knowledge to read new papers in 3 fields is really rather formidable.

Then as well as working forwards work backwards. For this you'll need journal access. Get a new paper you like the sound of, briefly look up anything you don't know and try to identify any recent seminal papers in the field.

Working backwards seems like a very nice idea, I think I need to develop a backgrounds in the basics though. I agree that concentrating on three different fields is rather formidable but I don't see it being so different to undergrads studying the basics in several subjects at the same time. I certainly feel more focused now than when I was an undergrad.

I just need a little guidance as the task seems very daunting and I need to get some kind of hook on it to get me going i.e. know where to start. I have spent a lot of time in the past couple of years learning about evolutionary biology, but that was simpler as it is a more 'compact' subject where it is easier to get to the frontier quickly.

As mentioned in Hooft's guide, getting a teacher is very handy and will speed things up. Not sure how you can do that...some sort of correspondence may be possible with a professor at a local Uni.

Not exactly sure a busy professor would appreciate a stanger getting in touch about something like that. Not sure though...

 

[reasonableman]

I should say that the purpose of working backwards is not to gain a complete understanding, just it will just highlight important things that you require a proper understanding of.

For example there are lots of differential equations in physics but learning to solve all the types is really an excercise in mathematics. Also a physicists definition of a solution is very different to that of a mathematician!

Regarding contacting a professor, provided you are not just saying 'I want to learn stuff - teach me' it shouldn't be too bad. Professors like to talk about their field. I have sent emails to professors saying things like:

Dear Prof. Megabrain,

I am currently trying to do/make 'a gizmo that uses a principle you know about'. I see from the Somewhere university's webpage you have research interests in this. I have made a certain amount of progress but need advice on the following - x, y and z.

Could you help or direct me to somewhere that might be able to?

They may just direct you to a textbook or one of their postdocs/PhD students but having someone to discuss things with is helpful.

 

[rstaylor]

I think I am still some time until I have the kind of understanding necessary to have a proper discussion. First I need to recapture some of my rapidly fading mathematical knowledge :-)

I suppose I just need a little guidance on what to concentrate on...

 

[reasonableman]

I suppose I just need a little guidance on what to concentrate on...

For that you really need a theoretical physicists advice...

Additional stuff you might want to try:

The Road to Reality by Roger Penrose. Supposed to take you to the state of the art of theoretical physics. But I've heard you need to read around it (it is quite cheap though).

For new papers, I think this journal open to all.

http://www.iop.org/EJ/journal/NJP

A quick browse found 'Hawking radiation of the Vaidya–Bonner–de Sitter black hole' Deyou Chen et al 2007 New J. Phys. 9 252. That's probably quite sexy, but I bet all the references are in expensive journals.