The mathematics required for mathematical physics (LQG, ST, FT, etc)?

[hadsed]

I realize this might be a little bit of academic advisement, but I think it'd be more useful to have some people who really know a lot about the subject post about this. I like math a lot, and specifically I was wondering about number theory or set theory and logic, which is interesting but I'm not interested enough in it to really try hard to master it. I'm not actually sure what field of mathematics or mathematical physics I'd like to do, but the mathematics I enjoy learning about tend to be in geometry and topology and such.

Really, if anyone could, I'd like to know what types of mathematics I could study in my spare time to better understand 'beyond the standard model' theories in mathematical physics. Of course I'll study what I like to, but I'd love to actually be able to understand sometime the actual mechanics of what goes on in these nutty theories like LQG, string theory, field theories and all of that.

Thanks.

 

[marcus]

... understand sometime the actual mechanics of what goes on in these nutty theories like LQG,...

LQG is not nutty. There is no stretch of credulity required. You don't have to believe in anything weird like extra dimensions.

It is a straightforward development of quantum geometry, starting with the geometric quantities we MEASURE. Quantum chunks of volume meeting each other at quantum faces of area. The geometry of the universe corresponds to a network of nodes (volume) joined by links (area) defining in this way a quantum state of geometry.

Of the universe or a small piece of the universe. If it is the whole universe then it has to be a very large network with many nodes. Or you have to think about taking a limit with more and more nodes in the network.

What you start with is the simplest most rudimentary way to determine the geometry of a region of space---by measuring a lot of areas and volumes. And you allow those to be UNCERTAIN. To fluctuate in and out of existence. So that the overall geometry with its resulting curvature is quantum/uncertain.

Space is not considered to be discrete (not made of little grains ), but our measurements of geometry are necessarily finite and discrete, so our geometric INFORMATION can be thought of as granular in a certain sense.

Tell me CAN YOU STAND TO WATCH THE FIRST 8 MINUTES OF http://pirsa.org/10110052/ ON YOUR COMPUTER?
I'm curious. Can you understand Eugene's speech. I can. I find it easy to understand him. It is just an Italian accent and I am used to that. How about you? He gives some basic intuition in the first 8 minutes.

If you know basic quantum mechanics, there is a hilbert space of quantum states of geometry and there are operators on that hilbertspace. Operators like measuring areas and volumes.
I have a link to some online textbook-like resources, and some other possibly helpful links, in the thread called "Introduction to LQG".
https://www.physicsforums.com/showthread.php?p=2904073#post2904073

THE BAD NEWS IS THERE IS as yet NO LQG TEXTBOOK, at least that I have read and can recommend.
There is a new textbook that comes out next month by Bojowald which starts with General Relativity---that is, from scratch. To understand LQG you should learn the Hamiltonian approach to GR. That is what Bojo's textbook teaches you.
Here's the amazon page:
https://www.amazon.com/dp/0521195756/?tag=pfamazon01-20
Here is the publisher page (it comes out in December 2010 in the UK)
http://www.cambridge.org/gb/knowledge/isbn/item5692826/?site_locale=en_GB
I can't recommend it because I haven't seen the book. The word "Canonical" in the title just means that it is the Hamiltonian approach to GR. The idea is to get that Hamiltonian GR as a good start and then crash on through to LQG and Loop cosmology.
I don't know if Bojo's pedagogical strategy will work. I have to see the book before I decide.

The state of the art in LQG is described in two short survey papers that came out this year.
April paper http://arxiv.org/abs/1004.1780
October paper http://arxiv.org/abs/1010.1939
My advice: don't just read the abstract, click on PDF and delve into both papers even if you don't understand the math. Read the parts you can understand. Don't worry if the hard parts don't mean anything to you. All about Lie Groups, Hilbertspaces and stuff.
Then look for textbooks and introductory stuff later.
The papers may seem impossibly hard and completely baffling. But you will be exposed to the hardest face of the real thing, and that teaches something.

 

[hadsed]

I meant nutty as in the difficulty, implications, methods, etc. are so unlike anything I've ever seen before. Maybe it was a bad choice of word.

Anyway, thank you very much, I'm pleasantly surprised that you took the time to give me the nitty gritty technical stuff. I appreciate it very much. I watched some of Bianchi's lecture, I can understand everything [as far as his accent and speech, that is]. It looks very interesting and I'll definitely watch the rest at a more reasonable hour when I can think properly...

I always try to force myself through ridiculously technical wiki articles and journal publications. Sometimes I wonder how much good it does, but I did in astrophysics I definitely know a lot more about it now, but I think it's a bit different since I don't quite have the mathematical background yet for these theories, whereas with astrophysics and cosmology I can get a good idea with basic knowledge. I think I will try to get through some mathematical physics papers though, since you've recommended it.

 

[marcus]

...
I always try to force myself through ridiculously technical wiki articles and journal publications. Sometimes I wonder how much good it does, but I did in astrophysics I definitely know a lot more about it now,...

You mentioned sleep. I'm on pacific time 3 hours earlier. I'm going to sleep too and maybe tomorrow can refine the above suggestions.
Not to force too hard. You can get something from the April and October papers just reading the easy parts for a general impression and skipping the technical parts. But after that taste of the current formulation, there has to be some introductory material on the SU(2) group, L₂(G) the space of square integrable functions on a group, group representations (simple classification in the case of SU(2)) and such-like things you need to understand those two harder papers. I can't think of the appropriate introductory material right now.

John Baez has some nice handwritten notes on spin networks. I'll look for the link tomorrow.

 

[MathematicalPhysicist]

Btw, mathematical physics is surely not constrained to only field theories and quantum gravity issues.
There's quantum information, chaos, etc.

 

[physeven]

try out chaos. a really interesting field for me. i have a couple of books on chaos that i read in my spare time.

 

[marcus]

Hadsed,
sorry to be a let-down. I thought a bit about trying to assemble online sources
(Lie groups, esp. SU2, group reps, Hilbertspace) and then felt old, lazy, forgetful and not very knowledgeable about online sources. Retired mathematician, well into retirement! Nice morning here, almost 10AM. Coffee. the garden right outside the open door.

Do you already know about the square integrable complex-valued functions forming a (Hilbert) a vector space with inner product?

I feel in my bones that you do :-D but can't rely on these feelings. Maybe you have already several of the resources you would need and if I could overcome my slothfulness it would be largely duplication of effort.

Keep in touch. Don't be discouraged.

 

[humanino]

If you want to learn the mathematics for BSM physics, you may want to try
http://www.math.ias.edu/node/96
As the title suggests, this is not about LQG. I guess you do not want just one approach, but rather sample a few to decide with your own taste. You may also want to try
Alain Connes downloads
The algebraic approach is even more mathematical.
Finally, there are nice lectures on
Space-time, Quantum Mechanics and Scattering Amplitudes

 

[marcus]

I told you last night I would try to be helpful. The LQG of the April and October papers is based on such simple math concepts! It's tempting to try to explain.

The usual American textbook notation for square integrable functions uses a superscript L² but for some reason Rovelli is writing it with a subscript L₂.

they don't have to be functions defined on the real line. they could be, eg. on a circle, or on the cartesian product of two circles: a donut surface.

those are examples of compact Lie groups. "compact" basically means doesn't go on forever. a compact Lie group has an integration-measure defined on it which is the uniform measure completely evenly smeared out by the groups own action. Mr. Haar gave it his name. Thank you Herr Doktor Haar!

Let's think of a compact group G with Haar measure abbreviated "dg" analogous to "dx" on the real line. Haar measure let's us unambiguously integrate functions defined on compact groups.
We can take averages of anything defined on the group.

This is the Hilbert space defined on a compact group G: L₂(G, dg)

To be more specific I have go find the damn integral sign.

∫|f(g)|²dg < ∞.

The first concept you meet in Rovelli's LQG is the graph Hilbert space. It is an interesting idea, just a hair more complicated than the Hilbert space defined on a cartesian product of copies of SU2.

I like the "graph Hilbert space" idea a lot. A graph Γ is simple, just some nodes and links. Two lists really, all it is. For each link you write down two nodes, it's start and finish. So you list the nodes and then you list the links (and say each one's source and target.)

Then you let the graph Γ tell you how to put together copies of SU2 into something like a cartesian product group. You define L2 on that and suddenly you have quantum states of geometry. The "graph Hilbert space" H_Γ

You can calculate with it, and when you begin to want more complexity you can enlarge the graph----let the number of nodes go to infinity. But the approximate theory is with the finite graph and that graph's Hilbert, that you can calculate with. The space of quantum states of simplified geometry. The graph symbolizes a finite set of measurements you are able to make on Nature, or a finite restricted world of information. If it is not enough you enrich the graph and enrich your world of geometric information---but always keep it finite.

See if you can read a little in the April paper. I don't know if this kind of hit and run lunatic introduction to the subject will work pedagogically. I'm not on anyone's payroll and I'm totally irresponsible. Tell me if any of this works for you

http://arxiv.org/abs/1004.1780
http://arxiv.org/abs/1010.1939

Copied from Arivero:
αβγδεζηθικλμνξοπρσςτυφχψω...ΓΔΘΛΞΠΣΦΨΩ...∏∑∫∂√ ...± ÷...←↓→↑↔~≈≠≡≤≥...½...∞...(⇐⇑⇒⇓⇔∴∃ℝℤℕℂ⋅)

 

[hadsed]

This is as difficult as it is interesting (one of the two should tell you about the other, heh). I can't say I understand most of it, but reading the papers has forced me to go wiki and google some more things, so indirectly I've learned some this way. Of course I still don't understand what the papers are really saying, and anything I do sort of put together leaves me later on. I think I just need more training in the maths, everything I know past calculus is self-taught.

I'll see about getting a few of those books, and thank you humanino for the lectures. Bianchi was pretty helpful, though I could only understand so much of the lecture (after about 10 minutes it was getting too technical). Marcus, you're a big help as always, thank you. I tried reading the Intro to LQG thread a long time ago but it was very difficult. I feel a bit better about it, maybe after some more reading I can try to go through that.

 

[marcus]

This is as difficult as it is interesting (one of the two should tell you about the other, heh). I can't say I understand most of it, but reading the papers has forced me to go wiki and google some more things, so indirectly I've learned some this way. Of course I still don't understand what the papers are really saying, and anything I do sort of put together leaves me later on. I think I just need more training in the maths, everything I know past calculus is self-taught.

I'll see about getting a few of those books, and thank you humanino for the lectures. Bianchi was pretty helpful, though I could only understand so much of the lecture (after about 10 minutes it was getting too technical). Marcus, you're a big help as always, thank you. I tried reading the Intro to LQG thread a long time ago but it was very difficult. I feel a bit better about it, maybe after some more reading I can try to go through that.

Here's a recent post from the Intro to LQG thread. It might help you make a study plan if you got a list of the courses being taught in this one-year Masters degree program, and the lecture notes or textbooks being used.

A one-year masters (MSc) program is being set up by John Barrett and colleagues at Nottingham. The first year of operation will start September 2011. We may be able to pick up some information about their curriculum, lecture notes, textbooks---get some clues from this.
http://johnwbarrett.wordpress.com/

==quote==
New MSc course
Here at Nottingham we are starting an MSc in Gravity, Particles and Fields. This is very specifically aimed at students interested in getting into relativity and particle physics research, in areas such as quantum gravity, cosmology, quantum information, etc.

The first run starts in September 2011, so we are open for applications now.
==endquote==

http://pgstudy.nottingham.ac.uk/postgraduate-courses/gravity-particles-and-fields-masters-msc_1163.aspx
==quote==
Gravity, Particles and Fields Masters (MSc)
Duration: 1 year full-time

The course provides an introduction to the physical principles and mathematical techniques of current research in general relativity, quantum gravity, particle physics, quantum field theory, quantum information theory, cosmology and the early universe.
The programme of study includes a taught component of closely-related modules in this popular area of mathematical physics. The course also includes a substantial project that will allow students to develop their interest and expertise in a specific topic at the frontier of current research, and develop their skills in writing a full scientific report.
The course will provide training in advanced methods in mathematics and physics which have applications in a wide variety of scientific careers and provide students with enhanced employability compared with undergraduate Bachelors degrees. In particular, it will provide training appropriate for students preparing to study for a PhD in the research areas listed above. For those currently in employment, the course will provide a route back to academic study.
Key facts
• The course is taught jointly by the School of Mathematical Sciences and the School of Physics and Astronomy.
• Dissertation topics are chosen from amongst active research themes of the Particle Theory group, the Quantum Gravity group, and the Quantum Information group
• In addition to the lectures on the course, there are several related series of research-level seminars to which Masters students are welcomed.
• The University of Nottingham is ranked in the top 1 per cent of all universities worldwide.
...
==endquote==

I would say check out the website now and then go back again in a month or two and see if they have more detail. A list of courses offered (and textbooks used) would be great.
What I highlighted in blue: the "taught component" must refer to the courses---maybe some of them are computer course modules, but there will have to be a teacher, so I assume lectures and problemsets etc.

It's something to find out about because how they think it should be done---how they prepare students to go into a PhD program in QG---could give you guidance about what to do on your own. Barrett is one of the top LQG people, as well as having done important research in Noncommutative Geometry (tied if not actually "scooping" Alain Connes in a major result---NCG implementation of the Standard Model.)