# Graduate School for Theoretical Physics

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First off, let me just say that I am unsure if this question is appropriate for this site, and if the community deems it necessary, the question should be closed.

So right now I am a fourth year mathematics and physics major whose research interests lie in quantum field theory / quantum gravity. In particular, I would like to go into theoretical physics. However, I have found that most physicists do not achieve a level of rigor I would strive for in my work, and so I am reluctant to apply for graduate school in physics. On the other hand, it seems that in most mathematics departments in the U.S., there are at most a couple of mathematicians working in what you might all theoretical physics, and I would not feel comfortable applying to a graduate school banking on the fact I could work with a specific one or two persons.

What possible routes are there that I could take if my interests lie in theoretical physics, but I wish to practice theoretical physics as a mathematician would? Furthermore, are there any (not necessarily American) universities with a joint mathematics-physics department (like the DAMTP at Cambridge), or, the next best thing, a mathematics and physics department that collaborate frequently? What about theoretical physicists working in quantum field theory/quantum gravity who don't shy away from rigor?

Any and all input is greatly appreciated!

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asked Oct 18, 2011
Could you give us a couple of examples of TP papers you feel are representative of the physics level of rigor? At present, I'm not really clear on where you draw the line.

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The best example I can think of a *good* level of rigor would be Wald's General Relativity book. An example of a *bad* level of rigor would be defining a Lie group to be a group whose elements can be "continuously" labeled by a finite set of parameters. If more specific examples would help, please feel free to ask.

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The reason I'm asking is that I think you are perhaps being disingenuous in your view of theoretical physicists. Many papers are themselves mathematically rigorous, but are build on less rigorous foundations (i.e. renormalization etc.). I'm not clear how you view such work. You say your interests lie in QFT, so this is quite important, since unless you are interested only in AQFT, you'll likely find yourself using well established but less rigorous approaches.

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Side note: maybe I'm off base, but my impression has been that you can and should apply to graduate schools with the expectation that you're going to be working with a specific researcher - of course you would have to get in touch with that person first and see if they have an opening for you and are interested in having you work for them. In a sense, you "apply" to the research group first, and then once "accepted," you officially apply to the university that group happens to be based at.

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Perhaps we have differing views of what it means to be rigorous? I have only skimmed serious TP research papers before, so I can only speak to the level of rigor I have experienced in taught classes. From that experience, however, perhaps all except for one or two (theoretical) professors I've had could even define a Hilbert space, Lie group, manifold, etc. For example, my current QFT professor openly denounces questions pertaining to how to make arguments precise (he says to stop being like a mathematician).

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Unfortunately, the department I'm in has given me the impression that this attitude is more or less typical of theoretical physicists.

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Perhaps I should also mention that I am more interested in placing QFT on a mathematically rigorous foundation, so that, in my research at least, I probably would not be working "on top" of things like renormalization.

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I don't think this is a particularly efficient way to make life decisions. Instead, maybe it is a good idea to speak in person to a few professors, postdocs and grad students (Chicago should have great people to talk to) and seek advice specific to you and your preferences. When you do that, bear in mind that physics/math research is nothing like UG courses, and perhaps it's not a good idea to have a priori ideas on how things are or should be, before you have more experience.

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I have to agree with Moshe's last comment. However, I did upvote the question, because I think it is one of the more important types of soft-question.

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@Moshe I don't understand. What is not an efficient way of making life decisions? I have spoken to as many people at Chicago as I can, and while must of their advice has been useful, I am still without a clear idea of good schools for theoretical physics are. I am excited about Cambrdige and the Perimeter Institute, but besides these places, I am afraid of going to grad school in math and not able to work on the problems I want to, or conversely, going to grad school in physics and not being able to work on the problems I want to in the way I want to. I just want the best of both worlds :)

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One comment on rigor, which is not my favorite subject. I think the natural level of rigor for any set of ideas has to be adapted to its particular stage of development. If you enjoy high level of rigor, and still want to be productive in the business of generating new knowledge, you have to opt for more mature subjects where the efficient and appropriate language was already discovered. Most of these will be studied in math departments. In physics we often deal with structures that are still under construction, and the language and methods are adapted to this less complete level of knowledge.

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For what it's worth, I would say that I am more interested in placing QFT on a firm foundation than "generating new knowledge". That is to say, I am interested in taking the knowledge that we currently have, and placing it on a rigorous mathematical footing.

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I think an experienced person who has one-on-one conversation with you is in a better position to tease out what is most important for you, and give you advice based on that as well as what exists in reality. It is hard to do that over the internet, at least for me.

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I would say talk to some professors to get their opinion, my random opinion is that: If you hold Wald up as an example of good rigor, I would say just go to graduate school in physics.

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@jonathan, if we had this in-person conversation, I'd probably ask you precisely what you enjoy in math and physics before getting into any specifics. Having an idea as specific as "putting QFT on firmer footing" at this stage of your career is probably premature. Just my two cents - I really hesitate giving any kind of advice in this inefficient and tone-deaf medium, so I think I will stop here. Good luck!

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@BenjaminHorowitz You seem to be implying that Wald does not conform to a typical mathematician's level of rigor. I personally have only read the first half of the book or so, but as far as I remember, in terms of rigor, it might as well have been written by a mathematician. Do you have a specific example where it does not hold up to these standards?

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@JonathanGleason you really need to start reading papers if you're this unclear on the differences in rigor. In particular, compare hep-th and math-ph; I think you'll find many of your preconceptions of what rigor means and what mathematical physics is are misplaced.

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@Domenic Could you please elaborate? It seems as if you are not the only one who thinks I have a misguided notion of what "rigor" is, means, and what it's role in theoretical physics is. Could you please explain?

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Not right before bedtime I can't :P. But, I think @Joe Fitzsimons's answer is a good start. And I really think if you compare papers across disciplines the difference will be apparent.

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@JonathanGleason, The issue isn't so much "level of rigor" as intended audience. Wald's book is a G.R. book, designed to be read by physicists. It isn't a "math book", persay. If you like that sort of book, you will find many more like it throughout graduate study in a normal Physics graduate school.

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That is true. Well, honestly, I love the fact that it is a *physics* book with a great amount of rigor, but do not like so much that it is not formatted like a mathematics book. I am probably biased, but I think it could have been organized/formatted in a much clearer way had it been written closer to the "math book" standard. In particular, I found that a lot of important content that could have stood out as 'Theorem' or 'Definition' "boxes" was hidden away in paragraphs of text.

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The reason I have been asking about your perception of rigor in the comments is because I believe it is very easy for undergraduates to be misled by questions like Gil's on the role of rigor. This is talking about something more than simply using correct mathematical reasoning, but instead about making all arguments and postulates fully precise, in a way which undergraduates are rarely exposed to.

Physics courses are not really a good indication of the level of rigor among theorists within a given department, but rather the level of rigor they expect undergraduates should be exposed to. In a mathematics department the expected level of rigor for students is higher, because mathematicians must be rigorous. In a physics department, however, the bar is often a bit lower. This is because the department is training not just theoretical physicists, but also experimentalists too (and indeed these generally make up the majority of the class). The is a lot of physics known, and so there is a trade-off that needs to be made between rigor and the amount of material covered. Additionally, some courses will be taught by theorists and some by experimentalists, and the lecturer will not always be teaching in their area of expertise.

As a PhD student, all of this changes, because the level of rigor you are working at will largely depend on your field, your approach and your supervisor. It won't be the sort of compromise that is necessary when dealing with large classes in a diverse department. For this reason I strongly agree with Moshe's comment above

I don't think this is a particularly efficient way to make life decisions. Instead, maybe it is a good idea to speak in person to a few professors, postdocs and grad students (Chicago should have great people to talk to) and seek advice specific to you and your preferences. When you do that, bear in mind that physics/math research is nothing like UG courses, and perhaps it's not a good idea to have a priori ideas on how things are or should be, before you have more experience.

To give you a concrete example of how rigor is not determined strictly by department, I have had positions either as a PhD student, postdoc or faculty in each of the following: Mathematical Physics, Materials Science, Computer Science, Combinatorics & Optimization and Physics. Many others are in the same boat, spending time in one department before moving to another, not because their research interest changed, but rather because there are multiple departments which consider their research "on-topic" so to speak.

Further, while Cambridge has DAMPT (which for the record is different from DPMMS, their pure maths department), Oxford has the Peierls Centre for TP within physics, and has a separate maths department. Do you think that there is a substantial difference in the type of faculty they hire?

There most certainly are different flavors of theoretical physicist, with varying concerns about the mathematical underpinnings of their field, but the type of rigor you talk about seems not well correlated with department.

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answered Oct 18, 2011 by (3,575 points)
+1; "There most certainly are different flavors of theoretical physicist" - cannot agree more. hept-th is not all theoretical physics there is...

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As a former Chicago undergraduate who also majored (you're not supposed to call them "concentrations" now?) in math and physics and also, for a while, felt mildly disgruntled about the lack of rigor in physics, I have to say: your opinions as an undergraduate might feel a little ill-considered when you're a little older. Theoretical physics is a really intricate structure with a lot of internal consistency checks. It is not yet rigorous mathematics; rigorous versions of QFT lag far behind what we actually know about QFT. Field theory and string theory are very rich subjects, and I don't think the lack of sharp definitions is really problematic most of the time. There is a lot that can be rigorously shown once we take for granted that the theories exist and have certain properties.

If you want to be a mathematician, that's fine, and there are things like topological field theory that you might be interested in. (Look up what's happening at the Simons Center at Stony Brook, for instance; that's a place where it seems like people are doing serious mathematical physics.) But if you want to be a theoretical physicist, at some point you'll have to learn to stop worrying and start making non-rigorous assumptions to make progress.

(Also, at the risk of offending people, you should be wary of overly-rigorous approaches to "quantum gravity"; from what I've seen, often the rigor is making it harder to notice that they're giving deeply unphysical results.)

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answered Oct 18, 2011 by (1,630 points)
Nope, can't say I've heard the term "concentration" since I've been here. Thanks for the input!

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I don't think it is "non-rigourous" to make assumptions and proceed from them. "Non-rigourous" (and annoying) is if it remains vague and unclear what is assumed for what reason.

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@Phira I don't think he means "assumptions" in the sense of axioms. More likely he means something along the lines of just assuming things work out, converge, can be made percise, etc., instead of "wasting" hours worrying about how to make precise sense of a particular integral, for example.

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Personally, I think someone with your interests would be more at home in a math department than a physics department. There has been a lot of interesting activity in recent years at the intersection of math and physics with things like topological field theory, noncommutative geometry, and the Langlands program. These mathematical topics are some of the most promising lines of research for people seriously interested in quantum gravity.

Here is a list of math departments that are very strong in mathematical physics (and some possible advisors):

Berkeley (Auroux, Borcherds, Frenkel, Reshetikhin, Teleman)

Caltech (Gukov, Marcolli, Simon)

Chicago (Beilinson, Drinfeld, Ginzburg)

Columbia (Greene, Khovanov)

Harvard (Gaitsgory, Jaffe, Lurie, Yau)

MIT (Etingof, Kac)

Penn (Donagi, Pantev)

Texas (Ben-Zvi, Freed)

Yale (Frenkel, Goncharov, Kapranov, Zuckerman)

Some others that aren't as difficult to get into are

Kansas State (Crane, Soibelman)

Riverside (Baez)

If you really want to work in a physics department, there are some places that take a more mathematical approach. I don't strongly recommend it, however, because you'll be signing onto a very specific research program (like string theory or loop quantum gravity) which might never pan out. Some possibilities are

Harvard (Vafa)

Penn State (Ashtekar, Bojowald)

Perimeter Institute (Smolin)

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answered Oct 18, 2011 by (140 points)
Ooh I forgot Gukov! Yes, he's excellent, although he does more "work on interesting mathematical problems that stem from modern physics" than the "work on interesting mathematical problems at the foundation of 1970s physics" that the OP seems to be asking for.

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Jones has left Berkeley. But Auroux & Teleman are there.

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I didn't know that about Jones. I have edited my answer accordingly.

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If, as you say in the comments, your research interests lay in placing QFT on a firm mathematical foundation, you likely do not want to do theoretical physics research, but instead mathematical physics research (algebraic QFTs and the like).

In this case I would browse prominent math department websites and find mathematical physicists who are doing the kind of work you are interested in.

Some specific places, based on my experiences applying to graduate school two years ago:

• The Perimeter Institute would probably not be a great fit as they are generally much more interested in generating new knowledge.
• UC Berkeley's mathematical physics program seems quite likely up your alley
• My undergrad research mentor, Matilde Marcolli at Caltech, does some great stuff in this area. She is unfortunately the only person in the Caltech math department that has similar interests, but it's worth checking out. Noncommutative geometry is awesome.

Also, one of my best friends from undergrad is in the math department at Chicago and had research interests very similar to yours; I'd be happy to introduce you two over email if you email me, and maybe he can provide a perspective that the internet can't.

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answered Oct 18, 2011 by (70 points)
Yes, I would be very interested in talking with him about this. Feel free to send me an e-mail if you would like.

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First, I should say that I'm in almost the same position as you, except I'm a third year undergraduate.

I think that if you're thinking about problems of quantum gravity, you should apply to a physics department for graduate school. Looking around the arXiv, it's easy to conclude that there's no single accepted level of rigor in the theoretical physics community - some theorists have more of it, and some theorists have less of it, and it doesn't seem to be correlated with the depth or profundity of ideas.

Also, as a side note, keep in mind that your interests will change. You say that you're into "putting QFT on firmer footing" now - who knows what will excite you a couple of years down the line?!

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answered Oct 19, 2011 by (30 points)

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