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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