Can you research outside your major at Princeton?
What did you have in mind?
The standard process, as you probably know, is that for your junior papers and your senior thesis/project you work with a professor in the department that you’re majoring in. Exceptions are possible, but I think in most cases there has to be some justification, such as a desire to do a multidiscplinary project. As a hypothetical example, if you’re a molecular biology major who is also interested in computer science, you might be able to do research with a CS prof – but the project would need to have a molecular biology connection to meet the JP/thesis requirement for a mo-bio major.
If you’re talking about research that’s not meant to fulfill the requirements for your major, I don’t think there’s anything stopping you from working with someone outside the department, other than that you’ll need to get a prof to agree.