Among all other fields of science, physics has arguably influenced math the most. This tradition goes back to Newton’s *Principia *where calculus was invented to solve problems in mechanics and gravity or even Euclid’s *Elements *where mathematical reasoning was applied to understand the physical nature of space and shape. Einstein’s theory of general relativity (c1910s) relied on but also spurred further developments in differential geometry, and the quantum theory (c1920s) spurred further developments in Hilbert space theory, distribution theory, and functional analysis.

Most recently, current research in string theory and field theory, by pioneers such as Ed Witten (at the IAS, the only physicist so far to have won the Fields Medal), is once again showing that physics can lead to new mathematical discoveries. Physicists are sometimes thought to be less rigorous with their reasoning, discovering new results through a special quality called “physical intuition.” Then the mathematician usually confirms the physicist’s intuition by putting it on more rigorous foundations.

Given Princeton’s proximity to the IAS as well as its strong focus on mathematical research surrounded essentially on recent physics (geometry, topology, analysis, etc.), Princeton is one of the best places to explore those deep interdisciplinary relations between math and physics–or perhaps even find that there is not so much of a hard line between the two fields.

Lastly, when Einstein had a choice between researching at Princeton or Harvard, he chose Princeton. Some joke that it’s because the president of Princeton at that time spoke to him in German whereas the president of Harvard spoke to him in French.

[Alex Chen ’21, published 2019]*Note: The author of the following guide (below) is unknown. If they would like to identify theirself please email the current webmaster. The below piece was published in 2018 and represents one student’s view.*

Contents

#### An Informal Guide for Undergraduates Interested in Mathematical Physics

This is written primarily for first and second-year undergraduates interested in studying mathematics and physics. There will be some advice on and description of mathematics and physics courses of potential interest.

Although taking lots of courses in both physics and mathematics is a great way to start exploring one’s interest in mathematical physics, this need not be the only route. There seems to be many different flavors of doing mathematical physics, and an extensive knowledge of physics is not always a strict prerequisite. For example, one can work on an analysis problem motivated by physics.

Therefore, this document offers some generalized comments on how a beginning undergraduate can get started in exploring and deepening his or her interest in mathematical physics. To receive more concrete (or precise) advice with regards to one’s subjective interests, one should contact a graduate student or a faculty member working in the field.

PHY 103/105 (Mechanics)

This is the first semester of the introductory physics sequence. PHY 105 is a more advanced version of PHY 103, and is highly recommended if you had some previous exposure to AP Physics or equivalent, particularly if you are interested in continuing beyond the introductory sequence.

Physics 104/106 (Electricity and Magnetism)

Both courses cover the fundamentals of the theory of electricity and magnetism. The more advanced PHY 106 is a great way to learn multivariate calculus with strong physical motivation. If you are unfamiliar with multivariate calculus, you might consider taking a course on it concurrently. Again, PHY 106 is recommended if you want to go further in physics.

Multivariable Calculus

One has option to take MAT 218 or MAT 203. It is important to take into consideration one can improve his or her calculation abilities in MAT 203, and it may be worthwhile to do so. Of course, MAT 218 can give a little taste for differential geometry for the interested student.

Differential Equations

Extremely important to refine your technique here. Learn differential equations and partial differential equations. See how other courses are related to this.

Differential Geometry

Can open one’s interest in general relativity.

Calculus of Variations

First introduction to the topic in PHY 205/207 in deriving the Euler Lagrange equations. Pay particular attention to it. Take MAT 451 too for relevance in quantum mechanics.

Fourier Analysis

Functional and Real Analysis

MAT 451

Probability Theory

#### An Informal Guide for Mathematics Majors Interested in Physics

This is written primarily for mathematics majors (concentrators) who also want to enrich their education with physics courses and focus on the more theoretical and conceptual side of physics. It may also benefit others with very strong mathematical backgrounds who wish to learn physics, but beware that the advice here is less useful for one who actually wants to fulfill the requirements for the physics concentration.

The first thing you should know is that the physics department is much more rigid than the mathematics department in a variety of ways. First and foremost, they want you to carefully follow the sequence that they’ve set out, and they make it difficult for students to do otherwise. The math department, on the other hand, makes it entirely the students’ responsibility to choose the level of their courses (but the department still provides guidance, of course). In addition, contrary to what you might be used to in some upper-level mathematics courses, physics professors tend to be quite strict about handing in homework on time.

That being said, different people come to Princeton with different backgrounds, and the standard physics sequence might be good for some students. What I encourage you to do, however, is to at least consider alternative sequences, and, most of all, not to be deterred when bureaucracy tries to prevent you from doing what’s right for you. That being said, I’ll get into a bit more detail.

I’m going to describe the sequence that I took, along with pieces of commentary on the different courses, as well as a bit of commentary on courses that I did not take. While the specific sequence that I took might not be for everyone, I hope that it will help you better understand the different options.

First, if you have little or no high school physics experience, you will need to take PHY 103/104/105/106. If you don’t like these courses (and some people don’t), you can try to learn physics on your own, but I know of no higher level physics course that can make up for basic physics knowledge.

On the other hand, if you do have a basic background in, say, AP Physics, I highly recommend that you not take PHY 103/104/105/106. While it’s true that you might be exposed to some slightly more difficult problems in those courses, you will be exposed to many interesting problems in later courses. Therefore, if you can handle higher-level courses, taking the lower-level courses can be a waste of time (and possibly even destroy your interest in physics!). Furthermore, most courses (as in mathematics) will review basic material in the first week or so. So, if you already know that material but need a refresher, you will be fine in a higher-level class.

My biggest recommendation is to take PHY 207 as your first course. First, you might ask, why not PHY 205? That might depend on your taste, but here’s why. PHY 207 is much broader and conceptual, while PHY 205 goes through less material but gives students more complicated problems (which doesn’t necessarily mean the same thing as a “difficult problem” does in the mathematics world). PHY 207 will introduce you to lots of topics you need to know – special relativity, Langrangians and Hamiltonians, waves, and even basic quantum physics. As a result, you will have a whole host of knowledge that will benefit you in moving on to higher-level courses. PHY 205 will take up more of your time (during which you could learn other, interesting things) and won’t even give you as broad a curriculum. Finally, as of this writing, PHY 207 is taught by Professor Verlinde, who is great. On the other hand, there have been complaints about the teaching in PHY 205.

Once you’ve taken this, you will be able to take three courses: PHY 304, PHY 505, and PHY 523.

First, I’ll discuss PHY 304. I had some experience with E&M – I got a 5 on the AP. But I learned most of it the two weeks before, and so while I had experience with the material, I really needed a refresher. You might have thought that PHY 106 was the right course. In fact, PHY 304 was just perfect. It reviewed all of the basic material from E&M that I didn’t entirely understand, but it didn’t spend an entire semester doing so (which I didn’t need). In addition, it developed all of the more advanced topics to give me a very good working knowledge of E&M. Having taken PHY 207, you’ll be fine with relativity, Langrangians, and waves, all of which show up during different parts of PHY 304. If you’re taking it with Professor Meyers, note that he is particularly strict about handing in homework on time (he’ll go over his policies).

Next, let’s discuss PHY 523. You might be in a bit of shock – directly from 200-level to 500-level? Well, I’ll explain why. First, note that this only makes sense if you have taken MAT 218 or equivalence and optionally have some experience with differential geometry. If you have this, then the course is perfect. It begins with a review of special relativity in its form that most naturally generalizes to general relativity. If you’ve taken PHY 207, you will have no problem with this, and having spent the first couple weeks reviewing special relativity, your special relativity will certainly be strong enough to then proceed and learn general relativity. To comment on the course a bit, Professor Pretorius likes to jump around a bit in the curriculum, something that might help with the intuition but which I found somewhat disorganized and confusing. I should add that I didn’t pay enough attention in the last few classes, which was not great for the final exam.

Finally, I’ll discuss PHY 505. First, here’s why you would want to take it. PHY 208 and PHY 305, the undergraduate introductions to quantum physics, are not so well-taught. Next, PHY 505 is totally accessible to someone coming out of PHY 207. PHY 207 has an introduction to quantum physics that gives some basic principles. PHY 505, while being a graduate course and, in particular, mathematically mature, starts from the basics of quantum physics and builds up. As with many courses, it can actually be easier to understand a course like PHY 505, which explains things at a sophisticated level. However, I should note that they usually don’t let undergraduates take it for credit, so it might not be possible. I sat in on the first few weeks but left due to time constraints.