But first of all, it should be emphasized that people who do physics should know why they want to study a certain field, and history is very important. Weinberg's books always talk about history first, and then sort out physics; Vilcek also emphasized the importance of paying attention to the development history of physics for studying physics in his popular science book. This is the personal experience of two people with the second gradient of the Nobel Prize in Physics. One example is why we should study quantum field theory, which is a problem left over from history. Negative energy is a starting point, the combination of relativity and quantum mechanics is a starting point, and secondary quantization is also a starting point. When we know the development history of quantum field theory, we naturally know what quantum field theory wants to talk about and what problems to solve.

Mathematics has always been regarded as the language tool of physics, but after the evolution of the last century, it gradually became the starting point of physics, and even led many physicists to be criticized by their peers for studying mathematics instead of physics. This group of people are ridiculed by mathematicians for their imprecise and confusing language. They only know why. This group of people are people who study the grand unified theory, not just string theory.

There are few students trained by physicists a hundred years ago now. From the first day of college, they have to study mathematics first, which to a great extent leads students to think that mathematics is the first in physics (of course, the first). Unfortunately, everyone has forgotten that the starting point of physics is to explain natural phenomena. Natural phenomena are very complex, and physics can only abstract the simplest models, such as ideal gas model and Ising model. The strict language for describing the model is mathematics.

Having said that, I just want to say that physics needs to learn mathematics. The following is about mathematics classified by mathematics in physics:

Complex variable function: In physics, imaginary number is widely used, and the introduction of imaginary number in Fourier transform eliminates many problems of trigonometric function simplification. But in fact, the most beautiful place of complex variable function is conformal transformation (* * * shape transformation). The most widely used in physics is the famous * * * shape field theory.

I feel it necessary to say something. One of the most beautiful treatments in physics is symmetry. Although symmetry can't directly solve physical problems, it gives physicists an excellent tool to simplify physical theories or models. By studying symmetry, people classified fields and particles, defined gauge field, found out how to give gauge field particles mass, that is, Higgs mechanism, and even created the concept of supersymmetry from the definition of symmetry, which solved many problems and revived the interaction between physics and mathematics. The mathematical theory of symmetry is group theory.

Differential geometry: note that what we are talking about here is "differential" geometry, which is actually generalized calculus from a physical point of view. Usually calculus in universities is done in Euclidean space, and there is no concept of divisibility. On differential manifolds, the first thing we need to define is the concept of locality. You can only do local calculus, but not the whole differential manifold. So in physics, the first thing to use differential geometry is the general theory of relativity, which connects space-time and geometry. Gauge field theory has a formula similar to general relativity to some extent, which originated from the relationship between gauge field theory and fiber bundle.

Symplectic geometry: poisson bracket is a very important concept in quantization, which is closely related to symplectic geometry in mathematics. I am not familiar with the content, so neither books nor reviews can give good recommendations. I just want to emphasize that this is a very serious direction of mathematical physics, which was done by mathematicians. It is rare for people with a physical background to do this.