JPBenowitz said:What mathematics are necessary for understanding and using minimal surfaces particularly in quantum field theory? As of now I have a very limited mathematical background as I will be taking Calc III, Diff Eq, and Linear Algebra next semester but I hope to get into a quantum field theory research group by the end of the summer.
chiro said:Hey JPBenowitz.
I'm taking a quick look at a book on Minimal Surfaces, and it looks like the pre-requisites include some differential geometry. This in the first chapter and afterwards they jump straight into the minimal surfaces.
JPBenowitz said:Do you think I could jump into Differential Geometry while doing Diff Eq or should I wait?
chiro said:You could if you have a good enough foundation in Multivariable and Vector calculus, but if it interferes with your DE course, I'd wait until the course is over.
If you plan on doing stuff with General Relativity, then I would wait until you've done some PDE's first and for that you need a solid background in DE's.
Maybe what you could do is first familiarize yourself with the tensor theory and get used to the notation and how the generalized co-ordinate system theory works before you look at differential geometry with the theorems and things like Gauss-Bonnet and curvature. You need to understand this before you touch the more formal stuff.
You should be able to do tensor theory with the Multivariable and Vector calculus background so if you are keen just get a good book on tensor theory: different people use it including mathematicians, physicists (and other scientists) as well as engineers so there are plenty of different perspectives that should suit you to choose from.
A minimal surface is a surface that has the smallest surface area possible for a given boundary. This means that any small change in the surface would result in an increase in surface area. In other words, minimal surfaces are critical points of a functional that measures surface area.
Minimal surfaces are used in the study of quantum field theory as a mathematical tool to represent the energy of a quantum field. They provide a way to calculate the energy of a system in terms of a minimal surface, making it easier to solve complex problems.
To understand minimal surfaces in the context of quantum field theory, one should have a solid foundation in mathematical topics such as differential geometry, calculus of variations, and functional analysis. Knowledge of quantum mechanics and field theory is also helpful.
Minimal surfaces are used in physics to study various phenomena, including black holes, phase transitions, and the behavior of fluids. They are also used in string theory to study the properties of strings and membranes.
Yes, minimal surfaces can be observed in real-life. Some examples include soap bubbles, soap films, and the surfaces of liquids in containers. They are also present in nature, such as in the structure of certain plants and the shape of soap bubbles formed by some species of insects.