What kind of mathematical disciplines would be most useful for physics?
I'm worldbuilding a story, where a famous string theorist hires a student of mathematics to try construct a new theory. For better drama, my premise of the story is that the student never learned more than high school physics.
What kind of mathematical disciplines would be most useful for someone who has never studied relativity nor quantum mechanics to understand it as quickly as possible?
I don't know how important it is, but the basic idea is as follows. A famous physicist visits a small-town college on invitation of an acquaintance who is a professor there. While giving a talk presenting his theory, a latecoming student finds a flaw in the underlying mathematics. The theory is based on a conjecture that is unproven but assumed to be correct. The suggested counterexample gains infamy for the student - something akin to finding counterexamples for the Riemann hypothesis.
Shaken by the latest string of bad news from CERN where experimentalists fail to find supersymmetric particles predicted by his theory, he decides to invite the student to help try a new approach.
I'm not looking for hard science; I'm just looking to tell a good story with some background in math and physics, something like Leonardo making tools in Assassin's Creed.
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1 answer
Let's assume that this student wants to begin by understanding the twin pillars of modern physics: quantum mechanics and general relativity. There are several major tools in the toolkit of anyone studying both of these theories at a basic level:
- Calculus (single-variable and multivariable)
- Differentiation
- Integration
- Operators such as divergence, gradient, curl, etc.
- Linear algebra
- Eigenvalues and eigenvectors
- Vector spaces, finite-dimensional and infinite-dimensional
- Tensors
- Differential equations, particularly partial differential equations
- Abstract algebra
- Group theory
- Lie algebras and Lie groups
- Representation theory
- Differential geometry
- Manifolds
- Riemannian geometry
- Metrics and notions of curvature
- Topology
These are merely starting points that would allow you to understand the basics of the two theories in requisite detail. A physics major might graduate from college with a solid grounding in the first three topics and potentially the basics of group theory. To truly work at the fundamentals of these disciplines may require additional topics such as functional analysis or algebraic geometry.
These tools pop up in most major subfields of physics, not just quantum mechanics and general relativity. They are truly the language of the subject. Most physics majors will walk out of college familiar with calculus, linear algebra and differential equations, regardless of where they go next. Specialization dictates what comes after. If someone goes on to work in relativity, representation theory, for example, may be completely unnecessary. Beyond the list I've given you, it's hard to say what sub-topics would be optimal for this prodigy.
Also important is knowing these tools in context. If you spend years studying group theory in detail, you may be well-prepared to understand the mathematics behind particular parts of quantum mechanics, but you might have little idea what the equations and mathematical structures mean. This is why physics students often learn these tools as they study specific topics. For example, I'm most familiar with the calculus of variations from its use in analytical dynamics, where it is used to derive the Euler-Lagrange equations. In short, it's not just enough to know the math - you must also know the physics.
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