I'm interested in cutting edge string theory studied by research physicist. I'm wonder what mathematics is needed and how far am I in terms of mathematics background needed and how much more mathematics do I need in order to study it properly and reach my goal. My mathematics level is at calculus/Linear algebra.

  • $\begingroup$ You have a very long way to go in terms of math knowledge. $\endgroup$ – Jim Jul 20 '15 at 17:06
  • $\begingroup$ tensor calculus, group theory, topology, etc. The list is long. $\endgroup$ – Jim Jul 20 '15 at 17:08
  • $\begingroup$ Why string theory? Even Sheldon knows better. $\endgroup$ – john Jul 20 '15 at 17:39
  • $\begingroup$ Related: physics.stackexchange.com/q/5495/2451 $\endgroup$ – Qmechanic Jul 20 '15 at 17:57

Some years ago, Gerard 't Hooft posted "How to Become a Good Theoretical Physicist", which is more inclusive than just string theory but which you'll probably still find a valuable list. Here's what he recommends for mathematics:

"Primary Mathematics":

  • Natural numbers: 1, 2, 3, …
  • Integers: …, -3, -2, -1, 0, 1, 2, …
  • Rational numbers (fractions): 1/2, 1/4, 3/4, 2379/1773, …
  • Real numbers: Sqrt(2) = 1.4142135… , π = 3.14159265… , e = 2.7182818…, …
  • Complex numbers: $2+3i$, $e^{ia}= \cos(a) + i \sin( a)$, … they are very important!
  • Set theory: open sets, compact spaces. Topology. You may be surprised to learn that they do play a role indeed in physics!
  • Algebraic equations. Approximation techniques. Series expansions: the Taylor series.
  • Solving equations with complex numbers. Trigonometry: sin(2x)=2sin x cos x, etc.
  • Infinitesimals. Differentiation. Differentiate basic functions (sin, cos, exp).
  • Integration. Integrate basic functions, when possible. Differential equations. Linear equations.
  • The Fourier transformation. The use of complex numbers. Convergence of series.
  • The complex plane. Cauchy theorems and contour integration (now this is fun).
  • The Gamma function (enjoy studying its properties).
  • Gaussian integrals. Probability theory.
  • Partial differential equations. Dirichlet and Neumann boundary conditions.

"Advanced Mathematics":

  • Group theory, and the linear representations of groups
  • Lie group theory
  • Vectors and tensors
  • More techniques to solve (partial) differential and integral equations
  • Extremum principle and approximation techniques based on that
  • Difference equations
  • Generating functions
  • Hilbert spaces
  • Introduction to the functional integral

There is almost nothing on these lists that's jumps out at me as unnecessary for string theory, with the possible exception of probability theory (and even then, it's so baked in to quantum mechanics that it'd be hard to leave it out.)

  • 1
    $\begingroup$ +1 because of how depressing it is to read this list and see how much more one has to learn to be a good physicist. $\endgroup$ – Jim Jul 20 '15 at 17:45
  • $\begingroup$ Update: I have learnt a lot since, I now have calculus, multivariable calculus, ODE,Group Theory, Real Analysis, Numerical Analysis,Fourier Analysis, Complex Analysis, PDE $\endgroup$ – ys wong Jan 30 '18 at 18:51
  • $\begingroup$ @yswong Have you studied QFT yet? $\endgroup$ – bolbteppa Jun 11 '19 at 14:38

It really depends on what you want to research within string theory, but it's one of most mathematically intensive areas within physics. List a mathematical discipline, and chances are you can apply it within string theory.

At a bare minimum, you'll need everything through quantum field theory and general relativity, which includes calculus of variations, complex analysis, group theory, PDEs, path integrals, differential geometry, maybe some topology and anything else I've forgotten. String theory usually builds on this with at the very least a little algebraic geometry. If you take cues from people on the nLab, category theory can be big in string theory. Pick any combination of differential/algebraic and geometry/topology and it will be useful in string theory. Number theory as well.

Pick some math and learn it and it's hard to go wrong.


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