• Question: What type of mathematics should I master in to become a theoretical physicist?

    Asked by rajathjackson to Chris, Dave, David, Fiona, Jack on 23 Jun 2013.
    • Photo: Jack Miller

      Jack Miller answered on 23 Jun 2013:


      Hi Rajath,

      It’s a very long list (and I’m not sure that anyone ever really masters mathematics…), but here’s a rough guide, in the order I remember coming across the different topics at school/university:

      — Calculus (differentials/integrals of a single variable)
      — Euclidean geometry
      — Matrix algebra
      — Multivariate calculus (functions of many variables)
      — Coordinate geometry & vector calculus (in 3D)
      — Complex algebra

      — Linear vector spaces
      — Differential equations (1D)
      — Partial differential equations (many-D)
      — Applications of Partial differential equations in many different situations: Classical electromagnetism, classical physics, the Schrödinger equation, and so on…
      — Fourier’s theorem and Fourier analysis
      — A good course on relativity

      — Groups, Rings & Fields (&Spectral analysis)
      — Complex analysis
      — Applications of Fourier’s theorem to solving PDEs
      — Integral transformations, such as the Laplace transformation
      — More differentiable geometry (geometry on curved, non-Euclidean spaces)

      — A good course on QM, including lots of bra-ket notation, Dirac delta functions, the Klein-Gordon & Dirac equations
      — Functional calculus
      — Any other random maths I’ve forgotten about

      — Field theory & the standard model.

      I can probably find a detailed syllabus for the physics course here if you’d like it!

      All the best,

      — Jack

    • Photo: David Freeborn

      David Freeborn answered on 23 Jun 2013:


      Hi rajathjackson,

      Gerard T’Hooft has a pretty good website on what you need to learn for theoretical physics here. This might be above your current level (it’s mostly undergraduate level) and some of the links are now broken, but it’s still good. It was definitely of help during my undergraduate degree:
      http://www.staff.science.uu.nl/~hooft101/theorist.html

      Really important things that stand out to me are:

      (1) Mastering vectors and matrices. This will lead on to linear algebra, differential equations, vector calculus and modelling systems in many dimensions.

      (2) Complex numbers, and later on complex analysis. These techniques are really, really important.

      (3) Functionals, and functional calculus, when you get to it. Also mastering tensors, when you cover them at University (not yet!)

      (4) Statistics and probability theory.

      I’ll write some more on the details of what I think you need when I get the time.
      Hope this helps for now!

    • Photo: Chris Mansell

      Chris Mansell answered on 23 Jun 2013:


      The others are giving you very comprehensive answers so here is a little anecdote.

      Heisenberg didn’t know what matrices were when he developed his matrix formulation of quantum mechanics. He just knew how to deal with abstract concepts and how to apply logic consistently. Heisenberg was Max Born’s assistant at the time. Born had to tell him that the things he had written down and manipulated in his (highly successful) attempts to understand quantum mechanics were matrices. (Heisenberg and Born are both winners of the physics Nobel prize.)

      I’m not trying to make any point with this story. I think it’s just fairly interesting.

    • Photo: Dave Farmer

      Dave Farmer answered on 24 Jun 2013:


      Hi Rajath,

      Jack and David’s answers are very comprehensive, and you should certainly pay attention to them. The one thing I would add is that while you need to master all these topics, don’t forget that at the core, you need to be very good a simple algebra i.e. simply manipulating equations. Fortunately this is normally just a matter of practice, and if you learn all the topics on the list above, that’ll probably happen anyway!

      On a side note, if you’re committed to theoretical physics than that’s fine, but I would strongly recommend not avoiding lab work entirely at university. It is very different to labs A-Level, and you might find yourself taking to it more than you’d expect, I certainly did!

      Dave

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