Quantum Homotopy Type Theory

The wall I keep hitting when trying to learn physics is vector calculus. I missed out on this as an undergraduate, and I'm struggling with finding a textbook I like now. I think the best thing to do is to write my own treatment. So: Before we start doing stuff in vector calculus, we should do: Analysis Linear algebra Topology But analysis and linear algebra both use the concepts of fields, and so we should build up to that through rings and groups. So I want to start by working through Aluffi's Algebra Chapter 0 . I think I will probably end up using some form of type theory as my foundation, so that we can consider the objects of a category as a type without an equality relation, but for now I will work semi-formally and leave the foundation vague.

Here is what the reading list looks like: History of Science Emilio Segre, From Falling Bodies to Radio Waves: Classical Physicists and Their Discoveries, W. H. Freeman, New York, 1981. Emilio Segre, From X-Rays to Quarks: Modern Physicists and Their Discoveries, W. H. Freeman, San Francisco, 1980. Robert P. Crease and Charles C. Mann, The Second Creation: Makers of the Revolution in Twentieth-Century Physics, Rutgers University Press, New Brunswick, NJ, 1996. Abraham Pais, Inward Bound: of Matter and Forces in the Physical World, Clarendon Press, New York, 1986. Physics M. S. Longair, Theoretical Concepts in Physics, Cambridge U. Press, Cambridge, 1986. Richard Feynman, Robert B. Leighton and Matthew Sands, The Feynman Lectures on Physics, 3 volumes, Addison-Wesley, 1989. Ian D. Lawrie, A Unified Grand Tour of Theoretical Physics, Adam Hilger, Bristol, 1990. Herbert Goldstein, Charles Poole, and John Safko, Classical Mechanics, Addison Wesley, San Francisco...

Read John Baez's reading list . Read the references in the Background section of D. J. Myers, H. Sati, U. Schreiber Topological Quantum Gates in Homotopy Type Theory . Read the references in the statement of the Yang-Mills problem . BLOCKED: Need to know what a Lorentz tensor is. Write a rigorous treatment of quantum field theory (as much as is possible today). Write a mathematically rigorous version of this Wikipedia page . Read the Geometry of Physics pages. Write a treatment of quantum field theory in Martin-Löf Type Theory. Write a treatment of quantum field theory in Homotopy Type Theory.

Here's the first version of the reading list, the references in the official statement of the problem , in the order in which they are referred to in the text: L. O’Raifeartaigh, The Dawning of Gauge Theory, Princeton University Press, 1997. C. N. Yang and R. L. Mills, Conservation of isotopic spin and isotopic gauge invariance, Phys. Rev. 96 (1954), 191–195. S. Weinberg, A model of leptons, Phys. Rev. Lett. 19 (1967), 1264–1266. A. Salam, Weak and electromagnetic interactions, in Svartholm: Elementary Particle Theory, Proceedings of The Nobel Symposium held in 1968 at Lerum, Sweden, Stockholm, 1968, 366–377. D. J. Gross and F. Wilczek, Ultraviolet behavior of non-abelian gauge theories, Phys. Rev. Lett. 30 (1973), 1343–1346. H. D. Politzer, Reliable perturbative results for strong interactions? Phys. Rev. Lett. 30 (1973), 1346–1349. M. Creutz, Monte carlo study of quantized \(SU(2)\) gauge theory, Phys. Rev. D21 (1980), 2308–2315. K. G. Wilson, Quarks ...

I've bounced off this project of understanding, well, the universe several times. I keep hitting a wall each approach I try (but getting a little bit further each time). I tend to go through this cycle: OK, let's read about quantum field theory! I need to learn a lot more basic physics and mathematics before I understand this. The way this basic mathematics is done in the standard textbooks is horrible, I want to see a category theoretic version. Oh, that would be an actual research project. OK, let's sit down and learn the classical version. Hmm, the basic textbooks have so little rigour that I can't really fully what's going on, but the rigorous textbooks assume that you've already read the basic textbooks. In particular I'm feeling the hurt from missing out on vector calculus and topology when I was an undergraduate. I'll probably have to catch up with those sooner or later. But here's how I'm going to do things, in the ...