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Seiberg-Witten theory video lectures
This post has been migrated from my old blog, the math-physics learning seminar. I found some lectures by Sara Pasquetti on Seiberg-Witten theory here: Lecture 1 Lecture 2 Lecture 3
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A toy model for effective theory from extra dimensions
This post has been migrated from my old blog, the math-physics learning seminar. I wanted to see how the Fourier transform can turn field theory into many-particle mechanics. This is just silly fooling around, so you shouldn’t take what follows too seriously (there are much better models of extra dimensions, to be sure!
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Generating functions
This post has been migrated from my old blog, the math-physics learning seminar. The original post can be found here. Method of Generating Functions Let \(X\) and \(Y\) be two smooth manifolds, and let \(M = T^\ast X, N = T^\ast Y\) with corresponding symplectic forms \(\omega_M\) and \(\omega_N\).
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KAM
This post has been migrated from my old blog, the math-physics learning seminar. In this post I want to sketch the idea of KAM, following these lecture notes. Integrable Systems I don’t want to worry too much about details, so for now we’ll define an integrable system to be a Hamiltonian system \((M, \omega, H)\) for which we can choose local Darboux coordinates \((I, \phi)\) with \(I \in \mathbb{R}^N\) and \(\phi \in T^N\), such that the Hamiltonian is a function of \(I\) only.
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Circle diffeomorphisms
This post has been migrated from my old blog, the math-physics learning seminar. This is the first of a series of posts based on these lecture notes on KAM theory. For now I just want to outline section 2, which is a toy model of KAM thoery.
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Gaussian integrals and Wick's theorem
This post has been migrated from my old blog, the math-physics learning seminar. We saw in the last update that the generating function \(Z[J]\) can be expressed as \[ Z[J] = e^{\frac{1}{2} J \cdot A^{-1} J} \]
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Introduction to Gaussian integrals
This post has been migrated from my old blog, the math-physics learning seminar. As a warm-up for more serious stuff, I’d like to discuss Gaussian integrals over \(\mathbb{R}^d\). Gaussian integrals are the main tool for perturbative quantum field theory, and I find that understanding Gaussian integrals in finite dimensions is an immense aid to understanding how perturbative QFT works.
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Geometry of curved spacetime, part 5
This post has been migrated from my old blog, the math-physics learning seminar. Background Following last time, we are almost ready to write down the Einstein equations. Before doing any math, let’s understand what we’re trying to do.
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Geometry of curved spacetime, part 4
This post has been migrated from my old blog, the math-physics learning seminar. Today I had to try to explain connections and curvature in local frames (as opposed to coordinates), and I really feel that Wald’s treatment of this is just awful (this is one of the few complaints I have with an otherwise classic textbook).
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Geometry of curved spacetime, part 3
This post has been migrated from my old blog, the math-physics learning seminar. Today, some numerology. The Riemann curvature tensor is a tensor \(R_{abcd}\) satisfying the identities: \(R_{abcd} = -R_{bacd}.
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