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Path integrals, part 3
This post has been migrated from my old blog, the math-physics learning seminar. Consider the heat equation \[ \frac{\partial \psi}{\partial t} = -\hat{H} \psi \] We find similarly \[ \langle x_N|U_t|x_0\rangle = \int \exp \sum_{j=0}^{N-1} ik_j(x_j - x_{j+1}) -\Delta t H(x_j, k_j) dx dk.
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Path integrals, part 3
This post has been migrated from my old blog, the math-physics learning seminar. In my previous posts on path integrals, I described (rather tersely) how the path integral, suitably defined and interpreted, can be used to compute the Schwartz kernel of the operators \(e^{iHt}\) (Lorentzian signature) and \(e^{-Ht}\) (Euclidean signature).
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Geometry of curved spacetime, part 2
This post has been migrated from my old blog, the math-physics learning seminar. Disclaimer: as before, these are (incredibly) rough notes intended for a tutorial. I may clean them up a bit later but for now it will seem like a lot of unmotivated equations (with typos!
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Geometry of curved spacetime, part 1
This post has been migrated from my old blog, the math-physics learning seminar. I’m TAing a course on general relativity this semester, and I’m covering some of the geometry background in tutorials.
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Path integrals, part 1
This post has been migrated from my old blog, the math-physics learning seminar. Consider the Hilbert space \(\mathcal{H} = L^2(\mathbb{R})\) with Lebesgue measure and a Hamiltonian \(H = T(k) + V(x)\) (a sum of kinetic and potential energy).
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Lattice quantum mechanics in 1d
This post has been migrated from my old blog, the math-physics learning seminar. For some reason I’ve been interested in lattice QFT recently, especially lattice gauge theory (note to self: a miniproject for the Christmas break is to understand the paper by Kogut and Susskind http://prd.
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Susskind on string theory
This post has been migrated from my old blog, the math-physics learning seminar. Found this and thought I’d share: Lectures on String Theory by Susskind This is a series of introductory lectures on string theory by cofounder Leonard Susskind.
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Classical mechanics, part 6
This post has been migrated from my old blog, the math-physics learning seminar. The original post can be found here. Last time we saw that a classical mechanical system which has a Lagrangian formulation can (under some mild assumptions) be repackaged as a symplectic manifold \((M, \omega)\) together with a smooth function \(H\) called the Hamiltonian.
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Classical mechanics, part 5
This post has been migrated from my old blog, the math-physics learning seminar. The original post can be found here. As we saw in the previous post, the equations of motion for a mechanical system can be cast into a 1st order form called Hamilton’s equations, which are naturally interpreted as describing a path in the phase space \(T^\ast M\) associated to the configuration space \(M\).
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Classical mechanics, part 4
This post has been migrated from my old blog, the math-physics learning seminar. The original post can be found here. Recall from last time that a classical mechanical system consists of a manifold \(M\) (the configuration space) and a function \(L\) on the tangent bundle \(TM\).
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