3/31/2023 0 Comments All for strings theory workbook 3![]() ![]() Such a path integral that looks like it should make sense is the path integral for a supersymmetric quantum mechanics system that gives the index of a Dirac operator. You get an interesting analog of the sphere toy model for any co-adjoint orbit of a Lie group G, with a path integral that should correspond to a quantum theory with state space the representation of G that the orbit philosophy associates to that orbit. I spent a lot of time thinking about this, one thing I wrote early on (1989) is available here. The Chern-Simons path integral is of a similar nature and should have similar problems. The sphere is a sort of phase space, and “phase space path integrals” have well-known pathologies. It turns out that since the “action” only has one time derivative, the paths are moving in phase space not configuration space. If you discretize, there’s nothing at all damping out contributions from paths for which position at time $t$ is nowhere near position at time $t \delta t$. If you think about this toy model, which looks like a nice simple version of a path integral, you realize that it’s very unclear how to make any sense of it. ![]() You could think of A as the vector potential for a monopole field, where the monopole was inside the sphere. ![]() Here $S^2$ is a sphere of radius 1, and A is locally a 1-form such that dA is the area 2-form on the sphere. Zee’s book is a worthy attempt to explain QFT intuitively without equations, but here I want to write about what it shares with the Quanta article (see chapter II.3): the idea that QM or QFT can best be defined and understood in term of the integral Two things recently made me think I should write something about path integrals: Quanta magazine has a new article out entitled How Our Reality May Be a Sum of All Possible Realities and Tony Zee has a new book out, Quantum Field Theory, as Simply as Possible (you may be affiliated with an institution that can get access here). ![]()
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