Jonathan Gleason
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 Aug 18 awarded Popular Question Jun 28 awarded Popular Question May 15 comment How do I find constraints on the Nambu-Goto Action? Honestly, there is probably a much more clever way to do this, but I personally don't think it is a wise use of time to try to find a better solution to something like this. May 15 comment How do I find constraints on the Nambu-Goto Action? Then you can just have Mathematica row-reduce this matrix so that you can just read-off the multiplicity of $0$ as an eigen-value. If you do this for several different values of $p$, you should see a pattern. This of course gives you the answer. To prove it, you would have to turn this calculation into a proof; I am quite confident that I never actually bothered to do that. May 15 comment How do I find constraints on the Nambu-Goto Action? Write out the formula for $\det (g)$ using the co-factor expansion along the first column (or row) to obtain an explicit formula for $L$. For concrete-ness, take the space-time dimension to be, say, $D=4$. This way, you will be able to write down an explicit formula for the matrix $\frac{\partial ^2L}{\partial (\partial _tX^\mu )\partial (\partial _tX^\nu )}$ (of course if you want a general proof you will need to do it for arbitrary $D$). The point of doing this is that now we will have a very explicit $4\times 4$ matrix that you can just plug-in to Mathematica (or some other CAS) . . . May 15 comment How do I find constraints on the Nambu-Goto Action? Actually, since it's been a little over two years now since I actually did the computation, I admittedly don't remember exactly what I did, and I can't say for sure unless I just re-do everything. I can, however, tell you the first thing I would try . . . May 7 awarded Favorite Question May 3 awarded Yearling Apr 26 awarded Popular Question Jan 22 awarded Popular Question Jan 20 awarded Nice Question Dec 2 accepted Fast and slow modes, and the vanishing of certain diagrams during re-normalization Nov 30 comment Fast and slow modes, and the vanishing of certain diagrams during re-normalization "Within a perturbative RG scheme, by which I mean the expansion parameter remaining small all the way to the fixed point and at the fixed point, engineering scaling dimension is enough to prove the irrelevance of the $\phi ^6$ vertex." ---- How do I actually go about proving this statement? With a proof of this, I believe my question will be completely answered. (Also, am I correct in presuming that this is not specific to $\phi ^6$, but is also true for general $\phi ^{2n}$ with $n\geq 3$?) Nov 30 comment Fast and slow modes, and the vanishing of certain diagrams during re-normalization In particular, am I correct in saying that we can ignore the diagram I am worried about, not because it vanishes, but because its contribution is irrelevant? On another note, how does one relate this to "momentum conservation"? Is the meaning of this just that one cannot produce vertices with an odd number of slow modes? Nov 30 revised Fast and slow modes, and the vanishing of certain diagrams during re-normalization added 1 character in body Nov 30 comment Fast and slow modes, and the vanishing of certain diagrams during re-normalization How do we know a priori that vertices of the form $\phi ^{2n}$ with $n\geq 3$ will be irrelevant. Once again, don't we actually need to complete the re-normalization procedure to verify this precisely? Sure, one can do dimensional analysis and obtain that the 'engineering degree' suggests that they will be irrelevant, but to actually prove this expectation is correct, do we not have to do the full re-normalization procedure? Nov 30 comment Fast and slow modes, and the vanishing of certain diagrams during re-normalization "Clearly, $E$ cannot be odd since $V$ and $I$ are positive integers." ---- Sure, but what about, e.g., $E=6$. For example, consider a diagram with two $\phi _{\text{s}}^3\phi _{\text{f}}$ vertices with the two $\phi _{\text{f}}$ legs contracted. This yields a diagram with $6$ external slow legs, no? In fact, it is this specific diagram which I have been struggling to show vanishes for the past couple of days now. Nov 30 comment Fast and slow modes, and the vanishing of certain diagrams during re-normalization "However, $\phi _s^3$ vertex is not generated because there is no process in a $\phi ^4$ theory that can produce a $\phi ^3$ vertex." ---- I feel as if this argument is circular . . . Part of what we would like to show is that, after a RG transformation, we do not generate any new terms in the action (at least to the order we are working). It is conceivable that we could generate a $\phi ^3$ term after re-normalization, in which case the theory obviously could produce a $\phi ^3$ vertex. Am I seriously mis-understanding something? Nov 28 awarded Inquisitive Nov 27 asked Fast and slow modes, and the vanishing of certain diagrams during re-normalization