This account is temporarily suspended to cool down. The suspension period ends on Aug 24 at 17:32.
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479185
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location New York City
age 40
visits member for 2 years, 11 months
seen 11 hours ago

I do not participate on this site any longer, except to respond to comments regarding my own text, if that text is unavailable in another form. I do not accept the political moderation atmosphere here, it is not compatible with open science.


Jul
7
comment Should linear algebra and vector calculus from traditional courses be replaced with `geometric algebra`?
The problem is that it is not simpler! Ask Dimension10 whether it was easier to learn Dirac matrices or Geometric Algebra. The matrices are concrete, computational, and you can immediately see how they work, and you can work with them with no prior intellectual labor. The abstraction is too abstract! The first rule of mathematics pedagogy: any student can easily reconstruct the abstract from the concrete, but it's much more difficult to go the other way. Dirac matrices are already pretty abstract, but a matrix representation is essential for when you get stuck, and students get stuck a lot.
Jul
7
awarded  Enlightened
Jul
7
awarded  Nice Answer
Jul
3
awarded  Nice Answer
Jul
1
awarded  Nice Question
Jun
29
comment Why Silver atoms were used in Stern-Gerlach experiment?
@hiki: They could have known, because the Sommerfeld quantization always gives odd numbers, so in addition to -1 and 1, you need 0.
Jun
29
comment On-shell symmetry from a path integral point of view
@drake: Oh, I see what you mean now about real part--- this is not necessarily fatal, you can transform the fields with an i too so that the determinant is non-real, but then perhaps there is a field redefinition in real and imaginary parts so that it becomes fields shifted by independent fields. Yes, it's an issue. The Lagrangian becomes imaginary. But perhaps one can see it as a smooth deformation to get to PT quantum mechanics? I don't know.
Jun
29
comment On-shell symmetry from a path integral point of view
@drake: Yes, I know about the reexponentiation, it doesn't matter for the case in the question where you are only shifting fields by other independent fields, since then the determinant is one. The problem with compensation is that the classical variation is order 1, while the determinant variation is order $\hbar$, so this only can happen if there is no classical limit. Maybe 2d Gross-Neveu type models (four Fermi), where Bosonization can reveal new symmetry, but in this case it's not going to be a determinant compensation either, but a symmetry which is only evident in one formulation.
Jun
29
awarded  Popular Question
Jun
28
comment On-shell symmetry from a path integral point of view
Yes, we agree... (I edited the comment too late, hence time travel!)
Jun
28
comment On-shell symmetry from a path integral point of view
Your Ward identity formula is not really general. what should the Ward identity be for the formal transformation on $S = \int |\partial \phi|^2 + |\partial \eta|^2 + \phi^2 \eta^2 d^4x$ under the formal non-symmetry transformation $\delta \phi = \epsilon \phi \eta$, $\delta \eta = - \epsilon \eta $? The reason I ask is because the general formula is false for this case, as the transformation is not a field shift, but a field-value dependent field shift. so there is a determinant in the measure.
Jun
28
comment On-shell symmetry from a path integral point of view
Excellent! Thanks. I got it. The formal current thing is the idea that I was missing.
Jun
28
comment On-shell symmetry from a path integral point of view
I have seen a million people ask this question about SUSY closure, why it is justified to use eqns of motion. I have seen people give glib or wrong answers, or claim it's a Ward identity. But I never saw the actual answer you have above, not this identity. The derivation of the Ward identity and the Schwinger Dyson equation with coinciding insertions are exactly the same, but the field transformation is of a different kind. That's why I am asking for a source, because I checked to see if it appears in Schwinger and Dyson's papers, no, and 6 reviews, no. There SD is just Eherenfest/Heisenberg.
Jun
28
comment On-shell symmetry from a path integral point of view
Why I am mystified: the Ward identity is a special sort of Schwinger Dyson equation, for shifting the fields by a symmetry, while the equation of motion identity is for just shifting the fields. The identity for equation of motion times a local operator is what I never saw in books, so I had to do myself. I see the transformations involved, they are similar but not the same. I am taking your word for it that this is called "Schwinger Dyson equation", but I am wondering why I never saw the identity for ${\delta S\over \delta\phi}O$ anywhere (I read the books you gave, no). Sohnius didn't too.
Jun
28
comment On-shell symmetry from a path integral point of view
The Ward identity is doing an infinitesimal symmetry transformation, that's what the $\delta \phi_i$ are in your comment, while the identity I derived is doing a shift of the field $\phi$ (not a symmetry). In papers and books, they always derive the Schwinger dyson equation as the operator equation ${\delta S\over \delta \phi} =0$, where $S = \int L(\phi) + J \phi$, including a source term J, and this is sufficient to derive perturbation theory (like Schwinger's/Dyson's papers), but it's not enough to say exactly when the equation of motion works with a coinciding insertion.
Jun
28
comment On-shell symmetry from a path integral point of view
I am not claiming originality, but I want to know who writes this, because it just doesn't commonly appear, but it appears on Wikipedia. It's not the Ward identity (although the derivation is nearly identical), it's something else, involving the question of equation of motion times local operators, and this is something that confused the heck out of me for years until I figured it out. Is it in a textbook? I skimmed a few just now and didn't find it. Is it in some review paper?
Jun
28
comment On-shell symmetry from a path integral point of view
While what you write is the Schwinger Dyson equation as it appears on Wikipedia, namely $\langle {\delta S\over\delta\phi}O\rangle = \langle {\delta O \over \delta \phi}\rangle$, trivially equivalent to my answer, I read the original articles of Schwinger and Dyson, and a lot of related literature, and please, for my sanity, could you tell me who writes this down first? I had to come up with it myself, and I didn't even know it was called Schwinger Dyson equation, SD was the simpler $\langle {\delta S\over \delta \phi} \rangle=0$ in all sources I ever read. I had to rederive it myself.
Jun
28
awarded  Necromancer
Jun
28
revised On-shell symmetry from a path integral point of view
final fixes
Jun
28
awarded  classical-mechanics