there is a quantity $\chi^\dagger \bf{\sigma} \chi$ in sakurai. he is stating here that, though it appears to be a vector, $\bf{\sigma}$ is not a vector, $\chi^\dagger \bf{\sigma} \chi$ actually obeys the transformation rule of vectors. but, i do not understand this quantity. what kind of product is being applied here? i thought $\chi$ is 2x1, and $\bf{\sigma}$ is 3x1 where each element is 2x2. what am i missing here ?
@SillyGoose I would rather say that the Kronecker and dot product are representations in specific bases. Both the inner product and the tensor product structures are unique in the sense that all inner product spaces of the same dimension are isometrically isomorphic and everything that fulfills the universal property of the tensor product is likewise isomorphic to anything else that does so in that sense you actually don't have any choices here
the statement is that conformal invariance implies scale invariance, hence the absence of preferred length scales. I don't know why the negation would hold. Conformal non-invariance does not necessarily imply scale non-invariance, and in turn scale non-invariance does not necessarily imply the presence of a preferred length scale
When does one use the WKB formulas with blowing up exponentials? I think I've only seen decreasing exponentials
More precisely, I only know the formulae in the first row of this table and the first row of the next, I wonder where I should use the second rows of each
The small number of "conceptually independent types of processes and calculations" is exactly a symptom of the theory's being fundamental! Even in classical physics, all calculations could have been mathematically reduced to the calculation of the final state that evolves from an initial state (o...
If you work out $|S_{fi}|^2$ for different numbers of incoming particles, you find more and more volume elements due to the normalization of your wave functions, so we need to get rid of them. For one particle it goes away, for two particles we are 'lucky' that we can re-write it in terms of an invariant cross section by introducing the crazy Moller flux factor $I = \sqrt{(p_1 . p_2)^2 - m_1^2 m_2^2}$, for three incoming particles I have no idea what to do with those extra volume elements
One can pinpoint the technical error in LQG explicitly:
To recall, the starting point of LQG is to encode the Riemannian metric in terms of the parallel transport of the affine connection that it induces. This parallel transport is an assignment to each smooth curve in the manifold between point...
i am thinking that this theory is wrong because it treats space and time differently. The correct theory should allow for arbitrary topologies of the time dimension. LQG treats time as a boring unitary evolution parameter
idk any strict meaning of covariance beyond "there should be at least one re-formation of the theory where space and time on the same footing in the equations"
As far as I'm aware, there are no quantum gravity theories that deal directly with closed timelike curves. Some of them (like canonical quantum gravity, causal dynamical triangulation and loop quantum gravity) forbid them outright, others merely seem to not discuss the topic. I've found quite a v...
in fact it is much more unclear to me how to write down a path integral over different topologies, since in contrast to the worldsheet integral in QM where I can sum over the genera of 2d manifolds, 4d manifolds have no simple enumerations
since i asked the question about classical probabilities on a time loop, i realised its solution : a classical probability distribution on a time-loop is a probability distribution over classical field solutions on the time loop
is there a reason we expect something with spin to behave interestingly in a magnetic field? why cant we apply any type of force? is it just that magnetic forces are much stronger than, say, gravitational forces?
in order to get gravitational effects you probably would have to claim that there's some quantum version of frame dragging, which I don't think is plausible to ever detect with realistic measurements
e.g. spin angular momentum of light is "real" in that you can do experiments to measure it but photons aren't charged, so don't start thinking that spin and charge/magnetism are instrinically linked
in effect you would want to "sum" over all reasonable cobordisms between reasonable spatial slices (hold them fixed? maybe, maybe not), but still that "sum" is far too large in more than 2d
the article doesn't define quantum fluctuations, so i went to the article on fluctuations. It's defined as energy time uncertainty principle and creation of virtual particles :P
this is not the rigorous meaning, right?
this article says virtual particles are directly undetectable. this article shouldn't be allowed
no, the rigorous meaning is just that some variances of some operators don't vanish :P
the energy-time uncertainty principle in the form it is often presented in doesn't exist and has nothing to do with virtual particles, which are their own rabbit hole
sometimes it's because they're doing pop-sci and they think it's a good idea to simplify it in this way (I disagree), sometimes it's because they've just read these phrases in another explanation and haven't really stopped to think about them carefully
everyone knows that energy is to time as momentum is to position (which is true), so of course you must have an uncertainty principle for them that's like the one with momentum and position
so while I point out these kinds of errors whenever I see them, I really wouldn't want to call the people talking like that "crackpots". It's very understandable how they arrived at that idea and usually it doesn't cause a lot of harm - after all, lots of physicists keep saying this stuff because it has no (or at least no negative) impact on their ability to do correct physics most of the time
but it is a particularly ripe source of confusion for the layman or student who knows enough to realize these popular pictures are wrong but not enough to see how else to interpret them
and since physics.SE's main asker population is people confused about physics we have to engage with this kind of confusion far more often than most other physicists
indeed, many of the people who use that language probably would say something like "oh, don't take it too literally, I just like the picture" when you'd ask them. But if you've just seen the statement in a lecture or read it in an article, you can't do that as such and physics.SE ends up with a lot of these questions
there's also the weekly question that confuses quantum fields with classical fields and thinks that particles are bumps in the field
i have had this confusion in the past. i thought QFT resolved the wave particle duality because a particle is just a bump in a classical field according to QFT
i think some physicists are genuinely coming up with theories involving virtual particles being physical
they write these ideas in books and articles
but i don't think that that confusion can be attributed to popular videos. The official phrase "particles are excitations of quantum field" misleads non physicists into thinking that particles are bumps in a classical field
it's related to an issue with "physical intuition" more broadly: I think people who think this is an useful thing to say have trained their intuition to work with this in a way that doesn't mislead them
I'm looking for a color that stands out well against black. I'm working on revisions to a manuscript and I need the revisions to easily stand out for my collaborators. Most of the colors on this page overleaf.com/learn/latex/Using_colours_in_LaTeX are very "mat" and I'm looking for something a little more "glossy".
and they think it is useful to talk like this so that other people can develop the same kind of intuition, but it is very hard to communicate "intuition" except by way of example
the xcolor documentation linked there has a lot more color palettes/series/whatever in the Examples section
I'm not sure I get what you mean by glossy but if you don't find it there you're perhaps trying to create an effect that isn't related to the actual color (I'd use "glossy" to describe how something reflects light but there's usually no light source in text documents ;) )
@ACuriousMind that one's easy: you are a professional programmer and I am not (or at least nowhere near your level). My student does all this fancy version control using Git but my collaborator and I prefer to do revisions using the "token" method. For now I have the token, so I'm in charge of making revisions until the next iteration where someone else will have the token.
Let $\mathcal{H}$ be a Hilbert space. Let $T$ be an isomorphism (unitary map) defined by $T: \mathcal{H} \rightarrow \otimes_{i=1}^n \mathcal{H}_i$. Let $H$ be a hamiltonian $H: \mathcal{H} \rightarrow \mathcal{H}$. I am wondering about intuition about why the first sentence of the following picture is true. What I am thinking right now is that $THT^{-1}$ is still a unitary transformation of $H$. Thus, inner products are preserved. Thus, for each eigenvector $|\psi\rangle$ of the hamiltonian $H$
the inner product $\langle \psi | H \psi \rangle = \lambda$ is preserved, so the spectrum is preserved?
but i feel my reasoning might not be sound because in like textbook qm when you are performing a unitary transform, say the rotation operator, you are still mapping vectors from $\mathcal{H}$ to $\mathcal{H}$, not from the space to a factorization of the space?
@SillyGoose for every eigenvector $v_\lambda$ with $Hv_\lambda = \lambda v_\lambda$ you have that $THT^{-1}(Tv_\lambda) = THv_\lambda = \lambda Tv_\lambda$, so $Tv_\lambda$ is an eigenvector of $THT^{-1}$ with the same eigenvalue, so the spectra of $H$ and $THT^{-1}$ are the same
because $T$ is an isomorphism and the argument works in the other direction as well - for every eigenvector $v'_\lambda$ of $THT^{-1}$, $T^{-1}v'_\lambda$ is one of $H$
Am I the only one who thinks some dubious jumps have been made? Like we start with a classification system and end up with a prediction system which in some sense predicts that metaphysics is meaningless. However, using empirical probability betting against metaphysics is a bad idea as it has been an aid historically. — More Anonymous2 hours ago
Where do you think Carnap got the idea that anything not empirically verifiable, like metaphysics, is meaningless (hint: TLP 6.53)? But TLP is not dispositive because Wittgenstein repudiated much of it in the late period. His late philosophical therapy is arguably even more limiting, it denies philosophy its own subject or method, and sets aside many traditional philosophical problems as pseudo-problems stemming from linguistic confusion. Equivalence principle would not be philosophy to Wittgenstein, it is science's business. — Conifold3 hours ago
So here's my personal philosophy. I start out with a symbols as an attempt to describe a system. This is a classification system. For example for an ideal gas I describe the number of molecules N. Okay so far so good. Now I get lucky and find 2 classifications systems which are not equivalent (like N =N) to describe the same system.
I use a symbol to describe this
ANd then I have symbolically moved from a classification system to a predictive/falsifiable system
Now a predictive system can be falsified. So when I say Carnap starts with a system which is a classification I dont see how he ever gets to make the claim it is falsifyable
@ACuriousMind At some point Im going to give examples and say its possible use words without defining them axiomatically
Like, why does the symbol have to be a "swiggle". If I acoustically talk to someone, am I communicating an idea in a meaningfully different way than writing it down?