@TobiasFünke just in case you need to hear it: if you find yourself feeling a bit drained by all these conversations, you might want to consider taking a break
@naturallyInconsistent Nothing advanced. Via canonical quantization the Heisenberg picture seems much more reasonable for me. Why would we use the Schrodinger picture than?
Was historically the Schrodinger picture first to come up?
@User198 no. Both pictures are available right from the start. Heisenberg, obviously, started matrix mechanics with Heisenberg picture. But if you want to consider wavefunctions as is naturally and normally done in wave mechanics, you have to think at least a little bit about Schrödinger picture
@DIRAC1930 I also read that long ago. Dirac never came to terms with renormalisation. He kept holding onto certain issues that he could never let go of, and this is one of them.
@SillyGoose No, isotropic just means that no direction is preferred, i.e. the quantities only depend on the relative angle between the two vectors and not e.g. on their angle to the z-axis or something like that
but the literal meaning of rotationally invariant to me is consider some object $O \in X$ and group action $a: SO(3) \times X \to X$, then $O$ is rotationally invariant iff $a(g, O) = O \quad \forall g \in SO(3)$
@SillyGoose The $f_{pp'}$ in your picture is rotationally invariant under $p\mapsto Rp,p'\mapsto Rp'$ for any rotation $R$, i.e. $f_{pp'} = f_{Rp,Rp'}$
"effective" is just the name you give a field theory that's not "fundamental", but it's not somehow that you use completely different techniques for effective theories than for non-effective ones or whatever
if you mean stuff like getting the Fermi theory of the weak interaction from the Standard Model, that's to me just part of learning QFT, not some seperate "effective field theory" topic :P
@SillyGoose It depends on what "kind" of renormalization you're doing. The Wilsonian interpretation of renormalization produces a bunch of "effective" theories by lowering an actual cutoff in the theory, sure.
@Slereah The only example i know is that topological phases can be described by an effective topological quantum field theory “in the IR”. I think this is an instance of RG producing an effective field theory
@ACuriousMind what is the role of interpretation in renormalization?
see e.g. this answer of mine for what I called there the "old" and "modern" view; these days I probably wouldn't insist on calling the Wilsonian viewpoint "modern"
Don't believe the hype. An "AI" that can do a narrow set of geometry problems should in no way make you think we're even remotely close to systems that can just "do math" or "do physics".
ACM, while I am also very skeptical, one should also not underestimate the (potential) power and ability of AI (in general, not related to the math/physics discussion here)
...also given the fact that things evolve very quickly
@TobiasFünke from where I'm standing everyone is constantly overestimating it :P
yes, there are real problems but they are often due to people just buying the hype and thinking it's actually "intelligent" and can do the jobs of humans or thinking it's close to that
@ACuriousMind u have studied some AI models. what is the feature of AI that makes u think it is fundamentally not like humans (in terms of things like : learning from experience and applying that knowledge)
i think the hypothesis that AI is like humans assumes that the human mind is a computation (which it is to a great extent). the only question is : how significant are the non computational aspects of the human mind
in america we had like 5 minutes of class consciousness when Luigi shot that CEO
middle class people just dont want to think they are anything like the poor. they want to think that they worked hard and they deserve their (relatively) cozy life while the poor people deserve the conditions they live in
when in reality they are much closer to the poor than our billionaire overlords
> If machines produce everything we need, the outcome will depend on how things are distributed. Everyone can enjoy a life of luxurious leisure if the machine-produced wealth is shared, or most people can end up miserably poor if the machine-owners successfully lobby against wealth redistribution. So far, the trend seems to be toward the second option, with technology driving ever-increasing inequality
this is Hawking's quote. i think AI automation will probably bring something even worse than capitalism
the problem isnt automation. the problem is billionaire greed
the way i preserve my peace of mind is to remember that those billionaires are soulless deeply unhappy people with a gaping void in their life that they need to fill through more power
this is overwhelmingly apparent in how many people try to bandwagon as being victims of genuinely horrible situations (when they're not) for their own short term benefit. this is particularly evident in recent race-related events and covid.
yes. and social media keeps people divided. we don't have too much time now. if everything gets automated, they may decide that they don't need the working class anymore
im guilty of this myself. i find it difficult to want to connect to people who hold the belief that my identity is invalid and that i dont deserve my (life-saving) healthcare
@User198 Sure, the matrix exponential for matrices on $\mathbb{C}^n$ maps the algebra of self-adjoint operators into the Lie group of unitary operators. You'll have to ask a more specific question to get more specific answers
> Where nearly all authors prior to the 1980s remained active more than a decade after their initial publication, this was the case for 50% or less by the 2010s. Consequently, the fraction of theoretical physicists who achieve full academic careers have diminished significantly in the last four decades.
> Additionally, we study correlations between author survivability and potential success factors, inferring that early author productivity and collaborative efforts have become increasingly strong determinants over time.
i know what they will do with it. but i am more confused about what this has to do with isotropic systems. if anything, this expression is what is wanted to precisely deal with anisotropic systems
i think perhaps there is just some strange verbiage being used because later in the book it is stated:
which is the expected result for an isotropic situation, that you only have the 00 spherical harmonic involved (the constant function)
completely separately, is it fair to argue that the concept of a "coordinate chart" is somehow more general than that of a "coordinate system", because you could define it without a choice of basis, by defining it as a chart from a subset of a manifold to a general vector space rather than $\mathbb{R}^n$? whereas a coordinate system is defined as the corresponding set of linear functions on $\mathbb{R}^n$, which means you must implicitly choose some basis e.g. the standard basis?
my guess as to what the book is saying is that the system is initially isotropic and that the changes in distribution $\delta n$ can be generically anything
The tangent spaces are vector spaces and have bases, sure. Every coordinate chart (to $\mathbb{R}^n$) induces a basis $\partial_\mu$ of the tangent spaces
Hmmm this kinda got started when I was reading the princeton companion to mathematics
It seems they do not want to privilege one basis. So, what they say is that a coordinate system is specified by ``a choice of basis for the linear functions on $\mathbb{R}^n$''.
If energy and matter are equivelent, is this statement logically correct? " The stuff in the universe is called Matter" or " the universe is made out of spacetime + matter"
Usually if you look up components of universe you find energy , matter, dark energy, dark matter, excluding the dark stuff, can we just equate energy and matter and call everything inside the universe "matter"
I'm a student of Physics self learning representation theory from Peter Woit's book and I am having a bit of trouble understanding a few concepts related to the dual space $V^{*}$ of a vector space $V$. I have seen dual spaces before but I am a bit rusty it seems.
I understand that it is the spac...
