I have a question... so I'm reading about describing a solid's deformation in terms of stress and strain, and using a Hooke's law type equation to describe the relationship of the two
That is, the strain components are linear functions of the stress components and vice versa
I'm just a little confused on what the convention is here for what we're talking about when we say stress
I understand that strain is the deformation of the material and stress is the force applied to the material (in my case, an infinitesimal volume element of the material)
But, for example, in the case where we describe stress in terms of the strain, what stress exactly are we talking about? Are we talking about the stress that the material would feel when it's deformed/strained in a certain way? Like the restoring force in Hooke's law? Or are we talking about the stress that needs to be applied to cause that deformation (described by the strain)
Similarly, when we talk about writing the strain as a linear function of the stress, are we talking about how much a material will deform under a specific stress?
The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields. This density and flux of energy and momentum are the sources of the gravitational field in the Einstein field equations of general relativity, just as mass density is the source of such a field in Newtonian gravity.
== Definition ==
The stress–energy ten...
The bulk modulus (
K
{\displaystyle K}
or
B
{\displaystyle B}
or
k
{\displaystyle k}
) of a substance is a measure of the resistance of a substance to bulk compression. It is defined as the ratio of the infinitesimal pressure increase to the resulting relative decrease of the volume.
Other moduli describe the material's response (strain) to other kinds of stress: the shear modulus describes the response to shear stress, and Young's modulus describes the response to normal...
from a statistical mechanics perspective, isn't is good that there exists non-unitarily equivalent irreducible representations of the CCR?
i.e., isn't it good that Haag's theorem is a thing? Moreover, isn't it crucial that Haag's theorem is a thing as it provides an organizing principle for symmetry breaking states?
@SillyGoose Haag's theorem and symmetry breaking are orthogonal: Haag's theorem says that inequivalent dynamics have inequivalent representations, symmetry breaking however is a case of the same dynamics having inequivalent representations since symmetry breaking is a choice of ground state, not a change in the Hamiltonian (or however else you specified the dynamics).
i do not understand how choice of representation is a distinct choice from dynamics
since a choice of representation is necessary to do any computations, so in my mind i would choose a representation for my quantum system at the beginning and hold it fixed for the entirety of time (perhaps switching between isomorphic representations, for convenience).
@SillyGoose Haag's theorem at its core isn't really about representations of the CCR - it just says that if you have two Wightman fields $\phi_1(t,x),\phi_2(t,x)$ that are unitarily equivalent at some time $t_0$, then, if one of them is a free field (obeys the equation of motion of the free field), the other is a free field, too. This is a statement about the dynamics - the equations of motion.
but in the case of SSB, the different inequivalent representations that result from different choices of the vacuum are not like that - the field operators obey the same equations of motion in those inequivalent representations
@DIRAC1930 The Laplacian is not conformally invariant in dimensions $\neq 2$, and Fourier decompositions are in terms of eigenfunctions of the Laplacian.
@DebanjanBiswas I'm not sure what you mean by "intuitive", but you can derive conservation is momentum from translation invariance without deploying the whole machinery of Noether's theorem, see physics.stackexchange.com/a/439249/50583, if that's what you're asking
@SillyGoose I mean imagine you didn't put the $1/N$ in there. Then, if one considered a subset, things that have inner product 1 in the full object with $N$ elements will have inner produce $\neq 1$ in a subobject with fewer elements
It is a "normalization factor" that allows you to consider subobjects - like subgroups in the case of groups - so that the natural inner product in the subobject matches with the inner product in the full object
hm is it that specialized though? the other case i was thinking of is $SU(N)$ where i used the normalized inner product to compare values across different $N$, which precisely matches with what you said above.
well, I'm talking about groups specifically because what you posted is the inner product of functions on groups :P
you have similar looking formulae in different contexts but the reason why there's the factor in front will be the same - it's an average, and you're taking an average because you want to compare this to other things without having to always think about the normalization explicitly
but this factor can mess things up if you are just working in "one setting". e.g. when doing a quantum problem one normalizes the domain of the inner product itself, not the inner product.
so i guess it is just weird to normalize the inner product in some cases (e.g. fourier transform) and not in others (in quantum) and then choose to do one of them when doing a fourier transform in quantum
there is one choice that makes both directions unitary, but you can also choose to put it all on one direction (because this will eliminate factors of $2\pi$ from your most frequent formulae)
it's just a conventional choice depending on which formulae you want to have factors of $2\pi$ show up in
Still recovering, but progress is slow. But at least I'm able to sleep for a few hours at a time now. It's exhausting when you can't sleep more than 30 or 40 minutes at a stretch...
From Scott Aaronson's blog, some news about an improved Quantum Fourier Transform algorithm that turned out to be "hiding" in the older literature. scottaaronson.blog/?p=8593
When you read "I think the language of bundles is only going to confuse things more" at the beginning of an answer and you already know it's bolbteppa :P
It sounds to me as if you don't vibe with the usual way formal math is taught, which is that you get a bunch of definitions with often unclear motivations upfront and you only really can understand why those were the right things to talk about long afterwards
good pedagogy of course would present good motivations for all the definitions, but a) that might take more time than the course has, b) not everyone is a good pedagogue, c) opinions about what's a good motivation can vary wildly
@ACuriousMind Sure. When learning a new maths topic, I like to go through it (at least) 3 times. First time, just read through to get a general feel of what's coming & how it connects together, without trying to understand any details.
Second time, read a bit slower & try to actually learn stuff, but if something's too hard, don't waste time on it, but take note of it. Third time, we're getting serious. :) Go slowly through everything, with pen in hand, proving stuff, doing calculations & exercises, etc.
"The realistic description of the scatteringof an electron at a spin-1 2 hadron has to take into account the internal structure and anomalous magnetic moment of the hadron."
What is the anomalous magnetic moment of a hadron ?
@TobiasFünke Yes, although I feel a bit of a hypocrite to criticize such things when I don't really have the knowledge of what I claim to be better. Anyways, since you asked, I would say that pretty much everything is flawed, but of course a mathematically rigorous description of everything would make even a physics BSc 10 years long, probably. ACM and Qmechanic are good examples of the math level I enjoy in physics, nonetheless a level far from my own. This is what I like.
What do I deem as right? I'm perfectly fine with keeping most pedagogy the way it is when it comes to contents and modalities, but I would prefer to have people state when we are moving on mathematically slippery grounds and we're proceeding in a formal fashion, so that there is a clear distinction of the mathematical (which are more of a technical issue) and the truly physical problems
Example: y biggest gripe that is that e.g. almost no introductory QFT book dares to mention the formality of the perturbation expansion, or that most QM courses deal with infinite dimensional spaces working formally as if they were finite-dimensional, which is fine operatively, but so many books and lectures just give you a formal proof without telling you it is formal
Now, I think this can lead to very dangerous situations: it's not just about being nitpicky. If a student is misguided in learning these things without noticing they are just formal (which is far from obvious), they may end up doing a mistake involving e.g. the differences between functional analysis and linear algebra
What does this cause? In my case, that after I felt "betrayed" once, I can't trust anyone anymore in lectures or books. Now, you might be tempted to say that it's a good feature for a science student, which indeed it is. Yet, there are only so many hours in a day, and if you really are to fall in every rabbit hole, you're cooked
Of course SE is a good help and so is the ability to search the right sources and references. The problem is not when you know there is a problem. It's when you haven't realized it yet.
If you are a student who's just realized that $1=\int dx\rvert x\rangle\langle x\lvert$ is not so solid as you thought, you can find the right references and maybe try to understand the problem. What if you don't know it's a delicate issue, though?
Yes, I agree. But the good thing here is that so many (good) questions are asked, which I'd probably never had asked myself. So these questions/answers etc. point to problems, even if you do not know about them beforehand. Of course, this requires checking threats here more actively
I'm just saying that our lives will always have a "$1=\int dx\rvert x\rangle\langle x\lvert$" around the corner, and you won't always have an SE answer highlighting it :P
sure... but now you know (or at least should know :p) that as long as you did something not rigorously, there is a chance it is on shaky grounds for one reason or the other...roughly speaking
I can live with that and I can live without ever coming to know the rigorous version. I can't live with people writing papers/books without bothering to write a footnote saying that
@bolbteppa Theres a book called Space-Time Structure by Schrodinger which in my opinion is a masterpiece (one has to ignore the last chapter though because he goes into his own theory)
@SignorFeynman To continue the boat analogy, QFT is like a boat with holes in it, and nobody knows (yet) how to patch those holes properly. You felt betrayed because before QFT you were taught science that is solid and reliable, so it came as a shock to learn that QFT has these holes in it. But the boat (kinda) works, and (currently) it's the only boat we've got...
I'm not shocked to know that QFT has holes that no one understands, I'm shocked that there are some holes that some people understand and almost no one bothers to at least bring up
Uhm, I'm not very strong with circuits. Isn't this the ground symbol?
@DIRAC1930 if its the Cartan stuff then I think the conclusion was the pure ones are just not general enough and its basically an explicit way of talking about clifford algebras at the end of the day
how exactly not having a structure is tied with gauge invariance, which in turn implies $j=\overline{\Psi}\gamma\Psi$ ? When I was introduced with $j=\overline{\Psi}\gamma\Psi$, we simply derived this from the Dirac equation
From the Dirac equation you derive that $j$ is the conserved current for free fermions, associated to the $\mathrm{U}(1)$ global symmetry of the Dirac Lagrangian
When you construct QED as a $U(1)$ gauge theory, it turns out that the structure necessary to have an invariant lagrangian is an interaction term of the form $jA$