for some reason the fact that total angular momentum squared depends on the maximal angular momentum state seems kind of strange... is there an intuitive way to understand this? for example for a spin 1/2 particle, the maximal spin the particle can have is 1/2$\hbar$ and the spin squared goes like (1/2)(1/2+1)$\hbar$
@SillyGoose this is how it works classically too. If total angular momentum squared is L^2, then the maximum projection of angular momentum along any axis can be L
@SillyGoose in QM, the spin operators dont commute. This is why its 1/2(1/2+1) instead of just (1/2) ^2
@SillyGoose I think you are looking at it backwards. In the derivation of this result, it is assumed that the total angular momentum has some eigenvalue. As a consequence of that, it is established that the eigenvalues of the individual spins are bounded like that
We are deriving a simultaneous basis of L^2 and L_z. We establish ladder operators that bump L_z 's eigenvalue by one, but let L^2's eigenvalue unchanged
Then we derive that L_z's eigenvalue cannot be laddered above a certain limit. This makes sense classically too because, assuming classical L^2 is fixed, the maximum value of L_z is bounded between $-\sqrt L$ to $\sqrt {L}$
Why do we say things like : "1. In the worldline path integral, the particle bounces backward and forward in time like and the backward paths correspond to anti-particles. 2. The backward time theta function term of the propagator corresponds to anti-particle?
Are these statements just popular folk tales? Or they have any physical interpretation /experimental evidence?
@RyderRude The technical meaning of statements like "particles moving backwards in time are antiparticles" is CPT symmetry
I would be highly sceptical of any statements that people try to derive from just the natural language formulation, but like "Feynman diagrams are pretty pictures that show you how particles move during an interaction" this is so good a story that people can't seem to resist re-telling it even if it's really not all that helpful or correct
"Green's function represents the probability that a particle travels from x to y, or an anti particle travels from y to x". This statement must also be a folk tale.
People should just accept that the Feynman Propagator and Feynman diagrams are just the end result of a long list of shortcut theorems like LSZ formula, Wick's theorem, Dyson Series, etc
Feynman propagator and diagrams are useful calculational tools. People should stop assigning folk tale stories to them. The underlying theory is still fundamentally describing a quantum field and wavefunctionals in general
@RyderRude No, that's pretty close - you have something like $\langle \phi(x)\phi^\dagger(y)\rangle + \langle \phi^\dagger(y)\phi(x)\rangle$ ($\theta$-functions omitted) and since $\phi^\dagger$ creates a particle from the vacuum and $\phi$ an antiparticle (or vice versa, whatever) this is the kind of amplitude "particle x->y + anti-particle y->x" - the only nitpick is that $\phi^\dagger(x)\lvert 0\rangle$ isn't literally a state localized at $x$ because localization in QFT sucks
What annoys me much more about "Green's functions" is that plenty of physicists use that as a synonym for "n-point function" when for $n>2$ those aren't Green's functions in the mathematical sense at all :P
Thats the nitpick that made me call it a folk tale
I think people are not letting go of the early history of QFT. Back then people were trying to make sense of negative energies to make this "backward time travel" idea work
But now we understand that fields are fundamental. But some of the folk stories from early QFT persist.
@ACuriousMind yeah, they should be called n-point functions. They just happen to equal the Green function for n=2
@RyderRude I wouldn't go overboard here: There's a difference between stories that are actually wrong and those that are just a bit imprecise - there are senses in which $\phi(x)\lvert 0\rangle$ (or $\phi(f)\lvert 0\rangle$ for narrow $f$ if we're being nitpicky, really) is a well-localized state, it's just that "localized" is a more complicated notion than in non-rel QM
@ACuriousMind Yeah, this story approximately works. I just find it an unnecessary story. No step in the derivation : LSZ formula to Dyson series to Wick's theorem uses this story. In the end of this derivation, we find that the Feynman propagator turns out to be useful for perturbative calculations. We then make up this story to interpret the propagator
I was really confused when they I first read that they had named this the "propagator". I thought the propagator would be the exponential of the field Hamiltonian. That would be consistent with non-relativistic qm
Also a thing to keep in mind in QFT when it comes to particles going backward in time to keep your sanity is that QFT does indeed have a privileged time direction
@ACuriousMind do you think that unitary evolution with a time parameter is not in the spirit of relativity? Wavefunctional evolution treats time as special
In scattering, we take the $t\rightarrow \infty$ limits which makes QFT look covariant
@RyderRude why would it not be "in the spirit"? Spatial translation operators are unitary, too, so why would treating time and space on equal footing change anything about the unitarity of time evolution?
the problem with time in relativistic QM isn't time evolution, it's that you really would want to have position/time operators but there are no perfect candidates for that
that journey starts with Pauli's theorem (time can't be like position because the Hamiltonian would have to have continuous and unbounded spectrum) and gets worse from there :P
Yeah, still i hear there are attempts at making time an operator
"wavefunctional is a state at a particular point of time but not at particular point of space". Is this an asymmetry between time and space? @ACuriousMind
however this doesn't mean spatial/temporal translation doesn't make sense: the momentum operators are well-defined, and they generate the respective translations after all
@RyderRude I generally don't think about wavefunctionals :P
@RyderRude the options are not "do scattering or use wavefunctionals"
plenty of non-scattering lattice QCD with no wavefunctionals in sight, plenty of non-eq QFT, too
the reason we want some $\psi(x,t)$ is that we want to describe the temporal sequence of states some observer will be able to measure
this isn'T an "asymmetry between time and space", it's just how physics works: We describe the world from the viewpoints of an observer in terms of a sequence of states $\psi(t)$, where $t$ is the subjective time of that observer
relativity may mean that you have to be more careful how to translate between that time for different observers etc., but nothing about this means our theory somehow is "assymmetric" in time
@RyderRude you can think about the way time works like this: When you choose a notion of time, i.e. a specific observer, this foliates the spacetime into surfaces of simultaneity. There is a space of states (classical or quantum, doesn't matter) associated to each of these spatial slices, and what we call "equations of motion" determines the maps between these spaces
when you switch to a different observer, you may get a different foliation, etc.
Yeah, that reasoning works. It's just that, in classical field theory, im able to treat time as just a dimension, not caring at all about its human interpretation
I think one problem is that space translations of phase space points are trivial. As in, they just look like a change of co ordinates
This treats time as "special" only in so far as time is something that observers undoubtedly experience and so we need to model it. You could also foliate the space into a bunch of worldlines as temporal slices and talk about spatially translating between those, but that is much less useful because observers experience motion in time, not in space (observers perceive themselves at rest pretty much by definition, after all)
this is a mystery only insofar as the notion of time is a mystery in general in physics, but nothing about this is really specific to QM or relativity - doing quantum and relativistic physics is just harder than classical non-rel. physics, but I don't see any fundamental difference in this treatment of time
QM is much more fundamentally about observations than classicla mechanics is, to the point that things like "measurement" And "observers" Are built into the definition of QM
I think this nature of QM causes clashes with GR
But there are formulations of QFT in curved spacetime
So maybe I should learn those before making these conclusions
I hope there is a fundamental incompatibility just so we get to witness an even crazier theory :P
But it seems like there isnt an incompatibility. It cant just be a co incidence that the metric tensor has just the right number of degrees of freedom to describe spin-2 particles
people tend to write a lot of profound-sounding things about quantum gravity but that's all it boils down to at a technical level, really: QFT+GR in the straightforward way doesn't work at all energy scales, so if you want a "fundamental" theory (i.e. one that works always) you need something else
When I had it a while back I was very annoyed that after the initial bout of fever I couldn't really sleep through it but couldn't really concentrate on anything for about a week, either. Fortunately that eventually went away, too, but it was the longest-lasting symptom
A conductor has a conductivity. The conductivity relates the electric field $E$ to a current $J$. $E$ and $J$ can be parallel. They can also be perpendicular. The latter case is called a Hall conductivity. What is a good word for the parallel case? Conventional conductivity? Ohmic conductivity?
