They state the $\langle N \rangle$ in equation (4), but it is rather unexplicit (it is just the series representation of the polylog of order 3)
I think the complexity in obtaining an exact expression for $\langle N \rangle$ is in dealing with the degeneracy factor for each energy level of an anisotropic harmonic well, which is why I think there are no exact expressions.
Also, if you open up pathria & beale (~page 208 in my edition), you find a discussion of BEC in an anisotropic harmonic well and they too split off the ground state contribution to $\langle N \rangle$ and then approximate the rest using an approximate density of states.
also the above paper i guess is treating an isotropic well
@User198 and for a finite-dimensional single-particle state, also the CAR/CCR on the corresponding Fock spaces are equivalent (given some mathematical conditions, similar to SvN)
Arai's book on inequivalent representations could be of interest
@Allie Have you not seen a coherent fuller derivation? The "real" system has a dispersion relation that looks like $\omega\propto\left|\sin ka\right|$ that only exists for $k$ between $\pm\dfrac\pi{2a}$ and nowhere else. The Debye continuum approximation is merely equivalent to taking the small $k$ linearisation limit and extrapolating it outwards. This means that, yes, this linearisation limit has to terminate abruptly.
@Allie No, the difference simply doesnt matter. Real world crystals have finite sizes and not infinite sizes.
...I meant CAR (not CCR), because for a finite-dimensional single-particle space, the Fock space is also finite-dimensional for fermions. CCR of course needs infinite-dimensional Hilbert spaces
Does this make any sense to you? If $J\to0$ the field approaches $v$ as it is stated. Of course the first term is zero because $\delta\Gamma/\delta\phi=0$, but then every term of the series is zero if $J=0$ because $J=0\implies\overline{\phi}=v$ and so $\overline{\phi}-v=0$
I mean, okay, if $J=0$ the $N=1$ is zero because of $(12.45)$, but so is every other term of $(12.44)$ because of the $[\overline{\phi}(x_i)-v]$ factors
it is confusing to me, but I don't know too much about vertex functions etc. Which book is this from?
@SillyGoose thanks, but I cannot open it right now. My point was that the derivation of the bosonic partition function in the GCE is standard textbook knowledge (as you seem to know?), but not necessarily so in the CE. In any case, I would not expect even for the HO in the GCE an analytical expression for most quantities. But as far as I've understood you, you have one for the chemical potential and or the PF?
In the canonical ensemble consisting of $N$ independent harmonic oscillators in $d$ spatial dimensions one has to evaluate sums like
$$ \sum_n \ldots e^{-\beta E_n} = \sum_E g_E \ldots e^{-\beta E} $$
where $n$ labels the states, $E$ labels energies and $g_E(N,d)$ is the degeneracy of the energy ...
> The movie. Banned in China: Upon its original release, Kansas banned the film from being shown in the state, explaining that cross-dressing was "too disturbing for Kansans".
I find it a bit distracting and I seldom care about the topics/questions on there. it seems that for me it got replaced by a list of recent badges and tags as well (?) not sure if they were always there... i don't think I noticed them before
@ACuriousMind Hm. I meant to ask: "Does there exist than some "relation" for finite-dimensional Hilbert spaces, or do they don't have one at all."
@TobiasFünke Ah ok. Thank you.
1.) How do we know the dimension of the Hilbert space we are working in? Is it the number of dimensions of the $\psi(x)$?
For $N$ dimensional Hilbert space $\psi$ will be a vector of dimension $N$ and when $\psi(x)$ is a function (i.e. a infinite-dimensional vector) than the dimension of Hilbert space will be also infinite?
2.) When we have $[A,H]=0$ than that operator $A$ commutes with the Hamiltonian and the quantity is conserved and also we can measure one without distrurbing the measurment of the other. Cool.
I found out of examples: $[x,p_y]=0$ $[L^2,H]=0$ - for Hydrogen atom (spherically symetric)
Do you know of any more examples (maybe less known) ?
@User198 without any judging: your questions seem sooo random ^^
1. What do you mean by "know the dimensions"?
There are vector spaces. There is the notion of dimensions. You can check if a specific vector space has a specific dimension (at least all examples I know can be checked ;))
But no, not all function spaces are infinite-dimensional. The vector space of polynomials of degree $n$ is $n$-dimensional (vector space operations are the canonical ones).
@User198 $[H,P]=0$ for translationally invariant systems... and there are many more examples...
I think I got it. The effective action is a functional of $\phi_c$, $\Gamma[\phi_c]$. The classical field $\overline{\phi}$, i.e. the one that reduces to $v$ in the zero current limit, is only the solution to the EoM, $\frac{\delta\Gamma}{\delta \phi_c}\bigg\rvert_{\phi_0}=-J(x)$. Now we're working with fixed current (i.e. $\phi_0$ is a function of $J$), so sending it to zero we get the desired result. $\phi_c$ is just an arbitrary variable
@User198 there is a variant of this view which gives absolute answers. e.g. u attribute a hierarchy of structures to reality, and these structures are what humans discover as "approximate models".
so the structures are out there. the layers of hierarchy smoothly blend into other layers (e.g. relativistic mech blends into Newtonian mech in the $c\to \infty$ limit)
to give an analogy, i would maybe think of fractals here
the structures are out there in the fractals. deeper theories describe deeper structures
@User198 my view is perhaps that the structures that humans have come up with are maybe not "out there"
as in, these structures are how humans describe their subjective experience, instead of an (approximate) description of reality
but i am also open to the other view where physics is giving an approximate description of reality itself
i think classical mech gives strong vibes of (at least approximately) describing reality itself, which is why tons of physicists still cling on to that view
but some interpretations of quantum mech (like relative collapse interpretations) are explicitly about modeling ur own observations of the world instead of modeling the universe
if these interpretations are somewhat correct, then my view would be : there is the physical entity that is you, and there is the physical entity that is the rest of reality. these two interact to form ur subjective experience. and physics describes this interface between the two
@RyderRude Ikd, it seems like some sort of solipsism. I don't know if that direction of thinking is productive.
@ACuriousMind Ah ok. I don't know. I wanted to see some nontrivial cases of where the operators commute. Maybe see some cases that are not present in CM.
@ACuriousMind Some situations that are exclusive to QM, because $[p,H]= 0$ is both for QM and CM systems that are translationally invariant, so its not so interesting.
@User198 solipsism is an attitude.... one is acknowleding an outer reality here, so the attitude is not solipsism. it is saying that physics, at the fundamental level, doesnt describe the outer reality
but it is just a speculation. and physics at an effective level does describe an outer reality
@User198 "the defining characteristic of QM" is a vgaue term. but u can look at Von Neumann postulates for some sort of defining characteristic. it doesnt reference CCR
I don't even really know what that means - to even write it down as the commutator of operators on a Hilbert space, you have to already have built the entire mathematical formalism of QM with Hilbert spaces and operators and so on.
The idea that observables should be linear operators on a Hilbert space instead of functions on a classical phase space is to me much more what "defines" quantum mechanics
i'm a bit confused about the purpose of a community wiki post; are these supposed to be questions+answers with a likelihood of broad appeal and hence contributions/edits by a large number of people?
@ACuriousMind Ah ok. I see. For that system (if we don't care about position) than it is obvious that $[{\hat {x}}_{i},{\hat {p}}_{j}]=i\hbar \delta _{ij}$ doesn't hold. Because we are not even considering $x$.
@qwerty They're essentially historical baggage: Early SO introduced Community Wiki before they introduced Suggested Edits, i.e. this used to be the only way by which you could make your posts editable by other people. See this from 14 years ago :P
@ACuriousMind i searched on meta and saw something like that... I suggested a question have CW status removed as it seemed super specific to that one person, but it was declined as the OP had asked for resource recs (imo naively)
@qwerty well, we have a local policy (cf. physics.meta.stackexchange.com/a/7041/50583 and links) that recommendation questions are CW, but the reason for that is only because CW posts don't yield rep
so the breadth of appeal of the question is irrelevant in that context
the D-V postulates do not require time evolution, so string theory is admissible as a quantum theory according to D-V. what is admissible as a quantum theory depends on how general ur definition is
it seemed in this instance that the recommendation wasn't that relevant since it seemed the op was only asking due to misinterpreting a phrase, but I guess it's not a big deal either way
I know that integrals over Grassmann variables only share the name with real variable integrals but I wondered if there is some similar enough structure to talk about a "domain" of integration, just like for the real line the domain is $\mathbb{R}$
I know that it's not necessary, I just wondered if there a way to see these two ideas as an application of a broader, more general concept
Professor Garret Moddel (emeritus) from the University of Colorado Boulder claims he can harvest Zero-Point energy. colorado.edu/faculty/moddel/research/… Curiously, nobody else is verifying his claims, or denying them. At least, I found virtually nothing on a quick Google search. He just got mentioned on Space.SE. space.stackexchange.com/q/67950/38535
"harvesting" zero-point energy (or "using the quantum vacuum energy" or whatever) is an extremely common crackpot claim; there is really no need to debunk every instance of it separately
@ACuriousMind Sure. It's a bit disturbing seeing it on a university site, though. And coming from a guy who's been working on engineering devices using QM all through his career.
or one can speculate a weaker hypothesis : an arbitrary amount of energy extraction. this may be achieved, e.g., if a system has no vacuum. because it means there are states that are arbitrarily far below in terms of energy
Let's face it: a lot of high energy physics in the mid 20th century managed to get funding because people were scared of the other side discovering something that was as big a game-changer as the atom bomb.
One cute plan for cheap energy I read in a scifi story decades ago used a device that rotates particles through the 4th dimension. Rotation shouldn't take much energy. But you end up with antimatter, which you can annihilate with regular matter to liberate a lot of energy.
@RyderRude Practically speaking, the measurement problem isn't relevant. A quantum computer doesn't care which interpretation of QM you believe in. OTOH, David Deutsch was inspired to "invent" quantum computing because of his hard-core belief in MWI. And a lot of quantum computing people are fans of MWI.
@PM2Ring yes. Deutsch is going around saying that not believing in MWI based on not having seen multiverses is like not believing in dinosaurs based on not having seen dinosaurs
@SillyGoose hello
@PM2Ring i think it is the influence of Deutsche. people r taking QC as the evidence for multiple universes
the very fact that quantum computing works is supposed to be evidence, when in fact it is no more an evidence than any other experiment we've had since the beginning of QM
Is there anything fundamentally different in numerically solving the differential equations of QM, than to those in classical physics, because they describe quantum systems and not classical?
Or it doesn't matter, and PDE is just a PDE and it depends on just how it itself is complicated mathematically?
I am just curious what it means to have a BEC in real life and what people practically do to predict existence of BEC and diagnose the presence of BEC in their experimental system.
"If you take nothing else from this blog: quantum computers won't solve hard problems instantly by just trying all solutions in parallel" - Scott Aaronson
Also, I don't understand the claim "phase transitions only exist in TD limit". I know the usual (mathematical) argument for this claim. But, shouldn't we (as physicists) really interpret the conclusion of the argument as phase transitions obviously exist in finite systems. taking the thermodynamic limit in theoretical analyses is purely a convenient theoretical device to emphasize the nature of the phase transition?
Or does this confusion reflect an absence of a precise definition of "phase transition" in physics?
@SillyGoose What do you mean "phase transitions obviously exist in finite systems"? Be careful not to confuse the theoretical definition of a phase (transition) with experimental statements like "a finite amount of water can become ice". Your claim is only true if from the latter observation you could somehow deduce the theoretical definition of a phase transition must have happened
@SillyGoose that you're confused is clear, but I'm not sure what exactly the confusion is: The definition of phase transitions in terms of singularities is only possible in infinite systems, as you've said. What is the question?
If you don't take the TD, you are just doing QM, what is a phase transition in a finite QM system? The partition function will be a finite sum of exponentials, it is analytic, only if its an infinite sum can it be non-analytic
There is empirical evidence that this is an inappropriate model of the world. Namely, I can make my system (in the real world) occupy a superconducting phase. Are you arguing that a system in the laboratory is composed of an infinite number of particles?
@RyderRude There isn't a word about MWI in any of Kaku's textbooks as far as I know, even if his popular books seem to wax lyrical about it (if he really is arguing for it)
@SillyGoose No, see my statement about water/ice above: The claim "I can make my system occupy a superconducting phase" is clearly false for any finite system with the theoretical definitions of "phase" and "phase transition"
When you have so many identical particles in a finite volume that the total energy spectrum is for all intents and purposes continuous and always will be continuous compared to the accuracy of your measuring device, you are working with a quantum mechanical system with a continuous spectrum in a finite volume, but every finite volume QM system has a discrete spectrum
@ACuriousMind Yes but we actually treat objects like point particles and call them such in the context of analyzing the world using classical mechanics.
there is no limiting procedure that we are taking. we are taking a ball as definition to be a point particle (its center of mass or what not)
I mean if you want to be fancy you can look at the process of computing the error between a discrete system of many degrees of freedom versus the equivalent system and then notice the size of the error is very smol
@SillyGoose there is a deep contradiction buried in what I just said about continuous spectrums above, if your measuring device can't distinguish that the spectrum is not continuous (like your eyes can't tell that the stove is not continuous) then you have to treat the system like it has a continuous energy spectrum, but this contradicts basic QM - the spectrum of a finite system should be discrete
