there's two ways i know how to get the trajectory of an object in motion. the first is to take a video and see where it is, frame by frame
in that case we have the notion of trajectory as position at successive times
which is very convenient
but, suppose i didn't have a camera
like, all i have is a stopwatch
while it's much more tedious, i could still do the following: measure the time to go from x=0 to 1, from x=0 to 2, from x=0 to 3, etc
the difference is that i'm not ever in the position of measuring the trajectory of a single experiment. the best i can do is run the same experiment over and over and use that to build up a trajectory
and that's closer to how velocity works for pilot wave. you give up on being able to track one motion successively over time, in favor of running the experiment repeatedly over different times and buliding up a trajectory from that
that doesn't render the pilot wave story trivial or obvious, but i don't think the trajectory story is absurd on its face.
Well this point we're talking about is what Bohm brought up as his way to suggest the beginnings of an alternative interpretation, so that's something :p
i'm specifically speaking about the pilot wave part, mind. i've seen plenty of debate about whether Bell's presentation of the older "no hidden variables" proofs is accurate
(my own reading is that what von Neumann shows is that, if you want the Hilbert space structure of observables to reign supreme, then there's really no place for hidden variables. pilot wave is happy to use hermitian operators to provide book-keeping for wavefunction statistics, but it cannot and does not grant them any higher status than that.)
which is funny given that Bell's own attitude toward the problem of reconciling SR with QM is to flip Einstein the bird and assume an absolute reference frame
tho you could view the Bell inequality as asserting that "Einstein was never going to like QM"
When this is what is meant by hidden variables, it's not a bad idea, I just think it simply means that everything which comes after it needs to be questioned from first principles again
(I don't know what he means about Einstein and von Neumann and co-existence)
I have a question about my Physics Stack Exchange post: Is sufficient to have some anti-gravity generator to make a Alcubierre drive?
Where should ask this question?
"Local conservation also implies global conservation; that the total amount of the conserved quantity in the Universe remains constant" is given in Wikipedia page of conservation laws.
But Energy is something which is locally conserved but not globally right?
@ManasDogra Physicists argue about this. Most physicists I know would say energy is not conserved in an expanding universe, however there is a alternative view that energy is conserved provided you calculate the energy in the right way. Philip Gibbs is the name I associate with this view. I have a link somewhere to his articles on the subject.
Note that although this is on Vixra Philip Gibbs is a mainstream and well respected physicist. He helped set up Vixra and published various articles on it in a desperate (and largely failed) attempt to make it respectable.
The large scale limit of any quantum gravity theory is going to be GR, and in GR energy is not conserved (or is conserved depending on how you look at it), so quantum gravity is not going to change the global properties.
I guess I should qualify this and say that as far as we know the large scale limit of QG will be GR. It could turn out to be MOND/TeVeS or some other theory that we don't know yet.
@ManasDogra The fundamental problem with that is that you need to write down a definition of "energy" and "conservation" that plays nice with the fact that "time" is a malleable notion in GR
the usual definition of energy in non-GR is as the Noether charge associated to time translation
but in GR you need to figure how to translate a) Noether's theorem and b) "time translation" to its settings, and that's where the controversy starts
at least pilot-wave gives you an alternative mathematical framework, which is always nice. But I can never seem to figure out whether this mathematical framework is actually equivalent to the standard one or not... or if it is known whether it is
the "simple" parallel to Noether's theorem in GR is via Killing vectors - you can associate uncontroversial "conserved quantities" to each Killing field, but there's no guarantee that your spacetime has a time-like Killing field that you could associate with energy (in fact FLRW doesn't and that's why many people say there's no conserved energy in an expanding universe). See e.g. physics.stackexchange.com/q/218121/50583 for more technical discussion of Killing fields and conservation
"Conversely, a pair of classic results, due to Barrett, show that there is a certain sense in which, given appropriate “holonomy data” on a manifold M, there always exists a principal bundle over M and a principal connection on that bundle such that the holonomy data arises as the holonomies of that connection, and that this bundle is, in a sense to be explained, unique."
@glS in the sense that many-worlds seems to insist that 'other universes' talk is supposed to be taken as 'real' rather than as merely figurative
@glS it is equivalent. you start with a time-dependent wavefunction, write down the probability current density $\vec{j}$ and probability density $\rho$, and define a velocity field $\vec{v}=\vec{j}\rho$. All you've added are symbols; everything that's in those definitions is just the standard formalism.
What you can then show: Suppose you start with some closed surface $S_0$ at time 0, and ask for the probability to find the particle within that surface at that time. That'll be some $P(S_0)$
bah, bad typo: $\vec{v}=\vec{j}/\rho$
now allow the points on that surface to flow along the velocity field $\vec{v}=\vec{j}/\rho$ for a time $t$. that'll bring the original surface $S_0$ to some new surface $S_t$
and now you can ask for the probability to find the particle within that new surface
what you can show is that the choice $\vec{v}=\vec{j}/\rho$ guarantees $P(S_0)=P(S_t)$
so this is a flow which preserves probabilities
the argument then is that this is what you'd expect if the particle really was moving along that velocity field, with some probability distribution on its initial position
and thus interpret that velocity field as determining 'real' trajectories for the particle
thing is: what you ultimately predict operationally is still based on wavefunction statistics
that part of the story hasn't changed
it's just that you've chosen to interpret some part of the theory as "real" which the orthodox view would just treat as "formal"
so to me the problem isn't really whether you can do so, but whether you should
I don't get it. For two particles, you say 6 dimensions because you assume 3 spatial dimensions per particle. But that just sounds like having a standard pair of velocities characterising the motion of each particle
"No one can understand this theory until he is willing to think of ψ as a real objective field rather than just a ‘probability amplitude’. Even though it propagates not in 3-space but in 3N-space."
I don't understand what's nonclassical about this. Don't you get the same description if you model, say, two classical particles using a configuration space?
so... in other words, the description above is just a standard classical way to describe the motion of a bunch of particles. What changes is their actual dynamics, which for the pilot wave follows completely different rules
there's not really a good way to make a phase space
the problem is that, unlike in classical mechanics, the velocity (and thus the momentum) can't be taken as an independent variable
in classical mechanics, it makes sense to say "suppose the particle starts at position x with momentum p"
in pilot wave shenanigans, though: if i tell you the wavefunction, and that the particle is found at position $x$ at time $t$, then the wavefunction determines the instantaneous velocity as $\vec{v}=\vec{j}(x,t)/\rho(x,t)$
which means i have absolutely no freedom to specify the momentum
you could still track the momentum of the particle, but it's subservient to the particle's configuration space trajectory
I mean, talking about $\vec j$ and $\vec v$ sounds essentially the same here, modulo standard shenanigans due to using a description with densities rather than exact positions. So the nontrivial aspect is the fact that the $\vec j$ (equivalently, $\vec v$) is determined by the position, you are saying?
so... maybe the culprit here is the fact that the probability current is defined from spatial first derivatives of the wavefunction/position? Whereas in the classical case you take the time-derivative, which thus adds an additional dof
i'll note, though, that the story I've been telling is Bell's approach to pilot wave theory
which very much emphasizes the role of the wavefunction as providing guidance for the particles. the usual statement is that Bell's approach is 'first-order' in time
whereas Bohm's story "looks" a lot closer to classical mechanics and seems like it's second-order...but i'd say that's deceptive
My own gloss on this: what QM faces you with is the failure of classical phase space
orthodox QM interprets that as having to sacrifice "phase space" in favor of momentum and position as operators
pilot wave whereas sacrifices phase space in order to preserve configuration space
but local in configuration space =/= local in real space
and it's kinda suspicious that one should take a story about trajectories in 3N-dimensional configuration space as real
there's nothing stopping you from doing so, and there are some nice mathematical features to doing so
anyways. there are definitely things to criticize about pilot wave, to be clear. but i don't count internal consistency nor compatibility with QM amongst them
i guess. i'm not shocked by the notion that "finding all cycles is hard", to be clear
it's that there'd be such a jump from "find one cycle" to "find two cycles"
there's also something suspicious going on: mathematica isn't having trouble finding (short) paths between two different vertices
it does that fast
but in that case it shouldn't be so slow to find other cycles: start at the origin, move to an adjacent vertex, and look for paths between the origin and said vertex
so the worst-case complexity for fixed length $k$ is $n^k$ (it takes $n^k$ to even list all cycles in that worst case), if you restrict to $k\leq 4$ you have worst-case complexity $n^4$
an obvious route would be to write down the adjacency matrix and multiply
though the fact that said matrix is 5040-by-5040 is...less appealing
hmm, it works better than i expected. probably because the complete graph on 5040 vertices (lol) has 13million edges, so 45k edges is very sparse in those terms
on that basis, seems there's nine 2-cycles (which i knew), zero 3-cycles (which i hadn't thought about but makes sense from the problem) and 177 4-cycles
@Semiclassical so wait, are you saying that the mysterious "potential" guiding particles in pilot-wave theory is just the wavefunction following Schrodinger? So solve for the wavefunction, then interpret squared modulus as density of particles, and current density as their velocities? I thought it was more complicated than this
so you can't reduce it to the case of point particles? I was understanding it like in classical EM etc: you can discuss of point particles and their velocities, or of densities $\rho$ and current $J$
yeah, the problem is that this is still supposed to work for a single particle
it's not an ensemble interpretation
to the extent that there's an ensemble, it's of hypothetical initial positions
not an ensemble of many particles
also, the "no" part of my statement above: in Bohm's formulation you write down the Hamilton-Jacobi equation with potential $V(x,t)-\frac{\hbar^2}{2m}\frac{\nabla^2 \sqrt{\rho}}{\sqrt{\rho}$
@Semiclassical I don't understand how you actually solve the dynamics. You solve for the wavefunction, that's fine. Then you set as "initial condition" the position of the particle, and determine the direction in which it goes via the probability current density, is this correct?
so the wavefunction determines in which direction the particles move. Which in fairness is compatible with calling it a "pilot wave theory". And I suppose wavefunction and particles would have to evolve simultaneously
that the interaction is basically one way: wavefunctions interact with each other, and wavefunctions tell particles where to go, but particles don't act on wavefunctions as such
@glS same as you'd treat discrete dof's for the Schrodinger equation: replace the scalar wavefunction with a vector-valued wavefunction
the definition of the velocity gets a bit more complicated now, lemme see
@Semiclassical yes, but in this particle picture measuring a photon in polarization H means the particle is in that state, right? How do you handle it changing its polarization state? You cannot talk of velocities there. Can particles jump?
@Semiclassical actually, the above doesn't even make that much sense. The particle would have a definite state in all spin directions, this being a hidden variable theory and all, right? How do you even define the configuration space in that case then?
the way spin measurement works in pilot wave theory is that the interaction makes the wavepacket separate into two pieces, one deflected up and one deflected down
again, no
in pilot wave theory, to say that a particle is "spin-up" is to say that the particle was measured to be in the wave packet that was deflected up
there's no assignment of a specific spin dof. (at least not in the Bell way of doing it, Bohm's way is different and i don't know it that well)
spin is very much contextual in pilot wave theory as i know it
there's no hidden variable that says "this particle is spin-up in the Z-direction" prior to measurement
yes. but that property is the position itself, at least if we're doing a one-particle measurement
suppose you start out with a wavepacket and you send it into a +z-oriented Stern-Gerlach device
the particle will be somewhere in that wavepacket, with a fifty-fifty chance of being in the top half or bottom half
the wavepacket passes through the device and splits into two pieces, one deflected up and the other deflected down
if the particle started out in the upper half of the wavepacket, it'll keep moving with that portion. if it's in the lower portion, it'll keep moving with that portion
so there is a property here, but it's only "where did the particle start out in the wave packet"
this is characteristic of pilot wave theory: position is noncontextual, but everything else is contextual
(which, if that seems exceptionally asymmetric, welcome to the party)
but think about the particle before entering the measuring device. There must be something that will interact with the measurement setting, in such a way that the particle will always go up if I'm measuring, say, in the Z direction, but will go 50/50 if I'm measuring in X
would this be a property of the particle, or the wavefunction?
I really struggle to see this. If the wavefunction has a discrete dof, doesn't that mean the particle itself can be in an "up" or "down" state? So there is a point in which it must "jump" from one to the other
there's really no 'jump' point, there's just a smooth crossover
but you won't have a useful measurement during said crossover, because the wavepackets are still overlapping
inasmuch as there's a 'jump', it's because (for practical reasons) you prepare the system to have overlapping wavepackets and only measure once they're no longer overlapped
@Semiclassical but if the wavefunction is like that, isn't the particle also in one the two states? I don't get it. You are saying that the discrete dof only exists for the wavefunction, not the particle's state?
@Slereah look at it this way: holonomies only tell you what happens when you parallel transport things around loops. But that doesn't tell you whether something happens when you move things along paths that aren't loops - there is clearly more information in the connection that there is the holonomies alone
lattice gauge theories actually think about connections that way - they don't work with a gauge field $A$ assigned values at the lattice points, they work with the "comparator" or "connector" that assigns group elements to the edges of the lattice
@Semiclassical ok.. so I guess the question becomes, how does the discrete dof of the wavefunction interact with the particle then? Because it must affect its velocity. Reading the Wikipedia page, it says the the probability current is given as $(\psi,D_k\psi)$ with $D_k=\nabla_k -iA$ and $A$ vector potential. But at this point I lost any sort of intuition
so treating the interaction with the S-G device as being very narrow in the longitudinal coordinate
with the usual Schrodinger equation $i\hbar\partial_t \Psi(x,t)=H\Psi(x,t)$, with the proviso that it's now a spinor wavefunction
so if you follow a $\binom{1}{0} \psi(x,t)$ wavepacket through there, then this wavepacket deflects up (b/c the interaction is $-\mu z\delta(y)$, so potential energy decreases in the z-direction)
whereas a $\binom{0}{1}\psi(x,t)$ wavepacket will deflect down (b/c the interaction is now $+\mu z\delta(y)$)
all of which conforms with how you understand a Stern-Gerlach device classically: you have some intense magnetic field gradient over a small region
and that serves to move magnetic moments in different directions depending on their orientation
if you now take one one of those wavepackets and send it through a different device, the interaction will now be different. but the way it changes is just "what direction does the particle come in from" and "what direction is the magnetic field gradient"
and that's all you need to get the wavepackets to move off in different directions
a slightly annoying bit here is how to define probability current density. gotta deal with the imaginary part of $\Psi^\dagger \cdot \nabla \Psi$
which isn't exactly nice but it's what you get from the Schrodinger equation
my own gloss would be that, when you work through it, you get a perfectly coherent account of one-particle Stern-Gerlach measurements
nice. and his section on spin directly cites Norsen
"Unlike standard quantum mechanics, spin is not considered as an intrinsic property of a particle, but a property of the wave function and will affect the trajectory of a particle via the guiding equation". (guidance equation is "velocity = current density / density")
@bolbteppa one thing i definitely would point out in that paper is how it deals with relativity: namely, you have to impose some definition of simultaneity for spacelike separated events
Interestingly the discussion yesterday on SR pointed out how this instantaneous property of non-relativistic potentials basically is one of the main differences between Galileo and Einstein, it's basically a very bad sign they have to do this tbh
@BalarkaSen no, the electric field is part of the curvature tensor - so two connections with different electric field cannot be gauge equivalent as their curvatures differ
I was trying to setup $A=0$ as a setting with trivial holonomies and some setting with purely electric fields as something with also trivial holonomies
but that doesn't work because of relativity - "all holonomies" means that I also have to consider loops that don't lie in the spatial slices associated with the frame where the field is purely electric, so I still get some non-trivial holonomies there
That's pretty shocking thinking about it, another thing that I can see going really wrong is the 'velocity' associated to the Dirac equation and Zitterbewegung
I'm not entirely sure how that meshes with my intuition that there should be some information in the transport along paths that's not captured by loops
@Slereah No! Gauge transformations don't change gauge-invariant quantities, and the electric and magnetic fields - the curvature tensor - are gauge-invariant!
In normal QM one is free to change the interpretation of the wave function to deal with relativity, if Bohm is supposed to reproduce normal qm it will have to as well, the problem is they boxed themselves in way too deep using non-relativistic thinking and calling it a 'real' wave
the problem is that there's no such thing as "electrostatic" - when we do EM as a gauge theory we're doing it relativistically, and then it's pointless to think about a given configuration as "electrostatic" - there's other frames where it's not, and there you'll have magnetic fluxes, hence non-trivial holonomies through loops that are purely spatial in that frame
I'm not sure how rigid that really is, I'd say it's easy to weaken these assumptions if the basic logic of the theory actually makes sense internally (the bigger question)
Ah OK, no I just don't know enough electrodynamics so this is a good opportunity that's all. I see, so the background $U(1)$-bundle is some symmetry of the underlying physics ("rotation"), the $E$ and $B$ are part of the curvature form in some frame
(on the math side of things i think the connection is the same information as an $A_\infty$-algebra homomorphism $\Omega M \to GL(n)$... the higher info is hidden in "$A_\infty$")
I have a question about my Physics Stack Exchange post: Is sufficient to have some anti-gravity generator to make a Alcubierre drive?
Where should ask this question?
