The h Bar - 2018-07-09 (page 1 of 4)

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Cows

00:10

~ "Zernike series" lol sounds hardcore

@Mikhail why Zernike series? why are you decomposing it? What information will that enable you to extract? What do you mean by more compact? I usually think of [], (] etc when thinking of such things

Why would the Fourier bases be less compact?

How do you get the $\rho $ for the specific problem you are working on

Actually generally . . . . just curious,. . . what physical problem are you working on?

Mikhail

00:29

I'm trying to invert the Hopkin's TCC coefficient model for image formation. In raw form, the problem is O(n^6) but I think I can solve it.

I know, that there is a lot of sparsity in a particular integral, so I suspect I can evaluate it only for certain basis elements.

heather

hello

Mikhail

hello

Cows

hello

for some reason I got logged out , had to log back in

hmm

at any rate looking at the TCC stuff now

I am mostly just looking at the math, but don't know anything about the physical problem.

It looks cool though

Mikhail

Its simple, correct, and numerically intractable. So, everybody does the numerical tractable thing, which is wrong. Including my own papers ;-)

Cows

I will think about it a bit, while exploring some other stuff , if I have any spark I'll ping you

hehe, I am sure it is a non-trivial problem

Mikhail

00:40

Basically, I need to decompose the actual tcc coefficients into a sparse basis. In principle one could choose anything, but to invert the system you'd need a good guess to the instruments coefficients. Key problem is that for a 3D image you can't store the coefficients (too much ram), compared to the 2D case.

Cows

ah

Mikhail

Its actually a really interesting problem because decades of recent research have gone into modeling this stuff with stochastic processes causing a lot of confusion regarding temporal and spatial coherence :-)

Mikhail

01:05

That feel when integral over $\left|\frac{1}{x}\right$

4 hours later…

Cows

04:44

Not sure what to study tonight. I was going to just code, but my schedule has freed up . . . . . May be I should review how to massage functionals

Cows

04:57

eh let me read a not so random paper from the arxiv

Secret

05:15

Last night dream. It is the year 2032 and there are 5 experiments being done in the intersection between quantum mechanics and time travel. In a book store with white shiny floor and bluish green coloring, I head to a bookshelf where there is a thick laminated sketch book like album which is actually a book about everything we knew about quantum mechanics and quantum computing. A couple of pages sits in

TBC
Asymmetric quantum mechanics, quantum contradiction causality, quantum time travel mechanics https://physics.stackexchange.com/questions/415666/how-to-define-dynamical-dimensions

2

Secret

05:26

isolation around the bookshelves. One of these talked about asymmetric quantum mechanics, and the electron density map of a C shaped molecule was printed where near its tip there is a depletion of electron density illustrating the hydrophobic effect of proteins showing that the two arms effectively has steric repulsion

Anorher page reads Quantum enta(contradiction) dynamics, showing a genuine realisation of the grandfather paradox in a tabletop setup. In the loop structure showed, two mutually exclusive events coexists in the same spatial location, thus giving a physical realisation of a genuine P and not P contradiction

Finally, a third page consists of many small annotated diagrams in panels which showed recent experimental and theoretical advancements in the quantum phenomenon in time travel experiments

The experiment setup showed include optical benches with linearly positioned mirrors and beam lines, some bench experiments involving two cubes and many more

Some moment later, my family is telling me time to leave, I then quickly grab the book and rush to them that I want to buy it, but I don't know how worthwhile it is (The full book costs $10000). During the indecision, I am being woke up by mum, thus I then woke up in frustration as I never get to read that amazing book in more detail

The dream is most likely inspired from that dynamical dimension PSE which makes my eyes open wide, and possibly the incomplete analysis of the direct counterfactual communication paper in 2017

2 hours later…

Sir Cumference

07:17

Yo this is a thing?

Product integral

A "product integral" is any product-based counterpart of the usual sum-based integral of classical calculus. The first product integral (Type I below) was developed by the mathematician Vito Volterra in 1887 to solve systems of linear differential equations. Other examples of product integrals are the geometric integral (Type II below), the bigeometric integral (Type III below), and some other integrals of non-Newtonian calculus. Product integrals have found use in areas from epidemiology (the Kaplan–Meier estimator) to stochastic population dynamics using multiplication integrals (multigrals)...

I'd heard of infinite products, but huh

1 hour later…

Slereah

08:30

Well it's there

So I guess it's a thing

1 hour later…

Anonymous

09:43

@JohnRennie You around?

John Rennie

@Blue for a few minutes ...

What's up?

Anonymous

I'm trying to update pip on Ubuntu using pip install --upgrade pip

Anonymous

But I get an error

Anonymous

`Collecting pip
Using cached https://files.pythonhosted.org/packages/0f/74/ecd13431bcc456ed390b44c8a6e917c1820365cbebcb6a8974d1cd045ab4/pip-10.0.1-py2.py3-none-any.whl
Installing collected packages: pip
Found existing installation: pip 9.0.1
Uninstalling pip-9.0.1:
Successfully uninstalled pip-9.0.1
Rolling back uninstall of pip
`

Rumplestillskin

10:06

Does anyone here have much experience with non-dimensionalising an equation?

1 hour later…

Avnish Kabaj

11:12

@Rumplestillskin what Happens if I guess your name :P

Rumplestillskin

11:32

@AvnishKabaj Hm?

Avnish Kabaj

"I will give you three days, time," said he, "if by that time you find out my name, then shall you keep your child."

Now, mistress queen, what is my name?"

At first she said, "Is your name Conrad?"

"No."

"Is your name Harry?"

"No."

"Perhaps your name is Rumpelstiltskin?"

2 hours later…

Secret

13:28

-1

Q: What happens to an object when it passes through a portal?

As in the game Portal, if we create two portals on two different surfaces and then join the two surfaces with the portals facing each other, as soon as an object passes through one of them, what would happen to that object? Where would it go, and what would be its state? And what would happen af...

science-based physics space-constructs games

You uh, really cannot squish the two mouths together because the object will be literally bumping against its own back

Semiclassical

14:02

@Secret ah, the portal censorship hypothesis: nature forbids a portal making out with itself

Secret

lol

Secret

14:56

marvelcinematicuniverse.wikia.com/wiki/Quantum_Realm

I really don't understand. Why do filmakers have the tendency to depict the incomprehensible as fractals?

Semiclassical

b/c fractals look cool yo

and tbh I think fractals are about as good an example as one could hope for: a simple rule leading to complex phenomena

Secret

huh I guess that's true

Semiclassical

on the other hand, associating fractals with QM seems pretty spurious to me

i mean, fractals and chaotic dynamics, sure. but chaotic dynamics are something I associate with classical systems, not quantum ones

though apparently 'quantum fractals' are a real subject. (how appropriate that definition is, I could not tell you)

Secret

15:14

arxiv.org/abs/nlin/0107039

Well, there's also quantum chaos, where chaotic and nonlinear dynamics get applied to quantum scale

Semiclassical

yeah

Secret

But frankly I don't understood much of it, I am still trying to make sense of counterfactual communication

Semiclassical

ikr

I've been doing some bell inequality stuff myself lately

mostly to understand the difference between classical correlations, quantum correlations, and non-signalling correlations. (each being a subset of the subsequent class)

I sorta get it.

Secret

Yeah, we have PR boxes $\supset$ non signalling $\supset$ Quantum discord $\supset$ Entanglement $\supset$ Classical correlations

Semiclassical

right

Secret

15:20

I think I sorta understood PR boxes by plotting the whole joint probability distribution, and noticed how both outcomes and inputs are correlated, which is an extra level of correlation absent in discord and entanglement

and that correlation is "scrambled" in such a way that no looking at individual pairs of input or output can distinguish between the two sets

similar to how in bell states (which are maximally entangled), the density matrix is 1/n at the diagonals thus the outcomes of which (anti) correlated states you get has the probability of 1/n so it becomes completely agnostic on which subsystem leads to what observable

Emilio Pisanty

@Slereah quick question - what is an orbifold?

Semiclassical

@Secret sounds right

John Rennie

@EmilioPisanty I'm not sure that has a quick answer :-)

Emilio Pisanty

I find the Wikipedia page ... less than illuminating

In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold (for "orbit-manifold") is a generalization of a manifold. It is a topological space (called the underlying space) with an orbifold structure (see below). The underlying space locally looks like the quotient space of a Euclidean space under the linear action of a finite group. Definitions of orbifold have been given several times: by Satake in the context of automorphic forms in the 1950s under the name V-manifold; by Thurston in the context of the geometry of 3-manifolds in the 1970s when he coined the...

I mean, I guess I can navigate the definitions given enough time, but... what is that thing?

in the same spirit that a branched manifold is an object that locally looks like a manifold except that the charts can diverge from each other and go into different branches, say

@JohnRennie I know, but it's still worth a try

CC @ACuriousMind @Danu @BalarkaSen

Secret

Very brief reading suggests it is a Hausdorff space whose open covers form a group structure

Balarka Sen

15:28

Aha! Some topology is stirring

Emilio Pisanty

@Secret that.... tells me next to nothing, I'm afraid =(

Balarka Sen

@EmilioPisanty An orbifold is a topological manifold with some points having "more symmetries" than others, more or less.

Emilio Pisanty

@BalarkaSen ah, that's a great start

Balarka Sen

For example, consider the 2-disk $D^2$. There is an action of $\Bbb Z/k\Bbb Z$ on $D^2$ as follows

Write it as a complex disk $D^2 \subset \Bbb C$, i.e., as $\{z \in \Bbb C : |z| < 1\}$.

Emilio Pisanty

hang on, $\Bbb Z/k\Bbb Z$ is just the cyclic group with $k$ elements?

Balarka Sen

15:32

Yup

Emilio Pisanty

k

Balarka Sen

Denote $\Bbb Z/k\Bbb Z = \{\overline{0}, \overline{1}, \cdots, \overline{k-1}\}$. The action $\Bbb Z/k\Bbb Z \curvearrowright D^2$ of the cyclic group is defined by $\overline{x} \cdot z := e^{2\pi i x/k} z$.

So basically, $\overline{1}$ acts by rotating the disk by angle $2\pi/k$.

And the rest of the elements acts by multiples of that angle.

The action is "rotation"

Emilio Pisanty

sure

Balarka Sen

Consider the quotient $D^2/\Bbb Z_k$ (I will use $\Bbb Z_k = \Bbb Z/k\Bbb Z$ 'cuz I am a bum).

What is that object?

Emilio Pisanty

a thin wedge of the disk

with special handling of the origin?

Balarka Sen

15:37

Aha, great! That's very close

You have to identify the two sides of the wedge, though, right?

Emilio Pisanty

yes

Balarka Sen

So it's a cone!

Emilio Pisanty

sure

not that it can tell where it's been glued, though, right?

Balarka Sen

One moment, I'll be back in a second.

Emilio Pisanty

sure

Balarka Sen

15:42

Ok back. Yes, you can no longer tell where it's been glued

It's simply a cone with an angle at the cone point being $2\pi/k$

@EmilioPisanty Note that TOPOLOGICALLY, it's just a disk. You can flatten the cone continuously to a disk

Emilio Pisanty

@BalarkaSen sure

Balarka Sen

But there is GEOMETRY at the cone point, which is keeping track of the angle $2\pi/k$

Emilio Pisanty

yes

Balarka Sen

Therefore, this is an orbifold

Emilio Pisanty

ugh

so it just warp-drives into abstract math at that point?

surely not

Balarka Sen

15:44

Well I gave you a concrete example of an orbifold :) Did that seem abstract to you?

Emilio Pisanty

@BalarkaSen oh, no, it's great

so in this example, the underlying space is $D$?

Balarka Sen

Yup, the 2-disk.

The cone point, or the origin of the underlying disk, is an "orbifold point" with a standard Z/k-action by rotation

Emilio Pisanty

so what does that mean exactly?

just that it is invariant?

Secret

hmm.... I don't think I am helping here:

user image

Any point under Z/k-action will be mapped to somewhere, except the cone point

Balarka Sen

@EmilioPisanty Here's the point, in full generality

In general a topological manifold $M$ (of dimension $n$, say) is something which admits an atlas $\{\mathcal{U}_i\}$ of open sets covering $M$ such that there are homeomorphisms $\varphi_i : \mathcal{U}_i \to \Bbb R^n$.

Emilio Pisanty

15:51

@BalarkaSen yes

Balarka Sen

An orbifold structure on $M$ is an "equivariant atlas". That is, an atlas $\{\mathcal{U}_i\}$ with not only a homeomorphism $\varphi_k : U_k \to \Bbb R^n$ associated to each $\mathcal{U}_k$ but also a group $G_k$ acting on $\Bbb R^n$.

Moreover these group actions "patch up" in the sense that on $U_{ij}:=U_i \cap U_j$, $\varphi_i \circ \varphi_j^{-1} : \Bbb R^n \to \Bbb R^n$ is a homeomorphism which preserves the action of $G_j$ on the left $\Bbb R^n$ and the action of $G_i$ on the right $\Bbb R^n$.

So $M$ should be thought of as something locally modeled on Euclidean spaces with an action of a group

That is what an orbifold is. Each chart has a geometry to it, in the sense of Klein

Emilio Pisanty

@BalarkaSen so you don't actually care about how $G_k$ acts on $M$, you just need to see its action on the coordinates via the charts?

Balarka Sen

Yup!

Emilio Pisanty

ugh

why not just include an action of $G_k$ on $M$?

Balarka Sen

Well, the first thing is $G_k$ does not act globally on $M$.

It only acts on the chart $U_k$

Emilio Pisanty

15:56

would it just make things harder for actual computations? or does it actually change the object?

enumaris

wham bam

Emilio Pisanty

@BalarkaSen well, it does in your example, right?

enumaris

freaking...I think they are keeping it pretty warm here at work...saving energy or something...

not comfortable tho

Balarka Sen

@EmilioPisanty Yeah. But here's a more general example. Consider a (Riemann, if you want) surface $\Sigma$ of some genus and consider the disk $D^2 \subset \Sigma$. Make $\Bbb Z/k\Bbb Z$ act on that disk $D^2$ by rotation-by-$2\pi/k$, and on the rest of the $\Sigma$, only the trivial group acts... trivially.

That is an orbifold.

Secret

Does an orbifold basically have regions that are locally euclidean, and then these regions are glued together with a group action?

Balarka Sen

15:59

It should be thought as $\Sigma$ with the disk $D^2 \subset \Sigma$ replaced by a cone with the same boundary as that disk, with a cone angle of $2\pi/k$.

(That's what the "quotient space" is)

Emilio Pisanty

@BalarkaSen wait, what?

that makes no sense

are just cutting out a disk, making it into a cone, and then gluing it back?

Balarka Sen

Yup.

Emilio Pisanty

ok, I don't see the details but I can see how that'd be doable

Balarka Sen

The details are icky to write down; you have to make sense of transition functions in your equivariant charts. That's what wikipedia does in the "Formal definition" section

Emilio Pisanty

so then you don't have a global action but you do have an action of $G$ on a restricted set of $M$?

Balarka Sen

16:02

Yup

Emilio Pisanty

ok, got it

@BalarkaSen icky indeed

but I think I see why now

enumaris

I'm sleepy...

Balarka Sen

You can also, like, take two disks $D, D' \subset \Sigma$, and make $\Bbb Z/k\Bbb Z$ act on one and $\Bbb Z/\ell\Bbb Z$ act on the other and the trivial group $\{0\}$ act on everything else.

The orbifold quotient now has two cone points, of different geometries/angles.

Emilio Pisanty

ugh

Semiclassical

this reminds me vaguely of stuff in liquid helium, where you need to use relative homotopy groups instead of the usual ones

with the point being that, at larger length scales, the relevant symmetry gets reduced

Balarka Sen

16:05

I think these guys should appear in physics. You have various actions on local coordinates that don't patch up globally in physics all the time.

Secret

this is so weird:

user image

Balarka Sen

@Secret That is indeed the right picture.

Secret

So, Z/k acts on one subset and Z/l acts on the other

Semiclassical

the thing that weirds me out about that is what's happening to the boundary of the two disks

is there some condition on the boundary of your disk(s) in order for this construction to make sense?

Emilio Pisanty

@BalarkaSen so in your single-cone example, is a single chart sufficient?

Secret

16:08

So say I am initially located at some point outside the disc, I basically cannot move anywhere (because the group action is trivial), but once I am inside say D, then I suddenly can move around by rotating around the centre of the disk?

Balarka Sen

@Semiclassical Topologically nothing is happening. The conical picture indicates that the geometry changes.

Semiclassical

hrm

i guess it just seems like this would do a heck of a lot of violence to the nice Riemann surface structure

Balarka Sen

@EmilioPisanty Well, you may need multiple charts to cover $\Sigma$. But there is only one chart on which there is a cyclic group acting nontrivially; on the rest of the chart the trivial group is acting trivially

Semiclassical

but this might go to the icky details

Balarka Sen

@Semiclassical Aha!

But it doesn't, because the action of $\Bbb Z_k$ on that disk that I mentioned is a holomorphic action

Semiclassical

16:11

Is that guaranteed to be true for a disk of a generic Riemann surface?

I mean, if that's true, great

Secret

Or is it that an orbifold is a manifold, but some subsets of it has extra geometric structures and that geometry is given by a group action in that subset?

enumaris

hmmm

Balarka Sen

@Semiclassical For a complex disk in a holomorphic chart, yeah. The action is just rotation - that's a biholomorphic map.

Semiclassical

it's biholomorphic within the disk, sure

Secret

That is, if something is just a topological manifold, are we expect all the sets in the atlas to have no relationship between each other except the gluing (global topology)?

Semiclassical

16:14

but in your example you want to have rotation inside the disk and an identity map outside

Balarka Sen

@Secret There is a relationship; the transition functions of the charts need to preserve the actions.

Semiclassical

and that doesn't seem like it's guaranteed to be holomorphic in general

for instance, take an elliptic curve $y^2=x^3+bx^2+cx+d$

Balarka Sen

@Semiclassical I see the point of confusion. The action is restricted to that open disk; it isn't happening outside.

Semiclassical

:/

Secret

What exactly does the action does. Is it only defines the geometry in those subsets, or can it physically move points around on the orbifold?

Balarka Sen

16:21

It defines the geometry on the fixed point locus of the action.

Like the origin of the disk, which is fixed under rotation action

It assigns a notion of angle at that point

Secret

Ah that makes sense

Balarka Sen

@EmilioPisanty If you want to read about orbifolds I can find you a reference (so that I can read about them as well :P)

Reference = accessible reference that isn't a mess of definitions

Emilio Pisanty

@BalarkaSen that would be absolutely fantastic =)

Secret

I always thought one need at least an inner product to define geometry, but it seems a group is sufficient. Now that makes me wonder, if I have a set of relations R between many pairs of points on a manifold, would that be sufficient to define a notion of geometry on a manifold (sorry that goes onto a tangent of this discussion...)

Semiclassical

i'm-a stick with stuff I can at least pretend to understand

Balarka Sen

16:25

I'll dig around and let you know soon. My university admission stuff is over and there's a little more than a week before the semester starts so I can fuck around with different things.

@Secret That group actions on spaces are "geometry" was already understood by Klein way before Riemann and Gauss thought about geometry in the analytic sense, using inner products, etc.

Erlangen program

The Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as Vergleichende Betrachtungen über neuere geometrische Forschungen. It is named after the University Erlangen-Nürnberg, where Klein worked. By 1872, non-Euclidean geometries had emerged, but without a way to determine their hierarchy and relationships. Klein's method was fundamentally innovative in three ways: Projective geometry was emphasized as the unifying frame for all other geometries considered by him. In particular, Euclidean geometry was...

Semiclassical

@Secret you might be at least a bit interested in what I'm playing with right now, given that it's about correlations

Secret

Well, it depends on the details. My current investigations on that matter, besides to better understand entanglement, is also for a philosophical reason on how much structure we can remove before we ceased to have geometry and mathematical space. It all ties back to trying to grasp spacelessness

Semiclassical

ah

Secret

which might be a bit too metaphysical to be relevant to physics. But I felt like if I want to understand more about quantum mechanics, string theory and so on, I need to find a way to remove myself from the prejudiced thinking about the concept of space.

So in order to better comprehend weird things, I need to make my thinking very weird first so I can withstand it

Semiclassical

my stuff is more like: Suppose someone gives you a quantum system with certain outputs and their probabilities. Can you impute values to unmeasured quantities in such a way that is consistent with those probabilities?

Secret

16:33

I am guessing if the probability is of the form e.g. 1 3 3 1, then you cannot do that classically

that's an entangled state for example

Semiclassical

p much

enumaris

Erlangen....

lol, there's a programming language called Erlang...I thought it had to do with that for a minute...

Semiclassical

I mean, the trickiness is that it depends on what values you allow yourself to use while doing the imputation

if you only allow $\pm$ outputs, then yeah, there are quantum correlations you can't impute data to

if you allow values beyond that, then things are a bit different. as best I can tell, any quantum correlation can be produced by allowing arbitrary values to be imputed.

by contrast, there are non-signalling correlations which are non-quantum which (as I understand it) you can't impute missing values to at all

that's my current hypothesis anyways.

Secret

For that last category, are they more correlated than entanglement correlations?

Semiclassical

ya

Secret

16:38

I think you might be dealing with things in the PR box territory, which forms a convex set, with the PR box state being the maximally correlated maximising the inequality bound to 4

Semiclassical

agreed

ACuriousMind

@EmilioPisanty I have no idea what I'm reading :D

Semiclassical

mmm, spaaaam

(I'm minnesotan, so praising spam is obligatory despite the fact that I don't think I've ever had it)

enumaris

freaking...my office is so hot I'm sweating in here...what is this, a sweatshop??

danielunderwood

@enumaris They have to make back the money they spent to hire an ML dev and get him a new computer :D

enumaris

16:42

my salary ain't even that high

no higher than a normal data scientist

Emilio Pisanty

@ACuriousMind yeah, that's a weird one, innit

enumaris

gonna ask for a raise lol

Emilio Pisanty

there doesn't seem to be anything special about the date

and dec 2013 is pretty late for let's-just-have-random-fun-on-meta threads

or maybe not and that thread is the evidence?

@BalarkaSen I'm sooooooooo close to posting a preprint with the work to which the orbifold stuff applies

I hope we can talk more then =)

Secret

https://www.inverse.com/article/46755-ant-man-and-the-wasp-elihas-starr-egghead-easter-egg-explained

Random Marvel movie inspired question here: Is it actually possible to make a quantum state of say an electron such that its position observable is so unstable that when you left it in a room full of air, the constant measurement by the electron bumping into the air moelcules causes it to end up in different gaussians of position eigenstates every second. I will think after enough bumps even if the electron don't end up get absorbed, it will eventually get stablised in position as it get in…

Balarka Sen

@EmilioPisanty Wow very nice

enumaris

16:49

very unfun beans

Secret

i.e., decoherence due to prolonged interaction of an electron with the surrounding air moelcules will prevent its position observable from changing after each measurement because the electron will eventually get stuck in some pointer states?

Secret

17:05

Or asking this question in another way: What is the closest physics analogue to the scifi notion that "an object is in a state of flux so it phase and in out of reality"?

enumaris

commandeered an office that has AC...much better

Semiclassical

The closest analogue I can think of would be an electron precessing in a magnetic field

e.g. a spin-up electron subject to a magnetic field in the y-direction

at t=0, it's an eigenstate of Sz and so has a definite value of mz

if I wait a little bit, then it'll rotate away from that and so there should be some uncertainty in the mz value

with that uncertainty increasing to a maximum (when it's precessed to be an eigenstate of Sx) then decreasing back to zero (when it's precessed back to being an eigenstate)

Secret

Sounds valid

Semiclassical

That sort of time-dependence in regards to which quantum number is 'real' is the best analogue I can see

Secret

So, the "molecular disequlibrium" conditon in the antiman movie is basically something that is kinda like that, except the uncertainty and the range in position grows with time, eventually the atoms end up so far apart that they no longer combined to form a human being and thus the person "dies"

The only problem here when it comes to reality check is that the uncertainty of any macroscopic thing will be arrested by decoherence, so a quantum state of any kind cannot progressively become increasingly uncertain with time

Semiclassical

17:16

so maybe a system where a coherent state evolves into a momentum eigenstate? i dunno

Secret

that could be a possibility, because a momentum eigenstate is highly uncertain in position

Semiclassical

right

though the problem with that idea is that a person isn't a single-particle phenomenon :P

Secret

either way, that's really a very painful way to die, feeling every of your atoms slowly get moved apart

Semiclassical

depends on how slow it is, really

Secret

and you then literally "dissolves into thin air"

Semiclassical

17:18

poof

Secret

but yeah, a person being multi particle, the decoherence will kick in in picoseconds, thus forcing all of them to become locked in pointer states and thus the uncertainty will drop to zero

So let's hope real world physics does not have a process that can suddenly bring a person into a momentum eigenstate faster than decoherence lol

Semiclassical

or at least that we never run into it personally

Secret

yeah, as a final joke, hopefully at least the planck constant is truly a constant, otherwise imagine some time in the far future some mad scientist decided to change the value of $\hbar$ to say 100 meters...

and then suddenly we are all at risk of becoming momentum eigenstates lol

Semiclassical

Everybody's gone to the wave existence

Loong

iupac.org/2018-global-survey-of-scientists

DanielSank

17:30

Can anyone give me a half-decent explanation of why we have operator algebra in quantum mechanics?

Planck figured out that electromagnetic mode occupations have to be discrete. How do we get from there to operators?

ACuriousMind

@DanielSank I'm not sure I understand the question, can you be a bit more specific?

Ah

DanielSank

I know a bunch of tidbits from analytical classical mechanics that look kind of "quantum-ish", and I'd like to make the link as strong as possible.

bolbteppa

Don't need operators, e.g. path integrals

DanielSank

That's a good point, but Heisenberg picture quantum looks a lot like classical mechanics.

Cup Fever

perhaps, to get mathematians interested :P

ACuriousMind

17:37

@DanielSank If you're interested in the historical development, you probably have to trace the history of Heisenberg's matrix mechanics

DanielSank

@ACuriousMind I'm more interested in understanding the mathematical analogies between classical and quantum mechanics.

vzn

@DanielSank have been working on that myself for many years. have collected a lot of misc refs/ items/ leads. its a huge ambitious prj. think your inquiry into SHOs is on the right track (have gone that way also).

bolbteppa

Basically because path integrals exist and can be taken as a starting point for QM, you simply cannot say quantum mechanics is a bunch of axioms about wave functions and operators, you need to ask where they both come from, and of course one can get them from the idea of expected value, which is 'expected' measurement, but why do you do that?

Well, Heisenberg's uncertainty principle says there is no path of a particle, destroying classical mechanics, however we know it holds in some weird 'classical' limit, so if you derive the necessity of wave functions and operators from expected values as your starting point, then you can impose on your wave functions the existence of a quasi-classical limit, which means $\psi \sim e^{iS/\hbar}$ should hold in this limit.

Well, this is why quantum mechanics looks like classical mechanics, operations on $\psi$ translate into operations on $S$ and e.g. $\partial S/\partial t = - H$, $\nabla S = p$ etc

DanielSank

> Heisenberg's uncertainty principle says...

ACuriousMind

@DanielSank Basically, "quantum algebra of operators" <-> "Poisson algebra of classical phase space functions".

bolbteppa

17:39

Thus $\partial \psi = - (iH)\hbar) \psi$ is Schrodinger

DanielSank

Ok but the uncertainty principle is just a random stipulation until it's motivated.

@ACuriousMind Where does the $i \hbar$ come from?

Surely there is a way to arrive there by making assumptions about quantized energies or something.

bolbteppa

Heisenberg is usually 'derived' from the assumption of wave functions and operators, but you can, as Bohr, Landau etc... did, take Heisenberg as your starting point

DanielSank

@bolbteppa Why would I happen upon the uncertainty principle? Where does that come from?

Semiclassical

i sorta doubt one is going to be able to 'motivate' the fact that $\hbar$ has the specific value it does

DanielSank

@Semiclassical Ah. I don't care about the value. I do care that there's a constant with dimensions of action.

Semiclassical

17:41

I figured.

DanielSank

I suspect there is a reasonable way to convince one's self that such a constant must exist, and that it should show up in the commutator.

bolbteppa

Because the action $S$ has dimensions of energy times time, if you formulate something like $\psi = e^{iS}$ clearly it makes no sense, we need to make the argument dimensionless, so you simply divide by some random constant with dimensions of energy times time, so $\psi \sim e^{iS/\hbar}$ is dimensionless, remember that actions are invariant up to scalar multiples, so you can choose the value of $\hbar$ to be the experimental value, main thing is the units

Semiclassical

one obvious point is that, if one lets $C=[x,p]$ and demands that $x,p$ be Hermitian operators, then $C^\dagger = - C$

DanielSank

@bolbteppa Why would I be lead to write $\psi = \exp(i S)$?

@Semiclassical That's a good point.

Semiclassical

or $(i C)^\dagger = i C$, i.e. $i[x,p]$ is a Hermitian operator

DanielSank

17:44

Iiiiiiiinteresting...

Semiclassical

So if one takes seriously that $x,p$ should be non-commuting Hermitian operators, then $[x,p]=i\hbar$ is really the simplest possible option

of course, that puts the cart substantially before the horse

DanielSank

@Semiclassical Yes, as long as you've decided that eigenvalues have something to do with observable quantities, which remains to be motivated.

Semiclassical

right

ACuriousMind

@DanielSank Well, the commutator $[x,p]$ is dimensionful and proportional to the identity. We call the proportionality constant $\mathrm{i}\hbar$.

Semiclassical

my main point is that the $i$ isn't sooo mysterious either

DanielSank

17:45

Sure, but at least we have a node in the graph now, so to speak. I'm ok to pick up the pieces out of order for now :-)

Semiclassical

the real mysterious point is that $[x,p]\neq 0$ in the first place

vzn

@Semiclassical there is a "derivation" for plancks constant ("state-dependent viscosity of the fluid") based on fluid mechanics in my latest physics blog. [g6] de Cordoba vzn1.wordpress.com/2018/05/25/fluid-paradigm-shift-2018 arxiv.org/abs/1409.7036

DanielSank

^

I suspect that if you want discrete energy levels, you'll come to $[x, p] \neq 0$.

somehow

Semiclassical

And at some level there's continuity with classical mechanics, insofar as $\{x,p\}\neq 0$ for the Poisson brakcet

DanielSank

Yeah, I've been trying to understand what a Poisson bracket is from a geometrical perspective...

bolbteppa

17:46

@DanielSank I am basically summarizing chapter 1 of Landau's QM so check that, basically experiments like electrons in gas chambers lead to the fact that classical mechanics does not describe their motion, experiments show the better you measure the position at a given instant the more likely it is to appear somewhere totally randomly the next moment,

in other words, a path does not exist, and paths are described by position and velocity and initial conditions. But these are literally the concepts upon which classical mechanics are founded upon, if position and velocity do not simultaneously exist you have literally destroyed the concepts upon which classical mechanics is founded.

ACuriousMind

@Semiclassical Well, the classical Poisson bracket is also non-zero, so that is not so mysterious. The one thing you need to convince yourself of is that we should be looking at more general representations of the abstract algebra of observables than as the Poisson algebra on a classical phase space

bolbteppa

The Heisenberg uncertainty principle stating that there is no such thing as the path of a particle is literally the statement that classical mechanics does not apply to the real world, if you don't specifically state that, and why it doesn't apply, hey, maybe it does apply!

ACuriousMind

@DanielSank It is the infinitesimal change of one of the functions under the Hamiltonian vector field induced by the other.

Semiclassical

@bolbteppa do not simultaneously exist as independent quantities, anyways

(c.f. our discussions re: how position/velocity aren't independent in bohmian mechanics, for better or worse)

DanielSank

@ACuriousMind Yes, I need to investigate that.

I imagine it's simple, but I haven't seen it for almost ten years.

ACuriousMind

17:48

The classical phase space evolution equation $\partial_t f = \{H,f\}$ says exactly that the Hamiltonian induces the vector field whose flow is time evolution.

bolbteppa

As to why $\psi = e^{iS/\hbar}$, Landau gives an explanation motivated by the geometric optics approximation to electromagnetism, I think you can derive it based on the fact that probabilities are multiplicative and the action is additive so if you are seeking, i.e. trying to define, a quasi-classical way of describing classical systems starting from a wave function, forcing your wave function, which encodes probabilities and hence is multiplicative, to represent an additive quantity

- mathematics says the exponential function does that,

ACuriousMind

The quantum von Neumann equation $\mathrm{i}\hbar \partial_t \rho = [H,\rho]$ says exactly that the Hamiltonian is the operator whose exponential is time evolution.

bolbteppa

@Semiclassical yeah

Secret

This conversation is very interesting, but its getting late thus I a, going to sleep now, so I will follow it up tomorrow

ACuriousMind

Both actions - "vector field -> flow" and "Hermitian operator -> unitary operator" - are manifestations of general "Lie integration" - associating a Lie algebra element with the element of a corresponding Lie group.

bolbteppa

17:51

I am absolutely not convinced there is a serious relationship between poisson brackets and commutators yet, maybe it's there, but I don't buy what I've read so far

Balarka Sen

@EmilioPisanty This, especially starting from chapter 2, seems to be a nice introduction.

Semiclassical

I think it's a bit suggestive to write the evolution equations as such:

$\dfrac{\partial f}{\partial t} = \dfrac{\{H,f\}}{\{x,p\}}$ classically

and $\displaystyle \frac{\partial \rho}{\partial t} = \frac{[H,\rho]}{[x,p]}$ quantum-mechanically

ACuriousMind

@DanielSank It's standard lore in symplectic geometry and a straightforward, if tedious, calculation.

DanielSank

@ACuriousMind Indeed. I imagine the demonstration is not actually difficult, and will now attempt it.

Semiclassical

in that sense the translation is pretty literal, with the only difference being what the bracket between $x$ and $p$ is

that and, y'know, the 'mere' fact that one is trading a function $f$ on phase space for a density matrix $\rho$

ACuriousMind

17:58

@Semiclassical There's another way to think about quantization - the Wigner-Weyl map - where you directly map between classical probability distributions on phase space and the QM density matrix. "True quantum" effects correspond to (classically forbidden) negative values of the distribution

Semiclassical

neat.

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