The h Bar - 2023-06-09

« first day (4600 days earlier) ← previous day next day → last day (323 days later) »

2:12 AM

I think there is some recent work out of oxford on logic in quantum mechanics but idk if related :P

From bob coeke

coecke*

2:31 AM

user image

$6 !!

oh wait that is equivalent to like $70 today apparently xD

3 hours later…

5:08 AM

Are there any physical systems that violate Lorentz invariance?

When I asked this question from an AI bot I got the following reply:

"Yes, there are physical systems in which Lorentz invariance is violated. These include condensed matter systems with background fields, such as those of superconductors and superfluids, and systems involving extra spatial dimensions."

But the Wikipedia article on Modern searches for Lorentz violation (https://en.wikipedia.org/wiki/Modern_searches_for_Lorentz_violation) says "No Lorentz violations have been measured thus far, and exceptions in which positiv…

5:43 AM

@vyali Lorentz invariance is violated when there is a background. In the condensed matter systems the background is the crystal lattice of whatever the condensed matter is. This background provides an absolute reference frame and that's what causes the Lorentz violation.

In a vacuum there is no background and hence no Lorentz violation.

3 hours later…

8:39 AM

"morphisms are suitable structure-preserving functions between these spaces."

could they be any less specific

8:53 AM

@JohnRennie I see. Thanks!

9:12 AM

@Slereah could they be any more specific without committing to a particular kind of space? :P

I mean it's for the cartesian space category

surely we can say something

although there are many apparently

10:04 AM

Hi

In idealist philosophies, what do they about what the substance of the universe is? Is there anything specific?

Because saying that the universe is consciousness is very vague. I want to know some specific properties of an idealist substance.

The materialist philosophy is very specific about the substance : bits of information

11:10 AM

Currently reading some reference on nonlinear formulation of quantum mechanics. Still trying to find one where they start from nonlinear measureable operators

Specific leads to testable, testable leads to experiments... experiments! Lead to falsifiability. Beware the specific

Relativisticcucumber

11:24 AM

hi -- i have two questions. 1) in sakurai, he treats the case of time dependent perturbation theory. in this treatment, he says "in general, the perturbation removes the degeneracy in the sense that there will be g perturbed eigenkets all with different energies". he previously stated that we are in a case with a g-fold degeneracy. my question is why the perturbation removes the degeneracy as stated in this quote?

2) he says that if the method laid out for time dependent perturbation theory is to work, the degeneracy needs to be removed completely to first order, otherwise, different methods are needed. i was wondering how much of a problem this is practically speaking? how likely does this method not suffice, or will it usually be fine?

@Relativisticcucumber degeneracy is equivalent to symmetry - the idea is that the unperturbed Hamiltonian has a symmetry (that's what makes it simpler to solve than the perturbed case!) that is broken by the perturbation, and so the degeneracy is lifted

@Relativisticcucumber it will usually be fine :P

11:53 AM

@ACuriousMind Is there some formal relation between the way most physicists write structures as tuples and the way it is more properly done in category theory

Like say I have some smooth manifold defined as $(M, \tau, \mathcal{A}, \mathcal{A}_S)$, for a set, topology, atlas and smooth atlas

Easy to apply a forgetful functor on it certainly

at least from right to left

@Amit i think specific does not lead to testable. For e.g. claiming that universe is created by an intelligent being is specific, but is not testable. I'm just looking for some specific claims of idealism about the substance of the universe

Becuz being specific would make the philosophy interesting, at least

Maybe we could take "universe is conscious" to mean that "universe tries to gain knowledge about itself". This would be something specific

how does one try without a consciousness

@RyderRude Of course... but being more specific is kinda the "unit vector" lol in the direction of being closer to testability

Like the dark side again. You don't go in one step

Yes. It is an essential step toward testability

But i dont think philosophies of ontology will ever be testable

They are just for interesting discussions

Becuz what we can test in measurements is always numbers. We cant test the substance of the universe

That's why maybe the most exciting things we often forget about is when we discover a way to measure stuff we never ever measured before

Like imagine the guy who accidentally found out x-rays...

He must have been ecstatic

And physics creates a warped picture about that imo... I mean, professionals know it's not true, but the popular opinion now I think is that in Physics improved measurements is only about more energy, or at best, about building bigger telescopes / space telescopes... but it's actually a lot more subtle than that I think. The discoveries in that area can come from a totally surprising direction

12:12 PM

I dont know much about measurement devices :P

But yes, building them shud involve creativity

It seems like idealism just says "the substance of the universe is something mysterious and unknown". It cant get more specific than that @Amit

But at the same time, it doesnt seem 2 b the fault of idealism

It is only natural to say that the substance of the universe is something incomprehensible to us

Democritus -- "Nature... has buried truth deep in the bottom of the sea"

It has to stay incomprehensible, otherwise it won't be interesting

Materialism does get completely specific by claiming the substance is bits. But this also makes it a "jumping to conclusions without any good reason" philosophy

Maybe that's why in some eastern philsophiles "enlightenment", "death" and "total knowledge" are sometimes put in one basket. You can't really be a regular living human being if you have perfect knowledge

Package deal that you don't really want, lol

Lol. But can also do drugs to be enlightened :P

It sometimes comes with death tho

Im just joking

lol, serious joke that

Do GR not drugs!

12:24 PM

I think eastern philosophies r upto something

Funny, I just accidentally stumbled upon a story mentioned in the chat history here, of Poincaré telling of how black coffee helped him to discover some class of functions I know nothing about

Qualia and our own consciousness are the only first hand glimpse we have about the substance of the universe

@RyderRude For sure they are... but all kinds of weird unnecessary stuff imo developed around that core truth. I think that all cultures destroy truths in some way, because they institutionalize... and then you need another sage and another prophet and another avatar ad infinitum to keep the core fresh

@RyderRude I think that's a conflict right there... we like our culture and art and science etc. because it's so removed from any kind of first hand experience. The cow is having a first hand experience! lol... I'm putting it crudely just to convey the point, why do we need the first hand experience? It's a kind of a nostalgic craving

12:50 PM

@Slereah Not sure what you mean: The "tuple definition" isn't particular to physics, that's how mathematics does it, too

How you do it in category theory that's not founded on this set-theoretic definition depends on the context, but e.g. the forgetful functor to the category of sets is usually given by a "functor of points": In the category of manifolds, you have the point $\ast$, and the forgetful functor to Set is $M\mapsto \mathrm{hom}(\ast,M)$, with the hom-set just considered as a set.

Is there a proper way that one would translate those tuples in the language of category theory though

Or is it just a vague statement

Actually didn't metamath have a set theory description of categories

Maybe they have such a thing

us.metamath.org/mpeuni/mmtheorems172.html#mm17156s

maybe related idk

hard to read so who knows

1:10 PM

Maybe I could check how the various objects of the tuple behave under the various morphisms of the categories

1:22 PM

I guess this deals with the vaguest part of category theory

Stuff, structure and properties

> It’s like a horse transforming into a person from the head down. First it’s a horse, then it’s a centaur, then it’s a faun-like thing that’s horse from the legs down, and finally it’s a person.

naturallyInconsistent

@vyali You have asked to separate questions on the main site on this same topic, and now you have asked it here. In all 3 times, we have told you that AI CANNOT BE TRUSTED. What part of this do you not understand? What part of the answers that you received, all of them saying the same thing, do you not understand?

@Relativisticcucumber Most textbooks treat the case of Rayleigh-Schrödinger perturbation method and never mention that there are any others. As if it is the gospel. In RS perturbation, the denominators of the correction terms are $E_n^(0)-E_m^(0)$, leading to infinite perturbation in every correction term if you cannot force those offending cases to strictly be zero.

That state of affairs is kinda sad, because even the visually similar Brillouin-Wigner perturbation method, the denominators are completely perturbed energies, and so as long as the degeneracy is lifted at any order, it would be picked up from that order onwards. The price to pay, of course, is to need horrible, potentially high order, root finding to get the correct results, most of the roots of which are nonsense.

1:39 PM

@Slereah The whole point of "categorial thinking" is that objects are not just sets with extra structure!

@ACuriousMind I know, but you can define them this way

the "tuple definitions" are exactly that: You start with a set, and then you add informotion

I just wish to bridge te two!

naturallyInconsistent

note that one does not have to fully solve the degeneracy problem. Finding orthogonalisation is enough.

Show that it at least has the same kind of properties as objects in a category in some way

1:41 PM

I'm not sure what you mean

you can start from the tuple definition of a manifold and define the category of manifolds by that

trying to figure out how to make the forgetting explicit using this method

You have some category of topological spaces with objects $(X, \tau)$ and forget it to sets via $(X)$

I already wrote down how to phrase this in purely categorial terms

It is common enough to define physical things using some kind of hierarchy like that so it would be nice if it could work out thus

naturallyInconsistent

@Amit It was not just him getting ecstatic. His wife contributed the famous ring picture, and soon after that, the entire world went insane. Every doctor wanted one; healing broken bones suddenly became much easier to see. Travelling snake oil businesses sprung up to give everybody unsafe doses. He got the first Nobel prize for this work.

No more lectures

7 exams ahead ::screams of despair

naturallyInconsistent

1:45 PM

hugs

3

@naturallyInconsistent Cool, yes I vaguely remembered it was sensational in more than one way, that's why I thought of it first probably

@Mr.Feynman wow.. good luck ^_^ do you at least like the subjects?

> Usually people don't bother to define "forgetful functor" very precisely - like pornography, you're just supposed to know a forgetful functor when you see it.

@Slereah but what is there to "work out"? If you have the definition in terms of tuples, just define the functor via $(X,\tau) \mapsto X$. If you have some other definition of topological spaces, use my functor of points from above

Yeah but that would just map the objects there

I guess I also need to do something with the morphisms as well

@Slereah $\mathrm {hom}(\ast,-)$ is a functor

1:50 PM

@naturallyInconsistent :D

@Amit Well, if I can get myself to like QFT again after the renormalization group, then maybe yes

it maps any morphism $f: A\to B$ to $\mathrm{hom}(\ast,f) : \mathrm{hom}(\ast, A)\to \mathrm{hom}(\ast, B), a\mapsto f\circ a$

conversely, in the tuple definition, you don't need to do anything because the morphisms are already defined to be maps on the underlying sets

Yeah but the tuple description it's not even necessarily sets?

I mean it always is I guess in a ZFC kind of way

@Slereah what

But is the set M and the metric g the same kind of object under such things?

can you give an example for such a "tuple definition" where the first element of the tuple is not a set?

1:59 PM

@Mr.Feynman They can't all be on QFT right?! 7?! :)

@Amit Nope. I won't do all of them this summer either. Right now I'm concerned with 2.5 QFTs, 1 GR and 0.5 stuff about phase transitions

Wait until you'll have to be concerned about the rent

@Slereah Now I'm only concerned about my rant

2:15 PM

Chapter 4 Introduction to Nonlinear Operators. (1980). Applications of Functional Analysis and Operator Theory, 108–137. doi:10.1016/s0076-5392(08)62881-7
Currently reading this to get some *really deep* intuition on nonlinear operators

let's see how this 1980 book chapter will perform

Relativisticcucumber

@ACuriousMind i see this happens with magnetic and electric fields from zeeman effect and stark shift, but is this a general principle?

> Regular readers of this blog will be familiar with the notions of property, structure, and stuff. Less well-known is an intermediate notion between property and structure called “property-like structure.”

2:35 PM

@Relativisticcucumber yes

(also physics.stackexchange.com/a/266593/50583)

why is that matrix censored

what are they hiding

in more elementary terms: If the Hamiltonian has degenerate energy levels, then any non-trivial operator that only acts within these energy eigenspaces commutes with the Hamiltonian, and so is a symmetry

@Mr.Feynman lol.. GR! Yay!

> Like Gravitation, the title can be taken to refer not only to the subject matter but also to the immense size and scope of the book itself. Like The Lord of the Rings, it consists of 6 parts arranged evenly into 3 volumes (but without appendices).

2:54 PM

@Slereah what's that referring to?

The book Sketch of an Elephant

3:10 PM

Is that scary stuff starring in your book too?

Relativisticcucumber

thanks @ACuriousMind

Probably not top much

It is big enough without involving toposes

Weave the topestry

I'm allowed one silly pun a day 🤣

3:27 PM

user image

Not expecting this will go anywhere
Formulating measurement procedure as a projection operator on the state seemed sound though

First moments of consciousness must feel so precious. It's when a baby is slowly gaining awareness but the process is only starting

This reminds me of the first Avatar movie. They have some very precious world mechanics

Nonlinear QM has been tried before

It did not produce much

Probability evolution has to be linear, i think

Apart from state reduction, i mean

naturallyInconsistent

part of the problem is that the usual notions of nonlinearity in classical physics can be cast into linearity in QM notation, so it is not clear how to map the notions of nonlinearity around.

Another problem is that superposition seems to require linearity

This is one of these solutions in search of a problem

like, "non-linear QM" might sound cool, but what problem is it trying to solve?

3:35 PM

It is one of the proposals to solve measurement problem @ACuriousMind

naturallyInconsistent

But the measurement process ought to be fully characterisable in QM...

ah, so it's just for people who don't understand the measurement problem, gotcha

@naturallyInconsistent yes, this is what i meant by "probability evolution has to be linear"

@naturallyInconsistent do u mean that the Schrodinger eqn is linear while the Operator field equations r non linear?

The dynamical observables have to follow non linear evolution, in both QM and CM. But probabilities have to evolve linearly

Do we have experimental evidence of some probablistic outcomes do not quite follow a linear evolution (e.g. noisy outcomes which barely fits a linear model)
There's a diffusion current treatment of the schrodinger equation, but it is not clear if it captures enough nonlinearity in the operators https://www.sciencedirect.com/science/article/abs/pii/037596019290061P

@Secret we do not have one shred of evidence that standard quantum theory would be insufficient

3:40 PM

I feel uncomfortable with non linear Schrodinger eqn. Too ugly

But what MWI does not wanna accept is that state reduction is as fundamental to probability evolution as linearity is

MWI just wants to go all linear

naturallyInconsistent

And MWI manages to get an internally self-consistent explanation of the physics

There is no problem.

There is no need for state reduction if you accept multiverses

Yes, there is no actual problem with the math and consistency of it

But apart from it being bad philosophy, idk how world duplication could be translated into equations. Has this been done?

You need some world duplication equation to suppement the Schordinger eqn to enforce the Born rule

what

naturallyInconsistent

Do you have maths to prove that it is bad philosophy?
And how many times must we cover these interpretation issues?
Have I not already typed up some of these maths for you, that you asked for?

@RyderRude WTF is this

I mean the Schrodinger eqn does not imply Born rule. MWI needs this postulate that the worlds are duplicated proportional to the Born rule probability @naturallyInconsistent

And this postulate needs to be written in math language

You have not given me the math for this

But you did say that MWI requires this world duplication

naturallyInconsistent

3:56 PM

@RyderRude By postulating it. One line. What else do you want?

4:07 PM

In MWI, the current state of the multiverse corresponds to the wavefunction supplemented with the number of duplicate worlds of each branch at each instant of time. So one would need to give a time-dependent eqn describing when exactly the world duplication happens during decoherence

naturallyInconsistent

No, that is not required

आर्यभट्ट

Can any one of u have some basic understanding related to quantum computers?

I saw some videos it clearly go bouncer how internet be quantizied

With very basis of sharing data using em wave in given frequency band how the quantum internet is different from normal internet

what on earth is the "quantum internet"

I assure you the internet you are using remains fully in the realm of classical computing

Quantum internet afaik just means a more securely encrypted one (using QKS)

QKD* it is more often called

Totally unnecessary considering no one really showed classical crypto is inadequate as such. Although the deployed pk algorithms (RSA and DH variants) will be compromised if a large scale quantum machine is ever built

That's specific to these algorithms not to classical crypto broadly

@Amit I gave some thought to what u brought up a few months ago. Now i think that quantum logic does underlie classical logic

4:22 PM

Does it? I thought it's a dead end / painted door :)

@RyderRude but do explain

I mean that quantum systems are physically responsible for any quasi-classical behaving systems that exist in our world

So quantum logic is the physical reason why classical logic holds in this universe. This is because quantum mechanics is physically a deeper theory

But in terms of human-organisation of math, i think classical logic is a better foundation of the human study of math

But is it really a violation of 'classical' logic or just of some other implicit assumptions?

Quantum mechanics is formulated within classical logic, so it is hard to pin what is deeper

I am making a distinction between two types of foundations

आर्यभट्ट

@ACuriousMind hi

The Quantum Internet

1. Physical foundations 2. Logical foundations

Logical foundations are how humans organise math. So here, set theory and classical logic come first

And quantum mechanics is formulated in this framework

But physical foundations are different. The laws of physics underlie every piece of mathematics humans have cooked up in their brains

So in this sense, quantum mechanics allows human brains, and hence, classical logic to exist

This distinction between physical foundations and logical foundations helps me see the interplay between math and physics @Amit

naturallyInconsistent

4:38 PM

@RyderRude ... then nobody would have needed to coin the term, quantum logic.

I mean that quantum logic can be axiomatized within ZFC @naturallyInconsistent

naturallyInconsistent

Then, one has to wonder, is ZFC even classical?

Yes, ZFC is a theory written in first order logic. It's all classical logic @naturallyInconsistent

naturallyInconsistent

@Amit Good, then. In that case, Amit's point here stands.

Amit's point does stand in one of the camps called" Logical foundations". It's the "physical foundations" where quantum logic is deeper than than classical logic @naturallyInconsistent

naturallyInconsistent

4:44 PM

That part is kinda obvious. We even have experimental evidence that human minds seem to conform to quantum logic.

Logical foundations are just the human organisation of math. Here, classical logic and set theory underlie quantum mechanics

naturallyInconsistent

In particular, A -> B, ~A ->B, together, the human brain sometimes fail to see the B is always true, because in quantum logic, it is possible to violate that.

@naturallyInconsistent i dont think humans fail to see this because of quantum logic. It's just a human failure to do logic.

naturallyInconsistent

At least one does not have the evidence otherwise to prove that humans obey classical logic fully.

Sry, im on the road, real life intruds lol, will respond later

4:51 PM

Yes, this is why i used the word "quasi-classical". The deeper laws of physics like quantum mechanics allow quasi-classical computers to physically exist. This is why quantum mechanics is responsible for the physical existence of almost- classical logic @naturallyInconsistent

@Amit ok :)

I believe physical foundations are more fundamental than logical foundations

Logical foundations are just the logical order in which humans would define math

But nature doesnt care about what is define-able first. In nature, quantum mechanics exists first.

1 hour later…

5:59 PM

@naturallyInconsistent I'm curious what you mean by that example. I mean, I am aware that people often find it difficult to grasp the truth table for $\implies$ because, roughly, "effects can be realized even without a specific cause" is not a very intuitive notion in normal language... when we say "if X then Y" in everyday use we many times ascribe some exclusivity to X

@RyderRude I agree in general but what bugs me is that it seems tautological to say that Quantum logic is more fundamental because it applies to Quantum phenomena... the logic we do find useful more generally is still the classical one... I think it is necessary to show exactly in what way "Quantum-logical" axioms are superior to the classical ones. And if it's only in the context of QM... well let's just call it QM and maybe we don't need to even invoke the word "logic" at all

6:33 PM

May 1 at 14:15, by Slereah

why must we be eternally cursed with talking about the intepretations of QM

Not gonna lie, quoting old Slereah and ACM's messages might be my favorite activity on the chat

6:48 PM

Lol

It may be the "duplicate flag" analog for the chat 🤣

Is it true that classical logic underlies qm?

I have been confused lately about the meaning of “results that theoretically could be determined”. In double slit it is said that if you make a measurement on the photon at one of the slits, you decohere the photon and get the “bullet pattern”. When you don’t make a measurement you get the interference pattern.

But i mean theoretically you could have made a measurement on one of the photons even in the case where you did not make a measurement at all

So i seem to be misunderstanding the meaning of what a result that could in theory be determined means.

The ultimate point of this question is to see whether this idea of outcomes being inherently probabilistic is really unintuitive or nit

The ultimate point of this question is to see whether this idea of outcomes being inherently probabilistic is really unintuitive or not

@SillyGoose I think it may be that classical logic is still the most general framework we can find. It you take a look at this example, it seems to show how classical logic fails to describe a simple superposition roughly due to the uncertainty principle... I think that maybe just logic wrongly applied, not a fault with logic

@SillyGoose What, exactly, do you mean by "classical logic"? :P

this isn't a trick question, there are at least two different meanings of "logic" in this context and the distinction matters

7:04 PM

I always assumed it's propositional logic we're talking about

@Amit sure, but are we talking about the logic that applies to physical propositions or to mathematical propositions

Oh i see @Amit

e.g. the theory of true statements about operators in Hilbert spaces lives completely in classical logic

I suppose applied to mathematical props @ACuriousMind

but the theory of physical propositions like "The particle has momentum $p_1$" and The particle has position $x_1$" does not obey "classical logic"

7:06 PM

Stuff like what Amit linked

@ACuriousMind Exactly, that's what I would say is propositional logic wrongly applied

Not sure what that means.

In any case, when people try to take quantum logic as fundamental, they don't mean to use it in the math instead of classical logic

they mean that they start with the rules of quantum logic applying to physical propositions, and consider the rest of QM to follow from that

there are some theorems (some of them pretty recent) showing that under certain assumptions about what exactly "quantum logic" is, you end up with every quantum logic being equivalent to the logical structure of projection operators on a complex Hilbert space, i.e. standard QM

Ok so that's it, I guess this other use of "logic" is very misleading in my mind. I always associate "logic" with the pure math logic

@ACuriousMind I guess it means that statements like the one in that Wiki article ".. failure of the propositional distributive law.." is misleading, or maybe I am not really getting what they're trying to convey by that

it's not misleading - within quantum logic, this law fails

but we're not throwing out classical logic and inventing some new kind of math entirely based in quantum logic, we're just applying quantum logic to the set of physical propositions. And really, calling it "quantum logic" is a bit silly, you should just think of this as an abstraction of the properties that the spectral projection operators for the observables have

(since every projection for a spectrum $\sigma$ corresponds to the question "Does the particle have a value for the observable inside $\sigma$?")

Yes, I think some people read stuff like that and think that somehow refutes the validity of classical logic. But both frameworks are just as arbitrary like all mathematical frameworks are

@ACuriousMind Yes, that's a much more useful way to think about it

7:47 PM

It's a shame that in classical E&M it is not (tipically) stressed that the Green's function of Maxwell's equations $\Box A^\mu$ is really a tensor $G_{\mu\nu}$ and not a scalar $G$

I noticed this for the first time comparing with the photon propagator in QED...

1 hour later…

8:52 PM

Folks,
I've got an optics questions.
I've got a ray of light which travels through medium 1 (air) and arrives at the interface with medium 2 (glass). The ray arrives at an angle which is smaller than the total internal reflection angle. So, a portion of the light will be refracted into the medium 2, and a portion of the light will be reflected back into medium 1.
How can I calculate the proportion between the refracted and reflected portions? What equations govern this?

@misk94555 Snell's law I think...

@misk94555 Sorry, you use Snell's law to calculate the angle of refraction and then apply the Fresnel equation apparently -- not something I knew about really :)

Apparently the reflected light will be s-polarized which you need to take into account too. But I'm just reading the Wiki now :)

@misk94555 I think this calculator can pretty much give you what you need!

1 hour later…

10:21 PM

user image

this became oddly specific once I starred it, lol

the number of hugs is not conserved

user image

what is meant by "local" here? The notion of local I am used to is loosely: interactions occur between things geometrically close.

@SillyGoose I've no idea, but I found this thingy

oh yes i've read that answer but i dont think i know enough about the physics talked about to interpret what local means still from the answer

Just read it myself, we're in the same boat

@SillyGoose roughly yes? all the ideas there relate a physical theory in a volume to another physical theory on the boundary of that volume, meaning that the idea that physical information is "localized" anywhere doesn't really hold there - the viewpoints that physics is happening in some small local subsets inside the volume and happening "smeared over" the boundary are complementary

however, let me remark that the usage of "evidence" there is close to deceptive: This is "theoretical evidence", none of these ideas have as of yet something to do with empirical evidence, they're merely patterns in the math

10:30 PM

hm i see

reminds me of that ER=EPR thing Susskind gave numerous lectures on (I saw only the popular ones)

so are the points made above what people are referring to when justifying a belief that the world is approximately local?

what does a belief that "the world is approximately local" entail, and who's justifying it?

eh, sry, not enough context

hm okay I guess I should simplify to: systems are allowed to be approximately local. and by approximately local, I mean

hm perhaps this requires too much context xD

but this paper describes their result that random Hamiltonians can be approximately 2-localized, i.e. put into 2-local form via optimizing some cost function: arxiv.org/pdf/2303.02782.pdf

user image

10:40 PM

without reading the paper I'm gonna say that's not the kind of "local" the quantum gravity folks are talking about

"local" is one of these words where you should never assume two people mean the same thing by it until you've seen an actual definition

hm i see

so a priori they should not have anything to do with another

i am interested in the k-local definition of local as given in the tensor product structures paper, which is the same definition as used in the random hamiltonians paper

I mean, they might, but not simply because both talk about things being "local" :P

@SillyGoose what exactly are you "interested in"? Isn't a k-local Hamiltonian just a Hamiltonian that's made out of a sum of operators that act non-trivially on at most k subsystems ?

well why one should allow non-local hamiltonians

@ACuriousMind yes

what do you mean "allow"

also, "non-local" doesn't really have a definition in this context, what do you mean by it?

hm okay I think this is what I mean

hm actually i think i am conflating different notions of locality

well i mean should one expect two subsystems located light years apart to have an interaction term in their composite system Hamiltonian?

10:48 PM

who's talking about systems separated by light years? :P

k-locality is more about having some sort of mess of a system given and trying to find a partitioning of it into subsystems so that it looks "neat" - where "neat" means that the Hamiltonian is k-local for k as small as possible

you should not think about this notion of "local" as being directly related to notions of distance or whatever

well just me :P but im trying to connect k-locality (and perhaps an upgraded definition which incorporates a notion of distance) to something physical

a 2-local operator is "very local" because it affects only 2 out of however many subsystems you have

doesn't matter how these subsystems are actually positioned in physical space or whatever

it's about "localizing" the interaction to a small subset of all the available subsystems

well i think my question actually boils down to: should we expect only systems sufficiently physically close together to interact

generally yes?

what are systems that are counterexamples to this?

10:55 PM

I mean, if I look up into the sky, stars that are light years away from me are "interacting" with me when I see their light, no?

this isn't really a well-defined question

what problem are you trying to solve here?

well so the receiving light from a star interaction would described by an interaction term in a Hamiltonian between two very physically distanced systems, right?

well someone had said that approximately local hamiltonians is probably more physical and exactly local hamiltonians. but i think perhaps the notion of local was not precisely defined

but i had thought that an approximately local hamiltonian means that non-local interactions are allowed to occur, which seems to be a false deduction; well just not even wrong because i think i am not using a single definition of local

@SillyGoose It seems straightforward to me that they just meant that when you have N systems that interact, it will be very rare that there are interactions that really only exactly involve 2 (or 3 or whatever) of these systems- But it will often be the case that just by modelling the 2-local and 3-local parts of the interaction, you can get a good approximation of the overall behavior of the system

hm well the conversation was in the context of speculating how locality may emerge from decoherence, but now i think i just don't know what locality means and implies :P

11:12 PM

In Wikipedia it is written (roughly, paraphrasing) that the "fundamental solution" for the 3d Laplacian is the familiar looking "potential" type that is proportional to $1/r$.. but I can't fathom what exactly is meant by a fundamental solution. I just want to understand, does it mean all solutions will have that proportionality? Was it proven?

@Amit have you considered asking Wikipedia? :P

Yes, I didn't get very far in trying to understand this one

It introduces dirac deltas, linear operators, and I don't see anything relating to uniqueness

what do you mean "uniqueness"?

uniqueness up to constants, that all solutions will be proportional to $1/r$... uniqueness of the family of solutions I suppose? :)

all solutions of what?

11:18 PM

the 3d Laplacian

the article shows that if you have a solution $F$ to $LF = \delta$, then the solution to $Lf = g$ is $F\ast g$

the usual uniqueness theorems about differential equations that show that $f$ is unique also show that $F$ is unique

(up to sets of zero measure that don't affect the convolution)

the actual mathematical arguments for uniqueness have to be tailored to the specific equation you're trying to solve

e.g. physics.stackexchange.com/a/368081/50583

The context here is just to understand the claim that, in the weak Newtonian approximation, since $g_{00}$ satisfies Laplace's equation, we can identify it with the familiar $1/r$ Newton's potential

@ACuriousMind Thanks, currently I'm not able to understand that... I need to brush up on my PDEs

« first day (4600 days earlier) ← previous day next day → last day (323 days later) »

Transcript for

Jun9

The h Bar

General chat for Physics SE (physics.stackexchange.com). For M...

join this room
about this room

00:00

06:00

12:00

18:00

all times are UTC

site design / logo © 2024 Stack Exchange Inc; legal

mobile