/sci/ - Science & Math

>>6461875
Non-locality isn't a logical impossibility, it's just a preferential trend that science has taken (presumably to detach itself from seemingly primitive views of 'magical effects', which in-and-of itself is mostly reasonable). Newtonian gravity was nonlocal, and most QM interpretations contain at least some notion of nonlocality (either implicit or explicit).

Recommended reading topics:
>Bell's Theorem
>The post-1960s Bell tests (i.e. the ones with angle correlations, not just up-down)
>QM foundations (a.k.a. "interpretations)

Assuming you're at an appropriate mathematical/physical level of familiarity, a strongly-recommended book with some discussion about this is Bell's "Speakable and Unspeakable in Quantum Mechanics", which I'm sure you can "acquire" a copy of.

(Search the /sci/ archives, someone keeps posting a dropbox link)

>So as we all know, if you know the value of a function on a given interval, then you can extrapolate the value of the function on a larger domain.

Why do people on the internet start their sentences with
>as we all know
Is this something you read in books?

PS: That sentence is pretty vague. Are we talking about continuous functions here, for a start?
I can extrapolate in many ways.

>When I learned about "analytic continuation" recently, my mind was blown, since I had always thought of functions as things that were inherently local.

You should explain why your mind was blown here and what it means to say a function is "inherently local".
I'd argue that the function which you can continuous analytically are the most local of all - you seem to imply the opposite here.
On the other hand, a recursively defined function like the factorial is "quite unlocal" since the corresponding code makes sense for all arguments in its domain.

>and math itself seems to be nonlocal.

Again, the analytical functions are pretty "local", their value is determined by local data.

>After all, the universe is supposed to be perfectly described by math
How so? I'd argue the opposite, but that's metaphysics.

>So my question is, could this imply that it is possible for the laws of physics and the universe itself to be nonlocal?
In what sense? Locality is one of the "few" axioms for algebraic quantum field theory. It's in there because otherwise, with difficult timely behavior, things get out of hand.

>>6461885
>>6461887 again. Just to comment on field theories and locality. Obviously there are tons of formulations of quantum field theories, never mind those pertaining to de Broglie-Bohm theory and the like. Workers in those fields would probably argue that QFT gets out of hand <span class="math">anyway[/spoiler]; but in any case, for as long as we're still missing full empirical union with GR the question of how (fundamentally) to approach QFTs seems to be open.

OP here.
>>6461885
>You should explain why your mind was blown here and what it means to say a function is "inherently local".
I had always believed, on an intuitive level, that whatever is "going on" with a function on one region doesn't affect the function's values at other regions.
Especially if the function isn't even defined on another region.
But apparently, functions like the Riemann zeta function have a UNIQUE analytic continuation, to a region where the original definition of the function doesn't even make sense! This is what really blows my mind.

>How so? I'd argue the opposite, but that's metaphysics.
I would be quite shaken if the universe wasn't described mathematically. How else could it be described?

That complex color plot doesn't make sense to me. Why is the zeta function constant for all positive real parts?

>>6461887
But doesn't nonlocality cause problems, like violation of causality?

>Non-locality isn't a logical impossibility
I guess it's not a LOGICAL impossibility, but it still feels "wrong" for the laws of physics to be able to violate causality.

>>6461885
I said "as we all know" in order to establish that I am not speaking didactically. Sorry if it came off as condescending.

>Again, the analytical functions are pretty "local", their value is determined by local data.

Could you explain this a little further?

>>6461895
It's asymptotic in the Re(s)>1 region, so it becomes "nearly constant."

>>6461893
the key word is
>analytic

analytic means roughly being equal everywhere to its taylor series. Then the function is basically such power series, which can be reconstructed from any small open patch of values. It's fundamentally a tautology. Of course in general functions can't be determined completely from a subset of values.

For analytic functions, check any introduction to complex analysis, it's easy and will explain the tricks behind all this.

Riemann Zeta is analytic, this is key.

Also, I want to clarify that there is not a unique analytic extension of the factorial.

>>6461875
>After all, the universe is supposed to be perfectly described by math, and math itself seems to be nonlocal.
Weeeeeeell. The usual idea is that the universe <span class="math">can[/spoiler] be describ<span class="math">able[/spoiler] by math; but the trick always has been and always will be choosing the right math, and ascribing mathematical abstracta to physical materia (reifying the latter). The universe <span class="math">being[/spoiler] math is a much different, and uncommon stance (I think it's called "Platonic realism"?).

Most scientists will pooh-pooh this as "mere philosophy", but the problem of course is that that just puts their own internalised notions of what things "are real" or "ought to be considered real" into a box that no-one is ever allowed to open or investigate. And when you can't investigate it, you can't spot the problems with it - and so you'll rarely stop to consider whether your <span class="math">tacit[/spoiler] ontology/epistemlogy is at fault for the difficulty your subject is experiencing.

>>6461897
>but it still feels "wrong" for the laws of physics to be able to violate causality.
Instantaneous communication can only break causality <span class="math">if[/spoiler] you take locality as a fundamental axiom, if you're referring to relativity. The idea for a nonlocal relativity would be that an <span class="math">actual[/spoiler] causal path would be defined in a single preferred frame, and the instantaneity would only occur in this frame. It's not the normal way to do relativity, but it does does give equivalent <span class="math">measured[/spoiler] predictions.

I've seen this explained much better somewhere else, I'll see if I can dig it up. The essense though is that causality is only directly linked to locality given certain axioms about the fundamental structure of spacetime.

OP again.

>>6461907
>there is not a unique analytic extension of the factorial.
Sorry if this sounds stupid, but isn't the gamma function supposed to satisfy this? Is the gamma function itself "not unique"?


>>6461908
>The idea for a nonlocal relativity would be that an actual causal path would be defined in a single preferred frame, and the instantaneity would only occur in this frame.
Whoa, I've never heard of this before. Is there a name for this kind of physical framework (so I can look it up)?

>>6461889
I didn't argue that it's a must, I'm literally just saying
>Locality is one of the "few" axioms for algebraic quantum field theory.
I was refering to
http://en.wikipedia.org/wiki/Local_quantum_field_theory
but I disagree with the guy above me that QM implements non-locality directly or indirectly.
The structure of observables remains causal, formally expressed via ugly stuff like
http://en.wikipedia.org/wiki/Causal_structure#Causal_structure
but one can also just say the relevant operators commute for spacelike seperation.

PS: I personally don't feel there is a need to unify QFT with gravitity, that's just more aestetic, but I know no logical argument why it should even work - though I'm pretty sure it does.

>>6461893
>I had always believed, on an intuitive level, that whatever is "going on" with a function on one region doesn't affect the function's values at other regions.
But that already works with Taylor expansions.

And you can do many continutation. E.g. consider
f(x):=sin(x)+(x^7+3)^2
defined on the inteval [-324,653]
Now externd the function on all of R by saying that outside of the interval the function takes a constant value and this must be done so that the function is contusion at -324 and at 653.
Voila, you have a condition for a UNIQUE continuation.
Analytical continuation is just the same, only with another requirements, namely complex differentiability.

>>6461893
>I would be quite shaken if the universe wasn't described mathematically. How else could it be described?
You can descibe the universe, but that doesn't necesserily imply you can "describe" every natural process in full details.

>>6461912
it's one solution, namely if you add a superconvexity requirement
http://en.wikipedia.org/wiki/Bohr%E2%80%93Mollerup_theorem

contusion is supposed to mean continuous (silly English word)

>>6461875
locality in physics (signals do not exceed the speed of light)

integration and differentiation are inherently nonlocal, except (weirdly?) differentiation at integer values

What does this have to do with the speed of light? Nothing.

>>6461908
>>6461897
>>6461912
Hmm, it's probably quickest if I point you at the relevant lecture from Mike Towler's pilot wave theory course.

http://www.tcm.phy.cam.ac.uk/~mdt26/PWT/lectures/bohm5.pdf

You might want to skim through the rest too, since it might be assuming you're familiar with (nonstandard!) content from earlier lectures. But basically, the Locality-Causality link is only valid in a strictly local Minkowski spacetime, which is not the single logically-possible (and empirically-consistent) structure.

(Yes, yes, GR; but as I've mentioned before, since we have so far basically failed to fully unify QM and GR, it seems obvious that at least <span class="math">something[/spoiler] about one is just flat-out wrong, so why not at least entertain the possibility that it's locality, if not just to eventually prove that <span class="math">that[/spoiler] part definitely has to stay?)

>>6461917
>I personally don't feel there is a need to unify QFT with gravitity, that's just more aestetic, but I know no logical argument why it should even work - though I'm pretty sure it does.
I think almost every single theoretical physicist ever would prefer that we had one grand unifying theory to describe our one singular universe, rather than having a mish-mash of theories with arbitrary 'cuts' between their applicability. Unless we could demonstrate that Nature itself arbitrarily 'cut' between domains of relevance, but then I'm sure that that would just end up being analysed further into one single set of (admittedly "interesting") principles.

>>6461926
I think OP was making the connection:

>If the structure of an arbitrarily small part of a function contains information about the entirety of the function across the entire domain, and if mathematics has validity at describing aspects of the physical world, then is it logically possible for some arbitrarily small region of space to contain information about the entire universe?

>>6461926
>[...] differentiation are inherently nonlocal, except (weirdly?) differentiation at integer values
?

differentiation is nonlocal how. And values of what?

>>6461927
>I think almost every single theoretical physicist ever would prefer that we had one grand unifying theory to describe our one singular universe
Not sure why you say this as a response to me saying I don't feel the need, but yeah, I agree that most physicist would prefer that and I would too, given that it's more aesthetic. It's just not required.

>>6461917
>Analytical continuation is just the same, only with another requirements, namely complex differentiability
But isn't complex differentiability the most important requirement for extending a function (assuming it's already differentiable)? I would consider analytic continuation the most "proper" continuation.

>>6461917
>I personally don't feel there is a need to unify QFT with gravitity, that's just more aestetic
that's ambiguous, do you consider a need for a qauntum gravity?

>>6461934
>But isn't complex differentiability the most important requirement for extending a function (assuming it's already differentiable)? I would consider analytic continuation the most "proper" continuation.
It's more useful (solves <span class="math">\Delta f(x)=0[/spoiler], gives you good line integral properties etc.), but what's your point?

>>6461926
I was talking about the fact that the structure of a function can be "extended" with knowledge of it on a small interval,
and since the universe itself has a mathematical structure, this should apply to the universe as well. Or does it? That's my question.

>>6461931
Yes, this is exactly the connection I was making.

>>6461917
>I personally don't feel there is a need to unify QFT with gravitity, that's just more aestetic
I want to unify QFT and gravity, but not for aesthetic reasons.
Instead, it's because I think that doing this would reveal a lot of physics beyond the Standard Model that we don't know yet. Modern physics, though quite advanced, seems woefully incomplete to me.

>>6461933
>differentiation is nonlocal how.
the derivative of a function at a point only depends on the neighborhood around the point
> And values of what?
values of n in <div class="math">\frac{\mathrm{d}^nf(x)}{\mathrm{dx}^n}</div>

>>6461945
Differentiation as an operation is local, but analytic continuation is obviously nonlocal.

>>6461875
Firstly the analytic continuation is done via a "local" process. The result of this local process creates a unique analytic extension.

You may be interested in the action principle (a global principle) implying the Euler-Lagrange equations (a local principle).

>>6461933
>It's just not required.
I think this would just end up devolving into an argument about personal preference for what we feel science "ought" to aim for, or our personal motivations for being involved with science. I'll have it with you in a pub, if you live nearby.

I'll just say that at least a small fraction of physicists feel that the "purpose" of science is to find out "how Nature really is", rather than just building up a book full of
>equations that happen to predict the results of different experiments, all special cases
>blueprints for "useful" technology
and would feel that those satisfied at just doing those things are (to put it politely) "engineers" rather than physicists. But that literally is just my opinion, man. I wonder if there's a correlation with what kind of scientist you self-identify as.

I really don't want this to end up being a shitty argument; but I can't bring myself to delete what I've typed: so have a cheerful picture.

>>6461955
>You may be interested in the action principle (a global principle) implying the Euler-Lagrange equations (a local principle).
Wow, I knew about Lagrangian mechanics, but I'd never thought of it that way before.

So does this mean that the difference between having locality and nonlocality is just arbitrary and a pointless distinction?
But it still has a significant impact on the laws of physics...

>>6461937
Sorry, I see how it's not clear what I ment. But I also don't know what you mean by "need".
I'll not starve if there is no quantum gravity. It would be nice to have - and more importantly: that's "an excuse" for theoretical physicists to work on it, but I don't feel an inner hunt which wants me to see it done. In fact, the progress in doing physics is the relevant part.
Do you know the Anderson paper "More is different"?
I'd rather have a good theory of renormalization than a working qantum gravity theory. But they may come together, of course ;)
Field theory is cool, and I don't percieve the tools as a bunch of things that don't work, but as a nicely developing subject of its own.
That being said, a theory of quantum gravity is overrated. And I say this even though I did my masters thesis in the subject.


>>6461954
Yeah, I'm not a realist, in any case.
Have a pic in return.

>>6461940
>and since the universe itself has a mathematical structure
That's your prejudice (depending on how literal you see the connection)

>Yes, this is exactly the connection I was making.
I still don't see it. I don't want to say the defining property is "random" (for the fact that it's natural and useful), but there is still no magic involved and certainly nothing you can induct to the natural world. It's a nice idea of course.

>>6461945
>>differentiation is nonlocal how.
>the derivative of a function at a point only depends on the neighborhood around the point
That's what's ment with local. Not nonlocal.

> And values of what?
values of n in
http://en.wikipedia.org/wiki/Fractional_calculus#Fractional_derivative_of_a_basic_power_function

>>6461912

>Sorry if this sounds stupid, but isn't the gamma function supposed to satisfy this? Is the gamma function itself "not unique"?

no. Just add any nonzero analytic function that is zero on the integers (they exist)

>>6461975
sin(2pi*n) I guess

>>6461969
>Fractional Calculus
Goddamnit, man, now I'm remembering the arse-end of my third-year Mathematics lecture course in my Physics degree, where we had a handful of lectures about really bizarre stuff "because it was interesting". I mean, yeah, it WAS interesting, but fractional derivatives? Non-integer dimensional linear algebra? We were convinced he was on drugs. He even spent an hour doing something which seemed to relate to integral equations, but where all the "maths" was just loads of circles with notches on which he was ADDING TOGETHER.

You can imagine the look on our faces when we found out that he was the lecturer for fourth-year Relativistic Quantum Mechanics. We're glad we came out alive.

In seriousness, though, he was a great lecturer. Too fast for some, but I liked him.

OP here,
So I guess what I should take away from this thread is that it is possible for the universe to be non-local?

B-but muh causality! And what about all the physics professors trying to drill it into my head that physics is local, physics is local, physics is local... could they be wrong?

>>6461984
well

http://en.wikipedia.org/wiki/Fractional_quantum_mechanics
http://en.wikipedia.org/wiki/Fractional_Schr%C3%B6dinger_equation
http://en.wikipedia.org/wiki/Fractional_dynamics
Similarly
http://en.wikipedia.org/wiki/Tsallis_entropy
and so on

Related:
>>6461298
catching up with physics in systems without the law of excluded middle.

>>6461996
>could they be wrong?
I think what's actually to take away is that there's currently no *absolutely, undeniably irrefutably compelling* reason to decide either way.

e.g. QM doesn't magically start giving incorrect predictions just by making nonlocal interactions/correlations explicit. Indeed, Bell's Theorem shows us that "local realism" and "counterfactual definiteness" (look up the original papers and elsewhere for definitions of these terms, trying to discuss them here will get VERY muddy, VERY fast) from whatever we say about QM. This is usually taken to mean that QM is nonlocal, but physicists tend to not elaborate further and clarify anything of what they mean for fear of being accused of "metaphysics"; hence few are able to take Bell's Theorem on its own terms.

I always wondered how related the Birch and Swinnerton-Dyer conjecture is to the Riemann zeta function. It's also about some value near its or some related functions singularity, no? Somebody know about it?