The physics of infinity

https://www.nature.com/articles/s41567-018-0238-1.epdf?shared_access_token=nhIyZJldj4QzWZs7LvZIttRgN0jAjWel9jnR3ZoTv0PMOaEWTfe0Iq_Ol3Eo9bd6Lh9xyPK-ya44kxWDxYi4IQo2Zqj-Ymd6yZVANNbW9FXmT1HwoVMnEtM00qpXT48gLDqpQXX3mvS3gRH22aRhLs-Cf_4dd6NkVcLZZP3rPbg%3D

by okket • 8 years ago

101 93 8 years ago

nature.com

techno_modus2• 8 years ago

David Hilbert famously argued that infinity cannot exist in physical reality.

This statement reduces the problem to the definitions of "existence" and "physical reality" which of course may have quite different interpretations. For example, the existence of particles (with finite mass and finite coordinates which can be "touched") and the existence of waves are rather different notions.

subroutine2• 8 years ago

What aspect of waves as they exist in our universe is infinite?

whatshisface5• 8 years ago

If waves are continuous then they are over the real numbers, and there are an infinite number of reals between any two points.

jamesrcole6• 8 years ago

It seems to me that something can only have existence for us if it can have a transitive causal influence upon us, and surely both particles and waves exist in that same sense.

whatshisface5• 8 years ago

If by existence you also mean, "something that continues to exist upon closer inspection," then neither particles nor waves exist, you can always design an experiment to refract something around a corner and meausre individual quanta of it when it gets to the detector.

openfuture2• 8 years ago

Then this is easy, we are discussing the concept infinity, hence it is having a causal influence on our actions and so by your definition it exists.

nobody_nowhere• 8 years ago

Statements like this always bring me back to Goedel's incompleteness theorem...

mannykannot• 8 years ago

I don't get the dual argument about zero, for which the impossibility of a completely empty region of spacetime is given as an example. That may be so, but a region empty of countable things like atoms seems physically conceivable. I am writing from a room containing zero elephants.

techno_modus2• 8 years ago

Your room and elephants are different objects. Your room has some size and it is not empty but it has zero elephants or a finite number of zero size elephants.

0 = nothing = non-existence.

whatshisface5• 8 years ago

The complete wave function of your room contains some elephant-like fluctuations in the distant decimal places.

qubex3• 8 years ago

I have met the Queen of England exactly zero times.

danpat• 8 years ago

A functioning brain is posited to spontaneously appear in about 10^10^50 years due to random fluctuations.

https://en.wikipedia.org/wiki/Boltzmann_brain

I'm not sure if a functioning elephant is more or less likely to be honest....

throwaway375854• 8 years ago

What's the rest mass of a photon? What's the electric charge of a neutrino? What's the spin of a Higgs boson?

okket2• 8 years ago

Rules of the macroscopic world do not apply to the subatomic (or even molecular) world. There are no two identical elephants in the world. On the other hand, you will have a hard time to tell two electrons or even two molecules of the same configuration apart from another.

A total empty region, void of any atoms etc., still contains 'quantum noise' and thus virtual particles. The world, the universe as we know it so far has become inconceivable without this observation, known as the uncertainty principle.

I find the argument quite convincing.

naasking2• 8 years ago

The world, the universe as we know it so far has become inconceivable without this observation, known as the uncertainty principle.

Not really, deterministic interpretations of QM don't require such mental contortions.

NihilumExNil4• 8 years ago

Old shaky hands aiming a looking glass at one of two moving elephants standing close by each other would tell a different story, perhaps.

I'm not sure how well established Virtual Quantum Noise is, but saying that an empty place contains this is just wrong. If it has an effect in the propagation of a real particle in to said space, it wouldn't be empty anymore. That's confusing, and it might be helpful to note that these quantum fluctuations happen on the boundaries of the space, not just everywhere (and if they did, they wouldn't matter any more than my imaginary friend the pink elephant Safr riding the fluctuations).

You are talking of black holes, by the way, aren't you?

nabla9• 8 years ago

region empty of countable things like atoms seems physically conceivable

As far as I know this is not physically conceivable, if physically conceivable means physical laws as we understand them.

Quantum vacuum state, or zero-point field is not completely void empty.

peter303• 8 years ago

Somewhere in an infinite number of multiverses there is a room with both you and an elephant.

Koshkin4• 8 years ago

There are even those in which you are the elephant.

Senderman• 8 years ago

My [lay] understanding is that a portion of spacetime has yet to be found that has 0 countable things. I also believe 'countable' gets tricky the more you zoom in, as it were.

bena• 8 years ago

I think that's the craziest thing, every time we get smaller, we find more stuff. "Atom" means "indivisible". It was supposed to be the smallest unit of matter.

Then we found things smaller. Sub-atomic particles. Literally makes no sense because it literally means smaller than the smallest thing.

The universe is apparently fractal af.

mikec30102• 8 years ago

How would you eliminate electromagnetic fields or inconceivably small particles like neutrinos from entering?

misja111• 8 years ago

I find the article a bit pointless. The concept of infinity is made up by men. Just like the concept of numbers or logic. There doesn't exist something like a 'four' in reality, this only exists in models that we have created.

Therefore it goes without saying that there is no 'infinity' in physical reality. On the other hand, nothing is keeping me from creating some model of reality that has infinity in it.

jacobmoe• 8 years ago

"4" is a concept, but you don't deny that the 4 turnips on your table do have physical reality, do you? Is the notion that there are an infinite number of turnips spread out through an infinite universe "just a concept" whose physical reality we don't have to bother ourselves with, in a way that those 4 turnips aren't?

zaarn• 8 years ago

I will not deny that a collection of atoms, mostly carbon, hydrogen and oxygen, with the rough weight of 4 turnips, is on the table and looks like 4 turnips.

The 4 turnips do have physical reality but on the other hand, the 4 in this scenario is not a realized concept, there is nothing physical about the 4 in this scenario since all the things in it are made of particles made of mostly nothing.

simonh4• 8 years ago

The point in contention isn't that infinities can be imagined, it is whether they are useful - specifically in the context of cosmology.

Numbers are clearly useful, otherwise it would be difficult to explain the importance we ascribe to the value of our bank balances. However just because you can formulate a model of reality including infinities it does not therefore follow that this model must be useful, or that infinities must be useful.

cjohansson• 8 years ago

usefulness is a pragmatic attribute, it does not by itself say that something really exist in reality, unless you believe that everything that is useful exists. I think the OP do not believe that everything that is useful exists in physical reality, therefore seeing four tables does not mean that fours exists in physical reality separated from tables or other objects. So you are probably using different metaphysics..

But sure, since the days of William of Occam pragmatic metaphysics have been popular, but again popularity does not necessary prove something is right. Just explaining both sides of the argument here and trying to resolve the potential missunderstandings

jerf• 8 years ago

"The point in contention isn't that infinities can be imagined, it is whether they are useful - specifically in the context of cosmology."

Going the other way may be more useful; is it possible to create a correct model of reality that does not at any point use infinities? You could have alternate correct models that do, but if you can create one without them you'd have a reasonable basis to call the infinities in the other models unnecessary and nonexistent.

(I would suggest before anyone rush to correct me that they read what I said carefully. For instance, "reasonable basis" != "undeniable proof", which experience leads me to believe is a difference a hypothetical replier might, in their rush to reply, overlook. Also note the two clauses in the question; if it is simply impossible to create a correct model of reality at all, we may never get to the "infinity" portion of the question.)

andyjohnson0• 8 years ago

From the article:

There is a duality between zero and infinity, expressed in the elementary identity 1/0 = ∞.

I understand what they're saying here - but I was always under the impression that, rather than yielding infinity, division by zero was undefined. Wikipedia [1] says that this is true "In ordinary arithmetic", but goes on to say that it is "sometimes useful" to think of a "formal calculation" involving division by zero as evaluating to infinity. A formal calculation seems to be one that ignores whether the result is well defined.

Would anyone care to comment on whether the mathematical perspective above carries over into the physics domain?

[1] https://en.wikipedia.org/wiki/Division_by_zero

throwawaymath3• 8 years ago

Division by 0 is not possible in fields [1], which are the algebraic structure most commonly associated with “ordinary arithmetic.” For example, the real numbers comprise an ordered and complete field. You cannot divide by 0 in a field essentially by definition, so it’s undefined.

However, you can define meaningful algebras in which division by 0 is not only trivially allowed, but also equal to infinity. This is possible by extending the complexes, for example. More generally, see [2].

Note I’m speaking strictly of the math here. I can’t comment on this usage in physics specifically. But there’s no issue with defining 1/0 = ∞ if the rest of the theory remains fundamentally consistent.

______

1. https://en.m.wikipedia.org/wiki/Field_(mathematics)

2. https://en.m.wikipedia.org/wiki/Wheel_theory

fusiongyro2• 8 years ago

I disagree, only because infinity is not an element of the real or the complex numbers. You can make extended number systems (the hyperreals, for instance) but it isn't commonly done. Infinity as a notation in the limit is not the same thing as infinity as an element of a set.

CraneWorm2• 8 years ago

the result is well defined in terms of a limit: lim 1/x as x -> 0 = ∞. I consider 1/0 = ∞ a practical shortcut notation.

empath752• 8 years ago

Except it’s also negative infinity.

okket2• 8 years ago

You could think of approaching this result by dividing 1 by an ever smaller and smaller number and the result is getting bigger and bigger, with no end in sight, as you can always come up with a smaller number, never reaching 0.

There are several places in physics where you encounter infinities, the most prominent case probably is the aforementioned uncertainty principle, which leads to 'loops' in QED and need to be countered by renormalization[0].

[0] https://en.wikipedia.org/wiki/Renormalization

gowld2• 8 years ago

You could say that is impossible to start from something finite and completely destroy it to 0, and this analogous to the impossibility of creating infinite pieces. But another option is that there is simply nothing to start with.

ljcn• 8 years ago

It's been a long time since I've done any physics and there is absolutely no rigour here, but the (extremely useful) Dirac delta function comes to mind. You can think of it as being infinitely tall and infinitely thin, but with unit area. If you tried to divide the "area" (1) by the "width" (0), you'd get the height (infinity).

foxes• 8 years ago

Division by zero can certainly be given meaning in other mathematical contexts. The real projective line is one example [0]. There are other algebraic structures in ring theory that also have a division by zero, for example a ring with only a zero element. So I would say it depends on your context if something makes sense to do. You aren't ignoring or doing something ill defined, its the opposite (in mathematics). You can do it if its consistent mathematics.

Physicists have very good intuition but sometimes making their statements mathematically rigorous requires extra work. I think a good example is in the history of field theory. Over time the intuition was made into something more formal. The article in the OP discusses renormalisation.

[0] https://en.wikipedia.org/wiki/Projectively_extended_real_lin...

singularity2001• 8 years ago

~$ node

1/0

Infinity

Q.E.D.

unless you want to claim that javascript is occasionaly illdefined.

mccosmos• 8 years ago

It is 0/0 that is undefined. For 1/0 the standard is either infinity (easily checkable in a language like JavaScript or C) or you get the usual talk that you can't mix infinity with numbers and the operation would be meaningless, etc.

foobar1962• 8 years ago

1/0 is not infinity, it’s undefined.

For the function 1/x as x->0 from the positive side the function tends to positive infinity, while as x->0 from the negative side the function goes to negative infinity.

The function cannot be both plus and minus infinity at the same time.

opmac• 8 years ago

While you are technically correct, that's pedantic and doesn't impact the conclusions of this article one bit, nor does it really add to the discussion. It's written for the layman, and chose to purposefully ignore technical details such as lim x->0+ 1/x = +inf and lim x->0- 1/x = -inf.

Koshkin4• 8 years ago

Sure there is an infinity without a sign.

j1vms• 8 years ago

AFAIK, p-adic quantum mechanics has shown some promise in treating aspects of the "physics of infinity".

It is the application of p-adic analysis to quantum mechanics.

[0] https://en.wikipedia.org/wiki/P-adic_quantum_mechanics

curlcntr• 8 years ago

"There is a duality between zero and infinity, expressed in the elementary identity 1/0 = ∞ . If one side of the duality does not occur in nature, also the other side ought not to."

It wasn't clear to me that the second sentence was sufficiently proved. At first glance it seems reasonable, but the authors went a bit too fast past that claim for me.

JackFr• 8 years ago

I don't think it's proved at all. "Ought" is an appeal to symmetry. At the same time I don't think that necessarily undermines the argument.

jkabrg2• 8 years ago

The dual numbers contain infinitesimal numbers but not infinite numbers.

Oppositely, the variable X of a polynomial behaves much like an infinity, but is not invertible.

Their argument depends crucially on whether infinitesimals/infinities are invertible.

monster_group• 8 years ago

The article doesn't load for me in Firefox or Chrome. It's stuck at 'Loading enhanced PDF...'.

maaark• 8 years ago

Doesn't load for me either. Stuck at 'pay us money'.

everdev• 8 years ago

Infinity in nature sounds like an untestable argument. If space was infinite, by definition we'd never be able to measure it's boundary to verify the claim. The closest we could come is "we still haven't discovered a boundary to the volume of space".

Basically the claims "beyond the farthest point you can measure there is more space" and "beyond the farthest point you can measure there is nothing" are equally plausable and untestable.

psychometry• 8 years ago

If you had read the article, you'd learn that the problem is much richer and more complicated than the way you stated it.

pphysch4• 8 years ago

The more sensible question, IMO, is "Is the universe fundamentally quantized?". Can you concretely say where the boundary between an apple and the surrounding air is? Is a particular electron on the boundary part of the apple or the air?

If the universe is not truly quantized, then there is nothing to count. By extension, mathematics is not related to physical reality, and it makes no sense to apply mathematical notions of infinity to the universe in such speculative ways.

CompelTechnic• 8 years ago

Even if the universe is not discretized, there is no reason to completely discount the value of mathematics. Continuous probability distributions of particle locations still have meaning, individual entities can still be disambiguated as local maxima of density gradients, counting can still be built up from raw axioms and set theory.

Even for people don't accept the "monism" (to use a metaphor to the monism/ dualism debate of the human mind and body) of physics and mathematics, these same people cannot avoid accepting that correctly applied theoretical manipulations of mathematical symbols jive perfectly well with the predicted, physical outcome they represent. Does claiming that the numbers don't mean anything mean anything?

swerveonem• 8 years ago

Well said, I wish I knew you in real life.

pphysch4• 8 years ago

I don't know where you read that I "completely discount the value of mathematics".

"...cannot avoid accepting that correctly applied theoretical manipulations of mathematical symbols jive perfectly well with the predicted, physical outcome they represent."

Close, except for "perfectly". Mathematics and other scientific models are approximations of reality and nothing more. They are fantastic tools but don't be misled into thinking they give a "perfect" picture.

throwaway375854• 8 years ago

If the universe is not truly quantized... mathematics is not related to physical reality

How does that follow? Mathematics is perfectly capable of modeling continuous quantities.

pphysch4• 8 years ago

Yes, math can model "continuous quantities", but is the idea of "quantity" anything more than a human abstraction? How do you define quantity, and can you do it without the notion of counting and discrete objects?

The physics of infinity

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