5 ms·

How does an infinite sum of integers appear in nature (physics)?

by dskloet 10y ago

fidget 10y ago
Length of the coast of Britain?
- dskloet 10y ago
  That assumes details can be arbitrarily small, which Planck refuted.
  - lake99 10y ago
    Not Planck, but by a few others after him. https://en.wikipedia.org/wiki/Quantum_spacetime https://en.wikipedia.org/wiki/Quantum_spacetime
  - andrepd 10y ago
    Refuted? I thought that hypothesis was far from proven.
  - Dylan16807 10y ago
    We don't have to get anywhere near the Plank scale. If you trace the 'edges' of atoms you'll end up with a finite multiple of a meter-scale or kilometer-scale measurement. And that's the end of the fractal.
    - jakobegger 10y ago
      Also, at an atomic scale, the word "coast" doesn't really make sense any more...
tinco 10y ago
Well I'm not a physicist, but I think at the very least we perceive time (locally) as proceeding in a linear fashion infinitely. My mountain example obviously couldn't happen in the physical world. I suppose in that case you might as well substitute the infinite value for an arbitrary large one. Which is not really what infinite values are about in mathematics, as they are more about describing the (imaginary?) limit of a divergent series. I guess my point is more that for a mathematician it would probably be obvious that when you talk about the limit of a divergent series it could be any imaginary or intermediary value. But for a layman infinite values and infinite series are interpreted as larger than any value you can come up with, and more than any repetition you can write down. So any explanation for this equality should, I think, involve first deconstructing that.
- dskloet 10y ago
  Are you suggesting that time extends infinitely into the past? Because if it doesn't, there will never be an infinite amount of time, so there won't be any point in time when this -1/12 becomes relevant to physics in this sense. > I guess my point is more that for a mathematician it would probably be obvious that when you talk about the limit of a divergent series it could be any imaginary or intermediary value. I'm a mathematician and I might agree (not entirely sure what you mean). But -1/12 is a concrete value so it doesn't apply here.
  - jjnoakes 10y ago
    Why can't time begin at a point in the past, but extend infinitely into the future?
    - tinco 10y ago
      Well it might, but in what calculation would the fact that time extends infinitely be a factor? You can say all past time resulted in the current moment, but as far as I know there's no such thing flowing from the future into now. So any physics calculation you do will deal with finite amounts of time.
      - jjnoakes 10y ago
        I won't pretend to know every possible calculation that might be desired, but I can imagine any sort of prediction or simulation about the future may want to use such an infinite view of time going forward.
inlineint 10y ago
It appears in some quantum field theory calculations, in particular for Casimir Effect. In simple words Casimir Effect consists of a force that emerges between two conductor planes that are parallel to each other. The force is proportional to sum of energies of all possible standing electromagnetic waves between the planes. In calculations for this force a divergent series of sum of all natural numbers (or their powers) appears and physicists use 1 + 2 + ... = -1/12 to calculate it (or continuation of zeta function in other points if appropriate). https://en.wikipedia.org/wiki/Casimir_effect#Derivation_of_Casimir_effect_assuming_zeta-regularization https://en.wikipedia.org/wiki/Casimir_effect#Derivation_of_C... and https://en.wikiversity.org/wiki/Quantum_mechanics/Casimir_effect_in_one_dimension https://en.wikiversity.org/wiki/Quantum_mechanics/Casimir_ef...
- cohomologo 10y ago
  I've never seen a physics book that treats this using the zeta function, except popular articles that try to present this calculation as mysterious. In practice, in my QFT class we needed to compute the sum lim \epsilon -> 0+ ( \sum_{n=1}^\infty n e^{- \epsilon n} + ...), that is the series was multiplied by a decaying exponential function with a rate of decay that goes to zero. This sum can easily be evaluated for small epsilon takes the form sum = 1/epsilon - 1/12 + O(epsilon). The 1/epsilon term (which goes to infinity) drops out of the final physical result when you do the calculation properly.
Manishearth 10y ago
It doesn't mean that there are an infinite number of real things that add up, it means that the mathematics we use to model physics comes up with infinite sums of abstract things. Nevertheless, there are infinite sums of "real" things in physics too. I have put "real" in scare quotes because it turns out they aren't real :) In quantum electrodynamics, the charge of an electron turns out to be infinite. And it turns out that in Real Life, the charge of an electron is indeed infinite. Ish. ... but we know it isn't, right? So what happens is that the real electron gets surrounded by positively charged "virtual" particles. Virtual particles are basically quantum probabilities of a particle appearing out of nowhere with its antiparticle (among other things). So you can say that with some probability, that particle is there. Since there's an electron nearby, the positively charged particle is attracted to the electron, while the negatively charged antiparticle is repelled. This screens the electron charge. With an infinite number of virtual particles, the electron's charge is screened enough to become finite again. Basically, we subtracted two infinities and got something finite. The subtraction done here is called renormalization -- and a similar thing is being done in the -1/12 sum. While mathematics tells us that divergent series can be rearranged to get any "sum", this trick is often used in physics -- provided you can justify that rearrangement. In fact, if you probe an electron hard enough (by bombarding it with other charged particles with tons of energy), its apparent charge increases since the particles used to measure its charge "pierce" the shielding. Of course, this is all really a fancy way of saying that charge itself is energy-dependent, and what we call charge is actually the 0-energy charge. But for modelling purposes, virtual particles work better, and thinking about things in those terms gives a physicist a cleaner abstraction boundary to deal with. You get infinities everywhere, though. This is basically an example of the pattern I'm talking about. Abstractions in the model may have all kinds of infinities popping up. In the real world, these don't really manifest themselves because they're not directly linked to observables. You can apply your model to your detection mechanism to get values for non-observables and say "hey, look, an infinity", but that's really circular logic. The "Real Charge" of an electron isn't something we see. Virtual particles aren't something we see; unless we make them into real particles, but you can't do that to the infinite virtual particles around, so you'll never see an infinity.
tamana 10y ago
Fourier transform, which is used in classical wave theory (optics and electricity) and in quantum mechanics

How does an infinite sum of integers appear in nature (physics)?

by dskloet 10y ago

How does an infinite sum of integers appear in nature (physics)?

fidget 10y ago

Length of the coast of Britain?

dskloet 10y ago
That assumes details can be arbitrarily small, which Planck refuted.
- lake99 10y ago
  Not Planck, but by a few others after him. https://en.wikipedia.org/wiki/Quantum_spacetime https://en.wikipedia.org/wiki/Quantum_spacetime
- andrepd 10y ago
  Refuted? I thought that hypothesis was far from proven.
- Dylan16807 10y ago
  We don't have to get anywhere near the Plank scale. If you trace the 'edges' of atoms you'll end up with a finite multiple of a meter-scale or kilometer-scale measurement. And that's the end of the fractal.
  - jakobegger 10y ago
    Also, at an atomic scale, the word "coast" doesn't really make sense any more...

tinco 10y ago

Well I'm not a physicist, but I think at the very least we perceive time (locally) as proceeding in a linear fashion infinitely. My mountain example obviously couldn't happen in the physical world. I suppose in that case you might as well substitute the infinite value for an arbitrary large one. Which is not really what infinite values are about in mathematics, as they are more about describing the (imaginary?) limit of a divergent series. I guess my point is more that for a mathematician it would probably be obvious that when you talk about the limit of a divergent series it could be any imaginary or intermediary value. But for a layman infinite values and infinite series are interpreted as larger than any value you can come up with, and more than any repetition you can write down. So any explanation for this equality should, I think, involve first deconstructing that.

dskloet 10y ago
Are you suggesting that time extends infinitely into the past? Because if it doesn't, there will never be an infinite amount of time, so there won't be any point in time when this -1/12 becomes relevant to physics in this sense. > I guess my point is more that for a mathematician it would probably be obvious that when you talk about the limit of a divergent series it could be any imaginary or intermediary value. I'm a mathematician and I might agree (not entirely sure what you mean). But -1/12 is a concrete value so it doesn't apply here.
- jjnoakes 10y ago
  Why can't time begin at a point in the past, but extend infinitely into the future?
  - tinco 10y ago
    Well it might, but in what calculation would the fact that time extends infinitely be a factor? You can say all past time resulted in the current moment, but as far as I know there's no such thing flowing from the future into now. So any physics calculation you do will deal with finite amounts of time.
    - jjnoakes 10y ago
      I won't pretend to know every possible calculation that might be desired, but I can imagine any sort of prediction or simulation about the future may want to use such an infinite view of time going forward.

inlineint 10y ago

It appears in some quantum field theory calculations, in particular for Casimir Effect. In simple words Casimir Effect consists of a force that emerges between two conductor planes that are parallel to each other. The force is proportional to sum of energies of all possible standing electromagnetic waves between the planes. In calculations for this force a divergent series of sum of all natural numbers (or their powers) appears and physicists use 1 + 2 + ... = -1/12 to calculate it (or continuation of zeta function in other points if appropriate). https://en.wikipedia.org/wiki/Casimir_effect#Derivation_of_Casimir_effect_assuming_zeta-regularization https://en.wikipedia.org/wiki/Casimir_effect#Derivation_of_C... and https://en.wikiversity.org/wiki/Quantum_mechanics/Casimir_effect_in_one_dimension https://en.wikiversity.org/wiki/Quantum_mechanics/Casimir_ef...

cohomologo 10y ago
I've never seen a physics book that treats this using the zeta function, except popular articles that try to present this calculation as mysterious. In practice, in my QFT class we needed to compute the sum lim \epsilon -> 0+ ( \sum_{n=1}^\infty n e^{- \epsilon n} + ...), that is the series was multiplied by a decaying exponential function with a rate of decay that goes to zero. This sum can easily be evaluated for small epsilon takes the form sum = 1/epsilon - 1/12 + O(epsilon). The 1/epsilon term (which goes to infinity) drops out of the final physical result when you do the calculation properly.

Manishearth 10y ago

It doesn't mean that there are an infinite number of real things that add up, it means that the mathematics we use to model physics comes up with infinite sums of abstract things. Nevertheless, there are infinite sums of "real" things in physics too. I have put "real" in scare quotes because it turns out they aren't real :) In quantum electrodynamics, the charge of an electron turns out to be infinite. And it turns out that in Real Life, the charge of an electron is indeed infinite. Ish. ... but we know it isn't, right? So what happens is that the real electron gets surrounded by positively charged "virtual" particles. Virtual particles are basically quantum probabilities of a particle appearing out of nowhere with its antiparticle (among other things). So you can say that with some probability, that particle is there. Since there's an electron nearby, the positively charged particle is attracted to the electron, while the negatively charged antiparticle is repelled. This screens the electron charge. With an infinite number of virtual particles, the electron's charge is screened enough to become finite again. Basically, we subtracted two infinities and got something finite. The subtraction done here is called renormalization -- and a similar thing is being done in the -1/12 sum. While mathematics tells us that divergent series can be rearranged to get any "sum", this trick is often used in physics -- provided you can justify that rearrangement. In fact, if you probe an electron hard enough (by bombarding it with other charged particles with tons of energy), its apparent charge increases since the particles used to measure its charge "pierce" the shielding. Of course, this is all really a fancy way of saying that charge itself is energy-dependent, and what we call charge is actually the 0-energy charge. But for modelling purposes, virtual particles work better, and thinking about things in those terms gives a physicist a cleaner abstraction boundary to deal with. You get infinities everywhere, though. This is basically an example of the pattern I'm talking about. Abstractions in the model may have all kinds of infinities popping up. In the real world, these don't really manifest themselves because they're not directly linked to observables. You can apply your model to your detection mechanism to get values for non-observables and say "hey, look, an infinity", but that's really circular logic. The "Real Charge" of an electron isn't something we see. Virtual particles aren't something we see; unless we make them into real particles, but you can't do that to the infinite virtual particles around, so you'll never see an infinity.

tamana 10y ago

Fourier transform, which is used in classical wave theory (optics and electricity) and in quantum mechanics