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It is an opinion I occasionally hear, and perhaps hold myself, that the resolution to the 'infinities' that crop up in various bits of physics are artefacts of the approximation that space-time is continuous. If at some base granularity things are discrete then there is a natural cut-off. This also appeals from an information perspective as continuity is usually expressed via some reference to the real numbers, most of which contain an infinite amount of information (I'm thinking here of the criticisms raised in Gisin's work without committing to his more radical proposals). Having fundamental discreteness avoids the criticism that some finite region could contain infinite amounts of information.

The same people who say this (including perhaps myself) also seem rather comfortable with the fundamentality of the Lorentz group (I here mean $SL(2;\mathbb{C})$) and more crucially its irreducible unitary representations. Unlike those of $SU(2)$ (i.e spins) which are indexed by natural numbers, the Lorentz irreps are indexed by $(\rho,m)$ where $\rho\in \mathbb{R}$ and $m\in \mathbb{Z}$. It would seem then that these irreps contain can contain 'an infinite amount of information' and so when seen as localised entities (I'm thinking of them as EPRL like spin-foam edges/vertices) they seem equally suspect.

Is there a reason why the appearance of the reals in the irreps of the Lorentz group should be thought of in a different way to the appearance of the real numbers in other areas of physics, particularly from a quantum information perspective?

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  • $\begingroup$ I can't quite grasp your question, but I will point out the the presence of infinities in a theory can also be the result of an incomplete theory or model. $\endgroup$
    – garyp
    Commented Jun 20, 2022 at 20:09
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    $\begingroup$ The question seems to be conflating technical problems related to divergences in physical models, a philosophical inquiry about the nature of real numbers (Gisin's work is about determinism), and the nature of information in physics. This makes it very open ended and with no clear actual subject, so I vote to close under being opinion-based. $\endgroup$
    – Stéphane Rollandin
    Commented Jun 21, 2022 at 7:29
  • $\begingroup$ I think there is a misunderstanding here, indeed Gisin's work is about determinism (this is where I mention his radical proposals) but he has a critique about the use of real numbers in terms of the ability encode them in a finite volume and that is what I'm referencing (see section 4 of arxiv.org/abs/1803.06824 - probably I should have been clearer). I also don't believe I'm conflating technical problems and philosophy, in the sense of an accidental error. I am alluding to the possible resolution of technical problems via a philosophical perspective, this is to my eyes quite concrete. $\endgroup$
    – East
    Commented Jun 21, 2022 at 16:53
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    $\begingroup$ "...this is to my eyes quite concrete." But, you need to make it concrete to our eyes, since we are the ones answering the question. Unless you don't want an answer. $\endgroup$
    – hft
    Commented Jun 28, 2022 at 18:10
  • $\begingroup$ No to your final paragraph punchline question. You failed to argue otherwise. $\endgroup$
    – Cosmas Zachos
    Commented Jul 4, 2022 at 19:28

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The UV infinities in perturbative QFT is a mathematical artifact coming from the indiscriminate use of distributions. In causal perturbation theory, where more careful use of distributions is made, these UV infinities vanish.

As Nlab point out:

A key idea of causal perturbation theory is that the interaction term $V$ is considered multiplied by some smooth function which has compact support on spacetime. This hence serves as a spacetime dependent "coupling constant" which "switches off" the interaction term outside a compact region, but not discontinuously as in many other schemes, but smoothly, hence "adiabatically" in terminology borrowed from thermodynamics...

... It may be argued, vividly so by Scharf [Finite Electrodynamics] ... that the notorious "infinities" that "plague" QFT in other approaches are nothing but the result of incorrectly dealing with the extension of distributions ...

The key idea in causal perturbation theory is causal locality and this in turn means that the adiabatic switching functions have spacelike separated supports.

I would suggest the idea of the perspective of information theory as being physically badly motivated, or at least very badly named. The universe is not a computer processing 'information'.

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    $\begingroup$ In fact, infinities are not the problem, the problem is that the use of local interactions gives rise to ambiguously defined operations with distributions. These ambiguities are the finite renormalization terms. It is not necessary going to infinity and coming back to finite to get them. $\endgroup$
    – Valter Moretti
    Commented Jun 28, 2022 at 16:50
  • $\begingroup$ @Valter Moretti: Why are you quoting back my answer to me? I already said that "the UV infinities ... come from the indiscriminate use of distributions". I find this baffling. Did you read my post? $\endgroup$
    – Mozibur Ullah
    Commented Jun 28, 2022 at 17:43
  • $\begingroup$ @Valter Moretti: Are you saying that I think that UV infinities are the problem when I manifestly said they weren't? $\endgroup$
    – Mozibur Ullah
    Commented Jun 28, 2022 at 17:59
  • $\begingroup$ No, I agree with you! I am saying with you that they are not the problem, just a complicated way to approach the problem…I wrote my comment in support to your answer…+1 $\endgroup$
    – Valter Moretti
    Commented Jun 28, 2022 at 18:28
  • $\begingroup$ Thanks for the answer and it is interesting to hear that closer attention to distributions can also potentially remove UV infinities. I do think however that the core of this question (though I accept I motivated it by QFT) is really about the use of the reals in the Lorentz irreps and how this relates to the info perspective. I also think that the 'information' perspective is misunderstood here. The universe doesn't need to be a computer for us to find that an infinite amount of information, which is physical and must be encoded in physical systems, in finite space is suspect. $\endgroup$
    – East
    Commented Jun 29, 2022 at 11:45
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Our current physics is founded upon continuous variables, and so can quantum information theory. It's really up to advocates of fundamental discreteness to show how they implement e.g. rotational symmetry, and everything else

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