Is continuity the fundamental issue in discrete space theories?

[bhanum]

TL;DR

I want to know whether the mathematical apparatus of calculus is preventing the widespread adoption of discrete / quantized space theories.

Hello

I seem to have come under the impression that discrete space, or the breakdown of physics at the so called quantum foam level, is a result of the math used to model physical reality. Am I incorrect in assuming this, or is there validity in the claim that calculus, requiring limits at infinity/zero, is the reason discreteness in the fundamental structure of space time is generally discredited?

Thanks
Bhanu

 

[Ibix]

The reason no theory has replaced GR is that we can't find any clear cases of GR making incorrect predictions in regimes we can test or observe, so we currently have no evidence to use to choose between the many possible next generation theories. In the case of a discrete-space theory, this means that the error from assuming that space is actually continuous seems to be so small that we can't detect it. So we carry on modelling it continuous until some measurement shows a discrepancy with theory.

 

[bhanum]

Ok. It is good to know that continuity can be ignored for the sake of calculus, provided the delta we are talking about is really small, say on the Planck length scale. If there are mathematicians in the community looking at this, would someone please care to comment on this issue as well. Particularly, would I need to look at some discrete version of calculus which has been adapted for breaks in the continuum or can I ignore the continuity aspect completely as long as there isn't any deviation from observation? This is just a theoretical question, which may be of interest to others as well.

Some quick questions:

• Would you say it is worth working on some theory if preliminary results pop out some numbers that seem to match observation?
• Have you come across many such dead ends where preliminary results may look promising, but then further work reveals inconsistencies that cannot be reconciled?

I ask this because I have started to come up with a discrete-space theory myself, reasoning it out with AI, and some numbers have popped out. I am trying to learn both the existing frameworks as well as see how things may be otherwise in my framework. What is the physics community's opinion on using AI for research?

Thanks
Bhanu

 

[Matterwave]

or is there validity in the claim that calculus, requiring limits at infinity/zero, is the reason discreteness in the fundamental structure of space time is generally discredited?

We have many methods to deal with discrete spaces.

A lot of math deals with discrete objects. The simplest example is the natural numbers or the integers. An advanced example is the Regge Calculus: https://en.wikipedia.org/wiki/Regge_calculus

At the end of the day, calculus is just a tool. It doesn't block us from using a different tool to deal with discrete spaces if we really think the universe is discrete. The physics leads the math here.

 

[Matterwave]

If discrete quantum gravity is your interest, you may check:

https://en.wikipedia.org/wiki/Loop_quantum_gravity
https://en.wikipedia.org/wiki/Causal_sets

The beyond the standard model forum might be more appropriate for this question.

 

[bhanum]

Is my understanding correct that Regge calculus is discretizing space from a GR perspective? Can it also be applied from a first principles point of view where GR hasn't been integrated yet? If, in some universe, discrete space existed, then what is the correct way to deal with continuous quantities in that universe? That is essentially what I'm asking in the other post.

I will take your cue and maybe post next time about such stuff in the beyond the standard model forum.

 

[PeroK]

Is my understanding correct that Regge calculus is discretizing space from a GR perspective? Can it also be applied from a first principles point of view where GR hasn't been integrated yet? If, in some universe, discrete space existed, then what is the correct way to deal with continuous quantities in that universe? That is essentially what I'm asking in the other post.

I will take your cue and maybe post next time about such stuff in the beyond the standard model forum.

Differential Calculus is the limit of small changes; and, the Integral calculus is the limit of large sums. In principle, you can do everything mathematically with sufficiently good approximations. In fact, most differential equations are solved using approximate "numerical" methods.

The problem is finding the physical theories. Not finding a way to do finite mathematics.

Moreover, even if a quantity is discrete, it can often be modelled by a continuous function. E.g. population or food prices.

 

[WWGD]

Is my understanding correct that Regge calculus is discretizing space from a GR perspective? Can it also be applied from a first principles point of view where GR hasn't been integrated yet? If, in some universe, discrete space existed, then what is the correct way to deal with continuous quantities in that universe? That is essentially what I'm asking in the other post.

I will take your cue and maybe post next time about such stuff in the beyond the standard model forum.

Are yóu trying to define Integrals, Derivatives in a discrete space?

 

[bhanum]

Differential Calculus is the limit of small changes; and, the Integral calculus is the limit of large sums. In principle, you can do everything mathematically with sufficiently good approximations. In fact, most differential equations are solved using approximate "numerical" methods.

The problem is finding the physical theories. Not finding a way to do finite mathematics.

Moreover, even if a quantity is discrete, it can often be modelled by a continuous function. E.g. population or food prices.

That is heartening. The problem does lie in finding the physical theories, and also one or rather anyone learning or trying to come up with their own theory must be familiar with a large array of mathematical tools.

While I do know that things like population are modeled using continuous functions, I think I need to take another look at how it works as per the definitions.

 

[bhanum]

Are yóu trying to define Integrals, Derivatives in a discrete space?

Not yet

 

[PeroK]

Are yóu trying to define Integrals, Derivatives in a discrete space?

The discrete equivalent of the integral is a sum.

 

[Dale]

I have started to come up with a discrete-space theory myself, reasoning it out with AI,

Do not post it here. We require that all posts be consistent with the professional scientific literature. All new theories must be published in the professional scientific literature before we can discuss them here.

 

[Dale]

It doesn't block us from using a different tool to deal with discrete spaces if we really think the universe is discrete.

You don’t even have to think the universe actually is discrete. All you have to do is show that your discrete math matches observation.