A Infinities in QFT (and physics in general)

[Fra]

Difficulty with Infinity as “just another math widget” aside, I wonder if continuum assumptions are a barrier to answers.

The first kind of difficulty is not what I think anyone else in thread referred to though, it was more the latter thing you mention.

Demystifier for example referred to this paper by Baez in post #19 with a discussion

Struggles with the Continuum
"We have seen that in every major theory of physics, challenging mathematical questions arise from the assumption that spacetime is a continuum.The continuum threatens us with infinities. Do these infinities threaten our ability to extract predictions from these theories—or even our ability to formulate these theories in a precise way? We can answer these questions, but only with hardwork. Is this a sign that we are somehow on the wrong track? Is the continuum as we understand it only an approximation to some deeper model of spacetime? Only time will tell. Nature is providing us with plenty of clues, but it will take patience to read them correctly."
-- https://arxiv.org/abs/1609.01421

/Fredrik

 

[martinbn]

I have to say that the discussion whether the derivative of a function can be defined using finite small differences is very strange. If you think so, then you are missing the whole point of analysis. If, on the other hand, you mean the rate of change of a physical quantity, then you may have a point. But derivatives!

 

[gentzen]

In an old conversation between Werner Heisenberg and Carl Friedrich von Weizsäcker, a compromise is offered: "die Vergangenheit ist in einem gewissen Sinne diskret, und die Zukunft ist kontinuierlich" (in a certain sense, the past is discrete, and the future is continuous). The subsequent elaboration makes is clear that they use the intuitionistic conception where the continuous future is only potentially infinite, instead of being a completed (uncountably) infinite set.

 

[physicsworks]

But continuity or differentiability at that scale is also pure fiction; the noise in the values may be much bigger and would render the quotient meaningless.

And how does the absence of these help you with your argument? Also, before we even touch on differentiability, to talk about continuity you need the definition of a limit, which on such scales: a) may not exist in principle (in the usual epsilon-delta sense) and thus may need to be replaced with something else; b) if the usual definition is applicable, the limit as such may not exist due to some selection rule.

I have to say that the discussion whether the derivative of a function can be defined using finite small differences is very strange. If you think so, then you are missing the whole point of analysis.

No one here, I believe, is talking about that. The point is that analysis may not be the right mathematical language to describe nature at very small scales. After all, if spacetime is doomed and, as some believe, quantum mechanics together with it will emerge from something more fundamental, why should analysis survive?

 

[A. Neumaier]

Noether's theorem and conservation laws exist also on lattices. See arxiv:1709.04788 (of course, this uses also real numbers, which nobody considers to be a problem.)

This paper is still based on differential equations since it keeps continuous time and uses lattices only for the spatial part. Thus your observation not in conflict with my statements

Both Noether's theorem and conservation laws presuppose differential equations, hence real numbers.

On the basis of the success of the continuum methods together with Ockham's razor, it is therefore safe to assume that for practical purposes, the laws of nature are based on differential equations. At least all know laws are formulated in this way for several centuries, and there is no sign that this would have to change.

 

[Interested_observer]

“Many physicists believe that the so-called strict definition of derivatives and integrals is not necessary for a good understanding of differential and integral calculus. I share their point of view."

That begs the question - what is a "good" understanding?

 

[PeroK]

my understanding of calculus is that it depends on the rules of convergence at infinite limits. I like the convergence part. I don’t like the introduction of something as as bizarre as “infinity” it’s like fundamentally undefinable except in the utterly abstract sense. Always found that frustrating.

Your understanding is wrong. The limit is defined using entirely the properties of finite numbers. That was the whole point of the rigorous mathematics developed in the 19th century: to remove any dependence on undefinable concepts.

 

[PeroK]

why should analysis survive?

Because mathematics, and analysis in particular, is not dependent on physical theories. If the universe turns out to be finite, that doesn't mean that there are only finitely many prime numbers, for example.

Take the mathematical models of the spread of the COVID virus. The number of people infected can only be a whole number; but, it can still be modeled effectively using differential equations and all the power of calculus.

 

[Demystifier]

That begs the question - what is a "good" understanding?

The one that achieves the goal, in this case the goal of making physics computations the predictions of which agree with observations. In that sense, Newton's understanding of calculus and Dirac's understanding of his delta function was good.

 

[A. Neumaier]

The one that achieves the goal, in this case the goal of making physics computations the predictions of which agree with observations.

This is only part of the goal. The real goal is to understand Nature in a way that allows us to make the best use of it. This needs much more than just making physics computations the predictions of which agree with observations.

 

[Demystifier]

This is only part of the goal. The real goal is to understand Nature in a way that allows us to make the best use of it.

What you call the "real" goal, is a part of the goal too. In fact, there is no such thing as the "real goal". Different people at different times have different goals. An engineer, an experimental physicist, a theoretical physicist, a mathematical physicist, a pure mathematician and a philosopher may have different goals.

 

[A. Neumaier]

In fact, there is no such thing as the "real goal". Different people at different times have different goals.

Everyone speaks for himself. For me, there is a real goal, and whatever I write here is my view.

 

[physicsworks]

because mathematics, and analysis in particular, is not dependent on physical theories.

Exactly! So in the current form it may not be the language that physicists use to describe nature at small scales, where we need a different physical theory. That's all I'm saying. Take the amplituhedron and similar promising constructions as an example.

 

[physicsworks]

Take the mathematical models of the spread of the COVID virus. The number of people infected can only be a whole number; but, it can still be modeled effectively using differential equations and all the power of calculus.

This is of course true, but irrelevant to the discussion of mathematical language of physical laws at fundamentally small scales, which was the context of the text you quoted. No one here is disputing the longevity of calculus or any other branches of mathematics.

 

[Jimster41]

Your understanding is wrong. The limit is defined using entirely the properties of finite numbers. That was the whole point of the rigorous mathematics developed in the 19th century: to remove any dependence on undefinable concepts.

Taken from wiki on “Limit of a function”.

“Formal definitions, first devised in the early 19th century, are given below. Informally, a function fassigns an output f(x) to every input x. We say that the function has a limit L at an input p, if f(x) gets closer and closer to L as x moves closer and closer to p. More specifically, when f is applied to any input sufficiently close to p, the output value is forced arbitrarily close to L. On the other hand, if some inputs very close to p are taken to outputs that stay a fixed distance apart, then we say the limit does not exist”

only in the wiki the words “sufficiently” and “arbitrarily” are italicized. Rightly so?

On the other hand see Planck et al.

 

[PeroK]

Taken from wiki on “Limit of a function”.

“Formal definitions, first devised in the early 19th century, are given below. Informally, a function fassigns an output f(x) to every input x. We say that the function has a limit L at an input p, if f(x) gets closer and closer to L as x moves closer and closer to p. More specifically, when f is applied to any input sufficiently close to p, the output value is forced arbitrarily close to L. On the other hand, if some inputs very close to p are taken to outputs that stay a fixed distance apart, then we say the limit does not exist”

only in the wiki the words “sufficiently” and “arbitrarily” are italicized. Rightly so?

That's explicitly an "informal" explanation. The formal definition makes things rigorous, explicit and dependent only on the properties of (finite) real numbers.

 

[Jimster41]

I get that you can keep the notion of “tangent point” on a curve without shrinking the length of the deltas all the way to 1/infinity by invoking “embedded affine sub spaces”… so the tool is well defined, and wildly useful. I am not suggesting it isn’t, or that people shouldn’t learn it fapp. But don’t those notions (foliations of the continuum etc.) have their own problems?

So, real question: what about two tangent builders converging across the surface of a sphere getting way more and more tiny step-wise heading toward 1/infinity but always avoiding it by setting up affine sub spaces as they go. Where do they meet? What non-infinitesimal real valued “point” What guarantees the resulting tangent point is a point? Is it a single affine sub space? If so how does it go from two to one? How come they don’t just end up in a fight to the death down to 1/infinity trying to own the tangent point? Or do they form a stable harmonic oscillator? Or does the sphere provide the discrete foliation at the end of the day, where they somehow meet for non-infinite coffee?

 

[PeroK]

A tangent to a curve is a line like any other line. Two lines intersecting in a single point is not mathematically problematic.

There's no issue with never having learned rigorous mathematics, but it is an issue if you try to tell those of us who have that we're crazy and imagining "flying elephants".

 

[Jimster41]

So are you asserting reality is a smooth, infinitely differentiable a continuum? Has that been proven?
My only problem is that lines and point have these “negligible widths and sizes” which sounds a lot like “infinity”.

 

[PeroK]

So are you asserting reality is a smooth, infinitely differentiable a continuum? Has that been proven?

Mathematics is not dependent on the fundamental nature of spacetime.

 

[Jimster41]

Interesting, and our brains are or are not?
I would argue the fact mathematics had to be invented to support detailed agreement between individuals is totally dependent on the fundamental nature of spacetime.

 

[PeroK]

You mean that in order for our brains to conceive of a spacetime continuum, spacetime must be continuous?

The problem is that we can conceive fundamentally incompatible systems. So, however nature is configured we have the capacity to conceive of it otherwise.

 

[Jimster41]

fair enough. Perhaps a dialectic oscillator.
For my part I think I wonder if the opposite of your first statement, if true, could help explain the second.

IOW if spacetime was continuous there would not be brains, just brain (or some barely conceivable “one”). I.e. Discrete spacetime could help explain discrete experience/phenomena in general. I find it hard to understand how a continuum got itself into such a big various mess.

 

[Interested_observer]

This is only part of the goal. The real goal is to understand Nature in a way that allows us to make the best use of it. This needs much more than just making physics computations the predictions of which agree with observations.

I would say that the real goal is to understand Nature, period! Then we would KNOW how to make the best use of our knowledge.