Infinities in QFT (and physics in general)

[valenumr]

I suggest you to read some introduction to numerical analysis. Roughly it's like ordinary analysis, except with finite ##\Delta x## instead of infinitesimal ##d x##. And there are no ##\varepsilon##'s and ##\delta##'s.

I became fascinated with this concept when I first encountered gabriel's horn in calc ii in high school. It was mind bending to think about. I picked up computer modeling at that point just to demonstrate it couldn't be possible. Then I learned more about precision and numerical analysis. I suppose this is a little off topic, but I think it is some what relatable.

 

[valenumr]

Every introduction to numerical analysis (including the book I wrote on this topic) assumes real calculus when discussing differential equations.

If you work with discrete space and discrete time only, do you scrap all conservation laws? (But you even need one for Bohmian mechanics...)

Or how do you prove that energy is conserved for a particle in a time-independent external field?

Any attempt to give a full exposition of physics without using real numbers and continuity is doomed to failure. Not a single physics textbook does it. Claiming that physics does not need real numbers is simply ridiculous.

Noether's theorem, some basic assumptions about the universality (space-wise and time-wise) of physical laws, plus a lot of observations put symmetry and conservation laws on solid ground.

 

[valenumr]

It depends on what do you mean by "need". Does human physicist needs a pen and paper? Yes she does. Does human physicist needs her brain? Yes she does. But she needs them in a different sense. The latter is absolutely necessary, while the former is very very useful but not absolutely necessary. The need for real numbers is of the former kind. I can imagine an advanced civilization with advanced theoretical physics which does not use real numbers at all.

Hmm... Most PDE systems are likely unsolvable analytically. So we are only left with numerical approximations. Maybe quantum computing will prove that wrong.

 

[A. Neumaier]

Noether's theorem, some basic assumptions about the universality (space-wise and time-wise) of physical laws, plus a lot of observations put symmetry and conservation laws on solid ground.

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

 

[Sunil]

Here's an example: We might replace ##\frac{\mathrm df(x)}{\mathrm dx}## by ##\frac{f(x + 0.01) - f(x)}{0.01}##. If we do this, we probably lose all continuum symmetries, but we have introduced an ambiguity: Why ##0.01## and not ##0.0000043##? (And many more!) This is a completely undesirable situation, even if the continuum symmetries are emergent in this formalism.

You think Nature or Mathematics have to care about what is desirable for you?

Of course, discrete mathematics is much more complicate and less symmetric. Continuum mathematics is a useful for simplification. Simplification is something which can be reached also by approximation. The approximation is simple, but wrong. So what? Once it is fine as an approximation, we can use it, it is better than nothing. Once the exact computation is too complicate, we have no better choice than to choose the approximation. So, to use approximations is reasonable and fine.

But one should not confuse something being fine as an approximation with something being true. That's all.

 

[Sunil]

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

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.)

 

[Fra]

I didn't notice this thread, but I share a lot of Demystifiers objections. I may have a difference route of reasoning, but as I try to reason from the perspective of an inside observer and reconstruction rules and laws from there, the notion of real numbers are not to be introduced lightly. For me it has to do with wether what is distinguishable from an insider observer. Also any inferences risk getting out of control if you introduce uncountable sets and loose track of limits. The apparent infinitites we see in physics today seems to be because order of things are lost and the work of cleaning it up via variois post-mess renormalization schemes is exactly what to me is ambigous.

I also see continuum mathematics as an "approximation" of the discrete systems in a high complexity limit. Working with combinatorics and permutation symmetries seems a lot more complicated for anything by small systems, so I think there is good reasons for the continuum mathematics, but in trying to understand some things, it is to me, not fundamental. Even cox and otehr reconstructions of probability that throws up a real number, puts me off.

My hunch is that once physics (and its measures) are reconstructed from an instrinci perspective, a lot of the infinities nevery should show up in the first place. Even though this is complicated and continuous symmetries may be understood as hypothetical large complexity limits of permutation symmetries, it may say us from resorting to ambigous renormalization methods. Intrisic constructions hopefully would come with natural regulators.

I find it pathlogical to think that an observer or agent can distinguish between the points in a contiuum. Even tough most people may agree on that, the embedding into real numbers makes it deceptive, as what is mathematical degrees of freeodom and what is of physical relevants. Especially in the foundations of a measurement theory because one one considers the physical entropy of a system, it's the embedding that defines the number.

/Fredrik

 

[Fra]

This conversation seems very unproductive to me. Of course, in the end, calculations are made on a computer with finite precision, but physics is about understanding the laws of nature and that's hardly possible without continnuum mathematics.

What if you (which most don't, but it's a separate question) consider the idea that the laws of nature must be inferrable from the perspective of a real FINITE observer/agent. In this perspective, the agent is the "computer" and the agents actions should likely reflect the imperfections from the continuum approximatein except when neglectable?

/Fredrik

 

[physicsworks]

This discussion reminded me of the famous debate between a prominent soviet physicst Yakov Zeldovich and no less prominent soviet mathematician Lev Pontryagin. As another great mathematician Vladimir Arnold remembers, it concerned a textbook "Higher Mathematics for Beginners and its Application to Physics" published by Zeldovich in the 60's. The textbook was heavily criticized by mathematicians and censors of math literature in the Soviet Union at the time. It contained, among other things, a definition of the derivative as a ratio of increments "under the assumption that they are small". This definition, although blatantly disrespectful and almost criminal from the point of view of orthodox mathematics, is completely justified physically. As Zeldovich argued, increments of a physical quantity smaller than, say, $10^{-100}$ are a pure fiction: the structure of spacetime on such scales may turn out to be very far from the mathematical continuum. Zeldovich continued to defend his position and the debate ended with his complete victory. In 1980, Pontryagin in his textbook on mathematical analysis for high school students wrote: “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."

 

[A. Neumaier]

As Zeldovich argued, increments of a physical quantity smaller than, say, $10^{-100}$ are a pure fiction: the structure of spacetime on such scales may turn out to be very far from the mathematical continuum.

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.

You can see this in finite precision arithmetic (where the structure of reals breaks down at the scale of a relative error of about ##10^{-16}##. Try computing the derivative of ##3x^2-1.999999x## at x=1/3 by a difference quotient with ##dx=2^{-k}## for ##k=10:70##, say. The mathematical value of the derivative is ##10^{-6}## but in standard IEEE arithmetic, ##k=55## gives the value ##-2##, and ##k>55## gives zero. The best approximation is obtained for ##k=33## and ##k=34##, giving the quite inaccurate value ##0.95347\cdot 10^{-6}##.

The only definition that makes sense physically is therefore one where ##dx## is small but not too small. Since small and too small cannot be quantified exactly, it yields a fuzzy number, not an exact value.

 

[Fra]

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.
...
Since small and too small cannot be quantified exactly, it yields a fuzzy number, not an exact value.

If physical interactions at their fundamental level is essentially "stochastic", the notion of continuity should not be needed to phrase the laws of physics. Ie. the laws of physics might not be cast in terms of differential equations, but in terms of random transitions between discrete states as guided random walks.

That's at least my vision, that the differential equations represent a nice way to represent things "on average" in certain limits. But it may be a potential fallacy ot make too strong deductions on physics using the power of continuum mathematics, because the continuum models may be the approximation, not the other way around.

/Fredrik

 

[A. Neumaier]

But it may be a potential fallacy ot make too strong deductions on physics using the power of continuum mathematics, because the continuum models may be the approximation, not the other way around.

This is an undecidable issue since we never know the true laws of physics down to the tiniest scales.

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.

 

[Fra]

This is an undecidable issue since we never know the true laws of physics down to the tiniest scales.

Yes, and this is also exactly why one argument is that one should start the constructions from the observer perspective. Such approch would not exlude continuum limits, but it would also not presume them.

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.

No need to argue about the success of current models, that is unquestionable. But the success of the current methods, is also why it's mentally difficult to release from it, so the questions are I think rational.

The question of unifying laws, is not so much a practical matter I think, it's more a principal matter that refers to explanation of current models. I think it's a matter of interpretation and choice of stance for future research, and I don't quite agree.

/Fredrik

 

[Jimster41]

But these states are chosen adaptively, depending on the problem. Moreover, the approximation accuracy depends on the number of states chosen, and chemists add states until convergence is observed. This is possible only in an infinite-dimensional Hilbert space. And the basis varies from problem to problem, which shows how nonphysical the discretized setting is.

But these states are chosen adaptively, depending on the problem. Moreover, the approximation accuracy depends on the number of states chosen, and chemists add states until convergence is observed. This is possible only in an infinite-dimensional Hilbert space. And the basis varies from problem to problem, which shows how nonphysical the discretized setting is.

Why not start with an infinite number of states and subtract?

 

[Jimster41]

Every introduction to numerical analysis (including the book I wrote on this topic) assumes real calculus when discussing differential equations.

If you work with discrete space and discrete time only, do you scrap all conservation laws? (But you even need one for Bohmian mechanics...)

Or how do you prove that energy is conserved for a particle in a time-independent external field?

Any attempt to give a full exposition of physics without using real numbers and continuity is doomed to failure. Not a single physics textbook does it. Claiming that physics does not need real numbers is simply ridiculous.

Doesn’t mean they exist.

 

[Jimster41]

O

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.

You can see this in finite precision arithmetic (where the structure of reals breaks down at the scale of a relative error of about ##10^{-16}##. Try computing the derivative of ##3x^2-1.999999x## at x=1/3 by a difference quotient with ##dx=2^{-k}## for ##k=10:70##, say. The mathematical value of the derivative is ##10^{-6}## but in standard IEEE arithmetic, ##k=55## gives the value ##-2##, and ##k>55## gives zero. The best approximation is obtained for ##k=33## and ##k=34##, giving the quite inaccurate value ##0.95347\cdot 10^{-6}##.

The only definition that makes sense physically is therefore one where ##dx## is small but not too small. Since small and too small cannot be quantified exactly, it yields a fuzzy number, not an exact value.

Interesting. It’s almost like you are making the opposite argument you are trying to make.
Also, sort of thought the idea of a discrete limit h (and or b) functioning like the speed limit of light was a key to @RUTA block world reconciliation of QM and GR.

I agree with him and Pontryagin. Calculus is a tool that depends on flying elephants. Still a good tool.

 

[Fra]

If you work with discrete space and discrete time only, do you scrap all conservation laws?

One possibility to see it from an agent perspective:

Instead of seeing it so that you just "scrap" all the continuum symmetries, and loose powerful constraints and deductive power, one might instead see it so from an agent perspective, that we reject the continuum symmetries as valid intrinsic rules of reasoning, on the grounds that they are typically not inferrable with 100% confidence by a finite agent (because ones seeks an instrinsic reconstruction).

Instead one could embrace the inferrable but subjective and approximate symmetries and see they as guiding the actions in the evolving context, as the agents are interacting. One then replaces the constraint notion of observer equivalence with just observer democracy, and are then facing the problem of showing that the democratic process is compatible with an equivalence asymptotically in the relevant cases, and hopefully also explain WHY the symmetries ARE spontaneously emergent, but also WHY they are not perfect? IF that suceeds (which may not be the case of course), then it seems we have "added value" and insight. This is as I see one possible rationale for the thinking.

/Fredrik

 

[Demystifier]

Calculus is a tool that depends on flying elephants.

Can you explain?

 

[Jimster41]

Can you explain?

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.

 

[Jimster41]

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

“Evolutionary Dynamics” by Novak, specifically the phenomenon of evolutionary drift as an analogue of spontaneous symmetry breaking left me wondering if evolution as a phenomenon isn’t a clue to the answer of whether the limit is continuous.

Not my ideas: got them esp from Tannoudji’s “Universal Constants in Physics” and Chaisson’s “The Life Era” among others. Most recently I was excited to see @RUTA ’s modest proposal that such constants along with c sort of dictate (describe for us) the finite constraints of the lattice. Still working through his book.

 

[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.