The Need of Infinity in Physics - Comments

[dextercioby]

dextercioby submitted a new PF Insights post

The Need of Infinity in Physics

Continue reading the Original PF Insights Post.

 

[bhobba]

Nice write up.

Indeed QM requires an infinite dimensional vector space. But I have to mention in modern times neither Martix Mechanics or Wave Mechanics are used. What's used is the more general transformation theory worked out by Dirac that incorporates both, plus his rather strange q numbers:
http://www.lajpe.org/may08/09_Carlos_Madrid.pdf

Thanks
Bill

 

[Helios]

The notion of infinity in cosmology was the proposition of Anaximander in the 6th century BC.

 

[Tabasko633]

Isn't one of the reasons that time and space are continuous?
So we need the concept of infinity to calculate dynamics in such a environment?

 

[davidbenari]

Nice write up.

Indeed QM requires an infinite dimensional vector space. But I have to mention in modern times neither Martix Mechanics or Wave Mechanics are used. What's used is the more general transformation theory worked out by Dirac that incorporates both, plus his rather strange q numbers:
http://www.lajpe.org/may08/09_Carlos_Madrid.pdf

Thanks
Bill

Hmm. Don't they use the wave formulation a lot in chemistry and matter physics?

 

[mathman]

As a mathematician, it seems to me that the insight is making a mountain out of a molehill. Infinity is a useful item to have when carrying out calculations. Until you get to Cantor, it doesn't have any deep meaning that needs to be probed.

 

[Greg Bernhardt]

Interesting counter article
http://blogs.discovermagazine.com/crux/2015/02/20/infinity-ruining-physics/

 

[Dr. Courtney]

Nice write up.

Indeed QM requires an infinite dimensional vector space. But I have to mention in modern times neither Martix Mechanics or Wave Mechanics are used.

Maybe not in your end, but I recall writing and running codes to diagonalize really big matrices (100k by 100k) in atomic physics.

 

[Tabasko633]

Interesting counter article
http://blogs.discovermagazine.com/crux/2015/02/20/infinity-ruining-physics/

So how did Achilles pass the turtoise? With a numerical error? ^^

 

[Jano L.]

dextercioby submitted a new PF Insights post

The Need of Infinity in Physics

Continue reading the Original PF Insights Post.

There’s always the feeling you get when you study physics really deeply that you’re doing no more than applied mathematics.
Unless you’re a modern Michael Faraday, i.e. a guy who works in a team who works in a (sometimes really big) laboratory from a (typically huge) facility or research institute like CERN or FermiLab, and your day-to-day job involves working with electronic equipment.

This paragraph is really weird. How can the author possibly label person working in team in a research facility/institute as "modern Michael Faraday"? The work they do in modern laboratories is rarely even close to what Michael Faraday was doing in his research. He did moderate-cost research of basic EM phenomena with a small-size self-made equipment (he studied EM induction with magnets and coils). In CERN, they do immense-cost research of subtle and exotic EM phenomena with expensive machinery which takes years to build (they study what detectors say happens after microscopic particles collide).

And why does the author suggest that everybody else is doing "no more than applied mathematics"?

I can say that whenever "I was thinking of physics deeply", how rare soever it was, I have never had a feeling like I'm doing applied mathematics. I am not sure what the author thinks doing applied mathematics means, but I guess it means you're not doing physics at all and you're either calculating consequences of a mathematical model given to you or you're developing such a model based on some mathematically formulated requirements.

I think that when you think about theoretical physics deeply, you're thinking about how the claims from professor, peers, textbook or paper are inconsistent either with themselves or with other physics known. You're trying to discern which ideas are experimental facts, which are questionable interpretations of such facts, which are just a popular way to think of them but not really necessary. You're thinking whether they can possibly be consistent with that or that theory and facts. Or you think about statements of a person who claims he solves some physics problem and you're trying to find whether he's right by analyzing the arguments and validity of the assumptions made. In many ways, deep thinking in theoretical physics is much like deep thinking in philosophy (it really originated in there). Applied mathematics is not a good name for such endeavour, I would say.

Do let a Heisenberg matrix be finite (Avogadro’s number of lines and columns) and you won’t have a quantum theory whatsoever. [as a side note: do let Planck’s constant be = 0 and you won’t have a quantum theory again].

It is true that the standard way to talk about Heisenberg matrices and Schroedinger operators is using the concept of infinity. However, neither matrices nor operators really are the core part of the theory that implies the predictions and explanations derived from it.

The core is the Schroedinger equation and the Born interpretation. The equation is a partial differential equation in coordinates and time.
This equation works with concepts of derivative and differentiable function, which are close to concept of infinity. But it can also be discretized and its solutions calculated in computer with no use of infinity. This can be done so it leads to predictions/explanations arbitrarily close to those you would get from the partial differential equation. The infinity has no more special significance for Schroedinger equation any more it has for the heat conduction equation or wave equation.

 

[jambaugh]

" It is beyond doubt that the notion of infinity lies somewhere near the core of all mathematics..." I dispute vehemently. The "core" of all mathematics is the rigor of deductive logic applied to axiomatics. Infinities manifest when it is improper or inconvenient to impose the actual finiteness we find in applications of mathematics.

The notion of infinity should, in physics, always and only be understood as a place-holder for an unspecified finite boundary.
Physically we never actualize infinities except possibly in the measure of ignorance which is always infinite in contrast to our finite knowledge. The corollary to this is finite information encoded in an infinitude of possible ways which underlies the mysteries of quantum mechanics.

This is not to say that we should discard the (*mathematical*) concept. Many people measuring distances each using distinct minimal units of precision would not have their measurements readily comparable unless we mapped them all into the "infinite precision" ideal of a continuum of measurements. We, also, may extrapolate well beyond the effects of given conditions in a certain application model and when doing so it is convenient to speak of "behavior at infinity" but this is simply short hand for "behavior beyond the significant influence of the aforementioned effects."

In short Infinity = Ignorance (as to where the boundary lies in some application of the theory).

If some construct, (such as the continuum of space-time) is necessarily infinite then we should always second guess any attempt to treat such a construct as manifestly physical. (Hence, do NOT take too seriously the ontological reality of the "space-time" manifold and its geometry.)

Or so I would assert.

 

[bhobba]

Hmm. Don't they use the wave formulation a lot in chemistry and matter physics?

Maybe not in your end, but I recall writing and running codes to diagonalize really big matrices (100k by 100k) in atomic physics.

The point was its not one or the other. They are both different aspects of an even more general theory.

Thanks
Bill

 

[bhobba]

Isn't one of the reasons that time and space are continuous? So we need the concept of infinity to calculate dynamics in such a environment?

We don't know one way or the other, but calculus is so powerful a tool you model it that way. In QM we don't know if an actual infinite dimensional space is needed, but powerful theorems from functional analysis such as Stones theorem can't be used if its not modeled that way.

Personally in QM I consider the physical realizable states to be finite dimensional, but perhaps of very large dimension, experimentally indistinguishable from an actual infinite one. One then, for mathematical convenience, and since we don't actually know the dimension, introduces states of actual infinite dimension so the powerful theorems of functional analysis can be used.

Thanks
Bill

 

[mma]

>Avogadro’s number is big enough to be considered the physicists’ true infinity.

I think that the notion of infinity in mathematics is quite different from "very big". If you add one mole of oxygen to one mole oxygen, the you get two mole oxygen. This is definitely more, than one mole. But if you add infinite number of elements to an infinite set, then the "size" of the set remains unchanged. This is the difference.

 

[Samy_A]

But if you add infinite number of elements to an infinite set, then the "size" of the set remains unchanged. This is the difference.

Actually, the "size" (or cardinality) of an infinite set can change when you add an infinite number of elements to an infinite set (example: add a set of the size (cardinality) of $\mathbb R$ to a countable infinite set).

 

[mma]

Actually, the "size" (or cardinality) of an infinite set can change when you add an infinite number of elements to an infinite set (example: add a set of the size (cardinality) of $\mathbb R$ to a countable infinite set).

You are right. I should have told countably infinite.

 

[mma]

Schrödinger’s operators for coordinate and momentum make sense only in an infinite dimensional Hilbert space, as a consequence of Stone-von Neumann’s theorem (1931).

This interesting fact was new to me. I found the details in Wikipedia:

Hermann Weyl observed that this commutation law was impossible to satisfy for linear operators p, x acting on finite-dimensional spaces unless ℏvanishes. This is apparent from taking the trace over both sides of the latter equation and using the relation Trace(AB) = Trace(BA); the left-hand side is zero, the right-hand side is non-zero. Further analysis[6] shows that, in fact, any two self-adjoint operators satisfying the above commutation relation cannot be both bounded.

 

[ShayanJ]

Personally in QM I consider the physical realizable states to be finite dimensional, but perhaps of very large dimension, experimentally indistinguishable from an actual infinite one. One then, for mathematical convenience, and since we don't actually know the dimension, introduces states of actual infinite dimension so the powerful theorems of functional analysis can be used.

What you say is very hazy and strange to me. Could you give a reference?

 

[bhobba]

What you say is very hazy and strange to me. Could you give a reference?

The dual of all the row vectors of finite dimension contains everything used in QM - its in fact the maximal space of a Gelfland tripple:
https://en.wikipedia.org/wiki/Rigged_Hilbert_space

All the spaces used in QM are a subset of this space.

Thanks
Bill

 

[ShayanJ]

The dual of all the row vectors of finite dimension contains everything used in QM - its in fact the maximal space of a Gelfland tripple:
https://en.wikipedia.org/wiki/Rigged_Hilbert_space

All the spaces used in QM are a subset of this space.

Thanks
Bill

I think what I actually don't understand is what definition of dimension you are using. Because for me dimension is a property of the space not the states!

 

[bhobba]

I think what I actually don't understand is what definition of dimension you are using. Because for me dimension is a property of the space not the states!

Sure - its simply a matter of context. A row vector of finite size by itself has dimension of its size because you assume its an element of the vector space of that size, but as an element of the space of all elements of finite size has infinite dimension.

Thanks
Bill

 

[pbuk]

I agree with bhobba - and I don't think this "Insight" offers anything useful.

If Physics needed "infinity" there would be no question as to whether the Universe is bounded. Come to that if Physics needed the "infinitesimal" there would be no question as to quantum foam (note that I am not saying this theory is correct, just that it is not inconsistent with any accepted theory or observation).

Indeed I assert the opposite of this "Insight" - in Physics we cannot currently distinguish between the infinite and the very large, or between the infinitesimal and the very small; furthermore it is unlikely that we will ever know if anything in Physics is infinite or simply very large.

 

[lavinia]

" It is beyond doubt that the notion of infinity lies somewhere near the core of all mathematics..." I dispute vehemently. The "core" of all mathematics is the rigor of deductive logic applied to axiomatics. Infinities manifest when it is improper or inconvenient to impose the actual finiteness we find in applications of mathematics.

The notion of infinity should, in physics, always and only be understood as a place-holder for an unspecified finite boundary.
Physically we never actualize infinities except possibly in the measure of ignorance which is always infinite in contrast to our finite knowledge. The corollary to this is finite information encoded in an infinitude of possible ways which underlies the mysteries of quantum mechanics.

This is not to say that we should discard the (*mathematical*) concept. Many people measuring distances each using distinct minimal units of precision would not have their measurements readily comparable unless we mapped them all into the "infinite precision" ideal of a continuum of measurements. We, also, may extrapolate well beyond the effects of given conditions in a certain application model and when doing so it is convenient to speak of "behavior at infinity" but this is simply short hand for "behavior beyond the significant influence of the aforementioned effects."

In short Infinity = Ignorance (as to where the boundary lies in some application of the theory).

If some construct, (such as the continuum of space-time) is necessarily infinite then we should always second guess any attempt to treat such a construct as manifestly physical. (Hence, do NOT take too seriously the ontological reality of the "space-time" manifold and its geometry.)

Or so I would assert.

"The "core" of all mathematics is the rigor of deductive logic applied to axiomatics"

Deductive logic is used in all reasoning, mathematical or not. Mathematics is the study of mathematical objects just as Botany is the study of plants

"The notion of infinity should, in physics, always and only be understood as a place-holder for an unspecified finite boundary."

Why? Because we can only make finitely many observations?

 

[dave bj]

I think what I actually don't understand is what definition of dimension you are using. Because for me dimension is a property of the space not the states!

Your really intelligent man. I am a welder and am looking to go back to school for engineering. Math has always been a favourite subject and a bad one. I have an easy time when I can visually put concepts mathematically together. I suffer with the abstract or the incomplete. All the theories and most widely accepted always threw me for a loop. I like the progress that physicists have made for technologies and advancements but there are to many variables that hold us back and we need to keep established proven formulas that push us forward but go back and start over with the concepts that started them. Dark matter, dark energy, gravity and quantum physics at the sub atomic level are really lacking to almost created to get us by for the moment. Gravity should be number 1 on the list completed theories but it isn't even close to being answered.

" It is beyond doubt that the notion of infinity lies somewhere near the core of all mathematics..." I dispute vehemently. The "core" of all mathematics is the rigor of deductive logic applied to axiomatics. Infinities manifest when it is improper or inconvenient to impose the actual finiteness we find in applications of mathematics.

The notion of infinity should, in physics, always and only be understood as a place-holder for an unspecified finite boundary.
Physically we never actualize infinities except possibly in the measure of ignorance which is always infinite in contrast to our finite knowledge. The corollary to this is finite information encoded in an infinitude of possible ways which underlies the mysteries of quantum mechanics.

This is not to say that we should discard the (*mathematical*) concept. Many people measuring distances each using distinct minimal units of precision would not have their measurements readily comparable unless we mapped them all into the "infinite precision" ideal of a continuum of measurements. We, also, may extrapolate well beyond the effects of given conditions in a certain application model and when doing so it is convenient to speak of "behavior at infinity" but this is simply short hand for "behavior beyond the significant influence of the aforementioned effects."

In short Infinity = Ignorance (as to where the boundary lies in some application of the theory).

If some construct, (such as the continuum of space-time) is necessarily infinite then we should always second guess any attempt to treat such a construct as manifestly physical. (Hence, do NOT take too seriously the ontological reality of the "space-time" manifold and its geometry.)

Or so I would assert.

I would like to hear your theory of our universe with gravity explained in detail.

 

[Neandethal00]

I think when humans can't figure out something, they give it a name.
And that's the end of the story.
From then on they refer to it by the 'name'.
Infinity is such a name.
May be 'God' is also a name.

 

[Mark Harder]

Hmm. Don't they use the wave formulation a lot in chemistry and matter physics?

The needs of chemistry are pretty simple compared to those of physics. In chemistry, almost universally, we need to explain phenomena involving the behavior of electrons in atoms and molecules. We have no need for fundamental particle theories, universal field theories, etc. Wave mechanics is adequate for almost all purposes. Possible exceptions are nuclear chemistry, concerned with the properties of elements and their isotopes and the stability of these. Another is the behavior of nonlinear optical materials. One must use QED to explain and predict nonlinear optical phenomena in materials. The creation of new optical materials is chemical in that methods of synthesizing these are chemical manipulations. I guess there's a lot of overlap with applied solid-state physics.

 

[Ez4u2cit]

As Galileo said "the language in which the universe is written is mathematics" and yet, it is still a language, a symbolic abstraction. To paraphrase Wittgenstein, we run the risk of confusing symbols (words) with reality. In mathematics, if we cut an apple in half, then the half in half and so on, we require infinite cuts to reduce the apple to zero. In reality, at some point we have reached the molecular level and the apple ceases to be an apple. We have confused the symbolic integer "1" with a real object. Conduct a similar thought experiment with anything in the real world. The problem is when the mathematics leads the physics. For example Max Tegmark in "Mathematical Universe" talks about infinite parallel universes with every possibility for every atom playing out, meaning every decision you have ever made forks off into another reality. I have no doubt that rigorous mathematics supports this conclusion but, in my humble opinion, is a result of an unwarranted faith in symbols, confusing the abstract with reality.

 

[indio007]

Infinity: does it exist?? A debate with James Franklin and N J Wildberger