# Why the Need of Infinity in Physics

Common Topics: infinity, infinite, mechanics, theory, mathematics

It is beyond doubt that the notion of infinity lies somewhere near the core of all mathematics, probably in a finite vicinity of zero. But where does it lie in physics? 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 institutes like CERN or Fermilab, and your day-to-day job involves working with electronic equipment.

Since physics has not lost its experimental side (and will never do so), I’m trying to mentally reconcile the mathematical notion of infinity with the experimental side of physics. I’ve always thought that man-made machinery cannot truly probe infinity, no matter how hard we tried, or how fancy and expensive a telescope/microscope can be. The universe (if only one) can be deemed infinite (if at all) only by a theorem, at least that’s what a theorist (blood-related or not to Stephen Hawking or Roger Penrose) should hope for. Infinity remains a part of mathematics we could or could not do without in the physical theories. But it cannot be felt.

A quick review of the elementary theories shows that: Newtonian mechanics has the distinct feature that any of its notions (labeled as physical quantities) can be freely made infinite, either time, distance, velocity/instantaneous speed, accelerations, forces, etc., even if we could never measure infinite time, or an orbit of infinite radius, for example. Special relativity puts a serious bound on speed, but time and distance can still flow from – to + infinity and the whole relativity (especially GR) has nice results when such limits are taken. Statistical mechanics is anchored in reality, there’s never an infinite number of particles, but Avogadro’s number is big enough to be considered the physicists’ true infinity.

I’ve left quantum physics last, because here, as I will show, infinities are at the heart of all the (standard) theory. If you’re doing experiments and you wish to measure photon energies in the visible light spectrum from a particular transition in the Hydrogen atom, you have the option to use Balmer’s formula (1885) to compare results against the theory. But you know in 2016 that the currently accepted explanation of Balmer’s formula uses either the matrix mechanics of Born, Heisenberg, and Jordan (1925), or the wave mechanics of Schrödinger (1926). There’s no infinity in Balmer’s formula (yes, that “n” can go as high as one wishes, but experiment really puts a bound on it), but there’s one not really hidden in either form of Quantum Mechanics: the matrices of Born, Heisenberg and Jordan are necessarily infinite (as noted first by Born and Jordan in “Zur Quantenmechanik” (Zeitschrift für Physik, 1925, p. 780). 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).

Let’s sum it up: numerical calculations performed by human-programmed computers use mathematics which in turn uses the notion of infinity. Man-made measuring apparatus never have infinite scales. A physicist cannot perceive infinity through his senses.

In Newtonian mechanics, you can replace the mathematical infinity with an arbitrarily large number (Avogadro’s number expressed in meters, seconds, km/s, Joules) and some integrals will need computer power to be performed. You can remove infinity from this theory at a conceptual level. Relativity is a form of mechanics, you can again let infinity be replaced by a huge number and get rid of it. 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].

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31 replies
1. jedishrfu says:

[QUOTE=”Ronie Bayron, post: 5352557, member: 576428″]It’s a pretty interesting subject and the bond between 0 and infinity goes together.
The logic that lies behind the subject is intuitive but, I don’t know, why facts on zero and infinite are not accepted in mathematics.
A simple concept goes like this.
Any number- n, divided by infinity is zero or n/inf=0, yet any number which represents a real number in the number line is equal to infinite x zero.
The insight to ponder is….are all things created and formed by zero and infinite?
Certainly it is, by the rule of basic mathematical operation (division & multiplication) and its not easy to dismiss,[/QUOTE]

Mathematicians will tell you that division by zero is undefined because it leads to contradictory or ambiguous results.

Also infinity is not a number so any arithmetic operation cannot generate it as a result.

Hence its wrong to associate zero and infinity in the manner that you have done.

[URL]https://en.wikipedia.org/wiki/Division_by_zero[/URL]

2. Mark Harder says:

[QUOTE=”davidbenari, post: 5342556, member: 511036″]Hmm. Don’t they use the wave formulation a lot in chemistry and matter physics?[/QUOTE]

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.

3. Neandethal00 says:

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.

4. dave bj says:

[QUOTE=”Shyan, post: 5342998, member: 160907″]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![/QUOTE]

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.
[QUOTE=”jambaugh, post: 5342709, member: 76054″][I]” It is beyond doubt that the notion of infinity lies somewhere near the core of all mathematics…”[/I] 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.[/QUOTE]

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

5. dave bj says:

[QUOTE=”dextercioby, post: 5342231, member: 1064″]dextercioby submitted a new PF Insights post

[URL=’https://www.physicsforums.com/insights/need-infinity-physics/’]The Need of Infinity in Physics[/URL]

[URL=’https://www.physicsforums.com/insights/need-infinity-physics/’]Continue reading the Original PF Insights Post.[/URL][/QUOTE]
[QUOTE=”dextercioby, post: 5342231, member: 1064″]dextercioby submitted a new PF Insights post

[URL=’https://www.physicsforums.com/insights/need-infinity-physics/’]The Need of Infinity in Physics[/URL]

[URL=’https://www.physicsforums.com/insights/need-infinity-physics/’]Continue reading the Original PF Insights Post.[/URL][/QUOTE]
squared sorry
meaning speed is finite in our observable universe
what if speed is a result of finite space and that our universe is finite
then we take a leap that energy at a level is being absorbed in all directions into to this finite space that was finite even in the presence of the big bang
this energy is being absorbed at an exponential rate
think of an egg in the ocean sinking to the bottom and the force energy applied to it unevenly but generally in all directions
inside the universe because of the moment of our understanding of the big bang but more like the steady exponential rate of energy applied
now our universe would have to condence this ever growing energy into tighter spaces inside this finite universe that exist with its own laws inside its finite space
as energy condences into what we define as mass units and atom densities the idea of time is born as we have to observe it
These energies are compressed into tighter spaces and absorb mor of the expanding energy
they absorb more at exponential rates that are calculated with time being that they grow faster then less energy mass accumulations
space would seemingly expand in all directions because the greater accumulation of energy mass forces time to move faster and pushing space inside tighter
areas of the universe that seemingly don’t expand as fast are because there absorbtion of mass is less but ever unit of energy we define as a mass a unit in space would take this absorbtion until it is unobservable.
Taking this a step further to quantum it would be because of our absorbtion rate is so significately larger then the quantum level and that we exhist as same level units of mass energy equal to the earth at the earths mass aborbtion in the space we condence that we observe the quantum universe as unaffected by time however this is not possible but more like extremely close to zero however infinite because of this energy absorbtion in the universe. this is my theory and would love to have a talk with a pysics master. physics sorry

6. MrAnchovy says:

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.

7. bhobba says:

[QUOTE=”Shyan, post: 5342998, member: 160907″]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![/QUOTE]

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

8. Shyan says:

[QUOTE=”bhobba, post: 5342971, member: 366323″]The dual of all the row vectors of finite dimension contains everthing used in QM – its in fact the maximal space of a Gelfland tripple:
[URL]https://en.wikipedia.org/wiki/Rigged_Hilbert_space[/URL]

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

Thanks
Bill[/QUOTE]

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!

9. bhobba says:

[QUOTE=”Shyan, post: 5342959, member: 160907″]What you say is very hazy and strange to me. Could you give a reference?[/QUOTE]

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

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

Thanks
Bill

10. Shyan says:

[QUOTE=”bhobba, post: 5342846, member: 366323″]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.[/QUOTE]
What you say is very hazy and strange to me. Could you give a reference?

11. mma says:

[quote]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).[/quote]

This interesting fact was new to me. I found the details in [URL=’https://en.wikipedia.org/wiki/Stone%E2%80%93von_Neumann_theorem’]Wikipedia[/URL]:

[quote][URL=’https://en.wikipedia.org/wiki/Hermann_Weyl’]Hermann Weyl[/URL] observed that this commutation law was [I]impossible to satisfy[/I] for linear operators p, x acting on [URL=’https://en.wikipedia.org/wiki/Finite-dimensional’]finite-dimensional[/URL] spaces unless ℏvanishes. This is apparent from taking the [URL=’https://en.wikipedia.org/wiki/Trace_(linear_algebra)’]trace[/URL] over both sides of the latter equation and using the relation Trace([I]AB[/I]) = Trace([I]BA[/I]); the left-hand side is zero, the right-hand side is non-zero. Further analysis[URL=’https://en.wikipedia.org/wiki/Stone%E2%80%93von_Neumann_theorem#cite_note-6′][6][/URL] shows that, in fact, any two self-adjoint operators satisfying the above commutation relation cannot be both [URL=’https://en.wikipedia.org/wiki/Bounded_operator’]bounded[/URL].[/quote]

12. mma says:

[QUOTE=”Samy_A, post: 5342930, member: 574914″]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).[/QUOTE]
You are right. I should have told countably infinite.

13. Samy_A says:

[QUOTE=”mma, post: 5342918, member: 44619″] But if you add infinite number of elements to an infinite set, then the “size” of the set remains unchanged. This is the difference.[/QUOTE]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).

14. bhobba says:

[QUOTE=”Tabasko633, post: 5342524, member: 554358″]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?[/QUOTE]

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 cant be used if its not modelled 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

15. bhobba says:

[QUOTE=”davidbenari, post: 5342556, member: 511036″]Hmm. Don’t they use the wave formulation a lot in chemistry and matter physics?[/QUOTE]
[QUOTE=”Dr. Courtney, post: 5342577, member: 117790″]Maybe not in your end, but I recall writing and running codes to diagonalize really big matrices (100k by 100k) in atomic physics.[/QUOTE]

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

Thanks
Bill

16. Jano L. says:

[QUOTE=”dextercioby, post: 5342231, member: 1064″]dextercioby submitted a new PF Insights post

[URL=’https://www.physicsforums.com/insights/need-infinity-physics/’]The Need of Infinity in Physics[/URL]

[URL=’https://www.physicsforums.com/insights/need-infinity-physics/’]Continue reading the Original PF Insights Post.[/URL][/QUOTE]

[QUOTE]
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.
[/QUOTE]

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.

[QUOTE]
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].
[/QUOTE]

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.
[SIZE=4]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.[/SIZE]

[QUOTE=”Greg Bernhardt, post: 5342566, member: 1″]Interesting counter article
[URL]http://blogs.discovermagazine.com/crux/2015/02/20/infinity-ruining-physics/[/URL][/QUOTE]

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

18. Dr. Courtney says:

[QUOTE=”bhobba, post: 5342257, member: 366323″]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. [/QUOTE]

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

19. Greg Bernhardt says:

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

20. mathman says:

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.

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?

22. bhobba says:

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:

Thanks
Bill

23. indio007 says:

Infinity: does it exist?? A debate with James Franklin and N J Wildbergerhttps://www.youtube.com/watch?v=WabHm1QWVCA

24. Ronie Bayron says:

Other wise, 4×4 is not 16 and 8 div 2 is not 4.  What do you think?

25. Ronie Bayron says:

It's a pretty interesting subject and the bond between 0 and infinity goes together.The logic that lies behind the subject is intuitive but, I don't know, why facts on zero and infinite are not accepted in mathematics.A simple concept goes like this.Any number- n, divided by infinity is zero or n/inf=0, yet any number which represents a real number in the number line is equal to infinite x zero.The insight to ponder is….are all things created and formed by zero and infinite?Certainly it is, by the rule of basic mathematical operation (division & multiplication) and its not easy to dismiss,

26. Ez4u2cit says:

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.

27. lavinia says:

"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?

28. mma says:

>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.

29. jambaugh says:

" 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.

30. davidbenari says:

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

31. Helios says:

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