The Need of Infinity in Physics

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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 institute 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 a 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 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 by an arbitrary 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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  1. jambaugh
    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.

  2. mma
    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.

  3. lavinia
    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?

  4. Ez4u2cit
    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.

  5. Ronie Bayron
    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,

  6. bhobba
    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:


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