Why do physicists use infinity?

This post continues from my last post. In this post, I deal with the question: Even if infinity does not physically exist, why do physicists use it?

The Thermometer Effect

Consider the task of measuring the temperature of a hot body. How do you do that?

The simplest instrument to use would be: a “mercury” thermometer (i.e. of the kind you use when you come down with fever). Heat flows from the hot object to the “mercury” in the thermometer, and so, the temperature of the “mercury” rises. When its temperature increases, the “mercury” expands and so, its level in the tube rises. This rise in the level of the “mercury” can be calibrated to read out the temperature—of the mercury. After a while, the temperatures of the hot object (say the human body) and the thermometer become equal (practically speaking), and so, the calibrated tube gives you a reading of the temperature of the hot object. … We all learnt this in high-school (and many of us understood it (or at least had it explained to us) before the topic was taught to us in the school). What’s new about it?

It’s this (though even this point wouldn’t be new to you; you would have read it in some popular account of quantum mechanics): In the process of measurement, the thermometer takes away some heat from the hot object, and in the process makes the latter’s temperature fall. Therefore, the temperature that is read out refers to the temperature of the {hot object + thermometer} system, not to the initial temperature of the hot object itself. However, we were interested in measuring the temperature only of the hot object, not of the {hot object + thermometer} system.

QM folks, and QM popularizers, habitually go on a trip from this point on. We let them. Our objective is something different, something (hopefully) new. Our objective is the objectivity of measurements in an objectively understandable universe, not a subjective trip into an essentially subjective/unknowable universe. How do we ensure that? How do we ensure the objectivity of the temperature readings?

Enter infinity. Yes. Let me show you how.

We realize that the bigger the thermometer, the bigger is the heat leaked out from the hot object to the thermometer, and therefore, the bigger is the error in the measurement. So, we try something practical and workable. We try a smaller thermometer. Let’s be concrete.

Suppose that the hot object  remains the same for each experiment in this series. Suppose that thermometer T1 holds 10 ml of the sensing liquid, and suppose that at the end of the measurement process, it registers a temperature of 99 C. To decrease the amount of heat leaking into the thermometer, we get a second thermometer, T2; it holds only 1 ml of the sensing liquid. Suppose it registers a temperature of 99.9 C. We know that as the thermometer becomes smaller and still smaller, the reading read off from it will grow ever more accurate. For instance, with development in technology, yet another thermometer T3 may be built. It contains only 0.1 ml of the sensing liquid! It is found to register a temperature of 99.99 C. Etc. Yet, a thermometer of 0 ml liquid is never going to work, in the first place!

So, we do something new. We decide not to remain artificially constrained by the limits of the available technology, because we realize that here we don’t actually have to. We realize that we can take an inductive leap into the abstract. We plot a graph of the measured temperature against the size of the sensing liquid, and find that this graph has begun to “plateau out”. Given the way it is curving, it is obvious that it is coming to a definite limit.

We translate the graph into algebraic terms, i.e., as an infinite sequence, formulated via a general `n’th term. Using calculus, we realize that as `n’ approaches infinity, the amount of the sensing liquid approaches zero, and the temperature T_n registered by the thermometer approaches 100 C.

We thus conclude that in the limit of vanishing quantity of the sensing liquid, the measured temperature would be the true i.e. the unfallen or the initial temperature—viz. 100 C.

Carefully go through the above example. It shows the essence of how physicists use infinity. Indeed, if they were not to do so, they could not tie the purely imagined notion of a true temperature with any actual measurement done with any actual thermometer. Their theories would be either lacking in any generalization concerning temperature measurement, or it would be severed from the physical reality. However, one sure way that they can reach reality-based abstractions is via the idea of infinity. Before the 20th century, they did.

The sound-meter effect

Suppose you have a stereo system installed in your living room. For the sake of argument, just one speaker is enough. The logic here is simply additive: it applies even if you have a stereo system (two speakers) or a surround sound system (5 speakers); just find the effect that a single speaker produces one at a time, and add them all together, that’s all! That’s why, from now on, we consider only a single speaker.

Suppose you play a certain test sound, something like a pure C tone (say, as emitted by a flute), at a certain sound level—i.e. the volume knob on your music system is turned to some specific position, and thus, the energy input to the speaker is some definite fixed quantity.

You want to find out what the intensity of the sound actually emitted by a speaker is, when the volume knob on the music system is kept fixed at a certain fixed position.

Now, you know what the phenomenon of sound is like. It exists as a kind of a field. The sound reaches everywhere in the room. Once it reaches the walls of the room, many things happen. Here is a simplified version: a part of the sound reaching the wall gets reflected back into the room, a part of it gets transmitted beyond the wall (that’s the physical principle on the basis which your neighbors always harass you, but you never ever disturb anyone else’s sleep because that’s what your “dharma,” of course, teaches you), and a part of it gets absorbed by the wall (this is the part that ultimately gets converted into heat, and thereby loses relevance to the phenomenon of sound as such).

Now, if you keep a decibel-meter at some fixed position in the room, then the sound-level registered by it depends on both these factors: the level of the sound directly received from the speaker, as well as the level of the reflected sound. (Yep, we are getting closer to the thermometer logic once again).

Our task here is to find the intensity of the sound emitted by the speaker. To do that, all we have is only the decibel-meter. But the decibel meter is sensitive to two things: the directly received sound, as well as the indirectly received sound, i.e., the reflected sound. What the decibel meter registers thus also includes the effect of the walls of the room.

For instance, if the walls are draped with large, thick curtains, then the effective absorptivity of the walls increases, and so, the intensity of the reflected sound decreases. Or, if the walls are thinner (think the walls of a tent), then the amount leaked out to the environment increases, and therefore, the amount reflected back to the decibel-meter decreases.

The trouble is: we don’t know in advance what kind of a wall it is. We don’t know in advance the laws that apportion the incident sound energy into the reflecting, absorbing and transmitting parts. And therefore, we cannot use the decibel meter to calculate the true intensity of the sound emitted by the speaker itself. Or so it seems.

But here is the way out.

Since we don’t precisely know the laws operative at the wall, altering the material or thickness of the wall is of no use. But what we can do is: we can change the size of the room. This part is in our hands, and we can use it—intelligently.

So, once again, we conduct a series of experiments. We keep everything else the same: the speaker, the position of the decibel-meter relative to the speaker, the position of the volume-level knob, the MP3 file playing the C-note, etc. They all remain the same. The only thing that changes is: the size of the room.

So, suppose we first conduct the experiment in our own living room, and register a decibel meter reading of, say, 110 (in some arbitrary units). Then, we go to a friend’s house; it has bigger rooms. We conduct our experiment there. Suppose the reading is: 108 units. Then, we take the permission of the college lab in-charge, and conduct our experiment in the big laboratory hall: 106 units. We go into an in-door stadium in our town: 102 units. We go into an in-door stadium in another town: 102.5 units. We go out in a big open field out of town, and conduct the experiment at late night: 101.5 units. We go to that open salt field in the rann of the Kutch: 101.1 units. We take the measurement at a high level in the rann: 100.91 units.

Clearly, the nature of the wall has always been effecting our measurements. Clearly, the wall has always been different in different places—and we didn’t have any control over the kind of a wall there may be, neither do we know the kind of laws it follows. And yet, the size of the room has clearly emerged as the trend-setter here.

If we plot the intensity of the sound vs. the size of the room, the trend is not as simple (or monotonic) as in the thermometer case. There are slight ups and downs: even for a room of the same size, different readings do result. Yet, the overall trend is very, very clear. As the size of the room increases, the measurements go closer and closer to: 100 units.

Why? It’s because, choosing a bigger room leads to one definite effect: the effect of the wall on the measured sound level goes on dropping. The drop may be different for different kinds of walls. Yet, as the size of the room becomes really large, whatever be the nature of the wall and whatever be the laws operating at that remote location, they begin to exert smaller and ever so smaller effect on our measurements.

In the limit that the size of the room approaches infinity, the measurement procedure tends to yield an unchanging datum for the intensity of the measured sound. Indeed, in this limit, it would be co-varying with the intensity of the emitted sound, in a most simple, direct, manner. We have, once again, arrived at a stable, orderly datum—even if there were so many things affecting the outcome. We have, once again, managed to reach a universal principle—even if our measurement procedures were constrained by all kinds of limits; all kinds of superfluous influences of the walls.

For the advanced student of science/engineering:

In case you know differential equations (esp. computational modeling), the use of infinity makes the influence of the boundary conditions superfluous.

For instance, take a domain, take the Poisson equation, and use various boundary conditions—absorbing, partially absorbing, periodic, whatever—to find the field strengths at a point within the domain (as controlled by the various boundary conditions). Now, enlarge the domain, and once again try out the same boundary conditions. Go on increasing the domain size. Observe the logic. In the limit that the domain size approaches infinity, the value of the field variable approaches a certain limit—and this limit is given, for the Poisson problem, by the simple inverse-square law!

The Infinity, and Philosophy of Physics:

Increasing the size of the domain to the infinitely large serves the same purpose as does decreasing the amount of the sensing liquid to the infinitely small. The infinitely large or the infinitely small does not exist—the notion has no physical identity. But the physical outcomes in definitely arranged sequences do, and, in fact, even an only imaginary infinite sequence of these does help establish the physical identity of the phenomenon under discussion.

In both cases, infinity allows physicists the formulation of universal laws even if all the preceding empirical measurements are made in reference only to finite systems.

That incidentally is the answer to the question with which we began this post: Even if infinity does not physically exist, why do physicists use it?

It’s because, the idea allows them to objectively isolate the universal phenomena from the local physical experiences.

Homework for you:

In the meanwhile, here are a few questions for you to think about, loosely grouped around two (not unrelated) themes:

Group I: The Argument from the Arbitrary:

Is the question: “Is the physical universe infinite?” invalid? Can it be answered in the yes/no (or true/false) terms alone? Does it involve any arbitrary idea (in Ayn Rand’s sense of the term)? Is the very idea of the arbitrary valid?

Is there any sense to the idea of a finite physical universe?

Is there any sense to the supposition that you could reach the end of the universe?

Group II: The Argument from the Unseen Universe:

Think whether you would refute the following argument, and if yes, how: We cannot rely on physics, because the entirety of physics has been derived only in reference to a finite portion of the universe. Therefore, physics does not represent a truly universal knowledge. Our knowledge, as illustrated by its most famous example viz. physics, has no significance beyond being of a severely limited practical art. Knowledge-wise, it’s not a true form of knowledge; it’s only nominal. Some day it is bound to all break down, as influences from the unseen portion of the universe finally reach us.

Additional homework for the student of quantum mechanics:

Find out the relevance of this post to your course in quantum mechanics, as is covered usually in the universities, (e.g. Griffith’s text).

A Hint: No, this is not a “philosophy” related homework.

Spoiler Alert: Jump to the next section (on songs) if you don’t want to read a further hint, a very loud hint, about this homework.

A Very Loud Hint: Copy paste the following text into a plain text editor (such as the Notepad):

The Sommerfeld radiation condition

The Answer: In the next post, of course!

A Couple of Songs I Don’t Particularly Like:

Both are merely passable.

[This song is calculus-based. Really. In the reel life, it makes a monkey of the dashing young hero (and also of his dog), just the way the calculus does of most any one, in the real life.]

(Hindi) “samundar, samundar, yahaan se wahaa tak…”
Music: S. D. Burman
Singer: Lata Mangeshkar
Lyrics: Anand Bakshi

[This song used to be loved by the Americans (and many others, including Indians) when I was at UAB—and also for some time thereafter. It, or the quotable phrase that its opening line had become, doesn’t find too much of a mention anywhere. … Just the way neither does the phrase: the brave new world!]

(English) “A whole new world…”
Singers: Brad Kane and Lea Salonga
Music: Alan Menken
Lyrics: Tim Rice

[I intended to finish this thread off right in this post, but it grew too big. Further, I will be preoccupied in teaching activities (the beginning time is always the more difficult time), and so, there may be some time before I come back for the next post—may be the next weekend, or possibly even later—even though my attempt always would be to try to wrap this thing off as soon as possible anyway.]


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