How many possible electromagnetic wavelengths are possible?

Question

Disclaimer: Please keep in mind that I am a young highschool student with no background in physics, this research was done in the course of an hour, and my reasoning could very well be wrong.

After a short conversation about color, I did some thinking, and read this question. The answer claims that there are $\infty^{\infty}$ colors

My first problem is the definition of colors, I'm going to simplify this (although it's not the same thing) as electromagnetic frequencies. This is what I mean when I refer to color in the rest of this question

The problem with $\infty^{\infty}$ colors is that the plank length exists. This causes two problems to arise with $\infty^{\infty}$. If $l_p$ is assumed to exist (as it is), this means the smallest distance in the universe is $l_p$. So, all wavelengths of light can be divided into a set, of the wavelengths produced by objects at $0^{\circ}$ Kelvin to $\infty^{\circ}$ Kelvin. According to this logic, there are simply $\infty$ colors.

Things are complicated further, when you introduce the Plank Temperature ($P_t$). $P_t$ is the theoretical limit temperature of the universe (at least for the purposes of light and color) Because an object hotter than $P_t$ will produce light with wavelengths shorter than $l_p$. This would introduce an upper limit on colors too.

Wouldn't this mean that the total number of possible colors is equal to however many plank lengths difference there is between an electromagnetic wave produced by a 0 degree object and a $l_p$ object? And therefor far less than $\infty$? Does this reasoning make sense at all?

Answer 1

I will give a speculative answer - open to suggestions for improvement:

While mathematicians understand the difference between $\infty$ and $\infty^{\infty}$, I am not sure such distinction is terribly helpful for physicists - or how you would prove one versus the other. And if you are worrying about the distinction, you are not a typical high school student...

Frequency of a photon is an ill-defined property: in order to measure the frequency to an accuracy $\Delta \omega$, I need to measure for a length of time $t=\frac{1}{\Delta \omega}$. Since the universe has a finite age, it is simply not possible to determine (or define) the energy of any photon to greater precision than that - which effectively means that the number of distinct frequencies that exist in the EM spectrum (in the sense that they could be distinguished) is limited.

To claim there are more possible frequencies is something that I don't believe could be proven... Note also that the question you linked in your question (and the answers thereto) touch on the fact that "color" as perceived is in principle any combination of wavelengths; and if you can have "any number of photons with any infinite number of wavelengths", you do indeed end up with $\infty^{\infty}$ combinations. But the question in your title is just about "possible wavelengths" - and if we accept that a given photon has just one wavelength (one energy - within the bounds of uncertainty) then you go back to "countably infinitely many".

Answer 2

Wavelength for electromagnetic waves is defined

the distance over which the wave's shape repeats

Distance is mapped one to one with real numbers so in classical electromagnetic waves there can be an infinity of wavelengths as there exists an infinity of real numbers between two real numbers.

Your question seems to concentrate on the Planck length:

$$l_P=\sqrt{\frac{\hbar G}{c^3}} \approx 1.616299(38) \times 10^{-35}m$$

This is a very small number but still in approaching it, between two very small numbers there is an infinity of numbers, therefore an infinity of wavelengths differing very little from each other but still there.

BUT note that $\hbar$ enters in the definition of Planck length. This means that one can no longer talk of a classical electromagnetic wave without considering the quantum mechanical frame of photons, which build up the classical wave.

At such small wavelengths the frequencies are enormous, and the photon is characterized by $E=h\nu$, where $\nu$ is the frequency of the classical electromagnetic wave which can be built up by such frequency photons.

There is currently no proven physical significance of the Planck length; it is, however, a topic of theoretical research. Since the Planck length is so many orders of magnitude smaller than any current instrument could possibly measure, there is no way of examining it directly. According to the generalized uncertainty principle (a concept from speculative models of quantum gravity), the Planck length is, in principle, within a factor of 10, the shortest measurable length – and no theoretically known improvement in measurement instruments could change that.

So until one has a definitive quantization of gravity, everything is speculations.

Nevertheless, in reply to your title, there is an infinite number of wavelengths because of the nature of real numbers, though the colors/frequenies will be mostly indistinguishable even in the low frequency spectra.

Answer 3

Be aware that current physics can not deal with temperatures above Planck temperature at which wave length will become Planck length. So maximum temperature are limited as far as minimum.