Planck's catastrophe?

Question

In deriving Planck's blackbody formula, the number density of normal modes (per unit frequency$^\dagger$) is found, given by $$ N(\omega)=\frac{V}{\pi^2c^3}\omega^2, $$ where $V$ is the volume of the blackbody. Then the average energy of a mode of frequency $\omega$ is calculated using Planck's quantization that energy of a normal mode with frequency $\omega$ can only be $n\hbar\omega$ for some whole number $n$.

But hold on a minute!

The energy of a linear quantum oscillator with frequency $\omega$ can be $(n+1/2)\hbar\omega$, and since in deriving Planck's formula, it is the analogy of normal modes of the electromagnetic waves inside the blackbody cavity with the normal modes of the linear oscillator is made, I'd expect that we take into account this "zero-point energy", $\epsilon_\omega = \hbar\omega/2$ for each normal mode. Note that this zero-point energy, $\epsilon_\omega$ depends on $\omega$.

But if I do take this into account then I get bizarre results!

The average energy of normal mode of frequency $\omega$ is then given by $$ \langle E_\omega \rangle = \frac{\hbar\omega}{e^{\hbar\omega/k_BT}-1} + \epsilon_\omega, $$ and this leads to the following energy density.

\begin{align} \rho(T, \omega) &:= \frac{1}{V} N(\omega)\langle E_\omega \rangle\\ &\;= \frac{\hbar}{\pi^2c^3}\frac{\omega^3}{e^{\hbar\omega/k_BT} - 1} + \underbrace{\frac{1}{\pi^2c^3}\omega^2\epsilon_\omega}_{\text{additional term}} \end{align}

Now, this result is catastrophic! Unless $\epsilon_\omega = 0$ for all $\omega$'s (in which case this will coincide with the "correct" Planck's formula), $\rho$ diverges as $\omega\to\infty$.

Questions: So what's the way out? Is the often-presented analogy with quantum oscillator plainly wrong? For electromagnetic radiation, is the zero-point energy exactly zero for all $\omega$'s?

---

$^\dagger$ By frequency, I mean angular frequency.

Answer 1

Thermodynamics is all about the study of change. For example, the first law $dU=PdV + TdS$ expresses the change of energy in terms of work and heat flow.

Note that the zero point energy depends only on frequency and fundamental constants. There is no external parameter we can vary to change the zero point energy. So the zero point energy is, thermodynamically speaking, decoupled from the rest of the system, and therefore can be ignored.

Now there are some caveats.

First, it's not quite true that the fluctuations do not couple to an external parameter. If you consider a box with movable, reflecting walls, then there is a difference in how much zero point energy is contained in the box, then there would be in an equivalent amount of empty space without the reflecting walls. The reason is that the reflecting walls imposes boundary conditions which remove some of the modes from the sum over $n$. This difference in energy leads to the Casimir force acting on the walls of the box. However, this is a very small effect.

Second, gravity should couple to these fluctuations. There is an infinite (or at least very large) amount of energy density sourced by the zero-point fluctuations that should be causing the Universe to accelerate at an infinite (or at least very large) rate, but we don't observe this. This is the cosmological constant problem, to which there is no universally agreed-upon solution.