Max Planck (1858 - 1947) & Walther Nernst (1864 - 1941)


Zero-point energy and the vacuum

This page records the results of some investigations I made in 2023 regarding the origin of the idea of zero-point energy in the electromagnetic field and its implications for the vacuum density. What I had believed for many years was as follows:

But recently I discovered that Walther Nernst, the thermodynamics pioneer, made the same density calculation in 1916. How was this possible, given that zero-point energy had yet to be invented? A little googling indicated that Nernst had taken the idea of ZPE from a 1911 paper by Planck. Both of these historic contributions to science exist only in obscure conference proceedings, so how to know what the arguments were? With much appreciated help from Malcom Longair and Heige Kragh respectively, I obtained scans of the original document; with the aid of online OCR and translation facilities, plus some manual tidying up of equations, I was able to obtain English versions of the papers. The documents are given below (only the first part of Nernst's paper), but the key arguments are as follows.

Planck's 1911 argument

In 1900, Planck considered "oscillators" interacting with radiation. He assumed the oscillators could only emit or absorb energy in units of h nu, leading to his formula for Black-body radiation.

But in 1911 he went back on this, because he felt that sufficiently weak radiation could then never be absorbed. Thus he argued that emission from oscillators would be quantised but absorption would be continuous. Then we have the following picture (actually taken from his 1913 textbook):

The energy in oscillators grows linearly with time, until a sudden drop by h nu. Thus if we observe an oscillator at a random time, we must catch it half-way through its growth between E = n h nu and (n+1) h nu, so that its energy will exceed an integral number of quanta by h nu / 2 *on average*. So this is not Heisenberg's unique ZPE for each oscillator, nor is the idea of continuous absorption correct - but it gives the right answer.

The idea of additional ZPE seemed to make experimental sense in molecular energy levels. Today, we would say the energy is proportional to n(n+1), where n is the total angular momentum quantum number (with no ZPE, since n=0 is allowed). But in calculating energy differences, one may as well use (n + 1/2)^2. So apparently half-integral quantum numbers, as suggested by Planck, were a real thing.

Nernst's 1916 argument

Now, Planck was talking about the energy of material oscillators only: he made no suggestion that electromagnetic wave modes would have a ZPE, so he did not change his formula for black-body radiation. But Nernst was very worried by the idea that an oscillator with E < h nu could not radiate, since classically it would (this is also a problem for atoms, dealt with in Bohr's 1912 paper, but Nernst does not mention this). To fix this, he proposes that the electromagnetic field must then also have a ZPE: the oscillator comes into equilibrium with this by both emitting and absorbing radiation. As Nernst puts it:

"An electron oscillating in this way does nothing other than bring itself into equilibrium with the zero-point radiation; the laws of electrodynamics are not only not violated, but the zero-point energy is established according to the laws of electrodynamics."

Remarkably, by insisting on classical electrodynamics in terms of interaction with electrons, Nernst deduces the very un-classical idea of EM waves having a ZPE. He then goes on do the same calculation as Zeldovich and show that the vacuum density is colossal.

Einstein's missed opportunity

Having deduced the vacuum density, Nerst does not comment on its gravitational implications. This is not so surprising, as it was only the following year that Einstein introduced the cosmological constant. But it is hard not to speculate on what Einstein would have written in 1917 if he had been aware of Nernst's argument - there is no reason why Zeldovich's 1968 argument could not then have been fully in place at that time. I know of no evidence that Einstein did read Nernst's argument; if he did do so, it would seem odd that he did not mention the calculation and the implications for the cosmological constant. In any case, it is hard to understand why a further 50+ years had to pass before Zeldovich put the two pieces of the puzzle together. Perhaps there was discussion of the issue in the intervening years - hidden and overlooked, as the arguments of Planck and Nernst were to me for so long.

Papers

Links