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# Overestimating the Zero-Point Energy

Epistemic note: I am only minimally knowledgable in matters related to cosmology. My impression is that the observation below is “well known to those who know it well”.

There’s a standard story of the zero-point energy of the standard model—that is, the energy density of the vacuum state of the standard model. We’re going to try to estimate it from theory, and then estimate it from cosmological observations, and the two results differ by $120$ orders of magnitude.

The estimate from theory isn’t complicated. Let’s pretend that the standard model consists of a single scalar particle of mass $m$. The energy is just the sum of energies in every available momentum mode. Recalling that the vacuum energy of a harmonic oscillator with gap $\omega$ is $\frac\omega 2$, we find: $\epsilon = \int d^3 k \frac{\sqrt{m^2 + k^2}}{2} \text.$ Alas, this integral diverges! What can we do? The standard approach here is to notice that eventually, the momenta $k$ get large enough that Planck-scale physics (presumably quantum gravity) takes hold. We don’t know what happens there, but the integral along the way up is approximately $\Lambda^4$, where $\Lambda$ is the Planck cutoff scale. So we’ll just use that as a first approximation, neglecting whatever happens at the Planck scale.

This results in an unrealistically enormous zero-point energy: $\sim 10^{100}\;\mathrm{g}\,\mathrm{m}^{-3}$. The estimate from cosmology is far smaller—a factor of $10^{120}$. (An easy mnemonic is that it’s comparable to what you would get by using the cosmic microwave background to set the energy scale instead. This is, hopefully, a coincidence.)

At this point we’re in the land of many excuses. Here I want to make one narrow point about the calculation above: just from the standpoint of quantum field theory (that is, without needing to worry about cosmology, or quantum gravity, or any other difficult subjects), the calculation is already suspicious enough to probably be wrong.

To see why, let’s re-do the calculation, but now with a massless field. The energy density is $\epsilon = \int d^3 k \;|k| \sim \int_0^\Lambda dk \; k^2 |k| \sim \Lambda^4 \text.$ This is the same as it was before—no surprise, since $m \ll \Lambda$. But now we have a far more immediate problem than some vague mumbo-jumbo about the cosmological constant. The free massless scalar field is a conformal field theory, meaning, among other things, that it has no intrinsic length scale. The true energy density of a free massless scalar field is $0$. In that divergent integral above, we know what the correct regularization is. The integral must evaluate to zero.

Do not be deceived: I am implicitly invoking a somewhat questionable principle. It’s true that the energy density of a massless scalar field, at zero temperature, must be $0$. But the real universe is not a CFT; the real universe has some sort of cutoff. How can we ignore that? Well, most properties of a continuum QFT are defined by a limiting process, whereby the cutoff scale $\Lambda$ is lifted, and the masses (for example) of the QFT are defined by the limit $\Lambda\rightarrow \infty$ of their values in the regularized theories. I am assuming that the same idea must also hold for the “true” energy density. In other words, the assumption is that when $\Lambda$ is far larger than all other scales, the theory must look exactly like the theory with no cutoff at all. As far as I know, you’re free to disbelieve this!

Assuming you accept that principle, the energy density is $0$ or (at finite $\Lambda$) very close. In fact every dimensionful quantity vanishes in a CFT, so we have $\epsilon = \frac{d\epsilon}{dm} = \frac{d^2\epsilon}{d^2m} = \frac{d^3\epsilon}{d^3m} = \frac{d^5\epsilon}{d^5m} = \cdots \text.$ The only one missing is the fourth derivative, which as a dimensionless quantity is permitted to be non-zero in a conformal field theory. As a result, we can integrate back the mass to approximate $\epsilon \sim -m^4$ This is in retrospect unsurprising: after all, we had only one mass scale to begin with. (The sign is perhaps surprising, but I have nothing useful to say about that.)

The Planck energy is about $10^{17}$ times the mass of the top quark, so this removes 68 of our 120 problematic orders of magnitude. In the context of the standard model and real-world cosmology, this resolves nothing. Any SM mass scale is still far too large to yield a sensible cosmological constant. It seems to me, though, that the oft-repeated story is considerably overstated.

# When Luck Isn't

Sometime around 75 BC, Gaius Julius Caesar was kidnapped by pirates who held him for ransom. Supposedly, Caesar was a remarkably chummy captive, going so far as to demand that the pirates ask for a higher ransom, to indicate his true worth. After being freed, Caesar raised a fleet, hunted down the pirates, and had them executed.

As far as I can tell, this was not normal behavior for pirate captives. After all, most ransomed captives are going to be relatively wealthy, but perhaps not wealthy enough to raise a fleet. Moreover, it requires a certain single-minded energy, and maybe military experience, to pull off the revenge. The pirates, in this case, got unlucky. There were many people they might have captured and held for ransom; of those many people, very few would have their names turned into words like Kaiser and Czar.

Two millennia later, the U.S. government accused Hsue-Shen Tsien of communist sympathies and stripped him of his security clearance. Over the course of a few months in 1950-1951, he was variously questioned, arrested, told that he would be deported, and forbidden from leaving the country (or indeed his county). He was permitted to leave the U.S. for China five years later, at which point his sympathies had, perhaps shifted. Tsien went on to be dubbed “The Father of Chinese Rocketry”, among other things.

Various high U.S. officials considered the treatment of Tsien to be a self-inflicted wound. Quoth the Secretary of the Navy: “It was the stupidest thing this country ever did. He was no more a communist than I was, and we forced him to go.” It cannot be said, however, that these events were unlucky from the perspective of the American government. In the last century, Tsien is far from the only person accused of inappropriate foreign sympathies under questionable evidence. Tsien is also not Caesar—there are many people whose defection would be a substantial harm to U.S. national security. It is therefore not at all surprising that something of this sort should happen. Tsien’s specific case is a matter of luck, but the general story was not improbable.

Straightforwardly: if you play Russian roulette exactly once, and lose, then you got unlucky. If you play five times, and lose on the fifth, it’s no longer appropriate to invoke “luck”. Yes, the specific final event, viewed in isolation, is unlucky: there was only a 17% chance of that happening. But with five opportunities, it was more likely than not that one of the rounds would be a loss.

The examples of Caesar and Tsien indicate a profound shift undergone by the world’s wealthier societies. Two thousand years ago, routinely mistreating strangers (perhaps chosen reasonably judiciously) carried with it no large probability of negative consequences. Caesar’s pirates were thoroughly punished, but this outcome was far from certain, so that it can be said they were unlucky. But the strategic loss suffered by the U.S. government after Tsien’s exile was not a matter of luck. It’s not the case that most people have the ability to exact an effective revenge, but a high enough percentage do that “accuse random Chinese people of being communist” is a reliably bad strategy.

It is important, by the way, that these disproportionately powerful people are powerful in illegible ways. Tsien was clearly brilliant, but one could not have reliably predicted in 1948 that he would be such a towering figure in China. Because these people are “hidden” (and their identities may not even be predetermined), they effectively serve as an umbrella, deterring even bad behavior not directed at them.

Which brings me, of course, to Gawker and Thiel. Briefly, Gawker outs Thiel as gay in 2007. Thiel turns out to be not just wealthy, but in possession of a Caesar-style demonic single-mindedness: over the next decade he spends millions of dollars to fund lawsuits against Gawker, finally bankrupting them in 2016. Was Gawker unlucky? No, as you can tell from the name, violating the privacy of relatively wealthy people was kind of their bread and butter. As before, this event looks unlucky when viewed in isolation, but really should be expected in the bigger picture. Thiel’s own understanding of this event seems in line with the idea of the umbrella: “It’s less about revenge and more about specific deterrence.”

Maybe this whole post is just “applied central limit theorem”. It’s always tempting, when looking at a rare event, to fixate on the specifics. From this vantage point, the U.S. government should have been more aware of the value of Tsien, and Gawker would have done better to tread carefully around Thiel (or perhaps Hulk Hogan). I think a better view is to accept that the Thiels and Tsiens of the world can’t really be predicted in advance. The FBI, and Gawker, were simply playing an extended game of Russian roulette—habitually mistreating people presumed powerless to do anything about it. If you want to avoid the losses, the solution is to not play the game.