# Bekenstein bound busted?

First, Bekenstein's bound has been found to hold in all physically-sensible relativistic quantum field theories for which it has been checked. See ref 1 for an example. It definitely is applicable to the quantum realm.

Second, Bekenstein's bound is a bound on the number of mutually orthogonal quantum states that have volume $$\leq \frac{4\pi}{3}R^3$$ and energy $$\leq E$$. We can also express this in terms of the number of bits of information that can be stored in that volume with that energy, but it's the same idea either way. It's the number of different ways the volume can be populated with physical things, if the energy is not allowed to exceed $$E$$.

Examples:

• If $$E=hf$$ with $$f=3$$ MHz, then nothing with energy $$\leq E$$ can fit in a sphere of radius $$R$$. In particular, a photon with energy $$\leq E$$ has a wavelength $$\gg R$$, so it doesn't fit. Neither does anything else with so little energy.

• If we increase the upper limit on the energy to, say, $$E=100$$ Joules, then Bekenstein's bound says that lots of mutually orthogonal quantum states have $$\leq$$ that much volume and $$\leq$$ that much energy.

A photon is a particle in the sense that photons can be counted, but a photon is no more pointlike than a classical EM wavepacket with the same wavelength. An optical photon is pointlike when compared to a human eye, but it's not pointlike at all when compared to an atom, and a $$3$$ MHz photon is not even as pointlike as a bus.

We can build a $$3$$ MHz $$100$$ Watt transmission antenna and point it into a $$1$$ meter spherical space, but radio waves with wavelengths $$\geq 100$$ meters can't produce features of size $$\sim 1$$ meter. If we do something to trap the EM radiation inside the sphere, then Bekenstein's bound tells us that we are no longer dealing with photons having energy $$hf$$ with $$f=3$$ MHz. In other words, if we actually solve this boundary-value problem for the EM field to see what cavity-modes are possible, we will discover that the minimum possible energy in one quantum of such a mode is $$\gg h\times 3$$ MHz.

Here's an excerpt from a review by Bekenstein himself (ref 2):

it is not always obvious in a particular example how the system avoids having too many states for given energy, and hence violating the bound. We analyze in detail several purported counterexamples of this type (involving systems made of massive particles, systems at low temperature, systems with high degeneracy of the lowest excited states, systems with degenerate ground states, or involving a particle spectrum with proliferation of nearly massless species), and exhibit in each case the mechanism behind the bound's efficacy.

1. Schiffer and Bekenstein (1989), "Proof of the quantum bound on specific entropy for free fields," Phys. Rev. D 39, 1109, https://journals.aps.org/prd/abstract/10.1103/PhysRevD.39.1109

2. Bekenstein (2005), "How does the entropy/information bound work?", https://arxiv.org/abs/quant-ph/0404042