The Thing Fits Inside The Thing

I was reading this Substack post, which argues that the laws of physics may be too complex to ever be known, because the mathematical object encoding them is too large to fit inside the universe. I like what is being gestured towards and found it interesting, but I also think the argument is wrong.

To be clear, the author is vastly better informed and credentialled than I am, quite possibly using this figuratively, and acknowledges that its more of a gesture toward the mystery of the possible than a claim. But it struck me when I first read it, and I didn’t know to what extent it was accurate, so I just had to check it for fun - I am not a physicist, I don’t know a shred of quantum mechanics, in fact I was once described as “barely literate” by Nobel Laureate Roger Penrose, but I do know how to count.

Following the piece, we can describe the state of a physical system by a list of \(N\) numbers (one number per possible configuration), recording how much “probability amplitude” that configuration carries. For the observable universe, \(N\) is big big, roughly \(N \sim 10^{10^{122}}\), apparently.

The laws of physics are encoded in a matrix called the Hamiltonian, which is basically a transition table telling you how the state evolves in time, the basic point here is it therefore has \(N^2\) entries, and \(N^2 \gg N\) so you may not be able to encode the laws in the state. But this is only true if the Hamiltonian is extremely incompressible, if its entries are completely arbitrary and independent.

If it has pretty much any structure at all, which is the only case in which trying to describe it even makes sense, then it almost certainly can be encoded in itself, and probably in a relatively small number of bits.

Think of a universe consisting of \(N\) platonic switches, each either on or off. A state of this universe, an assignment of on/off to each site, is a list of \(N\) bits. The Hamiltonian is a rule that says, for every pair of states \((i, j)\), how strongly the system “wants” to transition between them. Written as a matrix, entry \(H_{ij}\) encodes the transition amplitude from state \(j\) to state \(i\), and in full generality, specifying every \(H_{ij}\) independently requires \(N^2\) numbers.

But we do not live in such a world, the laws of physics contain structure, and even the simplest, most uncontroversial structure is enough to reduce the number of independent parameters enough to fit inside the universe. As far as we know, our universe has no preferred spatial location, the laws of physics work the same way in Ibiza as they do on Tau Ceti, and this translation invariance means the rule governing how site \(i\) interacts with site \(j\) depends only on the distance between them - not on where they actually are. In matrix terms: \(H_{ij} = h_{j-i}\) for some function \(h\).

A matrix with this property, where every diagonal contains the same repeated value, is called a circulant matrix. It looks like this:

Every row is just the previous row shifted by one position, so the entire \(N \times N\) matrix is determined by its first row alone. The first row has \(N\) numbers \(\to\) the laws fit in the state.

Kicker, this is not the only symmetry! So even if all of our specific theories are wrong, but just translation invariance is real, then this is definitely incorrect; and if enough other symmetries are also real, then maybe we can write down the laws of the universe in one screenshot of Club Penguin.¹

Not only that, but the original estimate for the size of the state space is based on the entropy of the portion of the universe we will ever observe, but this is wildly overcounting what we actually need to “explain”. A universe of a trillion fair coin flips has a lot of entropy, but the laws governing those coin flips are trivial, and assuming you’re comfortable with indeterminacy, you can write down the laws on 6 grains of sand with no dependence on the state or its size at all.