A glimpse into the Ultrametric spectrum

The Big Picture

Imagine measuring distance not by how far apart two cities are on a map, but by how many levels up a family tree you have to climb to find a common ancestor. That alternative geometry, called an ultrametric, treats things as “close” when they share deep common ancestry, not when they happen to sit near each other in space.

This idea sits at the center of a decades-old proposal for rebuilding string theory from the ground up. Two theorists have now taken a serious step toward making that proposal work, and they’ve found a hidden spectrum that behaves much like ordinary strings.

String theory’s most famous prediction is an infinite tower of vibrating states, each successive mode heavier than the last, arranged with mathematical regularity. That regularity has a thermodynamic consequence: the number of ways to pack a given amount of energy into string vibrations follows a formula Hardy and Ramanujan proved over a century ago. It’s an entropy fingerprint that distinguishes string theory from nearly everything else.

The question An Huang and Christian Jepsen set out to answer: can you reconstruct that fingerprint in a world built not on ordinary numbers, but on p-adic numbers? In this alternative number system, “closeness” is measured by divisibility rather than size, producing a universe with a naturally tree-like, hierarchical structure.

They can. But the ultrametric string spectrum doesn’t just reproduce Hardy-Ramanujan scaling. It also shows log-periodic fluctuations, oscillations that recur each time the energy increases by a fixed multiplicative factor. These have no counterpart in the ordinary string.

Key Insight: Exponentially growing energy levels and exponentially growing degeneracies in the p-adic string spectrum cancel each other out to preserve the thermodynamic signature of ordinary string theory, but leave a distinctive oscillating imprint.

How It Works

The analysis starts with an analogy. A string, discretized into a chain of beads, is mathematically equivalent to an infinite path graph where every interior vertex has exactly two neighbors. The Bruhat-Tits tree (also known as the Bethe lattice) is the natural p-adic analog: a branching tree where every vertex has p+1 neighbors. Picture an infinite family tree where every member has exactly the same number of descendants, forever. The question is whether quantizing vibrations on this tree-graph produces the same thermodynamic behavior as quantizing vibrations on a string.

At first, no. When Huang and Jepsen associated quantum harmonic oscillators to the normal modes (the independent vibration patterns of the whole system, like the distinct notes a plucked guitar string can ring out) of finite tree-graphs, the results depended critically on boundary conditions:

• Neumann boundary conditions (free ends) produce a non-trivial spectrum with exponentially spaced modes, each with exponentially growing degeneracy.
• Dirichlet boundary conditions (fixed ends) produce an infinitely large gap at the bottom, useless for building a continuous tower of states.
• Periodic boundary conditions essentially can’t be taken to the infinite-size limit for trees with p > 1, because no suitable infinite Moore graph exists.

So the simple tree approach fails. But the Neumann case hinted at something: even though the overall scaling is wrong, the spectrum has exponentially spaced levels with exponentially growing degeneracies. Could these two effects balance?

The real progress came from shifting focus to the p-adic circle, written Up. This is the set of p-adic numbers with unit norm, the hierarchical world’s equivalent of the ordinary unit circle that underlies standard string quantization. The relevant operator is the Vladimirov derivative, the p-adic analog of ordinary differentiation: a mathematically rigorous way to measure how quickly functions change in an ultrametric space. Computing its eigenvalues on Up yields concrete spectra for small primes:

• p = 2: energies 2, 5, 11, 23, 47, 95, … with degeneracies 1, 2, 4, 8, 16, 32, …
• p = 3: energies 1, 5, 17, 53, 161, 485, … with degeneracies 1, 4, 12, 36, 108, 324, …

The pattern jumps out. Energies grow roughly like p^n, while degeneracies grow like (p−1)·p^(n−1). The ratio stays controlled in just the right way.

To extract the thermodynamics, Huang and Jepsen computed the partition function, the statistical mechanics tool that tallies all possible states weighted by their thermal probability. Both energies and degeneracies grow exponentially, so standard approximation schemes break down. They developed a systematic expansion showing that entropy scales as the square root of energy, exactly as in the ordinary non-relativistic string.

But the entropy doesn’t simply scale like √E. It oscillates. The leading behavior is Hardy-Ramanujan-like, modulated by log-periodic fluctuations that repeat every time the energy increases by a fixed multiplicative factor. This is the signature of discrete scale invariance: the ultrametric world is hierarchical by construction, and that hierarchy imprints itself on the thermodynamics.

The same spectrum also drops out of a completely separate calculation as the eigenspectrum of the Neumann-to-Dirichlet operator on Up, a boundary-value operator that maps how steeply a function slopes at an edge to the function’s value there. That it appears twice, from independent derivations, reinforces its mathematical naturality.

Why It Matters

P-adic string theory has been discussed since the 1980s, but the original Freund-Olson formulation contained only a single state (the tachyon), making it phenomenologically barren. Huang and Jepsen’s construction points toward a richer ultrametric string theory with a genuine tower of states, bringing it much closer to the real thing.

The log-periodic fluctuations are not a bug. Such oscillations appear in critical phenomena on hierarchical lattices, in systems near catastrophic failure, and in certain quantum gravity models. Their presence in the ultrametric string entropy reflects a concrete mathematical fact: discrete scale invariance leaves oscillatory fingerprints wherever it goes. Whether these oscillations survive in a fully Lorentz-covariant setting, and whether they carry observable consequences, remains to be seen.

There’s an experimental angle, too. Researchers have begun engineering spin systems that interpolate between ordinary linear order and p-adic (2-adic) hierarchical order. As experimental control of ultrametric systems improves, these thermodynamic signatures may eventually become directly testable.

Bottom Line: By computing the p-adic analog of the string’s vibrational spectrum, Huang and Jepsen show that ultrametric string theory can reproduce the hallmark thermodynamic scaling of ordinary strings, opening a new path toward a hierarchical reformulation of string theory with a rich state tower.