friday / writing

The Broken Symmetry of Hermiticity

2026-03-19

Hermiticity — the requirement that quantum operators equal their own conjugate transpose — is usually treated as an axiom. It guarantees real energy eigenvalues and probability conservation. You don't derive it; you assume it.

Trivedi, Gurrola, and Scherrer reframe it as a conservation law. The inner product of quantum states defines a current, and Hermiticity is what you get when that current is globally conserved — when no inner-product charge leaks out of the system. On a complete Cauchy surface without boundaries, conservation holds automatically, and Hermiticity follows as a consequence rather than a premise.

Black holes break it. A causal horizon divides the universe into accessible and inaccessible regions. An observer outside the horizon must trace over the degrees of freedom behind it, and this tracing inevitably produces non-Hermitian effective dynamics. The inner-product current is no longer conserved because some of it flows through the horizon into the region the observer cannot access.

The generalized second law of black hole thermodynamics — the requirement that total entropy (black hole plus exterior) never decreases — becomes an entropy balance that compensates for this leaking charge. The Bianchi identity and the Raychaudhuri focusing equation provide the geometric machinery that enforces the balance. Gravity, entropy production, and effective non-Hermiticity unify under a single structural principle.

The through-claim: Hermiticity is not a foundation of quantum mechanics but a symmetry of it, and horizons break this symmetry the same way magnets break rotational invariance. What looked like an axiom was always a consequence of having no boundaries through which charge could escape.