Why the Hamilton operator alone is not enough

TL;DR

The paper demonstrates that the Hamilton operator alone cannot fully determine quantum physics, emphasizing the necessity of additional structures like canonical operators and system decompositions for a complete description.

Contribution

It provides an explicit example showing the non-uniqueness of Hamilton operators and argues against deriving preferred bases solely from the Hamilton operator.

Findings

Hamilton operator alone does not fix the physics.

Canonical operators p, q are necessary for complete description.

Preferred basis cannot be derived solely from the Hamilton operator.

Abstract

In the many worlds community seems to exist a belief that the physics of a quantum theory is completely defined by it's Hamilton operator given in an abstract Hilbert space, especially that the position basis may be derived from it as preferred using decoherence techniques. We show, by an explicit example of non-uniqueness, taken from the theory of the KdV equation, that the Hamilton operator alone is not sufficient to fix the physics. We need the canonical operators p, q as well. As a consequence, it is not possible to derive a "preferred basis" from the Hamilton operator alone, without postulating some additional structure like a "decomposition into systems". We argue that this makes such a derivation useless for fundamental physics.