One phrase, the canonical commutation relation, covers three different situations – where the Stone-von Neumann uniqueness appears, and where it disappears.
The third edition of Paper 54 showed that what Pauli’s theorem requires is not the canonical commutation relation but the Weyl relation, and stopped there. This paper writes the next line: what does the Weyl relation buy? The answer is uniqueness, and that uniqueness lives only inside a narrow fence. NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED. The Stone-von Neumann theorem (1930/31), Nelson’s counterexample (1959), the existence of inequivalent representations at infinitely many degrees of freedom, and Haag’s theorem (1955) are all standard.
First, one phrase covers three different situations. (a) Imposing only the commutation relation on a dense domain: several mutually inequivalent irreducible representations exist (Nelson’s counterexample, 1959). (b) Imposing the Weyl relation, with finitely many degrees of freedom, on a separable space, restricted to irreducible representations: exactly one (the Stone-von Neumann theorem, 1930/31). (c) Imposing the Weyl relation but with infinitely many degrees of freedom: uncountably many inequivalent representations reappear (quantum field theory; Haag’s theorem of 1955 is the physical consequence).
Second, and this is the core. Uniqueness lives only in the middle row. Weaken the relation and it is lost; let the degrees of freedom become infinite and it is lost again. Uniqueness appears only in a narrow region fenced on both sides. And what separates the second row from the third is not the relation imposed: both impose the Weyl relation. The only thing separating them is whether the degrees of freedom are finite or infinite. This is a shape the corpus has met repeatedly, content changing across a threshold (Papers 114, 105, and the three variables of Paper 126), instantiated here at the boundary between finite and infinite.
Third, why the Weyl form is needed can be checked in finite dimensions. Taking the trace of both sides of the commutation relation gives zero on the left, by cyclicity, and something proportional to the dimension on the right. Hence no finite matrices satisfy it (verified here). The operators must be unbounded; unbounded means domains. The Weyl form is written entirely in bounded unitaries and has no domain problem, which is exactly why uniqueness can be stated there.
Fourth, the address of Galapon’s counterexample can be pinned down. The construction cited in Paper 54, a bounded self-adjoint time operator canonically conjugate to a Hamiltonian bounded below (Galapon 2002), lives in the first row. It can exist because only the commutation relation is imposed and the Weyl relation is not. Impose the Weyl relation and Stone-von Neumann forces the Schrodinger representation, where Pauli’s argument runs unchanged. Galapon’s construction stands outside the Stone-von Neumann fence. It is not a contradiction; it merely stands where the theorem does not reach. This is what writing the address means (Paper 17). Saying that there is a counterexample to Pauli’s theorem is half the work; only by writing which condition the counterexample drops does the claim become checkable.
Fifth, this pairs with Paper 126. Paper 126 said that an existence theorem delivers existence and nothing more. What this paper treats is a uniqueness theorem: Stone-von Neumann does decide which. But count the conditions. Frobenius demands one, integrability; Stone-von Neumann demands four, the Weyl relation, finitely many degrees of freedom, irreducibility, and separability with strong continuity. Asserting existence is cheap; asserting uniqueness is expensive. And the range in which the higher price can be paid is exactly the inside of the fence. On hearing that a uniqueness theorem exists, first count its fence.
IMPORTANT (scope): a census of established standard material (Layer 2). The Stone-von Neumann theorem, Nelson’s counterexample, inequivalent representations at infinitely many degrees of freedom, and Haag’s theorem are all standard. What is not claimed: any new mathematical theorem (there is none) / any new physical law (there is none) / that the Stone-von Neumann theorem is in error (it is not) / that Pauli’s theorem fails (it does not; the counterexample stands outside it). What is added: placing the three situations in one census table, writing the fence within which uniqueness holds, pinning the address of Galapon’s counterexample, and filling the line Paper 54 left open. What remains: one word points at three situations, uniqueness holds in only one of them, and that one is fenced on both sides.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). One Phrase, “the Canonical Commutation Relation”, Covers Three Different Situations [Paper 127]. Zenodo.
DOI: 10.5281/zenodo.22079508
ORCID: 0009-0006-8587-3450