The Stone-von Neumann theorem makes canonical quantization unique only after one imposes strong continuity of the Weyl translation groups. We show that this continuity assumption cannot be certified by any finite-resolution interrogation of the Weyl relations. A finite protocol probes only finitely many phase-space displacements and bounded expectation values, thereby determining a protocol algebra \(C_S\) generated by a finitely generated subgroup of Weyl translations, but not the regularity of an extension to the full Weyl algebra. Using the polymer representation as a canonical non-regular representation, we prove that for every regular Weyl representation, every normal state, and every finite family of bounded protocol observables, a polymer state reproduces the same expectation values to arbitrary prescribed accuracy. Thus finite agreement with Schrödinger quantum mechanics does not operationally certify the regularity hypothesis entering uniqueness. The result does not assert physical equivalence between regular and polymer quantizations, it identifies where additional input, such as Hamiltonian dynamics, energy regularity, semiclassicality, or a continuum limit, is required to distinguish or select representations. In this precise sense, non-regular polymer kinematics are not excluded by finite Weyl data alone, a point of direct relevance to quantum-gravity motivated quantizations.
The Stone-von Neumann theorem makes canonical quantization unique only after one imposes strong continuity of the Weyl translation groups. We show that this continuity assumption cannot be certified by any finite-resolution interrogation of the Weyl relations. A finite protocol probes only finitely many phase-space displacements and bounded expectation values, thereby determining a protocol algebra (C_S) generated by a finitely generated subgroup of Weyl translations, but not the regularity of an extension to the full Weyl algebra. Using the polymer representation as a canonical non-regular representation, we prove that for every regular Weyl representation, every normal state, and every finite family of bounded protocol observables, a polymer state reproduces the same expectation values to arbitrary prescribed accuracy. Thus finite agreement with Schrödinger quantum mechanics does not operationally certify the regularity hypothesis entering uniqueness. The result does not assert physical equivalence between regular and polymer quantizations, it identifies where additional input, such as Hamiltonian dynamics, energy regularity, semiclassicality, or a continuum limit, is required to distinguish or select representations. In this precise sense, non-regular polymer kinematics are not excluded by finite Weyl data alone, a point of direct relevance to quantum-gravity motivated quantizations.
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. ----- 同じ「正準交換関係」という一語が、三つの別の状況を指している——ストーン=フォン・ノイマンの一意性は、どこで現れ、どこで消えるか。 論文54 第3版は、パウリの定理が要求するのは正準交換関係ではなくワイル関係であることを示し、そこで止めた。本稿はその続きを書く——ワイル関係は、何を与えるのか。答えは一意性であり、しかもその一意性は狭い柵の内側にしか存在しない。新しい数学定理も新しい物理法則も主張しない。ストーン=フォン・ノイマンの定理(1930/31)、ネルソンの反例(1959)、無限自由度における非同値表現の存在、ハーグの定理(1955)は、いずれも標準的な内容である。 第一に、一語が三つの状況を指している。(a)稠密領域上の交換関係だけを課す場合——同値でない既約表現が複数存在する(ネルソンの反例、1959)。(b)ワイル関係を課し、自由度が有限で、可分空間上の既約表現に限る場合——ちょうど一つ(ストーン=フォン・ノイマンの定理、1930/31)。(c)ワイル関係を課すが、自由度が無限の場合——非同値な表現が非可算個ふたたび現れる(場の量子論。ハーグの定理 1955 がその物理的帰結である)。 第二に、これが本稿の芯である。一意性は真ん中の行にしかない。関係を弱めれば失われ、自由度を無限にすればまた失われる。一意性は、両側を柵で挟まれた狭い領域にだけ現れる。そして第二行と第三行を分けているのは、課す関係ではない。どちらもワイル関係を課している。分けているのは自由度が有限か無限か、ただそれだけである。これは体系が繰り返し出会ってきた形——ある閾値を越えて内容が変わる(論文114・105、そして論文126 の「三変数」)——の、有限と無限の境界における実例である。 第三に、なぜ領域の問題が避けられないかは、有限次元で確かめられる。交換関係の両辺のトレースを取ると、トレースの巡回性から左辺はゼロ、右辺は次元に比例する量になる。したがって有限行列はこれを満たさない(本稿で確認)。作用素は非有界でなければならず、非有界であるとは定義域を持つということである。ワイル形式はすべて有界なユニタリ作用素で書かれており、定義域の問題を持たない。だからこそ、そこで一意性が述べられる。 第四に、ガラポンの反例の住所が特定できる。論文54 が引いた構成——下に有界なハミルトニアンに正準共役な、有界かつ自己共役な時間作用素(ガラポン 2002)——は、第一行に住んでいる。交換関係だけを課してワイル関係を課さないから存在できるのである。ワイル関係を課せば、ストーン=フォン・ノイマンがシュレーディンガー表現を強制し、そこではパウリの議論がそのまま走る。ガラポンの構成は、ストーン=フォン・ノイマンの柵の外に立っている。矛盾ではない。定理の届かないところに立っているだけである。これが「住所を書く」ということである(論文17)。「パウリの定理に反例がある」と言うのは半分であり、その反例がどの条件を落としているかを書いて初めて、主張は検査可能になる。 第五に、論文126 と対をなす。論文126 は、存在定理は存在しか与えないと述べた。本稿が扱うのは一意性定理である。ストーン=フォン・ノイマンは「どれか」を決める。ただし条件を数えると、フロベニウスが要求するのは可積分性ひとつ、ストーン=フォン・ノイマンが要求するのはワイル関係・有限自由度・既約性・可分性と強連続性の四つである。存在を主張するのは安く、一意性を主張するのは高い。そして高い代金を払える範囲が、そのまま柵の内側である。一意性定理があると聞いたら、まずその柵を数えよ。 重要(射程):確立された標準的事実の点呼(第2層)。ストーン=フォン・ノイマンの定理、ネルソンの反例、無限自由度の非同値表現、ハーグの定理は、いずれも標準的な内容である。示さないこと:新しい数学定理(一つもない)/新しい物理法則(一つもない)/ストーン=フォン・ノイマンの定理に誤りがあること(ない)/パウリの定理が破れていること(破れていない。反例は定理の外側に立っている)。加えたもの:三つの状況を一つの点呼表に並べたこと、一意性が成り立つ柵を書いたこと、ガラポンの反例の住所を特定したこと、論文54 が空けたままにしていた行を埋めたこと。残るもの:一つの語が三つの状況を指し、そのうち一つでだけ一意性が成り立ち、その一つは両側を柵で挟まれている。 作成にあたって:本稿の着想と内容は、著者自身の考察に基づくものです。文章の構成整理や英訳、数式の確認には AI(大規模言語モデル)の助力を得ました。最終的な内容の解釈や誤りがあれば、それらはすべて著者の責に帰します。お気づきの点があれば、ご教示いただければ幸いです。
"Polymer quantum mechanics" is the name given to a quantization scheme inspired by loop quantum gravity in which the configuration space of the theory is chosen to have a discrete topology. Polymer quantization yields a representation of the canonical commutation relations that is genuinely distinct from the conventional "Schrödinger" representation. In this paper, we summarize the main features of polymer quantum mechanics and investigate in detail the polymer quantization of systems with configuration spaces that are classically compact. We show explicitly how using the standard construction of polymer states leads to a Hilbert space of states defined on a finite graph of points. By way of example, we find the exact energy eigenvalues and eigenfunctions for a particle on a ring and a particle in a box defined on such lattices, and discuss similarities and differences from standard Schrödinger quantum mechanics. We also explore the continuum limit of states in these systems, and demonstrate in detail how the exact eigenfunctions in the position representation approach their continuum counterparts.
In this paper, a version of polymer quantum mechanics, which is inspired by loop quantum gravity, is considered and shown to be equivalent, in a precise sense, to the standard, experimentally tested, Schroedinger quantum mechanics. The kinematical cornerstone of our framework is the so called polymer representation of the Heisenberg-Weyl (H-W) algebra, which is the starting point of the construction. The dynamics is constructed as a continuum limit of effective theories characterized by a scale, and requires a renormalization of the inner product. The result is a physical Hilbert space in which the continuum Hamiltonian can be represented and that is unitarily equivalent to the Schroedinger representation of quantum mechanics. As a concrete implementation of our formalism, the simple harmonic oscillator is fully developed.
In this paper, a version of polymer quantum mechanics, which is inspired by loop quantum gravity, is considered and shown to be equivalent, in a precise sense, to the standard, experimentally tested, Schroedinger quantum mechanics. The kinematical cornerstone of our framework is the so called polymer representation of the Heisenberg-Weyl (H-W) algebra, which is the starting point of the construction. The dynamics is constructed as a continuum limit of effective theories characterized by a scale, and requires a renormalization of the inner product. The result is a physical Hilbert space in which the continuum Hamiltonian can be represented and that is unitarily equivalent to the Schroedinger representation of quantum mechanics. As a concrete implementation of our formalism, the simple harmonic oscillator is fully developed.
This thesis addresses the representations of the Weyl algebra of quantum mechanics with an explicitly Born reciprocal structure and the dynamics of physical systems in these representations. In this context, we discuss both the regular modular representations based on Aharonov's modular variables, as well as the irregular modular polymer representations inspired by the spin network representation in loop quantum gravity. \n \nWe introduce the modular space as the quantum configuration space corresponding to the modular representation and treat it as an alternative to the classical configuration space. The modular representations have a built-in Abelian gauge symmetry. We discuss the singular limits in which the modular representation converges to the Schrodinger and momentum representations, and show that these limits require two distinct fixings of the modular gauge symmetry. \n \nIn order to explore the propagation of quantum states in the modular space, we construct a Feynman path integral for the harmonic oscillator explicitly as the transition amplitude between two modular states. The result is a functional integral over the compact modular space, which shows novel features of the dynamics such as winding modes, an Aharonov-Bohm phase, and a new action on the modular space. We compare the stationary trajectories that extremize this modular action to the phase space solution of Hamilton's equations in the Schrodinger representation and find a new translation symmetry. We also identify the other symmetries of the harmonic oscillator in the modular action and show their correspondence to the classical case. \n \nWe generalize the relationship between the classical Hamiltonian function and the modular action by postulating a new modular Legendre transform. For demonstration we apply this transformation on the Kepler problem and reformulate it in terms of modular variables. \n \nWe then switch our focus to the representations of the Weyl algebra that violate the assumptions of the Stone-von Neumann theorem. In a similar fashion to the polymer representations studied as a toy model for quantum gravity, we polymerize the modular representation to obtain an inequivalent representation called the modular polymer (MP) representation. \n \nThe new MP representation lacks both position and momentum operators, therefore the Hamiltonian of the harmonic oscillator is regularized with the Weyl operators and the non-separable MP Hilbert space is split into superselection sectors. The solutions to the Schrodinger equation fall into two categories depending on whether the ratio between the modular scale and the polymeric regularization scale is a rational or irrational number. We find a discrete and finite energy spectrum in each superselection sector in the former case, and a continuous, bounded one in the latter.
We discuss the q deformation of Weyl-Heisenberg algebra in connection with the von Neumann theorem in Quantum Mechanics. We show that the q-deformation parameter labels the Weyl systems in Quantum Mechanics and the unitarily inequivalent representations of the canonical commutation relations in Quantum Field Theory. The von Neumann theorem in Quantum Mechanics (QM) states that for systems with finite number of degrees of freedom the representations of the canonical commutation relations (ccr), also called the Weyl representations, are each other unitarily equivalent. In the limit of infinite number of degrees of freedom, namely in Quantum Field Theory