Semiclassical Consistency of the Weil Representation on Heis₃(Z/qZ): Quantitative Sinc Embedding, Generator Convergence, and Aliasing Control

Overview

The Q5a Mosco convergence theorem establishes that the Weil representations $\rho_c^{(q)}$ on the finite Heisenberg group $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$ converge to the standard Schrödinger representation of the continuous Heisenberg group as $q \to \infty$, in the Mosco sense on $L^2(\mathbb{R})$.

H2 closes the last open hypothesis of this theorem. Hypothesis [H2] asserts strong operator convergence of the rescaled Weil generators: the discrete position and momentum operators $\hat{X}_q$ and $\hat{P}_q$ converge to their continuous counterparts $x$ and $-i\partial_x$ in the strong operator topology on $L^2(\mathbb{R})$.

The proof combines two main ingredients: a quantitative Poisson aliasing lemma controlling the $L^2$ error introduced by periodic aliasing in the discrete Fourier embedding, and a discrete Sobolev identity that relates the action of $\hat{P}_q$ to a finite-difference derivative whose strong limit is $-i\partial_x$.

Key consequence. After H2, all hypotheses of the Q5a large-$q$ Mosco convergence theorem are verified unconditionally on the admissible sector. The convergence of the Weil representation to Schrödinger mechanics is now complete.

Core contributions

Proof of [H2]: strong operator convergence $\hat{X}_q \to x$ and $\hat{P}_q \to -i\partial_x$ in the strong operator topology on $L^2(\mathbb{R})$, unconditionally on the admissible sector.
Quantitative Poisson aliasing lemma: explicit $L^2$ error bound for the periodic aliasing introduced by embedding discrete functions into $L^2(\mathbb{R})$ via sinc interpolation. The error is controlled uniformly in the $q \to \infty$ limit.
Discrete Sobolev identity: algebraic identity relating the action of the discrete momentum operator $\hat{P}_q$ to a finite-difference derivative, enabling the strong limit to be identified as $-i\partial_x$.
Quantitative sinc embedding: precise control of the sinc interpolation embedding $\ell^2(\mathbb{Z}/q\mathbb{Z}) \hookrightarrow L^2(\mathbb{R})$ with explicit convergence rates.
Completion of Q5a: H2 closes the last open hypothesis of the Q5a Mosco convergence theorem, establishing that the discrete-to-continuous convergence of the Weil representation is fully proved.

The Q5a Mosco convergence programme

The Q5a Mosco convergence theorem is the central result connecting the discrete Weil representation on the finite Heisenberg group to the continuous Schrödinger representation. It ensures that, as the prime $q \to \infty$, the discrete quantum mechanics on $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$ converges to standard quantum mechanics on $L^2(\mathbb{R})$.

This convergence is the mathematical foundation for the claim that quantum mechanics is not postulated in the Cosmochrony framework but emerges as the large-scale limit of the admissible projection on the Heisenberg substrate.

Hypothesis [H2] was the last remaining open condition. The proof given in H2 uses the sinc interpolation embedding — which maps finite discrete functions to band-limited functions in $L^2(\mathbb{R})$ — and shows that under this embedding, the discrete generators converge strongly to their continuous counterparts.

The Poisson aliasing lemma is the key technical tool: it provides quantitative control over the aliasing error that arises when a periodic discrete signal is embedded into the non-periodic $L^2(\mathbb{R})$. The discrete Sobolev identity then translates strong convergence of the difference quotient into strong convergence of the derivative.

Relation to the Cosmochrony programme

H2 completes the semiclassical consistency programme for the Weil representation. The Q5a theorem, now fully proved, is one of the central structural results of the Cosmochrony framework: it establishes that the finite Heisenberg substrate gives rise to continuous quantum mechanics in the admissible large-scale limit.

The paper W1 provided the analytic framework (Weil representation on the infinite Heisenberg group) used as the target of the convergence. Q5a formulated the Mosco convergence theorem and verified all hypotheses except [H2]. H2 closes the programme.

The result strengthens the foundational claim of the programme: the Heisenberg algebra $[\hat{X}, \hat{P}] = i\hbar$ is not postulated but emerges from the admissibility constraints on the discrete substrate, as the large-$q$ limit of the structural generators $\hat{X}_q$ and $\hat{P}_q$.

References

Jérôme Beau. Semiclassical Consistency of the Weil Representation on Heis₃(Z/qZ): Quantitative Sinc Embedding, Generator Convergence, and Aliasing Control. Preprint. doi:10.5281/zenodo.19962051