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arXiv:2609.28494

Quantization and Mirror Reduction Do Not Commute in Hamiltonian Embeddings of Nonreciprocal Dynamics

Gaurav Sarmah, Ramakrishna Podila

quant-phcond-mat.mtrl-sciSubmitted Sep 2, 2026• Announced Sep 4, 2026
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Abstract

Hamiltonian embeddings can represent dissipative nonreciprocal classical dynamics exactly on invariant manifolds of enlarged reciprocal systems. We show that canonical quantization of such an embedding need not commute with reduction to the target dynamics. For the mirror construction recently introduced for pairwise nonreciprocal interactions, the invariant manifold is Lagrangian. Exact quantum enforcement of the mirror condition therefore removes the dynamical sector rather than producing a quantum analogue of the reduced flow. If the constraint is imposed only semiclassically, contraction of the target dynamics generates inverse-transpose expansion in the conjugate mirror directions. Quantum uncertainty then gives $\ln D\geΞ(t)+n\ln[π/(σε)]$, where $Ξ=-\ln|\det M|$ is the accumulated contraction and $D$ the torus Hilbert-space dimension. Exact finite-dimensional Weyl evolution of one- and two-degree-of-freedom nonreciprocal models confirms the resulting logarithmic correspondence time, including its predicted change when the initial localization scales with $\hbar$. Thus the classical embedding is exact, but its direct canonical quantization is a singular semiclassical construction.

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14 pages, 4 figures

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Hamiltonian embeddings can represent dissipative nonreciprocal classical dynamics exactly on invariant manifolds of enlarged reciprocal systems. We show that canonical quantization of such an embedding need not commute with reduction to the target dynamics. For the mirror construction recently introduced for pairwise nonreciprocal interactions, the invariant manifold is Lagrangian. Exact quantum enforcement of the mirror condition therefore removes the dynamical sector rather than producing a quantum analogue of the reduced flow. If the constraint is imposed only semiclassically, contraction of the target dynamics generates inverse-transpose expansion in the conjugate mirror directions. Quantum uncertainty then gives ln⁡D≥Ξ(t)+nln⁡[π/(σε)]\ln D\geΞ(t)+n\ln[π/(σε)]lnD≥Ξ(t)+nln[π/(σε)], where Ξ=−ln⁡∣det⁡M∣Ξ=-\ln|\det M|Ξ=−ln∣detM∣ is the accumulated contraction and DDD the torus Hilbert-space dimension. Exact finite-dimensional Weyl evolution of one- and two-degree-of-freedom nonreciprocal models confirms the resulting logarithmic correspondence time, including its predicted change when the initial localization scales with ℏ\hbarℏ. Thus the classical embedding is exact, but its direct canonical quantization is a singular semiclassical construction.
Can classically branched dynamics associated with spontaneous time-translation-symmetry breaking be quantized without explicitly retaining the branches of the Hamiltonian? We investigate this question using a reduced phase space. formulation in which the Hamiltonian is single-valued, while the critical structure of the classical Legendre map is encoded in a singular, momentum-dependent symplectic geometry. Quantization of the reduced Poisson algebra yields a deformed position-momentum commutator whose singularities determine distinct regular momentum sectors. Using the positive measure induced by the reduced symplectic structure, we identify a bounded central sector and two unbounded outer sectors. In the central sector, states with vanishing mean momentum obey a nonzero lower bound on position uncertainty, subject to the operator-domain assumptions of the representation. We further identify a possible field-theoretic connection with k-essence, where nonlinear scalar-field kinetics can produce finite-velocity degeneracies of the Legendre map. The relationship between the resulting reduced quantum description and branch-matching quantization remains unresolved and requires a comparison of the corresponding Hilbert spaces and operator domains.
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