Quantum Algebras and Effective Interactions in Discrete Systems

The process algebra has been used successfully to provide a novel formulation of quantum mechanics in which non-relativistic quantum mechanics (NRQM) emerges as an effective theory asymptotically. The process algebra is applied here to the formulation of quantum field theory. The resulting QFT is intuitive, free from divergences and eliminates the distinction between particle, field and wave. There is a finite, discrete emergent space-time on which arise emergent entities which transfer information like discrete waves and interact with measurement processes like particles. The need for second quantization is eliminated and the particle and field theories rest on a common foundation, clarifying and simplifying the relationship between the two.The process algebra has been used successfully to provide a novel formulation of quantum mechanics in which non-relativistic quantum mechanics (NRQM) emerges as an effective theory asymptotically. The process algebra is applied here to the formulation of quantum field theory. The resulting QFT is intuitive, free from divergences and eliminates the distinction between particle, field and wave. There is a finite, discrete emergent space-time on which arise emergent entities which transfer information like discrete waves and interact with measurement processes like particles. The need for second quantization is eliminated and the particle and field theories rest on a common foundation, clarifying and simplifying the relationship between the two.

arXiv:1502.02346Feb 9, 20150 citations

Algebraic Formulation of the Operatorial Perturbation Theory. Part 2. Aplications

Ary W. Espinosa Müller, Adelio Matamala-V�squez

The algebraic approach to operator perturbation method has been applied to two quantum--mechanical systems ``The Stark Effect in the Harmonic Oscillator'' and ``The Generalized Zeeman Effect''. To that end, two realizations of the superoperators involved in the formalism have been carried out. The first of them has been based on the Heisenberg--Dirac algebra of $\hat{a}^\dagger$ , $\hat{a}$ , $\hat{1}$ operators, the second one has been based in the angular momemtum algebra of $\hat{L}_+$ , $\hat{L}_-$ and $\hat{L}_0$ operators. The successful results achieved in predicting the discrete spectra of both systems have put in evidence the reliability and accuracy of the theory.The algebraic approach to operator perturbation method has been applied to two quantum--mechanical systems ``The Stark Effect in the Harmonic Oscillator'' and ``The Generalized Zeeman Effect''. To that end, two realizations of the superoperators involved in the formalism have been carried out. The first of them has been based on the Heisenberg--Dirac algebra of $\hat{a}^\dagger$, $\hat{a}$, $\hat{1}$ operators, the second one has been based in the angular momemtum algebra of $\hat{L}_+$, $\hat{L}_-$ and $\hat{L}_0$ operators. The successful results achieved in predicting the discrete spectra of both systems have put in evidence the reliability and accuracy of the theory.

arXiv:quant-ph/9606013Jun 13, 19960 citations

Quantization of Physical Interaction Strengths via Singular Moduli

Prasoon Saurabh

Since the 2019 redefinition of the SI units, precision metrology has sought to anchor all physical quantities to fundamental constants and integer invariants. While the optical frequency comb revolutionized timekeeping by discretizing the continuum of light into countable teeth, and the Quantum Hall Effect standardized resistance via topological invariants, a comparable standard for interaction strength remains elusive. Currently, measuring the coupling constant ( $g$ ) between quantum systems is an estimation problem, inherently subject to drift, noise, and fabrication variance. Here, we introduce Interaction Metrology, a protocol that transforms the measurement of coupling strengths from an analog estimation into a topological counting problem. By engineering a specific class of algebraic catastrophe -- the Unimodal $X_9$ singularity -- in a driven-dissipative lattice, we prove that the system's interaction moduli are topologically forced to take discrete, quantized values, forming a "Geometric $k$ -Comb." We derive the universality class of this quantization, showing that it arises from the discrepancy between the Milnor ( $μ$ ) and Tjurina ( $τ$ ) numbers of the effective potential, a strictly non-Hermitian effect forbidden in standard quantum mechanics. Finally, we provide an ab-initio blueprint for a silicon nitride implementation, demonstrating that this quantization is robust against disorder levels exceeding current foundry tolerances. This discovery establishes a universal standard for force sensing and quantum logic gates, enabling the calibration of interaction strengths with topological certainty.Since the 2019 redefinition of the SI units, precision metrology has sought to anchor all physical quantities to fundamental constants and integer invariants. While the optical frequency comb revolutionized timekeeping by discretizing the continuum of light into countable teeth, and the Quantum Hall Effect standardized resistance via topological invariants, a comparable standard for interaction strength remains elusive. Currently, measuring the coupling constant ($g$) between quantum systems is an estimation problem, inherently subject to drift, noise, and fabrication variance. Here, we introduce Interaction Metrology, a protocol that transforms the measurement of coupling strengths from an analog estimation into a topological counting problem. By engineering a specific class of algebraic catastrophe -- the Unimodal $X_9$ singularity -- in a driven-dissipative lattice, we prove that the system's interaction moduli are topologically forced to take discrete, quantized values, forming a "Geometric $k$-Comb." We derive the universality class of this quantization, showing that it arises from the discrepancy between the Milnor ($μ$) and Tjurina ($τ$) numbers of the effective potential, a strictly non-Hermitian effect forbidden in standard quantum mechanics. Finally, we provide an ab-initio blueprint for a silicon nitride implementation, demonstrating that this quantization is robust against disorder levels exceeding current foundry tolerances. This discovery establishes a universal standard for force sensing and quantum logic gates, enabling the calibration of interaction strengths with topological certainty.

quant-pharXiv:2512.23741Dec 26, 2025

\hat{a}^\dagger

\hat{a}

\hat{1}

operators, the second one has been based in the angular momemtum algebra of

\hat{L}_+

\hat{L}_-

and

\hat{L}_0

operators. The successful results achieved in predicting the discrete spectra of both systems have put in evidence the reliability and accuracy of the theory.

g

) between quantum systems is an estimation problem, inherently subject to drift, noise, and fabrication variance. Here, we introduce Interaction Metrology, a protocol that transforms the measurement of coupling strengths from an analog estimation into a topological counting problem. By engineering a specific class of algebraic catastrophe -- the Unimodal

X_9

singularity -- in a driven-dissipative lattice, we prove that the system's interaction moduli are topologically forced to take discrete, quantized values, forming a "Geometric

k

-Comb." We derive the universality class of this quantization, showing that it arises from the discrepancy between the Milnor (

μ

) and Tjurina (

τ

) numbers of the effective potential, a strictly non-Hermitian effect forbidden in standard quantum mechanics. Finally, we provide an ab-initio blueprint for a silicon nitride implementation, demonstrating that this quantization is robust against disorder levels exceeding current foundry tolerances. This discovery establishes a universal standard for force sensing and quantum logic gates, enabling the calibration of interaction strengths with topological certainty.