The modified Langevin noise formalism (MLNF) models the interaction of the quantized electromagnetic field with an arbitrary lossy magneto-dielectric object placed in vacuum using three types of non-interacting bosonic polaritons: scattering, electric, and magnetic. These respectively represent free-space photons scattered by the object, and photons radiated by quantized electric and magnetic dipolar sources embedded within its volume. Recently [A. Ciattoni, Phys. Rev. A 110, 013707 (2024)], this formalism was justified from the canonical quantization of macroscopic electromagnetism (CQME) [Philbin, New J. Phys. 12, 123008 (2010)] in the Heisenberg picture. This was achieved by identifying the polariton operators within the formal solution of the macroscopic Maxwell equations, assuming they obey bosonic commutation relations to retrieve the canonical ones, and showing they diagonalize the CQME Hamiltonian. However, the explicit functional dependence of these polaritons on the underlying canonical field operators remained undetermined. In this paper, we derive the exact analytical expressions for the polariton operators in terms of the canonical CQME field operators. Using these mappings, we provide a direct and rigorous derivation of the MLNF from the canonical theory in the Schrödinger picture. Our derivation is structured in three foundational steps: 1) adopting the derived analytical expressions as the constitutive definitions of the polariton operators; 2) mathematically proving that these operators are strictly bosonic as a direct consequence of the canonical commutation relations; and 3) demonstrating that they exactly diagonalize the macroscopic CQME Hamiltonian.The modified Langevin noise formalism (MLNF) models the interaction of the quantized electromagnetic field with an arbitrary lossy magneto-dielectric object placed in vacuum using three types of non-interacting bosonic polaritons: scattering, electric, and magnetic. These respectively represent free-space photons scattered by the object, and photons radiated by quantized electric and magnetic dipolar sources embedded within its volume. Recently [A. Ciattoni, Phys. Rev. A 110, 013707 (2024)], this formalism was justified from the canonical quantization of macroscopic electromagnetism (CQME) [Philbin, New J. Phys. 12, 123008 (2010)] in the Heisenberg picture. This was achieved by identifying the polariton operators within the formal solution of the macroscopic Maxwell equations, assuming they obey bosonic commutation relations to retrieve the canonical ones, and showing they diagonalize the CQME Hamiltonian. However, the explicit functional dependence of these polaritons on the underlying canonical field operators remained undetermined. In this paper, we derive the exact analytical expressions for the polariton operators in terms of the canonical CQME field operators. Using these mappings, we provide a direct and rigorous derivation of the MLNF from the canonical theory in the Schrödinger picture. Our derivation is structured in three foundational steps: 1) adopting the derived analytical expressions as the constitutive definitions of the polariton operators; 2) mathematically proving that these operators are strictly bosonic as a direct consequence of the canonical commutation relations; and 3) demonstrating that they exactly diagonalize the macroscopic CQME Hamiltonian.

quant-pharXiv:2603.04336Mar 5, 2026

Canonical Quantization and its Generalization

Yoshio Ohnuki, S. Kamefuchi

Springer eBooksJan 1, 19820 citations

Vector Affine Quantization Can Create Valid Quantum Field Theories

John R. Klauder

Affine quantization, a parallel procedure to canonical quantization, needs to use its principal quantum operators, specifically D = (PQ+QP)/2 and Q ≠ 0, to represent appropriate kinetic factors, such as P2, which involves only one canonical quantum operator. The need for this requirement stems from path integral quantizations of selected problems that affine quantization can solve but canonical quantization fails to solve. This task is resolved for simple examples, as well as examples that involve scalar, and vector, quantum field theories.Affine quantization, a parallel procedure to canonical quantization, needs to use its principal quantum operators, specifically D = (PQ+QP)/2 and Q ≠ 0, to represent appropriate kinetic factors, such as P2, which involves only one canonical quantum operator. The need for this requirement stems from path integral quantizations of selected problems that affine quantization can solve but canonical quantization fails to solve. This task is resolved for simple examples, as well as examples that involve scalar, and vector, quantum field theories.

Journal of High Energy Physics Gravitation and CosmologyJan 1, 20220 citations

Canonical Quantization

Ray LaPierre

Undergraduate texts in physicsJan 1, 20222 citations

Canonical quantization of some dissipative systems and nonuniqueness of Lagrangians

Susumu Ôkubo

Some one-dimensional dissipative systems can be found to admit canonical-quantization procedure, contrary to opinions expressed occassionally in the literature. Also, a natural way of quantizing an equation $\stackrel{\ifmmode\ddot\else\textasciidieresis\fi{}}{q}+\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{V}(q)=0$ for a function $V(q)$ is found to use a nonstandard commutation relation $[q,\ensuremath{\pi}]=i\ensuremath{\hbar}\ensuremath{\pi}$ together with Hamiltonian $H=\ensuremath{\pi}+V(q)$. Here, $\ensuremath{\pi}=\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{q}$ stands for the velocity operator rather than the momentum operator. Finally, a quantization problem of a Newtonian geodesic equation of motion in nonflat space is discussed.Some one-dimensional dissipative systems can be found to admit canonical-quantization procedure, contrary to opinions expressed occassionally in the literature. Also, a natural way of quantizing an equation $\stackrel{\ifmmode\ddot\else\textasciidieresis\fi{}}{q}+\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{V}(q)=0$ for a function $V(q)$ is found to use a nonstandard commutation relation $[q,\ensuremath{\pi}]=i\ensuremath{\hbar}\ensuremath{\pi}$ together with Hamiltonian $H=\ensuremath{\pi}+V(q)$. Here, $\ensuremath{\pi}=\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{q}$ stands for the velocity operator rather than the momentum operator. Finally, a quantization problem of a Newtonian geodesic equation of motion in nonflat space is discussed.

Physical Review AJun 1, 198126 citations

Canonical quantization

Yannick Meurice

Institute of Physics eBooksFeb 1, 20211 citation

V(q)

is found to use a nonstandard commutation relation $[q,\ensuremath{\pi}]=i\ensuremath{\hbar}\ensuremath{\pi}$ together with Hamiltonian $H=\ensuremath{\pi}+V(q)$. Here, $\ensuremath{\pi}=\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{q}$ stands for the velocity operator rather than the momentum operator. Finally, a quantization problem of a Newtonian geodesic equation of motion in nonflat space is discussed.