Analytical and Canonical Formalism in Physics

Ulf Leonhardt, Joan A. Vaccaro, B. Böhmer +1 more

We derive relationships between canonical and measured phase distributions for quantum-oscillator states in the semiclassical regime. First, we extend the formalism for the canonical phase to include external measurement-induced uncertainty. We require that a phase shifter shifts a phase distribution while a number shifter does not change it. These axioms determine pure canonical phase distributions uniquely while a noisy distribution can be interpreted as a weighted average of pure phase distributions. As a second step, we show that measured phase distributions, i.e., s-parametrized phase distributions fulfill approximately the axioms of noisy canonical phase, and we derive simple analytical expressions for the corresponding weight functions. Our analysis thus bridges all three conceptions of quantum-optical phase (canonical phase, s-parametrized phase, phase from measurements) and provides important physical insight into the relationship between them.

Kazuhiko Nishijima, Масуд Чайчиан, Anca Tureanu

Sol Wieder

Naoki Sasakura

The canonical tensor model, which is a tensor model in the Hamilton formalism, can be straightforwardly quantized and has an exactly solved physical state. The state is expressed by a wave function with a generalized form of the Airy function. The simplest observable on the state can be expressed by a matrix model which contains non-pairwise index contractions. This matrix model has the same form as the one that appears when the replica trick is applied to the spherical $p$-spin model for spin glasses, but our case has different ranges of variables and parameters from the spin glass case. We analyze the matrix model analytically and numerically. We show some evidences for the presence of a continuous phase transition at the location required by a consistency condition of the canonical tensor model. We also show that there are dimensional transitions of configurations around the transition region. Implications of the results to the canonical tensor model are discussed.

Alessandro Ciattoni

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.

Y. B. Band, Y. Avishai