Operators of the phase. Fundamentals

Laarni B. Natividad, Job A. Nable

In this work, the statement and proof of a fundamental formula in the phase space representation of quantum systems will be carried out for the similitude group, 𝑆𝑖𝑚(2).This formula takes the form∫ 𝒂(𝒀)𝑷(𝒀)𝒅(𝒀) = 〈𝑨 ̂〉, where 𝑌 is the phase space variable and 〈𝐴 ̂〉 is a linear operator on Hilbert space representing a quantum dynamical observable.〈𝐴 ̂〉 is the quantum expected value of the observable in a state of the system.The focus on the similitude group is due to current interest in signal analysis, localization operators and pseudo-differential operators.The fundamental formula states that this may be computed in a classical manner, as an integral against a probability distribution.The formula is intimately related to the quantization-dequantization problem 𝑎(𝑌) ⟷ 𝐴 ̂ which assigns a quantum operator to the classical phase space function 𝑎(𝑌).

Hiroshi Iwasawa

For a long time, about 60 years, the phase as an observable of the quantized light fields has r emained a fundamental problem with considerable difficulties on quantum mechanics and also q uantum optics.Recently, however, the hermitian phase operator and its eigenstates in a finite H ilbert space are found by Pegg and Barnett.In this paper, we provide a historical review and t he background of the problem on the phase operator, explore the Pegg-Barnett formalism, and s how the recent developements of this fields and its application to various light fields.Further problems on this formalism and other approaches are also discussed.

Jonatan P. camargo

We present a variational formulation in which the phase is identified as the operational manifestation of the accumulated action, establishing a structural unification between different physical regimes. It is shown that the fundamental phase of any admissible system is given by \[ \phi = \frac{S}{\hbar}, \] from which it follows that the frequency — and, therefore, the operational content of time — emerges directly from the rate of action accumulation, \[ \omega = \frac{\dot S}{\hbar}. \] The analysis of the relative phase evolution between independent systems leads to the demonstration that interferential coherence is, in general, episodic, being characterized by a discrete set of instants of compatibility. This results in a precise formulation of the physical interaction as \[ E_{\mathrm{int}} = E_{\mathrm{esp}} \cap E_\phi, \] in which the occurrence of observable events is filtered by phase compatibility. It is further shown that massive and massless regimes correspond to different parameterizations of the same phase structure, eliminating the need for distinct ontologies for these cases. The introduction of linear classical operators, such as macroscopic electromagnetic systems, allows us to demonstrate that such operators act exclusively on the observed phase, \[ \phi_{\mathrm{obs}} = \phi_{\mathrm{fund}} + \phi_C, \] shifting the synchronization instants without altering the underlying variational dynamics. As a result, a rigorous distinction is established between the structural level and the observational level: the action governs the fundamental dynamics, while the observed phase corresponds to its operational projection under reading transformations. Finally, the possibility of an extended variational framework in which the experimental apparatus is incorporated into a global functional \(S_{\mathrm{tot}}\) is discussed, leaving this construction as a program for future work.