Analytical spectral density of dual quantum oscillators: The logarithmic mean postulate. The relationship between integral functions and entropy. Projection into information space.

This work develops the theory of the dual quantum oscillator (DQO). Preprints detailing the progress of the research were published in stages in the repository. This article presents the results of the subsequent stage. There has been a significant deepening of the very foundation of the theory - the postulate regarding the equal probability of conjugate dual states of the DQO under a bell-shaped gamma distribution. It has been found that this principle follows from another exact relation: two states of a bell-shaped gamma distribution are equally probable only when their logarithmic mean is equal to the thermal scale or the temperature of the system. Therefore, the main postulate retains its meaning but is formulated at a deeper and more precise level: “If the logarithmic mean of two states of a bell-shaped gamma distribution is equal to the thermal energy scale, then they are equally probable.” In previous works, spectral densities for Bose and Fermi oscillators within a DQO were derived based on these two postulates. At this stage, it has been found that they can be integrated analytically in elementary algebraic functions and are self-normed to unity. Their integral functions are directly related to entropy and information. Through the fluctuational nature of the spectral densities, their interrelation with classical Bose and Fermi distributions has been demonstrated. The constructed spectral densities were used to calculate the average energies over an ensemble of Bose oscillators, as a result of the convergence of their integrals over frequencies. This was regarded as a natural resolution of the problem of ultraviolet divergence of the vacuum energy. The connection between the integral functions of spectral densities and information allowed the study of the statistical theory of DQO to be continued within the framework of information geometry. Since the proposed DQO model is fundamental, it is expected that its theory will provide a powerful impetus for a deeper understanding of quantum physics and for even greater efficiency and reliability of quantum technologies in practical applications of the DQO model.