C-number

Abstract: We present a method of treating the assembly of interacting bosons under Bose‐Einstein condensation. Without applying the Bogoliubov approximation in which the creation and the annihilation operators of zero‐momentum particles are replaced by a c number, we keep the quantum nature of these operators—thus the title, ``Quantum Mechanics.'' The method is a quantum mechanical adaptation of the theory of small oscillation. The oscillation means the fluctuation of the number of condensed particles. The interaction between particles determines the stability of this oscillation. When it is stable and its amplitude is not macroscopic, the Bogoliubov approximation is valid. In this way, our method provides a validity criterion for the Bogoliubov approximation as well as an estimation of the errors thereby committed. We have to note that the excitations associated with the fluctuation of condensed particles can never be obtained within that approximation. Our method is applied to the Huang model, the assembly of bosons interacting through a hard core plus weak attractive potential. Having found that, within the physically accessible range of the particle density, the above‐mentioned oscillation is stable, we can conclude that Huang's treatment is well founded. We have discussed the mathematical background of our approximation by invoking the representation theory of canonical variables of an infinitely large system.

Abstract: In this paper, we develop the theoretical foundations of discrete Dirac mechanics, that is, discrete mechanics of degenerate Lagrangian/Hamiltonian systems with constraints. We first construct discrete analogues of Tulczyjew’s triple and induced Dirac structures by considering the geometry of symplectic maps and their associated generating functions. We demonstrate that this framework provides a means of deriving discrete Lagrange–Dirac and nonholonomic Hamiltonian systems. In particular, this yields nonholonomic Lagrangian and Hamiltonian integrators. We also introduce discrete Lagrange–d’Alembert–Pontryagin and Hamilton–d’Alembert variational principles, which provide an alternative derivation of the same set of integration algorithms. The paper provides a unified treatment of discrete Lagrangian and Hamiltonian mechanics in the more general setting of discrete Dirac mechanics, as well as a generalization of symplectic and Poisson integrators to the broader category of Dirac integrators.

Abstract: Dirac expected that his symbolic method (q-number theory) which could express the physical law in neat and concise way would probably get developed. In this paper, we show that the technique of integration within an ordered product of operators can fashion Dirac's symbolic method and develop representation theory and transformation theory of quantum mechanics.

Abstract: We describe the real tessarines or \split-complex numbers" anddescribe a novel instance where they arise in biomedical informat-ics. We use the split-complex numbers to give a mathematical def-inition of a Hyperbolic Dirac Network (HDN) | a hyperbolic ana-logue of a classical probabilistic graphical model that uses quantumphysics as an underlying heuristic.The methods of theoretical physics should be applicable to all those branches ofthought in which the essential features are expressible in numbers.| P. A. M. Dirac, Nobel Prize Banquet Speech 1933