Infinite Degrees Of Freedom Research Articles

Wigner-Weisskopf atom and quantum Stochastic dilations

Quantum stochastic processes are operator processes in Fock space adapted in a natural way with respect to the tensor product factorization of the space. One can describe the usual classical stochastic processes (e.g., Brownian, Poisson) as well as new variety of non-commuting processes in this language. In this framework, quantum stochastic calculus (like Ito calculus in the classical case) has been developed by Hudson and Parthasarathy. Using this calculus, one can rewrite the simple Wigner-Weisskopf model of a decaying atom. In this model, the system (atom) is coupled linearly to the environment (a bath of infinite degrees of freedom represented by a reducible bosonic quantum stochastic process at non-zero temperature). The quantum stochastic differential equation of evolution is solved, and the bath variables are averaged out by taking vacuum expectation values leading to a law of relaxation of the state of the system.

On the structure of Hamiltonian operators in the field theory

A complete description of the Hamiltonian operators is shown and a ⪡Darboux lemma⪢ is proved (for some values of the parameters) in the framework of systems with infinite degrees of freedom.

Chaos Caused by the Soliton-Soliton Interaction

Chaotic behavior is found to be generated through the interaction of nonlinear excitations in the non-integrable systems with infinite degrees of freedom. The perturbed sine-Gordon equation is investigated by using the numerical simulation, which shows that the chaos is caused by soliton-soliton collisions. The orbital instability generated at the collision spot propagates spatially. This mechanism is effective to the onset of the chaos both for energy conserved and non-conserved perturbation.

Canonical forms for quadratic Hamiltonians

Our analysis of the applicable representations of the group of Bogoliubov transformations shows that the diagonalization of a quadratic fermion Hamiltonian with arbitrary complex coefficients is equivalent to the reduction of a skew symmetric matrix to secondary diagonal form by an orthogonal transformation, which we construct explicitly. Similarly, the diagonalization of a positive definite boson Hamiltonian with complex coefficients is equivalent to Whittaker's diagonalization of a symmetric matrix by a symplectic transformatio. Both results are shown to follow from a general spectral theorem for indefinite inner product space, a recent extension of which allows us to block diagonalize a positive definite quadratic boson Hamiltonian with complex coefficients and infinite degrees of freedom and thereby provide a counterpart to Araki's result for Fermi fields.

Response of an Inflatable Offshore Platform to Surface Wave Excitations

The response to the ocean wave excitations of a neutrally buoyant inflatable offshore platform consisting of an array of three tapered inflatable legs attached to a central head, connected to a surface float by a cable is investigated. A Lagrangian formulation of the dynamics of the system is presented where the float and the central head are allowed to move vertically and the rotations of the array as well as the flexural displacements of the legs are superposed on this motion. The non-linear coupled system with infinite degrees of freedom is truncated by considering the first two terms in the admissible function expansion of each flexural displacement. This results in a set of ordinary differential equations which are converted to algebraic equations using Ritz's averaging technique. Representative plots showing the effect of system parameters on the platform response are presented.

Synthesis of reciprocal n‐ports with infinite degree of freedom

Synthesis of reciprocal n‐ports with infinite degree of freedom

Synthesis of Reciprocal n-Ports with Infinite Degree of Freedom

Synthesis of Reciprocal <i>n</i>-Ports with Infinite Degree of Freedom

Theoretical Studies on Impact-vibraton : 2nd Report, Periodic Solutions and Stability of Vibration System with Infinite Degree of Freedom

Theoretical Studies on Impact-vibraton : 2nd Report, Periodic Solutions and Stability of Vibration System with Infinite Degree of Freedom

Note on the Eigenvalue Problem in the Quantum Field Theory

. In the quantum mechanics of atoms and molecules, many eigenvalue problems are solved in good agreement with experiments. While, iil the quantum mechanics of wave fields, the problem is so complicated because of its infinite degrees of freedom that we had to abandon the hope of obtaining the solution. The recent vivid development in the quanlum electrodynamics had brought brilliant informations on such problems as scattering, production and capture. These calculations are €ssentially based on the S matrix formalism1)2) and shows the usefulness of the concept of S matrix, in which it is assumed that the interaction is switched on adiabatically in the past and switched off again adiabatically in the future. And as its result, the kineTic energy and momentum of the total system is conserved. But in some respect, the use of S matrix in the eigenvalue problem seems to be inadequate, since the truly conserved quantity should be the total energy including the potential (or interaction) energy, but not the mere kinetic energy. This fact seems to show us that the total energy is not an adiabatic invariant, and we should not use the concept of remote past or future in the eigenvalue problem. In this short note, we discuss how to formulate the problem, and how the qualitative nature of the eigenvalues will be. And in this problem, it is pointed out that we should always carefully doubt the properties of operators if they are surely hermitian, surely unitary.