Canonical transformation

About: Canonical transformation is a research topic. Over the lifetime, 1854 publications have been published within this topic receiving 38019 citations.

Adiabatic representation in the three-body problem with the Coulomb interaction. II. The effective two-level approximation

Abstract: For pt.I see ibid., vol.12, no.4, p.567 (1979). A canonical transformation of an infinite system of differential equations describing the motion of the three-body system in the adiabatic basis is suggested. This transformation allows one to reduce the original problem to the solution of a finite set of differential equations. As an example, a system of two differential equations is constructed which represents the infinite (or finite) system of equations within the accuracy (2M)-2, where M-1=mc/M0 is the ratio of the mass, mc, of the negative charged particle, c, to the reduced mass, M0, of two positively charged particles a and b. The physical meaning of the transformation and the solutions obtained is discussed.

Dynamical Properties of the Mukhanov-Sasaki Hamiltonian in the Context of Adiabatic Vacua and the Lewis-Riesenfeld Invariant

Abstract: We use the method of the Lewis-Riesenfeld invariant to analyze the dynamical properties of the Mukhanov-Sasaki Hamiltonian and, following this approach, investigate whether we can obtain possible candidates for initial states in the context of inflation considering a quasi-de Sitter spacetime. Our main interest lies in the question of to which extent these already well-established methods at the classical and quantum level for finitely many degrees of freedom can be generalized to field theory. As our results show, a straightforward generalization does in general not lead to a unitary operator on Fock space that implements the corresponding time-dependent canonical transformation associated with the Lewis-Riesenfeld invariant. The action of this operator can be rewritten as a time-dependent Bogoliubov transformation, where we also compare our results to already existing ones in the literature. We show that its generalization to Fock space has to be chosen appropriately in order to not violate the Shale-Stinespring condition. Furthermore, our analysis relates the Ermakov differential equation that plays the role of an auxiliary equation, whose solution is necessary to construct the Lewis-Riesenfeld invariant, as well as the corresponding time-dependent canonical transformation, to the defining differential equation for adiabatic vacua. Therefore, a given solution of the Ermakov equation directly yields a full solution of the differential equation for adiabatic vacua involving no truncation at some adiabatic order. As a consequence, we can interpret our result obtained here as a kind of non-squeezed Bunch-Davies mode, where the term non-squeezed refers to a possible residual squeezing that can be involved in the unitary operator for certain choices of the Bogoliubov map.

Mapping of the position-dependent mass Schrödinger equation under point canonical transformation

Abstract: In this paper, the three-dimensional radial position-dependent mass Schrodinger equation is exactly solved through mapping this wave equation into the constant mass Schrodinger equation with Coulomb potential by means of point canonical transformation. The wavefunctions here can be given in terms of confluent hypergeometric functions.

Functional integrals and canonical quantum gravity.

Abstract: The functional-integral sum-over-histories formulation of quantum gravity in the canonical Arnowitt-Deser-Misner formalism is examined. Reduced phase-space quantization (RPSQ) is contrasted with Dirac quantization (DQ). While it does not appear that RPSQ is even defined for gravity, there do exist minisuperspace models in which different identifications of time correspond to inequivalent RSPQ's none of which appears to correspond to DQ. While the Batalin-Fradkin-Vilkovisky Becchi-Rouet-Stora-Tyutin-invariant functional integral provides a representation for the propagators of DQ, its (assumed) independence of the choice of gauge function has been used (incorrectly) to demonstrate that the canonical limit corresponds to a functional integral in RPSQ. There is clearly something amiss. To identify the source of the inconsistency we use a minisuperspace model. In this way, we can demonstrate explicitly that the apparent coincidence is due, in part, to a mistaken counting of factors appearing in the respective measures. We also reexamine the Batalin-Vilkovisky theorem, demonstrating how it might break down when the two gauges compared are not infinitesimally related.

Design of robust H∞ fault detection filter for uncertain time-delay systems using canonical form approach

Abstract: In this paper, the problem of fault detection for an uncertain time-delay system is considered. Using a canonical form transformation, original time-delay system is transformed into a delay-free system. The fault detection scheme is devised using H ∞ model matching approach for the delay-free system and then implemented on original system. The so-called canonical transformation is found to be less conservative and offers reduced complexity in comparison with bicausal transformation, which can also be employed for the aforementioned purpose. An algorithm is proposed for design of such a fault detection system. Simulation results show the effectiveness of proposed scheme.