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Canonical transformation

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


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G. Ghandour1
TL;DR: An effective generating function F(q,Q) is introduced for any given pair of quantum-mechanical systems whose classical Hamiltonians are canonically equivalent, and the generalization to quantum field theory is possible using the Schrodinger wave-functional formalism.
Abstract: An effective generating function F(q,Q) is introduced for any given pair of quantum-mechanical systems whose classical Hamiltonians are canonically equivalent. Using ${e}^{\mathrm{iF}}$ as a kernel, an integral transform relates the wave functions of the corresponding quantum systems. The function F reduces in the classical limit (\ensuremath{\Elzxh}\ensuremath{\rightarrow}0) to the generating function of the classical transformation. Conversely, starting with the classical form, F can be calculated in a recurrent fashion, order by order in powers of \ensuremath{\Elzxh}. For the canonical transformation that relates a particle moving in an exponential (Liouville) potential to a free particle, the effective quantum generating function is identical to its classical counterpart. The generalization to quantum field theory is possible using the Schr\"odinger wave-functional formalism.

34 citations

Journal ArticleDOI
TL;DR: In this article, an algebraic semiclassical approach to the calculation of vibrational transition probabilities in inelastic collisions between molecules is presented, which leads to a set of linear differential equations for the parameters of the coherent state, coupled to the classical Hamilton equations.
Abstract: An algebraic semiclassical approach to the calculation of vibrational transition probabilities in inelastic collisions between molecules is presented. Translational motion is treated classically, while vibrational motion is described quantum mechanically using the generalized coherent state of a proper Lie algebra. This leads to a set of linear differential equations for the parameters of the coherent state, coupled to the classical Hamilton equations. Use is also made of a time dependent canonical transformation to simplify the algebraic structure. Two examples are treated explicitly: colinear collision of an atom and a diatom and a diatom–diatom collision. Good agreement with the exact quantum results is found.

34 citations

Journal ArticleDOI
TL;DR: In this article, the authors introduce the multisymplectic formalism for the study of integrable defects in 1 + 1 classical field theory, which is the coexistence of two Poisson brackets, one for each spacetime coordinate.
Abstract: We introduce the concept of multisymplectic formalism, familiar in covariant field theory, for the study of integrable defects in 1 + 1 classical field theory. The main idea is the coexistence of two Poisson brackets, one for each spacetime coordinate. The Poisson bracket corresponding to the time coordinate is the usual one describing the time evolution of the system. Taking the nonlinear Schrodinger (NLS) equation as an example, we introduce the new bracket associated to the space coordinate. We show that, in the absence of any defect, the two brackets yield completely equivalent Hamiltonian descriptions of the model. However, in the presence of a defect described by a frozen Backlund transformation, the advantage of using the new bracket becomes evident. It allows us to reinterpret the defect conditions as canonical transformations. As a consequence, we are also able to implement the method of the classical r matrix and to prove Liouville integrability of the system with such a defect. The use of the new Poisson bracket completely bypasses all the known problems associated with the presence of a defect in the discussion of Liouville integrability. A by-product of the approach is the reinterpretation of the defect Lagrangian used in the Lagrangian description of integrable defects as the generating function of the canonical transformation representing the defect conditions.

34 citations

Journal ArticleDOI
TL;DR: In this paper, the Hamiltonian dynamics and thermodynamics of spherically symmetric spacetimes within a one-parameter family of five-dimensional Lovelock theories are considered.
Abstract: We consider the Hamiltonian dynamics and thermodynamics of spherically symmetric spacetimes within a one-parameter family of five-dimensional Lovelock theories. We adopt boundary conditions that make every classical solution part of a black hole exterior, with the spacelike hypersurfaces extending from the horizon bifurcation three-sphere to a timelike boundary with fixed intrinsic metric. The constraints are simplified by a Kucha\ifmmode \check{r}\else \v{r}\fi{}-type canonical transformation, and the theory is reduced to its true dynamical degrees of freedom. After quantization, the trace of the analytically continued Lorentzian time evolution operator is interpreted as the partition function of a thermodynamical canonical ensemble. Whenever the partition function is dominated by a Euclidean black hole solution, the entropy is given by the Lovelock analogue of the Bekenstein-Hawking entropy; in particular, in the low temperature limit the system exhibits a dominant classical solution that has no counterpart in Einstein's theory. The asymptotically flat space limit of the partition function does not exist. The results indicate qualitative robustness of the thermodynamics of five-dimensional Einstein theory upon the addition of a nontrivial Lovelock term.

34 citations

Journal ArticleDOI
TL;DR: In this paper, the authors present sufficient conditions for strong local optimality in the generalized problem of Bolza, which can be applied to both the calculus of variations and to optimal control problems, as well as problems with nonsmooth data.
Abstract: This paper presents sufficient conditions for strong local optimality in the generalized problem of Bolza. These conditions represent a unification, in the sense that they can be applied to both the calculus of variations and to optimal control problems, as well as problems with nonsmooth data. Also, this paper brings to light a new point of view concerning the Jacobi condition in the classical calculus of variations, showing that it can be considered as a condition which guarantees the existence of a canonical transformation which transforms the original Hamiltonian to a locally concave-convex Hamiltonian.

34 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20237
202218
202158
202042
201932
201829