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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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TL;DR: In this paper, a deformation approach is proposed for the quantization of non-integrable Hamiltonian systems about a point of stable equilibrium, which resolves problems concerning the ordering of operators and the commutation of quantization with canonical transformation.
Abstract: A deformation approach resolves problems concerning the ordering of operators and the commutation of quantization with canonical transformation. This completes the programme of Robnik (1984) to develop a purely algebraic method for the quantization of non-integrable Hamiltonian systems about a point of stable equilibrium.
11 citations
TL;DR: In this article, a general non-abelian Nambu model (NANM) was proposed to provide spontaneous Lorentz symmetry breaking in terms of the corresponding Goldstone bosons.
Abstract: The emergence of gauge particles (e.g., photons and gravitons) as Goldstone bosons arising from spontaneous symmetry breaking is an interesting hypothesis which would provide a dynamical setting for the gauge principle. We investigate this proposal in the framework of a general $% SO(N)$ non-abelian Nambu model (NANM), effectively providing spontaneous Lorentz symmetry breaking in terms of the corresponding Goldstone bosons. Using a non-perturbative Hamiltonian analysis, we prove that the $SO(N)$ Yang--Mills theory is equivalent to the corresponding NANM, after current conservation together with the Gauss laws are imposed as initial conditions for the latter. This equivalence is independent of any gauge fixing in the YM theory. A substantial conceptual and practical improvement in the analysis arises by choosing a particular parametrization that solves the non-linear constraint defining the NANM. This choice allows us to show that the relation between the NANM canonical variables and the corresponding ones of the YM theory, $A_{i}^{a}$ and $E^{bj}$, is given by a canonical transformation. In terms of the latter variables, the NANM Hamiltonian has the same form as the YM Hamiltonian, except that the Gauss laws do not arise as first class constraints. The dynamics of the NANM further guarantees that it is sufficient to impose them only as initial conditions, in order to recover the full equivalence. It is interesting to observe that this particular parametrization exhibits the NANM as a regular theory, thus providing a substantial simplification in the calculations.
11 citations
TL;DR: In this article, the authors considered the specific problem of bringing to focus an axis-symmetric aberrating optical system by warping the output screen, and the solution for the surface curvature coefficients was given in terms of the Lie aberration coefficients of the system.
Abstract: The projection of optical images on warped screens is a canonical transformation of phase space between flat and warped evolution-parameter surfaces. In mechanics, the evolution parameter is time; in geometric optics it is the optical axis of coordinate space. We consider the specific problem of bringing to focus an axis-symmetric aberrating optical system by warping the output screen. The solution for the surface curvature coefficients is given in terms of the Lie aberration coefficients of the system; a linear optimization strategy applies.
11 citations
TL;DR: In this article, the dynamics of time-dependent coupled oscillator model for the charged particle motion subjected to a timedependent external magnetic field is investigated, and the full wave functions in the original system are derived from the inverse unitary transformation of the wave functions associated to the transformed system.
Abstract: The dynamics of time-dependent coupled oscillator model for the charged particle motion subjected to a time-dependent external magnetic field is investigated. We used canonical transformation approach for the classical treatment of the system, whereas unitary transformation approach is used when managing the system in the framework of quantum mechanics. For both approaches, the original system is transformed to a much more simple system that is the sum of two independent harmonic oscillators which have time-dependent frequencies. We therefore easily identified the wave functions in the transformed system with the help of invariant operator of the system. The full wave functions in the original system is derived from the inverse unitary transformation of the wave functions associated to the transformed system.
11 citations
TL;DR: The canonical transformation from the harmonic oscillator to the free particle is constructed and used as an example of transformations which cannot be maintained when going from classical to quantum systems in this paper.
Abstract: It is shown how to find some canonical transformations without solving the Hamilton–Jacobi equation. The canonical transformation from the harmonic oscillator to the free particle is constructed and used as an example of transformations which cannot be maintained when going from classical to quantum systems. The group of canonical transformations which can be carried over to quantum mechanics is discussed.
11 citations