Showing papers on "Canonical transformation published in 2012"
TL;DR: The present approach constructs a singularity-free Hamiltonian a priori, similarly to the so-called transcorrelated theory, while the use of the canonical transformation assures that the effective Hamiltonian is two-body and Hermite.
Abstract: An effective Hamiltonian perturbed with explicit interelectronic correlation is derived from similarity transformation of Hamiltonian using a unitary operator with Slater-type geminals. The Slater-type geminal is projected onto the excitation (and deexcitation) component as in the F12 theory. Simplification is made by truncating higher-body operators, resulting in a correlated Hamiltonian which is Hermitian and has exactly the same complexity as the original Hamiltonian in the second quantized form. It can thus be easily combined with arbitrary correlation models proposed to date. The present approach constructs a singularity-free Hamiltonian a priori, similarly to the so-called transcorrelated theory, while the use of the canonical transformation assures that the effective Hamiltonian is two-body and Hermite. Our theory is naturally extensible to multireference calculations on the basis of the generalized normal ordering. The construction of the effective Hamiltonian is non-iterative. The numerical assessments demonstrate that the present scheme improves the basis set convergence of the post-mean-field calculations at a similar rate to the explicitly correlated methods proposed by others that couple geminals and conventional excitations.
72 citations
TL;DR: In this paper, the construction of rationally-extended Morse potentials is analyzed in the framework of first-order supersymmetric quantum mechanics, and the existence of another family of extended potentials, strictly isospectral to VA+1, B(x), is pointed out.
Abstract: The construction of rationally-extended Morse potentials is analyzed in the framework of first-order supersymmetric quantum mechanics. The known family of extended potentials VA, B, ext(x), obtained from a conventional Morse potential VA-1, B(x) by the addition of a bound state below the spectrum of the latter, is reobtained. More importantly, the existence of another family of extended potentials, strictly isospectral to VA+1, B(x), is pointed out for a well-chosen range of parameter values. Although not shape invariant, such extended potentials exhibit a kind of "enlarged" shape invariance property, in the sense that their partner, obtained by translating both the parameter A and the degree m of the polynomial arising in the denominator, belongs to the same family of extended potentials. The point canonical transformation connecting the radial oscillator to the Morse potential is also applied to exactly solvable rationally-extended radial oscillator potentials to build quasi-exactly solvable rationally-extended Morse ones.
64 citations
TL;DR: In this article, the authors investigated the ambiguities in the Fock quantization of the scalar perturbations of a Friedmann-Lema-tre-Robertson-Walker model with a massive scalar field as matter content.
Abstract: We investigate the ambiguities in the Fock quantization of the scalar perturbations of a Friedmann-Lema\^{\i}tre-Robertson-Walker model with a massive scalar field as matter content. We consider the case of compact spatial sections (thus avoiding infrared divergences), with the topology of a three-sphere. After expanding the perturbations in series of eigenfunctions of the Laplace-Beltrami operator, the Hamiltonian of the system is written up to quadratic order in them. We fix the gauge of the local degrees of freedom in two different ways, reaching in both cases the same qualitative results. A canonical transformation, which includes the scaling of the matter-field perturbations by the scale factor of the geometry, is performed in order to arrive at a convenient formulation of the system. We then study the quantization of these perturbations in the classical background determined by the homogeneous variables. Based on previous work, we introduce a Fock representation for the perturbations in which: (a) the complex structure is invariant under the isometries of the spatial sections and (b) the field dynamics is implemented as a unitary operator. These two properties select not only a unique unitary equivalence class of representations, but also a preferred field description, picking up a canonical pair of field variables among all those that can be obtained by means of a time-dependent scaling of the matter field (completed into a linear canonical transformation). Finally, we present an equivalent quantization constructed in terms of gauge-invariant quantities. We prove that this quantization can be attained by a mode-by-mode time-dependent linear canonical transformation which admits a unitary implementation, so that it is also uniquely determined.
58 citations
TL;DR: In this paper, the authors consider the quantization of scalar fields in spacetimes such that, by means of a suitable scaling of the field by a time dependent function, the field equation can be regarded as that of a field with a time-dependent mass propagating in an auxiliary ultrastatic static background.
Abstract: We consider the quantization of scalar fields in spacetimes such that, by means of a suitable scaling of the field by a time dependent function, the field equation can be regarded as that of a field with a time dependent mass propagating in an auxiliary ultrastatic static background. For Klein-Gordon fields, it is well known that there exist an infinite number of nonequivalent Fock representations of the canonical commutation relations and, therefore, of inequivalent quantum theories. A context in which this kind of ambiguities arises and prevents the derivation of robust results is, e.g., in the quantum analysis of cosmological perturbations. In these situations, typically, a suitable scaling of the field by a time dependent function leads to a description in an auxiliary static background, though the nonstationarity still shows up in a time dependent mass. For such a field description, and assuming the compactness of the spatial sections, we recently proved in three or less spatial dimensions that the criteria of a natural implementation of the spatial symmetries and of a unitary time evolution are able to select a unique class of unitarily equivalent vacua, and hence of Fock representations. In this work, we succeed to extend our uniqueness result to the consideration of all possible field descriptions that can be reached by a time dependent canonical transformation which, in particular, involves a scaling of the field by a function of time. These kinds of canonical transformations modify the dynamics of the system and introduce a further ambiguity in its quantum description, exceeding the choice of a Fock representation. Remarkably, for any compact spatial manifold in less than four dimensions, we show that our criteria eliminate any possible nontrivial scaling of the field other than that leading to the description in an auxiliary static background. Besides, we show that either no time dependent redefinition of the field momentum is allowed or, if this may happen---something which is typically the case only for one-dimensional spatial manifolds---the redefinition does not introduce any Fock representation that cannot be obtained by a unitary transformation.
42 citations
TL;DR: In this paper, a variational method was proposed to study the bifurcation of growing cylinders with circular section, accounting for a constant axial pre-stretch, providing a locally isochoric mapping.
Abstract: Morphoelastic theories have demonstrated that elastic instabilities can occur during the growth of soft materials, initiating the transition toward complex patterns. Within the framework of non-linear elasticity, the theory of incremental elastic deformations is classically employed for solving stability problems with finite strains. In this work, we define a variational method to study the bifurcation of growing cylinders with circular section. Accounting for a constant axial pre-stretch, we define a set of canonical transformations in mixed polar coordinates, providing a locally isochoric mapping. Introducing a generating function to derive an implicit gradient form of the mixed variables, the incompressibility constraint for the elastic deformation is solved exactly. The canonical representation allows to transform a generic boundary value problem, characterized by conservative body forces and surface traction loads, into a completely variational formulation. The proposed variational method gives a straightforward derivation of the linear stability analysis, which would otherwise require lengthy manipulations on the governing incremental equations. The definition of a generating function can also account for the presence of local singularities in the elastic solution. Bifurcation analysis is performed for few constrained growth problems of biomechanical interests, such as the mucosal folding of tubular tissues and surface instabilities in tumor growth. In a concluding section, the theoretical results are discussed for clarifying how anisotropy, residual strains and external constraints can affect the stability properties of soft tissues in growth and remodeling processes.
40 citations
TL;DR: In this article, a canonical transformation is applied to a water wave equation to remove cubic nonlinear terms and to simplify fourth-order terms in the Hamiltonian, which is very suitable for analytical studies and numerical simulations.
Abstract: We apply a canonical transformation to a water wave equation to remove cubic nonlinear terms and to drastically simplify fourth-order terms in the Hamiltonian. This transformation explicitly uses the vanishing exact four-wave interaction for water gravity waves for a 2D potential fluid. After transformation, the well-known but cumbersome Zakharov equation is drastically simplified and can be written in X -space in a compact form. This new equation is very suitable for analytical studies and numerical simulations.
34 citations
TL;DR: A parallelized algorithm and implementation of the canonical transformation theory to handle large computational demands of the CT calculation, which has the same scaling as the coupled cluster singles and doubles theory.
Abstract: The canonical transformation (CT) theory has been developed as a multireference electronic structure method to compute high-level dynamic correlation on top of a large active space reference treated with the ab initio density matrix renormalization group method. This article describes a parallelized algorithm and implementation of the CT theory to handle large computational demands of the CT calculation, which has the same scaling as the coupled cluster singles and doubles theory. To stabilize the iterative solution of the CT method, a modification to the CT amplitude equation is introduced with the inclusion of a level shift parameter. The level-shifted condition has been found to effectively remove a type of intruder state that arises in the linear equations of CT and to address the discontinuity problems in the potential energy curves observed in the previous CT studies.
33 citations
TL;DR: In this paper, the authors construct the canonical transformation that controls the full dependence (local and non-local) of the vertex functional of a Yang-Mills theory on a background field.
Abstract: We construct explicitly the canonical transformation that controls the full dependence (local and non-local) of the vertex functional of a Yang-Mills theory on a background field. After showing that the canonical transformation found is nothing but a direct field-theoretic generalization of the Lie transform of classical analytical mechanics, we comment on a number of possible applications, and in particular the non perturbative implementation of the background field method on the lattice, the background field formulation of the two particle irreducible formalism, and, finally, the formulation of the Schwinger-Dyson series in the presence of topologically non-trivial configurations.
32 citations
TL;DR: In this paper, a solution to a spectral problem involving the Schrodinger equation for a particular class of multiparameter exponential-type potentials is presented, which is based on the canonical transformation method applied to a general second-order differential equation, multiplied by a function g(x), to convert it into aSchrodinger-like equation.
Abstract: The solution to a spectral problem involving the Schrodinger equation for a particular class of multiparameter exponential-type potentials is presented. The proposal is based on the canonical transformation method applied to a general second-order differential equation, multiplied by a function g(x), to convert it into a Schrodinger-like equation. The treatment of multiparameter exponential-type potentials comes from the application of the transformed results to the hypergeometric equation under the assumption of a specific g(x). Besides presenting the explicit solutions and their spectral values, it is shown that the problem considered in this article unifies and generalizes several former studies. That is, the proposed exactly solvable multiparameter exponential-type potential can be straightforwardly applied to particular exponential potentials depending on the choice of the involved parameters as exemplified for the Hulthen potential and their isospectral partner. Moreover, depending on the function g(x), the proposal can be extended to find different exactly solvable potentials as well as to generate new potentials that could be useful in quantum chemical calculations. © 2011
29 citations
TL;DR: In this paper, it was shown that the background-quantum splitting of SU(N) Yang-Mills theory is non-trivially deformed at the quantum level by a canonical transformation with respect to the Batalin-Vilkovisky bracket associated with the Slavnov-Taylor identity.
Abstract: We show that for a SU(N) Yang-Mills theory the classical background-quantum splitting is non-trivially deformed at the quantum level by a canonical transformation with respect to the Batalin-Vilkovisky bracket associated with the Slavnov-Taylor identity of the theory. This canonical transformation acts on all the fields (including the ghosts) and antifields; it uniquely fixes the dependence on the background field of all the one-particle irreducible Green's functions of the theory at hand. The approach is valid both at the perturbative and non-perturbative level, being based solely on symmetry requirements. As a practical application, we derive the renormalization group equation in the presence of a generic background and apply it in the case of a SU(2) instanton. Finally, we explicitly calculate the one-loop deformation of the background-quantum splitting in lowest order in the instanton background.
18 citations
TL;DR: In this article, the authors studied the coupling of N charged scalar particles plus the electromagnetic field to ADM tetrad gravity and its canonical formulation in asymptotically Minkowskian space-times without super-translations.
Abstract: We study the coupling of N charged scalar particles plus the electromagnetic field to Arnowitt–Deser–Misner (ADM) tetrad gravity and its canonical formulation in asymptotically Minkowskian space–times without super-translations. To regularize the self-energies, both the electric charge and the sign of the energy of the particles are Grassmann-valued. The introduction of the noncovariant radiation gauge allows reformulation of the theory in terms of transverse electromagnetic fields and to extract the generalization of the Coulomb interaction among the particles in the riemannian instantaneous 3-spaces of global noninertial frames, the only ones allowed by the equivalence principle. Then we make the canonical transformation to the York canonical basis, where there is a separation between the inertial (gauge) variables and the tidal ones inside the gravitational field and a special role of the eulerian observers associated with the 3+1 splitting of space–time. The Dirac hamiltonian is weakly equal to the we...
TL;DR: A symbolic algorithm for construction of a real canonical transformation that reduces the Hamiltonian determining motion of an autonomous two-degree-of-freedom system in a neighborhood of an equilibrium state to the normal form is discussed and the application of the algorithm to the restricted planar circular three-body problem is demonstrated.
Abstract: A symbolic algorithm for construction of a real canonical transformation that reduces the Hamiltonian determining motion of an autonomous two-degree-of-freedom system in a neighborhood of an equilibrium state to the normal form is discussed. The application of the algorithm to the restricted planar circular three-body problem is demonstrated. The expressions obtained for the coefficients of the Hamiltonian normal form substantiate results derived earlier by A. Deprit. Symbolic computations are performed in the computer algebra system Mathematica.
TL;DR: In this paper, the six-degree-of-freedom Hamiltonian problem is formulated as a perturbation of the Kepler motion and torque-free rotation, and a chain of canonical transformations is used to reduce the problem.
Abstract: The roto-translational dynamics of an axial-symmetric rigid body is discussed in a central gravitational field. The six-degree-of-freedom Hamiltonian problem is formulated as a perturbation of the Kepler motion and torque-free rotation. A chain of canonical transformations is used to reduce the problem. First, the elimination of the nodes reduces the problem to a system of four degrees of freedom. Then, the elimination of the parallax simplifies the resulting Hamiltonian, which is shaped as a radial intermediary plus a remainder. Some features of this integrable intermediary are pointed out. The normalized first order system in closed form is also given, thus completing the solution. Finally the full reduction of the radial intermediary is constructed using the Hamilton-Jacobi equation.
TL;DR: In this article, a new deformed Poisson bracket was proposed, which leads to the Fock coordinate transformation by using an analogous procedure as in Deformed Special Relativity, and a canonical transformation with which the new coordinates and momentum satisfy the usual Poisson brackets and therefore transform like the usual Lorentz vectors.
Abstract: In this paper, we propose a new deformed Poisson brackets which leads to the Fock coordinate transformation by using an analogous procedure as in Deformed Special Relativity. We therefore derive the corresponding momentum transformation which is revealed to be different from previous results. Contrary to the earlier version of Fock's nonlinear relativity for which plane waves cannot be described, our resulting algebra keeps invariant for any coordinate and momentum transformations the four dimensional contraction $p_{\mu} x^{\mu} $, allowing therefore to associate plane waves for free particles. As in Deformed Special Relativity, we also derive a canonical transformation with which the new coordinates and momentum satisfy the usual Poisson brackets and therefore transform like the usual Lorentz vectors. Finally, we establish the dispersion relation for Fock's nonlinear relativity.
TL;DR: In this paper, a new deformed Poisson bracket was proposed to obtain the Fock coordinate transformation, which is invariant to the four-dimensional contraction pμ xμ, allowing to associate plane waves for free particles.
Abstract: In this paper, we propose a new deformed Poisson brackets which leads to the Fock coordinate transformation by using an analogous procedure as in Deformed Special Relativity. We therefore derive the corresponding momentum transformation which is revealed to be different from previous results. Contrary to the earlier version of Fock's nonlinear relativity for which plane waves cannot be described, our resulting algebra keeps invariant for any coordinate and momentum transformations the four-dimensional contraction pμ xμ, allowing therefore to associate plane waves for free particles. As in Deformed Special Relativity, we also derive a canonical transformation with which the new coordinates and momentum satisfy the usual Poisson brackets and therefore transform like the usual Lorentz vectors. Finally, we establish the dispersion relation for Fock's nonlinear relativity.
TL;DR: In this article, a consistent, local coordinate formulation of covariant Hamiltonian field theory is presented, where the covariant canonical field equations are equivalent to the Euler-Lagrange field equations, and the canonical transformation rules for fields are derived from generating functions.
Abstract: A consistent, local coordinate formulation of covariant Hamiltonian field theory is presented. While the covariant canonical field equations are equivalent to the Euler-Lagrange field equations, the covariant canonical transformation theory offers more general means for defining mappings that preserve the action functional - and hence the form of the field equations - than the usual Lagrangian description. Similar to the well-known canonical transformation theory of point dynamics, the canonical transformation rules for fields are derived from generating functions. As an interesting example, we work out the generating function of type F_2 of a general local U(N) gauge transformation and thus derive the most general form of a Hamiltonian density that is form-invariant under local U(N) gauge transformations.
TL;DR: In this paper, it is shown how to extend the hydrodynamic Lagrangian-picture method of constructing field evolution using a continuum of trajectories to second-order theories.
Abstract: Using the wave equation as an example, it is shown how to extend the hydrodynamic Lagrangian-picture method of constructing field evolution using a continuum of trajectories to second-order theories. The wave equation is represented through Eulerian-picture models that are distinguished by their Lorentz transformation properties. Introducing the idea of the relativity of the particle label, it is demonstrated how the corresponding trajectory models are compatible with the relativity principle. It is also shown how the Eulerian variational formulation may be obtained by canonical transformation from the Lagrangian picture, and how symmetries in the Lagrangian picture may be used to generate Eulerian conserved charges.
TL;DR: The generalization of the classical Poisson sum formula, by replacing the ordinary Fourier transform by the canonical transformation, has been derived in the linear canonical transform sense.
Abstract: The generalization of the classical Poisson sum formula, by replacing the ordinary Fourier transform by the canonical transformation, has been derived in the linear canonical transform sense. Firstly, a new sum formula of Chirp-periodic property has been introduced, and then the relationship between this new sum and the original signal is derived. Secondly, the generalization of the classical Poisson sum formula to the linear canonical transform sense has been obtained.
TL;DR: In this paper, the authors analyze tunneling-induced quantum fluctuations in a single-level quantum dot with arbitrarily strong on-site Coulomb interaction, generating cotunneling processes and renormalizing system parameters.
Abstract: We analyze tunneling-induced quantum fluctuations in a single-level quantum dot with arbitrarily strong on-site Coulomb interaction, generating cotunneling processes and renormalizing system parameters. For a perturbative analysis of these quantum fluctuations, we remove off-shell parts of the Hamiltonian via a canonical transformation. We find that the tunnel couplings for the transitions connecting empty and single occupation and connecting single and double occupation of the dot renormalize with the same magnitude but with opposite signs. This has an important impact on the shape of the renormalization extracted, for example, from the conductance. Finally, we verify the compatibility of our results with a systematic second-order perturbation expansion of the linear conductance performed within a diagrammatic real-time approach.
TL;DR: In this article, the authors derived exactly solvable rationally extended quantum Hamiltonians which are non-Hermitian in nature and whose bound state wave functions are associated with Laguerre- or Jacobi-type X 1 exceptional orthogonal polynomials.
03 Sep 2012
TL;DR: Stability of three equilibrium positions for the majority of initial conditions in case of mass parameters of the system belonging to the domain of the solutions linear stability is proved, except for the points in the parameter plane for which the third and fourth order resonance conditions are fulfilled.
Abstract: We study stability of equilibrium positions in the spatial circular restricted four-body problem formulated on the basis of Lagrange's triangular solution of the three-body problem. Using the computer algebra system Mathematica, we have constructed Birkhoff's type canonical transformation, reducing the Hamiltonian function to the normal form up to the fourth order in perturbations. Applying Arnold's and Markeev's theorems, we have proved stability of three equilibrium positions for the majority of initial conditions in case of mass parameters of the system belonging to the domain of the solutions linear stability, except for the points in the parameter plane for which the third and fourth order resonance conditions are fulfilled.
TL;DR: In this paper, the Hamilton-Jacobi theory was used to show that all the Hamiltonian systems with n degrees of freedom are equivalent, i.e., there is a canonical transformation connecting two arbitrary Hamiltonian system with the same number of degrees offreedom.
Abstract: In this work we use the Hamilton-Jacobi theory to show that locally all the Hamiltonian systems with n degrees of freedom are equivalent. That is, there is a canonical transformation connecting two arbitrary Hamiltonian systems with the same number of degrees of freedom. This result in particular implies that locally all the Hamiltonian systems are equivalent to that of a free particle. We illustrate our result with two particular examples; first we show that the one-dimensional free particle is locally equivalent to the one-dimensional harmonic oscillator and second that the two-dimensional free particle is locally equivalent to the two-dimensional Kepler problem.
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10 Aug 2012
TL;DR: In this paper, the classical and quantum properties of position-dependent mass (PDM) oscillators are discussed and a parametric set for quasi-free PDM particles is obtained.
Abstract: We recycle Cruz et al.'s (Phys. Lett. A 369 (2007) 400) work on the classical and quantum position-dependent mass (PDM) oscillators. To elaborate on the ordering ambiguity, we properly amend some of the results reported in their work and discuss the classical and quantum mechanical correspondence for the PDM harmonic oscillators. We use a point canonical transformation and show that one unique quantum PDM oscillator Hamiltonian (consequently, one unique ordering-ambiguity parametric set j=l=-1/4 and k=-1/2) is obtained. To show that such a parametric set is not just a manifestation of the quantum PDM oscillator Hamiltonian, we consider the classical and quantum mechanical correspondence for quasi-free PDM particles moving under the influence of their own PDM force fields.
27 Feb 2012
TL;DR: In this article, a method extending the Munn-Silbey approach has been applied to obtain the temperature dependence of transport properties of a generalized Holstein model incorporating simultaneous diagonal and off-diagonal carrier-phonon coupling.
Abstract: A method extending the Munn-Silbey approach has been applied to obtain the temperature dependence of transport properties of a generalized Holstein model incorporating simultaneous diagonal and off-diagonal carrier-phonon coupling. The Hamiltonian is partially diagonalized by a canonical transformation, and optimal transformation coefficients are determined in a self-consistent manner. Effects of off-diagonal coupling on the optimal transformation coefficients and diffusion coefficients have been discussed in details. The off-diagonal coupling has been revealed as a localization factor as well as a transport mechanism. Moreover, momentum-space variation of the transformation coefficients are found to be responsible for enhanced transport due to off-diagonal coupling.
TL;DR: In this paper, a variational method of solution is proposed to remove the lack of uniqueness of the transformation in 3-D geometries by using a relabelling symmetry, which is a generalisation of ghost surfaces and quadratic-flux-minimising surfaces.
Abstract: Straight-field-line coordinates are very useful for representing magnetic fields in toroidally confined plasmas, but fundamental problems arise regarding their definition in 3-D geometries because of the formation of islands and chaotic field regions, ie non-integrability. In Hamiltonian dynamical systems terms these coordinates are a form of action-angle variables, which are normally defined only for integrable systems. In order to describe 3-D magnetic field systems, a generalisation of this concept was proposed recently by the present authors that unified the concepts of ghost surfaces and quadratic-flux-minimising (QFMin) surfaces. This was based on a simple canonical transformation generated by a change of variable $\theta = \theta(\Theta,\zeta)$, where $\theta$ and $\zeta$ are poloidal and toroidal angles, respectively, with $\Theta$ a new poloidal angle chosen to give pseudo-orbits that are a) straight when plotted in the $\zeta,\Theta$ plane and b) QFMin pseudo-orbits in the transformed coordinate. These two requirements ensure that the pseudo-orbits are also c) ghost pseudo-orbits. In the present paper, it is demonstrated that these requirements do not \emph{uniquely} specify the transformation owing to a relabelling symmetry. A variational method of solution that removes this lack of uniqueness is proposed.
TL;DR: In this paper, it was shown that, under small perturbations that preserve the Jordan structure, canonical transformation matrices have the Lipschitz property, and an analogous result is proved for matrices under additional structure, such as Hamiltonian, skew-Hamiltonian, and symplectic.
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TL;DR: In this article, an unconstrained Hamiltonian formulation of the SU(3) Yang-Mills quantum mechanics of spatially constant fields is given using the method of minimal embedding of SU(2) into SU( 3) by Kihlberg and Marnelius.
Abstract: An unconstrained Hamiltonian formulation of the SU(3) Yang-Mills quantum mechanics of spatially constant fields is given using the method of minimal embedding of SU(2) into SU(3) by Kihlberg and Marnelius. Using a canonical transformation of the gluon fields to a new set of adapted coordinates (a non-standard type polar decomposition), which Abelianizes the Non-Abelian Gauss law constraints to be implemented, the corresponding unconstrained Hamiltonian and total angular momentum are derived. This reduces the colored spin-1 gluons to unconstrained colorless spin-0, spin-1, spin-2 and spin-3 glueball fields. The obtained unconstrained Hamiltonian is then rewritten into a form, which separates the rotational from the scalar degrees of freedom. It is shown that the chromomagnetic potential has classical zero-energy valleys for two arbitrarily large classical glueball fields, which are the unconstrained analogs of the well-known ”constant Abelian fields”. On the quantum level, practically all glueball excitation energy is expected to go into the increase of the strengths of these two fields. Finally, as an outlook, the straightforward generalization to low energy SU(3) Yang-Mills quantum theory in analogy to the SU(2) case is indicated, leading to an expansion in the number of spatial derivatives, equivalent to a strong coupling expansion, with the SU(3) Yang-Mills quantum mechanics constituting the leading order.
01 Jan 2012
TL;DR: In this article, the authors harness modern methods of analytical perturbation theory to normalize the system dynamics about the circular restricted three-body problem and about one of the triangular Lagrange points.
Abstract: of Dissertation Analytical Methods and Perturbation Theory for the Elliptic Restricted Three-Body Problem of Astrodynamics The distinguishing characteristic of the elliptic restricted three-body problem is a pulsating potential field resulting in non-autonomous and non-integrable spacecraft dynamics, which are difficult to model using classical methods of analysis. The purpose of this study is to harness modern methods of analytical perturbation theory to normalize the system dynamics about the circular restricted three-body problem and about one of the triangular Lagrange points. The normalization is achieved through a canonical transformation of the system Hamiltonian function based on the Lie transform method introduced by Hori and Deprit in the 1960s. The classic method derives a near-identity transformation of a Hamiltonian function expanded about a single parameter such that the transformed system possesses ideal properties of integrability. One of the major contributions of this study is to extend the normalization method to two-parameter expansions and to non-autonomous Hamiltonian systems. The twoparameter extension is used to normalize the system dynamics of the elliptic restricted three-body problem such that the stability of the triangular Lagrange points may be determined using the Kolmogorov-Arnold-Moser theorem. Further dynamical analysis is performed in the transformed phase space in terms of local integrals of motion
TL;DR: In this article, a class of non-Hermitian Hamiltonians with velocity dependent potentials is introduced and studied, and it is shown that the energy spectra are real and bounded from below, which proves the stability of all members in the class.
Abstract: We introduce and study a class of non-Hermitian Hamiltonians which have velocity dependent potentials. Since stability cannot be advocated directly from the classical potential, we show that the energy spectra are real and bounded from below which proves the stability of the spectra of all members in the class. We find that the introduced class of non-Hermitian Hamiltonians do have a corresponding superpartner class of non-Hermitian Hamiltonians. We were able to introduce supercharges which in conjunction with the corresponding super Hamiltonians constitute a closed super algebra. Among the introduced Hamiltonians, we show that non- symmetric Hamiltonians can be transformed into their corresponding superpartner Hamiltonians via a specific canonical transformation while the -symmetric ones failed to be mapped to their corresponding superpartner Hamiltonians via the same canonical transformation. Since canonical transformations preserve the spectrum, we conclude that non- -symmetric Hamiltonians out of the introduced class of Hamiltonians have the same spectrum as the corresponding superpartner Hamiltonians and thus supersymmetry (Susy) is broken for such Hamiltonians. This kind of intertwining of -symmetry and Susy is new as all the so far discussed cases concentrate on Hamiltonians of broken -symmetry that have broken Susy too while we showed that Susy can be also broken for non- -symmetric and non-Hermitian Hamiltonians.
01 Jan 2012
TL;DR: In this paper, the authors analyze the Hamilton-Jacobi theory whose purpose it is to determine a set of canonical coordinates in which the Hamiltonian equations have such a simple form that we can obtain solutions without effort.
Abstract: We have repeatedly emphasized that determining the explicit solutions of Hamiltonian equations is a task beyond the capabilities of mathematics. This situation justifies all attempts to obtain information about these solutions. For instance, a knowledge of the first integrals allows us to localize the dynamical trajectories in the phase space M 2n ∗. In turn, Nother’s theorem proves that the presence of first integrals is strictly related to the existence of symmetries. In this chapter, we analyze the Hamilton–Jacobi theory whose purpose it is to determine a set of canonical coordinates in which the Hamiltonian equations have such a simple form that we can obtain solutions without effort. However, determining this set of coordinates requires solving a nonlinear partial differential equation. Although, in general, this problem could be more difficult than the direct integration of the Hamiltonian equations, there are some interesting cases in which we are able to exhibit its solution by adopting the method, proposed by Jacobi, of separated variables.