Showing papers on "Canonical transformation published in 2017"

On the Adiabatic Representation of Meyer-Miller Electronic-Nuclear Dynamics

Abstract: The Meyer-Miller (MM) classical vibronic (electronic + nuclear) Hamiltonian for electronically non-adiabatic dynamics-as used, for example, with the recently developed symmetrical quasiclassical (SQC) windowing model-can be written in either a diabatic or an adiabatic representation of the electronic degrees of freedom, the two being a canonical transformation of each other, thus giving the same dynamics. Although most recent applications of this SQC/MM approach have been carried out in the diabatic representation-because most of the benchmark model problems that have exact quantum results available for comparison are typically defined in a diabatic representation-it will typically be much more convenient to work in the adiabatic representation, e.g., when using Born-Oppenheimer potential energy surfaces (PESs) and derivative couplings that come from electronic structure calculations. The canonical equations of motion (EOMs) (i.e., Hamilton's equations) that come from the adiabatic MM Hamiltonian, however, in addition to the common first-derivative couplings, also involve second-derivative non-adiabatic coupling terms (as does the quantum Schrodinger equation), and the latter are considerably more difficult to calculate. This paper thus revisits the adiabatic version of the MM Hamiltonian and describes a modification of the classical adiabatic EOMs that are entirely equivalent to Hamilton's equations but that do not involve the second-derivative couplings. The second-derivative coupling terms have not been neglected; they simply do not appear in these modified adiabatic EOMs. This means that SQC/MM calculations can be carried out in the adiabatic representation, without approximation, needing only the PESs and the first-derivative coupling elements. The results of example SQC/MM calculations are presented, which illustrate this point, and also the fact that simply neglecting the second-derivative couplings in Hamilton's equations (and presumably also in the Schrodinger equation) can cause very significant errors.

Thiemann complexifier in classical and quantum FLRW cosmology

Abstract: In the context of loop quantum gravity (LQG), we study the fate of Thiemann complexifier in homogeneous and isotropic Friedmann-Lemaitre-Roberston-Walker (FLRW) cosmology. The complexifier is the dilatation operator acting on the canonical phase space for gravity and generates the canonical transformations shifting the Barbero-Immirzi parameter. We focus on the closed algebra consisting in the complexifier, the 3d volume and the Hamiltonian constraint, which we call the CVH algebra (for Complexier-Volume-Hamiltonian constraint algebra). In standard cosmology, for gravity coupled to a scalar field, the CVH algebra is identified as a $\mathfrak{su}(1,1)$ Lie algebra, with the Hamiltonian as a null generator, the complexifier as a boost and the $\mathfrak{su}(1,1)$ Casimir given by the matter density. The loop gravity cosmology approach introduces a regularization length scale $\ensuremath{\lambda}$ and regularizes the gravitational Hamiltonian in terms of SU(2) holonomies. We show that this regularization is compatible with the CVH algebra, if we suitably regularize the complexifier and inverse volume factor. The regularized complexifier generates a generalized version of the Barbero's canonical transformation which reduces to the classical one when $\ensuremath{\lambda}\ensuremath{\rightarrow}0$. This structure allows for the exact integration of the actions of the Hamiltonian constraints and the complexifier. This straightforwardly extends to the quantum level: the cosmological evolution is described in terms of SU(1, 1) coherent states and the regularized complexifier generates unitary transformations. The Barbero-Immirzi parameter is to be distinguished from the regularization scale $\ensuremath{\lambda}$, it can be rescaled unitarily and the Immirzi ambiguity ultimately disappears from the physical predictions of the theory. Finally, we show that the complexifier becomes the effective Hamiltonian when deparametrizing the dynamics using the scalar field as a clock, thus underlining the deep relation between cosmological evolution and scale transformations.

Two-body problem in Scalar-Tensor theories as a deformation of General Relativity : an Effective-One-Body approach

Abstract: In this paper we address the two-body problem in massless Scalar-Tensor (ST) theories within an Effective-One-Body (EOB) framework. We focus on the first building block of the EOB approach, that is, mapping the conservative part of the two-body dynamics onto the geodesic motion of a test particle in an effective external metric. To this end, we first deduce the second post-Keplerian (2PK) Hamiltonian of the two-body problem from the known 2PK Lagrangian. We then build, by means of a canonical transformation a ST-deformation of the general relativistic EOB Hamiltonian which allows to incorporate the Scalar-Tensor (2PK) corrections to the currently best available General Relativity EOB results. This EOB-ST Hamiltonian defines a resummation of the dynamics that may provide information on the strong-field regime, in particular, the ISCO location and associated orbital frequency and can be compared to other, e.g. tidal, corrections.

Equivalent and inequivalent canonical structures of higher order theories of gravity

Abstract: The canonical formulation of higher-order theories of gravity can only be accomplished by introducing additional degrees of freedom, namely, the extrinsic curvature tensor ${K}_{ij}$. Consequently, to match Cauchy data with the boundary data, terms in addition to the three-space metric ${h}_{ij}$ must also be fixed at the boundary. While in the Ostrogradsky, Dirac, and Horowitz formalisms the extrinsic curvature tensor is kept fixed at the boundary, a modified Horowitz formalism fixes the Ricci scalar $R$ instead. It has been taken for granted that the Hamiltonian structures corresponding to all of the formalisms with different end-point data are either the same or are canonically equivalent. In the present study, we show that this indeed is true, but only for a class of higher-order theories. However, for more general higher-order theories---e.g., dilatonic coupled Gauss-Bonnet gravity in the presence of a curvature-squared term---the Hamiltonian obtained following the modified Horowitz formalism is found to be different from the others, and is not related under canonical transformation. Further, it has also been demonstrated that only the modified Horowitz' formalism can produce a viable quantum description of the theory, since it only admits a classical analogue under an appropriate semiclassical approximation. Thus, fixing the Ricci scalar $R$ at the boundary appears to be a fundamental issue for a canonical formulation of higher-order theories of gravity.

Canonical transformation path to gauge theories of gravity II --- Spacetime coupling of massive spin particle fields

Abstract: The generic form of spacetime dynamics as a gauge theory has recently been derived, based on only the action principle and on the general principle of relativity. In the present paper, the physical consequences are discussed. The gauge coupling terms obtained imply that Einstein's theory holds only for structureless (spin zero) particles, and aggregations of them. For massive particles with spin, however, spacetime dynamics is to be described by an additional, Poisson-type equation describing the interaction of the particle's spin with the torsion of spacetime. This equation shows that torsion propagates with gravitational waves. Spin carrying matter is thus shown to couple with the torsion of spacetime. The proper source term for the spacetime dynamics is given by the canonical energy-momentum tensor---which embraces also the energy density furnished by microscopic internal spin. This changes the model of compact astrophysical objects and of relativistic collapse dynamics, with significant impact on the description of binary neutron star mergers and pulsar dynamics. The final generally covariant Hamiltonian must contain a term quadratic in the conjugate momenta of the gauge fields in order to yield a closed system of field equations---in analogy to all other Hamiltonian descriptions of field theories. The fact that the canonical gauge theory of gravity derived here requires that both, quadratic curvature tensors and canonical energy momentum tensors, enter the field equation for the spacetime dynamics leads to a qualitatively new framework for general relativity---and to a new understanding of Friedman cosmology and the cosmological constant problem.

The Quantum State as Spatial Displacement

Abstract: We give a simple demonstration that the Schrodinger equation may be recast as a self-contained second-order Newtonian law for a congruence of spacetime trajectories. This provides a pictorial representation of the quantum state as the displacement function of the collective whereby quantum evolution is represented as the deterministic unfolding of a continuous coordinate transformation. Introducing gauge potentials for the density and current density it is shown that the wave-mechanical and trajectory pictures are connected by a canonical transformation. The canonical trajectory theory is shown to provide an alternative basis for the quantum operator calculus and the issue of the observability of the quantum state is examined within this context. The construction illuminates some of the problems involved in connecting the quantum and classical descriptions.

Canonical formulation of Pais–Uhlenbeck action and resolving the issue of branched Hamiltonian

Abstract: Canonical formulation of higher order theory of gravity requires to fix (in addition to the metric), the scalar curvature, which is acceleration in disguise, at the boundary. On the contrary, for the same purpose, Ostrogradskis or Diracs technique of constrained analysis, and Horowitz formalism, tacitly assume velocity (in addition to the co-ordinate) to be fixed at the end points. In the process when applied to gravity, GibbonsHawking-York term disappears. To remove such contradiction and to set different higher order theories on the same footing, we propose to fix acceleration at the endpoints/ boundary. However, such proposition is not compatible to Ostrogradskis or Diracs technique. Here, we have modified Horowitzs technique of using an auxiliary variable, to establish a one-to-one correspondence between different higher order theories. Although, the resulting Hamiltonian is related to the others under canonical transformation, we have proved that this is not true in general. We have also demonstrated how higher order terms can regulate the issue of branched Hamiltonian.

Position-Dependent Mass Schrödinger Equation for the Morse Potential

Abstract: The position dependent mass Schrodinger equation (PDMSE) has a wide range of quantum applications such as the study of semiconductors, quantum wells, quantum dots and impurities in crystals, among many others. On the other hand, the Morse potential is one of the most important potential models used to study the electronic properties of diatomic molecules. In this work, the solution of the effective mass one-dimensional Schrodinger equation for the Morse potential is presented. This is done by means of the canonical transformation method in algebraic form. The PDMSE is solved for any model of the proposed kinetic energy operators as for example the BenDaniel-Duke, Gora-Williams, Zhu-Kroemer or Li-Kuhn. Also, in order to solve the PDMSE with Morse potential, we consider a superpotential leading to a special form of the exactly solvable Schrodinger equation of constant mass for a class of multiparameter exponential-type potential along with a proper mass distribution. The proposed approach is general and can be applied in the search of new potentials suitable on science of materials by looking into the viable choices of the mass function.

Hamiltonian consistency of the gravitational constraint algebra under deformations

Abstract: The importance of the first-class constraint algebra of general relativity is not limited just by its self-contained description of the gauge nature of spacetime, but it also provides conditions to properly evolve the geometry by selecting a gauge only once throughout the whole evolution of a gravitational system. This must be a property of all background independent theories. In this paper we consider gravitational theories which arise from deformations of the fundamental canonical variables of general relativity where the proposed deformations are inspired by modifications of gravity. These variable deformations result in new theories when the deformation is not a canonical transformation. The new theory must preserve the first-class structure of the algebra, which is a non-trivial restriction for generic deformations. In this vein we present a general deformation scheme along with consistency conditions, so that the algebra of constraints is still satisfied in the resulting theory. This is illustrated both in metric theory as well as in tetrad theory.

Input‐to‐state stability of stochastic port‐Hamiltonian systems using stochastic generalized canonical transformations

Abstract: Summary As a practically important class of nonlinear stochastic systems, this paper considers stochastic port-Hamiltonian systems (SPHSs) and investigates the stochastic input-to-state stability (SISS) property of a class of SPHSs. We clarify necessary conditions for the closed-loop system of an SPHS to be SISS. Moreover, we provide a systematic construction of both the SISS controller and Lyapunov function so that the proposed necessary conditions hold. In the main results, the stochastic generalized canonical transformation plays a key role. The stochastic generalized canonical transformation technique enables to design both coordinate transformation and feedback controller with preserving the SPHS structure of the closed-loop system. Consequently, the main theorem guarantees that the closed-loop system obtained by the proposed method is SISS against both deterministic disturbance and stochastic noise. Copyright © 2017 John Wiley & Sons, Ltd.

RATIONALLY EXTENDED SHAPE INVARIANT POTENTIALS IN ARBITRARY D DIMENSIONS ASSOCIATED WITH EXCEPTIONAL Xm POLYNOMIALS

Abstract: Rationally extended shape invariant potentials in arbitrary D-dimensions are obtained by using point canonical transformation (PCT) method. The bound-state solutions of these exactly solvable potentials can be written in terms of Xm Laguerre or Xm Jacobi exceptional orthogonal polynomials. These potentials are isospectral to their usual counterparts and possess translationally shape invariance property.

On Linear Hamiltonian Systems

Abstract: In this paper we consider the normalization of quadratic Hamiltonian We get the new method to find the generating function of the canonical transformation We obtain the solution of the system of matrix equations to find this transformation The corresponding Hamiltonian matrix has multiply eigenvalues

Asymptotic Stability Analysis of 2-D Discrete State Space Systems with Singular Matrix

Abstract: The aim of this paper is to establish, on the basis of Lagrange method for solving partial difference equations, conditions under an asymptotic stability analysis procedure to investigate conditions for the existence of a solution to 2-D (two dimensional) discrete system whose state space representation is composed by a non-singular matrix To accomplish it, the concept of generalized inverse of matrices and Jordan canonical transformation are applied on the original system and then Lagrange solutions to the transformed systems are pursued Once the conditions are determined on the grounds of the transformed system and the existence conditions of solutions for this system is accomplished, the conditions for the original system is obtained by back transformation A numerical example is given to show how the procedure works

Lorentzian condition in holographic cosmology

Abstract: We derive a sufficient set of conditions on the Euclidean boundary theory in dS/CFT for it to predict classical, Lorentzian bulk evolution at large spatial volumes. Our derivation makes use of a canonical transformation to express the bulk wave function at large volume in terms of the sources of the dual partition function. This enables a sharper formulation of dS/CFT. The conditions under which the boundary theory predicts classical bulk evolution are stronger than the criteria usually employed in quantum cosmology. We illustrate this in a homogeneous isotropic minisuperspace model of gravity coupled to a scalar field in which we identify the ensemble of classical histories explicitly.

Dirac's formalism for time-dependent Hamiltonian systems in the extended phase space

Abstract: The Dirac's formalism for constrained systems is applied to the analysis of time-dependent Hamiltonians in the extended phase space. We show that the Lewis invariant is a reparametrization invariant and we calculate the Feynman propagator using the extended phase description. We show that the quantum phase of the Feynman propagator is given by the boundary term of the canonical transformation of the extended phase space.

Second-order theory for the two-body problem with varying mass including periastron effect

Abstract: The model of varying mass function, including periastron effect, in terms of Delaunay variables will be expanded. The Hamiltonian of the problem is developed in the extended phase space by introducing a new canonical pair of variable ( $$q_4, Q_4$$ ). The first “ $$q_4 $$ ” is defined as explicit function of time and the initial mass of the system. The conjugate momenta “ $$Q_4$$ ” is assigned as the momenta raises from the varying mass. The short-period analytical solution through a second-order canonical transformation using “Hori’s” method developed by “Kamel” is obtained. The variation equations for the orbital elements are obtained too. The results of the effect of the varying mass and the periastron effect in the case $$n = 2$$ are analyzed.

The multiplicative Hamiltonian and its hierarchy

Abstract: The multiplicative Hamiltonian flow on the phase space for a system with 1 degree of freedom was constituted from infinite hierarchy Hamiltonian flows. A new type of canonical transformation associated with the multiplicative Hamiltonian was found and called the {\lambda}- extended class of the standard canonical transformations.

The multiplicative Hamiltonian and its hierarchy

Abstract: The multiplicative Hamiltonian flow on the phase space for a system with 1 degree of freedom was constituted from infinite hierarchy Hamiltonian flows. A new type of canonical transformation associated with the multiplicative Hamiltonian was found and called the {\lambda}- extended class of the standard canonical transformations.

Dirac-Bergmann Constraints in Physics: Singular Lagrangians, Hamiltonian Constraints and the Second Noether Theorem

Abstract: There is a review of the main mathematical properties of system described by singular Lagrangians and requiring Dirac-Bergmann theory of constraints at the Hamiltonian level. The following aspects are discussed: i) the connection of the rank and eigenvalues of the Hessian matrix in the Euler-Lagrange equations with the chains of first and second class constraints; ii) the connection of the Noether identities of the second Noether theorem with the Hamiltonian constraints; iii) the Shanmugadhasan canonical transformation for the identification of the gauge variables and for the search of the Dirac observables, i.e. the quantities invariant under Hamiltonian gauge transformations. Review paper for a chapter of a future book.

Boundary control of distributed port-hamiltonian systems via generalised canonical transformations

Abstract: This paper presents a novel approach for the development of boundary control laws for a class of linear, distributed port-Hamiltonian systems, with one dimensional spatial domain. The idea is to determine a control action able to map the initial system into a target one, characterised not only by a different Hamiltonian function, but also by new internal dissipative and power-preserving interconnection structures. The methodology consists of two main steps, each associated to a generalised canonical transformation. In the first one, a coordinate change (based on a combination of a linear mapping and a backstepping transformation) is employed to modify the internal structure of the system. Then, in the second step, a generalised canonical transformation capable of properly shaping the Hamiltonian function is introduced. The proposed approach is illustrated with the help of an example, the boundary stabilisation of a lossless transmission line.

Generalized Dicke Model of Graphene Cavity QED

Abstract: We present a theory of the cavity quantum electrodynamics of graphene cyclotron resonance. By employing a canonical transformation, we derive an effective Hamiltonian for the system comprised of two neighboring Landau levels dressed by the cavity electromagnetic field (integer quantum Hall polaritons). This generalized Dicke Hamiltonian, which contains terms that are quadratic in the electromagnetic field and respects gauge invariance, is then used to verify the impossibility of super-radiant instability.

Canonical Transformations, Quantization, Mutually Unbiased and Other Complete Bases

Abstract: Using ideas based on supersymmetric quantum mechanics, we design canonical transformations of the usual position and momentum to create generalized “Cartesian-like” positions, W, and momenta, Pw , with unit Poisson brackets. These are quantized by the usual replacement of the classical , x Px by quantum operators, leading to an infinite family of potential “operator observables”. However, all but one of the resulting operators are not Hermitian (formally self-adjoint) in the original position representation. Using either the chain rule or Dirac quantization, we show that the resulting operators are “quasi-Hermitian” relative to the x-representation and that all are Hermitian in the W-representation. Depending on how one treats the Jacobian of the canonical transformation in the expression for the classical momentum, Pw , quantization yields a) continuous mutually unbiased bases (MUB), b) orthogonal bases (with Dirac delta normalization), c) biorthogonal bases (with Dirac delta normalization), d) new W-harmonic oscillators yielding standard orthonormal bases (as functions of W) and associated coherent states and Wigner distributions. The MUB lead to W-generalized Fourier transform kernels whose eigenvectors are the W-harmonic oscillator eigenstates, with the spectrum (±1,±i) , as well as “W-linear chirps”. As expected, W, Pw satisfy the uncertainty product relation: ΔWΔPw ≥1/2 , h=1.

Dirac-Bergmann Constraints in Relativistic Physics: Non-Inertial Frames, Point Particles, Fields and Gravity

Abstract: There is a review of the physical theories needing Dirac-Bergmann theory of constraints at the Hamiltonian level due to the existence of gauge symmetries. It contains: i) the treatment of systems of point particles in special relativity both in inertial and non-inertial frames with a Wigner-covariant way of eliminating relative times in relativistic bound states; ii) the description of the electro-magnetic field in relativistic atomic physics and of Yang-Mills fields in absence of Gribov ambiguity in particle physics; iii) the identification of the inertial gauge variables and of the physical variables in canonical ADM tetrad gravity in presence of the electro-magnetic field and of charged scalar point particles in asymptotically Minkowskian space-times without super-translations by means of a Shanmugadhasan canonical transformation to a York canonical basis adapted to ten of the 14 first-class constraints and the definition of the Hamiltonian Post-Minkowskian weak field limit. Review paper for a chapter of a future book

Semiclassical analysis of a quasi-exactly solvable system: second harmonic generation G Alvarez and R F Alvarez-Estrada Third harmonic generation: complex canonical transformation and JWKB solution

Abstract: The usual second harmonic generation effective Hamiltonian is shown to be equivalent to a one-dimensional Schrodinger operator with a sextic polynomial potential. This operator belongs to the class of quasi-exactly solvable models, for which a finite number of eigenvalues and eigenvectors can be determined exactly. A Jeffreys-Wentzel-Kramers-Brillouin analysis provides accurate asymptotic expansions for these eigenvalues in the limit of large unperturbed energies.

Canonical Transformation of Potential Model Hamiltonian Mechanics to Geometrical Form I

Abstract: Using the methods of symplectic geometry, we establish the existence of a canonical transformation from potential model Hamiltonians of standard form in a Euclidean space to an equivalent geometrical form on a manifold, where the corresponding motions are along geodesic curves. The advantage of this representation is that it admits the computation of geometric deviation as a test for local stability, shown in previous studies to be a very effective criterion for the stability of the orbits generated by the potential model Hamiltonian. We describe here an algorithm for finding the generating function for the canonical transformation and describe some of the properties of this mapping under local diffeomorphisms. We give a convergence proof for this algorithm for the one-dimensional case, and provide a precise geometric formulation of geodesic deviation which relates the stability of the motion in the geometric form to that of the Hamiltonian standard form. We discuss the relation of bounded domains in the two representations for which Morse theory would be applicable. Numerical computations for some interesting examples will be presented in forthcoming papers.

Lie algebraic solution of the Kratzer oscillator in diatomic molecules

Abstract: The study of diatomic molecules plays a central role in the understanding of the chemical bond. For their simplicity, they serve as a model for the study of more complex molecular systems. In this article, we solve the rovibrational Schr\"odinger equation for diatomic molecules using the Kratzer oscillator, by means of so(2,1) Lie algebra. The energies and bound states for this simple model are obtained through a canonical transformation of the molecular Hamiltonian. The main contribution of the Lie-algebraic approach is that this allows us to reduce the degree of Schr\"odinger equation, obtaining a first-order differential equation whose resolution is considerably simpler than the original one. Additionally, we give the physical insight of the symmetry transformation of the SO(2,1) Lie group and show the relationship between this group and its associated Lie algebra. Finally, as an illustrative example, we calculated the selection rules for the vibrational quantum number by the use of transformation rules of SO(2,1) Lie group.

Special matrix transformations of essential nonlinear control systems

Abstract: Complex nonlinear control systems that are fundamentally non-linearizable (containing, for example, ambiguous nonlinearities), as a rule, can not be fully investigated analytically. On the other hand, the use of only computer technologies for their research often gives rise to doubts about the reliability of the results obtained. Therefore, methods based on the decomposition of the space of parameters of essentially nonlinear multidimensional systems acquire special significance. Describe the method for finding the matrix of a canonical transformation, under which the matrix of the linear part of the original system is reduced to the first natural normal form.