Showing papers on "Canonical transformation published in 2007"
TL;DR: The canonical transformation theory of Yanai and Chan as mentioned in this paper provides a rigorously size-extensive description of dynamical correlation in multireference problems, and a new formulation of the theory based on the extended normal ordering procedure of Mukherjee and Kutzelnigg is presented.
Abstract: The canonical transformation theory of Yanai and Chan [J. Chem. Phys. 124, 194106 (2006)] provides a rigorously size-extensive description of dynamical correlation in multireference problems. Here we describe a new formulation of the theory based on the extended normal ordering procedure of Mukherjee and Kutzelnigg [J. Chem. Phys. 107, 432 (1997)]. On studies of the water, nitrogen, and iron oxide potential energy curves, the linearized canonical transformation singles and doubles theory is competitive in accuracy with some of the best multireference methods, such as the multireference averaged coupled pair functional, while computational timings (in the case of the iron oxide molecule) are two to three orders of magnitude faster and comparable to those of the complete active space second-order perturbation theory. The results presented here are greatly improved both in accuracy and in cost over our earlier study as the result of a new numerical algorithm for solving the amplitude equations.
135 citations
TL;DR: A new formulation of the canonical transformation theory based on the extended normal ordering procedure of Mukherjee and Kutzelnigg is described, greatly improved both in accuracy and in cost over an earlier study as the result of a new numerical algorithm for solving the amplitude equations.
Abstract: The Canonical Transformation theory of Yanai and Chan [J. Chem. Phys. 124, 194106 (2006)] provides a rigorously size-extensive description of dynamical correlation in multireference problems. Here we describe a new formulation of the theory based on the extended normal ordering procedure of Mukherjee and Kutzelnigg [J. Chem. Phys. 107, 432 (1997)].
On studies of the water, nitrogen, and iron-oxide potential energy curves, the Linearised Canonical Transformation Singles and Doubles theory is competitive in accuracy with some of the best multireference methods, such as the Multireference Averaged Coupled Pair Functional, while computational timings (in the case of the iron-oxide molecule) are two-three orders of magnitude faster and comparable to those of Complete Active Space Second-Order Perturbation Theory. The results presented here are greatly improved both in accuracy and in cost over our earlier study as the result of a new numerical algorithm for solving the amplitude equations.
132 citations
14 Mar 2007
97 citations
TL;DR: This work provides a general expression and different classification schemes for the general two-dimensional canonical integral transformations that describe the propagation of coherent light through lossless first-order optical systems.
Abstract: We provide a general expression and different classification schemes for the general two-dimensional canonical integral transformations that describe the propagation of coherent light through lossless first-order optical systems. Main theorems for these transformations, such as shift, scaling, derivation, etc., together with the canonical integral transforms of selected functions, are derived.
88 citations
TL;DR: In this article, a new parametrization of the 3-metric allows to find explicitly a York map by means of a partial Shanmugadhasan canonical transformation in canonical ADM tetrad gravity.
Abstract: A new parametrization of the 3-metric allows to find explicitly a York map by means of a partial Shanmugadhasan canonical transformation in canonical ADM tetrad gravity. This allows to identify the two pairs of physical tidal degrees of freedom (the Dirac observables of the gravitational field have to be built in term of them) and 14 gauge variables. These gauge quantities, whose role in describing generalized inertial effects is clarified, are all configurational except one, the York time, i.e. the trace \(^{3}K(\tau ,\vec \sigma )\) of the extrinsic curvature of the instantaneous 3-spaces \(\Sigma_{\tau}\) (corresponding to a clock synchronization convention) of a non-inertial frame centered on an arbitrary observer. In \(\Sigma_{\tau}\) the Dirac Hamiltonian is the sum of the weak ADM energy \(E_{\rm ADM} = \int d^3\sigma\, {\mathcal{E}}_{\rm ADM}(\tau ,\vec \sigma )\) (whose density \({\mathcal{E}}_{\rm ADM}(\tau ,\vec \sigma )\) is coordinate-dependent, containing the inertial potentials) and of the first-class constraints. The main results of the paper, deriving from a coherent use of constraint theory, are: (i) The explicit form of the Hamilton equations for the two tidal degrees of freedom of the gravitational field in an arbitrary gauge: a deterministic evolution can be defined only in a completely fixed gauge, i.e. in a non-inertial frame with its pattern of inertial forces. The simplest such gauge is the 3-orthogonal one, but other gauges are discussed and the Hamiltonian interpretation of the harmonic gauges is given. This frame-dependence derives from the geometrical view of the gravitational field and is lost when the theory is reduced to a linear spin 2 field on a background space-time. (ii) A general solution of the super-momentum constraints, which shows the existence of a generalized Gribov ambiguity associated to the 3-diffeomorphism gauge group. It influences: (a) the explicit form of the solution of the super-momentum constraint and then of the Dirac Hamiltonian; (b) the determination of the shift functions and then of the lapse one. (iii) The dependence of the Hamilton equations for the two pairs of dynamical gravitational degrees of freedom (the generalized tidal effects) and for the matter, written in a completely fixed 3-orthogonal Schwinger time gauge, upon the gauge variable \({}^3K(\tau ,\vec \sigma )\) , determining the convention of clock synchronization. The associated relativistic inertial effects, absent in Newtonian gravity and implying inertial forces changing from attractive to repulsive in regions with different sign of \({}^3K(\tau ,\vec \sigma )\) , are completely unexplored and may have astrophysical relevance in the interpretation of the dark side of the universe.
62 citations
TL;DR: In this article, exact solutions of the Schrodinger equation are obtained for the Rosen-Morse and Scarf potentials with the position-dependent effective mass by appliying a general point canonical transformation.
Abstract: Exact solutions of the Schrodinger equation are obtained for the Rosen-Morse and Scarf potentials with the position-dependent effective mass by appliying a general point canonical transformation. The general form of the point canonical transforma- tion is introduced by using a free parameter. Two different forms of mass distributions are used. A set of the energy eigenvalues of the bound states and corresponding wave functions for target potentials are obtained as a function of the free parameter.
50 citations
TL;DR: In this article, a Hamiltonian model with two-photon interaction with a two-level atom and a degenerate parametric amplifier is considered, and an exact solution of the wave function in the Schrodinger picture is obtained.
47 citations
01 Jan 2007
TL;DR: In this article, the authors return to the Hamiltonian formulation of the NS model in order to discuss the basic transformation of the inverse scattering method from a Hamiltonian standpoint, and show that the integrals of the motion introduced in Chapter I are in involution.
Abstract: In this chapter we return to the Hamiltonian formulation of the NS model in order to discuss the basic transformation of the inverse scattering method
from the Hamiltonian standpoint. We shall describe the Poisson structure on the scattering data of the auxiliary linear problem induced through f from the initial Poisson structure defined in Chapter I. Under the rapidly decreasing or finite density boundary conditions, the NS model proves to be a completely integrable system, with f defining a transformation to action-angle variables. In particular, we will show that the integrals of the motion introduced in Chapter I are in involution. In these terms scattering of solitons amounts to a simple canonical transformation.
24 citations
TL;DR: In this paper, the authors present a new proof of the adiabatic invariance of a slowly varying quantity closely related to the action variable for one-dimensional classical time-dependent Hamiltonian systems with quasi-periodic orbits.
Abstract: We consider one-dimensional classical time-dependent Hamiltonian systems with quasi-periodic orbits. It is well known that such systems possess an adiabatic invariant which coincides with the action variable of the Hamiltonian formalism. We present a new proof of the adiabatic invariance of this quantity and illustrate our arguments by means of explicit calculations for the harmonic oscillator. The new proof makes essential use of the Hamiltonian formalism. The key step is the introduction of a slowly varying quantity closely related to the action variable. This new quantity arises naturally within the Hamiltonian framework as follows: a canonical transformation is first performed to convert the system to action-angle coordinates; then the new quantity is constructed as an action integral (effectively a new action variable) using the new coordinates. The integration required for this construction provides, in a natural way, the averaging procedure introduced in other proofs, though here it is an average in phase space rather than over time.
23 citations
27 Sep 2007
TL;DR: This paper recommends a canonical representation of the PH(3) class and presents a transformation procedure to obtain the canonical representation based on any (not only Markovian) vector-matrix representations of the distribution.
Abstract: The characterization and the canonical representation of order n phase type distributions (PH(n)) is an open research problem.
This problem is solved for n = 2, since the equivalence of the acyclic and the general PH distributions has been proven for a long time. However, no canonical representations have been introduced for the general PH distribution class so far for n > 2. In this paper we summarize the related results for n = 3. Starting from these results we recommend a canonical representation of the PH(3) class and present a transformation procedure to obtain the canonical representation based on any (not only Markovian) vector-matrix representation of the distribution.
Using this canonical transformation method we evaluate the moment bounds of the PH(3) distribution set and present the results of our numerical investigations.
22 citations
TL;DR: In this paper, a new symplectic variational approach is developed for modeling dissipation in kinetic equations, which yields a double bracket structure in phase space which generates kinetic equations representing coadjoint motion under canonical transformations.
27 Mar 2007
TL;DR: In this article, the authors propose a canonical transformation of phase space into two orthogonal sectors, the first of which includes canonically conjugate gauge invariant variables with free massive excitations, and the second sector, which decouples from the total Hamiltonian, is equivalent to the phase space description of the associated pure TFT.
Abstract: Abelian topologically massive gauge theories (TMGT) provide a topological mechanism to generate mass for a bosonic p-tensor field in any spacetime dimension. These theories include the 2+1 dimensional Maxwell-Chern-Simons and 3+1 dimensional Cremmer-Scherk actions as particular cases. Within the Hamiltonian formulation, the embedded topological field theory (TFT) sector related to the topological mass term is not manifest in the original phase space. However through an appropriate canonical transformation, a gauge invariant factorisation of phase space into two orthogonal sectors is feasible. The first of these sectors includes canonically conjugate gauge invariant variables with free massive excitations. The second sector, which decouples from the total Hamiltonian, is equivalent to the phase space description of the associated non dynamical pure TFT. Within canonical quantisation, a likewise factorisation of quantum states thus arises for the full spectrum of TMGT in any dimension. This new factorisation scheme also enables a definition of the usual projection from TMGT onto topological quantum field theories in a most natural and transparent way. None of these results rely on any gauge fixing procedure whatsoever. Comment: 1+25 pages, no figures
TL;DR: In this paper, the authors generalize the prescription realizing classical Poisson-Lie T-duality as canonical transformation to poisson-lie T-plurality, and present explicit formulae for realizing canonical transformation.
TL;DR: In this paper, the bound-state solutions and the su(1,1) description of the $d$-dimensional radial harmonic oscillator, the Morse and the radial Coulomb Schrodinger equations are reviewed in a unified way using the point canonical transformation method.
Abstract: The bound-state solutions and the su(1,1) description of the $d$-dimensional radial harmonic oscillator, the Morse and the $D$-dimensional radial Coulomb Schrodinger equations are reviewed in a unified way using the point canonical transformation method. It is established that the spectrum generating su(1,1) algebra for the first problem is converted into a potential algebra for the remaining two. This analysis is then extended to Schrodinger equations containing some position-dependent mass. The deformed su(1,1) construction recently achieved for a $d$-dimensional radial harmonic oscillator is easily extended to the Morse and Coulomb potentials. In the last two cases, the equivalence between the resulting deformed su(1,1) potential algebra approach and a previous deformed shape invariance one generalizes to a position-dependent mass background a well-known relationship in the context of constant mass.
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TL;DR: In this paper, a general point canonical transformation is applied by using a free parameter to obtain the energy eigenvalues of the bound states and corresponding wave functions for target potentials as a function of the free parameter.
Abstract: PT-symmetric solutions of Schrodinger equation are obtained for the Scarf and generalized harmonic oscillator potentials with the position-dependent mass. A general point canonical transformation is applied by using a free parameter. Three different forms of mass distributions are used. A set of the energy eigenvalues of the bound states and corresponding wave functions for target potentials are obtained as a function of the free parameter.
TL;DR: In this paper, different forms of the Hamiltonian formulations of linearized General Relativity/spin-two theories are discussed in order to show their similarities and differences, and it is demonstrated that non-covariant modifications to the initial covariant Lagrangian (similar to those modifications used in full gravity) are in fact unnecessary.
Abstract: The different forms of the Hamiltonian formulations of linearized General Relativity/spin-two theories are discussed in order to show their similarities and differences. It is demonstrated that in the linear model, non-covariant modifications to the initial covariant Lagrangian (similar to those modifications used in full gravity) are in fact unnecessary. The Hamiltonians and the constraints are different in these two formulations but the structure of the constraint algebra and the gauge invariance derived from it are the same. It is shown that these equivalent Hamiltonian formulations are related to each other by a canonical transformation which is explicitly given. The relevance of these results to the full theory of General Relativity is briefly discussed.
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TL;DR: In this paper, a class of semiclassical Fourier Integral Operators (FIOs) with complex phase associated to some canonical transformation of the phase space was constructed, based on some general boundedness assumptions on the symbol and the canonical transformation.
Abstract: In this work, a class of semiclassical Fourier Integral Operators (FIOs) with complex phase associated to some canonical transformation of the phase space $T^*\R^d$ is constructed. Upon some general boundedness assumptions on the symbol and the canonical transformation, their continuity (as operators) from the Schwartz class into itself and from $L^2$ into itself are proven.
TL;DR: The variational canonical transformation method has been applied to the Holstein model to obtain an efiective polaronic Hamiltonian, which is subsequently analyzed in the limit of a weak electron-electron inter-action as discussed by the authors.
Abstract: The variational canonical transformation method has been applied tothe Holstein model to obtain an efiective polaronic Hamiltonian, which issubsequently analyzed in the limit of a weak efiective electron{electron inter-action. A competition between the superconducting and charge-density wavephases has been studied in the light of strong polaronic efiects. The phasediagrams illustrating the system evolution from adiabatic to anti-adiabaticlimit are presented.PACS numbers: 71.38.{k, 71.45.Lr, 74.20.Mn
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TL;DR: An alternative version of Hamiltonian formalism for higher-derivative theories is presented in this paper, which is related to the standard Ostrogradski approach by a canonical transformation.
Abstract: An alternative version of Hamiltonian formalism for higher-derivative theories is presented. It is related to the standard Ostrogradski approach by a canonical transformation. The advantage of the approach presented is that the Lagrangian is nonsingular and the Legendre transformation is performed in a straightforward way.
TL;DR: In this paper, the authors propose a canonical transformation of phase space into two orthogonal sectors, the first of which includes canonically conjugate gauge invariant variables with free massive excitations, and the second sector, which decouples from the total Hamiltonian, is equivalent to the phase space description of the associated pure TFT.
Abstract: Abelian topologically massive gauge theories (TMGT) provide a topological mechanism to generate mass for a bosonic p-tensor field in any spacetime dimension. These theories include the 2+1 dimensional Maxwell-Chern-Simons and 3+1 dimensional Cremmer-Scherk actions as particular cases. Within the Hamiltonian formulation, the embedded topological field theory (TFT) sector related to the topological mass term is not manifest in the original phase space. However through an appropriate canonical transformation, a gauge invariant factorisation of phase space into two orthogonal sectors is feasible. The first of these sectors includes canonically conjugate gauge invariant variables with free massive excitations. The second sector, which decouples from the total Hamiltonian, is equivalent to the phase space description of the associated non dynamical pure TFT. Within canonical quantisation, a likewise factorisation of quantum states thus arises for the full spectrum of TMGT in any dimension. This new factorisation scheme also enables a definition of the usual projection from TMGT onto topological quantum field theories in a most natural and transparent way. None of these results rely on any gauge fixing procedure whatsoever.
TL;DR: In this paper, the one-dimensional Dirac equation with linear potential has been solved by the method of canonical transformation, and bound-state wavefunctions and corresponding energy spectrum have been obtained for all bound states.
Abstract: In this paper the one-dimensional Dirac equation with linear potential has been solved by the method of canonical transformation. The bound-state wavefunctions and the corresponding energy spectrum have been obtained for all bound states.
TL;DR: In this article, a formulation of quantum mechanics in terms of complex canonical variables is presented, and it is shown that these variables are governed by Hamilton's equations, and the action variables need to be quantized.
Abstract: A formulation of quantum mechanics in terms of complex canonical variables is presented. It is seen that these variables are governed by Hamilton's equations. It is shown that the action variables need to be quantized. By formulating a quantum Hamilton equation for the momentum variable, the energies for two different systems are determined. Quantum canonical transformation theory is introduced and the geometrical significance of a set of generalized quantization conditions which are obtained is discussed.
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TL;DR: In this paper, the authors explore fundamental constraints on the evolution of certain symplectic subvolumes possessed by any Hamiltonian phase space, and show that these constraints reduce to Liouville's Theorem, and present a preferred basis that has certain minimality properties with regards to the local volume expansion of phase space.
Abstract: In this paper we will explore fundamental constraints on the evolution of certain symplectic subvolumes possessed by any Hamiltonian phase space. This research has direct application to optimal control and control of conservative mechanical systems. We relate geometric invariants of symplectic topology to computations that can easily be carried out with the state transition matrix of the flow map. We will show how certain symplectic subvolumes have a minimal obtainable volume; further if the subvolume dimension equals the phase space dimension, this constraint reduces to Liouville's Theorem. Finally we present a preferred basis that, for a given canonical transformation, has certain minimality properties with regards to the local volume expansion of phase space.
TL;DR: In this paper, a canonical transformation of the integration variables can remove the chemical potential from the action, but the potential appears in the transformed boundary conditions and the final result for the perturbative Green's functions is unchanged.
Abstract: In the functional integral formulation of real-time thermal field theory, a time-dependent canonical transformation of the integration variables can remove the chemical potential from the action. The transformation eliminates the chemical potential from the differential equation satisfied by the propagator, but the chemical potential appears in the transformed boundary conditions and the final result for the perturbative Green's functions is unchanged.
TL;DR: In this article, it was shown that Egorov's theorem on the canonical transformation of symbols of pseudodifferential operators conjugated by Fourier integral operators can be sharpened.
Abstract: We prove that the theorem of Egorov, on the canonical transformation of symbols of pseudodifferential operators conjugated by Fourier integral operators, can be sharpened. The main result is that the statement of Egorov's theorem remains true if, instead of just considering the principal symbols in $S^m/S^{m-1}$ for the pseudodifferential operators, one uses refined principal symbols in $S^m/S^{m-2}$, which for classical operators correspond simply to the principal plus the subprincipal symbol, and can generally be regarded as the first two terms of its Weyl symbol expansion: we call it the principal Weyl symbol of the pseudodifferential operator. Particular unitary Fourier integral operators, associated to the graph of the canonical transformation, have to be used in the conjugation for the higher accuracy to hold, leading to microlocal representations by oscillatory integrals with specific symbols that are given explicitly in terms of the generating function that locally describes the graph of the transformation. The motivation for the result is based on the optimal symplectic invariance properties of the Weyl correspondence in ${\mathbb R}^n$ and its symmetry for real symbols.
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TL;DR: In this article, necessary and sufficient conditions for the integrability of geodesic flows on homogeneous spaces with invariant and central metrics were obtained, and a special canonical transformation in the space was proposed based on constructing the canonical coordinates on the orbits of the coadjoint representation and on the simplectic sheets of the Poisson algebra of invariant functions.
Abstract: We obtain necessary and sufficient conditions for the integrability in quadratures of geodesic flows on homogeneous spaces $M$ with invariant and central metrics. The proposed integration algorithm consists in using a special canonical transformation in the space $T^*M$ based on constructing the canonical coordinates on the orbits of the coadjoint representation and on the simplectic sheets of the Poisson algebra of invariant functions. This algorithm is applicable to integrating geodesic flows on homogeneous spaces of a wild Lie group.
TL;DR: In this paper, the non-relativistic limit of the linear wave equation for zero and unity spin bosons of mass $m$ in the Duffin-Kemmer-Petiau representation is investigated by means of a unitary transformation, analogous to the Foldy-Wouthuysen canonical transformation for a relativistic electron.
Abstract: The non-relativistic limit of the linear wave equation for zero and unity spin bosons of mass $m$ in the Duffin-Kemmer-Petiau representation is investigated by means of a unitary transformation, analogous to the Foldy-Wouthuysen canonical transformation for a relativistic electron. The interacting case is also analyzed, by considering a power series expansion of the transformed Hamiltonian, thus demonstrating that all features of particle dynamics can be recovered if corrections of order $1/m^{2}$ are taken into account through a recursive iteration procedure.
TL;DR: In this paper, the algebraic structure of the Feynman propagator with a general time-dependent quadratic Hamiltonian system was investigated using the Lie-algebraic technique.
Abstract: We investigate the algebraic structure of the Feynman propagator with a general time-dependent quadratic Hamiltonian system. Using the Lie-algebraic technique, we obtain a normal-ordered form of the time-evolution operator, and then the propagator is easily derived by a simple “integration within ordered product” technique. It is found that this propagator contains a classical generating function, which demonstrates a new correspondence between classical and quantum mechanics.