Showing papers on "Canonical coordinates published in 1993"
TL;DR: In this paper, it was shown that the quantum integrability of a system is equivalent to the existence of a sequence of elementary canonical transformations, and that the procedure for solving a differential equation can be viewed as a series of canonical transformations trivializing the super-Hamiltonian associated to the equation.
Abstract: Three elementary canonical transformations are shown both to have quantum implementations as finite transformations and to generate, classically and infinitesimally, the full canonical algebra. A general canonical transformation can, in principle, be realized quantum mechanically as a product of these transformations. It is found that the intertwining of two super-Hamiltonians is equivalent to there being a canonical transformation between them. A consequence is that the procedure for solving a differential equation can be viewed as a sequence of elementary canonical transformations trivializing the super-Hamiltonian associated to the equation. It is proposed that the quantum integrability of a system is equivalent to the existence of such a sequence.
88 citations
TL;DR: In this article, a canonical transformation is performed on the phase space of a number of homogeneous cosmologies to simplify the form of the scalar (or Hamiltonian) constraint, which is then easy to obtain explicit expressions of Dirac observables, i.e. phase-space functions which commute weakly with the constraint.
Abstract: A canonical transformation is performed on the phase space of a number of homogeneous cosmologies to simplify the form of the scalar (or Hamiltonian) constraint. Using the new canonical coordinates, it is then easy to obtain explicit expressions of Dirac observables, i.e. phase-space functions which commute weakly with the constraint. This, in turn, enables us to carry out a general quantization program to completion. We are also able to address the issue of time through “deparametrization” and discuss physical questions such as the fate of initial singularities in the quantum theory. We find that they persist in the quantum theory in spite of the fact that the evolution is implemented by a one-parameter family of unitary transformations. Finally, certain of these models admit conditional symmetries which are explicit already prior to the canonical transformation. These can be used to pass to the quantum theory following an independent avenue. The two quantum theories — based, respectively, on Dirac observables in the new canonical variables and conditional symmetries in the original ADM variables — are compared and shown to be equivalent.
79 citations
TL;DR: In this article, an analysis of all the consequences of the extended second Noether theorem in the second-order formalism is made, and an answer to the Dirac conjecture is given for this class of singular Lagrangians.
Abstract: After the definition of a class of well-behaved singular Lagrangians, an analysis of all the consequences of the extended second Noether theorem in the second-order formalism is made. The phase-space reformulation contains arbitrary first- and second-class constraints. An answer to the problem of the Dirac conjecture is given for this class of singular Lagrangians. By using the concepts of function groups and of the associated Shanmugadhasan canonical transformations, an attempt is made to arrive at a global formulation of the theorem, in which the original invariance under an “infinite continuous group” of transformations is replaced by weak quasi-invariance under an “infinite continuous group ,” whose algebra is an involutive distribution of Lie-Backlund vector fields generating the Noether transformations. Its phase-space counterpart is the involutive distribution associated with a special function group Ḡpm, which contains a function subgroup Ḡp connected (when in canonical form) to the Shanmugadhasan canonical transformations. Also, the various possible first-order formalisms are analyzed.
50 citations
TL;DR: In this paper, the generalized and canonical momentum operators in spherical polar coordinates are derived and discussed, and the radial operators applied to the evaluation of the Heisenberg uncertainty relation for bound states of hydrogenic atoms.
Abstract: The generalized and canonical momentum operators in spherical polar coordinates are derived and discussed, and the radial operators applied to the evaluation of the Heisenberg uncertainty relation for bound states of hydrogenic atoms. The two Pasternack relations, which connect the expectation values 〈rk〉 for values of k differing by an integer, are examined, as well as a generalization of the first relation, and the advantages which they hold for computational purposes discussed. An earlier calculation by the present author, of the electron impact ionization cross section for hydrogen atoms in the 2S metastable state, is compared with other experimental data from the literature. The calculation of the wavelengths of transitions between hydrogen‐like levels in highly excited ions possessing a few additional, tightly bound, electrons is described and compared with experimental data.
24 citations
01 Jan 1993
TL;DR: In this article, hidden and non-local symmetries of nonlinear ODEs were identified. But the hidden symmetry was not defined in terms of the order of the ODE, and it was not shown that ODE invariants can be reduced by linear nonlocal group generators.
Abstract: New results on hidden and nonlocal symmetries of nonlinear ordinary differential equations (NLODEs) are presented Two types of hidden symmetries have been identified A type I (II) hidden symmetry of an ODE occurs if a symmetry is lost (gained) when the order of the ODE is reduced Both type I and type II hidden symmetries are found in the reduction of a third-order NLODE invariant under a three-parameter nonsolvable Lie group Nonlocal group generators are determined of the exponential form and a new linear form The ODEs can be reduced by the nonlocal group generators until first-order ODEs are obtained where the procedure fails because canonical coordinates cannot be calculated in that case ODEs cannot be reduced by the linear nonlocal group generators
24 citations
TL;DR: In this article, the authors defined quantum canonical transformations as changes of the phase space variables which preserve the Dirac bracket structure and proposed a definition of quantum integrability in terms of canonical transformations, including systems which have fewer commuting integrals of motion than degrees of freedom.
23 citations
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TL;DR: The Hamiltonians underlying the Painleve transcendants are obtained by pulling back elements of the ring of spectral invariants as discussed by the authors, and these are shown to determine simple Hamiltonian systems within the underlying symplectic vector space.
Abstract: The isomonodromic deformations underlying the Painleve transcendants are interpreted as nonautonomous Hamiltonian systems in the dual $\gR^*$ of a loop algebra $\tilde\grg$ in the classical $R$-matrix framework. It is shown how canonical coordinates on symplectic vector spaces of dimensions four or six parametrize certain rational coadjoint orbits in $\gR^*$ via a moment map embedding. The Hamiltonians underlying the Painleve transcendants are obtained by pulling back elements of the ring of spectral invariants. These are shown to determine simple Hamiltonian systems within the underlying symplectic vector space.
21 citations
TL;DR: In this paper, a new class of spherically symmetric analytic solutions in canonical coordinates in general relativity is presented, which correspond to causal models of perfect fluid balls and Tolman's IV solution is a member of this class.
Abstract: In this paper is presented a new class of spherically symmetric analytic solutions in canonical coordinates in general relativity which corresponds to causal models of perfect fluid balls. Tolman’s IV solution is a member of this class. Two new solutions belonging to this class have been studied here in detail. These correspond to positive, monotonically decreasing expressions for pressure and density and realistic equations of state within causal fluid balls.
15 citations
TL;DR: In this paper, a concise account of the structure of the canonical transformation is given, in the lowest dimensional case, and a comparative study of the several different definitions of a canonical transformation can be found in the literature.
Abstract: A concise account of the structure of the canonical transformation is given, in the lowest dimensional case. This case is chosen because it offers a special clarity in several respects. In particular, the diversity of possible generating functions is illustrated by m any examples which are not available elsewhere. Many of these are of physical interest, and some of them are multivalued. These examples are used to inform a comparative study of the several different definitions of a canonical transformation to be found in the literature. The paper is pertinent to all those branches of mechanics which can be given a hamiltonian representation. These include not only the classical dynamics of particles and rigid bodies, but also some more recent studies in continuum mechanics, including geophysical fluid dynamics. An area of particular modern interest is that of symplectic integrators. These are numerical integrating algorithms which generate a solution to Hamilton’s equations via a sequence of canonical transformations, which preserve the hamiltonian structure in the numerical solution.
10 citations
TL;DR: In this paper, the integrability condition of a pair of commuting completely integrable isospectral Hamiltonian flows in the dual of the positive part of the twisted loop algebra is described.
Abstract: The two-dimensional massive Thirring model is described as the integrability condition of a pair of commuting completely integrable isospectral Hamiltonian flows in the dual\(\widehat{su}\)(2)+* of the positive part\(\widehat{su}\)(2)+ of the twisted loop algebra\(\widehat{su}\)(2). Action-angle coordinates corresponding to the spectral invariants are derived on rational coadjoint orbits and a linearization of the flows obtained in the Jacobi variety of the underlying invariant spectral curve through a Liouville generating function for canonical coordinates.
9 citations
TL;DR: In this paper, a canonical transformation is made to new canonical variables in which the new canonical momentum is energy and its conjugate coordinate is tempus, which is a function of the original generalized coordinate, the energy, and time t.
Abstract: In classical Hamiltonian dynamics for a system with a single degree of freedom a canonical transformation is made to new canonical variables in which the new canonical momentum is energy and its conjugate coordinate is called tempus. This canonical coordinate tempus conjugate to the energy is not necessarily the time t in which the system evolves, but is a function of the original generalized coordinate, the energy, and time t. For conservative systems tempus reduces to the time t, and the equations reduce to the Hamilton–Jacobi equation for Hamilton’s characteristic function. For periodic or almost periodic systems, the energy and tempus canonical variables act as a bridge to the action and angle canonical variables. Hamilton’s equations for the action and angle variables in the adiabatic limit involve a generalized Hannay (or geometrical) angle. A pendulum with a length varying in time is treated as an example.
TL;DR: In this article, the classical and quantum constraints that arise in a canonical hamiltonian formulation of a class of two-dimensional models of gravitation were analyzed and the closure of the classical constraint algebras was discussed.
TL;DR: In this paper, the problem of defining Lagrangian and Hamiltonian mechanics for a particle moving on a quantum plane Qq,p was examined, and two different differential calculi were constructed.
Abstract: We examine the problem of defining Lagrangian and Hamiltonian mechanics for a particle moving on a quantum plane Qq,p. For Lagrangian mechanics, we first define a tangent quantum plane TQq,p spanned by non-commuting particle coordinates and velocities. Using techniques similar to those of Wess and Zumino (1990), we construct two different differential calculi on TQq,p. These two differential calculi can, in principle, give rise to two different particle dynamics, starting from a single Lagrangian. For Hamiltonian mechanics, we define a phase space T*Qq,p spanned by non-commuting particle coordinates and momenta. The commutation relations for the momenta can be determined only after knowing their functional dependence on coordinates and velocities. Thus these commutation relations, as well as the differential calculus on T*Qq,p, depend on the initial choice of Lagrangian. We obtain the deformed Hamilton's equations of motion and the deformed Poisson brackets, and their definitions also depend on our initial choice of Lagrangian. We illustrate these ideas for two sample Lagrangians. The first system we examine corresponds to that of a non-relativistic particle in a scalar potential. The other Lagrangian we consider is first order in time derivatives and it is invariant under the action of the quantum group SLq(2). For that system, SLq(2) is shown to correspond to a canonical symmetry transformation.
TL;DR: A consistent light-cone canonical quantization of the four-dimensional heavy-fermion QED model is carried out using the Dirac algorithm for a system with first- and second-class constraints.
Abstract: Four-dimensional heavy-fermion QED is studied in light-cone coordinates with (anti)periodic field boundary conditions. We carry out a consistent light-cone canonical quantization of this model using the Dirac algorithm for a system with first- and second-class constraints. To examine the role of the zero modes, we consider the quantization procedure in the zero mode and the nonzero-mode sectors separately. In both sectors we obtain the physical variables and their canonical commutation relations. The physical Hamiltonian is constructed via a step-by-step exclusion of the unphysical degrees of freedom. An example using this Hamiltonian in which the zero modes play a role is the verification of the correct Coulomb potential between two heavy fermions.
TL;DR: In this article, the authors obtained all possible representations of the Wheeler-DeWitt equation that may arise from the development of the first-order formalism for the quantization of gravity.
Abstract: We obtain all possible representations of the Wheeler-DeWitt equation that may arise from the development of the first-order formalism for the quantization of gravity. They are the usual one, the modified extrinsic curvature representation and the intrisic and extrinsic time representations, which arise naturally in this formalism. We show that in the intrinsic time representation the factor-ordering problem is restricted to the pair consisting of the determinant of the metric of the spacelike hypersurfaces and its canonical momentum. Contrary to what was suggested in an earlier paper, no new representations can be obtained in this formalism or by considering actions differing by a total divergence.
Journal Article•
TL;DR: In this article, a general approach based on a set of postulates reflecting properties of physical configuration space is proposed for constructing a quantum model, consisting in canonical quantization of a related classical model, can be applied only if we know the classical model and its Hamiltonian formulation in advance.
Abstract: Conventional approach to constructing a quantum model, consisting in canonical quantization of a related classical model,can be applied only if we know the classical model and its Hamiltonian formulation in advance. Approach based on a set of postulates reflecting properties of physical configuration space is more general. As an example, quantum mechanics of a particle on a space consisting of just two points is constructed
TL;DR: The transformation theory is an important method to study the problem of analytical mechanics as well as a theory of dynamics as discussed by the authors, and one can use the transformation to make the equations simplified, integrable or to turn them into the form convenient for study without changing the structure of canonical equa-
Abstract: 1 Introduction The transformation theory is an important method to study the problem of analytical mechanics as well as a theory of dynamics. For the Hamiltonian systems, we always hope that one can use the transformation to make the equations simplified, integrable or to turn them into the form convenient for study without changing the structure of canonical equa-
TL;DR: The quantum-mechanical anomaly is shown to be equivalent to a term in the action, similar to a gauge field configuration, whose presence maintains self-adjointness of the Hamiltonian or corrects the phase-space measure of the path integral to reflect the bound-state spectrum of the original problem.
Abstract: Anomalies associated with canonical transformations in quantum-mechanical systems are examined in the context of the path-integral formalism, where they appear as nontrivial Jacobians. The quantum-mechanical anomaly, for the case of a transformation to cyclic coordinates, is shown to be equivalent to a term in the action, similar to a gauge field configuration, whose presence maintains self-adjointness of the Hamiltonian or corrects the phase-space measure of the path integral to reflect the bound-state spectrum of the original problem. Various nonperturbative aspects of quantum-mechanical transition amplitudes are calculated by using classical solutions to Hamilton's equations of motion combined with the anomaly generated by the canonical transformation
TL;DR: Working in the context of an ADM splitting into space plus time, this paper identifies a Hamiltonian formulation, i.e., cosymplectic structure, for the Vlasov-Maxwell system, as formulated in a fixed, albeit curved, background spacetime.
Abstract: Working in the context of an ADM splitting into space plus time, this paper identifies a Hamiltonian formulation, i.e., cosymplectic structure, for the Vlasov-Maxwell system, as formulated in a fixed, albeit curved, background spacetime. The fundamental arena of physics is an infinite-dimensional phase space, coordinatized by the distribution function f, the spatial vector potential A i , and the conjugate momentum Π i . This Hamiltonian formulation entails the identification of (i) Lie brackets , defined for functionals F[A i , Π i , f] and G[A i , Π i , f], and (ii) a Hamiltonian function H[A i , Π i , f], so chosen that the equations of motion take the form ∂ t F = , with ∂ t a coordinate time derivative
01 Jan 1993
TL;DR: In this article, the transition from a canonical set of Delaunay-like elements to another one of the DS type, both kinds of sets being applicable to point masses moving along hyperbolic trajectories in problems of orbital motion, is performed by means of a completely canonical transformation derived from a generating function.
Abstract: The transition from a canonical set of Delaunay-like elements to another one of the DS type, both kinds of sets being applicable to point masses moving along hyperbolic trajectories in problems of orbital motion, is performed by means of a completely canonical transformation derived from a generating function. The two-body Hamiltonian is simplified after the introduction of a fictitious time as the new independent variable; in particular, the arbitrariness in this reparametrizing transformation allows us to recover, as special cases, independent variables analogous to the corresponding classical anomalies in elliptic motion
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TL;DR: In this article, the canonical symmetry is used to look for the hierarchy of the Hamiltonian operators relevant to the system under consideration and it appears that only the invariance condition can be used to solve the problem.
Abstract: It is shown how the canonical symmetry is used to look for the hierarchy of the Hamiltonian operators relevant to the system under consideration It appears that only the invariance condition can be used to solve the problem
TL;DR: The generalized Hannay angle as mentioned in this paper is the classical counterpart of the generalized Berry phase in quantum theory, which is defined by making a distinction between the Hamiltonian and the energy, and it describes a harmonic oscillator with mass, spring and friction.
Abstract: The most general time-dependent Hamiltonian for a harmonic oscillator is both linear and quadratic in the coordinate and the canonical momentum. It describes in general a harmonic oscillator with mass, spring “constant,” and friction (or antifriction) “constant,” all of which are time dependent, that is acted on by a time-dependent force. A generalized Hannay angle, which is gauge invariant, is defined by making a distinction between the Hamiltonian and the energy. The generalized Hannay angle is the classical counterpart of the generalized Berry phase in quantum theory. When friction is present the generalized Hannay angle is nonzero. If the Hamiltonian is (incorrectly) chosen to be the energy, the generalized Hannay angle is different. Nevertheless, in the adiabatic case the same total angle is obtained.
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TL;DR: In this paper, the properties of the canonical symmetry of the nonlinear Schrodinger equation were investigated and the densities of local conservation laws for this system were shown to change under the action of canonical symmetry by total space derivatives.
Abstract: The properties of the canonical symmetry of the nonlinear Schrodinger equation are investigated The densities of the local conservation laws for this system are shown to change under the action of the canonical symmetry by total space derivatives
01 Jan 1993
TL;DR: In this paper, a 2 + 1-dimensional nonlinear system due to Stoker which arises in two-layer shallow water theory is shown to admit a symmetry reduction to a Ermakov system.
Abstract: Ermakov-type systems in 2 + 1-dimensions are introduced. Multi-wave solutions of a 2 + 1-dimensional Pinney equation and a modulated 2 + 1-dimensional sine-Gordon equation are constructed via the standard Ermakov system. Lie group analysis of the latter is undertaken to reveal its underlying linear structure in terms of canonical coordinates. In conclusion, a 2 + 1-dimensional nonlinear system due to Stoker which arises in two-layer shallow water theory is shown to admit a symmetry reduction to a Ermakov system.
TL;DR: In this article, a canonical transformation is made from the original set of canonical variables to a new set in which the new canonical momentum is energy E and the canonical coordinate conjugate to it is called tempus and denoted by T.
Abstract: For a Hamiltonian system with one degree of freedom a canonical transformation is made from the original set of canonical variables to a new set in which the new canonical momentum is energy E and the new canonical coordinate conjugate to it is called tempus and denoted by T. This new canonical coordinate T is a function of the original canonical coordinate q, the energy E, and the time t in which the system evolves. Hamilton's equations for energy and tempus canonical variables are obtained. To illustrate the method it is applied to a particle in one dimension with a non-conservative force quadratic in the velocity. The energy is obtained directly by solving Hamilton's equations for the new variables. The displacement q as a function of time is obtained by transforming back.
TL;DR: In this paper, it was shown that any canonical mapping in the complex domain can be labelled by two different functions, one corresponds to an observable in the sense of classical mechanics and the second one has special analytic properties and can be used to form quantities which are important in quantum statistical mechanics.
Abstract: We investigate the Lie series representation of the canonical transformations in a complex phase space. It is shown that any canonical mapping in the complex domain can be labelled by two different functions. One of these functions corresponds to an observable in the sense of classical mechanics. The second one has special analytic properties and can be used to form quantities which are important in quantum statistical mechanics. In particular we show that the entropy of ideal quantum gases generates a special canonical transformation and, moreover, the entropy itself can be represented as a Lie function formed by a special characteristic function.
TL;DR: In this paper, a canonical transformation is performed on the phase space of a number of homogeneous cosmologies to simplify the form of the scalar (or, Hamiltonian) constraint.
Abstract: A canonical transformation is performed on the phase space of a number of homogeneous cosmologies to simplify the form of the scalar (or, Hamiltonian) constraint. Using the new canonical coordinates, it is then easy to obtain explicit expressions of Dirac observables, i.e.\ phase space functions which commute weakly with the constraint. This, in turn, enables us to carry out a general quantization program to completion. We are also able to address the issue of time through ``deparametrization'' and discuss physical questions such as the fate of initial singularities in the quantum theory. We find that they persist in the quantum theory {\it inspite of the fact that the evolution is implemented by a 1-parameter family of unitary transformations}. Finally, certain of these models admit conditional symmetries which are explicit already prior to the canonical transformation. These can be used to pass to quantum theory following an independent avenue. The two quantum theories --based, respectively, on Dirac observables in the new canonical variables and conditional symmetries in the original ADM variables-- are compared and shown to be equivalent.
TL;DR: In this article, a canonical transformation of the collective variables of center of mass type which transforms the canonical generators of the Poincare algebra in one form of dynamics into the corresponding generators in another form is presented.
Abstract: In the context of nonquantum Hamiltonian formalism of the relativistic theory of direct interaction we construct a canonical transformation of the collective variables of center of mass type which transforms the canonical generators of the Poincare algebra in one form of dynamics into the corresponding generators in another form of dynamics.
TL;DR: In this paper, a method for integration of classical and quantum Hamiltonian systems, based on the use of approximate symmetries, is proposed, which is similar to the averaging method in classical mechanics; however, it does not use canonical transformations to the variables "action-angle".
Abstract: A method for integration of classical and quantum Hamiltonian systems, based on the use of approximate symmetries, is proposed in this paper. The proposed method is similar to the averaging method in classical mechanics; however, it does not use canonical transformations to the variables “action-angle.” This allows one in some cases to apply this method to quantum equations. A nontrivial example is analyzed in the paper.