Showing papers on "Canonical transformation published in 2013"
TL;DR: In this article, a one-parameter family of algebras FIO ( Ξ, s ), 0 ⩽ s⩽ ∞, consisting of Fourier integral operators is constructed, which is defined by the decay properties of an associated Gabor matrix around the graph of the canonical transformation.
77 citations
TL;DR: In this paper, it is shown that there is a non-conservative force quadratic in the velocity associated to the variable mass and the time-dependent first integrals of motion are obtained from the factorization of such a constant.
Abstract: We analyze the dynamical equations obeyed by a classical system with position- dependent mass. It is shown that there is a non-conservative force quadratic in the velocity associated to the variable mass. We construct the Lagrangian and the Hamiltonian for this system and find the modifications required in the Euler{Lagrange and Hamilton's equations to reproduce the appropriate Newton's dynamical law. Since the Hamiltonian is not time invariant, we get a constant of motion suited to write the dynamical equations in the form of the Hamilton's ones. The time-dependent first integrals of motion are then obtained from the factorization of such a constant. A canonical transformation is found to map the variable mass equations to those of a constant mass. As particular cases, we recover some recent results for which the dependence of the mass on the position was already unnoticed, and find new solvable potentials of the Poschl{Teller form which seem to be new. The latter are associated to either the su(1;1) or the su(2) Lie algebras depending on the sign of the Hamiltonian.
39 citations
TL;DR: In this paper, it was shown that Dirac's proof that a first-class primary constraint generates a gauge transformation, by comparing evolutions from similar initial data, cancels out and hence fails to detect the alterations made to the initial state.
Abstract: In Dirac-Bergmann constrained dynamics, a first-class constraint typically does not _alone_ generate a gauge transformation. Each first-class constraint in Maxwell's theory generates a change in the electric field E by an arbitrary gradient, spoiling Gauss's law. The secondary constraint p^i,_i=0 still holds, but being a function of derivatives of momenta (mere auxiliary fields), it is not directly about the observable electric field (a function of derivatives of A), which couples to charge. Only a special combination of the two first-class constraints, the Anderson-Bergmann-Castellani gauge generator G, leaves E unchanged. Likewise only that combination leaves the canonical action invariant---an argument independent of observables. If one uses a first-class constraint to generate instead a canonical transformation, one partly strips the canonical coordinates of physical meaning as electromagnetic potentials, vindicating the Anderson-Bergmann Lagrangian orientation of interesting canonical transformations. The need to keep gauge-invariant q,t-dH/dp=-E-p=0 supports using the gauge generator and primary Hamiltonian rather than the separate first-class constraints and the extended Hamiltonian.
Partly paralleling Pons's criticism, it is shown that Dirac's proof that a first-class primary constraint generates a gauge transformation, by comparing evolutions from _identical_ initial data, cancels out and hence fails to detect the alterations made to the initial state. It also neglects the arbitrary coordinates multiplying the secondary constraints _inside_ the canonical Hamiltonian. Thus the gauge-generating property has been ascribed to the primaries alone.
Hence the Dirac conjecture about secondary first-class constraints as generating gauge transformations rests upon a false presupposition about primary first-class constraints. Clarity about electromagnetism help with GR.
32 citations
TL;DR: In this article, the authors construct O(D,D) invariant actions for the RNS superstring and the bosonic string using Hamiltonian methods and ideas from double field theory.
Abstract: We construct O(D,D) invariant actions for the bosonic string and RNS superstring, using Hamiltonian methods and ideas from double field theory. In this framework the doubled coordinates of double field theory appear as coordinates on phase space and T-duality becomes a canonical transformation. Requiring the algebra of constraints to close leads to the section condition, which splits the phase space coordinates into spacetime coordinates and momenta.
30 citations
TL;DR: In this paper, the role of Jacobi's last multiplier in mechanical systems with a position-dependent mass is revealed, and the quantization of the Lienard II equation is carried out using the point canonical transformation method together with the Von Roos ordering technique.
Abstract: In this paper, the role of Jacobi’s last multiplier in mechanical systems with a position-dependent mass is unveiled. In particular, we map the Lienard II equation to a position-dependent mass system. The quantization of the Lienard II equation is then carried out using the point canonical transformation method together with the Von Roos ordering technique. Finally, we show how their eigenfunctions and eigenspectrum can be obtained in terms of associated Laguerre and exceptional Laguerre functions.
25 citations
TL;DR: In this paper, the atomic and the field entropies of a two-level atom, which is additionally driven by an external classical field, are investigated under a certain canonical transformation for the excited and ground states, the system is transformed into the usual JCM.
Abstract: The atomic and the field entropies of a two-level atom, which is additionally driven by an external classical field are investigated. Under a certain canonical transformation for the excited and ground states the system is transformed into the usual JCM. Using the equations of motion in the Heisenberg picture exact solutions for the time-dependent dynamical operators are obtained. The entanglement between atom-field system is studied by using the atomic and the field entropies. Also we use the concurrence to detect the sudden death phenomenon and the relationship between entropies and the concurrence of the entanglement are discussed. It is shown that the amount of entanglement, the atomic and the field entropies of the subsystem can be improved by controlling the external classical field.
24 citations
TL;DR: The Jacobi zeta function is given the physical interpretation as the generating function of the canonical transformation from the pendulum coordinates ϑ and p ≡ ∂ ϑ / ∂ t to the action-angle coordinates ( J, ζ ) for both the librating pendulum and the rotating pendulum.
21 citations
TL;DR: For a double quantum dot system in a parallel geometry, it is demonstrated that by combining the effects of a flux and driving an electrical current through the structure, the spin correlations between electrons localized in the dots can be controlled at will.
Abstract: For a double quantum dot system in a parallel geometry, we demonstrate that by combining the effects of a flux and driving an electrical current through the structure, the spin correlations between electrons localized in the dots can be controlled at will. In particular, a current can induce spin correlations even if the spins are uncorrelated in the initial equilibrium state. Therefore, we are able to engineer an entangled state in this double-dot structure. We take many-body correlations fully into account by simulating the real-time dynamics using the time-dependent density matrix renormalization group method. Using a canonical transformation, we provide an intuitive explanation for our results, related to Ruderman-Kittel-Kasuya-Yoshida physics driven by the bias.
19 citations
TL;DR: In this article, a general field-covariant approach to quantum gauge theories is developed, which extends the usual set of integrated fields and external sources to proper fields and sources, which include partners of the composite fields.
Abstract: We develop a general field-covariant approach to quantum gauge theories. Extending the usual set of integrated fields and external sources to “proper” fields and sources, which include partners of the composite fields, we define the master functional Ω, which collects one-particle irreducible diagrams and upgrades the usual Γ-functional in several respects. The functional Ω is determined from its classical limit applying the usual diagrammatic rules to the proper fields. Moreover, it behaves as a scalar under the most general perturbative field redefinitions, which can be expressed as linear transformations of the proper fields. We extend the Batalin–Vilkovisky formalism and the master equation. The master functional satisfies the extended master equation and behaves as a scalar under canonical transformations. The most general perturbative field redefinitions and changes of gauge-fixing can be encoded in proper canonical transformations, which are linear and do not mix integrated fields and external sources. Therefore, they can be applied as true changes of variables in the functional integral, instead of mere replacements of integrands. This property overcomes a major difficulty of the functional Γ. Finally, the new approach allows us to prove the renormalizability of gauge theories in a general field-covariant setting. We generalize known cohomological theorems to the master functional and show that when there are no gauge anomalies all divergences can be subtracted by means of parameter redefinitions and proper canonical transformations.
16 citations
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TL;DR: In this article, the sparsity of the Gabor-matrix representation of Fourier integral operators with a phase having quadratic growth was investigated, and it was shown that if the phase and symbol have a regularity of Gevrey type of order $s>1$ or analytic ($s=1$), the above decay is in fact sub-exponential or exponential, respectively.
Abstract: We investigate the sparsity of the Gabor-matrix representation of Fourier integral operators with a phase having quadratic growth. It is known that such an infinite matrix is sparse and well organized, being in fact concentrated along the graph of the corresponding canonical transformation. Here we show that, if the phase and symbol have a regularity of Gevrey type of order $s>1$ or analytic ($s=1$), the above decay is in fact sub-exponential or exponential, respectively. We also show by a counterexample that ultra-analytic regularity ($s<1$) does not give super-exponential decay. This is in sharp contrast to the more favorable case of pseudodifferential operators, or even (generalized) metaplectic operators, which are treated as well.
16 citations
TL;DR: This letter presents tracking control with only position measurements for rigid-joint robots, by applying the canonical transformation theory for port-Hamiltonian systems, and shows how the initial conditions of the controller can be tuned in order to improve transient performance.
Abstract: In this letter, we present tracking control with only position measurements for rigid-joint robots, by applying the canonical transformation theory for port-Hamiltonian systems. We show that besides giving the same results as presented in the literature for Euler-Lagrange systems, the canonical transformation theory also justifies a Coriolis matrix based on the desired velocities. Furthermore, we show how the initial conditions of the controller can be tuned in order to improve transient performance. Finally, we validate our results on a simple experimental setup.
TL;DR: In this paper, a mathematical technique for deriving dynamical invariants (i.e., constants of motion) in time-dependent gravitational potentials is proposed. But the method relies on the construction of a canonical transformation that removes the explicit time-dependence from the Hamiltonian of the system by referring the phase-space locations of particles to a coordinate frame in which the potential remains'static' and the dynamical effects introduced by the time evolution vanish.
Abstract: This paper explores a mathematical technique for deriving dynamical invariants (ie constants of motion) in time-dependent gravitational potentials The method relies on the construction of a canonical transformation that removes the explicit time-dependence from the Hamiltonian of the system By referring the phase-space locations of particles to a coordinate frame in which the potential remains `static' the dynamical effects introduced by the time evolution vanish It follows that dynamical invariants correspond to the integrals of motion for the static potential expressed in the transformed coordinates The main difficulty of the method reduces to solving the differential equations that define the canonical transformation, which are typically coupled with the equations of motion We discuss a few examples where both sets of equations can be exactly de-coupled, and cases that require approximations The construction of dynamical invariants has far-reaching applications These quantities allow us, for example, to describe the evolution of (statistical) microcanonical ensembles in time-dependent gravitational potentials without relying on ergodicity or probability assumptions As an illustration, we follow the evolution of dynamical fossils in galaxies that build up mass hierarchically It is shown that the growth of the host potential tends to efface tidal substructures in the integral-of-motion space through an orbital diffusion process The inexorable cycle of deposition, and progressive dissolution, of tidal clumps naturally leads to the formation of a `smooth' stellar halo
01 Jan 2013
TL;DR: In this paper, 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 \(\fancyscript{H}\) that is form-invariant under local \(U(N)\) gauge transformations.
TL;DR: In this paper, a model distribution function for relativistic bi-Maxwellian with drift is proposed, based on the maximum entropy principle and the relativism canonical transformation, which is compatible with existing distribution functions and has a relatively simple form as well as smoothness.
Abstract: A model distribution function for relativistic bi-Maxwellian with drift is proposed, based on the maximum entropy principle and the relativistic canonical transformation. Since the obtained expression is compatible with the existing distribution functions and has a relatively simple form as well as smoothness, it might serve as a useful tool in the research fields of space or high temperature fusion plasmas.
TL;DR: An iterative method is presented to construct an integrable approximation H(reg), which resembles the regular dynamics of a given mixed system H and extends it into the chaotic region of H.
Abstract: Generic Hamiltonian systems have a mixed phase space, where classically disjoint regions of regular and chaotic motion coexist. We present an iterative method to construct an integrable approximation ${H}_{\mathrm{reg}}$, which resembles the regular dynamics of a given mixed system $H$ and extends it into the chaotic region. The method is based on the construction of an integrable approximation in action representation which is then improved in phase space by iterative applications of canonical transformations. This method works for strongly perturbed systems and arbitrary degrees of freedom. We apply it to the standard map and the cosine billiard.
TL;DR: In this paper, a modification of a generalized translation operator was proposed by including a $q$exponential factor, which implies in the definition of a Hermitian deformed linear momentum operator and its canonically conjugate deformed position operator.
Abstract: We propose a modification of a recently introduced generalized translation operator, by including a $q$-exponential factor, which implies in the definition of a Hermitian deformed linear momentum operator $\hat{p}_q$, and its canonically conjugate deformed position operator $\hat{x}_q$. A canonical transformation leads the Hamiltonian of a position-dependent mass particle to another Hamiltonian of a particle with constant mass in a conservative force field of a deformed phase space. The equation of motion for the classical phase space may be expressed in terms of the generalized dual $q$-derivative. A position-dependent mass confined in an infinite square potential well is shown as an instance. Uncertainty and correspondence principles are analyzed.
TL;DR: In this paper, the Green function of the FV spinless particle in a noncommutative (NC) phase-space coordinates, where the Pauli matrices describing the charge symmetry are replaced by the Grassmannian odd variables, was constructed.
Abstract: In this paper, we have constructed the Green function of the Feshbach–Villars (FV) spinless particle in a noncommutative (NC) phase-space coordinates, where the Pauli matrices describing the charge symmetry are replaced by the Grassmannian odd variables. Subsequently, for the perform calculations, we diagonalize the Hamiltonian governing the dynamics of the system via the Foldy–Wouthuysen (FW) canonical transformation. The exact calculations have been done in the cases of free particle and magnetic field interaction. In both cases, the energy eigenvalues and their corresponding eigenfunctions are deduced.
TL;DR: In this paper, the exact solution of the problem of two two-level atoms interacting with a two-mode radiation field in the presence of a parametric converter term is presented.
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TL;DR: In this paper, a new unified formulation of the current algebra theory in terms of supergeometry is proposed, which takes a QP-structure, i.e. a differential graded symplectic structure on a graded cotangent bundle as a fundamental framework.
Abstract: We propose a new unified formulation of the current algebra theory in terms of supergeometry. We take a QP-structure, i.e. a differential graded symplectic structure, on a graded cotangent bundle as a fundamental framework. A Poisson bracket in a current algebra is constructed by the so called derived bracket of the graded Poisson structure induced from the above symplectic structure. By taking a canonical transformation on a QP-manifold, correct anomalous terms in physical theories are derived. A large class of current algebras with and without anomalous terms are constructed from this structure. Moreover, a new class of current algebras related higher structures are obtained systematically.
TL;DR: In this paper, the authors extend the applicability regime of the torus construction method by demonstrating that H can be based on a Stackel potential, the most general known form of an integrable potential.
TL;DR: In this paper, the cosmological constant of the hadron mass was derived within a Hamiltonian framework, and relations between the logarithmic gauge coupling derivative of the Hadron mass and cosmology constant were derived.
Abstract: We derive recently obtained relations, relating the logarithmic gauge coupling derivative of the hadron mass and the cosmological constant to the matter and vacuum gluon condensates, within a Hamiltonian framework. The key idea is a canonical transformation which brings the relevant part of the Hamiltonian into a suitable form. Furthermore we illustrate the relations within the Schwinger model and ${\cal N}=2$ super Yang Mills theory (Seiberg-Witten theory).
TL;DR: In this paper, the long-range residual force on the expectation value of observables in the nuclear ground states is evaluated by finding optimal values for the coefficients of the canonical transformation which connects the phonon vacuum state with the (quasi-)particle ground state.
Abstract: The action of the long-range residual force on the expectation value of observables in the nuclear ground states is evaluated by finding optimal values for the coefficients of the canonical transformation which connects the phonon vacuum state with the (quasi-)particle ground-state. After estimating the improvements over the predictions of the independent-particle approximation we compare the ground-state wave functions, obtained using the presented approach, with those, obtained using the conventional random phase approximation (RPA) and its extended version. The problem with overbinding of the nuclear ground state calculated using the RPA is shown to be removed if one sticks to the prescriptions of the present approach. The reason being that the latter conforms to the original variational formulation. Calculations are performed within the two-level Lipkin-Meshkov-Glick model in which we present results for the ground and first excited state energies as well as for the ground-state particle occupation numbers.
TL;DR: In this paper, a variational method of solution was proposed to remove the lack of uniqueness of the transformation in 3D geometries due to a relabelling symmetry, and the transformation was shown to be also ghost pseudo-orbits.
Abstract: Straight-field-line coordinates are very useful for representing magnetic fields in toroidally confined plasmas, but fundamental problems arise regarding their definition in 3D geometries because of the formation of islands and chaotic field regions, i.e. non-integrability. In Hamiltonian dynamical systems terminology these coordinates are similar to action-angle variables, but these are normally defined only for integrable systems. In order to describe 3D magnetic field systems, a generalization of this concept was proposed recently by the present authors that unified the concepts of ghost surfaces and quadratic-flux-minimizing (QFMin) surfaces. This was based on a simple canonical transformation generated by a change of variable ??=??(?, ?), where ? and ? are poloidal and toroidal angles, respectively, with ? a new poloidal angle chosen to give pseudo-orbits that are (a) straight when plotted in the ?, ? 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 this paper, it is demonstrated that these requirements do not 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, Chenciner et al. considered the planar restricted (N + 1 )-body problem, where the primaries are moving in a central configuration, and showed that the infinitesimal mass m 1 is arbitrarily close to a primary.
01 Jan 2013
TL;DR: In this article, the Hamilton-Jacobi theory is used to obtain the Hamilton function for nonholonomic constraints in addition to the equations of motion, which is then used to construct the wave function and to quantize these systems using the WKB approximation.
Abstract: The Hamilton-Jacobi theory is used to obtain the Hamilton function for nonholonomic constraints in addition to the equations of motion. The technique of separation of variables and canonical transformation is applied here to solve the Hamilton- Jacobi partial differential equation for nonholonomic systems. The Hamilton-Jacobi function is then used to construct the wave function and to quantize these systems using the WKB approximation.
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TL;DR: In this paper, the authors perform a LQC-quantization of the FRW cosmological model with nonminimally coupled scalar field and show that the classical singularity is replaced by a "mexican hat"-shaped bounce, joining the contracting and expanding branches.
Abstract: We perform a LQC-quantization of the FRW cosmological model with nonminimally coupled scalar field. Making use of a canonical transformation, we recast the theory in the minimally coupled form (Einstein frame), for which standard LQC techniques can be applied to find the physical Hilbert space and the dynamics. We then focus on the semiclassical sector, obtaining a classical effective Hamiltonian, which can be used to study the dynamics. We show that the classical singularity is replaced by a "mexican hat"-shaped bounce, joining the contracting and expanding branches. The model accommodates Higgs-driven inflation, with more than enough e-folding for any physically meaningful initial condition.
01 Jan 2013
TL;DR: In this paper, the Lagrangian and action for a single particle are discussed and the principle of least action is stated, and the Euler-Lagrange equations are derived.
Abstract: Lagrangian and Hamiltonian formulations of the dynamics of relativistic particles are presented. The Lagrangian and action for a single particle are first discussed. The principle of least action is stated, and the Euler–Lagrange equations are derived. The specific cases of a free particle and a particle in scalar, vector or second-rank tensor fields are treated. The Noether theorem is presented. The generalized four-momentum and the Hamiltonian are introduced for a single particle, and the canonical equations of Hamilton are written. Finally, systems of many particles are discussed by means of the Tetrode–Fokker action.
31 Dec 2013
TL;DR: In this paper, the star-product version of three basic quantum canonical transformations which are known as the generators of the full canonical algebra is studied, and it is shown that while the constructions of gauge and point transformations are immediate, generator of the interchanging transformation deforms this isomorphism.
Abstract: We study construction of the star-product version of three basic quantum canonical transformations which are known as the generators of the full canonical algebra. By considering the fact that star-product of c-number phase-space functions is in complete isomorphism to Hilbert-space operator algebra, it is shown that while the constructions of gauge and point transformations are immediate, generator of the interchanging transformation deforms this isomorphism. As an alternative approach, we study all of them within the deformed form. How to transform any c-number function under linear-nonlinear transformations and the intertwining method are shown within this argument as the complementary subjects of the text.