Showing papers on "Canonical transformation published in 2019"

Iterative Qubit Coupled Cluster approach with efficient screening of generators.

Abstract: An iterative version of the qubit coupled cluster (QCC) method [I.G. Ryabinkin et al., J. Chem. Theory Comput. 14, 6317 (2019)] is proposed. The new method seeks to find ground electronic energies of molecules on noisy intermediate-scale quantum (NISQ) devices. Each iteration involves a canonical transformation of the Hamiltonian and employs constant-size quantum circuits at the expense of increasing the Hamiltonian size. We numerically studied the convergence of the method on ground-state calculations for LiH, H$_2$O, and N$_2$ molecules and found that the exact ground-state energies can be systematically approached only if the generators of the QCC ansatz are sampled from a specific set of operators. We report an algorithm for constructing this set that scales linearly with the size of a Hamiltonian.

Polymer quantum cosmology: Lifting quantization ambiguities using a SL(2,R) conformal symmetry

Abstract: In this letter, we stress that the simplest cosmological model consisting in a massless scalar field minimally coupled to homogeneous and isotropic gravity has an in-built $\SL(2,\mathbb{R})$ symmetry. Protecting this symmetry naturally provides an efficient way to constrain the quantization of this cosmological system whatever the quantization scheme and allows in particular to fix the quantization ambiguities arising in the canonical quantization program. Applying this method to the loop quantization of the FLRW cosmology leads to a new loop quantum cosmology model which preserves the $\SL(2,\mathbb{R})$ symmetry of the classical system. This new polymer regularization consistent with the conformal symmetry can be derived as a non-linear canonical transformation of the classical FLRW phase space, which maps the classical singular dynamics into a regular effective bouncing dynamics. This improved regularization preserves the scaling properties of the volume and Hamiltonian constraint. 3d scale transformations, generated by the dilatation operator, are realized as unitary transformations despite the minimal length scale hardcoded in the theory. Finally, we point out that the resulting cosmological dynamics exhibits an interesting duality between short and long distances, reminiscent of the T-duality in string theory, with the near-singularity regime dual to the semi-classical regime at large volume. The technical details of the construction of this model are presented in a longer companion paper \cite{BenAchour:2019ywl}.

Dynamics of the generalized unimodular gravity theory

Abstract: The Hamiltonian formalism of the generalized unimodular gravity theory, which was recently suggested as a model of dark energy, is shown to be a complicated example of constrained dynamical system. The set of its canonical constraints has a bifurcation---splitting of the theory into two branches differing by the number and type of these constraints, one of the branches effectively describing a gravitating perfect fluid with the time-dependent equation of state, which can potentially play the role of dark energy in cosmology. The first class constraints in this branch generate local gauge symmetries of the Lagrangian action---two spatial diffeomorphisms---and rule out the temporal diffeomorphism which does not have a realization in the form of the canonical transformation on phase space of the theory and turns out to be either nonlocal in time or violating boundary conditions at spatial infinity. As a consequence, the Hamiltonian reduction of the model enlarges its physical sector from two general relativistic modes to three degrees of freedom including the scalar graviton. This scalar mode is free from ghost and gradient instabilities on the Friedmann background in a wide class of models subject to a certain restriction on time-dependent parameter $w$ of the dark fluid equation of state, $p=wϵ$. For a special family of models this scalar mode can be ruled out even below the phantom divide line $w=\ensuremath{-}1$, but this line cannot be crossed in the course of the cosmological expansion. This is likely to disable the generalized unimodular gravity as a model of the phenomenologically consistent dark energy scenario, but opens the prospects in inflation theory with a scalar graviton playing the role of inflaton.

Quadratic quantum Hamiltonians: General canonical transformation to a normal form

Abstract: A system of linearly coupled quantum harmonic oscillators can be diagonalized when the system is dynamically stable using a Bogoliubov canonical transformation. However, this is just a particular case of more general canonical transformations that can be performed even when the system is dynamically unstable. Specific canonical transformations can transform a quadratic Hamiltonian into a normal form, which greatly helps to elucidate the underlying physics of the system. Here, we provide a self-contained review of the normal form of a quadratic Hamiltonian as well as step-by-step instructions to construct the corresponding canonical transformation for the most general case. Among other examples, we show how the standard two-mode Hamiltonian with a quadratic position coupling presents, in the stability diagram, all the possible normal forms corresponding to different types of dynamical instabilities.

Conformal bridge between freedom and confinement

Abstract: We construct a nonunitary transformation that relates a given "asymptotically free" conformal quantum mechanical system $H_f$ with its confined, harmonically trapped version $H_c$. In our construction, Jordan states corresponding to the zero eigenvalue of $H_f$, as well as its eigenstates and Gaussian packets are mapped into the eigenstates, coherent states and squeezed states of $H_c$, respectively. The transformation is an automorphism of the conformal $\mathfrak{sl}(2,{\mathbb R})$ algebra of the nature of the fourth-order root of the identity transformation, to which a complex canonical transformation corresponds on the classical level being the fourth-order root of the spatial reflection. We investigate the one- and two-dimensional examples that reveal, in particular, a curious relation between the two-dimensional free particle and the Landau problem.

Dynamical Properties of the Mukhanov-Sasaki Hamiltonian in the Context of Adiabatic Vacua and the Lewis-Riesenfeld Invariant

Abstract: We use the method of the Lewis-Riesenfeld invariant to analyze the dynamical properties of the Mukhanov-Sasaki Hamiltonian and, following this approach, investigate whether we can obtain possible candidates for initial states in the context of inflation considering a quasi-de Sitter spacetime. Our main interest lies in the question of to which extent these already well-established methods at the classical and quantum level for finitely many degrees of freedom can be generalized to field theory. As our results show, a straightforward generalization does in general not lead to a unitary operator on Fock space that implements the corresponding time-dependent canonical transformation associated with the Lewis-Riesenfeld invariant. The action of this operator can be rewritten as a time-dependent Bogoliubov transformation, where we also compare our results to already existing ones in the literature. We show that its generalization to Fock space has to be chosen appropriately in order to not violate the Shale-Stinespring condition. Furthermore, our analysis relates the Ermakov differential equation that plays the role of an auxiliary equation, whose solution is necessary to construct the Lewis-Riesenfeld invariant, as well as the corresponding time-dependent canonical transformation, to the defining differential equation for adiabatic vacua. Therefore, a given solution of the Ermakov equation directly yields a full solution of the differential equation for adiabatic vacua involving no truncation at some adiabatic order. As a consequence, we can interpret our result obtained here as a kind of non-squeezed Bunch-Davies mode, where the term non-squeezed refers to a possible residual squeezing that can be involved in the unitary operator for certain choices of the Bogoliubov map.

Neural Canonical Transformation with Symplectic Flows

Abstract: Canonical transformation plays a fundamental role in simplifying and solving classical Hamiltonian systems. We construct flexible and powerful canonical transformations as generative models using symplectic neural networks. The model transforms physical variables towards a latent representation with an independent harmonic oscillator Hamiltonian. Correspondingly, the phase space density of the physical system flows towards a factorized Gaussian distribution in the latent space. Since the canonical transformation preserves the Hamiltonian evolution, the model captures nonlinear collective modes in the learned latent representation. We present an efficient implementation of symplectic neural coordinate transformations and two ways to train the model. The variational free energy calculation is based on the analytical form of physical Hamiltonian. While the phase space density estimation only requires samples in the coordinate space for separable Hamiltonians. We demonstrate appealing features of neural canonical transformation using toy problems including two-dimensional ring potential and harmonic chain. Finally, we apply the approach to real-world problems such as identifying slow collective modes in alanine dipeptide and conceptual compression of the MNIST dataset.

The Non-Relativistic Limit of the DKP Equation in Non-Commutative Phase-Space

Abstract: The non-relativistic limit of the relativistic DKP equation for both of zero and unity spin particles is studied through the canonical transformation known as the Foldy–Wouthuysen transformation, similar to that of the case of the Dirac equation for spin-1/2 particles. By considering only the non-commutativity in phases with a non-interacting fields case leads to the non-commutative Schrodinger equation; thereafter, considering the non-commutativity in phase and space with an external electromagnetic field thus leads to extract a phase-space non-commutative Schrodinger–Pauli equation; there, we examined the effect of the non-commutativity in phase-space on the non-relativistic limit of the DKP equation. However, with both Bopp–Shift linear transformation through the Heisenberg-like commutation relations, and the Moyal–Weyl product, we introduced the non-commutativity in phase and space.

Covariance in the Batalin–Vilkovisky formalism and the Maurer–Cartan equation for curved Lie algebras

Abstract: We express covariance of the Batalin–Vilkovisky formalism in classical mechanics by means of the Maurer–Cartan equation in a curved Lie superalgebra, defined using the formal variational calculus and Sullivan’s Thom–Whitney construction. We use this framework to construct a Batalin–Vilkovisky canonical transformation identifying the Batalin–Vilkovisky formulation of the spinning particle with an AKSZ field theory.

PDM creation and annihilation operators of the harmonic oscillators and the emergence of an alternative most simplistic PDM-Hamiltonian

Abstract: The exact solvability and impressive pedagogical implementation of the harmonic oscillator creation and annihilation operators make it a problem of great physical relevance and the most fundamental one in quantum mechanics. So would be the position dependent mass oscillator for the position dependent mass quantum mechanics. We, hereby, construct the position dependent mass creation and annihilation operators for the position dependent mass oscillator via two different approaches. First, via von Roos position dependent mass Hamiltonian and show that the commutation relation between the position dependent mass creation and annihilation operators not only offers a unique position dependent mass Hamiltonian but also suggests a position dependent mass deformation in the coordinate system. Next, we use a position dependent mass point canonical transformation of the textbook constant mass harmonic oscillator analog and obtain yet another set of position dependent mass creation and annihilation operators, hence an apparently new position dependent mass Hamiltonian is obtained. The new position dependent mass Hamiltonian turned out to be not only correlated but also represents an alternative and most simplistic user-friendly position dependent mass Hamiltonian that has never been reported before.

Anti-Newtonian Expansions and the Functional Renormalization Group

Abstract: Anti-Newtonian expansions are introduced for scalar quantum field theories and classical gravity. They expand around a limiting theory that evolves only in time while the spatial points are dynamically decoupled. Higher orders of the expansion re-introduce spatial interactions and produce overlapping lightcones from the limiting isolated world line evolution. In scalar quantum field theories, the limiting system consists of copies of a self-interacting quantum mechanical system. In a spatially discretized setting, a nonlinear “graph transform” arises that produces an in principle exact solution of the Functional Renormalization Group for the Legendre effective action. The quantum mechanical input data can be prepared from its 1 + 0 dimensional counterpart. In Einstein gravity, the anti-Newtonian limit has no dynamical spatial gradients, yet remains fully diffeomorphism invariant and propagates the original number of degrees of freedom. A canonical transformation (trivialization map) is constructed, in powers of a fractional inverse of Newton’s constant, that maps the ADM action into its anti-Newtonian limit. We outline the prospects of an associated trivializing flow in the quantum theory.

Energy transfer from spacetime into matter and a bouncing inflation from covariant canonical gauge theory of gravity

Abstract: Cosmological solutions for covariant canonical gauge theories of gravity are presented. The underlying covariant canonical transformation framework invokes a dynamical spacetime Hamiltonian consist...

Recursion operator in a noncommutative Minkowski phase space

Abstract: A recursion operator for a geodesic flow, in a noncommutative (NC) phase space endowed with a Minkowski metric, is constructed and discussed in this work. A NC Hamiltonian function \({\mathcal{H}}_{\mathrm{nc}}\) describing the dynamics of a free particle system in such a phase space, equipped with a noncommutative symplectic form ωnc is defined. A related NC Poisson bracket is obtained. This permits to construct the NC Hamiltonian vector field, also called NC geodesic flow. Further, using a canonical transformation induced by a generating function from the Hamilton–Jacobi equation, we obtain a relationship between old and new coordinates, and their conjugate momenta. These new coordinates are used to re-write the NC recursion operator in a simpler form, and to deduce the corresponding constants of motion. Finally, all obtained physical quantities are re-expressed and analyzed in the initial NC canonical coordinates.

A stochastic ordering based on the canonical transformation of skew-normal vectors

Abstract: In this paper, we define a new skewness ordering that enables stochastic comparisons for vectors that follow a multivariate skew-normal distribution. The new ordering is based on the canonical transformation associated with the multivariate skew-normal distribution and on the well-known convex transform order applied to the only skewed component of such canonical transformation. We examine the connection between the proposed ordering and the multivariate convex transform order studied by Belzunce et al. (TEST 24(4):813–834, 2015). Several standard skewness measures like Mardia’s and Malkovich–Afifi’s indices are revisited and interpreted in connection with the new ordering; we also study its relationship with the J-divergence between skew-normal and normal random vectors and with the Negentropy. Some artificial data are used in simulation experiments to illustrate the theoretical discussion; a real data application is provided as well.

Analytic orbit theory with any arbitrary spherical harmonic as the dominant perturbation

Abstract: In the gravitational potential of Earth, the oblateness term is the dominant perturbation, with its coefficient at least three orders of magnitude greater than that of any other zonal or tesseral spherical harmonic. Therefore, analytic orbit theories (or satellite theories) are developed using the Keplerian Hamiltonian as the unperturbed solution, oblateness term as the first-order and the remaining spherical harmonics as the second-order perturbations. These orbit theories are generally constructed by applying multiple near-identity canonical transformations to the perturbed Hamiltonian in conjunction with averaging to obtain a secular Hamiltonian, from which the short-period and long-period terms are removed. If the oblateness term is the only first-order perturbation, then the long-period terms appear only in the second- or higher-order terms of the single-averaged Hamiltonian, from which the short-period terms are removed. These second-order long-period terms are separated from the Hamiltonian by the first-order generating function using a second canonical transformation. This results in a secular Hamiltonian dependent only on the momenta. However, in the case of other gravitational bodies with more deformed shapes compared to Earth such as moons and asteroids, the oblateness coefficient may have the same order of magnitude as some of the higher spherical harmonic coefficients. If these higher harmonics are treated as the first-order perturbation along with the oblateness term, then the long-period terms appear in the first-order single-averaged Hamiltonian. These first-order long-period terms cannot be separated using the generating function in the conventional way. This problem occurs because the zeroth-order Hamiltonian, i.e., the Keplerian part, is degenerate in the angular momentum. In this paper, a new approach to the long-period transformation is proposed to resolve this issue and obtain a fully analytic orbit theory when for the perturbing gravitational body, any arbitrary zonal or tesseral harmonic is the dominant perturbation. The proposed theory is closed form in the eccentricity as well. It is applied to predict the motion of artificial satellites for the two test cases: a lunar orbiter and a satellite of 433 Eros asteroid. The prediction accuracy is validated against the numerical propagation using a force model with $$6\times 6$$ gravity field.

QES solutions of a two-dimensional system with quadratic nonlinearities

Abstract: We consider a one parameter family of a PT symmetric two dimensional system with quadratic non-linearities. Such systems are shown to perform periodic oscillations due to existing centers. We describe this systems by constructing a non-Hermitian Hamiltonian of a particle with position dependent mass. We further construct a canonical transformation which maps this position dependent mass systems to a QES system. First few QES levels are calculated explicitly by using Bender-Dunne (BD) polynomial method.

Broken Charges Associated with Classical Spacetime Symmetries under Canonical Transformation in Real Scalar Field Theory

Abstract: We know from Noether’s theorem that there is a conserved charge for every continuous symmetry. In General Relativity, Killing vectors describe the spacetime symmetries and to each such Killing vector field, we can associate conserved charge through stress-energy tensor of matter which is mentioned in the article. In this article, I show that under simple set of canonical transformation of most general class of Bogoliubov transformation between creation, annihilation operators, those charges associated with spacetime symmetries are broken. To do that, I look at stress-energy tensor of real scalar field theory (as an example) in curved spacetime and show how it changes under simple canonical transformation which is enough to justify our claim. Since doing Bogoliubov transformation is equivalent to coordinate transformation which according to Einstein’s equivalence principle is equivalent to turn on effect of gravity, therefore, we can say that under the effect of gravity those charges are broken.

Canonical Approach To Generate Multidimensional Potential Energy Surfaces.

Abstract: A new force-based canonical approach for the accurate generation of multidimensional potential energy surfaces is demonstrated. Canonical transformations previously developed for diatomic molecules are used to construct accurate approximations to the 3-dimensional potential energy surface of the water molecule from judiciously chosen (adopting the right coordinate system) 1-dimensional planar slices that are shown to have the same canonical shape as the classical Lennard-Jones potential curve. Spline interpolation is then used to piece together the 1-dimensional canonical potential curves, to obtain the full 3-dimensional potential energy surface of a water molecule with a relative error less than 0.01. This work provides an approach to greatly reduce the computational cost of constructing potential energy surfaces in molecules from ab initio calculations. The canonical transformation techniques developed in this work illuminate a pathway to deepening our understanding of chemical bonding.

Generalized momenta in constrained non-holonomic systems—Another perspective on the canonical equations of motion

Abstract: In this paper generalized momenta are defined for non-holonomic rigid body systems in a possibly novel formalism. The momenta are defined for Lagrangian/energy systems as well as projective d’Alembert systems, thus allowing canonical equations to be developed. The momenta can be defined in terms of quasi-variables or with ultimately an excess of the time derivatives of the generalized coordinates. This work may lay a framework to build non-holonomic generalizations to the Hamilton–Jacobi canonical transformation equations. In the sense that these equations are fundamental and minimalistic, they too can be considered canonical in their nature. For immediate practical consideration, demonstrated is significant reductions of the time required to numerically integrate the expository examples modeled with these techniques, indicating that this modeling formalism may have practical applicability

Generalized 't Hooft-Nobbenhuis complex transformation from a modified Lorentz transformation

Abstract: A generalized form of 't Hooft-Nobbenhuis Complex Transformation of space-time was derived as a canonical transformation of coordinates from a modified Lorentz Transformation equivalent to an Improper Lorentz Transformation. The modification of the Lorentz Transformation was done via a matrix transformation of the Lorentz Boost and imaginary transformation of the rapidity.

Thermodynamics of $d$-Dimensional Charged AdS (Anti-de Sitter) Black Holes: Hamiltonian Approach and Clapeyron Equation

Abstract: The study of thermodynamics in the view of the Hamiltonian approach is a newest tool to analyze the thermodynamic properties of the black holes. In this letter, we investigate the thermodynamics of $d$-dimensional ($d>3$) asymptotically AntideSitter black holes. A thermodynamic representation based on symplectic geometry is introduced in this letter. We extend the thermodynamics of $d-$dimensional charged AntideSitter black holes in the views of a Hamiltonian approach. Firstly, we study the thermodynamics in reduced phase space and correlate with the Schwarzschild solution. Then we enhance it in the extended phase space. In an extended phase space the thermodynamic equations of state are stated as constraints. We apply the canonical transformation to analyze the thermodynamics of said type of black holes. We plot $P$-$v$ diagrams for different dimensions $d$ taking the temperatures $T T_c$ and analyze the natures of the graphs and the dependencies on $d$. In theses diagrams, we point out the regions of coexistence. We also examine the phase transition by applying "Maxwell's equal area law" of the said black holes. Here we find the regions of coexistence of two phases which are also depicted graphically. Finally, we derive the "Clapeyron equation" and investigate the latent heat of isothermal phase transition.

Invariance of Witten's quantum mechanics under point canonical transformations

Abstract: We show that the supersymmetric algebra of Witten's quantum mechanics is invariant under a given point canonical transformation. It is shown that Witten's supersymmetric quantum mechanics can be isospectral or not to the seed Hamiltonian depending on the space coordinate you work on. We illustrate our results by generating a new class of exactly solvable supersymmetric partner Hamiltonians which are not isospectral to the seed Hamiltonian.