Canonical Phase Space Coordinates Research Articles

Quantization of non-Abelian Yang-Mills theories

A non-Abelian theory of fermions interacting with gauge bosons, the constrained system, is studied. The equations of motion for a singular system are obtained as total differential equations in many variables. The integrability conditions are investigated, and the set of equations of motion is integrable. The Senjanovic and the canonical methods are used to quantize the system, and the integration is taken over the canonical phase space coordinates.

Generalizations of Snyder model to curved spaces

We consider generalizations of the Snyder algebra to a curved spacetime background with de Sitter symmetry. As special cases, we obtain the algebras of the Yang model and of triply special relativity. We discuss the realizations of these algebras in terms of canonical phase space coordinates, up to fourth order in the deformation parameters. In the case of triply special relativity we also find exact realization, exploiting its algebraic relation with the Snyder model.

Path Integral Quantization of Doubly Supersymmetric Model

The Hamilton-Jacobi formalism is used to discuss the path integral quantization of the double supersymmetric models with the spinning superparticle in the component and superfield form. The equations of motion are obtained as total differential equations in many variables. The equations of motion are integrable, and the path integral is obtained as an integration over the canonical phase space coordinates.

Hamiltonians and canonical coordinates for spinning particles in curved space-time

The spin-curvature coupling as captured by the so-called Mathisson–Papapetrou–Dixon (MPD) equations is the leading order effect of the finite size of a rapidly rotating compact astrophysical object moving in a curved background. It is also a next-to-leading order effect in the phase of gravitational waves emitted by extreme-mass-ratio inspirals (EMRIs), which are expected to become observable by the LISA space mission. Additionally, exploring the Hamiltonian formalism for spinning bodies is important for the construction of the so-called effective-one-body waveform models that should eventually cover all mass ratios.The MPD equations require supplementary conditions determining the frame in which the moments of the body are computed. We review various choices of these supplementary spin conditions and their properties. Then, we give Hamiltonians either in proper-time or coordinate-time parametrization for the Tulczyjew–Dixon, Mathisson–Pirani, and Kyrian–Semerák conditions. Finally, we also give canonical phase-space coordinates parametrizing the spin tensor. We demonstrate the usefulness of the canonical coordinates for symplectic integration by constructing Poincaré surfaces of section for spinning bodies moving in the equatorial plane in Schwarzschild space-time. We observe the motion to be essentially regular for EMRI-ranges of the spin, but for larger values the Poincaré surfaces of section exhibit the typical structure of a weakly chaotic system. A possible future application of the numerical integration method is the inclusion of spin effects in EMRIs at the precision requirements of LISA.

Path Integral Quantization of Regular Lagrangian

Path integral formulation based on the canonical method is discussed. The Hamilton Jacobi function for regular Lagrangian is obtained using separation of variables method. This function is used to quantize regular systems using path integral method. The path integral is obtained as integration over the canonical phase space coordinates. One illustrative example is considered to demonstrate the application of our formalism.

Computer simulation of quantum dynamics in a classical spin environment

In this paper a formalism for studying the dynamics of quantum systems coupled to classical spin environments is reviewed. The theory is based on generalized antisymmetric brackets and naturally predicts open-path off-diagonal geometric phases in the evolution of the density matrix. It is shown that such geometric phases must also be considered in the quantum-classical Liouville equation for a classical bath with canonical phase space coordinates; this occurs whenever the adiabatics basis is complex (as in the case of a magnetic field coupled to the quantum subsystem). When the quantum subsystem is weakly coupled to the spin environment, non-adiabatic transitions can be neglected and one can construct an effective non-Markovian computer simulation scheme for open quantum system dynamics in classical spin environments. In order to tackle this case, integration algorithms based on the symmetric Trotter factorization of the classical-like spin propagator are derived. Such algorithms are applied to a model comprising a quantum two-level system coupled to a single classical spin in an external magnetic field. Starting from an excited state, the population difference and the coherences of this two-state model are simulated in time while the dynamics of the classical spin is monitored in detail. It is the author's opinion that the numerical evidence provided in this paper is a first step toward developing the simulation of quantum dynamics in classical spin environments into an effective tool. In turn, the ability to simulate such a dynamics can have a positive impact on various fields, among which, for example, nano-science.

PATH INTEGRAL QUANTIZATION OF SUPERSTRING

Motivated by the Hamilton–Jacobi approach of constrained systems, we analyze the classical structure of a four-dimensional superstring. The equations of motion for a singular system are obtained as total differential equations in many variables. The path integral quantization based on Hamilton–Jacobi approach is applied to quantize the system, and the integration is taken over the canonical phase space coordinates.

Semiclassical description of a triaxial rigid rotor

A triaxial rotor Hamiltonian is treated by a time-dependent variational principle, using a coherent state as a trial function. The critical points of the constant energy surface depend on the ordering relations satisfied by the three moments of inertia. All three orderings are considered and to each of them a specific wobbling frequency is derived. The transition between two distinct phases is discussed in terms of a cranking Hamiltonian. Four distinct pairs of complex canonical phase-space coordinates are pointed out. Each of them yields, after quantization, a distinct boson expansion for the angular-momentum components. The separation of the potential energy is treated for one of the four phase-space bases. One of the semiclassical descriptions is used for the yrast state energies of $^{158}\mathrm{Er}$ and an excellent agreement with experimental data is obtained up to very high angular momentum.

HAMILTON–JACOBI QUANTIZATION OF TWO-DIMENSIONAL GRAVITY WITH TORSION

Two-dimensional gravity with torsion is investigated using the Hamilton–Jacobi formalism. The equations of motion and the action integral are obtained as total differential equations in many variables. The integrabilty conditions, lead us to obtain the path integral quantization as an integration over the canonical phase-space coordinates.

Quantization of the O(N) Nonlinear Sigma Model

The Hamilton–Jacobi method of quantizing singular systems is discussed. The equations of motion are obtained as total differential equations in many variables. It is shown that if the system is integrable, one can obtain the canonical phase space coordinates and set of canonical Hamilton–Jacobi partial differential equations without any need to introduce unphysical auxiliary fields. As an example we quantize the O(2) nonlinear sigma model using two different approaches: the functional Schrödinger method to obtain the wave functionals for the ground and the exited state and then we quantize the same model using the canonical path integral quantization as an integration over the canonical phase-space coordinates.

The Hamilton-Jacobi treatment for an abelian Chern-Simons system

The abelian Chern-Simons system is treated as a constrained system using the Hamilton-Jacobi approach. The equations of motion are obtained as total differential equations in many variables. It is shown that their simultaneous solutions with the constraints lead to obtain canonical phase space coordinates and the reduced phase space Hamiltonian with out introducing Lagrange multipliers and with out any additional gauge fixing condition.

The Canonical Path Integral Quantization of CP1 Model

The Hamilton-Jacobi method of quantizing singular systems is discussed. The equations of motion are obtained as total differential equations in many variables. It is shown that if the system is integrable, then one can obtain the canonical phase space coordinates and the set of the canonical Hamilton-Jacobi partial differential equations without any need to introduce unphysical auxiliary fields. As an example we quantize the CP1 model using the canonical path integral quantization formalism to obtain the path integral as an integration over the canonical phase-space coordinates.