Canonical Quantization Method Research Articles

Fractional particle and sigma model

We introduce a classical fractional particle model in ℝn, extending the Newtonian particle concept with the incorporation of the fractional Laplacian. A comprehensive discussion on kinetic properties, including linear momentum and kinetic energy, is provided. We further derive the equations of motion and discuss the symmetries of the system. The Green’s function method is employed to solve the equations of motion in a general case. We illustrate the theory with three important examples: the free fractional particle, the fractional harmonic oscillator, and the charged fractional particle that interacts locally with the electromagnetic field. We use the results of the extension problem by Caffarelli and Silvestre, to construct the associated classical local sigma model for the fractional particle. The sigma model is then quantized using the canonical quantization method, and we compute the vacuum energy at the boundary.

Canonical quantization of dissipative systems

A consistent quantization procedure for a dissipative system is introduced. To this end Hamilton's least action principle compatible with initial value problems is exploited to obtain the Hamiltonian for the dissipative systems. The derivation of the Schrodinger equation for the dissipative system is presented. The canonical quantization method introduced is simple as well as general for linear damping. The canonical quantization procedure is employed to quantize the damped harmonic oscillator.

Quantum skyrmion crystals and the symmetry energy of dense matter

The canonical quantization method for collective coordinates in crystalline configurations of the generalized Skyrme model is applied in order to find the quantum ground state of skyrmion crystals and study the quantum corrections to the binding energy resulting from the isospin degrees of freedom. This leads to a consistent description of asymmetric nuclear matter within the Skyrme framework and allows us to compute the symmetry energy of the skyrmionic crystal as a function of the baryon density and to compare with recent observational constraints.

Consistent quantization of systems with positive and negative energy states

The Stueckelberg-Feynman (SF) treatment, where positive energy antiparticles are described as negative energy particles going backward in time, lies on the basis of particle physics, but it was inconsistent with quantum field theory, since led to a negative norm for negative energy states. In the paper a new consistent method of canonical quantization in SF treatment is presented, where norms of all states is positive, since changing the direction of time integration in the action function changes the sign of Lagrangian of antiparticles and momentum. Minimal Lagrangians for complex canonical variables do not lead to the zero-point energy, which partially solves the cosmological constant problem. Causal propagators and amplitudes appear as symmetric chronological products of field operators, which slightly modifies diagram technique. Modified microcausality conditions and proof of spin and statistics theorem are presented, applications to particle physics and condensed media are discussed.

Design of quantum sensor to duplicate European Robins navigational system

In this article, we design a quantum device to duplicate the European Robins procedure to precisely determine the migratory route. In the mentioned procedure, the important issue is the geomagnetic field effect on the magnetic momentum of the created radical pairs (triplet-singlet states) dancing with a special frequency. To duplicate the procedure, a quantum sensor consisting of two coincident tripartite systems is designed. Each tripartite system is independently excited with the entangled photons (signal and idler). The interesting point is that by manipulation of the system in the right condition, the microwave cavities modes separately affected by the entangled photons can be entangled. The entangled microwave photons play the same role as the triplet-singlet states present in the bird’s navigational system. The key point in the design of the quantum sensor is that the entanglement between microwave photons can be strongly affected by the external magnetic field. In fact, this is the criterion employed by the quantum sensor to sense the magnetic field intensity and the direction. To analyze the system, the canonical quantization (or microscopic) method is used to determine the sensor’s Hamiltonian, and also the system dynamics equations of motions are analytically derived using Heisenberg-Langevin equations.

Affine Quantization on the Half Line

The similarity between classical and quantum physics is large enough to make an investigation of quantization methods a worthwhile endeavour. As history has shown, Dirac's canonical quantization method works reasonably well in the case of conventional quantum mechanics over $\mathbb{R}^n$ but it may fail in non-trivial phase spaces and also suffer from ordering problems. Affine quantization is an alternative method, similar to the canonical quantization, that may offer a positive result in situations for which canonical quantization fails. In this paper we revisit the affine quantization method on the half line. We formulate and solve some simple models, the free particle and the harmonic oscillator.

Transverse Ward-Takahashi identities and full vertex functions in different representations of QED3 * * Supported in part by the National Natural Science Foundation of China (11475085, 11535005, 11690030), the National Major state Basic Research and Development (2016YFE0129300) and the Anhui Provincial Natural Science Foundation (1908085MA15)

We derive the transverse Ward-Takahashi identities (WTI) of N-dimensional quantum electrodynamics by means of the canonical quantization method and the path integration method, and subsequently attempt to prove that QED3 is solvable based on the transverse and longitudinal WTI, indicating that the full vector and tensor vertices functions can be expressed in terms of the fermion propagators in QED3. Further, we discuss the effect of different γ matrix representations on the full vertex function.

Quantization of electromagnetic modes and angular momentum on plasmonic nanowires**Project supported by the National Natural Science Foundation of China (Grant Nos. 11704058 and 11974069), the National Special Support Program for High-level Personnel Recruitment, China (Grant No. W03020231), Liaoning Revitalization Talents Program, China (Grant No. XLYC1902113), Program for Liaoning Innovation Team in University, China (Grant No. LT2016011), Science and Technique Foundation

Quantum theory of surface plasmons is very important for studying the interactions between light and different metal nanostructures in nanoplasmonics. In this work, using the canonical quantization method, the SPPs on nanowires and their orbital and spin angular momentums are investigated. The results show that the SPPs on nanowire carry both orbital and spin momentums during propagation. Later, the result is applied to the plasmonic nanowire waveguide to show the agreement of the theory. The study is helpful for the nano wire based plasmonic interactions and the quantum information based optical circuit in the future.

Analysis of Quantum Radar Cross-Section by Canonical Quantization Method (Full Quantum Theory)

This article investigates the difference between two quantum-based theories to calculate the radar cross-section (RCS). Quantum radar cross-section (QRCS) has been commonly analyzed using the dipole approximation method, and the related results show that it can improve the sidelobe of the interference pattern in contrast to the classical methods. This study, on the other hand, utilizes the canonical quantization (or microscopic) method, which is a more comprehensive theory than the dipole approximation method to calculate the radar cross-section. It is shown that there are some similarities between two methods; nonetheless, there are some crucial quantities and factors that have been ignored in the dipole approximation methods. The main difference arises due to the interaction Hamiltonian that two methods relied on. The theoretical calculation shows some critical points suggesting that the dipole approximation method cannot cover all aspects of the radar cross-section calculation. To verify the mentioned point, we establish a new method in which the radar cross-section is calculated by merging the quantum approach with the method of moment (MoM), called quantum-method of moment (QMoM). The simulation results show that the newly established method is in harmony with the canonical quantization method.

Superstatistics and canonical quantization of the damped harmonic oscillator

In this paper, we investigate the behavior of the energy eigenvalues of the Schrödinger equation by using the canonical quantization method. We obtain the Hamiltonian of the Schrödinger equation by the Lagrangian in terms of the new coordinates. Then we calculate the partition function by the eigenvalues and the thermodynamic properties of the system in the superstatistics formalism for the modified Dirac delta and the Gamma distributions. All results in the limiting cases satisfy that of the harmonic oscillator. Furthermore, the effects of the all parameters in the problem of energy eigenvalues and thermodynamic properties are calculated and shown graphically.

Exact derivation of the Hawking effect in canonical formulation

The Hawking effect is one of the most extensively studied topics in modern physics. Yet it remains relatively under-explored within the framework of canonical quantization. The key difficulty lies in the fact that the Hawking effect is principally understood using the relation between the ingoing modes which leave past null infinity and the outgoing modes which arrive at future null infinity. Naturally, these modes are described using advanced and retarded null coordinates instead of the usual Schwarzschild coordinates. However, null coordinates do not lead to a true Hamiltonian that describes the evolution of these modes. In order to overcome these hurdles in a canonical formulation, we introduce here a set of near-null coordinates which allows one to perform an exact Hamiltonian-based derivation of the Hawking effect. This derivation opens up an avenue to explore the Hawking effect using different canonical quantization methods such as polymer quantization.

Quantum massive conformal gravity

We first find the linear approximation of the second plus fourth order derivative massive conformal gravity action. Then we reduce the linearized action to separated second order derivative terms, which allows us to quantize the theory by using the standard first order canonical quantization method. It is shown that quantum massive conformal gravity is renormalizable but has ghost states. A possible decoupling of these ghost states at high energies is discussed.

Do Global Virtual Axionic Gravitons Exist?

Recently, the author has proposed the global one-dimensionality conjecture which, throughout making use of the method of the primary canonical quantization of General Relativity according to the Dirac-Arnowitt-Deser-Misner Hamiltonian formulation, formally avoids the functional differential nature which is the basic obstacle of quantum geometrodynamics, a model of quantum gravity based on the Wheeler-DeWitt equation and resutling from quantization of the Einstein field equations. The emergent quantum gravity model is straightforwardly integrable in terms of the Riemann-Lebesgue integral, has an unambiguous physical interpretation within quantum theory, and introduces a consistent concept of global graviton. In this article, the axionic nature of the global gravitons, that is the hypothetical particles responsible for gravitation in the global one-dimensional quantum gravity model, is shortly described and, making use of fundamental methods of quantum field theory, the virtual states of such gravitons are generated.

Decoherence dynamics of a charge qubit coupled to the noise bath

By virtue of the canonical quantization method, we present a quantization scheme for a charge qubit based on the superconducting quantum interference device (SQUID), taking the self-inductance of the loop into account. Under reasonable short-time approximation, we study the effect of decoherence in the ohmic case by employing the response function and the norm. It is confirmed that the decoherence time, which depends on the parameters of the circuit components, the coupling strength, and the temperature, can be as low as several picoseconds, so there is enough time to record the information.

Improved Canonical Quantization Method of Self Dual Field

In this paper, the improved canonical quantization method of the self dual field is given in order to overcome linear combination problem about the second class constraint and the first class constraint number maximization problem in the Dirac method. In the improved canonical quantization method, there are no artificial linear combination and the first class constraint number maximization problems, at the same time, the stability of the system is considered. Therefore, the improved canonical quantization method is more natural and easier accepted by people than the usual Dirac method. We use the improved canonical quantization method to realize the canonical quantization of the self dual field, which has relation with string theory successfully and the results are equal to the results by using the Dirac method.

FADDEEV–JACKIW FORMALISM FOR CONSTRAINED SYSTEMS WITH GRASSMANN DYNAMICAL FIELD VARIABLES

The Faddeev–Jackiw canonical quantization formalism for constrained systems with Grassmann dynamical variables within the framework of the field theory is reviewed. First, by means of a iterative process, the symplectic supermatrix is constructed and their associated constraints are found. Next, by taking into account the phase space of the system, the constraint structure is considered. It is found that, if there are no auxiliary dynamical field variables, the supermatrix whose elements are the Bose–Fermi brackets between the constraints associated with the independent dynamical field variables coincides with the symplectic supermatrix corresponding to these independent variables. An alternative procedure to obtain the first-class constraints is given. It is shown that for systems with gauge symmetries, by means of suitable gauge-fixing conditions, a nonsingular final symplectic supermatrix can be found. Then, two possible ways of calculating the Faddeev–Jackiw brackets are pointed out. The relation between the Faddeev–Jackiw and Dirac brackets is discussed. Throughout the previous developments, the Faddeev–Jackiw and Dirac algorithms are compared. Finally, the Faddeev–Jackiw canonical quantization method is applied to a simple model and the obtained results are compared with the ones corresponding to the use of the Dirac procedure on this model.

Concept of Fully Dually Symmetric Electrodynamics

It has been shown, that electromagnetic field (EM) has in general case quaternion structure, consisting of four independent fields, which differ each other by the parities under space inversion and time reversal. Generalized Maxwell equations for quaternion four-component EM-field are obtained. Invariants for EM-field, consisting of dually symmetric parts are found. It is shown, that the only one physical conserving quantity corresponds to both Rainich dual and additional hyperbolic dual symmetry of Maxwell equations. It is spin in general case and spirality in the geometry, when vectors E⃗, H⃗ are directed along coordinate axes in (E⃗, H⃗) functional space. It is the proof for quaternion four component structure of EM-field to be a single whole. Canonical Dirac quantization method is developed in two aspects. The first aspect is its application the only to observable quantities. The second aspect is the realization along with well known time-local quantization of space-local quantization and space-time-local quantization. It is also shown, that Coulomb field can be quantized in 1D and 2D systems, that is it is radiation field in given lowdimensional systems.

IMPROVED CANONICAL QUANTIZATION METHOD OF SPINOR ELECTRODYNAMICS WITH SINGULAR LAGRANGIAN

This paper analyzes the general interaction of spinor field and electromagnetic field, i.e. spinor electrodynamics with singular Lagrangian, and the improved canonical quantization method of spinor electrodynamics is given in order to overcome the problems of the second-class constraint linear combination and the first-class constraint number maximization brought by the traditional canonical quantization method. In the improved canonical quantization method, there are no problems of artificial linear combination and first-class constraint number maximization, at the same time, the stability of the system is considered. The improved canonical quantization method is more natural than the traditional way and we show that the results of the canonical quantization of spinor electrodynamics are equal to the results of the traditional way. For a general dynamical system with singular Lagrangian, one can extendingly use the improved canonical quantization method to quantize the system.

QUANTUM EFFECTS OF A DOMAIN WALL IN THE FERROMAGNETIC SYSTEM

Starting from the classical equation of the motion of a domain wall in the ferromagnetic systems, the quantum energy levels of the wall and the corresponding eigenfunctions in the case of considering damping term are given by using the canonical quantization method and unitary transformation. The quantum fluctuations of displacement and momentum of the moving wall has also been given as well as the uncertain relation.

Non-standard loop quantum cosmology

We present results concerning the nature of the cosmological big bounce(BB) transition within the loop geometry underlying loop quantum cosmology (LQC). Our canonical quantization method is an alternative to the standard LQC. An evolution parameter we use has clear interpretation both at classical and quantum levels. The physical volume operator has discrete spectrum which is bounded from below. The minimum gap in the spectrum defines a quantum of the volume. The spectra of operators are parametrized by a free parameter to be determined.