Canonical Quantization Method Research Articles

Bianchi I model of the universe in terms of nonstandard LQC

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.

Non‐standard loop quantum cosmology

AbstractWe 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.

QUANTIZATION FOR THE MESOSCOPIC RLC CIRCUIT AND ITS THERMAL EFFECT BY VIRTUE OF GHFT

The mesoscopic single RLC (resistance-inductance-capacitance) circuit and the RLC circuit including complicated coupling are quantized by employing Dirac's standard canonical quantization method. The thermal effects for the systems are investigated by virtue of GHFT (the generalized Hellmann–Feynman theorem). The results distinctly show the effect of temperature on the quantum fluctuation.

Energy scale of the Big Bounce

We examine the nature of the cosmological Big Bounce transition within the loop geometry underlying loop quantum cosmology at classical and quantum levels. Our canonical quantization method is an alternative to the standard loop quantum cosmology. An evolution parameter we use has a clear interpretation. Our method opens the door for analyses of spectra of physical observables like the energy density and the volume operator. We find that one cannot determine the energy scale specific to the Big Bounce by making use of the loop geometry without an extra input from observational cosmology.

Thermal effect and energy-level transition rule for a mesoscopic LC circuit with inductance–capacitance coupling

This paper reports that the mesoscopic inductance and capacitance coupling LC circuit is quantized by means of the canonical quantization method. Using the ‘invariant eigen-operator’ method, it deduces the energy-level transition rule when the system is disturbed by an external electromagnetic field. At the same time, the quantum fluctuations for the system at finite temperature are examined by virtue of the generalized Hellmann–Feynman theorem.

Ostrogradski approach for the Regge-Teitelboim type cosmology

We present an alternative geometric inspired derivation of the quantum cosmology arising from a brane universe in the context of geodetic gravity. We set up the Regge-Teitelboim model to describe our universe, and we recover its original dynamics by thinking of such field theory as a second-order derivative theory. We refer to an Ostrogradski Hamiltonian formalism to prepare the system to its quantization. Our analysis highlights the second-order derivative nature of the RT model and the inherited geometrical aspect of the theory. A canonical transformation brings us to the internal physical geometry of the theory and induces its quantization straightforwardly. By using the Dirac canonical quantization method our approach comprises the management of both first- and second-class constraints where the counting of degrees of freedom follows accordingly. At the quantum level our Wheeler-De Witt equation agrees with previous results recently found. On these lines, we also comment upon the compatibility of our approach with the Hamiltonian approach proposed by Davidson and coworkers.

Effects of capacitive coupling on the escape rate in intrinsic Josephson junction stacks

We formulate a macroscopic quantum theory that can describe the macroscopic quantum tunneling (MQT) in intrinsic Josephson junctions (IJJs). The capacitively coupled IJJ model is quantized in terms of the canonical quantization method. The multi-junction effect for the MQT to the first resistive branch is clarified. It is shown that the escape rate is greatly enhanced by the capacitive coupling between junctions.

Macroscopic quantum tunneling in a stack of capacitively-coupled intrinsic Josephson junctions

A macroscopic quantum theory for the phase dynamics in capacitively-coupled intrinsic Josephson junctions (IJJ’s) is constructed. We quantize the capacitively-coupled IJJ model in terms of the canonical quantization method. The multi-junction effect for the macroscopic quantum tunneling (MQT) to the first resistive branch is clarified. It is shown that the escape rate is greatly enhanced by the capacitive coupling between junctions. We also discuss the origin of the N2-enhancement in the escape rate observed in the uniformly switching in Bi-2212 IJJ’s.

Deformation quantization, quantization and the Klein–Gordon equation

The aim of this paper is to give a basic introduction to deformation quantization (DQ) to physicists. We compare DQ to canonical quantization and path integral methods. It is described how certain issues such as the roles of associativity, coordinate invariance, dynamics and operator orderings are understood in the context of DQ. Convergence issues in DQ are mentioned. Additionally, we formulate the Klein–Gordon (KG) equation in DQ. Original results are discussed which include the exact construction of the Fedosov star-product on the dS and AdS spacetimes. Also, the KG equation is written down for these spacetimes.

On quantization of time-dependent systems with constraints

The Dirac method of canonical quantization of theories with second-class constraints has to be modified if the constraints depend on time explicitly. A solution of the problem was given by Gitman and Tyutin. In the present work we propose an independent way to derive the rules of quantization for these systems, starting from the physical equivalent theory with trivial non-stationarity.

Krein space quantization in curved and flat spacetimes

We reexamine in detail a canonical quantization method a la Gupta-Bleuler in which the Fock space is built over a so-called Krein space. This method has already been successfully applied to the massless minimally coupled scalar field in de Sitter spacetime for which it preserves covariance. Here, it is formulated in a more general context. An interesting feature of the theory is that, although the field is obtained by canonical quantization, it is independent of Bogoliubov transformations. Moreover no infinite term appears in the computation of $T^{\mu\nu}$ mean values and the vacuum energy of the free field vanishes: $<0|T^{00}|0>=0$. We also investigate the behaviour of the Krein quantization in Minkowski space for a theory with interaction. We show that one can recover the usual theory with the exception that the vacuum energy of the free theory is zero.

Hamilton-Jacobi Quantization of Systems With Time-Dependent Constraints

Hamilton–Jacobi formalism is used to investigate time-dependent constraint systems. It is proved that the generalization of Dirac's canonical quantization method in the nonstationary case can be obtained naturally in Hamilton–Jacobi formalism. The example of the relativistic particle in a plane wave is analyzed in detail.

Modification of special relativity and formulation of convergent and invariant quantum field theory

Besides two fundamental postulates, (i) the principle of relativity and (ii) the constancy of the speed of light in all inertial frames of reference, the special theory of relativity uses another assumption. This other assumption concerns the Euclidean structure of gravity-free space and the homogeneity of gravity-free time in the usual inertial coordinate system. Introducing the primed inertial coordinate system, in addition to the usual inertial coordinate system, for each inertial frame of reference, we assume the Euclidean structures of gravity-free space and time in the primed inertial coordinate system and their generalized Finslerian structures in the usual inertial coordinate system. We combine the alternative assumption with the two postulates (i) and (ii) to modify the special theory of relativity. The modified special relativity theory involves two versions of the light speed, infinite c′ in the primed inertial coordinate system and finite c in the usual inertial coordinate system. It also involves the c′-type Galilean transformation between any two primed inertial coordinate systems and the localized Lorentz transformation between two corresponding usual inertial coordinate systems. Since all our experimental data are collected and expressed in the usual inertial coordinate system, the physical principle is: the c′-type Galilean invariance in the primed inertial coordinate system plus the transformation from the primed inertial coordinate system to the usual inertial coordinate system. This principle is applied to a reformulation of mechanics, field theory and quantum field theory. Relativistic mechanics in the usual inertial coordinate system is unchanged, while field theory is developed and divergence-free. Any c′-type Galilean-invariant field system can be quantized by using the canonical quantization method in the primed inertial coordinate system. We establish a transformation law for quantized field systems as they are transformed from the primed to the usual inertial coordinate system. It is shown that the modified special relativity theory, together with quantum mechanics, leads to a convergent and invariant quantum field theory, in full agreement with experimental facts. The formulation of this quantum field theory does not demand departures from the concepts such as local Lorentz invariance in the usual inertial coordinate system, locality of interactions, and local or global gauge symmetries.

Noncommutative Φ4 theory at two loops

We study perturbative aspects of noncommutative field theories. This work is arranged in two parts. First, we review noncommutative field theories in general and discuss both canonical and path integral quantization methods. In the second part, we consider the particular example of noncommutative $\Phi^4$ theory in four dimensions and work out the corresponding effective action and discuss renormalizability of the theory, up to two loops.

G/u(1)dparafermions from constrained WZNW theories

The conformal field theory based on the $g/u(1)^d$ coset construction is treated as the WZNW theory for the affine Lie algebra $\hat g$ with the constrained $\hat u(1)^d$ subalgebra.Using a modification of the generalized canonical quantization method generators and primary fields of an extended symmetry algebra are found for arbitrary d.

Quantum Wavefunctions and Fluctuations of Mesoscopic RLC Circuit

The quantum wavefunctions and the corresponding energy levels of a RLC (Resistance-Inductance-Capacity) electric circuit are obtained by using canonical quantization method and unitary transformation from the classical equation of motion. The quantum fluctuations of charge and current in an arbitrary eigenstate of the system have also been given as well as the uncertainty relation. It is showed that even if at 0 K charge and current in the circuit exhibit quantum fluctuations, which originates from fluctuations of zero point vibrations of the system.

Conformal anomaly in the two-dimensional nonlinear σ model

The conformal anomaly of the nonlinear \ensuremath{\sigma} model is investigated by the canonical quantization method. The relations built among the components of the energy-momentum tensor are discussed with the standard quantum field theory argumentation. A more general structure for the canonical quantization is furnished by considering the freedom of redefining the field variables in the Hamiltonian formalism. Special attention is paid to the analysis of the so-called Schwinger terms defined as anomalous terms generated at the quantum level. The conformal anomaly can be obtained by virtue of various conservation rules without preimposing reparametrization invariance. Our results not only coincide with the ones obtained from the \ensuremath{\beta}-functional approach to the two-loop level but also illustrate the transformation properties for the field configurations in terms of the so-called improvement terms. While it is more natural to deal with the canonical energy-momentum tensor than the usual stress-energy tensor, the difference between both is clarified. With extra consideration of the dilaton coupling, we furthermore show that the distinction between these two energy-momentum tensors is obvious. In addition, the dilaton effect on the conformal anomaly can be obtained with no reference to its explicit coupling pattern in the action.

A PROPOSAL FOR A GENERALIZED CANONICAL osp(1, 2) QUANTIZATION OF DYNAMICAL SYSTEMS WITH CONSTRAINTS

The aim of this letter is to consider the possibility of constructing a generalized canonical quantization method for arbitrary dynamical systems with first-class constraints based on the osp (1, 2) supersymmetry principle. This proposal can be considered as a counterpart to the osp (1, 2)-covariant Lagrangian quantization method introduced recently by Geyer, Lavrov and Mülsch. The gauge dependence of Green's functions is studied. It is shown that if the parameter m2 of the osp (1, 2) superalgebra is not equal to zero, then the vacuum functional and S-matrix depend on the gauge. In the limit m→0 the gauge independence of vacuum functional and S-matrix are restored. The Ward identities related to the osp (1, 2) symmetry are derived.

Yang–Mills theory on a cylinder coupled to point particles

A model of quantum Yang–Mills theory with a finite number of gauge invariant degrees of freedom is studied. The gauge field has only a finite number of degrees of freedom since it is assumed that space–time is a two-dimensional cylinder. The gauge field is coupled to matter, modeled by either one or two nonrelativistic point particles. These problems can be solved without any gauge fixing, by generalizing the canonical quantization methods of S. G. Rajeev [Phys. Lett. B 212, 203 (1988)] to the case including matter. For this, the geometry of the space of connections is used, which has the structure of a principal fiber bundle with an infinite-dimensional fiber. Both problems are reduced to finite-dimensional, exactly solvable, quantum mechanics problems. In the case of one particle, it is found that the ground state energy will diverge in the limit of infinite radius of space, consistent with confinement. In the case of two particles, this does not happen if they can form a color singlet bound state (‘‘meson’’).

Generalized covariance condition and quantization in curved configuration space.

We expound the generalized covariance condition in quantization. This condition is combined with the other standard conditions to determine the quantum-mechanical operator of a quadratic Hamiltonian ${\mathit{H}}_{2}$, represented by a quadratic function of momenta with a nonsingular coefficient matrix depending on coordinates, and the operator of an arbitrary quadratic quantity ${\mathit{F}}_{2}$ belonging to a system whose Hamiltonian is of the ${\mathit{H}}_{2}$ type. It is shown that the operator of ${\mathit{F}}_{2}$ depends also on ${\mathit{H}}_{2}$ and that H${\mathrm{^}}_{2}$ or F${\mathrm{^}}_{2}$ cannot contain a curvature term. We also present a modified ordering method with certain conditions under which the operator of a quadratic quantity can be found by replacing some linear functions of momenta with their operators. Finally, a path-integration formula is formed and shown to yield the expression of H${\mathrm{^}}_{2}$ that we derive by the canonical quantization method.