Usually the only difference between relativistic quantization and standard one is that the Lagrangian of the system under consideration should be Lorentz invariant. The standard approaches are logically incomplete and produce solutions with unpleasant properties: negative-energy, superluminal propagation etc. We propose a two-projections scheme of (special) relativistic quantization. The first projection defines the quantization procedure (e.g. the Berezin-Toeplitz quantization). The second projection defines a casual structure of the relativistic system (e.g. the operator of multiplication by the characteristic function of the future cone). The two-projections quantization introduces in a natural way the existence of three types of relativistic particles (with $0$, $\frac{1}{2}$, and $1$ spins). Keywords: Quantization, relativity, spin, Dirac equation, Klein-Gordon equation, electron, Segal-Bargmann space, Berezin-Toeplitz quantization. AMSMSC Primary: 81P10, 83A05; Secondary: 81R30, 81S99, 81V45
We show that non-relativistic and relativistic mechanical systems on a configuration space Q can be seen as the conservative Dirac constraint systems with zero Hamiltonians on different subbundles of the same cotangent bundle T^*Q. The geometric quantization of this cotangent bundle under the vertical polarization leads to compatible covariant quantizations of non-relativistic and relativistic Hamiltonian mechanics.
We provide geometric quantization of the vertical cotangent bundle V^*Q equipped with the canonical Poisson structure. This is a momentum phase space of non-relativistic mechanics with the configuration bundle Q -> R. The goal is the Schrodinger representation of V^*Q. We show that this quantization is equivalent to the fibrewise quantization of symplectic fibres of V^*Q -> R, that makes the quantum algebra of non-relativistic mechanics an instantwise algebra. Quantization of the classical evolution equation defines a connection on this instantwise algebra, which provides quantum evolution in …
For a relativistic particle under a constant force and a linear velocity dissipation force, a constant of motion is found. Problems are shown for getting the Hamiltoninan of this system. Thus, the quantization of this system is carried out through the constant of motion and using the quantization of the velocity variable. The dissipative relativistic quantum bouncer is outlined within this quantization approach.
A relativistic Hamiltonian mechanical system is seen as a conservative Dirac constraint system on the cotangent bundle of a pseudo-Riemannian manifold. We provide geometric quantization of this cotangent bundle where the quantum constraint serves as a relativistic quantum equation.
It was shown recently that stochastic quantization can be made into a well defined quantization scheme on (pseudo-)Riemannian manifolds using second order differential geometry, which is an extension of the commonly used first order differential geometry. In this letter, we show that restrictions to relativistic theories can be obtained from this theory by imposing a stochastic energy-momentum relation. In the process, we derive non-perturbative quantum corrections to the line element as measured by scalar particles. Furthermore, we extend the framework of stochastic quantization to massless s…
We study the reparametrization invariant system of a classical relativistic particle moving in (5+1) dimensions, of which two internal ones are compactified to form a torus. A discrete physical time is constructed based on a quasi-local invariant observable. Due to ergodicity, it is simply related to the proper time on average. The external motion in Minkowski space can then be described as a unitary quantum mechanical evolution.
In this contribution we study the Landau levels arising within the relativistic quantum dynamics of a neutral particle which possesses a permanent magnetic dipole moment interacting with an external electric field. We consider the Aharonov-Casher coupling of magnetic dipole to the electric field to investigate an an analog of Landau quantization in this system and solve the Dirac equation for two different field configurations. The eigenfunctions and eigenvalues of Hamiltonian in both cases are obtained.
We revisit the classical approach of comoving coordinates in relativistic hydrodynamics and we give a constructive proof for their global existence under suitable conditions which is proper for stochastic quantization. We show that it is possible to assign stochastic kinematics for the free relativistic spinless particle as a Markov diffusion globally defined on ${\sf M}^4$. Then introducing dynamics by means of a stochastic variational principle with Einstein's action, we are lead to positive-energy solutions of Klein-Gordon equation. The procedure exhibits relativistic covariance properties.
The path integral quantization method is applied to a relativistically covariant version of the Hopfield model, which represents a very interesting mesoscopic framework for the description of the interaction between quantum light and dielectric quantum matter, with particular reference to the context of analogue gravity. In order to take into account the constraints occurring in the model, we adopt the Faddeev-Jackiw approach to constrained quantization in the path integral formalism. In particular we demonstrate that the propagator obtained with the Faddeev-Jackiw approach is equivalent to th…
In the framework of Dirac quantization with second class constraints, a free particle moving on the surface of a $(d-1)-$dimensional sphere has an ambiguity in the energy spectrum due to the arbitrary shift of canonical momenta. We explicitly show that this spectrum obtained by the Dirac method can be consistent with the result of the Batalin-Fradkin-Tyutin formalism, which is an improved Dirac method, at the level of the first-class constraint by fixing the ambiguity, and discuss its physical consequences.
A series of successive quantizations is considered, starting with the quantization of a non relativistic or relativistic point particle: 1) quantization of a particle's position, 2) quantization of wave function, 3) quantization of wave functional. The latter step implies that the wave packet profiles forming the states of quantum field theory are themselves quantized, which gives new physical states that are configurations of configurations. In the procedure of quantization, instead of the Schrödinger first order equation in time derivative for complex wave function (or functional), the equiv…
The nonholonomic constrained system with second-class constraints is investigated using the Hamilton-Jacobi (HJ) quantization scheme to yield the complete equations of motion of the system. Although the integrability conditions in the HJ scheme are equivalent to the involutive relations for the first-class constrained system in the improved Dirac quantization method (DQM), one should elaborate the HJ scheme by using the improved DQM in order to obtain the first-class Hamiltonian and the corresponding effective Lagrangian having the BRST invariant nonholonomic constrained system.
Based on Lorentz invariance and Born reciprocity invariance, the canonical quantization of Special Relativity (SR) has been shown to provide a unified origin for the existence of Dirac's Hamiltonian and a self adjoint time operator that circumvents Pauli's objection. As such, this approach restores to Quantum Mechanics (QM) the treatment of space and time on an equivalent footing as that of momentum and energy. Second quantization of the time operator field follows step by step that of the Dirac Hamiltonian field. It introduces the concept of time quanta, in a similar way to the energy quanta …
In this paper, we analyze Dirac fermions in Gödel-type background space-times with torsion. We also consider the Gödel-type spacetimes embedded in a topological defect background. We show that relativistic bound states solutions to the Dirac equation can be obtained by dealing with three cases of the Gödel-type solutions with torsion, where a cosmic string passes through these three cases of the space-time. We obtain the relativistic energy levels for all cases of the Gödel-type solutions with torsion with a cosmic string, where we show that there exists an analogy with the Landau levels for D…
We investigate the two-dimensional motion of relativistic cold electrons in the presence of `strictly' spatially varying magnetic fields satisfying, however, no magnetic monopole condition. We find that the degeneracy of Landau levels, which arises in the case of the constant magnetic field, lifts out when the field is variable and the energy levels of spin-up and spin-down electrons align in an interesting way depending on the nature of change of field. Also the varying magnetic field splits Landau levels of electrons with zero angular momentum from positive angular momentum, unlike the const…
We present an exact quantization condition for the time independent solutions (energy eigenstates) of the one-dimensional Dirac equation with a scalar potential well that gives only two `effective' turning points (defined by the roots of $V(x)+mc^2=\pm E$) for a given energy $E$ and satisfies $\min V(x)+mc^2\geq 0$. This result generalizes the previously known non-relativistic quantization formula and preserves many physically desirable symmetries, besides, attaining the correct non-relativistic limit. Numerical calculations demonstrate the utility of the formula for computing accurate energy …
New approach to quantization of the relativistic Majorana field is presented. It is based on expansion of the field into eigenfunctions of the axial momentum -- a novel observable introduced recently. Relativistic invariance is used as the main guiding principle instead of canonical formalism. Hidden structure of the quantized Majorana field in the form of real Clifford algebra of Hermitian fermionic operators is unveiled. Generators of the Poincaré transformations in the Fock space are found as solutions of certain operator equations, without invoking the principle of correspondence with clas…
The approximated energy eigenvalues and the corresponding eigenfunctions of the spherical Woods-Saxon effective potential in $D$ dimensions are obtained within the new improved quantization rule for all $l$-states. The Pekeris approximation is used to deal with the centrifugal term in the effective Woods-Saxon potential. The inter-dimensional degeneracies for various orbital quantum number $l$ and dimensional space $D$ are studied. The solutions for the Hulthén potential, the three-dimensional (D=3), the $% s$-wave ($l=0$) and the cases are briefly discussed.
A notion of quantization is proposed that is independent of the original statistical interpretation of the distribution of energy in a photon gas or of the quantization of angular momentum in hydrogen atom. Such a procedutre implies the existance of finite space-time four-interval that any relativistic preparation and measurement of a physical event requires. That finite four-interval is also the epistemological source of QM uncertainty relations. Furthermore, the consetrvation of helicity in the propagation of a photon is a relativistic invariant, and is the origin of the "appearence" of a pa…
The diffculties of relativistic particle theories formulated my means of canonical quantization, such as Klein-Gordon and Dirac theories, ultimately led theoretical physicists to turn on quantum field theory to model elementary particle physics. The aim of the present work is to pursue a method alternative to canonical quantization that avoids these dfficulties. In order to guarantee this result, the present approach is constrained to be developed deductively from physical principles. The physical principles assumed for a free particle consist of the symmetry properties of the particle with re…
A systematic method for calculating higher-order corrections of the relativistic semiclassical fixed-energy amplitude is given. The central scheme in computing corrections of all orders is related to a time ordering operation of an operator involving the Van Vleck determinant. This study provides us a new viewpoint for quantization.
The relativistic effects of the integer-spin quantum field theory imply that the wave functions describing a fixed number of particles do not admit the usual probabilistic interpretation. Among several most popular interpretations of quantum mechanics applied to first quantization, the only interpretation for which this fact does not lead to a serious problem, and therefore the only consistent interpretation of first quantization, is the de Broglie-Bohm interpretation.
The properties of relativistic particles in the quasiclassical region are investigated. The relativistic semiclassical wave equation appropriate in the quasiclassical region is derived. It is shown that the leading-order WKB quantization rule is the appropriate method to solve the equation obtained.
The bound state energies and wave functions for a particle exposed to the Hulthén potential field in the D-dimensional space are obtained within the improved quantization rule for any arbitrary l state. The present approximation scheme used to deal with the centrifugal term in the effective Hulthén potential is systematic and accurate. The solutions for the three-dimensional (D=3) case and the s-wave (l=0) case are briefly discussed. Keywords: Hulthén potential, improved quantization rule, approximation schemes. 03.65.Ge, 12.39.Jh
Quantum computing has demonstrated the potential to revolutionize our understanding of nuclear, atomic, and molecular structure by obtaining forefront solutions in non-relativistic quantum many-body theory. In this work, we show that quantum computing can be used to solve for the structure of hadrons, governed by strongly-interacting relativistic quantum field theory. Following our previous work on light unflavored mesons as a relativistic bound-state problem within the nonperturbative Hamiltonian formalism, we present the numerical calculations on simulated quantum devices using the basis lig…
The relativistic 20-component wave equation, describing particles with spin 1/2 and two mass states, is analyzed. The projection operators extracting states with definite energy and spin projections, and density matrix are obtained. The canonical quantization of the field with two mass states in the formalism of the first order is performed and the chronological pairing of the 20-component operators was found.
The problem of ultraviolet divergences is analysed in the quantum field theory. It was found that it has common roots with the problem of cosmological singularity. In the context of fibre bundles the second quantization method is represented as a procedure of the quantization for vector bundle cross-section. It is shown to be quite a different quantization way called as a fibre quantization which leads to an idea on existence of the non-standard dynamical system, i.e. the relativistic be-Hamiltonian system. It takes place on supersmall distances and is well described by the mathematical appara…
It is argued that quantum and relativistic correlations can be described in a unified way, in that both assume free will as an axiom, and happen without any continuous connection in space-time. This description may contribute to a coherent definition of "space-time quantization" and highlights the importance of solving the "measurement problem".
Based on the Chetaev theorem on stable dynamical trajectories in the presence of dissipative forces, we obtain the generalized condition for stability of relativistic classical Hamiltonian systems (with an invariant evolution parameter) in the form of the Stueckelberg equation. As is known, this equation is the basis of a competing paradigm known as parametrized relativistic quantum mechanics (pRQM). It is shown that the energy of dissipative forces, which generate the Chetaev generalized condition of stability, coincides exactly with Bohmian relativistic quantum potential. We show that the sq…
Exact solutions of two-particle relativistic equations of quantum field theory describing the scattering $s$-states and the bound $s$-states are found in the cases of delta-shell potential and superposition of delta-shell potentials. Some properties of obtained relativistic wave functions, scattering amplitudes and quantization conditions are investigated. The resonance character of scattering processes is demonstrated by the behavior of amplitudes. It is shown that the non-relativistic limits of these relativistic values coincide with respective non-relativistic ones obtained from the Schrödi…
