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 Quantum Field Theory (QFT). An early connection is found allready in Feshbach's unified theory of nuclear reactions. Its possible relevance in current developments such as Feshbach resonances in the fields of cold atom systems, of Bose-Einstein condensates and in the problem of time in Quantum Gravity is noted. .
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.
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 discuss the deformation quantization approach for the teaching of quantum mechanics. This approach has certain conceptual advantages which make its consideration worthwhile. In particular, it sheds new light on the relation between classical and quantum mechanics. We demionstrate how it can be used to solve specific problems and clarify its relation to conventional quantization and path integral techniques. We also discuss its recent applications in relativistic quantum field theory.
A formalism is developed to obtain the energy eigenvalues of spatially confined quantum mechanical systems in the framework of The usual WKB and MAF methods. The technique is applied to three different cases,viz one dimensional Harmonic Oscillators,Quartic Oscillators and a boxed-in charged particle in electric field.
Basing on the Chetaev's theorem on stable trajectories in dynamics in the presence of dissipative forces we obtain a generalized stability condition for Hamiltonian systems that has the form of the Schrodinger equation. We show that the energy of the dissipative forces generating generalized Chetaev's stability condition exactly coincides with Bohm's "quantum" potential. Using the principle of least action of perturbation we prove that the Bohmian quantum mechanics complemented with the Chetaev's generalized theorem does not have hidden variables.
We derive a standard quantum limit for probing mechanical energy quantization in a class of systems with mechanical modes parametrically coupled to external degrees of freedom. To resolve a single mechanical quantum, it requires a strong-coupling regime -- the decay rate of external degrees of freedom is smaller than the parametric coupling rate. In the case for cavity-assisted optomechanical systems, e.g. the one proposed by Thompson et al., zero-point motion of the mechanical oscillator needs to be comparable to linear dynamical range of the optical system which is characterized by the optic…
In a previous paper, we obtained the functional form of quantum potential by a quasi-Newtonian approach and without appealing to the wave function. We also described briefly the characteristics of this approach to the Bohmian mechanics. In this article, we consider the quantization problem and we show that the 'eigenvalue postulate' is a natural consequence of continuity condition and there is no need for postulating that the spectrum of energy and angular momentum are eigenvalues of their relevant operators. In other words, the Bohmian mechanics predicts the 'eigenvalue postulate'.
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.
Liénard-type nonlinear one-dimensional oscillator is quantized using van Roos symmetric ordering recipe for the kinetic-like part of the new derived Hamiltonian. The corresponding Schrödinger equation is exactly solved in momuntum space via the approach of supersymmetric quantum mechanics (SUSYQM). The bound-states energy spectra and corresponding wave functions are given explicitly in terms of the ambiguity parameters. The limiting case of no deformation agrees exactly with the eigenenergies and eigenfunctions of the ordinary quantum harmonic oscillator.
In this work, we give the wave equations of relativistic and non-relativistic quantum mechanics which are different from the Schrödinger and Klein-Gordon equation, and we also give the new relativistic wave equation of a charged particle in an electromagnetic field
The results, different aspects and applications of our method of quantisation on configuration manifolds - called Borel Quantisation - were presented at meetings of the series `Symmetries in Science' and can be found in the published proceedings. The developments with numerous coauthors, on Borel quantisation and the related family of nonlinear Schrödinger equations called Doebner-Goldin equations, are reviewed and commented here.
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.
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 …
The star product formalism has proved to be an alternative formulation for nonrelativistic quantum mechanics. We want introduce here a covariant star product in order to extend the star product formalism to relativistic quantum mechanics in the proper time formulation.
A review of the symplectic tomographic approaches within the framework of star-product quantization is presented. The classical statistical mechanics within the framework of the tomographic representation is considered. The kernels of star-product of functions - symbols of operators in classical and quantum mechanics are presented.
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 …
We discuss the axiomatic basis of quantum mechanics and show that it is neither general nor consistent, since its axioms are incompatible with each other and moreover it does not incorporate the magnetic quantization as in the cyclotron motion. A general and consistent system of axioms is conjectured which incorporates also the magnetic quantization.
The covariant canonical method of quantization based on the De Donder-Weyl covariant canonical formalism is used to formulate a world-sheet covariant quantization of bosonic strings. To provide the consistency with the standard non-covariant canonical quantization, it is necessary to adopt a Bohmian deterministic hidden-variable equation of motion. In this way, string theory suggests a solution to the problem of measurement in quantum mechanics.
A quantum version of the action principle is considered in the case of a free relativistic particle. The classical limit of the quantum action is obtained.
In this paper, we investigate the connection between Classical and Quantum Mechanics by dividing Quantum Theory in two parts: - General Quantum Axiomatics (a system is described by a state in a Hilbert space, observables are self-adjoint operators and so on) - Quantum Mechanics properly that specifies the Hilbert space, the Heisenberg rule, the free Hamiltonian... We show that General Quantum Axiomatics (up to a supplementary "axiom of classicity") can be used as a non-standard mathematical ground to formulate all the ideas and equations of ordinary Classical Statistical Mechanics. So the ques…
We raise the possibility of developing a theory of constructing quantum dynamical observables independent from quantization and deriving classical dynamical observables from pure quantum mechanical consideration. We do so by giving a detailed quantum mechanical derivation of the classical time of arrival at arbitrary arrival points for a particle in one dimension.
In the paper is presented an invariant quantization procedure of classical mechanics on the phase space over flat configuration space. Then, the passage to an operator representation of quantum mechanics in a Hilbert space over configuration space is derived. An explicit form of position and momentum operators as well as their appropriate ordering in arbitrary curvilinear coordinates is demonstrated. Finally, the extension of presented formalism onto non-flat case and related ambiguities of the process of quantization are discussed.
The spin-statistics connection has been proved for nonrelativistic quantum mechanics (Jabs, A., 2010: Found. Phys., {\bf 40}, 776-792). The proof is extended here to the relativistic regime using the parametrized Dirac equation. A causality condition is not required.
Based on Lorentz invariance and Born reciprocity invariance, the canonical quantization of Special Relativity (SR) is shown to provide a unified origin for: i) the complex vector space formulation of Quantum Mechanics (QM); ii) the momentum and space commutation relations and the corresponding representations; iii) the Dirac Hamiltonian in the formulation of Relativistic Quantum Mechanics (RQM); iv) the existence of a self adjoint Time Operator that circumvents Pauli's objection.
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…
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.
Quantum systems with adiabatic classical parameters are widely studied, e.g., in the modern holonomic quantum computation. We here provide complete geometric quantization of a Hamiltonian system with time-dependent parameters, without the adiabatic assumption. A Hamiltonian of such a system is affine in the temporal derivative of parameter functions. This leads to the geometric Berry factor phenomena.
In this paper we investigate a possible energy scale dependence of the quantization rules and in particular, from a phenomenological point of view, an energy scale dependence of an effective $\hbar$. We set a bound on the deviation from the value of $\hbar$ at the muon scale from its usual value using measurements of the anomalous magnetic moment of the muon. Assuming that inflation took place, we can conclude that nature is described by a quantum theory at least up to an energy scale of about $10^{16}$ GeV.
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…
Using a new state-dependent, $λ$-deformable, linear functional operator, ${\cal Q}_ψ^λ$, which presents a natural $C^{\infty}$ deformation of quantization, we obtain a uniquely selected non--linear, integro--differential Generalized Schrödinger equation. The case ${\cal Q}_ψ^{1}$ reproduces linear quantum mechanics, whereas ${\cal Q}_ψ^{0}$ admits an exact dynamic, energetic and measurement theoretic {\em reproduction} of classical mechanics. All solutions to the resulting classical wave equation are given and we show that functionally chaotic dynamics exists.
Deformation quantization (sometimes called phase-space quantization) is a formulation of quantum mechanics that is not usually taught to undergraduates. It is formally quite similar to classical mechanics: ordinary functions on phase space take the place of operators, but the functions are multiplied in an exotic way, using the star product. Here we attempt a brief, pedagogical discussion of deformation quantization, that is suitable for inclusion in an undergraduate course.
The aim of the famous Born and Jordan 1925 paper was to put Heisenberg's matrix mechanics on a firm mathematical basis. Born and Jordan showed that if one wants to ensure energy conservation in Heisenberg's theory it is necessary and sufficient to quantize observables following a certain ordering rule. One apparently unnoticed consequence of this fact is that Schrödinger's wave mechanics cannot be equivalent to Heisenberg's more physically motivated matrix mechanics unless its observables are quantized using this rule, and not the more symmetric prescription proposed by Weyl in 1926, which has…
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…
On the basis of a 5-dimensional form of space-time transformations non-relativistic quantum mechanics is reformulated in a manifestly covariant manner. The resulting covariance resembles that of the conventional relativistic quantum mechanics.
In this article we show that the fundamental equations of relativistic Bohmian mechanics for a single particle can be derived from a scalar theory of curved space-time.
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…
We explore the group theoretical underpinning of noncommutative quantum mechanics for a system moving on the two-dimensional plane. We show that the pertinent groups for the system are the two-fold central extension of the Galilei group in $(2+1)$-space-time dimensions and the two-fold extension of the group of translations of $\mathbb R^4$. This latter group is just the standard Weyl-Heisenberg group of standard quantum mechanics with an additional central extension. We also look at a further extension of this group and discuss its significance to noncommutative quantum mechanics. We build un…
We propose a manifestly covariant canonical method of field quantization based on the classical De Donder-Weyl covariant canonical formulation of field theory. Owing to covariance, the space and time arguments of fields are treated on an equal footing. To achieve both covariance and consistency with standard noncovariant canonical quantization of fields in Minkowski spacetime, it is necessary to adopt a covariant Bohmian formulation of quantum field theory. A preferred foliation of spacetime emerges dynamically owing to a purely quantum effect. The application to a simple time-reparametrizatio…
I present a short overview of my recent achievements on the Bohmian interpretation of relativistic quantum mechanics, quantum field theory and string theory. This includes the relativistic-covariant Bohmian equations for particle trajectories, the problem of particle creation and destruction, the Bohmian interpretation of fermionic fields and the intrinsically Bohmian quantization of fields and strings based on the De Donder-Weyl covariant canonical formalism.
Second quantization of a classical nonrelativistic one-particle system as a deformation quantization of the Schrodinger spinless field is considered. Under the assumption that the phase space of the Schrodinger field is $C^{\infty}$, both, the Weyl-Wigner-Moyal and Berezin deformation quantizations are discussed and compared. Then the geometric quantum mechanics is also quantized using the Berezin method under the assumption that the phase space is $CP^{\infty}$ endowed with the Fubini-Study Kahlerian metric. Finally, the Wigner function for an arbitrary particle state and its evolution equati…
Phase Space is the framework best suited for quantizing superintegrable systems--systems with more conserved quantities than degrees of freedom. In this quantization method, the symmetry algebras of the hamiltonian invariants are preserved most naturally, as illustrated on nonlinear $σ$-models, specifically for Chiral Models and de Sitter $N$-spheres. Classically, the dynamics of superintegrable models such as these is automatically also described by Nambu Brackets involving the extra symmetry invariants of them. The phase-space quantization worked out then leads to the quantization of the cor…
Wigner's quasi-probability distribution function in phase-space is a special (Weyl) representation of the density matrix. It has been useful in describing quantum transport in quantum optics; nuclear physics; decoherence (eg, quantum computing); quantum chaos; "Welcher Weg" discussions; semiclassical limits. It is also of importance in signal processing. Nevertheless, a remarkable aspect of its internal logic, pioneered by Moyal, has only emerged in the last quarter-century: It furnishes a third, alternative, formulation of Quantum Mechanics, independent of the conventional Hilbert Space, or P…
In the context of our recently developed emergent quantum mechanics, and, in particular, based on an assumed sub-quantum thermodynamics, the necessity of energy quantization as originally postulated by Max Planck is explained by means of purely classical physics. Moreover, under the same premises, also the energy spectrum of the quantum mechanical harmonic oscillator is derived. Essentially, Planck's constant h is shown to be indicative of a particle's "zitterbewegung" and thus of a fundamental angular momentum. The latter is identified with quantum mechanical spin, a residue of which is thus …
Many wave phenomena are related to interactions. Considering once neglected interactions in some cases, states of large objects and Newton's idea about measurement, we attempt to modify some concepts and principles of non-relativistic quantum mechanics. Our modifications may help one to understand some typical quantum phenomena with near classical and intuitive ideas without changing past correct results, may mediate some contradictions logically, and then may partially perfect non-relativistic quantum mechanics. We also give a review of the argument between Einstein and Bohr about non-relativ…
This paper can be seen as an exercise in how to adapt quantum mechanics from a strict relativistic perspective while being respectful and critical towards the experimental achievements of the contemporary theory. The result is a fully observer independent relativistic quantum mechanics for N particle systems without tachyonic solutions. A remaining worry for the moment is Bell's theorem.
We construct a quantum mechanics based on the hypothesis of existing compact extra dimensions for a particle that wants to detect it. By introducing a probability function, we express the transition of particle to the extra 2d window. The general properties of this function has been examined and a length scale for occurrence of particle to extra window is given. By a diverse view point we consider that, the new length scale plays another quantum criteria for another quantization, beside the Planck constant. Canonical quantization of second class constrained systems, is our method for construct…
This paper proposes a basic theory on physical reality, and a new foundation for quantum mechanics and classical mechanics. It does not only solve the problem of the arbitrariness on the operator ordering for the quantization procedure, but also clarifies how the classical-limit occurs. It further compares the new theory with the known quantization methods, and proposes a self-consistent interpretation for quantum mechanics. It also provides the internal structure inducing half-integer spin of a particle, the sense of the regularization in the quantum field theory, the quantization of a phenom…
