Ambiguities arising in different approaches (canonical, quasiclassical, path integration) to quantization are discussed by an example of the mechanics of a point-like particle in the Riemannian space (the geodesic dynamics). A way to select a single rule of quantization is proposed by requirement of a consistency of the quantum mechanics' following from the canonical and quasiclassical approaches. This rule selects also a definition of the path integration. A geometric interpretation of the noncovariance of the canonical Hamilton operator of a particle with respect to the diffeomorphisms of the Riemannian configuration space is proposed.
We give an explicit operator realization of Dirac quantization of free particle motion on a surface of codimension 1. It is shown that the Dirac recipe is ambiguous and a natural way of fixing this problem is proposed. We also introduce a modification of Dirac procedure which yields zero quantum potential. Some problems of abelian conversion quantization are pointed out.
Combination of a construction of unambiguous quantum conditions out of the conventional one and a simultaneous quantization of the positions, momenta, angular momenta and Hamiltonian leads to the geometric potential given by the so-called thin-lay quantization.
We explore the quantization of a $(1+1)$-dimensional inhomogeneous scalar field theory in which Poincaré symmetry is explicitly broken. We show the `classical equivalence' between a scalar field theory on curved spacetime background and its corresponding inhomogeneous scalar field theory. This implies that a hidden connection may exist among some inhomogeneous field theories, which corresponds to general covariance in field theory on curved spacetime. Based on the classical equivalence, we propose how to quantize a specific field theory with broken Poincaré symmetry inspired by standard field …
A suitable deformation of the Hopf algebra of the creation and annihilation operators for a complex scalar field, initially quantized in Minkowski space--time, induces the canonical quantization of the same field in a generic gravitational background. The deformation parameter q turns out to be related to the gravitational field. The entanglement of the quantum vacuum appears to be robust against interaction with the environment.
The classical time of arrival in the interacting case is quantized by way of quantizing its expansion about the free time of arrival. The quantization is formulated in coordinate representation which represents ordering rules in terms of two variable polynomial functions. This leads to representations of the quantized time of arrival operators as integral operators whose kernels are determined by the chosen ordering rule. The formulation lends itself to generalization which allows construction of time of arrival operators that cannot be obtained by direct quantization using particular ordering…
To comply with the equivalence principle, fields in curved spacetime can be quantized only in the neighborhood of each point, where one can construct a freely falling M i n k o w s k i frame with z e r o curvature. In each such frame, the geometric forces of gravity can be replaced by a selfinteracting spin-2 field, as proposed by Feynman in 1962. At a n y fixed distance $R$ from a black hole, the vacuum in each freely falling volume element acts like a thermal bath of all particles with Unruh temperature T_U=\hbar GM/2πc R^2. At the horizon R=2GM/c^2, the falling vacua show the Hawking temper…
For particles constrained on a curved surface, how to perform quantization within Dirac's canonical quantization scheme is a long-standing problem. On one hand, Dirac stressed that the Cartesian coordinate system has fundamental importance in passing from the classical Hamiltonian to its quantum mechanical form while preserving the classical algebraic structure between positions, momenta and Hamiltonian to the extent possible. On the other, on the curved surface, we have no exact Cartesian coordinate system within intrinsic geometry. These two facts imply that the three-dimensional Euclidean s…
An overview of maximally superintegrable classical Hamitonians on spherically symmetric spaces is presented. It turns out that each of these systems can be considered either as an oscillator or as a Kepler-Coulomb Hamiltonian. We show that two possible quantization prescriptions for all these curved systems arise if we impose that superintegrability is preserved after quantization, and we prove that both possibilities are gauge equivalent.
We use a deformed differential structure to obtain a curved metric by a deformation quantization of the flat space-time. In particular, by setting the deformation parameters to be equal to physical constants we obtain the Friedmann-Robertson-Walker (FRW) model for inflation and a deformed version of the FRW space-time. By calculating classical Einstein-equations for the extended space-time we obtain non-trivial solutions. Moreover, in this framework we obtain the Moyal-Weyl, i.e. a constant non-commutative space-time, by a consistency condition.
Using azimuthally symmetrized cylindrical coordinates, we report the consequences of zero-energy quantal states on the von Roos Hamiltonian. A position-dependent mass M(ρ,ϕ,z)=bz^{j}ρ^{2\u{psion}+1}/2 is used. We show that the zero-energy setting not only offers an additional degree of freedom towards feasible separability for the von Roos Hamiltonian, but also manifestly yields auxiliary quantized ambiguity parametric constraints.
In this work, Schrödinger and Dirac equations will be examined in geometries that confine the particles to hypersurfaces. For this purpose, two methods will be considered. The first method is the thin layer method which relies on explicit use of geometrical relations and the squeezing of a certain coordinate of space (or spacetime). The second is Dirac's quantization procedure involving the modification of canonical quantization making use of the geometrical constraints. For the Dirac equation, only the first method will be considered. Lastly, the results of the two methods will be compared an…
Starting with the first-order singular Lagrangian describing the dynamical system with 2nd-class constraints, the noncommutative quantum mechanics on a curved space is investigated by the constraint star-product quantization formalism of the projection operator method. Imposing the additional constraints to eliminate the reduntant degrees of freedom, it is shown that the resultant noncommutative quantum system on the curved space is represented with two kinds of the constrained quantum systems, which are equivalent with each other. Then, it is shown that the resultant Hamiltonians contain the …
By requiring unambiguous symmetric quantization leading to the Dirac equation in a curved space, we obtain a special representation of the spin connections in terms of the Dirac gamma matrices and their space-time derivatives. We also require that squaring the equation give the Klein-Gordon equation in a curved space in its canonical from (without spinor components coupling and with no first order derivatives). These requirements result in matrix operator algebra for the Dirac gamma matrices that involves a universal curvature constant. We obtain exact solutions of the Dirac and Klein-Gordon e…
In Foldy-Wouthuysen representation, we deduce the effective quantum mechanics for a particle confined to a curved surface by using the thin-layer quantization scheme. We find that the spin effect caused by confined potential as the results of relativistic correction in the non-relativistic limit. Furthermore, the spin connection appeared in curved surface which depends on curvature contributes a Zeeman-like gap in the relativistic correction term. In addition, the confined potential also induces a curvature-independent energy shift, which is from the zitterbewegung effect. As an example, we ap…
This is an extended version of the talk given by the Author at the 40th Symposium on Mathematical Physics held in Torun, Poland, June 25-28, 2008. We review the methods of canonical quantization of free particle motion on curved submanifolds considered as a system with second class constraints. The work is based on our previous articles, quant-ph/0508044 and quant-ph/0508111. However, some new results are also presented.
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.
A gauge invariant quantization in a closed integral form is developed over a linear phase space endowed with an inhomogeneous Faraday electromagnetic tensor. An analog of the Groenewold product formula (corresponding to Weyl ordering) is obtained via a membrane magnetic area, and extended to the product of N symbols. The problem of ordering in quantization is related to different configurations of membranes: a choice of configuration determines a phase factor that fixes the ordering and controls a symplectic groupoid structure on the secondary phase space. A gauge invariant solution of the qua…
When a two-dimensional curved surface is conceived as a limiting case of a curved shell of equal thickness d, where the limit d\rightarrow0 is then taken, the well-known geometric potential is induced by the kinetic energy operator, in fact by the second order partial derivatives. Applying this confining procedure to the momentum operator, in fact to the first order partial derivatives, we find the so-called geometric momentum instead. This momentum is compatible with the Dirac's canonical quantization theory on system with second-class constraints. The distribution amplitudes of the geometric…
The full spectrum and eigenfunctions of the quantum version of a nonlinear oscillator defined on an N-dimensional space with nonconstant curvature are rigorously found. Since the underlying curved space generates a position-dependent kinetic energy, three different quantization prescriptions are worked out by imposing that the maximal superintegrability of the system has to be preserved after quantization. The relationships among these three Schroedinger problems are described in detail through appropriate similarity transformations. These three approaches are used to illustrate different feat…
A strengthened canonical quantization scheme for the constrained motion on a curved hypersurface is proposed with introduction of the second category of fundamental commutation relations between Hamiltonian and positions/momenta, whereas those between positions and moments are categorized into the first. As an $N-1$ ($N\geq2$) dimensional hypersurface is embedded in an N dimensional Euclidean space, we obtain the proper momentum that depends on the mean curvature. For the surface is the spherical one, a long-standing problem on the form of the geometric potential is resolved in a lucid and una…
Quantum-mechanical WKB-method is elaborated for the known quantum oscillator problem in curved 3-spaces models Euclid, Riemann, and Lobachevsky E_{3}, H_{3}, S_{3} in the framework of the complex variable function theory. Generalized generally covariant Schrödinger equation is considered. In all three space models, exact energy levels are found with the help of constructing special formal WKB-sieries.
In this work we show that the ordering ambiguity on quantization depends on the representation choice. This property is then used to solve unambiguously some particular systems. Finally, we speculate on the consequences for more involved cases.
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 metho…
For a free particle that non-relativistically moves on a curved surface, there are curvature-induced quantum potentials that significantly influence the surface quantum states, but the experimental results in topological insulators, whenever curved or not, indicate no evidence of such a potential, implying that there does not exist such a quantum potential for the relativistic particles, constrained on the surface or not. Within the framework of Dirac quantization scheme, we demonstrate a general result that for a Dirac fermion on a two-dimensional curved surface of revolution, no curvature-in…
A fundamental problem regarding the Dirac quantization of a free particle on an $N-1$ curved hypersurface embedded in $N$($\geq 2$) flat space is the impossibility to give the same form of the curvature-induced quantum potential, the geometric potential as commonly called, as that given by the Schrödinger equation method where the particle moves in a region confined by a thin-layer sandwiching the surface. We resolve this problem by means of previously proposed scheme that hypothesizes a simultaneous quantization of positions, momenta, and Hamiltonian, among which the operator-ordering-free se…
The product of quantum mechanics is defined as the ordinary multiplication followed by the application of superoperator that orders involved operators. The operator version of Poisson bracket is defined being the Lie bracket which substitutes commutator in the von Neumann equation. These result in obstruction free quantization, with the ordering rule which coincides with Weyl ordering rule.
We clearly refine the fundamental framework of the thin-layer quantization procedure, and further develop the procedure by taking the proper terms of degree one in $q_3$ ($q_3$ denotes the curvilinear coordinate variable perpendicular to curved surface) back into the surface quantum equation. The well-known geometric potential and kinetic term are modified by the surface thickness. Applying the developed formalism to a toroidal system obtains the modification for the kinetic term and the modified geometric potential including the influence of the surface thickness.
Quantization of free Dirac fields is formulated in terms of a reducible representation of CAR. Similarly to the bosonic case we arrive at field operators which are indeed operators and not operator valued distributions. Observables such as four-momentum and charge can be defined without any need of normal ordering.
Compatibility of symmetric quantization of the Dirac equation in a curved space with general covariance gives a special representation of the spin connections in which their dot product with the Dirac gamma matrices becomes equal to the "covariant divergence" of the latter. Requiring that the square of the equation gives the conventional Klein-Gordon equation in a curved space results in an operator algebra for the Dirac gamma matrices that involves the "covariant derivative" connections and the Riemann-Christoffel connections. In 1+1 space-time with static metric, we obtain exact solutions of…
In the spirit of the thin-layer quantization approach, we give the formula of the geometric influences of a particle confined to a curved surface embedded in three-dimensional Euclidean space. The geometric contributions can result from the reduced commutation relation between the acted function depending on normal variable and the normal derivative. According to the formula, we obtain the geometric potential, geometric momentum, geometric orbital angular momentum, geometric linear Rashba and cubic Dresselhaus spin-orbit couplings. As an example, a truncated cone surface is considered. We find…
We consider one particle confined to a deformed one-dimensional wire. The quantum mechanical equivalent of the classical problem is not uniquely defined. We describe several possible hamiltonians and corresponding solutions for a finite wire with fixed endpoints and non-vanishing curvature. We compute and compare the disparate eigenvalues and eigenfunctions obtained from different quantization prescriptions. The JWKB approximation without potential leads precisely to the square well spectrum and the coordinate dependent stretched or compressed box related eigenfunctions. The geometric potentia…
By introducing the s-parameterized generalized Wigner operator into phase-space quantum mechanics we invent the technique of integration within s-ordered product of operators (which considers normal ordered, antinormally ordered and Weyl ordered product of operators as its special cases). The s-ordered operator expansion formula of density operators is derived, and the s-parameterized quantization scheme is thus completely established.
By considering the spin connection, we deduce the effective equation for a spin-1/2 particle confined to a curved surface with the non-relativistic limit and in the thin-layer quantization formalism. We obtain a pseudo-magnetic field and an effective spin-orbit interaction generated by the spin connection. Geometrically, the pseudo-magnetic field is proportional to the Gaussian curvature and the effective spin-orbit interaction is determined by the Weingarten curvature tensor. Particularly, we find that the pseudo-magnetic field and the effective spin-orbit interaction can be employed to separ…
The quantum space-time and the phase space with fuzzy structure is investigated as the possible quantization formalism. In this theory the state of nonrelativistic particle corresponds to the element of fuzzy ordered set (Foset) - fuzzy point. Due to Foset partial (weak) ordering, particle's space coordinate x acquires principal uncertainty dx. It's shown that Shroedinger formalism of Quantum Mechanics can be completely derived from consideration of particle evolution in fuzzy phase space with minimal number of axioms.
