This is the first of two related papers analyzing and explaining the origin, manifestations and parodoxical features of the quantum potential (QP) from the non-relativistic and relativistic points of view. The QP arises in the quantum Hamiltonian under various procedures of quantization of natural systems, i.e., those whose Hamilton functions are positive-definite quadratic forms in momenta with coefficients depending on the coordinates in (n-dimensional) configurational space V n thus endowed with a Riemannian structure. The result of quantization may be considered as quantum mechanics (QM) of a particle in V n in the normal Gaussian coordinate system in the globally static space-time V 1,n . Contradiction of the QP to the General Covariance and Equivalence principles is discussed. It is found that actually the historically first Hilbert space-based quantization by E. Schrödinger (1926), after revision in the modern framework of QM, also leads to a QP in the form that B. DeWitt found 26 years later. Efforts to avoid the QP or to reduce its drawbacks are discussed. The general conclusion is that some form of QP and a violation of the principles of general relativity which it induces are inevitable in the non-relativistic quantum Hamiltonian. It is also shown that Feynman (path-integral) quantization of natural systems singles out two versions of the QP, which both determine two bi-scalar (independent of a choice of coordinates) propagators fixing two different algorithms of path integral calculation. The accompanying paper under the same general title and the subtitle “The Relativistic Point of View” (published in the same issue of the journal and referred to as Paper II) considers a relation of the nonrelativistic QP to the quantum theory of a scalar field non-minimally coupled to the curved space-time metric.
In this short review paper the detailed analysis of six two-dimensional quantum {\it superintegrable} systems in flat space is presented. It includes the Smorodinsky-Winternitz potentials I-II (the Holt potential), the Fokas-Lagerstrom model, the 3-body Calogero and Wolfes (equivalently,
G2 rational, or
I6) models, and the Tremblay-Turbiner-Winternitz (TTW) system with integer index
k. It is shown that all of them are exactly-solvable, thus, confirming the Montreal conjecture (2001); they admit algebraic forms for the Hamiltonian and both integrals (all three can be written as differential operators with polynomial coefficients without a constant term), they have polynomial eigenfunctions with the invariants of the discrete symmetry group of invariance taken as variables, they have hidden (Lie) algebraic structure
g(k) with various
k, and they possess a (finite order) polynomial algebras of integrals. Each model is characterized by infinitely-many finite-dimensional invariant subspaces, which form the infinite flag. Each subspace coincides with the finite-dimensional representation space of the algebra
g(k) for a certain
k. In all presented cases the algebra of integrals is a 4-generated
(H,I1,I2,I12≡[I1,I2]) infinite-dimensional algebra of ordered monomials of degrees 2,3,4,5, which is a subalgebra of the universal enveloping algebra of the hidden algebra.
We show that the list of analytically solvable potentials in nonrelativistic quantum mechanics can be considerably enlarged. In particular, we show that those noncentral potentials for which the Schrödinger equation is separable are analytically solvable provided the separated problem for each of the coordinates belongs to the class of exactly solvable one dimensional problems. As an illustration, we discuss in detail two examples, one in two and the other in three dimensions. A list of analytically solvable noncentral potentials in spherical polar coordinates is also given. Extension of these ideas to other standard orthogonal coordinate systems as well as to higher dimensions is straightforward.
The number of exactly solvable eigenvalue problems has been rapidly increasing since the advent of quantum mechanics and there are many efficient ways to treat them. As a general rule, it is possible to write exactly solvable abstract eigenvalue equations in terms of the generators of finite-dimensional Lie algebras and solve them by means of purely algebraic methods. Matrix elements are commonly necessary for the application of perturbation theory and variational principles, among other approximate methods. The chapter discusses examples of three-dimensional Lie algebras with well-known quantum-mechanical realizations. Typically, exactly solvable quantum-mechanical problems are either onedimensional models or lead to separable equations in many dimensions. In the coordinate representation both cases reduce to ordinary differential equations of second order that determine the eigenvalues and separation constants compatible with the physical boundary conditions. In its purest form, a Lie algebraic method is representation-independent. Extensive application of algebraic methods to selected physical problems is already available elsewhere.
The maximally-decoupled method has been considered as a theory to apply an basic idea of an integrability condition to certain multiple parametrized symmetries.The method is regarded as a mathematical tool to describe a symmetry of a collective submanifold in which a canonicity condition makes the collective variables to be an orthogonal coordinate-system.For this aim we adopt a concept of curvature unfamiliar in the conventional time-dependent (TD) self-consistent field (SCF) theory.Our basic idea lies in the introduction of a sort of Lagrange manner familiar to fluid dynamics to describe a collective coordinate-system.This manner enables us to take a one-form which is linearly composed of a TD SCF Hamiltonian and infinitesimal generators induced by collective variable differentials of a canonical transformation on a group.The integrability condition of the system read the curvature C = 0. Our method is constructed manifesting itself the structure of the group under consideration.To go beyond the maximaly-decoupled method, we have aimed to construct an SCF theory, i.e., υ (external parameter)-dependent Hartree-Fock (HF) theory.Toward such an ultimate goal, the υ-HF theory has been reconstructed on an affine Kac-Moody algebra along the soliton theory, using infinite-dimensional fermion.An infinite-dimensional fermion operator is introduced through a Laurent expansion of finite-dimensional fermion operators with respect to degrees of freedom of the fermions related to a υ-dependent potential with a Υ-periodicity.A bilinear equation for the υ-HF theory has been transcribed onto the corresponding τ -function using the regular representation for the group and the Schur-polynomials.The υ-HF SCF theory on an infinite-dimensional Fock space F ∞ leads to a dynamics on an infinite-dimensional Grassmannian Gr ∞ and may describe more precisely such a dynamics on the group manifold.A finite-dimensional Grassmannian is identified with a Gr ∞ which is affiliated with the group manifold obtained by reducting gl(∞) to sl(N ) and su(N ).As an illustration we will study an infinite-dimensional matrix model extended from the finite-dimensional su(2) Lipkin-Meshkov-Glick model which is a famous exactly-solvable model.