From the known coordinate representation of these operators, a unified treatment of the abstract operators for curvilinear coordinates and their canonically conjugate momenta is given for systems in three dimensions. A configuration representation, corresponding to classical configuration space, exists in which description is simplified; the three-dimensional ket space factors into a direct product of one-dimensional spaces. Four cases are examined, according to the range of the continuous curvilinear coordinate. In addition to normalization of momentum eigenstates to the Kronecker delta for (...) finite range of coordinate and normalization to the Dirac delta for range on the entire real axis, normalization occurs to δ (or δ+) for range on the positive (or negative) part of the real axis. The domain of Hermiticity of curvilinear coordinate and conjugate momentum operators in each case is the entire three-dimension ket space. The fundamental commutation relations hold in all representations. Statements to the contrary for the radial and azimuth coordinates in spherical polar coordinates are seen to be erroneous. (shrink)

An error in Gruber's paper, “Quantization in Generalised Coordinates,” and the reason for this kind of error are pointed out. A partial answer to the problem posed by the paper is stated.

Classical contact transformation theory is reconstructed from the concept of explicit rather than implicit transformation equations. This proves the existence of contact transformations from any given Hamiltonian to any prescribed Hamiltonian (with the same number of degrees of freedom).

There is no unique way to generalize the mass operator 0 2 −p · p to curved spacetimes. The possible generalizations using either an analytic or an algebraic (group-theoretic) approach are discussed. We investigate in detail three possibilities: (i) the generalized D'Alembertian $$- \tilde \square = - (1/\sqrt { - g} )(\partial /\partial x^v )(\sqrt { - g} g^{\mu v} {\text{ }}\partial /\partial x^\mu$$ (ii) the operator ( $- (\tilde \square + \tfrac{1}{6}{\text{R)}}$ , which is conformally invariant in the massless (...) case, and (iii) an operator suggested by Meggs for Robertson-Walker spacetimes. A criterion for choosing among the different possibilities is proposed. It is used to eliminate the Meggs operator, but is not stringent enough to decide between the other two. (shrink)

In connection with another article by the author, we show how it might be possible to travel faster than the speed of light. We show that for clocks and rods moving faster than the speed of light, we get instead of “time dilation” and “Lorentz contraction,” respectively, “time contraction” and “Lorentz expansion,” respectively. It is shown that this paper is in confirmation with earlier articles dealing with this subject.

In an article by Margenau and Cohen various correspondence principles were described in connection with Weyl, Born-Jordan, and symmetrized ordering of quantum mechanical operators. In this article we make an interesting comparison between the aforementioned ordering process and our previous prescriptions.

In a recent “Rejoinder” by Landau,(8) he indicates that there is an error in a previous article of mine. In fact, this was corrected in an immediately subsequent article(2) of mine, and the “intriguing problem” to which Landau refers is solved in this subsequent article. In the present paper, I consolidate these and other ideas on the subject. In particular, I show that by discussing generalized coordinates in quantum mechanics one achieves much new insight, both philosophical and physical, in understanding (...) the transition of classical to quantum mechanics. (shrink)

The aim of this paper is to contribute to the clarification of concepts usually found in books on quantum mechanics, aided by knowledge from the field of the theory of operators in Hilbert space. Frequently the basic distinction between bounded and unbounded operators is not established in books on quantum mechanics. It is repeatedly overlooked that the condition for an unbounded operator to be symmetric (Hermitian) is not sufficient to make it self-adjoint. To make things worse, nearly all operators in (...) quantum mechanics are unbounded. Often one finds statements such as: For any linear operator A we can write a Hermitian operator HA=(A+A+)/2, where Hermitian is thought to mean self-adjoint. Along these lines, self-adjointness of the momentum operator in generalized coordinates, taken from that expression, is questioned. In particular, the redescription in terms of spherical polar coordinates and its implications for the eventual loss of self-adjointness of the momenta conjugate to them are studied. (shrink)