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)

The formalism of (±)-frequency parts , previously applied to solution of the D'Alembert equation in the case of the electromagnetic field, is applied to solution of the Klein-Gordon equation for the K-G field in the presence of sources. Retarded and advanced field operators are obtained as solutions, whose frequency parts satisfy a complex inhomogeneous K-G equation. Fourier transforms of these frequency parts are solutions of the central equation, which determines the time dependence of the destruction/creation operators of the field. The (...) retarded field operator is resolved into kinetic and dissipative components. Correspondingly, the energy/stress tensor is resolved into three components; the power/force density, into two—a kinetic and a dissipative component. As in the analogous electromagnetic case the dissipation theorem is derived according to which work done by the dissipative power/force is negative: energy/momentum is dissipated from the sources to the K-G field. Boson quantization conditions are satisfied by the kinetic component but not by the dissipative component of the retarded K-G field. (shrink)

It is shown that in every gauge the potential of the electromagnetic field in the presence of sources is resolved by an extension of the Helmholtz theorem into a solenoidal component and an irrotational component irrelevant for description of the field. Only irrotational components are affected by gauge transformations; in Coulomb gauge the irrotational component vanishes: the potential is solenoidal. The method of solution of the wave equation by use of positive- and negative-frequency parts is extended to solutions of D'Alembert's (...) equation, and applied to equations satisfied by the potential in Coulomb gauge and the electric and magnetic vectors. Fourier transforms of potentials specifying destruction/creation operators become time dependent in the presence of sources. Our central equation states this time dependence. Frequency parts of Maxwell's equations are obtained. When the retarded potential in Coulomb gauge is resolved into kinetic and dissipative components, the latter is shown to be in radiation gauge. Correspondingly, the energy/stress tensor is resolved into three components; the power/force density, into two: a kinetic and a dissipative component. Work done by the latter component is negative: energy and momentum are dissipated from matter to radiation. Boson quantization conditions are satisfied by the kinetic component of the retarded potential, but commutators of the dissipative component are determined by the current sources. The energy/stress tensor and Hamiltonian of the field in the presence of sources are derived from the classical Lagrangian density. The relation between the Hamiltonian and the energy is shown to agree with the time dependence of the destruction/creation operators in Heisenberg picture. (shrink)

Publication in this journal recently of a paper by A. Grünbaum has induced me to submit a treatment of the same subject. Like Grünbaum, I shall use the framework of special relativity, but my analysis will be different. Some of his arguments, I believe, are inadmissable. I shall pose the problem in a form different from his, but the solution to be described will be applicable to his case as well.

The linear vector space for the quantum description of a physical system is formulated as the intersection of the domains of Hermiticity of the observables characterizing the system. It is shown that on a continuous interval of its spectrum every Hermitian operator on a Hilbert space of one degree of freedom is a generalized coordinate with a conjugate generalized momentum. Every continuous spectral interval of a Hermitian operator is the limit of a discrete spectrum in the same interval. This result (...) is applied to description of the state of a system by a statistical operator p( $\hat q$ ) following measurement of the probabilities of the eigenvalues of an observable $\hat q$ . Each continuous interval in the spectrum of $\hat q$ is replaced by a discrete spectrum in the same interval with a parameterK; the limitK → ∞ in which the spectrum becomes continuous is not attainable physically. In the linear vector space of a quantum mechanical system every continuous interval in the spectra of the observables is similarly replaced by a discrete spectrum with a finite parameter: it is a space of discrete bases and a set of such parameters. The case of degenerate spectrum is also discussed. (shrink)

The Einstein law of velocity addition is well-known. Consider three inertial reference frames S, K, and M, with X axes parallel in the same sense. Let K have velocity v1 along the X axis of S, and M have velocity v2 along the X axis of K. Then, according to the Einstein formula, the velocity v3 of M along the X axis of S is If v2 is the velocity of an object at rest in M as measured by an (...) observer in K, then v3 is the velocity of the same object as measured by an observer in S. The formula is valid for a change in reference frame of the observer. (shrink)