Foundations of Physics

doi: 10.1007/BF00708428pmid: N/A

Prugovečki, Eduard

doi: 10.1007/BF00708429pmid: N/A

The measurement of one or more observables can be considered to yield sample points which are in general fuzzy sets. Operationally these fuzzy sample points are the outcomes of calibration procedures undertaken to ensure the internal consistency of a scheme of measurement. By introducing generalized probability measures on σ-semifields of fuzzy events, one can view a quantum mechanical state as an ensemble of probability measures which specify the likelihood of occurrence of any specific fuzzy sample point at some instant. These sample points are the possible outcomes of any infinitely rapid succession of measurements at that instant of any sequence of observables of the system.

Lavenda, B.

doi: 10.1007/BF00708430pmid: N/A

The thermodynamics of averaged motion treats the asymptotic spatiotemporal evolution of nonlinear irreversible processes. Dissipative and conservative actions are associated with short and long spatiotemporal scales, respectively. The motion of asymptotically stable systems is slow, monotonic, and continuous, so that the microscopic state variable description of rapid motion can be supplanted by an analysis of the macroscopic variable equations of motion of amplitude and phase. Rapid motion is associated with instability, and the direction of system motion is determined by thermodynamic criteria, in place of an analysis of the microscopic equations of motion. The characteristic structural configurations, deduced from the extremum principles of partial differential equations, are compared with the thermodynamic criteria. As a result of the nature of asymptotic motion, variational principles exist which characterize the asymptotic states of the system.

Witriol, Norman

doi: 10.1007/BF00708431pmid: N/A

Problems related to the operator form of the generalized canonical momenta in quantum mechanics are resolved by use of the general quantum mechanical canonical point transformation method. This method can be applied to any general canonical point transformation irrespective of the relationship between the domains of the original and transformed variables. The differential representation of the original canonical momenta pi in the original coordinate space is −i $$\begin{array}{*{20}c} / \\ h \\ \end{array}$$ ∂/∂x i and of the transformed canonical momentap i ′ in the transformed coordinate space is −i $$\begin{array}{*{20}c} / \\ h \\ \end{array}$$ ∂/∂x i ′. Relationships are derived between the eigenvalues of the original and transformed momenta in either space. The ordering problem for general point transformations is discussed and is shown to be solved. As an example of the generality of the method, it is demonstrated on the point transformation in three dimensions from Cartesian rectilinear to spherical rectilinear coordinates.

Čapek, Milič

doi: 10.1007/BF00708432pmid: N/A

The merging of space and time proposed by Minkowski in 1908 is still sometimes misinterpreted as a sort of four-dimensional hyperspace of which time is the fourth dimension, analogous to the other, spatial dimensions. An inevitable consequence of this view is that the future events somehow exist prior to, and independently of, human awareness and that what we call “becoming” is “merely a coming into our awareness” (A. Grünbaum). However, an attentive inspection of the space-time diagram and of Minkowski's formula for the constancy of the world interval shows that the events contained in the absolute future of any frame of reference areintrinsically unobservable not only within this system, but also by any other conceivable observer: consequently, there is no reason to postulate their existence.

Frost, Authur

doi: 10.1007/BF00708433pmid: N/A

The two-component spinor theory of van der Waerden is put into a convenient matrix notation. The mathematical relations among various types of matrices and the rule for forming covariant expressions are developed. Relativistic equations of classical mechanics and electricity and magnetism are expressed in this notation. In this formulation the distinction between time and space coordinates in the four-dimensional space-time continuum falls out naturally from the assumption that a four-vector is represented by a Hermitian matrix. The indefinite metric of tensor analysis is a derived result rather than an arbitrary ad hoc assumption. The relation to four-component spinor theory is also discussed.

Edmonds, James

doi: 10.1007/BF00708434pmid: N/A

The basic structure of a second quantized relativistic quantum theory is outlined. The vector space is over the ring of complex quaternions instead of the usual field of complex numbers. This is motivated by the simple quaternion structure of the Dirac equation.

Kobus, J.

doi: 10.1007/BF00708435pmid: N/A

Radial motion of a small point mass in the gravitational field of a large point mass is investigated for the law of gravitationR 44 =0. When geodesic equations are expressed in terms of components of acceleration, it is found that the normally “attractive force” of gravitation gradually weakens as the large mass is approached, and becomes “repulsive” inside a critical nonsingular radius close to the origin of coordinates. A particle requires an infinite time to reach the origin, regardless of its initial distance. Gravitational collapse, or at least violent collapse, is thus precluded.