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The Euler-Lagrange equation derived from Schwinger's action principle (1951) has been shown by Kianget al. (1969) and Linet al. (1970) to lead to inconsistencies for quadratic lagrangians of the form $$\bar L(\dot q,q) = \tfrac{1}{2}\dot q^j g_{jk} (q)\dot q^k - V(q)$$ except in the Euclidean caseg jk =δ jk . This inadequacy is linked to Schwinger's specification that the variations of operators bec-numbers. We reformulate the action principle by introducing the concept of ‘proper’ Gauteaux variation of operators to find the most general class of admissible variation consistent with the postulated quantisation rules. This new action principle, applied to the LagrangianL, yields a quantum Euler equation consistent with the Hamilton-Heisenberg equations.

Generalizing from a classical application of the paradigm of Elementary Measurement discussed elsewhere (Leiter, 1969), we consider a non-linear, spinor wave-mechanical field theory of Elementary Measurement. In this theory, charged particles are represented by complex spinorc-number fields interacting through their associated electromagnetic fields in space-time. The paradigm of Elementary Measurement implies that the particle fields, and their associatedc-number electromagnetic fields, are interdependent degrees of freedom in an action principle associated with the measurement interaction, and are not elementary in themselves. Making the action stationary with respect to the interacting field degrees of freedom gives the equations of motion of the measurement. The application of this model theory to atomic hydrogen yields the result that the inherent ‘limit cycle solutions’ (LCS) of the non-linear measurement equations correspond to the quantum levels of conventional relativistic Dirac quantum mechanics of hydrogen, in the approximation that the nucleus has infinite mass. Superpositions of these Dirac-LCS solutions have the property of collapsing (reduction of the wave packet) into one of the LCS in the superposition,in a characteristic time which is identical to the ‘lifetime’ of the associated atomic levels as calculated from conventional quantum mechanics. Hence, in thisc-number electromagnetic theory,both spontaneous and induced transitions can be accounted for. ‘Photons’, in this theory, are not elementary particles, but instead are associated with the secondary dynamics related to the inherent nonlinear structure in the elementary measurement equations of motion. The ‘hidden variable’ characteristics of this measurement theory (as seen from the point of view of ordinary quantum mechanics), in describing a universe made up of such hydrogen atoms, is discussed. Within this context, a consistent derivation of the Planck blackbody radiation formula is given, in which the associated electromagnetic fields arec-numbers and arenot second quantized. Finally, a generalization of this prototype model theory, to a more consistent form which can account for the presence of ‘vacuum interaction processes’ and negative energy states, is suggested.

If one supposes a quantum logicL to be a σ-orthocomplete, orthomodular partially ordered set admitting a set of σ-orthoadditive functions (called states) fromL to the unit intervals [0, 1] such that these states distinguish the ordering and orthocomplement onL, then the observables onL are identified withL-valued measures defined on the Borel subsets of the real line. In this structure (and without the aid of Hilbert space formalism) the author shows that (1) the spectrum of an observable can be completely characterised by studying the observable (A−λ)−1, and (2) corresponding to every observableA there is a spectral resolution uniquely determined byA and uniquely determiningA.

Sudarshan's semi-classical treatment of correlation functions is applied to the study of quantum corrections to the Van Hove function. In its generalised form, it enables one to choose the best correlation function for a given potential. Higher-order correlations are also sketched briefly and the details are similar to those considered by Oppenheim and Bloom.

Parastatistics (parafields) has been used in relation to several models of physical systems like the quark and the nuclear shell models. However, the physics of parafields is not completely clear. If classical para-Bose or para-Fermi variables could be constructed, then because of the correspondence principle some traces of the corresponding quantum properties could be found at the classical limit. In this way, by studying the simplestc-number systems some hints for the quantum of parafields could be expected.

This work shows that a certain class of classical dynamical formalisms, characterised by non-singular Lie structures more general than the usual (Poisson) one, are derivable from ordinary constrained dynamical formalisms. As a consequence, the Lie brackets considered are special cases of suitably chosen Dirac brackets. Both unconstrained and constrained generalised dynamical formalisms are considered. The relations of our results with the problem of constructing classical analogues of generalised quantum systems are stressed.

We prove that there are quantum mechanical observables which are sensitive to the type of state-vector (first type or second type) describing two correlated physical systems, in the sense that the expectation value of these ‘sensitive observables’ is measurably different in the two cases. The proof centers around Bell's inequality since we show that in quantum mechanics forall state-vectors of the second type (and only for them) sensitive observables exist in the absence of super-selection rules. Experimental verification of the existence of sensitive observables rules out local hidden variables.

The usual treatment of decaying non-relativistic particles by means of a non-unitary irreducible representation of the Galilei group is deduced from a suitable formulation of symmetry principles. In such a formulation time translation is distinguished from time evolution; this point is crucial to obtain the irreversible behaviour of unstable particles.

The problem of resonance pressure broadening of spectral lines in monatomic gases is discussed using a resolvent operator formalism. A differential equation is developed to determine the resolvent, and it is shown how its solution for a limiting case yields the familiar classical path approximation for the translational motion of the atoms, and how quantum corrections may be systematically studied. Commonly used limiting cases within the classical path approximation (two-body static and impact approximations) are also exhibited as limiting cases, with methods for systematic evaluation of corrections. Closed form solutions are obtained for the two-body static and impact cases. The results are compared with available experimental data, and generally satisfactory agreement is obtained. Of some theoretical interest is the formalism, which embraces all the usual approximations and permits them to be studied together with corrections to them from a unified point of view. New results of more practical interest are the closed form solutions for the limiting cases, and the estimation of the lowest-order quantum corrections, which are appreciable under some experimental conditions.

The author recently introduced a new theory of motion which resolves several problems in the philosophy of physics and mathematics by replacing the continuum with nonvanishing time. Nonvanishing time means that the magnitude of an instant does not have the functional properties of zero, even though it may be very small. As a result, joint functions of time, such ass=vt, vary as the other terms even when time is instantaneous. Hence size (vdt or instantaneous position) is increasing as velocity. This paper presents some quantitative solutions to the above for the case in which a point of massm is resolving in a perfect circle with uniform speedv=ωr. The magnitude of an instantdt — the minimum time for real events—is found by obtaining the magnitude of the arcds along which the rotating point is distributed overdt time. Under the Heisenberg uncertainty principle this turns out to beds 2=h(mω)−1. If the de Broglie equation and Einstein's two equations for photon energy are introduced, then in generalds 2=λr. Under the quantum mechanical definition of orbital angular momentum λr −1=2πn −1, wheren is the integral scalar forħ. Hence there is an asymmetry on the right side of the last expression fords, and the ratio ofds to the circumference of the circle is (2πn)−1/2.