MASSLESS SCALAR FIELD PROPAGATOR IN A QUANTIZED SPACETIME

We consider in detail the analytic behaviour of the non-interacting massless scalar field two-point function in H. S. Snyder's discretized non-commuting spacetime. The propagator we find is purely real on the Euclidean side of the complex p2 plane and goes like 1/p2 as p2→0 from either the Euclidean or Minkowski side. The real part of the propagator goes smoothly to zero as p2 increases to the discretization scale 1/a2and remains zero for p2>1/a2. This behaviour is consistent with the termination of single-particle propagation on the ultraviolet side of the discretization scale. The imaginary part of the propagator, consistent with a multiparticle-state spectral function branch discontinuity, is finite and continuous on the Minkowski side, slowly falling to zero when 1/a2<p2<∞. The multi-particle aspect of this spectral function within the Källen-Lehmann representation of the propagator leads to the interpretation that the propagation of free-fields in a quantized spacetime is analogous to propagation of interacting fields in a continuous spacetime.

The Sigma Term: Leading Non-Analytic Behaviour Of Decuplet Contributions

We examine the analytic structure, in powers of the quark mass (m), of the chiral loop corrections to the pion-nucleon sigma term arising from coupling to the lowest mass baryon decuplet. The leading non-analytic term is found to go like m2 In(m/2a) (with a the delta-nucleon mass splitting), and to be quite significant numerically.

Gravitational radiation near spatial and null infinity

Taking Bondi’s approach to an extreme by adding a 1/ u -expansion to the usual 1/ r -expansion, one can study the effects of the presence of mass at space-like infinity. Assuming convergence of the formal expansion one finds: (1) the limit of the Bondi mass agrees with the ADM-mass of certain space-like hypersurfaces; (2) relations between expansion coefficients on space-like hypersurfaces and the radiation field on J + can be given; (3) analytic properties in J + lead to non-analytic behaviour on space-like hypersurfaces.

On the possible transition of the Gaussian model of an imperfect gas

It is known that the model of an imperfect gas for which the Mayer function is the Gaussian function - A exp ( - r 2 / a 2 ) of the distance r between two molecules is reducible to a problem in graph theory. It is shown that if the irreducible graphs occurring in the virial series are grouped according to their cyclomatic number we can expand the virial series in powers of the parameter A . The radius of convergence of the first few sub-series is the same. For the two-dimensional gas in the limiting case of A small, we find that, at a certain density, well outside the radius of convergence of the virial series, the third derivative of pressure vanishes. It is suggested that this may be the indication of a phase-transition of the model, and that its analytic behaviour is similar to that of lattice-type antiferromagnetic models.

On the threshold branch-point of many-body channels

The analytic behaviour of the elastic and break-up scattering amplitudes in a soluble one-dimensional model has been examined by Nussenzveig. It was found that the break-up threshold gave rise to rather curious cube-root branch-points in the scattering amplitude. In this paper, we shall examine the scattering amplitude in a soluble model which reduces Nussenzveig’s model to the special case where the incident and ionized particles have equal masses. It will be shown that the threshold branch-point is a function of the ratio of the masses of the two particles and that the cube-root occurs only when the masses are equal. In general, there are an infinite number of Riemann sheets associated with the threshold branch-point. An examination into the physical origin of such a threshold behaviour will also be made to determine if a more complicated branch-point than a simple square-root may exist for a more realistic potential model.

The coupling between electromagnetic waves and excitation waves in a molecular crystal

Analysis is given of some of the approximations involved in the electrostatic calculation of the dipole-dipole forces between molecules in an excited state of a molecular crystal. It is shown that there is a close analogy between the quantum mechanical theory of these excited states and the classical theory of the vibrations of an ionic crystal. The theory of long waves developed by Born & Huang for an ionic crystal is adapted for a molecular crystal which has one molecule in each unit cell. This enables the effects due to the coupling of the transverse excitation waves and the electromagnetic waves, the retardation of the forces, the molecular polarizability and the non-analytic behaviour of the energy bands to be included in the theory. A further modification of the theory allows for the finite life time of the excited state and produces changes in the energy bands, the extinction coefficient and the reflectivity.