scholarly journals Analytic Behaviour of Competition among Three Species

2006 ◽  
Vol 13 (4) ◽  
pp. 535-548 ◽  
Author(s):  
PGL Leach ◽  
J Miritzis
Keyword(s):  
Analytic Behaviour

On the threshold branch-point of many-body channels

1963 ◽  
Vol 276 (1367) ◽  
pp. 492-501
Keyword(s):  
Branch Point ◽  
Scattering Amplitude ◽  
Cube Root ◽  
Branch Points ◽  
Many Body ◽  
Realistic Potential ◽  
Break Up ◽  
Analytic Behaviour ◽  
The Masses ◽  
Special Case

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.

scholarly journals On the Analytic Behaviour of Dyson Transformation Function

1952 ◽  
Vol 7 (2) ◽  
pp. 171-184 ◽  
Author(s):  
H. Suura ◽  
Y. Mimura ◽  
T. Kimura
Keyword(s):  
Transformation Function ◽  
Analytic Behaviour

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

1977 ◽  
Vol 357 (1690) ◽  
pp. 345-353 ◽  
Keyword(s):  
Gaussian Function ◽  
Gaussian Model ◽  
Radius Of Convergence ◽  
Virial Series ◽  
Imperfect Gas ◽  
Cyclomatic Number ◽  
Analytic Behaviour ◽  
Irreducible Graphs ◽  
Limiting Case ◽  
Third Derivative

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.

Real and virtual processes in quantum electrodynamics

1952 ◽  
Vol 48 (4) ◽  
pp. 640-651 ◽  
Author(s):  
J. Hamilton
Keyword(s):  
Matrix Element ◽  
Power Series ◽  
Quantum Electrodynamics ◽  
Coupling Constant ◽  
Threshold Behaviour ◽  
Physical Processes ◽  
Matrix Formulation ◽  
The Matrix ◽  
Multiple Integrals ◽  
Analytic Behaviour

The S-matrix formulation of quantum electrodynamics, as developed by Feynman (3) and Dyson (1), expresses the matrix element for any process as a power series in the coupling constant, the coefficients of the series being, in general, rather complicated multiple integrals. These integrals contain singularities in their integrands, and in certain circumstances the coincidence of such singularities gives the S-matrix a non-analytic behaviour as a function of, for example, the total energy. The threshold behaviour of the S-matrix in the neighbourhood of energy values at which this phenomenon occurs has been investigated by Eden (2), who shows that the non-analytic behaviour is connected with the possible commencement of new physical processes, such as the creation of a particle.

scholarly journals MASSLESS SCALAR FIELD PROPAGATOR IN A QUANTIZED SPACETIME

2007 ◽  
Vol 16 (01) ◽  
pp. 51-57 ◽  
Author(s):  
V. ELIAS ◽  
T. G. STEELE ◽  
K. TANAKA
Keyword(s):  
Scalar Field ◽  
Spectral Function ◽  
Imaginary Part ◽  
Single Particle ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Point Function ◽  
Particle Propagation ◽  
Free Fields ◽  
Analytic Behaviour

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 coupling between electromagnetic waves and excitation waves in a molecular crystal

1962 ◽  
Vol 270 (1341) ◽  
pp. 285-294 ◽  
Keyword(s):  
Excited State ◽  
Excited States ◽  
Ionic Crystal ◽  
Electromagnetic Waves ◽  
Molecular Crystal ◽  
Life Time ◽  
Energy Bands ◽  
Quantum Mechanical Theory ◽  
Transverse Excitation ◽  
Analytic Behaviour

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.

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