Norman Dombey

Seventy years of the Klein paradox

The Klein paradox is examined. Its explanation in terms of electron–positron production is reassessed. It is shown that a potential well or barrier in the Dirac equation can produce positron or electron emission spontaneously if the potential is strong enough. The vacuum charge and lifetime of the well/barrier are calculated. If the well is wide enough, a seemingly constant current is emitted. These phenomena are transient whereas the tunnelling first calculated by Klein is time-independent. Furthermore, tunnelling without exponential suppression occurs when an electron is incident on a high barrier, even when it is not high enough to radiate. Klein tunnelling is therefore a property of relativistic wave equations and not necessarily connected to particle emission. The Coulomb potential is investigated in this context: it is shown that a heavy nucleus of sufficiently large Z will bind positrons. Correspondingly, it is expected that as Z increases the Coulomb barrier will become increasingly transparent to positrons. This is an example of Klein tunnelling.

Pion electroproduction and photoproduction including PCAC

Abstract The PCAC constraints on pion electroproduction and photoproduction are considered in order to construct models applicable to these processes at low and high energies. In particular the roles of the nucleon axial vector form factor and the pion electromagnetic form factor are investigated.

Supercriticality and transmission resonances in the dirac equation

It is shown that a Dirac particle of mass m and arbitrarily small momentum will tunnel without reflection through a potential barrier V = U(c)(x) of finite range provided that the potential well V = -U(c)(x) supports a bound state of energy E = -m. This is called a supercritical potential well.

The hydrino and other unlikely states

We discuss the tightly bound (hydrino) solution of the Klein–Gordon equation for the Coulomb potential in 3 dimensions. We show that a similar tightly bound state occurs for the Dirac equation for the Coulomb potential in 2 dimensions. These states are unphysical since they disappear if the nuclear charge distribution is taken to have an arbitrarily small but non-zero radius.

Electromagnetic Properties of the

The contribution of the fermion triangle diagram responsible for the ABJ anomaly to theZZγ vertex is calculated paying particular attention to the symmetries which must be satisfied. Contrary to previous calculations no static electric dipole moment of theZ is found. Two otherP-violating but CP-conserving couplings are demonstrated as is a new anomaly condition.

Z

It is shown that the amplitude for reflection of a Dirac particle with arbitrarily low momentum incident on a potential of finite range is −1 and hence the transmission coefficient T = 0 in general. If, however, the potential supports a half-bound state at momentum k = 0 this result does not hold. In the case of an asymmetric potential the transmission coefficient T will be nonzero whilst for a symmetric potential T = 1. Therefore in some circumstances a Dirac particle of arbitrarily small momentum can tunnel without reflection through a potential barrier.

Boson. 1.

A modified Sommerfeld‐Watson transformation is established for the singular potential V(r) = g2r−4. The positions of the Regge poles and their residues are obtained for a general class of singular potentials and the analyticity properties of the scattering amplitude f(k, cos θ) are discussed.

Low momentum scattering in the Dirac equation

Abstract The special nature of the Coulomb coupling of a charged particle suggests that non-analytic terms are present in the Regge theory of photoproduction. This provides a natural explanation of the sharp forward peaks observed in charged pion photoproduction off nucleons. A high energy theorem for the forward photoproduction of pseudoscalar mesons follows.

ANALYTICITY PROPERTIES OF THE SCATTERING AMPLITUDE FOR SINGULAR POTENTIALS.

Abstract Using the Proca equations, which are appropriate when the photon has a finite mass m, the force between two perfectly conducting parallel slabs, each of width N, and separated by a distance 2L, is calculated. The approach is through the L-dependence of the combined quantum zero-point energies (ZPE) of all the normal modes of the system. The general results are evaluated in a regime with λ = mL ⪡ 1 and ν = mN ⪡ 1. The two leading finite-mass corrections are of relative order λ2 and λ4 log λ; both stem from essentially kinematic corrections to the modes already present in the Maxwell case m = 0, and having a discrete spectrum between the slabs. There are further corrections of relative order λ4 and (for N ⪢ L) λ 4 log ( N L ) , the last stemming from dynamically new (penetrating) modes present only for m ≠ 0, to which even perfect conductors are almost transparent, and which possess a continuous spectrum. The calculation has byproducts which may prove more fruitful than the results themselves. These are: (i) A complete analysis of the Proca normal-mode structure for parallel-plane geometry, the first such complete analysis, as far as it is known, for any system. A special role is played by the component A2 of the vector potential normal to the slabs, which is unique amongst the potentials and fields in that not only A2 but also ∂A 2 ∂z are continuous across the surfaces; this governs the classification of the modes, and effectively reduces the analysis of the penetrating modes to that for a scalar field. (ii) A clearer understanding of the way in which the total ZPE of continuum modes varies with system parameters like L and N. (iii) Requisite for (ii), a statement of Levinsons theorem in one dimension, which for even-parity modes displays unfamiliar features.

Coulomb coupling, gauge invariance and the Regge theory of photoproduction

We consider the Dirac equation in one space dimension in the presence of a symmetric potential well. We connect the scattering phase shifts at E= +m and E= -m to the number of states that have left the positive energy continuum or joined the negative energy continuum, respectively, as the potential is turned on from zero.