Volume 38, Issue 1
January 1992, pages 1-90
pp 1- January 1992 Rapid Communication
Transition from order to chaos in SU(2) Yang-Mills-Higgs system
Time-dependent spherically symmetricSU(2) Yang-Mills-Higgs system is shown to be chaotic near the ’t Hooft-Polyakov monopole solution by calculating the maximal Lyapunov exponents. A phase transition like behaviour from order to chaos is observed as a parameter depending on the self interaction constant of scalar fields increases.
pp 1-10 January 1992
Relation between Green’s functions for different potentials
S S Vasan M Seetharaman K Raghunathan
A general relation between the energy-dependent Green’s functions for different potentials is derived in a simple and direct manner. This interesting connection enables the eigenstates of one physical system to be deduced from those of a related system. The derivation is based on the Schrödinger equation and provides an independent justification for the technique of path-dependent time transformation used in path integration.
pp 11-20 January 1992
BRST invariance and the conical pendulum
The Hamiltonian formulation of the BRST method for quantizing constrained systems developed recently by Nemeschanskyet al is applied to the well-known problem of the conical pendulum in classical mechanics. The similarity of the system to a gauge theory wherein the two constraints serve as generators of local Abelian gauge transformations is also pointed out. The definition of the physical states of the system as a gauge theory and also as a BRST invariant theory is then discussed in some detail.
pp 21-29 January 1992
The general dynamical equations for perfect fluid filled spheres with an outward flux of photons are derived. The vital role played by the energy density of the free gravitational field in accelerating photon production has been emphasized. It is pointed out that even when the material energy density is finite, the energy density of the free gravitational field can take infinitely large values resulting in vanishing surface area of the star. A generalized Schwarzschild interior solution with conformally flat geometry but with photon emission has been obtained. It is pointed out that the interior conformal coordinate system bears a strong resemblance to the exterior Krushkal coordinates. It is shown that for spherical star the invariant velocity of the fluid particles, falling towards the centre, is proportional to its radius suggesting that the outer envelopes collapse at a faster rate than the core part. It is shown that the interior radiating solution can be matched with generalized Schwarzchild exterior solution.
pp 31-35 January 1992
Energy and momentum in Vaidya spacetime
The components of the energy-momentum pseudotensors of Einstein, Tolman, Landau and Lifshitz, and Møller are evaluated for the Vaidya radiating spacetime. These pseudotensors are found to be traceless for this spacetime. The pseudotensors of Einstein and Tolman give exactly same result for all their components. Unlike in the case of the Kerr-Newman field, the pseudotensor of Møller gives the same energy as given by that of the Einstein, Tolman or Landau and Lifshitz.
pp 37-49 January 1992
Field localization and particle confinement effects on pair creation by Schwinger’s mechanism
Solutions of the Dirac equation in the presence of a static uniform electric fieldɛ in thez-direction and a linear confining potentialAz, are obtained. Generalized reflection and transmission coefficients are derived for such divergent potentials forɛ >A/e. The eigenspectrum and corresponding localized eigenfunctions forɛ <A/e are obtained from the reflection coefficient and the continuum solutions respectively. The rate for the electric field to decay into pairs is derived from the transmission coefficient. Neglecting nonabelian effects in quantum chromodynamics we identify the fieldɛ with a colour electric field and the produced particles with a quark and an antiquark. By considering a cylindrical geometry, we thus obtain a generalization of Schwinger’s formula, for the fieldɛ in a finite spatial region with the quark (antiquark) being confined in thez direction by the linear potentialAz and in the perpendicular direction by the MIT bag boundary condition. The result is used to qualitatively study Schwinger’s mechanism of quark-gluon plasma (QGP) formation in ultrarelativistic heavy ion collisions. It is found that the critical strength of the field required to create$$q\bar q$$ pairs is enhanced,ɛ_{c}(A) >ɛ_{c}(A = 0). The rate of pair creation for constantɛ, decreases for non-zeroA, implying longer QGP formation times. Because ofɛ_{c}(A) >ɛ_{c}(0), QGP is predicted to be formed in the early stages of the nuclear collision. The finite size effects and the MIT bag boundary condition effects on QGP formation are also discussed.
pp 51-75 January 1992
Bound state solutions of the two-electron Dirac-Coulomb equation
We present a variational method for solving the two-electron Dirac-Coulomb equation. When the expectation value of the Dirac-Coulomb Hamiltonian is made stationary for all possible variations of the different components of a well-behaved trial function one obtains solutions representative of the physical bound state wave functions. The ground state wave function is derived from the application of a minimax principle. Since the trial function remains well-behaved, the method remains safe from the twin demons of variational collapse and continuum dissolution.
The ground state wave function thus derived can be interpreted as a linear combination of different configurations. In particular, the admixing of intermediate states having one (two) electron(s) deexcited to a negative-energy orbital (orbitals) contributes a second-order level shiftE_{0−}^{(2)} which can be identified with the second-order shift due to the Pauli blocking of the production of one (or two) virtual electron-positron pair(s). Thus the minimax solution corresponds to the renormalized ground state in quantum electrodynamics, with deexcitations to negative-energy orbitals taking the place of the avoidance of virtual pairs.
If one extends the relativistic configuration interaction (RCI) treatment by additionally including negative-energy and mixed-energyeigenvectors of the Dirac-Hartree-Fock hamiltonian matrix in the two-electron basis, the calculated energy will be shifted from the conventional RCI value by an amount that is much smaller thanE_{0−}^{(2)}. For two-electron atoms, we have derived expressions for the all-spinor limit (δE) and thes-spinor limit (δE_{s}) of this shift in leading orders. The all-spinor limit (δE) is of orderα^{4}Z^{4 1/3} whereas thes-spinor limit (δE_{s}) is of orderα^{4}Z^{3 2/3}. leading components are related to the 1-pair component ofE_{0−}^{(2)} in a simple way, and the relationships offer the possibility of computing energy due to virtual pairs. Numerical results are discussed.
pp 77-90 January 1992
Effect of distortion in electron impact excitation in Coulomb-projected Born approximation
We have calculated total and differential cross-sections for 1s →ns (n = 2, 3, 4) electron impact excitation of hydrogen and hydrogenic ions at various energies in Coulomb-projected Born approximation. Distortion due to static interactions, target polarization and exchange effects has been incorporated in the initial channel. The present calculations have been compared with other theoretical and experimental results.
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