15 research outputs found
Massive Jackiw-Rebbi Model
In this paper we analyze a generalized Jackiw-Rebbi (J-R) model in which a
massive fermion is coupled to the kink of the model as a
prescribed background field. We solve this massive J-R model exactly and
analytically and obtain the whole spectrum of the fermion, including the bound
and continuum states. The mass term of the fermion makes the potential of the
decoupled second order Schrodinger-like equations asymmetric in a way that
their asymptotic values at two spatial infinities are different. Therefore, we
encounter the unusual problem in which two kinds of continuum states are
possible for the fermion: reflecting and scattering states. We then show the
energies of all the states as a function of the parameters of the kink, i.e.
its value at spatial infinity () and its slope at (). The
graph of the energies as a function of , where the bound state
energies and the two kinds of continuum states are depicted, shows peculiar
features including an energy gap in the form of a triangle where no bound
states exist. That is the zero mode exists only for larger than a
critical value . This is in sharp contrast to the
usual (massless) J-R model where the zero mode and hence the fermion number
for the ground state is ever present. This also makes the origin of
the zero mode very clear: It is formed from the union of the two threshold
bound states at , which is zero in the massless J-R
model.Comment: 10 pages, 3 figure
An Investigation of the Casimir Energy for a Fermion Coupled to the Sine-Gordon Soliton with Parity Decomposition
We consider a fermion chirally coupled to a prescribed pseudoscalar field in
the form of the soliton of the sine-Gordon model and calculate and investigate
the Casimir energy and all of the relevant quantities for each parity channel,
separately. We present and use a simple prescription to construct the
simultaneous eigenstates of the Hamiltonian and parity in the continua from the
scattering states. We also use a prescription we had introduced earlier to
calculate unique expressions for the phase shifts and check their consistency
with both the weak and strong forms of the Levinson theorem. In the graphs of
the total and parity decomposed Casimir energies as a function of the
parameters of the pseudoscalar field distinctive deformations appear whenever a
fermionic bound state energy level with definite parity crosses the line of
zero energy. However, the latter graphs reveal some properties of the system
which cannot be seen from the graph of the total Casimir energy. Finally we
consider a system consisting of a valence fermion in the ground state and find
that the most energetically favorable configuration is the one with a soliton
of winding number one, and this conclusion does not hold for each parity,
separately.Comment: 13 pages, 8 figure
Vacuum Polarization and Casimir Energy of a Dirac Field Induced by a Scalar Potential in One Spatial Dimension
We investigate the vacuum polarization and the Casimir energy of a Dirac
field coupled to a scalar potential in one spatial dimension. Both of these
effects have a common cause which is the distortion of the spectrum due to the
coupling with the background field. Choosing the potential to be a symmetrical
square-well, the problem becomes exactly solvable and we can find the whole
spectrum of the system, analytically. We show that the total number of states
and the total density remain unchanged as compared with the free case, as one
expects. Furthermore, since the positive- and negative-energy eigenstates of
the fermion are fermion-number conjugates of each other and there is no
zero-energy bound state, the total density and the total number of negative and
positive states remain unchanged, separately. Therefore, the vacuum
polarization in this model is zero for any choice of the parameters of the
potential. It is important to note that although the vacuum polarization is
zero due to the symmetries of the model, the Casimir energy of the system is
not zero in general. In the graph of the Casimir energy as a function of the
depth of the well there is a maximum approximately when the bound energy levels
change direction and move back towards their continuum of origin. The Casimir
energy for a fixed value of the depth is a linear function of the width and is
always positive. Moreover, the Casimir energy density (the energy density of
all the negative-energy states) and the energy density of all the
positive-energy states are exactly the mirror images of each other. Finally,
computing the total energy of a valence fermion present in the lowest fermionic
bound state, taking into account the Casimir energy, we find that the lowest
bound state is almost always unstable for the scalar potential.Comment: 16 pages, 7 figure