5,359 research outputs found
Isomorphic Hilbert spaces associated with different Complex Contours of the -Symmetric Theory
In this work, we stress the existence of isomorphisms which map complex
contours from the upper half to contours in the lower half of the complex
plane. The metric operator is found to depend on the chosen contour but the
maps connecting different contours are norm-preserving. To elucidate these
features, we parametrized the contour considered in
Phys.Rev.D73:085002 (2006) for the study of wrong sign theory. For the
parametrized contour of the form , we found that there
exists an equivalent Hermitian Hamiltonian provided that is taken to
be real. The equivalent Hamiltonian is -independent but the metric operator
is found to depend on all the parameters , and . Different values of
these parameters generate different metric operators which define different
Hilbert spaces . All these Hilbert spaces are isomorphic to each other even for
parameters values that define contours with ends in two adjacent wedges. As an
example, we showed that the transition amplitudes associated with the contour
are exactly the same as those calculated using the contour
, which is not -Symmetric and has ends in two
adjacent wedges in the complex plane.Comment: In this version, we have added many details and omitted confusing
statements. Also, the way we present the main idea has been changed and added
two figure
New Ansatz for Metric Operator Calculation in Pseudo-Hermitian Field Theory
In this work, a new ansatz is introduced to make the calculations of the
metric operator in Pseudo-Hermitian field theory simpler. The idea is to assume
that the metric operator is not only a functional of the field operators
and its conjugate field but also on the field gradient .
Rather than the locality of the metric operator obtained, the ansatz enables
one to calculate the metric operator just once for all dimensions of the
space-time. We calculated the metric operator of the scalar field
theory up to first order in the coupling. The higher orders can be conjectured
from their corresponding operators in the quantum mechanical case available in
the literature. We assert that, the calculations existing in literature for the
metric operator in field theory are cumbersome and are done case by case
concerning the dimension of space-time in which the theory is investigated.
Moreover, while the resulted metric operator in this work is local, the
existing calculations for the metric operator leads to a non-local one. Indeed,
we expect that the new results introduced in this work will greatly lead to the
progress of the studies in Pseudo-Hermitian field theories where there exist a
lack of such kind of studies in the literature. In fact, with the aid of this
work a rigorous study of a -symmetric Higgs mechanism can be
reached.Comment: In this version, for a more illustrative presentation, we used the
i\phi^3 theory to show that the new ansatz introduced is applicabl
Vacuum Stability of the -Symmetric Scalar Field Theory
In this work, we study the vacuum stability of the classical unstable scalar field potential. Regarding this, we obtained the
effective potential, up to second order in the coupling, for the theory in
and space-time dimensions. We found that the obtained effective
potential is bounded from below, which proves the vacuum stability of the
theory in space-time dimensions higher than the previously studied case.
In our calculations, we used the canonical quantization regime in which one
deals with operators rather than classical functions used in the path integral
formulation. Therefore, the non-Hermiticity of the effective field theory is
obvious. Moreover, the method we employ implements the canonical equal-time
commutation relations and the Heisenberg picture for the operators. Thus, the
metric operator is implemented in the calculations of the transition
amplitudes. Accordingly, the method avoids the very complicated calculations
needed in other methods for the metric operator. To test the accuracy of our
results, we obtained the exponential behavior of the vacuum condensate for
small coupling values, which has been obtained in the literature using other
methods. We assert that this work is interesting, as all the studies in the
literature advocate the stability of the theory at
the quantum mechanical level while our work extends the argument to the level
of field quantization.Comment: 20 pages, 4 figures, appendix added and more details have been added
to
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