765 research outputs found
Comment on ``New ansatz for metric operator calculation in pseudo-Hermitian field theory''
In a recent Brief Report by Shalaby a new first-order perturbative
calculation of the metric operator for an scalar field theory is
given. It is claimed that the result is an improvement on a previous
calculation by Bender, Brody and Jones because it is local. Unfortunately
Shalaby's calculation is not valid because of sign errors.Comment: 2 pages, no figures. This comment replaces the previous comment on
the Brief Report by Shalaby. In the previous comment we pointed out that
Shalaby's calculation failed in all but 2 space-time dimensions. We have
subsequently found additional errors which mean that the calculation is not
valid even in that cas
Must a Hamiltonian be Hermitian?
A consistent physical theory of quantum mechanics can be built on a complex
Hamiltonian that is not Hermitian but instead satisfies the physical condition
of space-time reflection symmetry (PT symmetry). Thus, there are infinitely
many new Hamiltonians that one can construct that might explain experimental
data. One would think that a quantum theory based on a non-Hermitian
Hamiltonian violates unitarity. However, if PT symmetry is not broken, it is
possible to use a previously unnoticed physical symmetry of the Hamiltonian to
construct an inner product whose associated norm is positive definite. This
construction is general and works for any PT-symmetric Hamiltonian. The
dynamics is governed by unitary time evolution. This formulation does not
conflict with the requirements of conventional quantum mechanics. There are
many possible observable and experimental consequences of extending quantum
mechanics into the complex domain, both in particle physics and in solid state
physics.Comment: Revised version to appear in American Journal of Physic
Faster than Hermitian Quantum Mechanics
Given an initial quantum state |psi_I> and a final quantum state |psi_F> in a
Hilbert space, there exist Hamiltonians H under which |psi_I> evolves into
|psi_F>. Consider the following quantum brachistochrone problem: Subject to the
constraint that the difference between the largest and smallest eigenvalues of
H is held fixed, which H achieves this transformation in the least time tau?
For Hermitian Hamiltonians tau has a nonzero lower bound. However, among
non-Hermitian PT-symmetric Hamiltonians satisfying the same energy constraint,
tau can be made arbitrarily small without violating the time-energy uncertainty
principle. This is because for such Hamiltonians the path from |psi_I> to
|psi_F> can be made short. The mechanism described here is similar to that in
general relativity in which the distance between two space-time points can be
made small if they are connected by a wormhole. This result may have
applications in quantum computing.Comment: 4 page
Bound states of PT-symmetric separable potentials
All of the PT-symmetric potentials that have been studied so far have been
local. In this paper nonlocal PT-symmetric separable potentials of the form
, where is real, are examined.
Two specific models are examined. In each case it is shown that there is a
parametric region of the coupling strength for which the PT symmetry
of the Hamiltonian is unbroken and the bound-state energies are real. The
critical values of that bound this region are calculated.Comment: 10 pages, 3 figure
Quantum counterpart of spontaneously broken classical PT symmetry
The classical trajectories of a particle governed by the PT-symmetric
Hamiltonian () have been studied in
depth. It is known that almost all trajectories that begin at a classical
turning point oscillate periodically between this turning point and the
corresponding PT-symmetric turning point. It is also known that there are
regions in for which the periods of these orbits vary rapidly as
functions of and that in these regions there are isolated values of
for which the classical trajectories exhibit spontaneously broken PT
symmetry. The current paper examines the corresponding quantum-mechanical
systems. The eigenvalues of these quantum systems exhibit characteristic
behaviors that are correlated with those of the associated classical system.Comment: 11 pages, 7 figure
Semiclassical analysis of a complex quartic Hamiltonian
It is necessary to calculate the C operator for the non-Hermitian
PT-symmetric Hamiltonian H=\half p^2+\half\mu^2x^2-\lambda x^4 in order to
demonstrate that H defines a consistent unitary theory of quantum mechanics.
However, the C operator cannot be obtained by using perturbative methods.
Including a small imaginary cubic term gives the Hamiltonian H=\half p^2+\half
\mu^2x^2+igx^3-\lambda x^4, whose C operator can be obtained perturbatively. In
the semiclassical limit all terms in the perturbation series can be calculated
in closed form and the perturbation series can be summed exactly. The result is
a closed-form expression for C having a nontrivial dependence on the dynamical
variables x and p and on the parameter \lambda.Comment: 4 page
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