425 research outputs found
Introduction to PT-Symmetric Quantum Theory
In most introductory courses on quantum mechanics one is taught that the
Hamiltonian operator must be Hermitian in order that the energy levels be real
and that the theory be unitary (probability conserving). To express the
Hermiticity of a Hamiltonian, one writes , where the symbol
denotes the usual Dirac Hermitian conjugation; that is, transpose and
complex conjugate. In the past few years it has been recognized that the
requirement of Hermiticity, which is often stated as an axiom of quantum
mechanics, may be replaced by the less mathematical and more physical
requirement of space-time reflection symmetry (PT symmetry) without losing any
of the essential physical features of quantum mechanics. Theories defined by
non-Hermitian PT-symmetric Hamiltonians exhibit strange and unexpected
properties at the classical as well as at the quantum level. This paper
explains how the requirement of Hermiticity can be evaded and discusses the
properties of some non-Hermitian PT-symmetric quantum theories
Faster than Hermitian Time Evolution
For any pair of quantum states, an initial state |I> and a final quantum
state |F>, in a Hilbert space, there are many Hamiltonians H under which |I>
evolves into |F>. Let us impose the constraint that the difference between the
largest and smallest eigenvalues of H, E_max and E_min, is held fixed. We can
then determine the Hamiltonian H that satisfies this constraint and achieves
the transformation from the initial state to the final state in the least
possible 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. The minimum value of \tau can be made
arbitrarily small because for PT-symmetric Hamiltonians the path from the
vector |I> to the vector |F>, as measured using the Hilbert-space metric
appropriate for this theory, can be made arbitrarily 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: This is a contribution to the Proc. of the Seventh International
Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007,
Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
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