10,007 research outputs found
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
PT-symmetric quantum field theory in D dimensions
PT-symmetric quantum mechanics began with a study of the Hamiltonian
. A surprising feature of this non-Hermitian
Hamiltonian is that its eigenvalues are discrete, real, and positive when
. This paper examines the corresponding
quantum-field-theoretic Hamiltonian
in
-dimensional spacetime, where is a pseudoscalar field. It is shown
how to calculate the Green's functions as series in powers of
directly from the Euclidean partition function. Exact finite expressions for
the vacuum energy density, all of the connected -point Green's functions,
and the renormalized mass to order are derived for .
For the one-point Green's function and the renormalized mass are
divergent, but perturbative renormalization can be performed. The remarkable
spectral properties of PT-symmetric quantum mechanics appear to persist in
PT-symmetric quantum field theory.Comment: 8 page
A Class of Exactly-Solvable Eigenvalue Problems
The class of differential-equation eigenvalue problems
() on the interval
can be solved in closed form for all the eigenvalues and
the corresponding eigenfunctions . The eigenvalues are all integers and
the eigenfunctions are all confluent hypergeometric functions. The
eigenfunctions can be rewritten as products of polynomials and functions that
decay exponentially as . For odd the polynomials that are
obtained in this way are new and interesting classes of orthogonal polynomials.
For example, when N=1, the eigenfunctions are orthogonal polynomials in
multiplying Airy functions of . The properties of the polynomials for all
are described in detail.Comment: REVTeX, 16 pages, no figur
PT-Symmetric Representations of Fermionic Algebras
A recent paper by Jones-Smith and Mathur extends PT-symmetric quantum
mechanics from bosonic systems (systems for which ) to fermionic systems
(systems for which ). The current paper shows how the formalism
developed by Jones-Smith and Mathur can be used to construct PT-symmetric
matrix representations for operator algebras of the form ,
, , where
. It is easy to construct matrix
representations for the Grassmann algebra (). However, one can only
construct matrix representations for the fermionic operator algebra
() if ; a matrix representation does not exist for the
conventional value .Comment: 5 pages, 2 figure
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