Model of supersymmetric quantum field theory with broken parity symmetry

Recently, it was observed that self-interacting scalar quantum field theories having a non-Hermitian interaction term of the form $g(i\phi)^{2+\delta}$, where $\delta$ is a real positive parameter, are physically acceptable in the sense that the energy spectrum is real and bounded below. Such theories possess PT invariance, but they are not symmetric under parity reflection or time reversal separately. This broken parity symmetry is manifested in a nonzero value for $$, even if $\delta$ is an even integer. This paper extends this idea to a two-dimensional supersymmetric quantum field theory whose superpotential is ${\cal S}(\phi)=-ig(i\phi)^{1+\delta}$. The resulting quantum field theory exhibits a broken parity symmetry for all $\delta>0$. However, supersymmetry remains unbroken, which is verified by showing that the ground-state energy density vanishes and that the fermion-boson mass ratio is unity.Comment: 20 pages, REVTeX, 11 postscript figure

WKB Analysis of PT-Symmetric Sturm-Liouville problems

Most studies of PT-symmetric quantum-mechanical Hamiltonians have considered the Schroedinger eigenvalue problem on an infinite domain. This paper examines the consequences of imposing the boundary conditions on a finite domain. As is the case with regular Hermitian Sturm-Liouville problems, the eigenvalues of the PT-symmetric Sturm-Liouville problem grow like $n^2$ for large $n$. However, the novelty is that a PT eigenvalue problem on a finite domain typically exhibits a sequence of critical points at which pairs of eigenvalues cease to be real and become complex conjugates of one another. For the potentials considered here this sequence of critical points is associated with a turning point on the imaginary axis in the complex plane. WKB analysis is used to calculate the asymptotic behaviors of the real eigenvalues and the locations of the critical points. The method turns out to be surprisingly accurate even at low energies.Comment: 11 pages, 8 figure

Quantum tunneling as a classical anomaly

Classical mechanics is a singular theory in that real-energy classical particles can never enter classically forbidden regions. However, if one regulates classical mechanics by allowing the energy E of a particle to be complex, the particle exhibits quantum-like behavior: Complex-energy classical particles can travel between classically allowed regions separated by potential barriers. When Im(E) -> 0, the classical tunneling probabilities persist. Hence, one can interpret quantum tunneling as an anomaly. A numerical comparison of complex classical tunneling probabilities with quantum tunneling probabilities leads to the conjecture that as ReE increases, complex classical tunneling probabilities approach the corresponding quantum probabilities. Thus, this work attempts to generalize the Bohr correspondence principle from classically allowed to classically forbidden regions.Comment: 12 pages, 7 figure

Multiple-Scale Analysis of the Quantum Anharmonic Oscillator

Conventional weak-coupling perturbation theory suffers from problems that arise from resonant coupling of successive orders in the perturbation series. Multiple-scale perturbation theory avoids such problems by implicitly performing an infinite reordering and resummation of the conventional perturbation series. Multiple-scale analysis provides a good description of the classical anharmonic oscillator. Here, it is extended to study the Heisenberg operator equations of motion for the quantum anharmonic oscillator. The analysis yields a system of nonlinear operator differential equations, which is solved exactly. The solution provides an operator mass renormalization of the theory.Comment: 12 pages, Revtex, no figures, available through anonymous ftp from ftp://euclid.tp.ph.ic.ac.uk/papers/ or on WWW at http://euclid.tp.ph.ic.ac.uk/Papers/papers_95-6_.htm

Vector Casimir effect for a D-dimensional sphere

The Casimir energy or stress due to modes in a D-dimensional volume subject to TM (mixed) boundary conditions on a bounding spherical surface is calculated. Both interior and exterior modes are included. Together with earlier results found for scalar modes (TE modes), this gives the Casimir effect for fluctuating ``electromagnetic'' (vector) fields inside and outside a spherical shell. Known results for three dimensions, first found by Boyer, are reproduced. Qualitatively, the results for TM modes are similar to those for scalar modes: Poles occur in the stress at positive even dimensions, and cusps (logarithmic singularities) occur for integer dimensions $D\le1$. Particular attention is given the interesting case of D=2.Comment: 20 pages, 1 figure, REVTe

Level Crossings in Complex Two-Dimensional Potentials

Two-dimensional PT-symmetric quantum-mechanical systems with the complex cubic potential V_{12}=x^2+y^2+igxy^2 and the complex Henon-Heiles potential V_{HH}=x^2+y^2+ig(xy^2-x^3/3) are investigated. Using numerical and perturbative methods, energy spectra are obtained to high levels. Although both potentials respect the PT symmetry, the complex energy eigenvalues appear when level crossing happens between same parity eigenstates.Comment: 9 pages, 4 figures. Submitted as a conference proceeding of PHHQP

Exact PT-Symmetry Is Equivalent to Hermiticity

We show that a quantum system possessing an exact antilinear symmetry, in particular PT-symmetry, is equivalent to a quantum system having a Hermitian Hamiltonian. We construct the unitary operator relating an arbitrary non-Hermitian Hamiltonian with exact PT-symmetry to a Hermitian Hamiltonian. We apply our general results to PT-symmetry in finite-dimensions and give the explicit form of the above-mentioned unitary operator and Hermitian Hamiltonian in two dimensions. Our findings lead to the conjecture that non-Hermitian CPT-symmetric field theories are equivalent to certain nonlocal Hermitian field theories.Comment: Few typos have been corrected and a reference update

Eigenvalue Integro-Differential Equations for Orthogonal Polynomials on the Real Line

The one-dimensional harmonic oscillator wave functions are solutions to a Sturm-Liouville problem posed on the whole real line. This problem generates the Hermite polynomials. However, no other set of orthogonal polynomials can be obtained from a Sturm-Liouville problem on the whole real line. In this paper we show how to characterize an arbitrary set of polynomials orthogonal on $(-\infty,\infty)$ in terms of a system of integro-differential equations of Hartree-Fock type. This system replaces and generalizes the linear differential equation associated with a Sturm-Liouville problem. We demonstrate our results for the special case of Hahn-Meixner polynomials.Comment: 28 pages, Latex, U. Texas at Austin/ Washington University preprin

On the eigenproblems of PT-symmetric oscillators

We consider the non-Hermitian Hamiltonian H= -\frac{d^2}{dx^2}+P(x^2)-(ix)^{2n+1} on the real line, where P(x) is a polynomial of degree at most n \geq 1 with all nonnegative real coefficients (possibly P\equiv 0). It is proved that the eigenvalues \lambda must be in the sector | arg \lambda | \leq \frac{\pi}{2n+3}. Also for the case H=-\frac{d^2}{dx^2}-(ix)^3, we establish a zero-free region of the eigenfunction u and its derivative u^\prime and we find some other interesting properties of eigenfunctions.Comment: 21pages, 9 figure

Calculation of the Hidden Symmetry Operator for a \cP\cT-Symmetric Square Well

It has been shown that a Hamiltonian with an unbroken \cP\cT symmetry also possesses a hidden symmetry that is represented by the linear operator \cC. This symmetry operator \cC guarantees that the Hamiltonian acts on a Hilbert space with an inner product that is both positive definite and conserved in time, thereby ensuring that the Hamiltonian can be used to define a unitary theory of quantum mechanics. In this paper it is shown how to construct the operator \cC for the \cP\cT-symmetric square well using perturbative techniques.Comment: 10 pages, 2 figure