arXiv subjects

Carl M. Bender

Publications and source records attributed to Carl M. Bender.

At least 73 records · Page 4Linked to original sources

arXiv2012

Universal spectral behavior of $x^2(ix)^ε$ potentials

The PT-symmetric Hamiltonian $H=p^2+x^2(ix)^ε$ ($ε$ real) exhibits a phase transition at $ε=0$. When $ε\geq0$, the eigenvalues are all real, positive, discrete, and grow as $ε$ increases. However, when $ε<0$ there are only a finite number of real eigenvalues. As $ε$ approaches -1 from above, the number of real eigenvalues decreases to one, and this eigenvalue becomes infinite at $ε=-1$. In this paper it is shown that these qualitative spectral behaviors are generic and that they are exhibited by the eigenvalues of the general class of Hamiltonians $H^{(2n)}=p^{2n}+x^2(ix)^ε$ ($ε$ real, n=1, 2, 3, ...). The complex classical behaviors of these Hamiltonians are also examined.

hep-th↗

arXiv2012

Complex Trajectories in a Classical Periodic Potential

This paper examines the complex trajectories of a classical particle in the potential V(x)=-cos(x). Almost all the trajectories describe a particle that hops from one well to another in an erratic fashion. However, it is shown analytically that there are two special classes of trajectories x(t) determined only by the energy of the particle and not by the initial position of the particle. The first class consists of periodic trajectories; that is, trajectories that return to their initial position x(0) after some real time T. The second class consists of trajectories for which there exists a real time T such that $x(t+T)=x(t) \pm2 π$. These two classes of classical trajectories are analogous to valence and conduction bands in quantum mechanics, where the quantum particle either remains localized or else tunnels resonantly (conducts) through a crystal lattice. These two special types of trajectories are associated with sets of energies of measure 0. For other energies, it is shown that for long times the average velocity of the particle becomes a fractal-like function of energy.

hep-th↗

arXiv2012

Negative-energy PT-symmetric Hamiltonians

The non-Hermitian PT-symmetric quantum-mechanical Hamiltonian $H=p^2+x^2(ix)^ε$ has real, positive, and discrete eigenvalues for all $ε\geq 0$. These eigenvalues are analytic continuations of the harmonic-oscillator eigenvalues $E_n=2n+1$ (n=0, 1, 2, 3, ...) at $ε=0$. However, the harmonic oscillator also has negative eigenvalues $E_n=-2n-1$ (n=0, 1, 2, 3, ...), and one may ask whether it is equally possible to continue analytically from these eigenvalues. It is shown in this paper that for appropriate PT-symmetric boundary conditions the Hamiltonian $H=p^2+x^2(ix)^ε$ also has real and {\it negative} discrete eigenvalues. The negative eigenvalues fall into classes labeled by the integer N (N=1, 2, 3, ...). For the Nth class of eigenvalues, $ε$ lies in the range $(4N-6)/3<ε<4N-2$. At the low and high ends of this range, the eigenvalues are all infinite. At the special intermediate value $ε=2N-2$ the eigenvalues are the negatives of those of the conventional Hermitian Hamiltonian $H=p^2+x^{2N}$. However, when $ε\neq 2N-2$, there are infinitely many complex eigenvalues. Thus, while the positive-spectrum sector of the Hamiltonian $H=p^2+x^2(ix)^ε$ has an unbroken PT symmetry (the eigenvalues are all real), the negative-spectrum sector of $H=p^2+x^2(ix)^ε$ has a broken PT symmetry (only some of the eigenvalues are real).

hep-th↗

arXiv2012

WKB Analysis of PT-Symmetric Sturm-Liouville problems. II

In a previous paper it was shown that a one-turning-point WKB approximation gives an accurate picture of the spectrum of certain non-Hermitian PT-symmetric Hamiltonians on a finite interval with Dirichlet boundary conditions. Potentials to which this analysis applies include the linear potential $V=igx$ and the sinusoidal potential $V=ig\sin(αx)$. However, the one-turning-point analysis fails to give the full structure of the spectrum for the cubic potential $V=igx^3$, and in particular it fails to reproduce the critical points at which two real eigenvalues merge and become a complex-conjugate pair. The present paper extends the method to cases where the WKB path goes through a {\it pair} of turning points. The extended method gives an extremely accurate approximation to the spectrum of $V=igx^3$, and more generally it works for potentials of the form $V=igx^{2N+1}$. When applied to potentials with half-integral powers of $x$, the method again works well for one sign of the coupling, namely that for which the turning points lie on the first sheet in the lower-half plane.

hep-th↗

arXiv2012

Asymptotic Analysis of the Boltzmann Equation for Dark Matter Relics

This paper presents an asymptotic analysis of the Boltzmann equations (Riccati differential equations) that describe the physics of thermal dark-matter-relic abundances. Two different asymptotic techniques are used, boundary-layer theory, which makes use of asymptotic matching, and the delta expansion, which is a powerful technique for solving nonlinear differential equations. Two different Boltzmann equations are considered. The first is derived from general relativistic considerations and the second arises in dilatonic string cosmology. The global asymptotic analysis presented here is used to find the long-time behavior of the solutions to these equations. In the first case the nature of the so-called freeze-out region and the post-freeze-out behavior is explored. In the second case the effect of the dilaton on cold dark-matter abundances is calculated and it is shown that there is a large-time power-law fall off of the dark-matter abundance. Corrections to the power-law behavior are also calculated.

hep-th↗

arXiv2012

Matrix representation of the time operator

In quantum mechanics the time operator $Θ$ satisfies the commutation relation $[Θ,H]=i$, and thus it may be thought of as being canonically conjugate to the Hamiltonian $H$. The time operator associated with a given Hamiltonian $H$ is not unique because one can replace $Θ$ by $Θ+ Θ_{\rm hom}$, where $Θ_{\rm hom}$ satisfies the homogeneous condition $[Θ_{\rm hom},H]=0$. To study this nonuniqueness the matrix elements of $Θ$ for the harmonic-oscillator Hamiltonian are calculated in the eigenstate basis. This calculation requires the summation of divergent series, and the summation is accomplished by using zeta-summation techniques. It is shown that by including appropriate homogeneous contributions, the matrix elements of $Θ$ simplify dramatically. However, it is still not clear whether there is an optimally simple representation of the time operator.

hep-th↗

arXiv2012

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.

hep-th↗

arXiv2012

Ordinary versus PT-symmetric $ϕ^3$ quantum field theory

A quantum-mechanical theory is PT-symmetric if it is described by a Hamiltonian that commutes with PT, where the operator P performs space reflection and the operator T performs time reversal. A PT-symmetric Hamiltonian often has a parametric region of unbroken PT symmetry in which the energy eigenvalues are all real. There may also be a region of broken PT symmetry in which some of the eigenvalues are complex. These regions are separated by a phase transition that has been repeatedly observed in laboratory experiments. This paper focuses on the properties of a PT-symmetric $igϕ^3$ quantum field theory. This quantum field theory is the analog of the PT-symmetric quantum-mechanical theory described by the Hamiltonian $H=p^2+ix^3$, whose eigenvalues have been rigorously shown to be all real. This paper compares the renormalization-group properties of a conventional Hermitian $gϕ^3$ quantum field theory with those of the PT-symmetric $igϕ^3$ quantum field theory. It is shown that while the conventional $gϕ^3$ theory in $d=6$ dimensions is asymptotically free, the $igϕ^3$ theory is like a $gϕ^4$ theory in $d=4$ dimensions; it is energetically stable, perturbatively renormalizable, and trivial.

hep-th↗

arXiv2011

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.

hep-th↗

arXiv2011

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 $V(x,y)=iε[U(x)U(y)-U(-x)U(-y)]$, where $U(x)$ 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 $ε$ for which the PT symmetry of the Hamiltonian is unbroken and the bound-state energies are real. The critical values of $ε$ that bound this region are calculated.

hep-th↗

arXiv2011

PT symmetry in relativistic quantum mechanics

In nonrelativistic quantum mechanics and in relativistic quantum field theory, time t is a parameter and thus the time-reversal operator T does not actually reverse the sign of t. However, in relativistic quantum mechanics the time coordinate t and the space coordinates x are treated on an equal footing and all are operators. In this paper it is shown how to extend PT symmetry from nonrelativistic to relativistic quantum mechanics by implementing time reversal as an operation that changes the sign of the time coordinate operator t. Some illustrative relativistic quantum-mechanical models are constructed whose associated Hamiltonians are non-Hermitian but PT symmetric, and it is shown that for each such Hamiltonian the energy eigenvalues are all real.

hep-th↗

arXiv2011

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 $T^2=1$) to fermionic systems (systems for which $T^2=-1$). 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 $η^2=0$, $\barη^2=0$, $η\barη+\bar η =α1$, where $\bar{eta}=η^{PT} =PT ηT^{-1}P^{-1}$. It is easy to construct matrix representations for the Grassmann algebra ($α=0$). However, one can only construct matrix representations for the fermionic operator algebra ($α\neq0$) if $α= -1$; a matrix representation does not exist for the conventional value $α=1$.

hep-th↗

arXiv2011

Response to Shalaby's Comment on "Families of Particles with Different Masses in PT-Symmetric Quantum Field Theory"

In a recent Comment [arXiv: 1101.3980] Shalaby criticised our paper "Families of Particles with Different Masses in PT-Symmetric Quantum Field Theory" [arXiv:1002.3253]. On examining his arguments, we find that there are serious flaws at almost every stage of his Comment. In view of space and time considerations, we point out the major flaws that render his arguments invalid. Essentially Shalaby is attempting to obtain our results from a variational principle and to find a physical interpretation of his calculation. The variational procedure that he uses is inapplicable, and his description of the physics is wrong. We thus refute his criticism on all levels.

hep-th↗

arXiv2011

Periodic orbits for classical particles having complex energy

This paper revisits earlier work on complex classical mechanics in which it was argued that when the energy of a classical particle in an analytic potential is real, the particle trajectories are closed and periodic, but that when the energy is complex, the classical trajectories are open. Here it is shown that there is a discrete set of eigencurves in the complex-energy plane for which the particle trajectories are closed and periodic.

math-ph↗

arXiv2010

PT-symmetric quantum state discrimination

Suppose that a system is known to be in one of two quantum states, $|ψ_1 > $ or $|ψ_2 >$. If these states are not orthogonal, then in conventional quantum mechanics it is impossible with one measurement to determine with certainty which state the system is in. However, because a non-Hermitian PT-symmetric Hamiltonian determines the inner product that is appropriate for the Hilbert space of physical states, it is always possible to choose this inner product so that the two states $|ψ_1 > $ and $|ψ_2 > $ are orthogonal. Thus, quantum state discrimination can, in principle, be achieved with a single measurement.

hep-th↗

arXiv2010

Extending PT symmetry from Heisenberg algebra to E2 algebra

The E2 algebra has three elements, J, u, and v, which satisfy the commutation relations [u,J]=iv, [v,J]=-iu, [u,v]=0. We can construct the Hamiltonian H=J^2+gu, where g is a real parameter, from these elements. This Hamiltonian is Hermitian and consequently it has real eigenvalues. However, we can also construct the PT-symmetric and non-Hermitian Hamiltonian H=J^2+igu, where again g is real. As in the case of PT-symmetric Hamiltonians constructed from the elements x and p of the Heisenberg algebra, there are two regions in parameter space for this PT-symmetric Hamiltonian, a region of unbroken PT symmetry in which all the eigenvalues are real and a region of broken PT symmetry in which some of the eigenvalues are complex. The two regions are separated by a critical value of g.

hep-th↗

arXiv2010

Quantum counterpart of spontaneously broken classical PT symmetry

The classical trajectories of a particle governed by the PT-symmetric Hamiltonian $H=p^2+x^2(ix)^ε$ ($ε\geq0$) 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 $ε$ for which the periods of these orbits vary rapidly as functions of $ε$ and that in these regions there are isolated values of $ε$ 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.

quant-ph↗

arXiv2010

Almost zero-dimensional PT-symmetric quantum field theories

In 1992 Bender, Boettcher, and Lipatov proposed in two papers a new and unusual nonperturbative calculational tool in quantum field theory. The objective was to expand the Green's functions of the quantum field theory as Taylor series in powers of the space-time dimension D. In particular, the vacuum energy for a massless ϕ^{2N} (N=1,2,3,...) quantum field theory was studied. The first two Taylor coefficients in this dimensional expansion were calculated {\it exactly} and a set of graphical rules were devised that could be used to calculate approximately the higher coefficients in the series. This approach is mathematically valid and gives accurate results, but it has not been actively pursued and investigated. Subsequently, in 1998 Bender and Boettcher discovered that PT-symmetric quantum-mechanical Hamiltonians of the form H=p^2+x^2(ix)^ε, where ε\geq0, have real spectra. These new kinds of complex non-Dirac-Hermitian Hamiltonians define physically acceptable quantum-mechanical theories. This result in quantum mechanics suggests that the corresponding non-Dirac-Hermitian D-dimensional ϕ^2(iϕ)^εquantum field theories might also have real spectra. To examine this hypothesis, we return to the technique devised in 1992 and in this paper we calculate the first two coefficients in the dimensional expansion of the ground-state energy of this complex non-Dirac-Hermitian quantum field theory. We show that to first order in this dimensional approximation the ground-state energy is indeed real for ε\geq0.

hep-th↗