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Carl M. Bender

Publications and source records attributed to Carl M. Bender.

At least 109 records · Page 6Linked to original sources
arXiv2007

Does the complex deformation of the Riemann equation exhibit shocks?

The Riemann equation $u_t+uu_x=0$, which describes a one-dimensional accelerationless perfect fluid, possesses solutions that typically develop shocks in a finite time. This equation is $\cP\cT$ symmetric. A one-parameter $\cP\cT$-invariant complex deformation of this equation, $u_t-iu(iu_x)^ε= 0$ ($ε$ real), is solved exactly using the method of characteristic strips, and it is shown that for real initial conditions, shocks cannot develop unless $ε$ is an odd integer.

hep-th↗
arXiv2007

Complexified Dynamical Systems

Many dynamical systems, such as the Lotka-Volterra predator-prey model and the Euler equations for the free rotation of a rigid body, are PT symmetric. The standard and well-known real solutions to such dynamical systems constitute an infinitessimal subclass of the full set of complex solutions. This paper examines a subset of the complex solutions that contains the real solutions, namely, those having PT symmetry. The condition of PT symmetry selects out complex solutions that are periodic.

hep-th↗
arXiv2007

No-ghost theorem for the fourth-order derivative Pais-Uhlenbeck oscillator model

Contrary to common belief, it is shown that theories whose field equations are higher than second order in derivatives need not be stricken with ghosts. In particular, the prototypical fourth-order derivative Pais-Uhlenbeck oscillator model is shown to be free of states of negative energy or negative norm. When correctly formulated (as a $\cP\cT$ symmetric theory), the theory determines its own Hilbert space and associated positive-definite inner product. In this Hilbert space the model is found to be a fully acceptable quantum-mechanical theory that exhibits unitary time evolution.

hep-th↗
arXiv2007

Geometry of PT-symmetric quantum mechanics

Recently, much research has been carried out on Hamiltonians that are not Hermitian but are symmetric under space-time reflection, that is, Hamiltonians that exhibit PT symmetry. Investigations of the Sturm-Liouville eigenvalue problem associated with such Hamiltonians have shown that in many cases the entire energy spectrum is real and positive and that the eigenfunctions form an orthogonal and complete basis. Furthermore, the quantum theories determined by such Hamiltonians have been shown to be consistent in the sense that the probabilities are positive and the dynamical trajectories are unitary. However, the geometrical structures that underlie quantum theories formulated in terms of such Hamiltonians have hitherto not been fully understood. This paper studies in detail the geometric properties of a Hilbert space endowed with a parity structure and analyses the characteristics of a PT-symmetric Hamiltonian and its eigenstates. A canonical relationship between a PT-symmetric operator and a Hermitian operator is established. It is shown that the quadratic form corresponding to the parity operator, in particular, gives rise to a natural partition of the Hilbert space into two halves corresponding to states having positive and negative PT norm. The indefiniteness of the norm can be circumvented by introducing a symmetry operator C that defines a positive definite inner product by means of a CPT conjugation operation.

hep-th↗
arXiv2007

Making Sense of Non-Hermitian Hamiltonians

The Hamiltonian H specifies the energy levels and time evolution of a quantum theory. A standard axiom of quantum mechanics requires that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that time evolution is unitary (probability-preserving). This paper describes an alternative formulation of quantum mechanics in which the mathematical axiom of Hermiticity (transpose + complex conjugate) is replaced by the physically transparent condition of space-time reflection (PT) symmetry. If H has an unbroken PT symmetry, then the spectrum is real. Examples of PT-symmetric non-Hermitian quantum-mechanical Hamiltonians are H=p^2+ix^3 and H=p^2-x^4. Amazingly, the energy levels of these Hamiltonians are all real and positive! In general, if H has an unbroken PT symmetry, then it has another symmetry represented by a linear operator C. Using C, one can construct a time-independent inner product with a positive-definite norm. Thus, PT-symmetric Hamiltonians describe a new class of complex quantum theories having positive probabilities and unitary time evolution. The Lee Model is an example of a PT-symmetric Hamiltonian. The renormalized Lee-model Hamiltonian has a negative-norm "ghost" state because renormalization causes the Hamiltonian to become non-Hermitian. For the past 50 years there have been many attempts to find a physical interpretation for the ghost, but all such attempts failed. Our interpretation of the ghost is simply that the non-Hermitian Lee Model Hamiltonian is PT-symmetric. The C operator for the Lee Model is calculated exactly and in closed form and the ghost is shown to be a physical state having a positive norm. The ideas of PT symmetry are illustrated by using many quantum-mechanical and quantum-field-theoretic models.

hep-th↗
arXiv2007

Spontaneous Breaking of Classical PT Symmetry

The classical trajectories of the family of complex PT-symmetric Hamiltonians $H=p^2+x^2(ix)^ε$ ($ε\geq0$) form closed orbits. All such complex orbits that have been studied in the past are PT symmetric (left-right symmetric). The periods of these orbits exhibit an unusual dependence on the parameter $ε$. There are regions in $ε$ of smooth behavior interspersed with regions of rapid variation. It is demonstrated that the onset of rapid variation is associated with strange new kinds of classical trajectories that have never been seen previously. These rare kinds of trajectories are not PT symmetric and occur only for special rational values of $ε$.

hep-th↗
arXiv2007

Optimal Shape of a Blob

This paper presents the solution to the following optimization problem: What is the shape of the two-dimensional region that minimizes the average L_p distance between all pairs of points if the area of this region is held fixed? [The L_p distance between two points ${\bf x}=(x_1,x_2)$ and ${\bf y}=(y_1,y_2)$ in $\Re^2$ is $(|x_1-y_1|^p+|x_2-y_2|^p)^{1/p}$.] Variational techniques are used to show that the boundary curve of the optimal region satisfies a nonlinear integral equation. The special case p=2 is elementary and for this case the integral equation reduces to a differential equation whose solution is a circle. Two nontrivial special cases, p=1 and p=\infty, have already been examined in the literature. For these two cases the integral equation reduces to nonlinear second-order differential equations, one of which contains a quadratic nonlinearity and the other a cubic nonlinearity.

math-ph↗
arXiv2006

Nonlinear Integral-Equation Formulation of Orthogonal Polynomials

The nonlinear integral equation P(x)=\int_alpha^beta dy w(y) P(y) P(x+y) is investigated. It is shown that for a given function w(x) the equation admits an infinite set of polynomial solutions P(x). For polynomial solutions, this nonlinear integral equation reduces to a finite set of coupled linear algebraic equations for the coefficients of the polynomials. Interestingly, the set of polynomial solutions is orthogonal with respect to the measure x w(x). The nonlinear integral equation can be used to specify all orthogonal polynomials in a simple and compact way. This integral equation provides a natural vehicle for extending the theory of orthogonal polynomials into the complex domain. Generalizations of the integral equation are discussed.

math-ph↗
arXiv2006

PT-Symmetric Extension of the Korteweg-de Vries Equation

The Korteweg-de Vries equation u_t+uu_x+u_{xxx}=0 is PT symmetric (invariant under space-time reflection). Therefore, it can be generalized and extended into the complex domain in such a way as to preserve the PT symmetry. The result is the family of complex nonlinear wave equations u_t-iu(i u_x)^epsilon+u_{xxx}=0, where epsilon is real. The features of these equations are discussed. Special attention is given to the epsilon=3 equation, for which conservation laws are derived and solitary waves are investigated.

math-ph↗
arXiv2006

Complex Trajectories of a Simple Pendulum

The motion of a classical pendulum in a gravitational field of strength g is explored. The complex trajectories as well as the real ones are determined. If g is taken to be imaginary, the Hamiltonian that describes the pendulum becomes PT-symmetric. The classical motion for this PT-symmetric Hamiltonian is examined in detail. The complex motion of this pendulum in the presence of an external periodic forcing term is also studied.

math-ph↗
arXiv2006

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.

quant-ph↗
arXiv2006

Equivalence of a Complex $\cP\cT$-Symmetric Quartic Hamiltonian and a Hermitian Quartic Hamiltonian with an Anomaly

In a recent paper Jones and Mateo used operator techniques to show that the non-Hermitian $\cP\cT$-symmetric wrong-sign quartic Hamiltonian $H=\half p^2-gx^4$ has the same spectrum as the conventional Hermitian Hamiltonian $\tilde H=\half p^2+4g x^4-\sqrt{2g} x$. Here, this equivalence is demonstrated very simply by means of differential-equation techniques and, more importantly, by means of functional-integration techniques. It is shown that the linear term in the Hermitian Hamiltonian is anomalous; that is, this linear term has no classical analog. The anomaly arises because of the broken parity symmetry of the original non-Hermitian $\cP\cT$-symmetric Hamiltonian. This anomaly in the Hermitian form of a $\cP\cT$-symmetric quartic Hamiltonian is unchanged if a harmonic term is introduced into $H$. When there is a harmonic term, an immediate physical consequence of the anomaly is the appearance of bound states; if there were no anomaly term, there would be no bound states. Possible extensions of this work to $-ϕ^4$ quantum field theory in higher-dimensional space-time are discussed.

hep-th↗
arXiv2006

Combinatorics and field theory

For any given sequence of integers there exists a quantum field theory whose Feynman rules produce that sequence. An example is illustrated for the Stirling numbers. The method employed here offers a new direction in combinatorics and graph theory.

quant-ph↗
arXiv2006

Classical Trajectories for Complex Hamiltonians

It has been found that complex non-Hermitian quantum-mechanical Hamiltonians may have entirely real spectra and generate unitary time evolution if they possess an unbroken $\cP\cT$ symmetry. A well-studied class of such Hamiltonians is $H= p^2+x^2(ix)^ε$ ($ε\geq0$). This paper examines the underlying classical theory. Specifically, it explores the possible trajectories of a classical particle that is governed by this class of Hamiltonians. These trajectories exhibit an extraordinarily rich and elaborate structure that depends sensitively on the value of the parameter $ε$ and on the initial conditions. A system for classifying complex orbits is presented.

math-ph↗
arXiv2006

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.

quant-ph↗
arXiv2005

PT-Symmetric Versus Hermitian Formulations of Quantum Mechanics

A non-Hermitian Hamiltonian that has an unbroken PT symmetry can be converted by means of a similarity transformation to a physically equivalent Hermitian Hamiltonian. This raises the following question: In which form of the quantum theory, the non-Hermitian or the Hermitian one, is it easier to perform calculations? This paper compares both forms of a non-Hermitian $ix^3$ quantum-mechanical Hamiltonian and demonstrates that it is much harder to perform calculations in the Hermitian theory because the perturbation series for the Hermitian Hamiltonian is constructed from divergent Feynman graphs. For the Hermitian version of the theory, dimensional continuation is used to regulate the divergent graphs that contribute to the ground-state energy and the one-point Green's function. The results that are obtained are identical to those found much more simply and without divergences in the non-Hermitian PT-symmetric Hamiltonian. The $\mathcal{O}(g^4)$ contribution to the ground-state energy of the Hermitian version of the theory involves graphs with overlapping divergences, and these graphs are extremely difficult to regulate. In contrast, the graphs for the non-Hermitian version of the theory are finite to all orders and they are very easy to evaluate.

hep-th↗
arXiv2005

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+\halfμ^2x^2-λx^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 μ^2x^2+igx^3-λx^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 λ.

quant-ph↗