Tag Carl M. Bender

Unbounded C-symmetries and their nonuniqueness

Carl M. Bender, Sergii Kuzhel

It is shown that if the C operator for a PT-symmetric Hamiltonian with simple eigenvalues is not unique, then it is unbounded. Apart from the special cases of finite-matrix Hamiltonians and Hamiltonians generated by differential expressions with PT-symmetric point interactions, the usual situation is that the C operator is unbounded. The fact that the C operator is unbounded is significant because, while there is a formal equivalence between a PT-symmetric Hamiltonian and a conventionally Hermitian Hamiltonian in the sense that the two Hamiltonians are isospectral, the Hilbert spaces are inequivalent. This is so because the mapping from one Hilbert space to the other is unbounded. This shows that PT-symmetric quantum theories are mathematically distinct from conventional Hermitian quantum theories.

Quantum Physics (quant-ph); Mathematical Physics (math-ph)

PT phase transition in multidimensional quantum systems

Carl M. Bender, David J. Weir

Non-Hermitian PT-symmetric quantum-mechanical Hamiltonians generally exhibit a phase transition that separates two parametric regions, (i) a region of unbroken PT symmetry in which the eigenvalues are all real, and (ii) a region of broken PT symmetry in which some of the eigenvalues are complex. This transition has recently been observed experimentally in a variety of physical systems. Until now, theoretical studies of the PT phase transition have generally been limited to one-dimensional models. Here, four nontrivial coupled PT-symmetric Hamiltonians, \(H=p^2/2+x^2/2+q^2/2+y^2/2+igx^2y\), \(H=p^2/2+x^2/2+q^2/2+y^2+igx^2y\), \(H=p^2/2+x^2/2+q^2/2+y^2/2+r^2/2+z^2/2+igxyz\), and \(H=p^2/2+x^2/2+q^2/2+y^2+r^2/2+3z^2/2+igxyz\) are examined. Based on extensive numerical studies, this paper conjectures that all four models exhibit a phase transition. The transitions are found to occur at \(g\approx 0.1\), \(g\approx 0.04\), \(g\approx 0.1\), and \(g\approx 0.05\). These results suggest that the PT phase transition is a robust phenomenon not limited to systems having one degree of freedom.

Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph)

Observation of PT phase transition in a simple mechanical system

Carl M. Bender, Bjorn K. Berntson, David Parker, E. Samuel

If a Hamiltonian is PT symmetric, there are two possibilities: Either the eigenvalues are entirely real, in which case the Hamiltonian is said to be in an unbroken-PT-symmetric phase, or else the eigenvalues are partly real and partly complex, in which case the Hamiltonian is said to be in a broken-PT-symmetric phase. As one varies the parameters of the Hamiltonian, one can pass through the phase transition that separates the unbroken and broken phases. This transition has recently been observed in a variety of laboratory experiments. This paper explains the phase transition in a simple and intuitive fashion and then describes an extremely elementary experiment in which the phase transition is easily observed.

Mathematical Physics (math-ph); High Energy Physics – Theory (hep-th); Quantum Physics (quant-ph)

Resolution of Inconsistency in the Double-Scaling Limit

Carl M. Bender, Moshe Moshe, Sarben Sarkar

The conventional double-scaling limit of a quartic quantum field theory is inconsistent because the critical coupling constant is negative. Thus, at the critical coupling the Lagrangian appears to define a quantum theory whose energy is complex. Worse yet, the functional integral for the partition function of the theory does not exist. It is shown that one can avoid these difficulties if one approaches this correlated limit in a PT-symmetric fashion. The partition function is calculated explicitly in the double-scaling limit of an zero-dimensional O(N)-symmetric quartic model.

High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

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

Carl M. Bender, Daniel W. Hook

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

High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Complex Trajectories in a Classical Periodic Potential

Alexander G. Anderson, Carl M. Bender

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 \(x(t+T)=x(t) \pm2 \pi\). 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.

High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Negative-energy PT-symmetric Hamiltonians

Carl M. Bender, Daniel W. Hook, S. P. Klevansky

The non-Hermitian PT-symmetric quantum-mechanical Hamiltonian \(H=p^2+x^2(ix)^\epsilon\) has real, positive, and discrete eigenvalues for all \(\epsilon\geq 0\). These eigenvalues are analytic continuations of the harmonic-oscillator eigenvalues \(E_n=2n+1\) (n=0, 1, 2, 3, …) at \(\epsilon=0\). 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 \(H=p^2+x^2(ix)^\epsilon\) 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, \(\epsilon\) lies in the range \((4N-6)/3<\epsilon<4N-2\). At the low and high ends of this range, the eigenvalues are all infinite. At the special intermediate value \(\epsilon=2N-2\) the eigenvalues are the negatives of those of the conventional Hermitian Hamiltonian \(H=p^2+x^{2N}\). However, when \(\epsilon\neq 2N-2\), there are infinitely many complex eigenvalues. Thus, while the positive-spectrum sector of the Hamiltonian \(H=p^2+x^2(ix)^\epsilon\) has an unbroken PT symmetry (the eigenvalues are all real), the negative-spectrum sector of \(H=p^2+x^2(ix)^\epsilon\) has a broken PT symmetry (only some of the eigenvalues are real).

High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

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

Carl M. Bender, Hugh F. Jones

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 \(V=ig\sin(\alpha x)\). 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 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.

High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Ordinary versus PT-symmetric φ^3 quantum field theory

Carl M. Bender, V. Branchina, Emanuele Messina

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 ig\phi^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 g\phi^3 quantum field theory with those of the PT-symmetric ig\phi^3 quantum field theory. It is shown that while the conventional g\phi^3 theory in d=6 dimensions is asymptotically free, the ig\phi^3 theory is like a g\phi^4 theory in d=4 dimensions; it is energetically stable, perturbatively renormalizable, and trivial.

High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Bound states of PT-symmetric separable potentials

Carl M. Bender, Hugh F. Jones

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

High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)