Carl M. Bender, Mariagiovanna Gianfreda

In 1980 Englert examined the classic problem of the electromagnetic self-force on an oscillating charged particle. His approach, which was based on an earlier idea of Bateman, was to introduce a charge-conjugate particle and to show that the two-particle system is Hamiltonian. Unfortunately, Englert’s model did not solve the problem of runaway modes, and the corresponding quantum theory had ghost states. It is shown here that Englert’s Hamiltonian is PT symmetric, and that the problems with his model arise because the PT symmetry is broken at both the classical and quantum level. However, by allowing the charged particles to interact and by adjusting the coupling parameters to put the model into an unbroken PT-symmetric region, one eliminates the classical runaway modes and obtains a corresponding quantum system that is ghost free.

http://arxiv.org/abs/1409.3828

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

Carl M. Bender, Daniel W. Hook, Nick E. Mavromatos, Sarben Sarkar

Logarithmic time-like Liouville quantum field theory has a generalized PT invariance, where T is the time-reversal operator and P stands for an S-duality reflection of the Liouville field \(\phi\). In Euclidean space the Lagrangian of such a theory, \(L=\frac{1}{2}(\nabla\phi)^2−ig\phi \exp(ia\phi)\), is analyzed using the techniques of PT-symmetric quantum theory. It is shown that L defines an infinite number of unitarily inequivalent sectors of the theory labeled by the integer n. In one-dimensional space (quantum mechanics) the energy spectrum is calculated in the semiclassical limit and the \(m\)th energy level in the \(n\)th sector is given by \(E_{m,n}∼(m+1/2)^2a^2/(16n^2)\).

http://arxiv.org/abs/1408.2432

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

Carl M. Bender, Daniel W. Hook

Complex trajectories for Hamiltonians of the form H=p^n+V(x) are studied. For n=2 time-reversal symmetry prevents trajectories from crossing. However, for n>2 trajectories may indeed cross, and as a result, the complex trajectories for such Hamiltonians have a rich and elaborate structure. In past work on complex classical trajectories it has been observed that turning points act as attractors; they pull on complex trajectories and make them veer towards the turning point. In this paper it is shown that the poles of V(x) have the opposite effect — they deflect and repel trajectories. Moreover, poles shield and screen the effect of turning points.

http://arxiv.org/abs/1402.3852

Mathematical Physics (math-ph)

Bo Peng, Sahin Kaya Ozdemir, Fuchuan Lei, Faraz Monifi, Mariagiovanna Gianfreda, Gui Lu Long, Shanhui Fan, Franco Nori, Carl M. Bender, Lan Yang

Optical systems combining balanced loss and gain profiles provide a unique platform to implement classical analogues of quantum systems described by non-Hermitian parity-time- (PT-) symmetric Hamiltonians and to originate new synthetic materials with novel properties. To date, experimental works on PT-symmetric optical systems have been limited to waveguides in which resonances do not play a role. Here we report the first demonstration of PT-symmetry breaking in optical resonator systems by using two directly coupled on-chip optical whispering-gallery-mode (WGM) microtoroid silica resonators. Gain in one of the resonators is provided by optically pumping Erbium (Er3+) ions embedded in the silica matrix; the other resonator exhibits passive loss. The coupling strength between the resonators is adjusted by using nanopositioning stages to tune their distance. We have observed reciprocal behavior of the PT-symmetric system in the linear regime, as well as a transition to nonreciprocity in the PT symmetry-breaking phase transition due to the significant enhancement of nonlinearity in the broken-symmetry phase. Our results represent a significant advance towards a new generation of synthetic optical systems enabling on-chip manipulation and control of light propagation.

http://arxiv.org/abs/1308.4564

Optics (physics.optics); Materials Science (cond-mat.mtrl-sci); Mathematical Physics (math-ph); Classical Physics (physics.class-ph); Quantum Physics (quant-ph)

Carl M. Bender, Sarben Sarkar

In an earlier paper it was argued that the conventional double-scaling limit of an O(N)-symmetric quartic quantum field theory is inconsistent because the critical coupling constant is negative and thus the integral representing the partition function of the critical theory does not exist. In this earlier paper it was shown that for an O(N)-symmetric quantum field theory in zero-dimensional spacetime one can avoid this difficulty if one replaces the original quartic theory by its PT-symmetric analog. In the current paper an O(N)-symmetric quartic quantum field theory in one-dimensional spacetime [that is, O(N)-symmetric quantum mechanics] is studied using the Schroedinger equation. It is shown that the global PT-symmetric formulation of this differential equation provides a consistent way to perform the double-scaling limit of the O(N)-symmetric anharmonic oscillator. The physical nature of the critical behavior is explained by studying the PT-symmetric quantum theory and the corresponding and equivalent Hermitian quantum theory.

http://arxiv.org/abs/1307.4348

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

Carl M. Bender, Mariagiovanna Gianfreda

The inspiration for this theoretical paper comes from recent experiments on a PT-symmetric system of two coupled optical whispering galleries (optical resonators). The optical system can be modeled as a pair of coupled linear oscillators, one with gain and the other with loss. If the coupled oscillators have a balanced loss and gain, the system is described by a Hamiltonian and the energy is conserved. This theoretical model exhibits two PT transitions depending on the size of the coupling parameter \epsilon. For small \epsilon the PT symmetry is broken and the system is not in equilibrium, but when \epsilon becomes sufficiently large, the system undergoes a transition to an equilibrium phase in which the PT symmetry is unbroken. For very large \(\epsilon\) the system undergoes a second transition and is no longer in equilibrium. The classical and the quantized versions of the system exhibit transitions at exactly the same values of \(\epsilon\).

http://arxiv.org/abs/1305.7107

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

Carl M. Bender, Mariagiovanna Gianfreda

The C operator in PT-symmetric quantum mechanics satisfies a system of three simultaneous algebraic operator equations, \(C^2=1\), \([C,PT]=0\), and \([C,H]=0\). These equations are difficult to solve exactly, so perturbative methods have been used in the past to calculate C. The usual approach has been to express the Hamiltonian as \(H=H_0+\epsilon H_1\), and to seek a solution for C in the form \(C=e^Q P\), where \(Q=Q(q,p)\) is odd in the momentum p, even in the coordinate q, and has a perturbation expansion of the form \(Q=\epsilon Q_1+\epsilon^3 Q_3+\epsilon^5 Q_5+\ldots\). [In previous work it has always been assumed that the coefficients of even powers of $\epsilon$ in this expansion would be absent because their presence would violate the condition that \(Q(p,q)\) is odd in p.] In an earlier paper it was argued that the C operator is not unique because the perturbation coefficient \(Q_1\) is nonunique. Here, the nonuniqueness of C is demonstrated at a more fundamental level: It is shown that the perturbation expansion for Q actually has the more general form \(Q=Q_0+\epsilon Q_1+\epsilon^2 Q_2+\ldots\) in which {\it all} powers and not just odd powers of \(\epsilon\) appear. For the case in which \(H_0\) is the harmonic-oscillator Hamiltonian, \(Q_0\) is calculated exactly and in closed form and it is shown explicitly to be nonunique. The results are verified by using powerful summation procedures based on analytic continuation. It is also shown how to calculate the higher coefficients in the perturbation series for Q.

http://arxiv.org/abs/1302.7047

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

Carl M. Bender, V. Branchina, Emanuele Messina

It was shown recently that a PT-symmetric \(i\phi^3\) quantum field theory in \(6-\epsilon\) dimensions possesses a nontrivial fixed point. The critical behavior of this theory around the fixed point is examined and it is shown that the corresponding phase transition is related to the existence of a nontrivial solution of the gap equation. The theory is studied first in the mean-field approximation and the critical exponents are calculated. Then, it is examined beyond the mean-field approximation by using renormalization-group techniques, and the critical exponents for \(6-\epsilon\) dimensions are calculated to order \(\epsilon\). It is shown that because of its stability the PT-symmetric \(i\phi^3\) theory has a higher predictive power than the conventional \(\phi^3\) theory. A comparison of the \(i\phi^3\) model with the Lee-Yang model is given.

http://arxiv.org/abs/1301.6207

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

Carl M. Bender, Stefan Boettcher

The paper by Bowen, Mancini, Fessatidis, and Murawski (2012 Phys. Scr. {\bf 85}, 065005) demonstrates in a dramatic fashion the serious difficulties that can arise when one rushes to perform numerical studies before understanding the physics and mathematics of the problem at hand and without understanding the limitations of the numerical methods used. Based on their flawed numerical work, the authors conclude that the work of Bender and Boettcher is wrong even though it has been verified at a completely rigorous level. Unfortunately, the numerical procedures performed and described in the paper by Bowen et al are incorrectly applied and wrongly interpreted.

http://arxiv.org/abs/1210.0426

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

A.S. Rodrigues, K. Li, V. Achilleos, P.G. Kevrekidis, D.J. Frantzeskakis, Carl M. Bender

In this work we analyze PT-symmetric double-well potentials based on a two-mode picture. We reduce the problem into a PT-symmetric dimer and illustrate that the latter has effectively two fundamental bifurcations, a pitchfork (symmetry-breaking bifurcation) and a saddle-center one, which is the nonlinear analog of the PT-phase-transition. It is shown that the symmetry breaking leads to ghost states (amounting to growth or decay); although these states are not true solutions of the original continuum problem, the system’s dynamics closely follows them, at least in its metastable evolution. Past the second bifurcation, there are no longer states of the original continuum system. Nevertheless, the solutions can be analytically continued to yield a new pair of branches, which is also identified and dynamically examined. Our explicit analytical results for the dimer are directly compared to the full continuum problem, yielding a good agreement.

http://arxiv.org/abs/1207.1066

Pattern Formation and Solitons (nlin.PS); Quantum Gases (cond-mat.quant-gas); Mathematical Physics (math-ph)