arXiv subjects

Carl M Bender

Publications and source records attributed to Carl M Bender.

5 recordsLinked to original sources

arXiv2021

$\PT$ Symmetry and Renormalisation in Quantum Field Theory

Quantum systems governed by non-Hermitian Hamiltonians with $\PT$ symmetry are special in having real energy eigenvalues bounded below and unitary time evolution. We argue that $\PT$ symmetry may also be important and present at the level of Hermitian quantum field theories because of the process of renormalisation. In some quantum field theories renormalisation leads to $\PT$-symmetric effective Lagrangians. We show how $\PT$ symmetry may allow interpretations that evade ghosts and instabilities present in an interpretation of the theory within a Hermitian framework. From the study of examples $\PT$-symmetric interpretation is naturally built into a path integral formulation of quantum field theory; there is no requirement to calculate explicitly the $\PT$ norm that occurs in Hamiltonian quantum theory. We discuss examples where $\PT$-symmetric field theories emerge from Hermitian field theories due to effects of renormalization. We also consider the effects of renormalization on field theories that are non-Hermitian but $\PT$-symmetric from the start.

hep-th↗

arXiv2018

Asymptotic Analysis of the Local Potential Approximation to the Wetterich Equation

This paper reports a study of the nonlinear partial differential equation that arises in the local potential approximation to the Wetterich formulation of the functional renormalization group equation. A cut-off-dependent shift of the potential in this partial differential equation is performed. This shift allows a perturbative asymptotic treatment of the differential equation for large values of the infrared cut-off. To leading order in perturbation theory the differential equation becomes a heat equation, where the sign of the diffusion constant changes as the space-time dimension $D$ passes through $2$. When $D<2$, one obtains a forward heat equation whose initial-value problem is well-posed. However, for $D>2$ one obtains a backward heat equation whose initial-value problem is ill-posed. For the special case $D=1$ the asymptotic series for cubic and quartic models is extrapolated to the small infrared-cut-off limit by using Padé techniques. The effective potential thus obtained from the partial differential equation is then used in a Schrödinger-equation setting to study the stability of the ground state. For cubic potentials it is found that this Padé procedure distinguishes between a $PT$-symmetric $igϕ^3$ theory and a conventional Hermitian $gϕ^3$ theory ($g$ real). For an $igϕ^3$ theory the effective potential is nonsingular and has a stable ground state but for a conventional $gϕ^3$ theory the effective potential is singular. For a conventional Hermitian $gϕ^4$ theory and a $PT$-symmetric $-gϕ^4$ theory ($g>0$) the results are similar; the effective potentials in both cases are nonsingular and possess stable ground states.

hep-th↗

arXiv2011

Resonances in Extreme Mass-Ratio Inspirals: Asymptotic and Hyperasymptotic Analysis

An expected source of gravitational waves for future detectors in space are the inspirals of small compact objects into much more massive black holes. These sources have the potential to provide a wealth of information about astronomy and fundamental physics. On short timescales the orbit of the small object is approximately geodesic. Generic geodesics for a Kerr black hole spacetime have a complete set of integrals and can be characterized by three frequencies of the motion. Over the course of an inspiral, a typical system will pass through resonances where two of these frequencies become commensurate. The effect of the resonance will be to alter significantly the rate of inspiral for the duration of the resonance. Understanding the impact of these resonances on gravitational wave phasing is important to detect and exploit these signals for astrophysics and fundamental physics. Two differential equations that might describe the passage of an inspiral through such a resonance are investigated. These differ depending on whether it is the phase or the frequency components of a Fourier expansion of the motion that are taken to be continuous through the resonance. Asymptotic and hyperasymptotic analysis are used to find the late-time analytic behavior of the solution for a system that has passed through a resonance. Linearly growing (weak resonances) or linearly decaying (strong resonances) solutions are found depending on the strength of the resonance. In the weak-resonance case, frequency resonances leave an imprint (a resonant memory) on the gravitational frequency evolution. The transition between weak and strong resonances is characterized by a square-root singularity, and as one approaches this transition from above, the solutions to the frequency resonance equation bunch up into families exponentially fast.

gr-qc↗

arXiv2008

Conjecture on the analyticity of PT-symmetric potentials and the reality of their spectra

The spectrum of the Hermitian Hamiltonian $H=p^2+V(x)$ is real and discrete if the potential $V(x)\to\infty$ as $x\to\pm\infty$. However, if $V(x)$ is complex and PT-symmetric, it is conjectured that, except in rare special cases, $V(x)$ must be analytic in order to have a real spectrum. This conjecture is demonstrated by using the potential $V(x)=(ix)^a|x|^b$, where $a,b$ are real.

math-ph↗

arXiv2005

Solvable model of quantum microcanonical states

This letter examines the consequences of a recently proposed modification of the postulate of equal {\it a priori} probability in quantum statistical mechanics. This modification, called the {\it quantum microcanonical postulate} (QMP), asserts that for a system in microcanonical equilibrium all pure quantum states having the same energy expectation value are realised with equal probability. A simple model of a quantum system that obeys the QMP and that has a nondegenerate spectrum with equally spaced energy eigenvalues is studied. This model admits a closed-form expression for the density of states in terms of the energy eigenvalues. It is shown that in the limit as the number of energy levels approaches infinity, the expression for the density of states converges to a $δ$ function centred at the intermediate value $(E_{\rm max}+E_{\rm min})/ 2$ of the energy. Determining this limit requires an elaborate asymptotic study of an infinite sum whose terms alternate in sign.

quant-ph↗