Explanipedia

Complex phases in quantum mechanics Open

Carl M. Bender, Daniel Hook · 2024

Schrödinger's equation is a local differential equation and boundary conditions are required to determine the solution uniquely. Depending on the choice of boundary conditions, a given Hamiltonian may describe several different physically …

Complex phases in quantum mechanics Open

Carl M. Bender, Daniel Hook · 2024

Hamilton's equations of motion are local differential equations and boundary conditions are required to determine the solution uniquely. Depending on the choice of boundary conditions, a Hamiltonian may thereby describe several different p…

New Classes of Solutions for Euclidean Scalar Field Theories Open

Carl M. Bender, Sarben Sarkar · 2024

This paper presents new classes of exact radial solutions to the nonlinear ordinary differential equation that arises as a saddle-point condition for a Euclidean scalar field theory in D-dimensional spacetime. These solutions are found by …

PT-symmetric quantum mechanics Open

Carl M. Bender, Daniel Hook · 2023

It is generally assumed that a Hamiltonian for a physically acceptable quantum system (one that has a positive-definite spectrum and obeys the requirement of unitarity) must be Hermitian. However, a PT-symmetric Hamiltonian can also define…

Dyson-Schwinger equations in zero dimensions and polynomial approximations Open

Carl M. Bender, Christos Karapoulitidis, S. P. Klevansky · 2023

The Dyson-Schwinger (DS) equations for a quantum field theory in $D$-dimensional space-time are an infinite sequence of coupled integro-differential equations that are satisfied exactly by the Green's functions of the field theory. This se…

New classes of solutions for Euclidean scalar field theories Open

Carl M. Bender, Sarben Sarkar · 2023

This paper presents new classes of exact radial solutions to the nonlinear ordinary differential equation that arises as a saddle-point condition for a Euclidean scalar field theory in $D$-dimensional spacetime. These solutions are found b…

-symmetrictheory Open

Wen-Yuan Ai, Carl M. Bender, Sarben Sarkar · 2022

The scalar field theory with potential $V(\varphi)=\textstyle{\frac{1}{2}} m^2\varphi^2-\textstyle{\frac{1}{4}} g\varphi^4$ ($g>0$) is ill defined as a Hermitian theory but in a non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric framework it…

Underdetermined Dyson-Schwinger equations Open

Carl M. Bender, Christos Karapoulitidis, Sandra Klevansky · 2022

This paper examines the effectiveness of the Dyson-Schwinger (DS) equations as a calculational tool in quantum field theory. The DS equations are an infinite sequence of coupled equations that are satisfied exactly by the connected Green's…

$\mathcal{P}\mathcal{T}$-symmetric $-gφ^4$ theory Open

Wen-Yuan Ai, Carl M. Bender, Sarben Sarkar · 2022

The scalar field theory with potential $V(φ)=\textstyle{\frac{1}{2}} m^2φ^2-\textstyle{\frac{1}{4}} gφ^4$ ($g>0$) is ill defined as a Hermitian theory but in a non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric framework it is well defined, …

Experimentally-realizable $\mathcal{PT}$ phase transitions in reflectionless quantum scattering Open

Micheline B. Soley, Carl M. Bender, A. Douglas Stone · 2022

A class of above-barrier quantum-scattering problems is shown to provide an experimentally-accessible platform for studying $\mathcal{PT}$-symmetric Schrödinger equations that exhibit spontaneous $\mathcal{PT}$ symmetry breaking despite ha…

Towards perturbative renormalization ofquantum field theory Open

Alexander Felski, Carl M. Bender, S. P. Klevansky, Sarben Sarkar · 2021

In a previous paper it was shown how to calculate the ground-state energy density $E$ and the $p$-point Green's functions $G_p(x_1,x_2,...,x_p)$ for the $PT$-symmetric quantum field theory defined by the Hamiltonian density $H=\frac{1}{2}(…

PT symmetry and renormalisation in quantum field theory Open

Carl M. Bender, Alexander Felski, S. P. Klevansky, Sarben Sarkar · 2021

Quantum systems governed by non-Hermitian Hamiltonians with symmetry are special in having real energy eigenvalues bounded below and unitary time evolution. We argue that symmetry may also be important and present at the level of Her…

PT -symmetric classical mechanics Open

Carl M. Bender, Daniel Hook · 2021

This paper reports the results of an ongoing in-depth analysis of the classical trajectories of the class of non-Hermitian -symmetric Hamiltonians H = p 2 + x 2 ( ix ) ε ( ε ⩾ 0). A variety of phenomena, heret…

Fourth Painlevé Equation and $PT$-Symmetric Hamiltonians Open

Carl M. Bender, Javad Komijani · 2021

This paper is an addendum to earlier papers \cite{R1,R2} in which it was shown that the unstable separatrix solutions for Painlevé I and II are determined by $PT$-symmetric Hamiltonians. In this paper unstable separatrix solutions of the f…

$PT$-symmetric classical mechanics Open

Carl M. Bender, Daniel Hook · 2021

This paper reports the results of an ongoing in-depth analysis of the classical trajectories of the class of non-Hermitian $PT$-symmetric Hamiltonians $H=p^2+ x^2(ix)^\varepsilon$ ($\varepsilon\geq0$). A variety of phenomena, heretofore ov…

PT -symmetric quantum field theory Open

Carl M. Bender · 2020

-symmetric quantum theory began with an analysis of the strange-looking non-Hermitian Hamiltonian H = p 2 + x ( ix ) ε . This Hamiltonian is symmetric and the eigenvalues Hamiltonian are discrete, real, and positive when ε ≥ 0. In this …

PT-symmetric potentials having continuous spectra Open

Zichao Wen, Carl M. Bender · 2020

One-dimensional PT-symmetric quantum-mechanical Hamiltonians having continuous spectra are studied. The Hamiltonians considered have the form $H=p^2+V(x)$, where $V(x)$ is odd in $x$, pure imaginary, and vanishes as $|x|\to\infty$. Five PT…

Relativistic -symmetric fermionic theories in and dimensions Open

Alireza Beygi, S. P. Klevansky, Carl M. Bender · 2019

Relativistic PT-symmetric fermionic interacting systems are studied in 1+1\nand 3+1 dimensions. The objective is to include non-Hermitian PT-symmetric\ninteraction terms that give {\\it real} spectra. Such interacting systems could\ndescri…

Nonlinear eigenvalue problems for generalized Painlevé equations Open

Carl M. Bender, Javad Komijani, Qinghai Wang · 2019

Eigenvalue problems for linear differential equations, such as\ntime-independent Schr\\"odinger equations, can be generalized to eigenvalue\nproblems for nonlinear differential equations. In the nonlinear context a\nseparatrix plays the ro…

Operator-valued zeta functions and Fourier analysis Open

Carl M. Bender, Dorje C. Brody · 2019

The Riemann zeta function Ϛ(s) is defined as the infinite sum \n∑^∞_n=1n^-s, which \nconverges when Re s ˃ 1. The Riemann hypothesis asserts that the nontrivial zeros \nof Ϛ(s)

View article: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi mathvariant="script">P</mml:mi><mml:mi mathvariant="script">T</mml:mi></mml:mrow></mml:math>-symmetric quantum field theory in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>D</mml:mi></mml:math> dimensions

-symmetric quantum field theory in dimensions Open

Carl M. Bender, Nima Hassanpour, S. P. Klevansky, Sarben Sarkar · 2018

PT-symmetric quantum mechanics began with a study of the Hamiltonian H=p2+x2(ix)μ. A surprising feature of this non-Hermitian Hamiltonian is that its eigenvalues are discrete, real, and positive when μ≥0. This paper examines the correspond…

View article: Scattering off <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="script">P</mml:mi><mml:mi mathvariant="script">T</mml:mi></mml:mrow></mml:math>-symmetric upside-down potentials

Scattering off -symmetric upside-down potentials Open

Carl M. Bender, Mariagiovanna Gianfreda · 2018

The upside-down $-x^4$, $-x^6$, and $-x^8$ potentials with appropriate\nPT-symmetric boundary conditions have real, positive, and discrete\nquantum-mechanical spectra. This paper proposes a straightforward macroscopic\nquantum-mechanical s…

View article: PT Symmetry

PT Symmetry Open

Carl M. Bender, Patrick Dorey, Clare Dunning, Andreas Fring, Daniel Hook , et al. · 2018

The ideas of PT symmetry were originally introduced in the context of quantum mechanics, but in recent years they have led to rapid developments in the apparently unconnected field of classical optics ...

View article: Asymptotic analysis of the local potential approximation to the Wetterich equation

Asymptotic analysis of the local potential approximation to the Wetterich equation Open

Carl M. Bender, Sarben Sarkar · 2018

This paper reports a study of the nonlinear partial differential equation\nthat arises in the local potential approximation to the Wetterich formulation\nof the functional renormalization group equation. A cut-off-dependent shift of\nthe p…

View article: Two- and four-dimensional representations of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">PT</mml:mi></mml:math>- and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">CPT</mml:mi></mml:math>-symmetric fermionic algebras

Two- and four-dimensional representations of the - and -symmetric fermionic algebras Open

Alireza Beygi, S. P. Klevansky, Carl M. Bender · 2018

Fermionic systems differ from their bosonic counterparts, the main difference with regard to symmetry considerations being that $T^2=-1$ for fermionic systems. In PT-symmetric quantum mechanics an operator has both PT and CPT adjoints. Fer…

View article: Asymptotic analysis on a pseudo-Hermitian Riemann-zeta Hamiltonian

Asymptotic analysis on a pseudo-Hermitian Riemann-zeta Hamiltonian Open

Carl M. Bender, Dorje C. Brody · 2018

The differential-equation eigenvalue problem associated with a recently-introduced Hamiltonian, whose eigenvalues correspond to the zeros of the Riemann zeta function, is analyzed using Fourier and WKB analysis. The Fourier analysis leads …

View article: Physics in the Complex Domain

Physics in the Complex Domain Open

Carl M. Bender · 2018

first_page settings Order Article Reprints Font Type: Arial Georgia Verdana Font Size: Aa Aa Aa Line Spacing:    Column Width:    Background: Open AccessAbstract Physics in the Complex Domain † by Carl M. Bender Physics Department, W…

View article: Winding in non-Hermitian systems

Winding in non-Hermitian systems Open

Stella T. Schindler, Carl M. Bender · 2017

This paper extends the property of interlacing of the zeros of eigenfunctions\nin Hermitian systems to the topological property of winding number in\nnon-Hermitian systems. Just as the number of nodes of each eigenfunction in a\nself-adjoi…

View article: Two-Dimensional Pulse Propagation without Anomalous Dispersion

Two-Dimensional Pulse Propagation without Anomalous Dispersion Open

Carl M. Bender, Francisco J. Rodríguez‐Fortuño, Sarben Sarkar, Anatoly V. Zayats · 2017

Anomalous dispersion is a surprising phenomenon associated with wave propagation in an even number of space dimensions. In particular, wave pulses propagating in two-dimensional space change shape and develop a tail even in the absence of …

View article: Nonlinear eigenvalue problems and<i>PT</i>-symmetric quantum mechanics

Nonlinear eigenvalue problems and<i>PT</i>-symmetric quantum mechanics Open

Carl M. Bender · 2017

Semiclassical (WKB) techniques are commonly used to find the large-energy behavior of the eigenvalues of linear time-independent Schrödinger equations. In this talk we generalize the concept of an eigenvalue problem to nonlinear differenti…

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