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PT Symmetry
in Quantum and
Classical Physics
Carl M. Bender
Washington University in St. Louis, USA
With contributions from
Patrick E. Dorey, Clare Dunning, Andreas Fring, Daniel W. Hook,
Hugh F. Jones, Sergii Kuzhel, Géza Lévai, and Roberto Tateo
World Scientific
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Library of Congress Cataloging-in-Publication Data
Names: Bender, Carl M., 1943– author. | Dorey, P. (Patrick), author.
Title: PT symmetry : in quantum and classical physics / by Carl M. Bender
(Washington University in St Louis, USA) ; contributions by Patrick E. Dorey
(Durham University, UK) [and seven others].
Description: Singapore ; Hackensack, NJ : World Scientific Publishing Co. Pte. Ltd., [2018]
Identifiers: LCCN 2018022857| ISBN 9781786345950 (hc ; alk. paper) |
ISBN 1786345951 (hc ; alk. paper) | ISBN 9781786346681 (pbk. ; alk. paper)
Subjects: LCSH: Symmetry (Physics)--Mathematics. | Space and time--Mathematics. |
Quantum theory--Mathematics. | Mechanics--Mathematics. | Kreĭn spaces.
Classification: LCC QC174.17.S9 B46 2018 | DDC 539.7/25--dc23
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Preface
“Doubt is the necessary tool of knowledge.”
—Paul Tillich
The condition that a Hamiltonian be PT symmetric is a generalization
of the condition that it be Hermitian. Hamiltonians that are PT sym-
metric but not Hermitian can describe actual physical systems. These
PT-symmetric systems display a rich array of unexpected features and
behaviors and such systems often do not require great knowledge and so-
phistication to understand. They offer the opportunity to study and ex-
plore new and interesting physical theories at both the theoretical and the
experimental level. Since 1998, research activity on non-Hermitian PT-
symmetric systems has intensified dramatically. As of this writing, over
3500 papers on PT symmetry have been published and PT papers have
beensubmittedtotwodozencategoriesofthearXiv. Therehavebeenwell
over 40 international conferences and symposia devoted to PT symmetry.
The term PT symmetry refers to combined space-time reflection sym-
metry. This term first appeared in [Bender and Boettcher (1998b)], but
some of the ideas behind PT symmetry were foreshadowed in earlier work
by [Dyson (1956); Cardy and Sugar (1975); Brower et al. (1978); Caliceti
et al. (1980); Harms et al. (1980); Scholtz et al. (1992); Hollowood (1992);
Buslaev and Grecchi (1993); Hatano and Nelson (1996)], and others.
My first exposure to a PT-symmetric Hamiltonian came about in the
summerof1992inSaclay. Inthecourseofaninformaldiscussion,D.Bessis
told me that he and Zinn-Justin had considered the quantum-mechanical
analog of the conformal φ3 field theory associated with the Yang-Lee edge
singularity. That quantum-mechanical theory is defined by the peculiar
non-HermitianHamiltonianHˆ =pˆ2+ixˆ3. Thesurpriseformewaslearning
vii
viii PT Symmetry — In Quantum and Classical Physics
that on the basis of numerical work, they believed that some (and perhaps
even all) of the eigenvalues of Hˆ might be real [Bessis and Zinn-Justin
(1992)]. Interestingly, but unknown to us, the numerical reality of the
eigenvalues of Hˆ had already been discovered twelve years earlier in stud-
ies of quantum-mechanical analogs of Reggeon field theory [Harms et al.
(1980)]. Furthermore, in the same year the mathematical properties of
perturbation expansions associated with cubic Hamiltonians had been ex-
amined [Caliceti et al. (1980)].
I would never have worked on PT symmetry had it not be for my
previous research on two subjects. First, in collaboration with Wu, Banks,
Turbiner,andothers,Istudiedtheanalyticcontinuationofeigenvalueprob-
lems;thatis,thebehaviorofeigenvaluesasfunctionsofacomplexcoupling
constant. This work explained the divergence of perturbation expansions
and led to techniques for summing divergent series. Second, in collabo-
ration with Jones, Moshe, and others, I developed the delta expansion, a
perturbative technique for solving nonlinear problems where the perturba-
tion parameter δ measures the degree of nonlinearity of the problem rather
than the strength of the coupling. For example, using the delta expansion
√
to solve the Thomas-Fermi equation y(cid:48)(cid:48)(x)=[y(x)]3/2/ x we consider the
problemy(cid:48)(cid:48)(x)=y(x)[y(x)/x]δ, andtosolveaquarticscalarquantumfield
theory we consider a φ2(φ2)δ field theory.1 We would then seek a pertur-
bationseriesinpowersofδ. Thisunconventionalperturbationprocedureis
easy to perform and yields great numerical accuracy [Bender et al. (1989)].
In 1997 I realized that while the Hamiltonian H = p2 + ix3 is not
Hermitian, it is PT invariant; that is, it remains invariant under x → −x
and i → −i. A possible drawback of a conventional perturbation expan-
sion is that it may violate an invariance (such as gauge invariance) of a
Hamiltonian. However, by following the delta expansion and considering
the Hamiltonian Hˆ = pˆ2 +xˆ2(ixˆ)ε, where ε is real, the PT invariance of
the Hamiltonian Hˆ is preserved for all ε.2 To my surprise, Boettcher and I
foundthatorderbyorderinpowersofε,theeigenvaluesofthisHamiltonian
remain real. This was the beginning of the formal study of PT symmetry.
Initial research on PT-symmetric systems was mathematical. This
research made use of complex-variable theory, differential equations,
and asymptotics. However, since 2009 there has been an avalanche of
1In this work φ2 and not φ is raised to a power δ in order to avoid the appearance of
complexnumberswhenthefieldφisnegative. Amazingly,asexplainedinthisbook,it
isbesttoraiseiφtothepowerδ.
2ThecubicHamiltonianHˆ =pˆ2+ixˆ3 isaspecialcaseofthisHamiltonianforε=1.
Preface ix
beautiful experimental work, mostly but not exclusively in optics. Some
of the pioneering optics work is described in [El-Ganainy et al. (2018)]
and in a special issue of Nature Photonics [Feng et al. (2017a); Pile (2017);
Limonovetal.(2017);Horiuchi(2017);Fengetal.(2017b)]. Intensivestud-
ies have been done on PT-symmetric photonic lattices and graphene [Sza-
meitet al.(2011);Regensburgeret al.(2012);Zhenet al.(2015);Weimann
et al. (2016); Kim et al. (2016); Cerjan et al. (2016); Zhang et al. (2017)],
PT-symmetric lasers, coherent perfect absorbers, and unidirectional in-
visibility [Chong et al. (2011); Baranov et al. (2017); Hsu et al. (2016);
Wong et al. (2016); Jahromi et al. (2017)], PT-symmetric metamaterials
[Rechtsmanet al.(2012);AlaeianandDionne(2014);Alaeianet al.(2016);
Weimann et al. (2016)], PT-symmetric acoustics [Cummer et al. (2016)],
PT-symmetric topological insulators [Fleury et al. (2016)], PT-symmetric
quantum critical phenomena [Ashida et al. (2017)], and PT-symmetric ex-
citons and polaritons [Gao et al. (2015)].
This experimental work has made the theoretical concepts connected
withPT symmetryclearerandeasiertoexplain. Moreover,ithasledtothe
development of new kinds of metamaterials and devices having remarkable
practical applications, such as enhanced sensing [Chen et al. (2017)] and
wireless power transfer [Assawaworrarit et al. (2017)].
To grasp the distinction between PT symmetry and Hermiticity, we
observe that all but one of the axioms of conventional Hermitian quan-
tum theory can be expressed in the language of physics (causality, locality,
Lorentzinvariance,vacuum-statestability,andsoon). However,oneaxiom
is notably different from the others in that it is phrased in the language of
mathematics. This is the requirement that the Hamiltonian be Hermitian.
In elementary terms, this axiom states that a Hamiltonian matrix Hˆ re-
mains invariant under Hermitian conjugation †, Hˆ = Hˆ†, where † denotes
combined matrix transposition and complex conjugation.
While Hermitian conjugation sounds more like a mathematical require-
ment than a physical requirement, the axiom of Hermiticity has significant
physical consequences. It guarantees that the eigenvalues of the Hamilto-
nianareallreal,andthisiscrucialbecauseameasurementoftheenergyof
aphysicalsystemwillreturnoneoftheeigenvaluesoftheHamiltonianthat
describes that system, and such a measurement must have a real outcome.
To show that the eigenvalues of a Hermitian Hamiltonian are real, we
take the Hermitian conjugate of the eigenvalue equation Hˆ|E(cid:105) = E|E(cid:105),
(cid:104)E|Hˆ† =(cid:104)E|E∗,andmultiplythisequationontherightby|E(cid:105). Thisgives
E(cid:104)E|E(cid:105)=E∗(cid:104)E|E(cid:105),
