Quantum Mechanics Based on Non-Hermitian Hamiltonians (Recent Trends in Exponential Asymptotics)

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64 Quantum

Mechanics Based on Non-Hermitian

Hamiltonians

Carl M. Bender*

Blackett Laboratory, Imperial College, London SW7 2BZ,

UK

October 16,

2004

Abstract

Aphysicaltheory of quantummechanicscanbebasedonacomplexHamiltonian that is

not Hermitian butinsteadsatisfies the physical condition ofspace-timereflectionsymmetry

($P\mathcal{T}$symmetry). Thus,there areinfinitelymany newHamiltonians that one can construct

that might explain experimental data. One would think that anon-HermitianHamiltonian

wouldgiveaquantumtheorythat violatesunitarity. However,if$P\mathcal{T}$symmetry is not broken,

it is possible to use aphysical symmetryof the Hamiltonianto constructaninner product

whose associated norm is positivedefinite. This construction is general and worksfor any

$P\mathcal{T}$-symmetric Hamiltonian. The dynamics is governed by

unitary time evolution. This

formulation does not conflict with the requirements of conventional quantum mechanics.

There aremany possible observable and experimentalconsequences ofextending quantum

mechanics into the complex domain,both in particle physicsandinsolid statephysics.

1 Introduction

In this paper wediscuss analternative astandard axiom ofquantum mechanics; namely, that

theHamiltonian$H$, which incorporates thesymm etriesandspecifiesthedynamicsofaquantum

theory, must be Hermitian: $H=H\dagger$

.

_{(The symbol \dagger represents} _{Dirac Hermitian conjugation;}

that is, transposeandcomplex conjugate.) It iscommonly believed that the Hamiltonian must

be Herm itian in orderto

ensure

that theenergyspectrum (the eigenvaluesof theHamiltonian)is

real and that the time evolution ofthetheoryisunitary (probabilityis conservedintime). This

axiom is sufficient to guarantee these desired properties, but

we

argue that it is not necessary.

Webelieve that the condition of Hermiticityis

a

mathematicalrequirement whosephysicalbasis

is somewhat obscure. We demonstrate that the more physical alternative axiom of space-time

reflectionsymmetry ($P\mathcal{T}$symmetry),$H=H^{P\mathcal{T}}$, allows for the possibility of

non-Hermitianand

complexHamiltonians but still leads to a consistent theory of quantummechanics.

We also showthat because$P\mathcal{T}$symmetry isanalternative toHermiticity

itis

now

possible to

construct infinitely many newHarniltonians that wouldhavebeen rejectedin thepast because

they are not Hermitian. An example of such

a

Hamiltonian is $H=p^{2}+ix^{3}$

.

It _{should be}

emphasized that we do not regard Hermiticity

as

wrong. Rather, $P\mathcal{T}$ symmetry offers the

possibility of studying

new

and interestingquantumtheories.

Letusrecallthepropertiesofthespacereflection(parity) operator$P$and the time-reflection

operator$\mathcal{T}$. Theparity operator_$P$

is linear and has the effect$parrow-p$and$xarrow-x$

.

The

time-reversal operator $\mathcal{T}$ is antilinear and has the

effect $parrow– p,$ $\arrow x$, and $\mathrm{i}arrow-\mathrm{i}$

.

Notethat $\mathcal{T}$

’Permanent address: Department ofPhysics, Washington University, St. Louis,MO 63130,USA.

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changesthlesign of2 because, like theparityoperator, itpreservesthefundamentalcommutation

relationof quantummechanics, $[x,p]=\mathrm{i}$, known asthe Heisenberg

algebra.l

It iseasy to construct infinitely many Hamiltonians that arenot Hermitian but do possess

$P\mathcal{T}$symmetry. For example, consider the one-parameter familyofHamiltonians

$H=p^{2}+x^{2}(\mathrm{i}x)^{\epsilon}$ ($\epsilon$real). (1)

While $H$ in (1) is not symm etric under $P$ or $\mathcal{T}$ separately, it is invariant under their

com-binedoperation. We say that such Hamiltonians possess space time reflection symmetry. Other

Ham iltonians having $P\mathcal{T}$ symmetryare$H=p^{2}+x^{4}(\mathrm{i}x)^{\epsilon},$ $H=p^{2}+x^{6}(\mathrm{i}x)^{\epsilon}$, andso on $[2].2$

The$P\mathcal{T}$-symmetric Hamiltonians consideredhere, which for simplicity

are

also symmetric,

is larger thanandincludes real symmetric Hermitians because anyreal symmetric Hamiltonian

is automatically $P\mathcal{T}$-symmetric. For example, consider the real symm etric Hamiltonian $H=$

$p^{2}+x^{2}+2x$

.

This Hamiltonianis time reversalsymmetric, but according to the usual definition

of space reflection for which $\arrow-\,$ this Hamiltoniam appears not to have $P\mathcal{T}$ symmetry.

However, the parity operator is defined only up to unitary equivalence. In this example, the

Hamiltonian has the form $H=p^{2}+(x+1)^{2}$ - 1 and it is evident that $H$ is $P\mathcal{T}$ symmetric,

provided that the parity operator perform $\mathrm{s}$ a space reflection about the point $x=-1$ rather

than$x=0$. See Ref. [1] for theconstruction of the relevant parityoperator.

In 1998 it was discovered that with properly defined boundary conditions thespectrum of

the Hamiltonian $H$ in (1) is real andpositive when $\epsilon\geq 0[3]$

.

The spectrum is partly real and

partly complex when $\epsilon<0$

.

The eigenvalues have been computed numerically to very high

precision, and therealeigenvalues

are

plotted

as

functions of$\epsilon$ in Fig. 1.

We say that the $P\mathcal{T}$ symmetry ofaHam iltonian $H$ is unbroken if allofthe eigenfunctions

of$H$

are

simultaneously eigenfunctions of$P\mathcal{T}^{3}$. It iseasy to show that if the $P\mathcal{T}$symmetry of

a Ham iltonian $H$ is unbroken, then the spectrumof$H$ isreal. The proofisshort and goes as

folows: AssumethataHamiltonian$H$possesses PTsymmetry (that is,that $H$

comm

utes with

thePToperator), and that if$\phi$isaneigenstateof$H$with eigenvalue$E$, then it issimultaneously

an

eigenstate of$P\mathcal{T}$with eigenvalue $\lambda$:

$H\phi=E\phi$ and 7’7 $$=\lambda\phi$

.

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We beginby showing that the eigenvalue A is

a

pure phase. Multiplying$P\mathcal{T}\phi=\lambda\phi$on the

left by $P\mathcal{T}$and usingthe fact that $P$ and $\mathcal{T}$commute and that $P^{2}=\mathcal{P}=1$ weconclude that

$\phi=\lambda^{*}\lambda\phi$ and thus $\lambda=e^{i\alpha}$ for

some

real $\alpha$

.

Next,

we

introduce the convention that is used

throughout this paper. Without loss of generalitywereplacethe eigenstate$\phi$ by $e^{-i\alpha/2}\phi$so that

its eigenvalue undertheoperator $P\mathcal{T}$is unity:

$P\mathcal{T}\phi=\phi$

.

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1_{The Heisenberg-Weyl algebra}_is_a_real_{three-dimensional}_{Lie algebra}_{whose generators satisfy}_the_commutation

relations $[e\iota, e_{2}]=e_{3},$ $[e_{1}, e\mathrm{s}]=[e_{2}, e\mathrm{s}]=0$. To recover the Heisenberg comrnutation relationswe set $e_{1}=$

$\mathrm{i}(\hslash)^{-1/2}p$, _e2$=\mathrm{i}(\hslash)^{-1/2}x$,and$e_{3}=\mathrm{i}$

.

2Theseclasses of Hamiltoniansare all_differerbt. For example,theHamiltonian obtainedby continuing$H$in(1)

along the path$\epsilon$: $0arrow 8$hasadifferent spectrum bom the Hamiltonianobtainedbycontinuing$H=p^{2}+x^{6}\langle ix)^{\epsilon}$

along the path$\epsilon$: $0\prec 4$. Thisis becausethe boundary conditionsontheeigenfunctionsaredifferent.

$31\mathrm{f}$anequationpossessesadiscretesymmetry,thesolutiontothisequation need mat exhibitthatsymmetry. For

example, thedifferential equation$\ddot{y}(t)=y(t)$issymmetricunder the discrete time reversalsymmetry$tarrow-t$. The

solutions$y(t)=e^{t}$ and$y(t$}$\cdot=e^{-\ell}$ do not exhibitthis timereversal symmetry while the solution$y(t)=$ cash(t)

is timereversal symmetric. Thesame is true ofa system whoseHarniltonian is $P\mathcal{T}$symmetric. Even if the

Schr\"odinger equation and the corresponding boundary conditions are PT symmetric, the wavefunction that solves the Schr\"odlnger eqA.ation boundary value problem may not be symmetric under space-time refiection.

Whenthesolutionexhibits$P\mathcal{T}$symmetry,wesay that thePTsymmetryis unbrokenand if the solutiondoes not

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ee

Next, we multiply the eigenvalue equation $H\phi=E\phi$ on the left by$P\mathcal{T}$and use the fact that

$[P\mathcal{T}, H]=0$ to obtain $E\phi=E^{*}\phi$

.

Hence, $E=E^{*}$ and the eigenvalue$E$ is real.

The crucial assumption in this argument is that $\phi$ is simultaneously an eigenstate of $H$

and $P\mathcal{T}$

.

In quantum mechanics ifa linear operator _$X$ commutes with the Hamiltonian $H$,

then the eigenstates of$H$

are

also eigenstates of$X$

.

However,

we

emphasize that the operator

PT is not linear (it is antilinear) and thus we must make the extra assumption that the $P\mathcal{T}$

symmetry of$H$ isunbroken; that is, that $\phi$ issimultaneously

an

eigenstate of$H$and$P\mathcal{T}$

.

This

extra assumption is montrivial because it is not easy to determine a priori whether the $P\mathcal{T}$

symmetry of

a

particularHamiltonian$H$ is brokenor umbroken. For the Hamiltonian$H$ in (1)

the PT symmetry is unbroken vhen $\epsilon\geq 0$ and it is broken when $\epsilon<0$

.

The conventional

HermitianHamiltonian forthe quantum mechanicalharm onic oscilJator liesat theboundaryof

the unbrokenand the brokenregimes. Dorey et

at.

proved rigorously that thespectrum of$H$

in (1) is real and positive [4] in theregion $\epsilon\geq 0$

.

Many other$P\mathcal{T}$-symmetric Hamiltonians for

whichspace-timereflection symmetry is not broken have been investigated, and the spectra of

tlese Ham iltonians havealso been shownto be real and positive [5],

It is useful to show that a given non-Hermitian $P\mathcal{T}$-symrnetric Hamiltonian operator has

a positive real spectrum, but the urgent question that must be answered is whether such

a

Hamiltonian defines a physical theory ofquantum mechanics. By a physical theory we mean

that there is a Hilbert space of state vectors and that this Hilbert space has an innerproduct

19 17 15 13 $\mathrm{u}’ 11\triangleright$ $\#\mathrm{h}$ $\mathrm{R}\mathrm{r}$ 9 7 5 3 1 $\epsilon$

Figure 1: Energy levels of the Hamiltonian$H=p^{2}+x^{2}(\mathrm{i}x)^{\epsilon}$

as

a functionofthe parameter $\epsilon$.

There

are

three regions: When$\epsilon\geq 0$, the spectrum is real and positive and the energy levels

rise with increasing $\epsilon$

.

The lower bound of this region, $\epsilon=0$, corresponds

to the harmonic

oscillator, whoseenergy levels are $E_{n}=2n+1$

.

_When $-1<\epsilon<0$, _thereare a finite _number

of real positive eigenvalues and

an

infinite number of complex comjugate pairs ofeigenvalues.

As $\epsilon$ decreases from 0 to -1, the nunber

ofreal eigenvalues decreases; when $6\leq-0.57793$

,

_the

only real eigenvalue is the ground-state energy.

As

$\epsilon$ approaches $-1^{+}$, the ground-state energy

(4)

with a positive norm. In the theory ofquantum mechanics we interpret the norm ofa state

as a probability and this probabilitymust be positive. Furthermore,

we

must show that the

time evolution ofthe theory is unitary. This

means

that

as

a state vector evolves in timethe

probabilitydoesnot leakaway.

It is not obvious whethera Hamiltomian such

as

$H$ in (1) gives rise to aconsistent quantum

theory. Indeed, while early investigations of this Ham iltonian have shown that the spectrumis

entirely real and positive when $\epsilon\geq 0$, it appeared that one inevitably encountered the

severe

problem of dealing with Hilbert spaces endowed with indefinite metrics [6]. We will identify

here a new symmetry that a1J$P\mathcal{T}$-symmetric Ham iltonianshaving

an

unbroken$P\mathcal{T}$-symmetry

possess. We denote the operator representing this symmetry by $\mathrm{C}$

because the properties of

this operator resemble those of the charge conjugation operator in particle physics. This will

allow

us

to introduceaninner product structure associated with$\mathrm{C}P\mathcal{T}$conjugationforwhich the

normsofquantumstates

are

positivedefinite. We willseethat$\mathrm{C}P\mathcal{T}$symmetryisanalternative

to the conventional Hermiticity requirement; it introduces the

new

concept of a dynamically

determined innerproduct (one that isdeftnedby theHam iltonianitself). Asaconsequence,

we

will extend theHamiltonian and its eigenstatesinto the complexdomain

so

that the associated

eigenvalues

are

real and the underlyingdynam ics isunitary.

2 Construction

of

the

C

Operator

We begin by summarizing the mathematical properties of the solution to the Sturm-Liouville

differential equationeigenvalueproblem

$-\phi_{n}’’(x)+x^{2}(\mathrm{i}x)^{\epsilon}\phi_{n}(x)=E_{n}\phi_{n}(x)$ (4)

associated with the Ham iltonian $H$ in (i). The differential equation (4) must be imposed

on

an

infinite contour 1n the complex-x plane. For large $|x|$ this contour lies in wedges that are

placed symmetrically with respect to the imaginary-z axis [3]. The boundary conditions

on

the eigenfunctions are that $\phi(x)$ $arrow 0$ exponentially rapidly as $|x|arrow\infty$ on the contour. For

$0\leq\epsilon<2$, thecontour may be taken to be the realaxis,

When $\epsilon\geq 0$, the Hamiltonian has an unbroken $P\mathcal{T}$ symmetry. Thus, the eigenfunctions $\phi_{n}(x)$ are simultaneously eigenstates of the$P\mathcal{T}$operator: $P\mathcal{T}\phi_{n}(x)=\lambda_{n}\phi_{n}(x)$

.

As

we

argued

above, A$n$ isa purephase and, without loss of generality, for each$n$ this phase

can

be absorbed

into $\phi_{n}(x)$ by amultiplicative rescalingso that the

new

eigenvalue is unity:

$P\mathcal{T}\phi_{n}(x)=\phi_{n}^{*}(-x)=\phi_{n}(x)$

.

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There is strongevidencethat, whenproperly normalized, the eigenfunctions$\phi_{n}(x)$ are

com-plete. Thecoordinate-spacestatement of completeness (forreal$ andy) reads

$\sum_{n}(-1)^{n}\phi_{n}(x)\phi_{n}(y)=\delta(x-y)$

.

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Thisisanontrivialresult that has been verified numerically toextremelyhighaccuracy (twenty

decimalplaces) $[7, 8]$

.

The unusual factor of$(-\mathrm{l})^{}$ in thesumdoes not appear inconventional

quantum mechanics. The presence of this factor is explained in the following discussion of

orthonormality [see (8)].

Aproblem associated withnon-Hermitian$P\mathcal{T}$-symmetric Hamiltonians arises because there

seems

to bea natural wayto definethe inner product of two functions $f(x)$ and$g(x)$:

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68

where $P\mathcal{T}f(x)=[f(-x)]^{*}$ and the integral is taken over the contour described above in the

complex-x plane. The apparent advantage of this inner product is that the associated

norm

$(f, f)$isindependentof the overall_phase_of$f(x)$and is cormerved in time. Phase independence is

desired because in quantum mechanics one

uses

a spaceof raysto represent quantummechanical

states. Withrespect to this innerproduct the eigenfunctions $\phi_{m}(x)$ and $\phi_{n}(x)$ of$H$ in (1) are

orthogonal for$n\neq m$. However, when_$m=n$ the

norm

is not positive:

$(\phi_{m}, \phi_{n})=(-1)^{n}\delta_{mn}$

.

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This result is apparentlytrue for all values of$\epsilon$ in (4) and it has been verified numerically to

extremely high precision, Because the

norms

ofthe eigenfunctions alternate in sign, the Hilbert

space metric associated with the $P\mathcal{T}$ innerproduct $(\cdot, \cdot)$ is indefinite. Thissplit signature (sign

alternation) is a generic feature ofthe $P\mathcal{T}$inner product. Extensive numerical

calculations

verify that theformula in (8) holds for all $\epsilon\geq 0$

.

Despite the nonpositivity of the innerproduct,weproceedwith the usual analysis that

one

wouldperformfor any Sturm-Liouville problemof the form $H\phi_{n}=E_{n}\phi_{n}$

.

First, we use (8) to

verify that (6) is therepresentation ofthe unityoperator, That is, weverify that

$\oint dy\delta(x-y)\delta(y-z)=$C5(r_$-z$_). ₍₉₎

Second, we reconstruct the parityoperator $P$in terms of the eigenstates. The parity operator

in positionspaceis $P(x, y)=\delta(x+y)$, so_ffom (6)

we

get

$P(x, y)= \sum_{n}(-1)^{n}\phi_{n}(x)\phi_{n}(-y)$, (10)

By virtue of(8) the square of the parity operator isunity: $P^{2}=1$

.

Third,

we

reconstruct the Hamiltoniam$H$ in coordinatespace:

$H(x, y)= \sum_{n}(-1)^{n}E_{n}\phi_{n}(x)\phi_{n}(y)$

.

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Using (6) - (8) it iseasy to see that this Hamiltomian satisfies _{$H\phi_{n}(x)$ $=E_{n}\phi_{n}\{x$}). Fourth,

we

construct the coordinate-space Green’s function$G(x,y)$_:

$G(x, y)= \sum_{n}(-1)^{n}\frac{1}{E_{n}}\phi_{n}(x)\phi_{n}(y)$

.

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The Green’s functionis the functionalinverse of theHam iltonian; that is, $G$satisfies

$\int dyH(x, y)G(y, z)=[-\frac{d^{2}}{dx^{2}}+x^{2}(\mathrm{i}x)^{\epsilon}]G(x, z)=\delta(x-z)$

.

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While_{the time independent Schr\"odinger equation (4) cannot} _be _solved _{analytically,} _the

differ-entialequationfor$G(x, z)$ in (13) can_{be solved exactly and}_{in closed}_form _[8]. _The_{technique is}

to considerthe

case

$0<\epsilon<2$

so

that we maytreat $x$

as

real and then to decompose the_$x$axis

intotwo regions,$x>z$ and$x<z$. We

carz

solve thedifferentialequation ineachof these regions

in term $\mathrm{s}$ of Bessel functions. Then, using this coordinate-space

representation of the Green’s

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inverses of the energy eigenvalues). To do so we set $y=$ in $G(x,y)$ and

use

(8) to integrate

over $x$. For all $\epsilon>0$ weobtain [8]

$\sum_{n}\frac{1}{E_{n}}=[1+\frac{\cos(\frac{3\epsilon\pi}{2\epsilon+\mathrm{S}})\sin(\frac{\pi}{4+\epsilon})}{\cos(\frac{\epsilon\pi}{4+2\epsilon})\sin(\frac{3\pi}{4+\epsilon})}]\frac{\Gamma(\frac{1}{4+\epsilon})\Gamma(\frac{2}{4+\epsilon})\Gamma(\frac{\epsilon}{4+e})}{(4+\epsilon)^{\frac{4+2\epsilon}{4+\epsilon}}\Gamma(\frac{1+\epsilon}{4+e})\Gamma(\frac{2+\epsilon}{4+\epsilon})}$

.

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Having presented these general Sturm-Liouville constructions,

we

discuss the question of

whethera$P\mathcal{T}$-symm etric Hamiltoniandefinesaphysically viablequantummechanicsorwhether

it merely provides an intriguing Sturm-Liouville eigenvalue problem. The apparent difficulty

with form ulating a quantum theory is that the vector space of quantum states is spannedby

energy eigenstates, of which half have

norm

+1 and half have

norm

-1. Becausethe normof

the states carriesa probabilistic interpretation instandard quantum theory, theexistenceofan

indefinite metric in (8)

seems

to bea serious obstacle.

Thesituation here1n which half of the energyeigenstateshave positive normandhalf have

negative norm is analogous to the problem that Dirac encountered in form ulating the spinor

wave

equation in relativistic quantum theory [9]. Following Dirac’s approach,

we

attack the

problemof

an

indefinite normby finding

a

physicalinterpretationforthe negativenormstates.

We claim that in any theory having an unbroken $P\mathcal{T}$symmetry there exists a symmetry of

the Hamiltonian connected with the fact that there are equal numbers of positive-norm and

negative-norm states. To describe thissymmetry

we

construct

a

linear operator denoted by$\mathrm{C}$

andrepresented in position spaceas asum

over

the energyeigenstates ofthe Hamiltonian [10]:

$\mathrm{C}(x, y)=\sum_{n}\phi_{n}(x)\phi_{n}(y)$. (15)

As stated earlier, the properties of this

new

operator $\mathrm{C}$ are nearly identical to those of the

charge conjugation operator in quantum field theory. For example, we can use equations (6)

-(S) to verifythat the squareof$\mathrm{C}$ is unity $(C^{2}=1)$:

$\oint dyC(x,y)C(y, z)=\delta(x-z)$

.

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Thus, the eigenvalues of$C$ are 81. Also,$\mathrm{C}$ commutes with the Hamiltonian$H$

.

Therefore, since

$C$ islinear, the eigenstates of$H$ have definite values ofC. Specifically, if the energy eigenstates

$\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\theta(8)$, then

we

have $\mathrm{C}\phi_{n}=(-1)^{n}\phi_{n}$ because

$C \phi_{n}(x)=\int dyC(x,y)\phi_{n}(y)=\sum_{m}\phi_{m}(x)\int dy\phi_{m}(y)\phi_{n}(y)$

.

We then

use

$\int dy\phi_{m}(y)\phi_{n}(y)=(\phi_{m}, \phi_{n})$ according to

our

convention. We conclude that $C$ is

the operatorthat represents the measurement of the signature of the$P\mathcal{T}$ norm ofa state.

The operators $P$ and $C$ are distinct square roots ofthe unity operator $\delta(x-y\rangle$

.

That is,

while$P^{2}=1$ and$C^{2}=1,$ $P$and$C$arenot identical. Indeed, the parity operator$P$ isreal,while

$\mathrm{C}$ is$\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}1\mathrm{e}\mathrm{x}^{4}$

.

Furtherm ore, these two operators do not

commute:

(CP)$(x, y)= \sum_{n}\phi_{n}(x)\phi_{n}(-y)$ but (PC)$(x, y)= \sum_{n}\phi_{n}(-x$

}

$\phi_{n}(y),$(17) $4\mathrm{T}\mathrm{h}\mathrm{e}$parityoperator incoordinate space is explicitlyreal$P(x,y)=\delta(x+y)j$theoperator$C(x,y)$iscomplex

because it isasumof products of complex functions,as weseein(15). Thecomplexityof the$C$operatorcanbe

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70

which shows that $\mathrm{C}P=(P\mathrm{C})^{*}$

.

However, $C$ doescommute with$P\mathcal{T}$

.

Finally, having obtained the operator $\mathrm{C}$ we define a

new

inner product structure having

positive

_definite

signatureby

$\langle f|g\rangle\equiv l_{\mathrm{C}}^{dx}[CP\mathcal{T}f(x)]g(x)$

.

(18)

Like the $P\mathcal{T}$inner product (7), this innerproduct is phase independentand conserved intime.

Thisisbecause thetime evolutionoperator,just as in ordinary quantummechanics,is$e^{iHt}$

.

The

fact that $H$ commutes with the $P\mathcal{T}$ andthe $CP\mathcal{T}$operators implies that both inner products,

(7) and (18), remain time independent as the states evolve in time. However, unlike (7), the

inner product (18) is positive definite because $C$ contributes -1 when it acts on states with

negative$P\mathcal{T}$ norm. In terms of the$\mathrm{C}P\mathcal{T}$ conjugate, the completenesscondition (4) reads

$\sum_{n}\phi_{n}(x)[CP\mathcal{T}\phi_{n}(y)]=\mathit{5}(x-y)$

.

(19)

Unlike the inner product of conventional quantum mechanics, the $\mathrm{C}P\mathcal{T}$ inner product (19) is

dynamically deterrreined; it dependsimplicitly

on

the choiceofHam iltonian.

Theoperator$C$doesnot existas adistinctentityinconventional quantummechanics. Indeed,

ifweallow the parameter$\epsilon$in (1) to tendtozero, the operator$C$ in this limit becomes identical

to$P$

.

Thus,in this limit the$\mathrm{C}P\mathcal{T}$operator becomes$\mathcal{T}$, which is just complex conjugation. Asa

consequence, theinnerproduct (18) definedwithrespect to the$CP\mathcal{T}$conjugationreducesto the

complexconjugate innerproduct ofconventional quantum mechanicswhen$\epsilonarrow 0$

.

Similarly,in

this limit (19) reducesto the usualstatement ofcompleteness $\sum_{n}\acute{\varphi}_{n}(x)\phi_{n}^{*}(y)=\delta(x-y)$

.

The $CP\mathcal{T}$inner-product (18) is independent of the choice ofintegration contour $\mathrm{C}$ so long

as $\mathrm{C}$ lies inside the asymptotic wedges associated with the boundary conditionsfor the

Sturm-Liouvilleproblem (2). Pathindependencefollows fromCauchy’s theorem andtheanalyticityof

theintegrand. In conventional quantummechanics, wherethe innerproduct is $fdxf^{*}(x)g(x)$,

theintegralmust be takenalongthereal axis and the path of the integration cannot be deformed

into thecomplex plane because the integrand is not

analytic.s

The_PTinnerproduct (7) shares

with (i8) the advantage of analyticity and path independence, but suffers from nonpositivity.

We find itsurprising that apositive-definitemetric can be constructed using $\mathrm{C}P\mathcal{T}$conjugation

without disturbingthe path independence of the inner-product integral,

Finally,weexplain why$P\mathcal{T}$-symmetrictheoriesareunitary. Tim$\mathrm{e}$evolutionis determined by

theoperator $e^{-iHt}$, whether the theory isexpressed in termsofa$P\mathcal{T}$-symmetric Hamiltonianor

just an ordinaryHermitian Hamiltonian. To establish theglobal unitarity ofa theory

we

must

show that as a state vector evolves its

norm

does not change in time. If$\psi_{0}(x)$ is

a

prescribed

initial

wave

function belonging to the Hilbertspace spannedby theenergy eigenstates, then it

evolves into the state $\psi_{t}(x)$at time$t$ according to

$\psi_{t}(x)=e^{-\mathrm{i}Ht}\psi_{0}(x)$

.

With respect to the$CP\mathcal{T}$innerproduct defined in (18), the

norm

of the vector _{$\psi_{t}(x)$} does not

change in time,

$\langle\psi_{t}|\psi_{t}\rangle=\langle\psi_{0}|\psi_{0}\rangle$,

$\epsilon$

Note that ifafunctionsatisfiesa linearorciinarydifferentialequation,then the function is analytic wherever

the coefficient functions of thedifferential equation areanalytic. TheSchr\"odingerequation (4) islinear and its

coefficientsareanalyticexceptforabranch cut atthe origin;this branch cutcanbe takentorunuptheimaginary

axis. Wechoosethe integration contourfor theinnerproduct(8)sothat itdoes notcrossthe positiveimaginary

(8)

because the Ham iltonian $H$ commutes with the $CP\mathcal{T}$ operator. Establishing unitarity at a

local level is more difficult. Here,

we

must show that in coordinate space, there exists a local

probability density that satisfies a continuity equation

so

that the probability does not leak

away. This is asubtle result because theprobability current flows about in the complex plane

rather than along the real axis

as

in conventional Hermitian quantummechanics, _Preliminary

numerical studies indeed indicate that the continuityequationisfulfilled [14].

3 Illustrative

Example:

A

$2\cross 2$

Matrix Hamiltonian

Wewillnow illustrate the above resultsconcerning$P\mathcal{T}$-symmetricquantummechanics inavery

simple context. To do so

we

will consider systems characterized by finite dirnensional matrix

Ham iltonians. Infinite-dim ensionalsystems the $P,$ $\mathcal{T}$, and$\mathrm{C}$ operatorsappear, but there is

no

analogue oftheboundaryconditions associated with coordinate-space Schr\"odinger equations.

Let

us

consider the $2\cross 2$matrix Hamiltonian

$H=(\begin{array}{ll}re^{i\theta} ss re^{-i\theta}\end{array})$ , (20)

where the three parameters$r,$ $s$, and$\theta$

are

real. This Ham iltonian is not Hermitian in theusual

sense, but it is $P\mathcal{T}$symmetric, where theparity operator isgiven by [15]

$P=(\begin{array}{ll}0 11 0\end{array})$ (21)

and$\mathcal{T}$performs complex conjugation.

There are two parametric regions for this Hamiltonian, _When $s^{2}<r^{2}\sin^{2}\theta$, the energy

eigenvalues form

a

complex conjugate pair, This isthe regionofbroken$P\mathcal{T}$ symmetry. Onthe

otherhand, if$s^{2}\geq r^{2}\sin^{2}\theta$, then the eigenvalues$\epsilon\pm=r\cos\theta\pm\sqrt{s^{2}-r^{2}\sin^{2}\theta}$

are

real. Thisis

the region of unbrokenPT symmetry. In the unbroken region the simultaneous eigenstates of

the operators $H$andPT

are

givenby

$| \epsilon_{+}\rangle=\frac{1}{\sqrt{2\cos\alpha}}(\begin{array}{l}e^{\dot{\mathrm{z}}\alpha}/2e^{-i\alpha/2}\end{array})$ aanndd $|\epsilon_{-}$) $= \frac{\mathrm{i}}{\sqrt{2\cos\alpha}}(\begin{array}{l}e^{-i\alpha}/2\backslash -e^{i\alpha/2}\end{array})$, (22)

wherewe set $\sin\alpha=(r/s)\sin\theta$. It iseasilyverified that $(\epsilon\pm, \epsilon\pm)=\pm 1$ and that $(\epsilon\pm,\epsilon_{\mp})=0$,

recallingthat $(u, v)=(P\mathcal{T}u)\cdot v$

.

Therefore,with respecttothe$P\mathcal{T}$inner product, the resulting

vector space spanned by energy eigenstates has a metric of signature ($+,$$-\rangle$

.

The condition

$s^{2}>r^{2}\sin^{2}\theta$

ensures

that $P\mathcal{T}$ symmetry is not broken. If this condition is violated, the states

(22) are nolongereigenstates of$P\mathcal{T}$because czbecomes $\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{r}\mathrm{y}^{6}$.

Next,

we

construct theoperator $C$:

$C= \frac{1}{\cos\alpha}$

(

$\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{n}\alpha$

$-\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{n}\alpha$

).

(23)

Notethat $\mathrm{C}$ is distinct from $H$and $\prime \mathrm{p}$ and has the key property that

$C|\epsilon_{\pm}\rangle=\pm|\epsilon_{\pm}\rangle$

.

(24)

The operator $\mathrm{C}$ commutes with $H$ and satisfies $\mathrm{C}^{2}=1$

.

Theeigenvaluesof

$\mathrm{C}$

are

precisely the

signs of the$P\mathcal{T}$normsofthe corresponding eigenstates.

$\epsilon$

(9)

72

Using theoperator$\mathrm{C}$

we

construct the newinnerproductstructure

$\langle u|v\rangle=(CP\mathcal{T}u)\cdot v$

.

(25)

This innerproductis positive definite because $\langle\epsilon\pm|\epsilon\pm\rangle=1$

.

Thus, the $\mathrm{t}\mathrm{w}\mathrm{o}rightarrow \mathrm{d}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}$Hilbert

spacespanned by $|\epsilon\pm\rangle$,withinnerproduct$\langle\cdot|\cdot\rangle$, has

a

Herm itianstructure withsignature $(+,$$+\rangle$

.

Let us demonstrate that the$\mathrm{C}P\mathcal{T}$

norm

of any vector is positive. We choose the arbitrary

vector$\psi=(\begin{array}{l}ab\end{array})$, where$a$ and $b$areany complex numbers. Then$\mathcal{T}\psi=(\begin{array}{l}ab^{*}\end{array}),$ $P\mathcal{T}\psi=(_{a}^{b}:)$, and

$\mathrm{C}P\mathcal{T}\psi=\frac{1}{\cos\alpha}(\begin{array}{ll}a^{*}+ib^{*} \mathrm{s}\mathrm{j}\mathrm{n}\alpha b^{*}-\iota a^{*} \mathrm{s}\mathrm{i}\mathrm{n}\alpha\end{array}).$ Thus, $\langle\psi|\psi\rangle=(CP\mathcal{T}\psi)\cdot\psi=$ $\frac{1}{\cos\alpha}[a^{*}a+b^{*}b+i(b^{*}b-a^{*}a)\sin\alpha]$

.

Nowlet $a=x+\mathrm{i}y$ and $b=u+\mathrm{i}v$, where _$x,$ _$y,u,$ and$v$

are

real$.$ Then

$\langle\psi|\psi\rangle=\frac{1}{\cos\alpha}$

(

$x^{2}+v^{2}+2xv\sin\alpha$ $+y^{2}+u^{2}-2yu\sin\alpha$

),

(26)

which isexpiicitlypositive and vanishesonlyif$x=y=u=v=0$

.

Recalling that $\langle$$u|$ denotes the$CP\mathcal{T}$-conjugateof $|u\rangle$, thecompleteness conditionreads

$|\epsilon_{+}\rangle\langle\epsilon_{+}|+|\epsilon_{-}\rangle\langle\epsilon_{-}|=(\begin{array}{ll}1 00 1\end{array})$

.

(27)

Furthermore,usingthe$\mathrm{C}P\mathcal{T}$conjugate (gg$|$,we canexpress$\mathrm{C}$in theform$C=|\epsilon_{+}\rangle$$(\epsilon_{+}|-|\epsilon_{-}\rangle\langle\epsilon_{-}|$,

as

opposed to therepresentationin (15), which

uses

the PTconjugate.

An observable in thistheoryisrepresentedby

a

linearoperator $A$that satisfies the equation

$\mathrm{C}P\mathcal{T}ACP\mathcal{T}=A^{\mathrm{T}}$, where$A^{\mathrm{T}}$

isthetransposeof$A$

.

If$\mathrm{C}\mathcal{P}\mathcal{T}$symmetry isunbroken, the

eigenval-ues

of$A$

are

real. Theoperator$\mathrm{C}$

satisfiesthis requirement, and hence it is an observable. For

the two-state system, ifweset $\theta=0$, then the Hamiltonian (20) becomes Hermitian. However,

the operator $\mathrm{C}$

then reduces to the parity operator $P$. As

a

consequence, the above condition

satisfied byanoperatorreducesto the standard condition of Hermiticity,namely, that $H=H^{*}$

.

This is whythe hidden symmetry $C$ was not noticed previously. The operator$\mathrm{C}$ emerges

only

whenweextenda realsymmetricHamiltonianinto the complex domain.

We have calculated the$\mathrm{C}$operatorin many kinds of quantum mechanical and quantum

field

theoretic models. Foran$x^{2}+\mathrm{i}x^{3}$potential,$C$

can

beobtained fromthesummation in (15) using

perturbativemethods [11]. For

an

$x^{2}-x^{4}$potential , $C$

can

becalculatedusingnonperturbative

WKB methods [12]. Quantum field theoretic calculations of$\mathrm{C}$

are

reported

in Ref. [13].

4 _{Applications and}

_{Possible Observable}

_Consequences

We have described here

an

alternative to the axiom ofHermiticityin quantum mechanics. In

quantum field theory, Hermiticity,

Lorentz

invariance, and

a

positive spectrum

are

crucial for

estabIishing$CP\mathcal{T}$

invariance

[16]. Here, we haveestablished the converse of

the$CP\mathcal{T}$ theorem

in the followinglimited sense: We

assume

that the Hamiltonian possessesspace-timereflection

symm etry, and that this symm _{etry is not broken.} $\mathrm{R}\mathrm{o}\mathrm{m}$ these assumptions, we

know that the

spectrumisreal_{and positive and we construct}

_an

_operator$\mathrm{C}$ that is

like the charge conjugation

operator. _{Quantum states in this theory have}_positive_norms_with _respect_to $\mathrm{C}P\mathcal{T}$conjugation.

Ineffect,wereplacethe mathematical condition ofHerm iticitybythe physicalconditionof

space-time and charge-conjugation symmetry. Thesesymmetries

ensure

therealityofthespectrumof

the Hamiltonian in complexquantum theories.

Could non-Herm itian, $P\mathcal{T}$-symmetric Ham iltonians be

used to describeexperimentally

ob-servable phenomena?

Non-Hermitian

Hamiltonians have already been used to describe

(10)

hard spheres is describedbyanon-Hermitian Hamiltonian [17]. Wufound that theground-state

energy ofthis system isreal andconjecturedthat all of theenergylevelswerereal. In1992,

Hol-lowood showedthat

even

though the Hamiltomianofa complexToda lattice is non-Hermitian,

the energylevels are real [18]. Non-Hermitian Hamiltonians ofthe form $H=p^{2}+\mathrm{i}x^{3}$ arise in

various Reggeon fieldtheory models that exhibit real positivespectra [19]. These cubic

Hamil-tonians also arise in thestudyofthe Lee Yangedge singularity [20]. Ineach ofthese casesthe

fact that a non-Hermitian Hamiltonianhad a real spectrumappeared mysterious at the time,

but

now

theexplanationis simple: In each ofthesecasesthemop-HermitianHamiltonian is$P\mathcal{T}-$

symmetric. Thatis, the Hamiltonian in each

case

is constructed

so

that the position operator

$x$or the field operator $\phi$is always multipliedby$\mathrm{i}$

.

An experimental signal of

a

complexHamiltonianmight befoundin thecontext of condensed

matter physics. Consider thecomplexcrystal lattice whose potentialis givenby $V(x)=\mathrm{i}\sin x$

.

While the Hamiltonian$H=p^{2}+i\sin x$ is not Hermitian, it is $P\mathcal{T}$-symmetric, and all of the

energy bandsare real. However, at theedgeofthebands thewavefunctionof

a

particle in such

a

lattice is always bosonic ($2\pi$-periodic) and, unlike the case ofordinary crystal lattices, the

wave

function is

never

fermionic ($4\pi$-periodic) [21]. Direct observation ofsucha bandstructure

would give unambiguous evidence ofa$P\mathcal{T}$-symmetricHamiltonian.

There are many opportunities for the use ofnon-Hermitian Hamiltonians in the study of

quantum field theory. For example, ascalar quantum fieldtheory withacubic self-interaction

describedby theLagrangian $L= \frac{1}{2}(\nabla\varphi)^{2}+\frac{1}{2}m^{2}\varphi^{2}+g\varphi^{3}$ isphysically unacceptable because the

energy spectrum is not bounded below. However, the cubic scaiar quantumfield theory that

correspondsto $H$in (1) with $\epsilon=1$ is given by the Lagrangian density $\mathcal{L}=\frac{1}{2}(\nabla\varphi)^{2}+\frac{1}{2}m^{2}\varphi^{2}+$

$\mathrm{i}g\varphi^{3}$

.

This is a new, physically acceptable quantum field theory. Moreover, the theory that

corresponds to $H$ in (1) with $\epsilon=2$ isdescribedby theLagrangian density

$\mathcal{L}=\frac{1}{2}(\nabla\varphi)^{2}+\frac{1}{2}m^{2}\varphi^{2}-\frac{1}{4}g\varphi^{4}$

.

(28)

What is remarkable about this “wrong-sign” fteld theory is that, in addition to the energy

spectrumbeingrealand positive, theone point Green’sfunction(the

vacuum

expectation value

of the field $\varphi$) is

nonzero

[22]. Furthermore, the field theory is

renorm

alizable, and in four

dimensions is asymptoticalJy free (and thus nontrivial) [23]. Based on these features of the

theory,

we

believethat the theorymay providea useful setting to describe the dynamicsof the

Higgs sector in the standard model.

Other fieldtheory models whose Hamiltonians

are

non-Hermitian and $P\mathcal{T}$-symmetric have

also been studied. For example, $P\mathcal{T}$-symmetric electrodynamics is particularly interesting

be-cause

itisasymptotically free(unlike ordinary electrodynamics) andbecause the direction of the

Casimirforceis thenegative of that inordinary electrodynamics [24]. Thistheoryis remarkable

because it

can

determine its

own

coupling constant. Supersymmetric $P\mathcal{T}$-symm etric quantum

field theories have also beenstudied [25].

These$P\mathcal{T}$-symmetricquantum theories exhibit unexpected phenomena. For example, when

$g$ issufficiently small,the $-g\varphi^{4}$ theorydescribedbythe Lagrangian (28) possessesbound states

(theconventional$g\varphi^{4}$theorydoes not because thepotentialisrepulsive). Boundstates

occur

for

all dimensions $0\leq D<3[26]$, but for purposesofillustrationwedescribe thebound states in

thecontext of one-dimensional quantum fieldtheory (quantum mechanics). Fortheconventional

anharmonicoscillator, which is describedby the Hamiltonian

$H= \frac{1}{2}p^{2}+\frac{1}{2}m^{2}x^{2}+\frac{1}{4}gx^{4}$ $(g>0)$, (29)

the small-y Rayleigh-Schr6dinger perturbation series for the$k\mathrm{t}\mathrm{h}$energy level$E_{k}$ is

(11)

74

where$\nu=g/(4m^{3})$

.

Therenormalizeimass$M$is defined asthe firstexcitationabovetheground

state: $M\equiv E_{1}-E_{0}\sim m[1+3\nu+\mathrm{O}(\nu^{2})]$

as

$\nuarrow 0^{+}$

.

To determineif thetwo-particle stateisbound, weexamine the secondexcitation above the

ground stateusing (30), We define

$B_{2}\equiv E\mathit{2}$ $-E_{0}\sim m[2+9\nu+\mathrm{O}(\nu^{2})]$ $(\nuarrow 0^{+})$

.

(31)

If$B_{2}<2M$,thenatwo-particleboundstate exists and the(negative) bin dingenergyis$B_{2}-2M$.

If$B_{2}>2M$, then the second excitation above the

vacuum

is interpreted as an unbound

two-particlestate. We

see

ffom (31) that in the smalI-coupling region, whereperturbation theory

is valid, the conventional anharmonic oscillator does not possess a bound state. Indeed, using

WKB, variationalmethods,

or

numericalcalculations, one

can

showthat there isnotwo-particle

bound state foranyvalue of$g>0$

.

Because there isno boundstate the$gx^{4}$interaction maybe

considered to represent a repulsive$\mathrm{f}o\mathrm{r}\mathrm{c}\mathrm{e}.7$

We obtainthe perturbation series for the man-Hermitian,$P\mathcal{T}$-symmetric Hamiltonian

$H= \frac{1}{2}p^{2}+\frac{1}{2}m^{2}x^{2}-\frac{1}{4}gx^{4}$ $(g>0)$, (32)

from theperturbation series for the conventional anharmonic oscillator by replacing $\nu$ with$-\nu$

.

While the conventional anharmonic oscillator does not possess

a

two particle boundstate, the

$P\mathcal{T}$-symmetric oscillator does possess such astate. We

measure

thebindingenergy of this state

in un its of the renormalizedmass $M$ and we define the dimensionless binding energy $\Delta_{2}$ by

$A_{2}$ $\equiv\frac{B_{2}-2M}{M}\sim-3\nu+\mathrm{O}(\nu^{2})$ $(\nuarrow 0^{+})$

.

(33)

Thisbound statedisappearswhen$\nu$increasesbeyond_$\nu=$ 0.0465.

. ..

As$\nu$continuestoincrease,

A2

reaches

a

maximum of0.427at $\nu=0.13$ and then_approaches _the_value _0.28 as $\nuarrow\infty$.

In the $P\mathcal{T}$-symmetric anharmonic oscillator, there

are

not only two-particle

bound states

for smallcoupling constant but also A-particle bound states for all $k\geq 2$

.

The dimensionless

bindingenergiesare

$\Delta_{k}\equiv(B_{k}-kM)/M\sim-3k(k-1)\nu/2+0(\nu^{2})$ ₍$\nuarrow 0+\rangle$. (34)

Thekey feature of this equation is that the coefficient of$\nu$ is negative. Sincethe dimensionless

bindingenergy becom es negative

as

$\iota/$ increases from 0, there isa $k$-particle boundstate. The

higher multiparticlebound states

cease

to be bound for smaller values of &; starting with the

three-particle bound state, the binding energy of these states becomes positive as $\nu$ increases

past 0.039, 0.034, 0.030, and0.027.

For anyvalue of$\nu$there

are

always

a

finite number of bound states andaninfinite number

of unbound states. The number of bound states decreases withincreasing $\nu$ until there are

no

bound states at all. There isarange of$\nu$for which thereareonlytwo- and three particlebound

states. This situationis analogous to the physical world in which

one

observes only states of

two and three bound quarks. Inthis rangeof$\nu$ ifonehas

an

initial state containing

a

numberof

particles (renormalized masses). theseparticles willclumptogether into bound states, releasing

7Ingeneral, arepulsiveforcein aquantumfieldtheoryisrepreseuted byanenergydependence in which the

energy ofatwo-particlestate decreases with separation. Theconventional anharmonic oscillator Hamiltonian

corresponds toa fieldtheory inonespac -time dimension, wherethere cannot beanyspatial dependence. Inthis

casethe repulsivenatureof theforceis understood tomeanthat theenergy$B_{2}$neededto create twoParticlesat

(12)

energy in the process. Depending on $\nu$, the final state will consist either of two- or of

three-particlebound states, whichever is energeticallyfavored. Also, there is

a

specialvalue of$\nu$ for

whichtwo- and three-particle bound states

can

exist in thermodynamicequilibrium.

How doesa$g\varphi^{3}$theorycomparewitha$g\varphi^{4}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{y}^{7}$A$g\varphi^{3}$ theoryhasanattractive force. The

bound statesthat ariseas

a

consequence ofthis force

can

befoundby using theBethe-Salpeter

equation. However, the $g\varphi^{3}$ fieldtheory is unacceptablebecause the spectrum is not bounded

below. Ifwereplace9by $\mathrm{i}g$, the spectrum becomes real and positive, butnowtheforce becomes

repulsiveandthereare noboundstates. The

same

istrue foratwo-scalar theory withinteraction

ofthe form $\mathrm{i}g\varphi^{2}\chi$

.

This latter theoryis amacceptablemodel of scalarelectrodynam $\mathrm{i}\mathrm{c}\mathrm{s}$, but has

noanalog of positronium.

5 Concluding

Remarks

We have argued in this paper that there is an alternative to the axiom of standard quantum

mechanics that the Hamiltonian must be Hermitian. We have shown that the axiom of

Her-miticity maybereplaced by the

more

physical condition of$P\mathcal{T}$(space-time reflection) symmetry.

Space-time reflection symm etryis distinct

&om

the condition ofHermiticity,

so

it is possible to

considernew kinds ofquantum theories, such

as

quantum field theories whose self-interaction

potentials

are

$ig\varphi^{3}$ or $-g\varphi^{4}$

.

Such theories have previously been thought to be

mathemati-callyandphysically unacceptable because thespectrum mightnot be real and because thetime

evolution might not be unitary.

Thesenewkindsof theories canbethought ofasextensions ofordinaryquantum mechanics

intothecomplex plane; thatis, continuations of real symmetric Hamiltonianstocompiex

Hamil-tonians. The idea ofanalyticallycontinuing

a

Hamiltonian

was

first discussed in1952by Dyson,

who arguedheuristicallythat perturbationtheoryforquantumelectrodynamicsisdivergent [27].

Dyson’sargument involvesrotatingthe electriccharge$e$into the complexplane$earrow \mathrm{i}e$

.

Applied

to the quantum anharm onic oscillator, whose Hamiltonian is given in (29), Dyson’s argument

would go as follows: If thecouplingconstant $g$ is continuedin the complex-y planeto $-g$, then

thepotential isnolonger boundedbelov,so the resultingtheoryhasnoground state. Thus, the

ground-stateenergy $E0(g)$ hasan abrupttransitionat $g=0$

.

Ifwe represent$E_{0}(g)$ as a seriesin

powersof$g$, thisseries must havea zeroradim ofconvergencebecause$E0(g)$ hasasingularity at

$g=0$ in the complex-coupling-constant plane. Hence, theperturbationseries must diverge for all$g\neq 0$

.

While the perturbation series does indeed diverge, this heuristic argument is flawed

because the $\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}^{r}\mathrm{u}\mathrm{m}$ of the Hamiltonian (32) that is obtained remains ambiguous until the

boundaryconditions that the wave functions must satisfyarespecified. The spectrum depends

cruciallyon how this Hamiltonian with

a

negative coupling constant is obtained.

Therearetwo waysto obtain the Hamiltonian (32). First,

one can

substitute $g=|g|e^{\mathrm{i}\theta}$ into

the Hamiltonian (29) and rotate fiiom $\theta=0$ to $\theta=\pi$

.

Under this rotation, the ground-state

energy$E_{0}(g)$ becomes complex. Evidently, $E_{0}(g)$ is real and positivewhen$g>0$ and complex

when$g<08$. Second, one can obtain (32) asa limit of the Ham iltonian

$H= \frac{1}{2}p^{2}+\frac{1}{2}m^{2}x^{2}+\frac{1}{4}gx^{2}(\mathrm{i}x)^{\epsilon}$ $(g>0)$ (35)

as $\epsilon$ : $0arrow 2$. The spectrum of this Hamiltonian is real, positive, and discrete. The spectrum

of the limitingHamiltonian (32) obtained in this manner 1s similar in structure to that of the

Hamiltonian in (29).

8Rotatingfrom$\theta=0$to $\theta=-\pi$,weobtain thesameHamiltonianasin (32)butthe spectrum is the complex

(13)

76

HowcantheHam iltonian(32)possess two differentspectra? Theanswerlies intheboundary

conditions satisfied by the

wave

functions $\phi_{n}(x)$

.

Inthefirst case, in which$\theta=\arg g$ isrotated

in thecomplex-g planefrom 0to$\pi,$ $\psi_{n}(x)$vanishesin thecomplex-r plane

as

$|x|arrow\infty$insidethe

wedges $-\pi/3<\arg x<0$ and $-4\pi/3<\arg x<-\pi$

.

Inthe second case, in which the exponent

$\epsilon$ ranges from 0 to 2, $\phi_{n}(x)$ vanishes in the complex-z plane

as

$|x|arrow\infty$ inside the wedges

$-\pi/3<\arg x<0$ and $-\pi<\arg x<-2\pi/3$. In this second

case

the boundary conditions hold

in wedges that are symmetric with respect to the imaginary axis; these boundary conditions

enforce thePT symmetryof$H$ andareresponsiblefor the realityofthe energyspectrum.

Apart from thespectra, there is another striking difference between the two theories

corre-spondingto$H$in (32). Theone-point Green’sfunction$G_{1}(g)$ is definedastheexpectation value

of the operator$x$ in the ground-state

wave

function$\phi_{0}(x)$,

$G_{1}(g)=\langle 0|x|0$

}

$/ \langle 0|0\rangle\equiv\int_{C}dxx\psi_{0}^{2}(x)/\int_{C}dx\psi_{0}^{2}(x)$, (36)

where $C$ is a contour that lies ₀₁ the asymptotic wedges described above. The value of$G_{1}(g)$

for$H$ in (32) dependson the limiting process by which

we

obtain$H$

.

If

we

substitute$g=g_{0}e^{\iota\theta}$

into the Hamiitonian (29) and rotate from $\theta=0$ to $\theta=\pi$, we find that _{$G_{1}(g)=0$} for all _$g$

on the semicircle in the complex-g plane. Thus, this rotation in the $g$ plane preserves parity

symmetry $(xarrow-x)$

.

However, ifwe_define$H$ in (32) by using the Hamiltonian in (35) and by

allowing$\epsilon$ to go from 0 to 2, wefind that $G_{1}(g)\neq 0$

.

Indeed, $G_{1}(g)\neq 0$for all values of_$\epsilon>0$

.

Thus, in thistheory$P\mathcal{T}$symmetry (reflectionabout theimaginary axis, _{$xarrow-x”$}) is

preserved,

butparity symmetryispermanently broken. This

means

that

one

mightbe ableto describe the

dynamicsoftheHiggs sectorbyusing a $-g\varphi^{4}$ quantum field theory,

Acknowledgement

I thank the Theoretical Physics Group at Imperial College for its hospitality and the $\mathrm{U}.\mathrm{K}$

.

Engineering and Physical Sciences Research Council, the JohnSimon Guggenheim Foundation,

and the$\mathrm{U}.8$. Department of Energy for financialsupport.

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