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arXiv:hep-th/0703038v1 5 Mar 2007

Cryptoreality of nonanticommutative Hamiltonians

E.A. Ivanov1, A.V. Smilga2,

1Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow Region,

Russia

eivanov@theor.jinr.ru,

2SUBATECH, Universit´ e de Nantes, 4 rue Alfred Kastler, BP 20722, Nantes 44307,

France∗

smilga@subatech.in2p3.fr

Abstract

We note that, though nonanticommutative (NAC) deformations of Minkowski su-

persymmetric theories do not respect the reality condition and seem to lead to non-

Hermitian Hamiltonians H, the latter belong to the class of “cryptoreal” Hamiltoni-

ans considered recently by Bender and collaborators. They can be made manifestly

Hermitian via the similarity transformation H → eRHe−Rwith a properly chosen

R. The deformed model enjoys the same supersymmetry algebra as the undeformed

one, though being realized differently on the involved canonical variables. Besides

quantum-mechanical models, we treat, along similar lines, some NAC deformed field

models in 4D Minkowski space.

1Introduction

Supersymmetric models with nonanticommutative (NAC) deformations [1] have recently

attracted a considerable interest. The main idea is that the odd superspace coordinates

θαand¯θ˙ αare not treated as strictly anticommuting anymore, but involve non-vanishing

anticommutators [2]1. In original Seiberg’s paper and in many subsequent works (see

e.g. [3,4] and references therein), the deformation is performed in Euclidean rather than

Minkowski space-time. The reason is that in Minkowski space it seems impossible to

preserve both supersymmetry and reality of the action after deformation, still retaining

simple properties of the corresponding ⋆-product (e.g., associativity and nilpotency) [5].

As discussed in [1], Euclidean NAC theories are of interest in stringy perspectives

An interesting question is whether NAC theories are meaningful by themselves, leaving

aside the issue of their relationships with string theory. In other words — whether it is

2.

∗On leave of absence from ITEP, Moscow, Russia.

1In other words, the original Grassmann algebra of the odd coordinates is deformed into a Clifford

algebra.

2The stringy origin of such deformations [6] was actually the main motivation of their consideration

in [1] (see also [4,7]).

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possible to consistently define them in Minkowski space, introduce a Hamiltonian with

real spectrum and find a unitary evolution operator.

We argue that the answer to this question is positive. Our consideration is mostly

based on the analysis of an interesting 1–dimensional NAC model constructed in a recent

paper of Aldrovandi and Schaposnik [8]. In that work, NAC deformations of the conven-

tional Witten’s supersymmetric quantum mechanics (SQM) model [9] were studied in the

chiral basis. In this case, the deformation operator commutes with the supercharge Q,

but does not commute with¯Q. However, Aldrovandi and Schaposnik noticed the presence

of the second supercharge¯Q that commutes with the Hamiltonian. On the other hand, Q

and¯Q seem not to be Hermitian conjugate to each other and the deformed Hamiltonian

also seemingly lacks the Hermitian property.

Our key observation is that, in spite of having a complex appearance, this Hamiltonian

is actually Hermitian in disguise. One can call it “crypto-Hermitian” (or “cryptoreal”).

It belongs to the class of Hamiltonians studied recently by Bender and collaborators [10].

The simplest example is

H =

p2+ x2

2

+ igx3.(1)

In spite of the manifestly complex potential, it is possible to endow the Hamiltonian (1)

with a properly defined Hilbert space such that the spectrum of H is real. The clearest

way to see this is to observe the existence of the operator R such that the conjugated

Hamiltonian

˜H = eRHe−R

(2)

is manifestly self-adjoint [11]. The explicit form of R for the Hamiltonian (1) is3

R = g

?2

3p3+ x2p

?

− g3

?64

15p5+20

3p3x2+ 4px4− 6p

?

+ O(g5) .(3)

The rotated Hamiltonian is

˜H =

p2+ x2

2

+ g2

?

3p2x2+3x4

2

−1

2

?

+ O(g4) .(4)

The (real) spectrum of˜H (and H) can be found to any order in g in the perturbation

theory, and also non-perturbatively.

We will see that in the case of the Aldrovandi-Schaposnik Hamiltonian, there also

exists the operator R making the Hamiltonian Hermitian.

eRQe−Rand eR¯Qe−Rare Hermitian-conjugated.

We start in Section 2 by constructing the operator R for certain non-supersymmetric

Hamiltonians. In particular, we discuss holomorphic deformations (adding to the Hamil-

tonian a holomorphic function of a complex dynamic variable). In Section 3, we present

The rotated supercharges

3Actually, what is written here is the Weyl symbol of the operator R. The expression for a contri-

bution to the quantum operator corresponding to a monomial ∼ pnxnin its Weyl symbol is a properly

symmetrized structure, px → (1/2)(ˆ px + xˆ p), x2p → (1/3)(x2ˆ p + ˆ px2+ xˆ px), etc.

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the Aldrovandi-Schaposnik model, find the corresponding operator R, as well as the ro-

tated Hamiltonian and supercharges. Also we briefly consider a NAC deformation of the

SQM model with two sorts of chiral supermultiplets. In Section 4, we discuss possible

generalizations to field theory.

2Cryptoreality: some comments

• First, about the term “cryptoreality”. In the original papers [10], the Hermiticity of

the Hamiltonian (1) and its relatives was deduced from a certain special symmetry

of this Hamiltonian, the PT -symmetry. Indeed, the Hamiltonian (1) is invariant

with respect to the combination of the parity transformation (which changes the

sign of x) and the time reversal transformation (which changes i to −i). The PT -

symmetry of the Hamiltonian might be a sufficient condition for the existence of

the operator R such that the conjugated Hamiltonian (2) is manifestly Hermitian,

but, as we will see later, it is not a necessary condition. In Ref. [11], the term

“pseudo-Hermiticity” was used. To our mind, however, there is nothing “pseudo”

about it, the Hamiltonian (1) is simply Hermitian (in the properly defined Hilbert

space), but its Hermiticity is hidden, not immediately obvious. That is why the term

“crypto-Hermiticity” (or “cryptoreality”) seems to us somewhat more appropriate.

• The conjugation (2) acts upon all operators including the operators p,x. The Weyl

symbols of the transformed operators p′,x′are

p′

x′

= p + 2igxp + g2(2p3− px2) + ...

= x − ig(x2+ 2p2) − g2(x3− 2xp2) + ... .(5)

One can actually obtain the expression (4) for the Weyl symbol of the rotated

Hamiltonian by simply expressing H in terms of p′, x′. The commutator [p,x] is not

changed after conjugation, that means that the Moyal bracket {p′,x′}M.B.is equal

to one. The Moyal bracket is defined as [12]

{A,B}M.B. = 2 sin

?1

2

?

∂2

∂p∂X−

∂2

∂P∂x

??

A(p,x)B(P,X)

????

p=P,x=X

(6)

The expansion starts with the Poisson bracket, but, generically, there are also higher

terms. In particular, {p′,x′}M.B.differs from {p′,x′}P.B.by the terms of order ∼

g4and higher. But that means that (5) is not a canonical transformation. And

this means that the classical dynamics of H(p,x) and H(p′,x′) are different. The

quantum dynamics of the original and conjugated Hamiltonians is, however, the

same.

• One can rotate away not only imaginary pieces in the potential, but also other

unfriendly looking terms in the Hamiltonian. For example, one can consider the

Hamiltonian

p2+ x2

2

H =

+ gx3

(7)

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and conjugate it with the operator R coinciding with the expression in Eq.(3) mul-

tiplied by the factor −i. The conjugated Hamiltonian coincides with (4), with the

sign of g2being reversed. The spectrum of the Hamiltonian (7) can be found by

the same token as for the Hamiltonian (1). Actually, an exact mapping relating

the system (7) to the system (1) exists. Indeed, for any eigenfunction Ψn(x) of the

Hamiltonian (1) with eigenvalue En, the function Ψn(−ix) is an eigenfunction of

the Hamiltonian (7) with the eigenvalue −En.

The appearance of complex values of x may be somewhat unusual, but it is actu-

ally an inherent feature of the crypto-Hermitian systems. The eigenfunctions of the

Hamiltonian are required to behave well (be not singular and die out for large abso-

lute values of the argument) in a certain domain in the complex x-plane that might

or might not include the real axis [10]. The relevant domains for the Hamiltonians

(1) and (7) are shown in Fig. 1. One is rotated with respect to the other by the

angle π/2.

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x

a)

b)

x

Figure 1: Wave functions for a) the Hamiltonian (1) and b) Hamiltonian (7) asymptoti-

cally die out in the dashed sectors.

Another unusual feature of the Hamiltonian (7) is the absence of the ground state -

the state with the lowest energy. In this case, the spectrum has an upper rather than

lower bound. But the overall sign of energy is in fact a matter of book-keeping. For

all physical purposes, the dynamics of the Hamiltonian (1) in the region in Fig.1a

and the dynamics of the Hamiltonian (7) in the region in Fig.1b are equivalent.

Consider now the Hamiltonian

H =¯ ππ + ¯ zz + gz3.(8)

Remarkably, by conjugating it with the operator

R = −ig

?

¯ πz2+2

3¯ π3

?

,(9)

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one can rotate away the cubic term in the potential without trace such that the

conjugated Hamiltonian H′= ¯ π′π′+ ¯ z′z′+ gz′3is simply ¯ ππ + ¯ zz. Hence the

spectrum of the Hamiltonian (8) coincides with the spectrum of a 2-dimensional

oscillator, En,m= 1 + n + m. The wave functions of the original Hamiltonian (8)

are obtained from the oscillator wave functions by conjugation Ψ = e−R˜Ψ. For

example, the ground state wave function is

Ψ0 ∼ exp

?

−gz3

3

− ¯ zz

?

. (10)

It decays exponentially in the three sectors in the complex plane of z shown in Fig.

2, and the Hilbert space where the crypto-Hermitian Hamiltonian (8) is well defined

is formed by the functions sharing this property.

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z

Figure 2: The same for the Hamiltonian (8).

By the same token, one can rotate away without trace any holomorphic term in the

potential. For example, for the Hamiltonian ¯ ππ + ¯ zz + gz5, this is done with the

operator

?

Generally, the operator rotating away the term gzNin the potential has the form

R = −ig¯ πz4+4

3¯ π3z2+8

15¯ π5

?

.

RN= −ig¯ πzN−1fN

?¯ π

z

?

,

with fN(r) satisfying the equation

[1 − r2(N − 1)]fN+ r(1 + r2)f′

N= 1 .(11)

When N is odd, the solution represents a polynomial. For even N, it is more

complicated. For example,

f2(r) =1

2

?1 + r2

r

arctanr + 1

?

.(12)

5