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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.

1 Introduction

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.

2 Cryptoreality: 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)

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• Cryptoreal Hamiltonians for the systems with continuum number of degrees of free-

dom also exist. Bender, Brody, and Jones found the proper conjugation operator

for the system described by the Lagrangian [10]

L =

1

2(∂µφ)2−µ2φ2

2

− igφ3, (13)

φ is a real scalar field. In the lowest order in g, it is given by a nonlocal expression

R =

? ? ?

dxdydz[Mxyzpxpypy+ Nxyzφxφypz], (14)

where pxare canonical momenta, px= −i∂/∂φx, and the kernels Mxyz,Nxyzhave

a complicated, but explicit form.

We want to notice that the system of the complex scalar field ϕ with the interaction

Hamiltonian ∼ ϕ3is also cryptoreal, and the corresponding conjugation operator is

given, again, by the expression (14) with ¯ πx= −i∂/∂ ¯ ϕxbeing substituted for px.

This operator rotates the interaction term away without trace by the same token as

the operator (9) rotates it away in the QM case.

Actually, the pattern is quite general. Any holomorphic interaction term can be

entirely rotated away simply because the proper conjugation operator R involves

in this case only the momentum operators ¯ πx rather than πx, and¯∂f = 0 for

holomorphic functions.

• Finally, let us reproduce here the arguments of [10] displaying the reality of the

spectrum of a PT -symmetric Hamiltonian. The operator PT commutes with the

Hamiltonian, and it is reasonable to assume that a basis of the states representing

the eigenstates of both PT and H can be chosen4. Let Ψ be an eigenstate of both

PT and H,

PT Ψ = λΨ,H Ψ = EΨ . (15)

Applying the operator PT to the second equality and using [PT ,H] = 0 and

PT (EΨ) = E∗PT (Ψ), we conclude that E = E∗Q.E.D. Note also that applying

PT to the first equality and using (PT )2= 1, one can show that λλ∗= 1 and hence

λ = eiα. By going from Ψ to Ψe−iα/2, one can set λ = 1.

The norm of some eigenstates may happen to be negative. However, this can be

mended [10] if redefining inner product by including in its definition the action of

the “charge conjugation” operator C that commutes with both H and PT and is

defined as

C(x,y) =

?

n

Ψn(x)Ψ∗

n(y) .(16)

4Were PT a linear operator, it would be trivial, but PT involves complex conjugation and is not

linear. Hence, the existence of such basis is, indeed, an assumption and the reasoning given here cannot

be regarded as a formal proof.

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The operator C is in fact closely related to the operator R rotating the Hamiltonian

to the manifestly Hermitian form, as discussed above,

C = e−2RP . (17)

3 Aldrovandi-Schaposnik model

The simplest SQM model [9] involves a real supervariable

X(θ,¯θ,t) = x(t) + θψ(t) +¯ψ(t)¯θ + θ¯θF(t) . (18)

The action is

S = −

?

dtd2θ

?1

2(DX)(¯DX) + V (X)

?

, (19)

with the convention?d2θθ¯θ = 1. Here V (X) is the superpotential and D,¯D are covariant

basis representation

derivatives. Bearing in mind the deformation coming soon, we will choose their left chiral

D =

∂

∂θ− 2i¯θ∂

∂t,

¯D = −∂

∂¯θ.

(20)

Here t = τ − iθ¯θ and τ is the real time coordinate of the central basis. Asymmetry

between D and¯D makes the Lagrangian following from (19) complex,

L = −i˙ xF −∂V (x)

∂x

F +1

2F2+ i¯ψ˙ψ +∂2V (x)

∂x2

¯ψψ , (21)

but one can easily make it real, rewriting it in terms of˜F = F − i˙ x and subtracting a

total derivative. This corresponds to going over to the central basis from the chiral one.

The deformation is introduced by postulating non-vanishing anticommutators

{θ,θ} = C,

{¯θ,¯θ} =¯C,

{θ,¯θ} =˜C . (22)

The deformed action involves star products,

S = −

?

dtd2θ

?1

2(D ⋆ X) ⋆ (¯D ⋆ X) + V⋆(X)

?

, (23)

where

X ⋆ Y = exp

?

−C

2

∂2

∂θ1∂θ2

−

¯C

2

∂2

∂¯θ1∂¯θ2

−

˜C

2

?

∂2

∂θ1∂¯θ2

+

∂2

∂¯θ1∂θ2

??

X(1)Y (2)

?????

1=2

(24)

and V⋆(X) is obtained from V (X) =?

Weyl ordering of any product of the θ monomials such that

ncnXnby substituting X2→ X2

⋆≡ X ⋆X, X3→

X3

⋆≡ X ⋆ X ⋆ X, etc in its Taylor expansion. The star product in (23) just ensures the

θ ⋆ θ =C

2,

¯θ ⋆¯θ =

¯C

2,θ ⋆¯θ = θ¯θ +

˜C

2,

¯θ ⋆ θ =¯θθ +

˜C

2

,

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in accordance with the basic relation (22). The star product is associative.

The component expression for the deformed Lagrangian is the same as in Eq. (21),

with V (x) being substituted by [8,13]

˜V (x,F) =

?1/2

−1/2

dξ V (x + ξcF) , (25)

where

c2=˜C2− C¯C (26)

is the relevant deformation parameter5. If¯C is conjugate to C and˜C is real, c2is also

real. Note, however, that one may, generally speaking, lift the condition that θ and¯θ are

conjugate to each other, in which case C,¯C and˜C can take arbitrary values. We still

require the reality of c2. The crypto-Hermiticity of the deformed Hamiltonian discussed

below is fulfilled under this condition.

In the simplest nontrivial case, V (X) = λX3/3,

˜V (x,F) =

λx3

3

+λc2xF2

12

. (27)

The corresponding canonical Hamiltonian is

H =

p2

2+ i∂˜V

∂xp −∂2˜V

∂x2¯ψψ , (28)

with p = −iF. The deformed Lagrangian and Hamiltonian look inherently complex.

Obviously, the complexities now cannot be removed by simply going from the chiral to

the central basis.

In the chiral basis, the supercharges are represented by the following superspace dif-

ferential operators,

Q =

∂

∂θ,

¯Q = −∂

∂¯θ− 2iθ∂

∂t.

(29)

Note that the star product operator (24) still commutes with Q (in other words, the

Leibnitz rule Q⋆ (X ⋆ Y ) = (Q⋆ X)⋆ Y +X ⋆(Q⋆ Y ) still holds), but not with¯Q. That

means that the deformed action (23) is still invariant with respect to the supersymmetry

transformations generated by Q, but not¯Q. The Q-invariance implies the existence of

the conserved N¨ other supercharge whose component phase space expression is simply

Q = ψp .(30)

As was observed in [8], there is another Grassmann-odd operator commuting with the

Hamiltonian. It reads

?

¯Q =¯ψp + 2i∂˜V

∂x

?

. (31)

5The relation (25) can be easily derived by keeping the term ∝ θ¯θ in the products Xn

associativity and the identity (θ¯θ) ⋆ (θ¯θ) = c2/4. Note the correct sign of c2in (26) as compared to the

wrong one in the definition of c2in [8].

⋆, with using

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The standard SUSY algebra

Q2=¯Q2= 0,

{Q,¯Q} = 2H (32)

holds, but, naively,¯Q is not adjoint to Q and H is not Hermitian.

Let us show now that the Hamiltonian (28) is in fact cryptoreal. Consider for simplicity

only the case (27). We have6,

H =

p2

2+ iλpx2− iβp3− 2λx¯ψψ ,(33)

where β = λc2/12.

It is convenient to treat λ and β on equal footing and to get rid of the complexities

∼ ipx2and ∼ ip3simultaneously. The operator R doing this job is

R = −λx3

3

+ βxp2− 2λβx2¯ψψ + ... ,(34)

where the dots stand for the terms of the third and higher order in λ and/or β. The

conjugated Hamiltonian is

˜H = eRHe−R=

p2

2− 2λx¯ψψ +1

2[λ2x4+ 3β2p4] +1

2λβ + O(λ3,β3,λ2β,λβ2). (35)

It is Hermitian. The rotated supercharges are

˜Q = eRQe−R= ψ[p − i(λx2− βp2) + λβx2p − β2p3+ ...],

˜ ¯Q = eR¯ Qe−R=¯ψ[p + i(λx2− βp2) + λβx2p + 3β2p3+ ...].(36)

We observe that they are still not adjoint to each other. To make them mutually adjoint

to the considered order in β,λ, one should add to the operator R one more term

R ⇒ ˆR = R − 2β2p2¯ψψ.(37)

It is easy to see that this modification does not change the rotated Hamiltonian in the

considered order, but ensures the rotated supercharges to be manifestly adjoint to each

other

ˆQ = e

ˆ ¯Q = eR¯Qe−R=¯ψ[p + i(λx2− βp2) + λβx2p + β2p3+ ...].

ˆRQe−ˆR= ψ[p − i(λx2− βp2) + λβx2p + β2p3+ ...],

(38)

By construction, the operatorsˆQ,ˆ ¯Q and˜H satisfy the standard algebra (32). We see

that the requirement of the mutual adjointness of supercharges is to some extent more

fundamental than that of the Hermiticity of the Hamiltonian — the latter does not strictly

fix the rotation operator R while the former does.

6Note that this Hamiltonian is not PT -, but just T -symmetric.

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One can be convinced, order by order in β,λ, that complexities in H can be success-

fully rotated away also in higher orders (with simultaneously restoring the mutual conju-

gacy of the supercharges), and this is also true for higher powers N > 3 in V (X) ∼ XN

and hence for any analytic superpotential7.

As the last topic of this Section, we shall consider NAC deformations of some other

N = 1 SQM models8.

Besides the N = 1 multiplet with the off-shell content (1,2,1), there also exist chiral

N = 1 multiplets (2,2,0) and (0,2,2), having, correspondingly, even and odd overall

Grassmann parity. They are described, respectively, by the chiral superfields Φ(θ,¯θ,t)

and Ψ(θ,¯θ,t):

¯DΦ = 0

⇒

Φ = z(t) + θχ(t),

¯DΨ = 0

⇒

Ψ = ω(t) + θh(t), (39)

where as before t = τ − iθ¯θ, z is a complex bosonic field, ξ and ω are complex fermionic

fields and h is a complex bosonic auxiliary field.

NAC deformation preserving the 1D chirality and anti-chirality corresponds to the choice

¯C =˜C = c2= 0,C ?= 0 in (22). Then the action of Φ,

It was shown in [8] that the only

SΦ= −

?

dtd2θ

?1

4DΦ¯D¯Φ + K(Φ,¯Φ)

?

=

?

dt

?

˙ z˙¯ z +i

2¯ χ ˙ χ + ...

?

, (40)

remains undeformed after replacing all products by the relevant ⋆ products [8].

Actually, the same is true for the action of Ψ

SΨ= −

?

dtd2θ

?1

4Ψ¯Ψ + β¯θΨ −¯βθ¯Ψ

?

=

?

dt

?i

2¯ ω ˙ ω −1

4h¯h + βh +¯β¯h

?

,(41)

where β is a complex constant. However, while considering mutual couplings of Ψ and

Φ, there arise new possibilities. Prior to switching on any deformation, such couplings

provide potential terms for the (2,2,0) multiplet which do not exist within the pure Φ

system (the “potential” term K(Φ,¯Φ) in (40) produces only a Wess-Zumino type term

∼ (˙ z¯ z −˙¯ zz) + ....). In particular, one can consider the action

SΦ+Ψ= −

?

dtd2θ

?1

4DΦ¯D¯Φ +1

4Ψ¯Ψ +¯θΨF(Φ) − θ¯Ψ¯ F(¯Φ)

?

, (42)

which gives rise to a non-trivial scalar potential for z, ¯ z upon eliminating the auxiliary

fields h,¯h by their equations of motion. For instance, choosing F = a + bΦ + dΦ2, one

obtains after elimination of h,¯h the following on-shell component action

SΦ+Ψ=

?

dt

?

|˙ z|2+i

2(¯ χ ˙ χ + ¯ ω ˙ ω) + 4|a + bz + dz2|2+ Yukawa ω,χ couplings

?

. (43)

Nevertheless, once again, the direct (anti)chirality-preserving deformation of (42) does

not yield nothing new. The reason is that the terms proportional to the deformation

7It would be worth being aware of the full analytic proof of this.

8We denote by N the number of complex supercharges.

10

Page 11

constant C either never appear (in the antiholomorphic part ∼¯Ψ) or vanish after doing

the Berezin integral (in the holomorphic part ∼ Ψ).

There still exists an interesting mechanism of generating new potential terms via the

deformation. It is based on the observation that, while Ψ2= 0 because of the Grassmann

character of Ψ, this nilpotency property is not longer valid for Ψ ⋆ Ψ and higher-order

star products. Indeed, we find

Ψ ⋆ Ψ =C

2h2, Ψ ⋆ (Ψ ⋆ Ψ) =C

2

?ωh2+ θh3?, Ψ ⋆ (Ψ ⋆ Ψ ⋆ Ψ) =C2

4h4, etc. (44)

The star products of¯Ψ coincide with the ordinary ones and so are identically zero. Let

us then e.g. add to the Lagrangian in (42), with F = a + bz as the simplest choice, the

term a1¯θΨ ⋆ (Ψ ⋆ Ψ). The bosonic part of the corresponding component Lagrangian is

given by the following expression

L = |˙ z|2−1

4h¯h + h(a + bz) +¯h(¯ a +¯b¯ z) −a1C

2

h3. (45)

Here we cannot longer treat¯h as a conjugate of h: both these fields should now be treated

as independent complex ones. Eliminating¯h by its equation of motion, we obtain

L = |˙ z|2+ 4|a + bz|2− 32a1C(¯ a +¯b¯ z)3. (46)

The additional term is holomorphic; by the same token as in Section 2 we conclude that

the corresponding term in the quantum Hamiltonian can be rotated away without trace!

So the modified system proves to be physically equivalent to the undeformed system (has

the same quantum spectrum) in spite of an apparent difference in their Lagrangians.

The star product deformation breaks a half of supersymmetries and the modified action

is manifestly invariant only under the holomorphic half of the original supersymmetry.

Since after rotation we reproduce the original system, the modified system should also

respect some additional hidden supersymmetry of the opposite holomorphy, like in the

(1,2,1) system of Ref. [8] discussed above.

4 Field theories

The first example of an anticommutative deformation of a supersymmetric field theory

was considered in Ref. [1]. Seiberg took the standard Wess-Zumino model

L =

?

d4θ¯ΦΦ +

??

d2θ

?mΦ2

2

+λΦ3

3

?

+ c.c

?

(47)

(where now?d2θ(θαθα) = 1) and deformed it by introducing the nontrivial anticommu-

{θα,θβ} = Cαβ,

tator

(48)

11

Page 12

Cαβ= Cβα, in the assumption that all other (anti)commutators vanish,

{¯θ˙ α,¯θ

˙β} = {θα,¯θ

˙β} = [θα,xL

µ] = [¯θ˙ α,xL

µ] = [xL

µ,xL

ν] = 0 .(49)

Note that this all was written in the chiral basis, xL

xµwas assumed to be Euclidean. We will work in Minkowski space, however, and will not

be scared by the appearance of complexities at intermediate steps. The Minkowski space

deformation (48), (49) is analogous to the SQM deformation (22) with¯C =˜C = 0.

The anticommutator (48) introduces a constant self-dual tensor which explicitly breaks

Lorentz invariance. However, the deformed Lagrangian expressed in terms of the compo-

nent fields proves still to be Lorentz invariant. Indeed, it is easy to find that the kinetic

term?d4θ¯ΦΦ is undeformed and the only extra piece comes from

λ

3

µ= xcentral

µ

+iθσµ¯θ. In Ref. [1], the space

?

d2θΦ3→

?

d2θΦ ∗ Φ ∗ Φ = F(mφ + λφ2) −λ

3det?C?F3. (50)

It depends only on the scalar det?C? and is obviously Lorentz invariant. Adding¯F(m¯φ+

λ¯φ2) from?d2¯θ¯Φ3and F¯F from the kinetic term, and expressing F and¯F via φ and¯φ, we

∼ (m¯φ + λ¯φ2)3.

We have seen, however, that such a holomorphic deformation can be rotated away

without trace! In other words, the deformation (48) does not change the dynamics (the

spectrum of the Hamiltonian etc) of the Wess-Zumino model in Minkowski space9.

The final example is the deformed N = 2 gauge theory [7,14]. There exists in this

case a natural Lorentz invariant deformation [7],

see that the undeformed potential |mφ+λφ2|2acquires an extra holomorphic contribution

{θα

i,θβ

j} =

1

4Jǫαβǫij,(51)

i,j = 1,210. The Lagrangian of the deformed N = 2 supersymmetric U(1) theory is [7,14]

L = Lφ+ LΨ+ LA,

Lφ= −1

2?¯φ

(52)

?

φ +JAmAm

1 + J¯φ

+1

4

J3∂m¯φ∂m¯φ

1 + J¯φ

?

, (53)

LΨ= i

?

Ψiα+JAm(σm)α

˙ α¯Ψi ˙ α

1 + J¯φ

?

(σn)α˙β∂n

?

¯Ψ

˙β

i

1 + J¯φ

?

,(54)

LA=1

4(1 + J¯φ)2?

fmn= ∂m

fmnfmn+ fmn˜fmn

?

, (55)

?

1

1 + J¯φAn

?

− ∂n

?

1

1 + J¯φAm

?

.(56)

9To avoid a misunderstanding, we would like to point out that even in Minkowski space, the fields φ

and¯φ (as well as F and¯F) after deformation should be treated as complex fields which are not conjugate

to each other. The standard complex conjugacy requirements can be consistently imposed on the rotated

fields and their canonical momenta.

10The deformation parameter J is related to the original one I [7] as J = 4I .

12

Page 13

The Lagrangian (52) was derived originally in Euclidean space. In Minkowski space,

it is clearly complex. Bearing in mind the previous discussion, it is natural to suggest,

however, that the corresponding Hamiltonian is cryptoreal. Leaving the issue of crypto-

reality of the full field theory Hamiltonian for future study, let us disregard the fermion

part of (52) and consider the 1D reduction of what is left. We will show that the resulting

quantum-mechanical model is cryptoreal and actually amounts to the free model. The

reduction goes as

? → ∂2

t, ∂mφ∂m¯φ →˙φ˙¯φ, AnAn→ −A0A0+?A?A

(57)

and we obtain

LQM

bos=1

2

˙φ˙¯φ +1

2

˙?A˙?A −1

24

J4(˙¯φ)4

(1 + J¯φ)2+J

2

¨ ¯φ

1 + J¯φA2

0. (58)

The corresponding canonical Hamiltonian, in the obvious notation, is as follows

H = 2P¯P +1

2

?P?P +2

3J4

P4

(1 + J¯φ)2− 2J2A2

0

P2

(1 + J¯φ)2. (59)

Making the rotation

H′= eRHe−R,R = −i

3

J3P3

1 + J¯φ+ iA2

0

JP

1 + J¯φ,

(60)

we find that

H′= 2P¯P +1

2

?P?P ,

(61)

i.e.

our quantum-mechanical system is reduced to the free one. In the full 4-dimensional

case the situation is more subtle due to the presence of the term ∼ εmnrqthat vanished

after reduction. Our simple 1D consideration shows that the corresponding dynamics in

Minkowski space is expected to be “almost trivial”. Nevertheless, we do not see reasons

why the deformation in this case can be entirely rotated away. Rather, the situation

should be similar to what we observed in the Aldrovandi-Schaposnik model. To get a

deeper insight into these issues, it would be instructive to analyze, from a similar point

of view, the deformations of the nonabelian N = 2 gauge theories [7] and the models

involving hypermultiplets [15], which are not free in the undeformed limit J = 0.

the deformation is rotated away without trace, like in the examples above, and

5 Discussion

Our main result is that NAC deformations of supersymmetric theories are well defined

not only in Euclidean, but also in Minkowski space. In spite of its unfriendly looking

complex appearance, the deformed theory can be endowed with a Hilbert space where

the Hamiltonian is Hermitian and its spectrum is real. In many cases (in particular in

13

Page 14

the case of the deformed Wess-Zumino model considered in Seiberg’s original paper), the

deformed Hamiltonian is actually physically equivalent to the undeformed one. Extra

contributions stemming from nonanticommutativity have holomorphic structure and can

be “rotated away” without trace, as was explained in the text of the paper. For some

other NAC theories, the new Hamiltonian is not equivalent to the old one and deformation

brings about nontrivial changes in dynamics.

We discussed at length a one-dimensional SQM example due to Aldrovandi and Scha-

posnik. While going to 4D field theories, the requirement that Lorentz invariance is kept

after deformation dictates the undeformed theory to possess at least N = 2 supersymme-

try [see Eq.(51)]. The Lagrangian of the deformed N = 2 gauge theory was constructed

before. We have not proven, but argued that it is cryptoreal (i.e. the Hamiltonian can be

made Hermitian) but is not equivalent to the undeformed Lagrangian. A thorough study

of this interesting question is a problem for the future.

Another interesting direction of study, not related to nonanticommutativity, but re-

lated to cryptoreality is the following. In Ref. [16], we constructed a gauge theory in six

dimensions which is superconformal at the classical level. It is renormalizable, and the

variant of the theory involving interaction with a hypermultiplet [17] is anomaly free [18].

However, this theory involves higher derivatives, which may in principle lead to the loss

of unitarity due to the presence of ghosts. In particular, the theory involves scalar fields

D of canonical dimension 2 with the potential ∼ D3. Naively, such a potential means

vacuum instability and the associated loss of unitarity. We have seen, however, that

the QM models with the potentials V (x) ∼ ix3or V (x) ∼ x3can be meaningful since

their Hamiltonians can be made Hermitian. It is not excluded that this is also the case

for certain higher-derivative field theories and, in particular, for the models constructed

in [16,17].

Acknowledgements

E.I. acknowledges a support from RFBR grant, project No 06-02-16684, NATO grant

PST.GLG.980302, the grant INTAS-05-7928, the FRBR-DFG grant 06-0204012, the DFG

grant No 436 RUS 113/669/0-3 and a grant of the Heisenberg-Landau program. He thanks

B. Zupnik for useful discussions and the SUBATECH for the warm hospitality in Nantes.

A.S. is grateful to C. Bender for the interest in the work and valuable correspondence.

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