N on.,.anticommutativity as Cliffordization and PT-hermitian Hamiltonians

Francesco Toppan

CBI'F, Rua Dr. Xavier Bigaud 150, cep 22290-180, Rio de Janeiro (RJ), Brazil [email protected]

Abstract It is shown that uon-anticommutative can be described through a Cliffordization of the supcrspace fermionic coordinates. A Berezin-like calculus for odd Clifford variables is introduced. The Fermionic covariant derivatives do not satisfy the graded Leibniz rule. PT-symmetric hamiltonians can he encountered when solving the equations of motion of the auxiliary fields.

1 Introduction We discuss in this work the main results and approaches of [l] to the formulation of Non-Anticommutative (NAC) supersymmetry in terms of a Cliffordization of the fermionic superspacc coordinates. Most of the existing works in the literature concerning the formulation of Nonanti­ commutative supersymmctry adopt (see e.g. [2]) the viewpoint of the deformation theory, as discussed in [3]. It was shown, e.g., in (4] that a Moyal-star deformation of the Grass­ mann algebra produces a . Needless to say, Grassmann generators can be regarded as fermionic parameters entering a superspace. The deformation results in an overall Cliffordization of the superspace. Several papers, either analyzing specific mod­ els [5- 7], or the general properties of the deformed supersymmetric theories [3, 9, 10], were produced within this framework. In [l], a different viewpoint was advocated. Instead of introducing nonanticommu­ tativity from a starting undeformed algebra, a top-down approach, much in the spirit of the Inonii-Wigner contraction, was proposed. From the very beginning, the super­ group associated to a superLie algebra can be constructed through "exponentiation" of the super- generators, by relaxing the condition that their associated odd pa­ rameters are of Grassmann type. In the particular case of the supergroup associated to the one-dimensional supersymmetric quantum mechanics, the odd-parameters can be assumed to satisfy a Clifford-algebra, whose normalization depends on (the inverse of) a mass-scale parameter. In the limit for the mass-scale going to infinity one recovers, by contraction, the standard supersymmetry. The construction of [1] is based on a further feature, the introduction of a an extension of the ordinary Berezin calculus (derivation and integration), which can be made consis­ tent for odd Clifford variables (whose square is non-vanishing). A superspace, spanucd by odd Clifford variables, can be defined and the fermionic covariant derivatives can be com­ puted with a standard procedure. The fermionic covariant derivatives, in the oclcl-Clifforcl

140 variables superspace, do not satisfy, in general, the graded Leibniz rule. While this fea­ ture is problematic in dealing with theories presenting chiral supcrfidds (the product ol" chiral supr,rficlds is no longf!l' a chiral supcrfield, if the graded Leibniz rule is violated), unconstrained superfields do not pose a problem. We show that the simplest non-trivial c~xample where this framework can be applied consists of an N = 2 non-anticomrnutative supersyrnmetric quantum mechanical system for the real (1, 2, 1) bosonic superfield with constant kinetic term and a trilincar superpotcntial. The theory is derived in terms of the component fields. It is further shown that its bosonic sector, divided into phys­ ical and auxiliary fields (the latter satisfy pmely algebraic equations of motion) is such t.hat, depending on the type of non-anticommutative deformation and coupling constants, the purely real algebraic equations of motion of the auxiliary fields can admit complex (imaginary) solutions. Inserting these solution into the equations of motion of the bosonic dynamical fields, one is naturally led to Bender-Boettcher [11 -·13] PT-symmetric pseudo­ hermitian hamiltonians. This feature appears naturally in the Clifford approach to NAC supersymmetry and is due to a very basic property. The auxiliary fields entering the odd-Clifford valued superfields satisfy algebraic equations of motion which are no longer linear, as in the ordinary supersymmetry case, but of higher order. Even if the algebraic equation admits real coefficient, the explicit solution is not necessarily real.

2 Supergroups with odd Clifford parameters: the one-dimensional N = 1, 2 supersymmetry Lie superalgebras are Zrgraded algebras whose generators, split into even and odd sectors, satisfy (anti)commutation relations (see [15-17]). For ordinary Lie groups the elements connected with the identity are obtained through "exponentiation" of the Lie algebra generators. Similarly, the elements connected to the identity of the Lie super­ groups are obtained by exponentiating the generators. In the standard construction ( [15, 16, 18]), Lie supergroups are locally expressed in terms of bosonic parameters associated to the even generators of the Lie superalgebra, while fermionic, Grassmann-number parameters are associated to the odd generators (being Grassmann, in particular, their square is assumed to vanish). A non-anticommutative supersymmetry can be obtained by relaxing the Grassmann condition for the odd parameters. Let's take Nodd parameters (Ji (i = 1, ... , N); we can assume their anticommutators (once conveniently normalized) being expressed through (1) where T/ij is a diagonal with p elements + 1, q elements -1 and r zero elements in the diagonal (therefore N = p + q + r). The subsector of r (};'s with vanishing square is still Grassmann. The procedure of replacing Grassmann variables with the (1) relation corresponds to the "Cliffordization" of the supergroup elements. The N-extended supersymmetry algebra underlying the Supersymmetric Quantum Mechanics is explicitly given in terms of a single even generator H (the hamiltonian, in physical applications) and Nodd generators Qi (i = 1, ... , N), satisfying (anti)commutation relations {Q;, Qj} [H,Qi] (2)

141 The bosonic parameter associated with H is the "time" t. An element g of the unitary N = 1 supergroup is expressed as

g (3)

We assumed the reality condition

(}* B. (4)

In mass-dimension, we have

[H] = 1, [t] = -1, (5) [Q] = ~ . [B] = - !. Since (J2 must have the correct mass-dimension, it should be expressed in terms of some positive mass scale M. We can distinguish three cases, up to a normalization factor. We can set

(}2 = € (6) M' with € = 0 {the Grassmann case), € = +1 or € = -1. Three N = 1 supergroups are associated with the N = 1 superalgebra {2). By setting

(7)

we obtain, for€= 0, 1, -1: i) € = 0, the rational case,

g = e-iHt(l + )..(}Q), {8)

ii) € = 1, the trigonometric case,

g = e-iHt(cosX +lsinX), (9)

where I= BQ/!i- and 12 = -1; iii) € = -1, the hyperbolic case,

g = e-iHt(coshX + JsinhX), (10)

where J = BQ/!i- and J2 = 1. The ordinary Grassmann case can be recovered from both Clifford cases in the special limit M --+ oo. The extension of the above procedure for arbitrary N is straightforward. For N = 2 we introduce B1 , B2 (B;* = B;), satisfying B;2 = *'where both €i, €2 can assume the three values 0, +1 and -1 {it is convenient to define "€" as€= €1€2 ; € = 1 corresponds to the Euclidean deformation, while € = -1 corresponds to the Lorentzian deformation of the N = 2 Grassmann supersymmetry).

142 3 A Berezin-like calculus for odd-Clifford variables The Berezin calculus sets the rules for the derivation and the integration of odd Grass­ rnann variables [14]. For our purposes we need to introduce a calculus which substitutes the Berezin calculus in the case of odd variables of Clifford type. For a single Grassrnann variable() the Berezin calculus states that the derivative 80 is normalized s.t. 80() = 1, while giving vanishing results otherwise. The Berezin integration I d(} coincides with the Berez in derivation (f d() = ao). The extension of the Berezin calculus to an arbitrary number of Grassmann variables is straightforward. Notice that, if() has mass-dimension [BJ = - ~ as in the previous Section, then 80 has mass-dimension [80] = !· We establish now the rules for an analogous calculus in the case of an odd () s.t. 2 () = iJ f' 0. We introduce an odd derivation 80 for the Clifford B by assuming that it has the same mass-dimension as the Berezin derivation and coincides with it in the M -+ oo limit. Therefore

(11)

The application of 80 to the powers (Jn, for integral values n > 1, is determined under the assumption that 80 satisfies a graded Leibniz rule. Let Ji, / 2 be two functions of grading deg(/1), deg(fz) respectively (deg(/) = 0 for a bosonic function f, while deg(!) = 1 for a fermionic function); we assume

(12)

As a consequence, the application of 80 to even and odd powers of () is respectively given by

(13)

By requiring the integral of a total derivative to be vanishing we can unambiguously fix

Id() ()2k = 0. (14)

The rule for the integration over odd powers of() can be set by requiring, as in the Berezin case, that the integration coincides with the derivation 80 . Therefore

(15)

4 Fermionic covariant derivatives for Clifford-valued superfields The construction of supersymmetric fermionic covariant derivatives for odd-Clifford variables mimics the [19) (see also [20]) construction in the Grassmann case. Let's discuss for simplicity the N = 1 supersymmetry algebra (2). Its supergroup element is given by g, introduced in (3). The left (right) action of the supersymmetry generator Q on g (Qg

143 and, respectively, gQ) induces the operator Qi (QR), determined in terms oft, 0 and their derivatives, s.t.

Qg Qig, gQ QRg, (16)

where

{Qi,Qd -H, {Qi, QR} 0, {QR,QR} H. (17)

Qi is the covariant fermionic derivative, also denoted as "D", while QR= Q. In the odd-Clifford case (for f = ±1) the (17) algebra is reproduced by D Qi, Q =QR (with dropped .X-dependence), given by

D 89 - i08t + i ~89 8t. Q Bo+ i08t - i ~80 81. (18)

The N = 2 case produces two fermionic covariant derivatives Dj and the two supersym­ metry generators Qi given by

Be; - i0i8t + i z.ao;at,

80; + i0i8t - i z.ae;at. {19)

They satisfy the algebra

{D;,Dj} -o;iH, {D;, Qi} 0, {Q;,Qj} O;jH. (20)

Let us denote with "*" the symmetrized (in the bosonic case) product of odd-Clifford variables expanded superfields A, B. The fermionic covariant derivatives D 1 , D 2 do not satisfy a graded Leibniz property, namely

.6.;(A, B) = D;(A * B) - (D;A * B + (-1)-deg(A) A* D;B) # 0. (21)

5 AN= 2 NAC Supersymmetric Quantum Mechanics model Without loss of generality, the most general N = 2 supersymmetric quantum me­ chanical model described by a (odd-Clifford expanded) real bosonic N = 2 superfield =

1 SN=2 = - -/ dt d01dB2(D1 * D2<1> * + a). (22) 2m j +~*6

144 Expressed in terms of component fields, the trilinear potential V is given by

(23)

where o = o1 o2 . For o = 0, the case of the Grassmann supersyrnmetry, the equation of motion of the auxiliary f is a linear equation. For o =f. 0, the equation of motion for f is an algebraic equation with several branches of solutions. Let us specialize the discussion to the Euclidean-deformed (o = 1) a= ~ case. In this

case we have that, setting the fermionic component fields 1/! 1 , ·t/;2 equal to zero, and solving the algebraic equation of motion for the auxiliary fields f, the purely bosonic effective action S for

(24)

This action corresponds to a Bender-Boettcher (11-13] PT-symmetric hamiltonian H (with p = ~)

12 i3 12 1 H -p - -

p

It is worth stressing the fact that our original odd-Clifford N = 2 supersymmetric action for ¢>, 1/!1, 1/!2 , f (no matter which Clifford deformation and which real value of the a pa­ rameter are taken) satisfies the reality condition. It's only after solving the equation of motion for the auxiliary field f that the imaginary unit i appears (for the above-discussed cases) in the reduced action. What we succeeded here is to directly link a PT-symmetric hamiltonian with a Non-anticommutative N = 2 supersymmetric quantum mechanical system.

6 Conclusions Non-Anticommutative supersymmetry was vastly studied in recent year after the work of (2]. In (6] it was pointed out that the most convenient non-trivial setting to investigate its main properties is provided by the non-anticommutative supersymmetric quantum mechanics. The models studied in (6] is based on a type of non-anticommutative de­ formation which is different from the one here investigated. We adopted the point of view that the non-anticommutative deformation is encoded in the presence of fermionic covariant derivatives which do not satisfy the graded Leibniz rule. This type of deforma­ tion can be applied to systems involving unconstrained superfields (i.e. superfields not involving chiral constraints defined in terms of the fermionic covariant derivatives). We discussed the example of the real N = 2 (1, 2, 1) superfield with trilinear superpotential. Our approach made use of the notion of supergroups parametrized by odd parameters which are no longer Grassmann, but satisfy instead a Clifford algebra. This viewpoint

145 should be regarded as complemementary w.r.t. the non-anticommutative deformations of the Grassmmann algebra defined in terms of a fermionic version of a Moya! star product (see [4]). One important issue is that our system produces iu a natural way a pseudo-hermitian hamiltonian of Bender-Boettcher type. This feature is understood as follows. The bosonic sector of supersymrnetric theories involves both dynamical and auxiliary.fields. In partic­ ular the auxilia1y fields satisfy algebraic equations of motion. In the standard (anticom­ mutative) supersymmetry, the algebraic equations of motion satisfied by the auxiliary fields are linear. This is no longer the case for non-anticommutative supersymmetry. Depending on the particular combination of both deformation parameters arid coupling constants entering the action, the algebraic equations of motion, even if expressed in terms of real coefficients, can admit complex (imaginary) solutions. Inserting back these solutions into the equations of motion of the dynamical bosonic fields produces the equa­ tions of motion of pseud-hermitian, PT-symmetric hamiltonians, with real spectrum of eigenvalues. The first paper pointing out a connection (in a different framework than the one here discussed) between Non-anticommutative supersymmetry and PT-symmetric hamiltonian is [7). Concerning the nature of Non-anticommutative supersymmetry and the failing of the graded Leibniz rule, we point out that this feature is also present in the works, [8-10), where a Drinfeld twist deformation of the Hopf algebra structure of a (given) supersymmetry algebra is discussed. This twist deformation implies, in partic­ ular, a deformed coproduct .0. for the fermionic generators Q; of the superalgebra, s.t . .0.(Q;) = Q; ® 1+1 ® Q; + ( ... ), where ( ... ) denotes the extra terms arising from the deformation. The coproduct for the even generators is left undeformed. We recall that the physical interpretation of the coproduct is in the construction of tensored multipar­ ticle states. The breaking of the graded Leibniz rule follows automatically from this construction. In the present work we discussed one-dimensional Non-anticommutative supersym­ metric quantum mechanics. The extension of our framework to, let's say, superPoincare algebras in higher dimension is straightforward. It is sufficient to introduce, e.g., for the anticommutator of fermionic Majorana coordinates Oa, (}fJ with spinorial indices Q, (3, the conjugation matrix Ca/J· This construction requires the symmetry of CafJ under the exchange a tt (3.

Acknowledgement I am grateful to the organizers for the invitation. This work reports results of a collaboration with Z. Kuznetsova and M. Rojas.

References

[1) Z. Kuznetsova, M. Rojas and F. Toppan, arXiv:0705.4007[hep-th). To appear in IJMPA.

[2) N. Seiberg, JHEP 0306, 010 (2003) 010.

[3] F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, Ann. Phys. (N.Y.) 111, 61 (1978).

146 [4] A. C. Hirshfeld and P. Henselder, Ann. Phys. 302, 59 (2002) .

[5] L. Alvarez-Gaume and M. A. Vazquez-Mozo, JHEP 0504, 007 (2005).

[6) L. G. Aldrovandi and F. A. Schaposnik, JHEP 0608, 081 (2006)

[7) E. A. Ivanov and A. V. Smilga, hep-th/0703038.

[8) Y. Kobayashi and S. Sasaki, Int. J. Mod. Phys. A 20, 7175 (2005).

[9) B. Zupnik, Phys Lett. B 627, 208 (2005).

(10) M. Ihl and C. Siimann, JHEP 0607, 027 (2006).

(11) C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).

(12] C. M. Bender, S. Boettcher and P. N. Meisinger, J. Math. Phys. 40, 2201 (1999).

(13] C. M. Bender, Contemp. Phys. 46, 77 (2005).

(14] F. A. Berezin, "Introduction to Algebra and Analysis with Anticommutative Vari­ ables" (in Russian), Izd. Mosk. Gos. Univ., Moscow (1983).

(15] V. G. Kac, Adv. Math. 26 (1977) 8.

(16] V. G. Kac, Comm. Math. Phys. 53 (1977) 31.

(17] L. Frappat, A. Sciarrino and P. Sorba, "Dictionary on Lie Superalgebras", hep­ th/9607161.

(18] F. A. Berezin and V. N. Tolstoy, Comm. Math. Phys. 78 (1981), 409.

(19] A. Salam and J. Strathdee, Nucl. Phys. B 76 (1974) 477.

(20) J. Wess and J. Bagger, "Supersymmetry and Supergravity" 2nd ed., Princeton Un. Press, Princeton (1992).

147