N on.,.anticommutativity as Superspace 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 supersymmetry 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 Clifford algebra. 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-Lie algebra 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 <I> 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 Lie superalgebra 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 matrix 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.