Twist Deformations of the Supersymmetric Quantum Mechanics
aa r X i v : . [ h e p - t h ] D ec Twist Deformations of theSupersymmetric Quantum Mechanics
P. G. Castro ∗ , B. Chakraborty † , Z. Kuznetsova ‡ and F. Toppan §∗†§ CBPF, Rua Dr. Xavier Sigaud 150,cep 22290-180, Rio de Janeiro (RJ), Brazil. † S. N. Bose National Center for Basic Sciences,JD Block, Sector III, Salt-Lake, Kolkata-700098, India. ‡ UFABC, Rua Catequese 242, Bairro Jardim,cep 09090-400, Santo Andr´e (SP), Brazil.
November 19, 2018
Abstract
The N -extended Supersymmetric Quantum Mechanics is deformed via an abeliantwist which preserves the super-Hopf algebra structure of its Universal EnvelopingSuperalgebra. Two constructions are possible. For even N one can identify the1 D N -extended superalgebra with the fermionic Heisenberg algebra. Alternatively,supersymmetry generators can be realized as operators belonging to the UniversalEnveloping Superalgebra of one bosonic and several fermionic oscillators.The deformed system is described in terms of twisted operators satisfying twist-deformed (anti)commutators.The main differences between an abelian twist defined in terms of fermionicoperators and an abelian twist defined in terms of bosonic operators are discussed. CBPF-NF-012/09 ∗ e-mail: [email protected] † e-mail: [email protected] ‡ e-mail: [email protected] § e-mail: [email protected] Introduction
In this paper we investigate the abelian twist-deformation of the fermionic Heisenberg alge-bra, introduced in [1], in application to the deformation of the Supersymmetric QuantumMechanics. We remark that the abelian twist deformation of the fermionic Heisenbergalgebra gives rise to the Cliffordization of the Grassmann variables, recovering, in a moregeneral setting, the results of [2]. The connection of the fermionic Heisenberg algebrawith the graded algebra underlying the Supersymmetric Quantum Mechanics is twofold.Indeed, the N -extended, one-dimensional superalgebra is isomorphic to the fermionicHeisenberg algebra for an even number N of (odd) supercharges. On the other hand,the same superalgebra can be realized, essentially, in terms of operators belonging to theUniversal Enveloping Algebra of one bosonic Heisenberg algebra and several copies of thefermionic Heisenberg algebras. In the following we point out the differences of the twoschemes. Within the second scheme we can recover, on the module, a lower dimensionalversion of the Cliffordization of [3] and [4].We further point out that in all previous studies in the literature graded undeformedbrackets or graded, Moyal-type, brackets were used along with the deformed coproducts.In our approach, on the other hand, motivated by the considerations in [5] (restricted,in that paper, to bosonic algebras), we use twist-deformed brackets, as in [1]. The twist-deformed brackets, which implement twist-deformed adjoint action, are directly associatedwith the twist-deformed coproduct.The abelian character of the (fermionic) twist deformation implies that the fermionicsupercharges are deformed. On the other hand, the bosonic charges remain undeformed.Nevertheless, even in the bosonic sector, the theory gets modified since the coproductapplied to bosonic operators gets deformed and can therefore affect multi-particle bosonicsystems. In order to stress the difference between a “fermionic” abelian twist and a“bosonic” abelian twist (in the first case the exponential defining the twist is given by atensor product of fermionic generators while, in the second case, it is a tensor product ofbosonic generators), we apply the abelian twist deformation to (an enlarged version of)the bosonic Heisenberg algebra for a particularly simple example. We show that in thiscase even the bosonic operators get deformed, not just their coproducts. In particular,an operator which gets deformed is the Hamiltonian of the harmonic oscillator.Returning to Supersymmetric Quantum Mechanics, in the second framework (gener-ators realized as operators in a Universal Enveloping Superalgebra), besides the super-charges, the fermionic derivative operators can be constructed (and deformed). The factthat their coproduct is deformed can be interpreted as a breaking of the (graded) Leibnizrule, in accordance with several results obtained in the literature, by using a variety ofdifferent methods [6, 7, 8].Non-anticommutative supersymmetric theories have been investigated by assuming,either as a mathematical possibility or in the string context, the spinorial coordinates tobe non-anticommutative (see [9, 10, 11, 12, 13, 14, 15, 16]). The particularly influential[16] paper introduced non-anticommutative supersymmetry in a 4-dimensional Euclideansuperspace. It provided motivations for studying lower-dimensional non-anticommutativesupersymmetric models [17, 18, 19, 20, 21, 22]. In one time-dimension, in particular, (i.e.non-relativistic Supersymmetric Quantum Mechanics) non-anticommutative deformations2ere studied in [23, 24, 7].It comes as no surprise that the majority of the works on non-anticommutative super-symmetry found inspiration from the related works of non-commutative deformations ofthe ordinary bosonic theories (see [25] for a review), whose recent upsurge is essentiallydue to the seminal works of [26] and [27]. The notion of the Drinfeld twist (of the Uni-versal Enveloping Algebra of the Poincar´e algebra) was first applied in [28] to restore theLorentz symmetry which would be otherwise spoiled in a relativistic non-commutativetheory. In the supersymmetric context, the Drinfeld twist has been investigated in the al-ready recalled papers [4, 3, 6, 8], while a Jordanian twist for the osp (1 |
2) SuperconformalQuantum Mechanics was also considered in [29].The scheme of this paper is as follows. In the next Section we introduce the abeliantwist of the fermionic Heisenberg algebra, presenting all the relevant formulas (twist-brackets and so on). In Section we point out the isomorphism between fermionicHeisenberg algebra and the one-dimensional N -Extended superalgebra for even valuesof N . The twist-deformation of the Supersymmetric Quantum Mechanics in terms of itssuperspace representation is investigated in Section . We introduce the relevant Univer-sal Enveloping Algebras for the construction. The specializations to N = 2 and N = 4are detailed. Non-upper triangular supersymmetry generators obtained from twist aregiven. The abelian twist in the bosonic case is presented in Section . The Grassmann algebra is generated by N anticommuting coordinates θ α . These coor-dinates, together with their fermionic, Berezin, derivatives ∂ α , define a Lie superalgebrawith 2 N odd generators (the Grassmann coordinates and their derivatives) and a singleeven generator, a central charge z . The (anti)-commutation relations are given by { θ α , θ β } = { ∂ α , ∂ β } = 0 , { ∂ α , θ β } = δ αβ z, [ z, ∂ α ] = [ z, θ α ] = 0 . (1)The central charge has to be introduced in order to form a Lie superalgebra.The (1) algebra is the fermionic Heisenberg algebra and will be denoted, see [1], as h F ( N ). Its Universal Enveloping Algebra U ( h F ( N )) has the structure of a Hopf super-algebra. The central charge z is treated at par with the other generators θ α and ∂ α ∗ ,so its coproduct is ∆( z ) = z ⊗ + ⊗ z and the antipode is S ( z ) = − z . In the unde-formed case the coproduct of θ α and ∂ α reads, respectively, ∆( θ α ) = θ α ⊗ + ⊗ θ α and∆( ∂ α ) = ∂ α ⊗ + ⊗ ∂ α . The corresponding antipodes are given as S ( θ α ) = − θ α and S ( ∂ α ) = − ∂ α ∗ This is analogous to considering the mass parameter of the Galileo algebra as a Lie generator, so thatone can use non-projective representation [30, 31]. The same prescription was used in [1] to deform theHeisenberg algebra. θ α ] = − , [ ∂ α ] = ,[ z ] = 0.An abelian twist F (for a review, see [32] and references therein) given by F := f α ⊗ f α = exp ( C αβ ∂ α ⊗ ∂ β ) , F − := f α ⊗ f α = exp ( − C αβ ∂ α ⊗ ∂ β ) (2)can be introduced in terms of the diagonal matrix C αβ = 1 M η αβ , (3)where M is a mass-parameter and η αβ is a non-dimensional matrix which admits p positivediagonal elements +1, q negative diagonal elements − r zero elements ( p + q + r = N ).(the Sweedler notation has been used).Following the notations and conventions of [1], the only deformed coproducts corre-spond to θ α (the others remain undeformed):∆ F ( θ α ) = F ∆( θ α ) F − = ∆( θ α ) + C αβ ( ∂ β ⊗ z − z ⊗ ∂ β ) , (4)where ∆( θ α ) = θ α ⊗ + ⊗ θ α is the undeformed coproduct introduced earlier.Since χ = f α S ( f α ) = exp ( − C αβ ∂ α ∂ β ) = , (5)the antipode is also undeformed.Among the generators, only the θ α ’s get deformed. We have θ F α := f β ( θ α ) f β = θ α + C αβ ∂ β z. (6)The deformed generators differ from the original ones in terms of a shift (the fermioniccounterpart of the Bopp shift, see also [33]).The universal R -matrix is simply F − , so that∆ F ( θ F α ) = θ F α ⊗ + ⊗ θ F α + 2 C αβ ∂ β ⊗ z. (7)The antipodes are S ( θ F α ) = − θ α + C αβ z∂ β = − θ F α + 2 C αβ z∂ β . (8)The deformed brackets (cid:2) u F , v F (cid:9) F = X k ( u F ) k v F ( − | v F || ( u F ) k | S ( u F ) k of the deformed generators coincide with the original (1) algebra: (cid:8) θ F α , ∂ F β (cid:9) F = δ αβ z F , (cid:8) θ F α , θ F β (cid:9) F = 0 , (cid:8) ∂ F α , ∂ F β (cid:9) F = 0 , (cid:2) ∂ F α , z F (cid:3) F = (cid:2) θ F α , z F (cid:3) F = 0 . (9)4he ordinary brackets of the deformed quantities yield a nonlinear algebra (cid:8) θ F α , ∂ F β (cid:9) = δ αβ z, (cid:8) θ F α , θ F β (cid:9) = 2 C αβ z , (cid:8) ∂ F α , ∂ F β (cid:9) = 0 , (cid:2) z F , θ F α (cid:3) = (cid:2) z F , ∂ F α (cid:3) = 0 . (10)The multiplication m on the module [1] acts as follows m ( θ α ⊗ θ β ) = θ α · θ β , m ( θ α ⊗ z ) = θ α · z, m ( z ⊗ z ) = z . (11)The deformed multiplication between elements belonging to the module (using the Sweedlernotation) is given by a ⋆ b ≡ m F ( a ⊗ b ) = ( m ◦ F − )( a ⊗ b ) = X α ( − | ¯ f α || a | ¯ f α ( a ) ¯ f α ( b ) . (12)Defining [ a, b } ⋆ ≡ a ⋆ b + ( − | a || b | b ⋆ a , we have { θ α , θ β } ⋆ = 2 C αβ z , [ z, θ α ] ⋆ = 0 . (13)The above fermionic Moyal-brackets, together with the relations { ∂ α , θ β } ⋆ = δ αβ z, { ∂ α , ∂ β } ⋆ = 0 , [ z, ∂ α ] ⋆ = 0 , (14)makes the ⋆ -brackets isomorphic to the ordinary brackets of the deformed quantities. D N -Extended Superalgebra In this section, we are going to basically establish the isomorphism between the fermionicHeisenberg algebra introduced in the previous section and the one-dimensional N -extendedsuperalgebra for even values of N . To this end, consider the supersymmetry algebra withodd generators b Q I ( I, J = 1 , . . . , N ) and an even generator H , given by { b Q I , b Q J } = δ IJ H, h H, b Q I i = 0 . (15)For N even, we can split the odd sector into a chiral and an antichiral set: Q i = b Q i + i b Q i + N ,Q i = b Q i − i b Q i + N , (16)5ith i = 1 , . . . , N .The algebra can be reexpressed as (cid:8) Q i , Q j (cid:9) = 2 δ ij H, { Q i , Q j } = (cid:8) Q i , Q j (cid:9) = 0 , [ H, Q i ] = (cid:2) H, Q i (cid:3) = 0 . (17)It is isomorphic to (1) if we identify Q i with θ α , Q i with ∂ α and 2 H with z .We shall deform the algebra (17) by means of the Abelian twist F = exp (cid:18) C ij Q i ⊗ Q j (cid:19) , (18)with C ij = η ij M , where η ij is a non-dimensional diagonal matrix admitting p positive +1 , q negative -1 and r zero entries ( p + q + r = N ).This deformation coincides with (2). In particular the deformed coproduct of Q i reads∆ F ( Q i ) = ∆( Q i ) + C ij ( Q j ⊗ H − H ⊗ Q j ) . (19)The antipode is undeformed due to χ = f α S ( f α ) = exp (cid:18) − C ij Q i Q j (cid:19) = . (20)The only deformed generators are the Q i : Q F i = Q i + C ij Q j H. (21)The universal R -matrix is F − , so∆ F ( Q F i ) = Q F i ⊗ + ⊗ Q F i + 2 C ij Q j ⊗ H. (22)The antipodes are S ( Q F i ) = − Q i + C ij Q j H = − Q F i + 2 C ij Q j H. (23)Now the deformed brackets are: n Q F i , Q F j o F = δ ij H F = δ ij H, n Q F i , Q F j o F = 0 , (cid:8) Q F i , Q F j (cid:9) F = 0 , (cid:2) Q F i , H F (cid:3) F = h Q F i , H F i F = 0 . (24)The ordinary brackets of the deformed quantities are n Q F i , Q F j o = δ ij H F , n Q F i , Q F j o = 0 , (cid:8) Q F i , Q F j (cid:9) = 2 C ij ( H F ) , h H F , Q F i i = (cid:2) H F , Q F i (cid:3) = 0 . (25)6 The superspace representation
To make the connection with the superspace we need to introduce the Grassmann variables θ I and their derivatives ∂ θ I which (along with the central extension z ) satisfy the h F ( N )algebra, as well as the bosonic parameter t and its derivative ∂ t which (along with thecentral extension ~ ) satifies the bosonic Heisenberg algebra h B (1), obtaining, in principle,the h B (1) ⊕ h F ( N ) algebra. We can now identify the two central extensions ( z = ~ ),thereby obtaining an algebra which we shall call h (1 , N ). Throughout this section we willbe working with its universal enveloping algebra, U ( h (1 , N )). An explicit realization ofthe N -extended supersymmetry algebra (15) is obtained in terms of composite operatorsbelonging to U ( h (1 , N )) ( N ≡ N ) † . Explicitly, b Q I = ∂ θ I + i ~ θ I ∂ t ,H = i∂ t , (26)with I = 1 , . . . , N .Since we are working with U ( h (1 , N )), it is now possible to apply the twist (2) F = exp( C IJ ∂ θ I ⊗ ∂ θ J ) (27)to the supersymmetry generators.We obtain that the deformed coproduct of b Q I is∆ F ( b Q I ) = ∆( b Q I ) + C IJ ( ∂ θ J ⊗ H − H ⊗ ∂ θ J ) . (28)The deformed generators are b Q F I = b Q I + C IJ H∂ θ J , (29)the deformed coproduct of the deformed generators being∆ F ( b Q F I ) = b Q F I ⊗ + ⊗ b Q F I + 2 C IJ ( ∂ θ J ⊗ H ) . (30)The ordinary brackets of the deformed generators read { b Q F I , b Q F J } = δ IJ H + 2 C IJ H , (31)whereas the deformed brackets read just { b Q F I , b Q F J } F = δ IJ H. (32)In a different context, non-linear supersymmetry such as encountered in (31) or in ther.h.s. of (25) was discussed in [34]. † We allow Laurent-expansion of the central element ~ . Concerning the undeformed coproduct of b Q I introduced below, it is assumed to coincide with the undeformed coproduct of a fermionic primitiveelement, i.e., ∆( b Q I ) = b Q I ⊗ + ⊗ b Q I , so that one never encounters ambiguous expressions like ∆( ~ )in (26). .1 The N = 2 case Let us now consider the U ( h (1 , N = 2 supersymmetrygenerators b Q = ∂ θ + i ~ θ ∂ t , b Q = ∂ θ + i ~ θ ∂ t . (33)These can be further augmented by the fermionic covariant derivatives D = ∂ θ − i ~ θ ∂ t ,D = ∂ θ − i ~ θ ∂ t . (34)They satisfy the so-called N = (2 ,
2) pseudo-supersymmetry algebra { b Q I , b Q J } = δ IJ H, { D I , D J } = − δ IJ H, { D I , b Q J } = 0 , h H, b Q I i = [ H, D I ] = 0 . (35)To deform it we can now apply any twist F ∈ U ( h (1 , ⊗U ( h (1 , F = exp (cid:16) ǫM Q ⊗ Q + ηM D ⊗ D (cid:17) , (36)where Q = b Q − i b Q ,D = D − iD (37)and ǫ , η are numbers which can be normalized, without loss of generality, to be +1, − is recovered for η = 0, C = ǫM ).This twist trivially satisfies the cocycle condition since { Q, Q } = { Q, D } = { D, D } = 0.The deformations of the generators are given by b Q F = b Q + ǫM ( b Q − i b Q ) H, b Q F = b Q − iǫM ( b Q − i b Q ) H,D F = D + ηM ( D − iD ) H,D F = D − iηM ( D − iD ) H. (38)8ogether with H F = H . The deformed coproducts are∆ F ( b Q ) = ∆( b Q ) + ǫM ( b Q ⊗ H − H ⊗ b Q ) − iǫM ( b Q ⊗ H − H ⊗ b Q ) , ∆ F ( b Q ) = ∆( b Q ) − iǫM ( b Q ⊗ H − H ⊗ b Q ) − ǫM ( b Q ⊗ H − H ⊗ b Q ) , ∆ F ( D ) = ∆( D ) − ηM ( D ⊗ H − H ⊗ D ) + iηM ( D ⊗ H − H ⊗ D ) , ∆ F ( D ) = ∆( D ) + iηM ( D ⊗ H − H ⊗ D ) + ηM ( D ⊗ H − H ⊗ D ) . (39)The antipode does not get deformed since χ = f α S ( f α ) = exp (cid:16) − ǫM Q − ηM D (cid:17) = , (40)so that the antipodes are S ( b Q F ) = − b Q F + 2 ǫM ( b Q − i b Q ) ,S ( b Q F ) = − b Q F − iǫM ( b Q − i b Q ) ,S ( D F ) = − D F − ηM ( D − iD ) ,S ( D F ) = − D F + 2 iηM ( D − iD ) . (41)The universal R -matrix is R = F − = exp h − (cid:16) ǫM Q ⊗ Q + ηM D ⊗ D (cid:17)i . (42)With those, we are able to work out the deformed coproducts of the deformed quantities.They are ∆ F ( b Q F ) = b Q F ⊗ + ⊗ b Q F + 2 ǫM ( b Q − i b Q ) ⊗ H, ∆ F ( b Q F ) = b Q F ⊗ + ⊗ b Q F − iǫM ( b Q − i b Q ) ⊗ H, ∆ F ( D F ) = D F ⊗ + ⊗ D F − ηM ( D − iD ) ⊗ H, ∆ F ( D F ) = D F ⊗ + ⊗ D F + 2 iηM ( D − iD ) ⊗ H. (43)These yield, as expected, the deformed brackets of the deformed quantities: { b Q F I , b Q F J } F = δ IJ H F , { D F I , D F J } F = − δ IJ H F , { D F I , b Q F J } F = 0 . (44)9e now want to study the deformed multiplication on a module consisting of thespace of functions of Grassmann variables θ , θ . The ordinary multiplication m acts asthe usual Grassmann product, that is m ( θ I ⊗ θ J ) = θ I · θ J . (45)The action of b Q I and D I is { b Q I , θ J } = { D I , θ J } = δ IJ . (46)Now we define the star product to be θ I ⋆ θ J = m F ( θ I ⊗ θ J ) = ( m ◦ F − )( θ I ⊗ θ J ) (47)and proceed to calculate explicitly θ ⋆ θ = − ǫM − ηM ,θ ⋆ θ = ǫM + ηM ,θ ⋆ θ = − iǫM − iηM + θ θ , (48)so that the star-anticommutators are { θ , θ } ⋆ = − (cid:16) ǫM + ηM (cid:17) , { θ , θ } ⋆ = 2 (cid:16) ǫM + ηM (cid:17) , { θ , θ } ⋆ = − i (cid:16) ǫM + ηM (cid:17) . (49)If we now go to the chiral coordinates θ = θ + iθ , ¯ θ = θ − iθ , (50)the star-anticommutators are { θ, θ } ⋆ = − (cid:16) ǫM + ηM (cid:17) , { ¯ θ, ¯ θ } ⋆ = 0 , { θ, ¯ θ } ⋆ = 0 , (51)which is the Cliffordization in half of the coordinates (in the chiral sector) as obtained in[3] and [4].It might appear at first that the bosonic sector does not get deformed at all, but thisis not the case. Consider the bosonic Hermitian operator10 = i b Q b Q − b Q b Q ) . (52)We now proceed to deform it, setting η = 0 for simplicity. Since[ Q, W ] = − HQ, we obtain that W F = W , so that W undergoes no deformation. However, its coproductexhibits nontrivial deformation:∆ F ( W ) = ∆( W ) − ǫM ( Q ⊗ QH + QH ⊗ Q ) . (53) We can also approach the question of supersymmetry deformation within the frameworkof the factorization method, which is an useful method for generating classes of solvablepotentials for a Hamiltonian. It was devised by Infeld and Hull [35] after pioneering worksby Dirac (factorization of the Hamiltonian of the harmonic oscillator) and Schr¨odinger(factorization of the radial part of the Coulomb Hamiltonian) and further generalized byMielnik [36]. The method (on its first-order version) consists of factorizing the Hamilto-nian by introducing intertwining operators A = (cid:18) ddx + α ( x ) (cid:19) , A + = (cid:18) − ddx + α ( x ) (cid:19) , (54)where α turns out to satisfy a Riccati differential equation (for a recent review see [37]).Supersymmetry algebra can be built up in this setting by writing Q = b Q + i b Q and Q = b Q − i b Q used earlier in this section as Q = (cid:18) A + (cid:19) , Q = (cid:18) A (cid:19) , (55)The Hamiltonian is H = 12 { Q, Q } = 12 (cid:18) A + A AA + (cid:19) = (cid:18) H + H − (cid:19) . (56)If we proceed to the same twist as before (setting η = 0), we obtain Q F = (cid:18) A +2 ǫM AA + A (cid:19) , Q F = (cid:18) A (cid:19) , (57)which is a non-upper triangular form for the supersymmetry generator, still satisfying { Q F , Q F } = (cid:18) A + A AA + (cid:19) = 2 H. (58)11 .2 The N = 4 case We now turn to the N = 4 supersymmetry algebra { b Q I , b Q J } = δ IJ H, h H, b Q I i = 0 , (59)( I, J = 1 , , ,
4) and apply the twist F = exp (cid:16) η ij M Q i ⊗ Q j (cid:17) , (60)where η ij is diagonal and Q = b Q − i b Q ,Q = b Q − i b Q . (61)From now on we set η = ǫ and η = η . Being Abelian, this twist trivially satisfiesthe cocycle condition.A similar procedure as before yields the deformation of the generators: b Q F = b Q + ǫM ( b Q − i b Q ) H, b Q F = b Q − iǫM ( b Q − i b Q ) H, b Q F = b Q + ηM ( b Q − i b Q ) H, b Q F = b Q − iηM ( b Q − i b Q ) H. (62)The deformed coproducts are∆ F ( b Q ) = ∆( b Q ) + ǫM ( b Q ⊗ H − H ⊗ b Q ) − iǫM ( b Q ⊗ H − H ⊗ b Q ) , ∆ F ( b Q ) = ∆( b Q ) − iǫM ( b Q ⊗ H − H ⊗ b Q ) − ǫM ( b Q ⊗ H − H ⊗ b Q ) , ∆ F ( b Q ) = ∆( b Q ) + ηM ( b Q ⊗ H − H ⊗ b Q ) − iηM ( b Q ⊗ H − H ⊗ b Q ) , ∆ F ( b Q ) = ∆( b Q ) − iηM ( b Q ⊗ H − H ⊗ b Q ) − ηM ( b Q ⊗ H − H ⊗ b Q ) . (63)The universal R -matrix is simply F − , allowing us to calculate the deformed coprod-12cts of the deformed quantities:∆ F ( b Q F ) = b Q F ⊗ + ⊗ b Q F + 2 ǫM ( b Q − i b Q ) ⊗ H, ∆ F ( b Q F ) = b Q F ⊗ + ⊗ b Q F − iǫM ( b Q − i b Q ) ⊗ H, ∆ F ( b Q F ) = b Q F ⊗ + ⊗ b Q F + 2 ηM ( b Q − i b Q ) ⊗ H, ∆ F ( b Q F ) = b Q F ⊗ + ⊗ b Q F − iηM ( b Q − i b Q ) ⊗ H. (64)The antipodes are S ( b Q F ) = − b Q F + 2 ǫM ( b Q − i b Q ) ,S ( b Q F ) = − b Q F − iǫM ( b Q − i b Q ) ,S ( b Q F ) = − b Q F + 2 ηM ( b Q − i b Q ) ,S ( b Q F ) = − b Q F − iηM ( b Q − i b Q ) , (65)so that the deformed brackets are { b Q F I , b Q F J } F = δ IJ H F , h H F , b Q F I i F = 0 . (66)We shall now find out the action of the deformed multiplication m F on a moduleconsisting of Grassmann variables θ I , I = 1 , . . . ,
4, such that { b Q I , θ J } = δ IJ .Defining the star product as previously, we obtain that θ ⋆ θ = − ǫM ,θ ⋆ θ = ǫM ,θ ⋆ θ = − ηM ,θ ⋆ θ = ηM ; (67)all the other products coincide with the ordinary product.If we now go to chiral coordinates ζ = θ + iθ ,ζ = θ − iθ ,ζ = θ + iθ ,ζ = θ − iθ , (68)13he star-anticommutators are { ζ I , ζ J } ⋆ = − η IJ M , { ζ I , ζ J } ⋆ = 0 , { ζ I , ζ J } ⋆ = 0 , (69)which is again the Cliffordization of the unbarred chiral coordinates as in [3] and [4].We proceed analogously and introduce the bosonic Hermitian operators W = i b Q b Q − b Q b Q ) ,W = i b Q b Q − b Q b Q ) . (70)Their algebra with the Q i ’s is[ Q i , W j ] = − HQ i δ ij (no sum on i) , (71)so that we obtain no deformation at the algebraic level, i.e, W F i = W i . On the otherhand, their deformed coproducts read∆ F ( W i ) = ∆( W i ) − η ij M ( Q j ⊗ Q j H + Q j H ⊗ Q j ) . (72) We start with Heisenberg algebra h B ( N ) , with generators x i , p i , ~ ( i = 1 , , . . . , N ) andnon-vanishing commutation relations given by[ x i , p j ] = iδ ij ~ . (73)We enlarge it by introducing the elements K ij = p i p j ~ ,M ij = x i p j ~ ,N ij = p i x j ~ ,V ij = x i x j ~ , (74)which are now declared to be primitive elements of an enlarged algebra.The thus enlarged algebra satisfies the relations14 K ij , x k ] = − iδ ik p j − iδ jk p i , [ M ij , x k ] = − iδ jk x i , [ N ij , x k ] = − iδ ik x j , [ M ij , p k ] = iδ ik p j , [ N ij , p k ] = iδ jk p i , [ V ij , p k ] = iδ ik x j + iδ jk x i , [ V ij , K kl ] = iδ jk M il + iδ jl M ik + iδ ik N lj + iδ il N kj , [ V ij , M kl ] = iδ il V jk + iδ jl V ik , [ V ij , N kl ] = iδ ik V jl + iδ jk V il , [ K ij , M kl ] = − iδ ik K jl − iδ jk K il , [ K ij , N kl ] = − iδ il K jk − iδ jl K ik , [ M ij , N kl ] = iδ ik M lj − iδ jl M ik . (75)Now let us consider the Hamiltonian given by H = X i p i ω X i x i λ (cid:0) K ii + ω V ii (cid:1) , (76) λ being a suitable dimensional normalization constant.We now twist it by applying F = exp( iα ij p i ⊗ p j ) , (77)with α ij = − α ji .The deformed Hamiltonian will be H F = H − λω ~ α ij M ij + λω ~ α ij α ik K jk . (78)The deformed coproduct of the Hamiltonian is∆ F ( H ) = ∆( H ) − λω α ij ( p i ⊗ x j − x j ⊗ p i ) + λω α ij α kj ( ~ K ik ⊗ ~ − ~ ⊗ ~ K ik ) , (79)while the deformed coproduct of the deformed Hamiltonian is∆ F ( H F ) = H F ⊗ + ⊗ H F − λω α ij ( p i ⊗ x j ) + 2 λω α ij α kj ( ~ K ik ⊗ ~ ) . (80) In this work we investigated the consequences of the simplest deformation of the N -extended Supersymmetric Quantum Mechanics, realized by an abelian twist of its un-derlying Universal Enveloping Superalgebra. We pointed out that two constructions arepossible. For even values of N , the 1 D N -extended superalgebra can be identified withthe fermionic Heisenberg algebra. Alternatively, a realization of the 1 D superalgebra can15e obtained in terms of operators belonging to the Universal enveloping algebra gener-ated by one bosonic and several fermionic oscillators. We defined the deformed theoriesin terms of twist-deformed generators and twist-deformed brackets. We recovered, in amore general setting, the Cliffordization results of [3, 4]. The bosonic sector of the theorygets deformed in its multi-particle sector, even if the bosonic operators themselves getundeformed, due to their deformed coproduct. The difference between a “bosonic” versusa “fermionic” abelian twist has been pointed out.The models under considerations admit fermionic derivatives which do not obey thegraded Leibniz rule. Acknowledgments
Z. K. and F. T. are grateful to the S.N. Bose National Center for Basic Sciences ofKolkata for hospitality. B. C. acknowledges a TWAS-UNESCO associateship appointmentat CBPF and CNPq for financial support. P. G. C. acknowledges financial support fromCNPq. The work was supported by Edital Universal CNPq, Proc. 472903/2008-0 (P.G.C.,Z.K., F.T).
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