aa r X i v : . [ h e p - t h ] J u l N = 2 supersymmetric odd-orderPais-Uhlenbeck oscillator Ivan Masterov
Laboratory of Mathematical Physics, Tomsk Polytechnic University,634050 Tomsk, Lenin Ave. 30, Russian Federation
E-mail: [email protected]
Abstract
We consider an N = 2 supersymmetric odd-order Pais-Uhlenbeck oscillator with dis-tinct frequencies of oscillation. The technique previously developed in [Acta Phys.Polon. B 36 (2005) 2115; Nucl. Phys. B 902 (2016) 95] is used to construct a familyof Hamiltonian structures for this system. PACS numbers: 11.30.-j, 11.25.Hf, 02.20.SvKeywords: Pais-Uhlenbeck oscillator, ghost problem, supersymmetry . Introduction
A systematic way to construct a Hamiltonian formulation for nondegenerate higher-derivativemechanical systems is based on Ostrogradsky’s approach [1]. Canonical formalism for degen-erate higher-derivative models can be obtained with the aid of Dirac’s method for constrainedsystems [2] or by applying the Faddeev-Jackiw prescription [3].However, some higher-derivative models are multi-Hamiltonian. The simplest example ofsuch systems is the one-dimensional fourth-order Pais-Uhlenbeck (PU) oscillator [4]. Ostro-gradsky’s Hamiltonian of this system is unbounded from below. As a consequence, quantumtheory of the model faces ghost-problem (see, e.g., a detailed discussion in Ref. [5]). Fordistinct frequencies of oscillation, this Hamiltonian can be presented as a difference of twoharmonic oscillators by applying an appropriate canonical transformation [4, 6]. This repre-sentation provides two functionally independent positive-definite integrals of motion. As wasobserved in [7] (see also Ref. [8]), a linear combination involving arbitrary nonzero coeffi-cients of these constants of motion can also play a role of a Hamiltonian for the fourth-orderPU oscillator . Thus, for positive coefficients, the alternative Hamiltonian is positive-definiteand consequently is more relevant for quantization than Ostrogradsky’s one.For arbitrary odd and even orders, the PU oscillator with distinct frequencies of oscillationcan also be treated by the technique employed in Ref. [7]. This fact has been established in[10, 11] (see also Ref. [12]) where the corresponding families of Hamiltonian structures havebeen constructed. The main advantage of the alternative Hamiltonian formulation obtainedin such a way is that this may correspond to a positive-definite Hamiltonian.The even-order PU oscillator with distinct frequencies of oscillation admits an N = 2supersymmetric extension [13]. This generalization is invariant under the time translations.However, the Noether charge associated with this symmetry can be presented as a sum of N = 2 supersymmetric harmonic oscillators which alternate in a sign [13] (see also Ref.[14]). A canonical formalism with regard to a such Hamiltonian brings about trouble withghosts upon quantization [13]. This problem is reflected in the fact that the quantum statespace of the model contains negative norm states, while a ground state is absent. In Ref.[10] an alternative Hamiltonian formulation for an N = 2 supersymmetric even-order PUoscillator has been constructed so as to avoid these nasty features.For a particular choice of oscillation frequencies, an N = 2 supersymmetric extensionof the odd-order PU oscillator has been derived in Ref. [15]. It has been shown thatthis extension accommodates conformal symmetry provided frequencies of oscillation form acertain arithmetic sequence. Any other aspects related with the N = 2 supersymmetric odd-order PU oscillator remain completely unexplored. In particular, a canonical formulation ofthis model has not yet been considered. The purpose of the present work is to construct aHamiltonian formulation for an N = 2 supersymmetric odd-order PU oscillator with distinctfrequencies of oscillation by applying the technique previously developed in Refs. [7, 10].The paper is organized as follows. In the next section we consider the odd-order PU An alternative Hamiltonian formulation for the fourth-order PU oscillator has been also constructed inpaper [9]. N = 2 supersymmetricextension of this model. A Hamiltonian formulation for an N = 2 supersymmetric third-order PU oscillator is constructed in Sect. 3, while the general case is treated in Sect.4. In Sect. 5, a quantum version of the N = 2 supersymmetric odd-order PU oscillatoris considered. We summarize our results and discuss further possible developments in theconcluding Sect. 6. Some technical details are given in Appendix. Throughout the worksummation over repeated spatial indices is understood, unless otherwise is explicitly stated.Both a superscript in braces and a number of dots over spatial coordinates designate thenumber of derivatives with respect to time. Complex conjugation of a function f is denotedby f ∗ . Hermitian conjugation of an operator ˆ a is designated as (ˆ a ) † .
2. The model
Symmetries of the PU oscillator have recently attracted some attention [16]-[23]. Theinterest was motivated by the desire to realize the so-called l -conformal Newton-Hooke al-gebra [24]-[26] in this model. As was shown in [21] (see also Ref. [27]), the (2 n + 1)-orderPU oscillator, which accommodates the l -conformal Newton-Hooke symmetry, is describedby the action functional S = 12 Z dt ǫ ij x i n Y k =1 (cid:18) d dt + k ω (cid:19) ˙ x j , (1)where ǫ ij is the Levi-Civit´a symbol with ǫ = 1.Symmetry structure intrinsic to the model (1) allows one to construct an N = 2 super-symmetric generalization of this system with the aid of Niederer-like coordinate transforma-tions [15, 35]. The action functional associated with this extension reads S = 12 Z dt ǫ ij x i n Y k =1 (cid:18) d dt + k ω (cid:19) ˙ x j − ψ i (cid:18) ddt + inω (cid:19) n − Y k =1 (cid:18) d dt + k ω (cid:19) ˙¯ ψ j −− ¯ ψ i (cid:18) ddt − inω (cid:19) n − Y k =1 (cid:18) d dt + k ω (cid:19) ˙ ψ j − z i n − Y k =1 (cid:18) d dt + k ω (cid:19) ˙ z j ! . (2)The configuration space of this model involves the real bosonic coordinates x i , the fermioniccoordinates ψ i , ¯ ψ i , which are complex conjugates of each other ¯ ψ i = ψ ∗ i , and real extrabosonic coordinates z i . The model (2) is invariant under the supersymmetry transformationsof the form [15] δx i = iψ i α + i ¯ ψ i ¯ α, δz i = (cid:16) − ˙ ψ i + inωψ i (cid:17) α + (cid:16) ˙¯ ψ i + inω ¯ ψ i (cid:17) ¯ α,δψ i = ( ˙ x i + inωx i − iz i ) ¯ α, δ ¯ ψ i = ( ˙ x i − inωx i + iz i ) α, (3) Some aspects of the third-order PU oscillator have been studied in [28]-[30] (see also Refs. [31]-[34]). α and ¯ α are odd infinitesimal parameters.It is evident that the model (1) can be generalized to the case of arbitrary distinctoscillation frequencies. For this purpose, the action (1) can be transformed to the form [11] S = 12 Z dt ǫ ij x i n − Y k =0 (cid:18) d dt + ω k (cid:19) ˙ x j . (4)For definiteness, we assume that 0 < ω < ω < .. < ω n − . On the other hand, a possibility togeneralize the model (2) along similar lines is less obvious, because we must simultaneouslychange both the action functional (2) and the supersymmetry transformations (3). Byanalogy with the analysis in Ref. [13], let us abandon the conformal invariance and modifythe action functional (2) as follows S = 12 Z dt ǫ ij x i n − Y k =0 (cid:18) d dt + ω k (cid:19) ˙ x j − iψ i n − Y k = − n +1 (cid:18) ddt + iω k (cid:19) ˙¯ ψ j −− i ¯ ψ i n − Y k = − n +1 (cid:18) ddt − iω k (cid:19) ˙ ψ j − z i n − Y k =1 (cid:18) d dt + ω k (cid:19) ˙ z j ! , (5)where, for convenience, we denoted ω − k = − ω k . The dynamics of this model is governed bythe equations of motion n X k =0 σ n, k x (2 k +1) i = 0 , n − X k =0 ( − i ) n − k − σ nk ψ ( k +1) i = 0 , n − X k =0 i n − k − σ nk ¯ ψ ( k +1) i = 0 , n − X k =0 σ n, k z (2 k +1) i = 0 , where σ n,sk , σ nk are elementary symmetric polynomials defined by σ n,sk = n − X i
Q, Q } = 0 , [ H, Q } = 0 , [ Q, ¯ Q } = − iH, [ H, ¯ Q } = 0 , [ ¯ Q, ¯ Q } = 0 , (11)with respect to the bracket (8). So, the model (5) is an N = 2 supersymmetric extension ofthe odd-order PU oscillator (4). N = 2 supersymmetric third-order PU oscillator According to the analysis in Ref. [11], a Hamiltonian formulation of the odd-orderPU oscillator (4) is not unique. Let us generalize this result to the case of an N = 2supersymmetric third-order PU oscillator. For n = 1, the Hamiltonian of the model (4) canbe presented as a difference of two one-dimensional harmonic oscillators [11]. This can beachieved by using the coordinates q k = 1 √ ω (cid:18) ˙ x + ( − k ω ¨ x (cid:19) , p k = r ω (cid:18) ˙ x + ( − k +1 ω ¨ x (cid:19) , y k = 1 ω (¨ x k + ω x k ) . (12)With respect to the supersymmetry transformations (9), the variables q k and y k are trans-formed as follows δq k = ϑ k α + ¯ ϑ k ¯ α, δy k = θ k α + ¯ θ k ¯ α, where we denoted ϑ k = 1 √ ω (cid:16) ˙ ψ + i ( − k ˙ ψ (cid:17) , θ k = i ˙ ψ k + ω ψ k , ¯ ϑ k = ( ϑ k ) ∗ , ¯ θ k = ( θ k ) ∗ . (13)5he nonvanishing structure relations between the coordinates (12), (13) read[ q k , p m } = δ km , [ y k , y m } = − ǫ km , [ ϑ k , ¯ ϑ m } = i ( − k δ km , [ θ k , ¯ θ m } = ω ǫ km , (no sum) . Using the variables (12), (13), the Hamiltonian (6) and supercharges (10) for n = 1 maybe rewritten as H = 12 ( p + ω q + 2 ω ϑ ¯ ϑ ) −
12 ( p + ω q + 2 ω ϑ ¯ ϑ ) , (14) Q = ϑ ( p − iω q ) + ϑ ( p + iω q ) − ǫ ij θ i ( y j − z j ) , ¯ Q = ( Q ) ∗ . (15)So, the Hamiltonian of an N = 2 supersymmetric third-order PU oscillator can be presentedas a difference of two one-dimensional N = 2 supersymmetric harmonic oscillators. At firstsight it may appear that an N = 2 supersymmetric odd-order PU oscillator is dynamicallyequivalent to a set of two decoupled N = 2 supersymmetric harmonic oscillators. This is nottrue because the phase spaces of these systems are not isomorphic. In addition to oscillatorcoordinates ( q i , p i , ϑ i , ¯ ϑ i ), the phase space of the N = 2 supersymmetric odd-order PUoscillator involves variables a i = { y i , z i , θ i , ¯ θ i } whose dynamics are trivial ˙ a i = 0. This alsocan be illustrated by rewriting the action functional (5) as follows (up to a total derivativeterm) S = 12 Z dt h(cid:16) ˙ q − ω q + iϑ ˙¯ ϑ + i ¯ ϑ ˙ ϑ − ω ϑ ¯ ϑ (cid:17) + ǫ ij ( y i ˙ y j − z i ˙ z j ) −− (cid:16) ˙ q − ω q + iϑ ˙¯ ϑ + i ¯ ϑ ˙ ϑ − ω ϑ ¯ ϑ (cid:17) + 1 ω ǫ ij ( θ i ˙¯ θ j − ¯ θ i ˙ θ j ) (cid:21) . Let us construct an alternative Hamiltonian formulation for an N = 2 supersymmetricthird-order PU oscillator by applying the approach previously developed in Ref. [7]. To thisend, we must deform both the Hamiltonian (14) and the corresponding Poisson bracket (8) insuch a way that the equations (7) will be preserved. Let us choose the following deformationof the Hamiltonian (14) H = γ p + ω q + 2 ω ϑ ¯ ϑ ) + γ p + ω q + 2 ω ϑ ¯ ϑ ) , (16)where γ and γ are arbitrary nonzero coefficients. With the change H → H , the equations(7) are satisfied provided the graded Poisson structure relations have the form[ x i , ¨ x j } = − γ − ǫ ij , [ x i , ˙ x j } = 1 ω γ + δ ij , [ ψ i , ˙¯ ψ j } = iγ − ǫ ij − γ + δ ij , [ z i , z j } = ǫ ij , [ ˙ x i , ˙ x j } = γ − ǫ ij , [ ˙ x i , ¨ x j } = ω γ + δ ij , [ ˙ ψ i , ¯ ψ j } = − iγ − ǫ ij + γ + δ ij , [¨ x i , ¨ x j } = ω γ − ǫ ij , [ ψ i , ¯ ψ j } = − iω γ + δ ij , [ ˙ ψ i , ˙¯ ψ j } = − ω γ − ǫ ij − iω γ + δ ij , (17) Note that the supersymmetry algebra (11) does not change when the supercharges are redefined asfollows Q → Q + ǫ ij θ i ( y j − z j ), ¯ Q → ¯ Q + ǫ ij ¯ θ i ( y j − z j ). γ ± = 12 (cid:18) γ ± γ (cid:19) . This Poisson structure is degenerate when γ = γ . By this reason, in what follows weexclude this case from our consideration.Let us introduce the new variables q k = p | γ k | q k , p k = ( − k +1 sign( γ k ) p | γ k | p k , y k = 1 p | γ − | y k , Ψ k = p | γ k | ϑ k , ¯Ψ k = p | γ k | ¯ ϑ k , Θ k = 1 p | γ − | θ k , ¯Θ k = 1 p | γ − | ¯ θ k . (no sum) (18)Under the bracket (17), these coordinates obey the following nonvanishing relations[ q k , p m } = δ km , [ y k , y m } = − sign( γ − ) ǫ km , [Ψ k , ¯Ψ m } = − i sign( γ k ) δ km , [Θ k , ¯Θ m } = ω sign( γ − ) ǫ km , (no sum) (19)Here and in what follows sign( x ) denotes the standard signum function. The Hamiltonian(16) in terms of the variables (18) takes the form H = sign( γ )2 ( p + ω q + 2 ω Ψ ¯Ψ ) + sign( γ )2 ( p + ω q + 2 ω Ψ ¯Ψ ) . (20)Along with this alternative Hamiltonian, the full formulation of an N = 2 supersymmet-ric third-order PU oscillator involves supercharges. According to the analysis in Ref. [10],one may try to find these by using an auxiliary action functional. Taking into account therelations (19), in our case such an action can be chosen in the form S = 12 Z dt sign( γ )( ˙ q − ω q + i Ψ ˙¯Ψ + i ¯Ψ ˙Ψ − ω Ψ ¯Ψ ) + sign( γ − ) ǫ ij y i ˙ y j − ǫ ij z i ˙ z j ++ sign( γ )( ˙ q − ω q + i Ψ ˙¯Ψ + i ¯Ψ ˙Ψ − ω Ψ ¯Ψ ) + sign( γ − ) ω ǫ ij (cid:16) Θ i ˙¯Θ j − ¯Θ i ˙Θ j (cid:17) . This action is invariant under the transformations δ q k = Ψ k α + ¯Ψ k ¯ α, δ Ψ k = ( − i ˙ q k + ω q k ) ¯ α, δ Θ k = ω ( y k − sign( γ − ) z k ) ¯ α,δ y k = δz k = Θ k α + ¯Θ k ¯ α, δ ¯Ψ k = ( − i ˙ q k − ω q k ) α, δ ¯Θ k = − ω ( y k − sign( γ − ) z k ) α, which yield the following Noether integrals of motion Q = Ψ ( p − i sign( γ ) ω q ) + Ψ ( p − i sign( γ ) ω q ) − ǫ ij Θ i (sign( γ − ) y j − z j ) , ¯ Q = ( Q ) ∗ . H , Q} = 0 , [ Q , ¯ Q} = − i H + (1 − sign( γ − )) ǫ ij Θ i ¯Θ j , [ H , ¯ Q} = 0 , [ Q , Q} = (1 − sign( γ − )) ǫ ij Θ i Θ j , [ ¯ Q , ¯ Q} = (1 − sign( γ − )) ǫ ij ¯Θ i ¯Θ j , Thus, for positive γ − , we have an appropriate supercharges Q and ¯ Q . Moreover, if we put0 < γ < γ then the corresponding alternative Hamiltonian becomes a direct sum of twoone-dimensional N = 2 supersymmetric harmonic oscillators.
4. The general case
Let us consider an N = 2 supersymmetric PU oscillator of arbitrary odd order. Toconstruct an alternative Hamiltonian formulation for this system, one should obtain a moreappropriate representation for the Hamiltonian (6). According to the analysis in Ref. [11],a Hamiltonian of the model (4) can be represented as a direct sum of the third-order PUoscillators which alternate in a sign. This can be achieved with the aid of the so-calledoscillator variables [4, 11]˜ x k,i = q ρ n, k n − Y m =0 m = k (cid:18) d dt + ω m (cid:19) ˙ x i , z ,i = 1 n − Q s =0 ω s n − Y m =0 (cid:18) d dt + ω m (cid:19) x i , (21)where k = 0 , , .., n −
1; the coefficients ρ n,sk are given by ρ n,sk = ( − k + sn − Q m = sm = k ( ω m − ω k ) , k = s, s + 1 , .., n − . Taking into account the results of Refs. [10, 13], let us introduce similar variables for theremaining coordinates ψ p,i = p ρ np n − Y m = − n +1 m = p (cid:18) ddt − iω m (cid:19) ˙ ψ i , θ i = i n − Q s =1 ω s n − Y m = − n +1 (cid:18) ddt − iω m (cid:19) ψ i , ¯ ψ p,i = ( ψ p,i ) ∗ , ˜ x − k,i = q ρ n, k n − Y m =1 m = k (cid:18) d dt + ω m (cid:19) ˙ z i , z ,i = 1 n − Q s =1 ω s n − Y m =1 (cid:18) d dt + ω m (cid:19) z i , ¯ θ i = ( θ i ) ∗ , (22)where k = 1 , , .., n − p = − n + 1 , − n + 2 , .., n −
1; the coefficients ρ np are defined by ρ np = ( − n + p − n − Q m = − n +1 m = p ( ω m − ω p ) = ω p + ω ω p ρ n, | p | . x ± k,i , ψ ± k,i , and ¯ ψ ± k,i ( k = 1 , , .., n − δ ˜ x ± k,i = ( µ ± k ψ k,i ± µ ∓ k ψ − k,i ) α + ( µ ± k ¯ ψ k,i ± µ ∓ k ¯ ψ − k,i ) ¯ α,δψ ± k,i = ( − µ ± k ( i ˙˜ x k,i ∓ ω k ˜ x k,i ) ∓ µ ∓ k ( i ˙˜ x − k,i ∓ ω k ˜ x − k,i )) ¯ α,δ ¯ ψ ± k,i = ( − µ ± k ( i ˙˜ x k,i ± ω k ˜ x k,i ) ∓ µ ∓ k ( i ˙˜ x − k,i ± ω k ˜ x − k,i )) α, with µ ± k = r ω k ± ω ω k . (23)This motivates us to perform one more change of the bosonic coordinates x − k,i = µ − k ˜ x k,i − µ + k ˜ x − k,i , x ,i = ˜ x ,i , x k,i = µ + k ˜ x k,i + µ − k ˜ x − k,i . (24)The supersymmetry transformations (23) then become δx ± k,i = ψ ± k,i α + ¯ ψ ± k,i ¯ α, δψ ± k,i = − ( i ˙ x ± k,i ∓ ω k x ± k,i ) ¯ α, δ ¯ ψ ± k,i = − ( i ˙ x ± k,i ± ω k x ± k,i ) α. The Hamiltonian (6) and the supercharges (10) in terms of x k,i , ψ k,i , ¯ ψ k,i , and z s,i maybe represented as follows H = n − X k = − n +1 ( − k +1 ǫ ij ( x k,i ˙ x k,j − iψ k,i ¯ ψ k,j ) ,Q = n − X k = − n +1 ( − k ω k ǫ ij ψ k,i ( i ˙ x k,j + ω k x k,j ) − ǫ ij θ i ( z ,j − z ,j ) , ¯ Q = ( Q ) ∗ . So, we have shown that the Hamiltonian of an N = 2 supersymmetric (2 n + 1)-order PUoscillator can be presented as a direct sum of (2 n − N = 2 supersymmetric third-orderPU oscillators which alternate in a sign. This fact correlates with the analysis in Ref. [11]for the model (4).By analogy with (12), (13), let us introduce the coordinates q k,s = 1 p | ω k | (cid:18) x k, + ( − s | ω k | ˙ x k, (cid:19) , p k,s = ( − k r | ω k | (cid:18) x k, + ( − s +1 | ω k | ˙ x k, (cid:19) ,ϑ k,s = 1 p | ω k | ( ψ k, + i ( − s sign( ω k ) ψ k, ) , ¯ ϑ k,s = 1 p | ω k | ( ¯ ψ k, − i ( − s sign( ω k ) ¯ ψ k, ) . (25)Given the bracket (8), these variables obey[ q k,s , p m,j } = δ km δ sj , [ ϑ k,s , ¯ ϑ m,j } = i ( − k + s δ km δ sj . (no sum)The existence of these coordinates automatically implies that (8) possesses standard prop-erties of a graded Poisson bracket.In terms of the variables (25), the Hamiltonian (6) takes the form H = n − X k = − n +1 ( − k (cid:20)(cid:18) p k, + ω k q k, + ω k ϑ k, ¯ ϑ k, (cid:19) − (cid:18) p k, + ω k q k, + ω k ϑ k, ¯ ϑ k, (cid:19)(cid:21) . H = n − X k = − n +1 γ | k | , (cid:18) p k, + ω k q k, + ω k ϑ k, ¯ ϑ k, (cid:19) + γ | k | , (cid:18) p k, + ω k q k, + ω k ϑ k, ¯ ϑ k, (cid:19) , (26)where γ , , γ , , γ , , .., γ n − , are arbitrary nonzero coefficients. It is straightforward to verify(for technical details see Appendix) that the equations (7), where H → H , are satisfiedprovided the following graded Poisson structure[ x ( s ) i , x ( m ) j } [ z ( s ) i , z ( m ) j } s = m = 0 0 0 s + m − odd ( − s − m +12 n − P k =0 ρ n, k ω s + m − k γ + k δ ij ( − s − m +12 n − P k =1 ρ n, k ω s + m − k γ + k δ ij s + m = 0 − even ( − s − m n − P k =0 ρ n, k ω s + m − k γ − k ǫ ij ( − s − m n − P k =1 ρ n, k ω s + m − k γ − k ǫ ij [ ψ ( s ) i , ¯ ψ ( m ) j } s = m = 0 − i n − P k =0 ρ n, k ω − k γ + k δ ij s + m − odd ( − s − m − (cid:18) ω n − P k =0 ρ n, k ω s + m − k γ + k δ ij − i n − P k =0 ρ n, k ω s + m − k γ − k ǫ ij (cid:19) s + m = 0 − even ( − s − m − (cid:18) i n − P k =0 ρ n, k ω s + m − k γ + k δ ij + ω n − P k =0 ρ n, k ω s + m − k γ − k ǫ ij (cid:19) (27)has been chosen. Above we denote γ ± k = (cid:16) γ k, ± γ k, (cid:17) . This structure is degenerateprovided g n, = 0 and/or g n, = 0, where g n,s = n − X k = s ρ n,sk ω k (cid:18) γ k, − γ k, (cid:19) = n − X k = s ρ n,sk γ − k ω k . By this reason, we restrict our consideration only to the case when g n, = 0, g n, = 0.The generalization of the coordinates (18) reads q k,i = q | γ | k | ,i | q k,i , z s,i = 1 n − Q m = s ω m p | g n,s | z s,i , p k,i = ( − k + i +1 sign( γ k,i ) q | γ | k | ,i | p k,i , Ψ k,i = q | γ | k | ,i | ϑ k,i , Θ i = 1 n − Q m =0 ω m p | g n, | θ i , ¯Ψ k,i = (Ψ k,i ) ∗ , ¯Θ i = (Θ i ) ∗ . (no sum) (28)With respect to the Poisson structure (27), these variables obey the relations[ q k,i , p m,j } = δ km δ ij , [ z s,i , z m,j } = − sign( g n,s ) δ sm ǫ ij , [Θ i , ¯Θ j } = ω sign( g n, ) ǫ ij , [Ψ k,i , ¯Ψ m,j } = − i sign( γ | k | ,i ) δ km δ ij . (no sum) (29)10hen the alternative Hamiltonian (26) may be rewritten as H = n − X k = − n +1 sign( γ | k | ,i )2 (cid:0) p k,i + ω k q k,i + 2 ω k Ψ k,i ¯Ψ k,i (cid:1) . (30)To find supercharges corresponding to this alternative Hamiltonian, let us introduce thefollowing auxiliary action functional S = 12 Z dt n − X k = − n +1 sign( γ | k | ,i )( ˙ q k,i − ω k q k,i + i Ψ k,i ˙¯Ψ k,i + i ¯Ψ k,i ˙Ψ k,i − ω k Ψ k,i ¯Ψ k,i ) ++ X s =0 sign( g n,s ) ǫ ij z s,i ˙ z s,j + sign( g n, ) ω ǫ ij (cid:16) Θ i ˙¯Θ j − ¯Θ i ˙Θ j (cid:17) , which is invariant under the transformations δ q k,i = Ψ k,i α + ¯Ψ k,i ¯ α, δ Ψ k,i = ( − i ˙ q k,i + ω k q k,i ) ¯ α, δ Θ i = ω ( z ,i + sign( g n, g n, ) z ,i ) ¯ α,δ z k,i = Θ i α + ¯Θ i ¯ α, δ ¯Ψ k,i = ( − i ˙ q k,i − ω k q k,i ) α, δ ¯Θ i = − ω ( z ,i + sign( g n, g n, ) z ,i ) α. The Noether charges associated with these symmetries read Q = n − X k = − n +1 Ψ k,i ( p k,i − i sign( γ | k | ,i ) ω k q k,i ) − ǫ ij Θ i (sign( g n, ) z ,j + sign( g n, ) z ,j ) , ¯ Q = ( Q ) ∗ . These integrals of motion, together with the Hamiltonian (30), obey the following relations[ H , Q} = 0 , [ Q , ¯ Q} = − i H − (sign( g n, ) + sign( g n, )) ǫ ij Θ i ¯Θ j , [ H , ¯ Q} = 0 , [ Q , Q} = − (sign( g n, ) + sign( g n, )) ǫ ij Θ i Θ j , [ ¯ Q , ¯ Q} = − (sign( g n, ) + sign( g n, )) ǫ ij ¯Θ i ¯Θ j , under the bracket (29). Thus, we have one more condition on the coefficients γ k,i sign( g n, ) = − sign( g n, ) . (31)It is evident that infinitely many possible sets of parameters γ k,i obey this restriction.
5. Quantization
To quantize an N = 2 supersymmetric odd-order PU oscillator with the Hamiltonian(30), let us introduce hermitian bosonic operators ˆ q k,i , ˆ p k,i ˆ z s,i as well as fermionic operatorsˆΨ k,i , ˆ¯Ψ k,i = ( ˆΨ k,i ) † , ˆΘ i , ˆ¯Θ i = ( ˆΘ i ) † . According to (29) and (31), they obey the followingnonvanishing (anti)commutation relations[ˆ q k,i , ˆ p m,j ] = i ~ δ km δ ij , [ˆ z s,i , ˆ z m,j ] = − i ( − s ~ sign( g n, ) δ sm ǫ ij , { ˆΘ i , ˆ¯Θ j } = i ~ ω sign( g n, ) ǫ ij , { ˆΨ k,i , ˆ¯Ψ m,j } = ~ sign( γ | k | ,i ) δ km δ ij , (no sum) (32)11here {· , ·} and [ · , · ] stand for the anticommutator and commutator, respectively. ~ is thereduced Planck constant.As the next step, we may introduce the creation ¯ a k,i , ¯ c k,i and annihilation a k,i , c k,i oper-ators, which correspond to oscillator coordinates ( q k,i , p k,i , Ψ k,i , ¯Ψ k,i ) a k,i = r | ω k | ~ ˆ q k,i + i p | ω k | ~ ˆ p k,i , c k,i = 1 √ ~ ˆΨ k,i , ¯ a k,i = r | ω k | ~ ˆ q k,i − i p | ω k | ~ ˆ p k,i , ¯ c k,i = 1 √ ~ ˆ¯Ψ k,i , ⇒ [ a k,i , ¯ a m,j ] = δ km δ ij , { c k,i , ¯ c m,j } = sign( γ | k | ,i ) δ km δ ij . Thus, for negative values of γ k,i , we have { c k,i , ¯ c m,j } = − δ km δ ij . Taking into account theanalysis in Refs. [13, 36], these relations bring about negative norm states. To avoid thisfeature, we set all coefficients γ k,i to be positive.For the variables z s,i , Θ i , and ¯Θ i , the creation ¯ b s , ¯ d s and annihilation b s , d s operatorsmay be defined as follows [37] b s = 1 √ ~ (ˆ z s, − i ( − s sign( g n, )ˆ z s, ) , d s = 1 √ ~ ω ( ˆΘ + i ( − s sign( g n, ) ˆΘ ) , ¯ b s = 1 √ ~ (ˆ z s, + i ( − s sign( g n, )ˆ z s, ) , ¯ d s = 1 √ ~ ω ( ˆ¯Θ − i ( − s sign( g n, ) ˆ¯Θ ) , s = 0 , . These operators obey the following nonvanishing relations[ b s , ¯ b p ] = δ sp , { d s , ¯ d p } = ( − s δ sp . Unfortunately, the relation { d , ¯ d } = −
6. Conclusion
To summarize, in this work we have introduced an N = 2 supersymmetric generaliza-tion for the odd-order PU oscillator with distinct frequencies of oscillation. This system isinvariant under the time translations. We have observed that the corresponding integral ofmotion can be presented as a direct sum of one-dimensional N = 2 supersymmetric har-monic oscillators which alternate in a sign. This representation has allowed us to constructa family of Hamiltonian structures for an N = 2 supersymmetric odd-order PU oscillator.Unfortunately, quantization of the system revealed the presence of negative norm states inthe corresponding Fock space.Turning to further possible developments, it is worth constructing various generalizationsof an N = 2 supersymmetric odd-order PU oscillator which are compatible with the alterna-tive Hamiltonian formulation. In particular, it would be interesting to generalize deformedodd-order PU oscillator introduced in paper [11] as well as higher-derivative field theories12onsidered in [38] to an N = 2 supersymmetric case. A construction of N = 2 supersym-metric many particle higher-derivative systems is also of interest. In this context it is worthstudying higher-derivative generalizations of N = 2 supersymmetric many body modelsconstructed in papers [39]-[42]. The odd-order PU oscillator with weak supersymmetry [43]has been introduced in paper [44]. It is also worth investigating a Hamiltonian formulationof this system. These issues will be studied elsewhere. Acknowledgements
We would like to thank D. Chow for pointing out reference [45]. We also thank ananonymous Referee for useful comments. This work was supported by the MSE programNauka under the project 3.825.2014/K, and RFBR grant 15-52-05022.
Appendix:
List of identities
When verifying the fact that the equations (7) are satisfied with respect to the Hamilto-nian structures introduced in both Sects. 2 and 4, the following identities P n,s k = n − X p = s ( − n + p +1 ω n +2 k − s − p ρ n,sp , k = − n + s + 1 , − n + s + 2 , .. ; n − X k = s ( − k + s ( − ω k ) r σ n,sp,k ρ n,sk = ( δ rp , r = 0 , , .., n − s − − σ n,sp , r = n − s ; n − X r =0 ( − r ω r σ n, p,r ρ n, r = σ n, p +1 n − Q k =0 ω k ; n − s − X r =0 ( − r ω rq σ n,sr,k = ( − k + s ρ n,sk δ qk ; σ n,sp,k = n − p − s − X r =0 ( − r ω rk σ n,sp + r +1 , k = s, s + 1 , .., n − σ n,sm,k = n − X i
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