Interaction via reduction and nonlinear superconformal symmetry

We show that the reduction of a planar free spin-1/2 particle system by the constraint fixing its total angular momentum produces the one-dimensional Akulov-Pashnev-Fubini-Rabinovici superconformal mechanics model with the nontrivially coupled boson and fermion degrees of freedom. The modification of the constraint by including the particle's spin with the relative weight $n\in \N$, $n>1$, and subsequent application of the Dirac reduction procedure (`first quantize and then reduce') give rise to the anomaly free quantum system with the order $n$ nonlinear superconformal symmetry constructed recently in hep-th/0304257. We establish the origin of the quantum corrections to the integrals of motion generating the nonlinear superconformal algebra, and fix completely its form.


Introduction
The superconformal mechanics was introduced twenty years ago by Akulov and Pashnev [1], and by Fubini and Rabinovici [2] as a supersymmetric analog of the conformal mechanics of De Alfaro, Fubini and Furlan [3]. It was examined in different aspects in [4,5], and nowadays the interest to the conformal and superconformal mechanics is related mainly to the AdS/CFT correspondence conjecture and to the integrable models [6]- [16].
Recently, the superconformal mechanics model [1,2] was generalized in [17] following the ideas of nonlinear supersymmetry [18]- [25]. In comparison with the original model [1,2], the model [17] is characterized at the classical level by the n-fold fermion-boson coupling constant, that gives rise to a radical change of the symmetry: instead of the osp(2|2) Lie superalgebraic structure, the modified system possesses the order n nonlinear superconformal symmetry. The latter is generated by the boson integrals which form, as in the linear osp(2|2) case, the Lie subalgebra so(1, 2) ⊕ u(1), while the set of 2(n + 1) fermion integrals of motion anticommutes for the order n polynomials of the even integrals. The two essential moments of the construction left, however, unclarified. First, though the quantum analogs of the integrals of motion were found from the requirement of preservation of the symmetry at the quantum level, the origin of the quantum corrections to the integrals remained to be completely unclear. Second, the quantum analog of the classical nonlinear superconformal symmetry algebra was determined entirely only for the simplest case of n = 2, whereas for the general case of n ∈ N only a part of the anticommutators of the odd integrals was fixed.
In the present paper, we shall clarify the origin of the quantum corrections to the integrals of motion of the model [17], and shall fix completely the form of the quantum nonlinear superconformal algebra of an arbitrary order n. This will be done by the reduction of a planar free spin-1 2 particle system to the surface of the constraint fixing linearly the orbital angular momentum in terms of the spin.
The paper is organized as follows. In Section 2 we first show how the one-dimensional model [1,2] with the nontrivial boson-fermion coupling can be obtained via reduction from the planar system of a free nonrelativistic particle in which spin and translation degrees of freedom are completely decoupled. In Section 3 we consider a modification of the reduction procedure which leads to the generalized model [17], and fix the form of the order n nonlinear superconformal algebra. In section 4 we shortly summarize the results and discuss possible applications and generalizations of the proposed method of introducing the boson-fermion interaction via a reduction procedure.

Superconformal symmetry: linear case
Let us consider a free nonrelativistic spin-1 2 particle on the plane (i = 1, 2), Its Hamiltonian coincides with that of a free 2D nonrelativistic spinless particle, H = 1 2 p 2 i , and so, is the set of the integrals of motion, linear in the phase space variables of the system. The integrals (2.2) form a superextended 2D Heisenberg algebra, and odd, The total angular momentum, J = L+Σ, commutes with all these quadratic scalar integrals, and (2.7)-(2.12) is identified as the osp(2|2) ⊕ u(1) superalgebra with the u(1) corresponding to the centre J. The non-Abelian part of the bosonic subalgebra so(1, 2) ⊕ u(1) of the conformal superalgebra osp(2|2) ∼ = su(1, 1|1) is generated here by the integrals (2.4), and its Abelian u(1) subalgebra is associated with the linear combination Σ + J. The boson and fermion degrees of freedom in the system (2.1) are completely decoupled. The interaction between them can be introduced without violating the superconformal symmetry osp(2|2) in the following manner. Since J is the centre, it can be fixed without changing the structure of the superalgebra (2.7)-(2.12). Let us make this by introducing the classical constraint where α is a real parameter. The physical variables are those which commute (in the sense of the Poisson brackets) with the constraint, and they can immediately be identified with the scalars. Having in mind the quantization and further generalization, it is more convenient first to pass over to the polar coordinates and then to identify the observables. Let us introduce the two orthonormal vectors n (1) i = (cos ϕ, sin ϕ), n (2) i = −ǫ ij n (1) j = (− sin ϕ, cos ϕ), (2.14) in terms of which The transformation (2.14), (2.15) to the new variables is canonical, but it is well defined only for r = x 2 i > 0. This will be essential for the quantum theory. The scalar variables commute with the constraint (2.13), and any function of them is an observable.
On the surface of the constraint (2.13), the orbital angular momentum is given in terms of Σ = −iψ 1 ψ 2 , L = α − Σ. Then the reduction of the Hamiltonian H = 1 2 (p 2 r + r −2 L 2 ) to the surface (2.13) produces the nontrivial boson-fermion interaction in the resulting onedimensional system: The Hamiltonian (2.17) coincides with that of the classical superconformal model [1,2], and the quantities H, K, D, Σ+α, Q a and S a being rewritten in terms of observables (2.16), take the form of the generators of the osp(2|2) superconformal symmetry for the system (2.17) (see below). The direct quantization of the classical one-dimensional system (2.17) on the half-line q > 0 reproduces the quantum superconformal model [1,2]. However, having in mind that the two procedures -'first reduce and then quantize' and 'first quantize and then reduce' -generally give different results [26], let us consider shortly the latter procedure. It is this method that will reproduce all the quantum corrections in the quantum analogs of the classical integrals necessary for preserving the nonlinear superconformal symmetry. In ref. [17] the corresponding corrections were introduced by hands, and so, their origin was unclear.
In terms of the polar coordinates the scalar product is Here Ψ(r, ϕ) is a two-component (spinor) wave function, on which the quantum analogs of the Grassmann variables ψ a act as the Pauli matrices: In what follows we put = 1. With respect to the scalar product (2.18) the operatorŝ are Hermitian. The quantum analog of the constraint (2.13) specifies the physical subspace of the system: In (2.20) the second term corresponds to the particle's spinΣ = − i 2 [ψ 1 ,ψ 2 ]. Taking into account the 2π-periodicity of the wave functions, Ψ(r, ϕ) = +∞ l=−∞ Φ l (r)e ilϕ , we find that eq. (2.20) has nontrivial solution of the form only when the parameter α takes a half-integer value, where k is a fixed integer number, k ∈ Z. The redefinition of the radial wave functions according to Φ(r) → φ(r) = √ 2πrΦ(r), and integration in the angular variable reduce (2.18) to the scalar product on the half-line, The spinor wave functions φ(q) are subject now to the boundary condition φ(q)| q→0 = 0, and the action of the operatorp r is reduced on them to the operatorp = −i∂/∂q. Having in mind the quantum relationsp and (2.20), the reduced quantum Hamiltonian takes the form of the Hamiltonian of the one-dimensional quantum superconformal mechanics model [1,2], However, in accordance with relation (2.21), here the quantized parameter α takes only half-integer values. In order it could take any real value, it is necessary to start with a free particle on the punctured plane. In this case the orbital angular momentum operator is changed forL Effectively such a change eliminates the restriction on the α (for the details, see ref. [26]). Note that physically the particle on the punctured plane with the angular momentum (2.25) corresponds to the system of a point charged particle in a field of the singular magnetic flux placed at x i = 0 [27,28], and as for the 3D charge-monopole system [29,30], (2.25) is the total angular momentum of the particle and electromagnetic field. Further on we shall suppose that the parameter α can take any real value. Before passing over to the generalization of the construction for the case of nonlinear superconformal symmetry, we note that the quantum constraint (2.20) can be represented equivalently in the formĴ where the term 1 2 is of the quantum origin (it includes the factor = 1), and Π + = 1 2 (σ 3 + 1) being a projector, Π 2 + = Π + , is the fermion quantum number operator, Π + =ψ +ψ− ,ψ ± = 1 2 (σ 1 ± iσ 2 ).

Nonlinear superconformal symmetry
Now we are in position to be able to generalize the construction for the nonlinear superconformal symmetry case. We start, again, from the system of a free spin-1 2 particle on the (punctured) plane, but instead of (2.26), we postulate the quantum constraint where n is an arbitrary integer, which for definiteness is supposed to be positive. The formal sense of the change of the quantum condition (2.26) for (3.1) is clear: eq. (2.26) singles out the two eigenstates ofL with the eigenvalues shifted for 1, while theL-eigenvalues of the upper and lower components of the spinor satisfying eq. (3.1) are shifted relatively in n. Let us demonstrate that the nonlinear superconformal symmetry is realized in the system reduced by the quantum equation (3.1). To identify the symmetry generators, we begin with the analysis of the corresponding classical system. The classical analog of the quantum equation (3.1) is the constraint where with L 0 given by eq. (2.1), as the unique (primary) constraint, while H = 1 2 p 2 i is generated by (3.3) as the canonical Hamiltonian.
(3.7) At n=1 these are the linear combinations of the odd integrals (2.6).
In terms of the polar coordinates (2.14), (2.15), we have the relations while the variables ψ ± = ξ ± e ∓inϕ , {ψ + , ψ − } = −i, (3.13) are the odd observables commuting with the constraint (3.2), which at n = 1 are transformed to the linear combinations of the odd variables defined by eq. (2.15). Using the notation q = r, p = p r , and the constraint (3.2), we obtain the reduced 1D classical Hamiltonian, the even, and the odd, integrals of motion, generating the nonlinear generalization of the superconformal symmetry osp(2|2). Therefore, the reduction of the nonrelativistic 2D free spin-1 2 particle system by the constraint (3.2) produces the 1D classical system of ref. [17] with nontrivially coupled boson and fermion degrees of freedom, which possesses the nonlinear superconformal symmetry.
In ref. [17], it was showed that the quantum nonlinear superconformal symmetry is generated by the set of quantum operatorŝ with a n = α 2 n + 1 4 (n 2 − 1), b n = −nα n , α n = α − 1 2 (n − 1), (3.15) where The second terms in a n and α n in eq. (3.15) (proportional to (n 2 −1) and (n−1)) include the quantum factors 2 and , respectively, while the term γ q in (3.18) includes the factor (= 1). These quantum corrections in the quantum analogs of the corresponding classical quantities were found in [17] from the requirement of preservation of the nonlinear superconformal symmetry. However, their origin remained to be completely unclear. Now we shall show that the application of the reduction procedure 'first quantize and then reduce' to the system (2.1), (3.2) produces exactly the anomaly free quantum system with the nonlinear superconformal symmetry generators given by eqs. (3.14)-(3.18).
In ref. [17] the quantum analog of the nonlinear superconformal algebraic relations (3.11) was obtained in a complete form only for the particular case n = 2 by a direct calculation of the anticommutation relations of the operators (3.17). The knowledge of the origin of the odd operators (3.17) allows us to fix the form of the nonlinear superconformal symmetry in general case of n ∈ N. To this end, we proceed from the quantum generators presented in the form corresponding to the classical expressions (3.5)-(3.7). It is obtained via a direct substitution of the classical quantities X ± , P ± and ξ ± for their quantum analogs satisfying the nontrivial (anti)commutation relations Then a simple calculation gives the following nontrivial commutation relations where Π ± = 1 2 ± Σ, min(a, b) = a (or, b) when a ≤ b (or, b ≤ a), C s l = l! s!(l−s)! , P k (z) is a polynomial of order k, P 0 (z) = 1, P k (z) = z(z + 2) . . . (z + 2(k − 1)), k > 0, and c s = α + 3 2 + n − 2(m + s), d s = −α + 1 2 + 2(l − s).
In (3.23) we suppose m ≥ l, while the case corresponding to m ≤ l is obtained from it by the Hermitian conjugation. To calculate the anticommutator (3.23), we have used the relation and the analogous relation with theX + andP − exchanged in their places and with the i changed for −i. Besides, the productX −P+ =D + i 2 (L + 1) has been presented in the equivalent form using the relationL = α + 1 2 − nΠ + following from the equation of the quantum constraint (3.1).
Eqs. (3.19)-(3.21), (3.23) give a general form of the (anti)commutation relations of the order n nonlinear generalization of the superconformal algebra osp(2|2), which can be denoted as osp(2|2) n . From them, in particular, it is easy to reproduce the simplest nonlinear case of the osp(2|2) 2 superalgebra found in ref. [17].

Discussion and outlook
We have showed that the one-dimensional models corresponding to the cases of linear, osp(2|2), and nonlinear, osp(2|2) n , superconformal symmetries can be obtained by the reduction of the planar free spin-1 2 particle system by the constraint (3.1) with n = 1 or n > 1. The reduction produces the nontrivial coupling of the boson and fermion degrees of freedom with conservation of the corresponding (linear or nonlinear) superconformal symmetry of the initial system. This method not only has given a natural explanation of the origin of the quantum corrections necessarily to be included in the generators of the osp(2|2) n with n > 1 for preservation of the symmetry at the quantum level, but also has allowed us to fix the form of the quantum osp(2|2) n superalgebra.
The reduction procedure with the constraint (3.1) can alternatively be treated as a reduction of the planar free spin-n 2 particle system. Indeed, the quantum constraint (3.1) can be changed for the system of the quantum equations L +Π and 1 2 σ k 3 , k = 1, . . . , n, corresponding to the set of the n independent spins of the value 1 2 . Then eq. (4.1) fixes the value of the total angular momentum, while the set of the equations (4.2) prescribes the constituent spins to be polarized in the same direction. With taking into account the quantum constraints (4.2), the operator in eq. (4.1) reduces to the operator L + nΠ 1+ − α − 1 2 , and after changing the notation σ 1 3 → σ 3 , eq. (4.1) takes a form of eq. (3.1). The corresponding classical Lagrangian for such a planar system is Here v j , j = 2, . . . n, is the set of Lagrange multipliers, ξ k i , k = 1, . . . , n, is the set of n planar Grassmann vectors, and ξ k ± = 1 √ 2 (ξ k 1 ± iξ k 2 ). To conclude, we enumerate shortly possible applications and generalizations of the results. It would be interesting to generalize the described method of introduction of the bosonfermion coupling for other systems. We hope that this, on the one hand, could clarify the nature of the nontrivial quantum corrections appearing generally under attempt of quantization of the systems possessing nonlinear supersymmetry [19,20,24]; on the other hand, the method could be useful for the analysis of the quasi exactly solvable systems [31,32,33], to which the nonlinear supersymmetry is intimately related [20]- [24]. If the method admits a generalization for higher dimensions, it could be applied to investigation of the supersymmetric many-particle integrable systems.