Quantum Mechanics and Hidden Superconformal Symmetry

Solvability of the ubiquitous quantum harmonic oscillator relies on a spectrum generating osp(1|2) superconformal symmetry. We study the problem of constructing all quantum mechanical models with a hidden osp(1|2) symmetry on a given space of states. This problem stems from interacting higher spin models coupled to gravity. In one dimension, we show that the solution to this problem is the Plyushchay family of quantum mechanical models with hidden superconformal symmetry obtained by viewing the harmonic oscillator as a one dimensional Dirac system, so that Grassmann parity equals wavefunction parity. These models--both oscillator and particle-like--realize all possible unitary irreducible representations of osp(1|2).


I. INTRODUCTION
The quantum harmonic oscillator is solvable because the ladder operators generate the spectrum. This is perhaps the simplest example of the Lie superalgebra osp(1|2): to see this, one treats the ladder operators as supercharges [28] S + := a † , S − := a .
Then defining the sp(2) generators [29] Q ++ = (a † ) 2 , Q +− = H , Q −− = a 2 , the five generators {S ± , Q ±± , Q +− } generate the algebra [30] osp(1|2): Strangely enough, here one assigns the ladder operators a Grassmann odd grading, even though these are the standard complex combinations of position and momentum given in Eq. (1). Thus, the fermion number operator F that grades the osp(1|2) algebra counts one for odd powers of ladder operators (and zero for even powers) and therefore labels wavefunction parity [31].
The basic question we address is the existence of operator quintuples acting on the harmonic oscillator Fock space obeying the osp(1|2) Lie superalgebra. The solution to this problem is a class of quantum mechanical models that have been studied in detail by Plyushchay [1]. We also answer this question for generalized particle models with plane wave-normalizable spectra for which the osp(1|2) algebra acts as a generalized one dimensional superconformal symmetry.
Our study is motivated by a proposal of Bars et al [2], who suggested that the space of operators obeying an sp(2) algebra and acting on functions of a d + 2 dimensional spacetime with two times, could describe gravitating, interacting higher spin theories. We have shown [3,4] that this proposal is intimately linked to the study of d dimensional conformal geometries in terms of a d + 2 dimensional ambient space initiated by Fefferman and Graham [5]. The inclusion of fermions in such models leads to an osp(1|2) generalization of Bars' theory [6] (see also [7]). The study of the detailed spectra, interactions, ultraviolet and unitarity properties of such models is a complicated problem commensurate with that of string field theories, as one is dealing with field equations for operator-valued fields. Although the solution we find in one dimension is largely controlled by orthosymplectic representation theory, the existence of a mathematically well-defined answer in this setting is an important first step towards analyzing models in d + 2 dimensions, for which the solution space already includes all ddimensional conformal geometries. Moreover D'Hoker and Vinet [9] have analyzed a hidden osp(1|2) symmetry of the Dirac equation in monopole backgrounds, which indicates tractability for models in higher dimensions.
Our analysis begins in Section II with the "master" equations of motion and gauge symmetries for the supercharges S ± . Sections III-VII are devoted to solving these equations on a harmonic oscillator Fock space while Sections VIII-X focus on particle models with hidden superconformal symmetry. Appendix B reviews osp(1|2) representations.

II. EQUATIONS OF MOTION
To answer the question posed in the introduction, we view the supercharges S ± as the fundamental "fields" and study "equations of motion" for these that guarantee that the algebra osp(1|2) of Eq. (2) holds. These have been formulated in [6]; the result is [32]: The statement here is that if the pair of operators S ± obey these equations, then the operator quintuple S ± , Q ±± = S ± S ± , Q +− = 1 2 [S + S − + S + S − ] satisfies the osp(1|2) Lie superalgebra (2).
Clearly, if S ± solve Eqs. (3), then so too do U −1 S ± U for any invertible operator U . Linearizing U around the identity U ≈ Id + ε gives the gauge invariance of the equations of motion (3). Here the gauge parameter ε is itself also an operator. The problem of solving Eqs. (3) for operators S ± is not defined without specifying the state space H on which these operators act [33]. The set of possible choices for an underlying Hilbert space H is clearly enormous. We commence with perhaps the simplest case in which H is the harmonic oscillator Fock space.

III. THE SPACE OF OPERATORS
We now let H equal the quantum harmonic oscillator Hilbert space with Fock basis {|n : n ∈ Z ≥0 }. We employ the slightly non-standard normalization m|n = n! δ mn for states |n , since this allows us to identify |n with the monomial z n and in turn study wavefunctions given by polynomials, or more generally suitable analytic functions, in z [34]. Thus we study operators where s ± i (z) are analytic functions of z in a neighborhood of the origin. In terms of ladder operators, this amounts to studying operators given by sums of normal ordered products of a's and a † 's. More precisely, we are looking for the most general set of formal power series in ladder operators obeying the osp(1|2) superalgebra.

IV. GAUGE CHOICES
To simplify our problem we fix a gauge using the freedom in Equation (4). A propitious choice is To verify gauge reachability, we consider Then a short computation gives Thus by solving for ǫ 1 (z), ǫ 2 (z),... we can bring S + = z to an operator of the general form (5) by a gauge transformation (4). The function ǫ 0 (z) remains undetermined because there are still residual gauge transformations, respecting our choice S + = z, of the form S ± → 1/U (z) S ± U (z). The beauty of the gauge choice (6) is that the first equation of motion in (3) is now linear.

V. THE LINEAR EQUATION
The linear equation for S − reads Using the identity we can solve this order by order for S − and find In the above A(z) and B(z) are arbitrary functions. Defining the number operator N := z ∂ ∂z , and denoting normal ordering by : • : (e.g., : N 2 : = z 2 ∂ 2 ∂z 2 = N (N − 1)), the above display becomes [35] Using the identity we have The normal ordered operator in the above expression is related to the Klein operator of [12]. It has an interesting action on number operator eigenstates i.e., it vanishes on the space of even number operator eigenstates B and is unity on the space of odd number operator eigenstates F . This means that the operator 1/z appearing in Eq. (8) is well-defined. Also, the operator in the above display is the fermion number operator This obeys {F, z} = z and [F, z 2 ] = 0. In addition to providing a Z 2 grading of the Hilbert space we may demand that F also coincides with the Z 2 grading of the Lie superalgebra osp(1|2) = sp(2) ⊕ R 2 . In the following we focus on the case where the two gradings coincide, since it leads quickly to the solution space; we prove that this yields the most general solution in Appendix A.

VI. HARMONIC OSCILLATOR SOLUTION
Requiring coincidence of Z 2 gradings in conjunction with the solution to the linear equation (8) forces us to consider an ansatz of the form where α(z) and β(z) are both odd with respect to the Z 2 grading (i.e., even and odd functions of z). Here we also used that (−1) F = 1 − 2F . It remains to solve the second, non-linear equation in (3) which we rewrite as where the Hamiltonian is easily computed from Eq. (9): The above leads to the relation for some constant c. Requiring that S − acting on the Fock space H (and in particular on the vacuum |0 ) is well-defined we set where A(z) is analytic and odd. Thus First observe that since F |0 = 0, the operator 1 z F is, as promised, well-defined. Moreover, since A(z) is odd, the function U (z) = exp − z A(z) is even and thus Altogether then, we find a one parameter family of solutions Although the Hamiltonian H = Q +− only receives a shift in its zero point energy. The commutator of the deformed oscillators S ± is easily calculated to be This is exactly the model proposed by Plyushchay [1] (although basic quantum commutators were already studied in [10]). The operator S − is a Yang-Dunkl type operator [11]. The osp(1|2) representation obeyed by this model was analyzed in [12] (and also recently discussed in [13]), this is summarized in the next section [36].

VII. OSCILLATOR ORTHOSYMPLECTIC REPRESENTATION
Our solution (11) obeys the osp(1|2) Lie superalgebra and therefore provides a representation thereof. To analyze this we start by searching for states annihilated by S − so consider ψ(z) subject to which we decompose as where the two terms on the right hand side are analytic and even/odd respectively. Since S − is odd we must separately have Thus ψ + (z) = 1 = |0 , the standard Fock vacuum. There is in addition the possibility of a second solution ψ − (z) = z 1−2E0 . Because ψ − (z) is analytic and odd this occurs Thus |0 is always a highest weight state subject to while |2n + 1 is a singular vector when E 0 = −n and then obeys H|2n + 1 = (n + 1)|2n + 1 .

At the harmonic oscillator value
Hilbert space is then the unitary irreducible representation S(1/2) = D(1/2) ⊕ D(3/2) given by a direct sum of two discrete series unitary irreducible sp(2) representations. Indeed, unlike sp (2), which also has supplementary and principal series representations, the Lie superalgebra osp(1|2) only has discrete series unitary irreducible representations [16] (see Appendix B for further details).
When E 0 > 0, these precisely match the Casimirs of the discrete series representations D(E 0 ) and D(E 0 + 1). Moreover, the direct sum of these representations yields the osp(1|2) discrete series representation S(E 0 ). Indeed, the orthosymplectic Casimir on the harmonic oscillator state space H = B ⊕ F . However, when E 0 = 1/2, the operators Q −− and S − are no longer the hermitean conjugates of Q ++ and S + with respect to the standard Fock space inner product. But, since the osp(1|2) action on the harmonic oscillator Fock space is isomorphic to that of the orthosymplectic discrete series, there exists a corresponding inner product with respect to which this is a unitary representation. This inner product can be computed as follows: First observe that with respect to the Fock norm the state |E 0 , n = (S + ) n |0 = |n obeys |E 0 , n 2 Fock = 0|a n (a † ) n |0 = n! 0|0 = n! . However, with respect to the unitary discrete series norm, Here we have employed the standard Pochhammer notation and used the identity (valid for n ∈ Z ≥1 ) The operator version of this identity is given in (12). Importantly, the above derivation uses only the osp(1|2) algebra. Hence we have the relation between Fock and discrete series inner products [37] for the complete set of states We would like to encode this using an operator built from the Casimir and number operators, and therefore note that Thus, by virtue of the identity (13), we introduce the operator where the operator-valued Pochhammer and double factorial are defined by expanding in eigenstates of N , while the operatorÊ 0 returns E 0 on all states and can be expressed in terms of the Casimir via (14). By construction aI|E 0 , n = IS − |E 0 , n whence Thus, the discrete series unitary inner product ·, · osp between states Ψ = |ψ and Φ = |φ then reads Hence, when E 0 > 0 we have found a realization of the unitary orthosymplectic discrete series representations S(E 0 ) in terms of the harmonic oscillator state space.

VIII. SUPERCONFORMAL QUANTUM MECHANICS
We now want to repurpose our harmonic oscillator analysis for a study of novel superconformal theories. For that we will modify our Hilbert space such that the operator − 1 2 Q −− is self-adjoint and plays the rôle of the Hamiltonian H. We may then view osp(1|2) as a conformal superalgebra: Here, because osp(1|2) imposes the operator Q is the SUSY generator. Also D and K correspond to dilations and conformal boosts while S is the conformal SUSY charge. We now need to build the Hilbert space on which H and Q act. For that we begin by studying the space of wavefunctions ψ(x) on the line R. Since the de Rham cohomology of this space is trivial, we will assume that the abelian gauge field A appearing in Eq. (10) can be gauged away in the following analysis. Thus the SUSY charge is while half its square gives the Hamiltonian In the above displays, the fermion occupation number F equals unity on odd wavefunctions ψ − (x) = −ψ − (−x) and vanishes on even wavefunctions ψ + (x) = ψ + (−x). The remaining osp(1|2) generators are obtained by the replacement z → x in the solution given in Eq. (11). Observe, that the Z 2 grading osp(1|2) = B ⊕ F with B = span{Q ±± , Q +− } and F = span{S ± } still holds when F is defined by wavefunction parity. The inverse square potential in the above Hamiltonian is typical of conformal quantum mechanical models [8]. Supersymmetry charges and Hamiltonians of this type were also studied by Plyushchay [1,12,17]. Our next task is to develop an inner product with respect to which they are self-adjoint. This will require a careful analysis of the space of self-adjoint extensions for these operators [38].

IX. THE INNER PRODUCT
Our first task is to ensure definite hermiticity for the supercharge Q (thereafter we will examine its selfadjointness). For that, first observe that acting on odd functions iQ simply acts as ∂ ∂x + 2E0−1 x . Therefore it is convenient to define so that we have the identity Note that E 0 is an, a priori arbitrary, complex number. Firstly let decompose wavefunctions into even and odd parts according to and then use that the information of ψ is stored by ψ ± on the positive half line x > 0. On the whole line we thus define [39] ψ ± (x) := ψ ± (x) , x > 0 , ±ψ ± (−x) , x < 0 .
Using the parametrization (15) for the odd part we may thus define the inner product For E 0 ∈ R, this inner product is positive definite and sesquilinear, but restricts the allowed behavior of wavefunctions at x = 0, ∞. In particular ψ ± must be square integrable with respect to the measure x 2E0−1 on R. In particular this requires that for small x, the fastest decay behavior of ψ ± is ψ ± ∼ x a± , with We denote the space of functions with square integrable behavior at large x and decay rate at the origin satisfying the above bound by H a+,a− . We next examine the SUSY charge on these spaces. Now since iQψ = ψ ′ + + x 1−2E0ψ′ (primes denote x derivatives), it follows that A wavefunction ψ sits inside the domain dom(Q) of Q provided it has the following small-x behavior The operator Q is hermitian, since In the above, the condition (18) guarantees cancellation of the boundary term, which only requires the (weaker) condition a + + a − > 1 − 2E 0 . Thus the SUSY charge is hermitean (indeed we chose the inner product (16) precisely for this reason). It remains to examine whether Q is (essentially) self-adjoint, or more precisely whether it admits self-adjoint extensions. The following analysis is standard and follows classical work by Von Neumann [19]. The space dom(Q) is dense in H so Q possibly has self-adjoint extensions. The dimension of the space of extensions equals the dimensions of [ran(Q ± iλ)] ⊥ for λ real and positive-if these dimensions differ for ±λ the operator Q has no self-adjoint extensions-these dimensions are known as deficiency indices. It is, of course, equivalent to compute the dimensions of ker(Q ± iλ) and the condition Qψ = −iλψ amounts to These can be reduced to a pair of modified Bessel equations: We call y = λx and ψ ± (x) = x 1−E0 u ± (y) and feed the two equations into one another which gives where α + = E 0 − 1 and α − = E 0 .
Equations (21) are identical for both ±λ, so that the deficiency indices are equal. Solutions to (21) are modified Bessel functions (I α , K α ) with indices α ± . Of these solutions only K α (λx) has a good behavior at x → ∞. On the other hand, for small, positive, x it behaves (up to a non-zero coefficient) as [40] Hence, solutions to the kernel condition (20) are in H if the above exponents satisfy the condition (17), which amounts to In other words, when the parameter E 0 satisfies the above condition both deficiency indices are unity and there is a one-parameter set of self-adjoint extensions [41]. On the other hand if E 0 does not satisfy (22) there is a unique extension. Since 2H = Q 2 , it follows that the Hamiltonian also has a unique self-adjoint extension in the latter case. Moreover, we immediately learn that the spectrum of H is bounded below by zero. This can also be seen by explicitly computing the expectation value of the Hamiltonian for some state ψ = ψ + + ψ − := ψ + + x 1 2 −E0 χ: Here we have used that ψ is in the domain of H to kill boundary terms at the origin generated by integrations by parts in the above computation. The final result is manifestly positive for all E 0 (even though the Hamiltonian has non-positive potential term for E 0 < 1 2 acting on odd wavefunctions).

X. THE SPECTRUM
To compute the spectrum of the model we diagonalize the SUSY charge Q in order to solve the Schrödinger equation Hψ = Eψ. The "BPS" states obeying Qψ = 0 are constants which are not finite norm. This indicates that we expect to find a plane wave normalizable spectrum, just as for the free particle on a line. Indeed, we may recycle our deficiency index computation to solve Hψ = Eψ by replacing λ → i √ 2E. We Here we have chosen ν ± ≥ 0 in order that we get plane wave normalizable solutions. Thus we have where the complex constants β ± multiply the even/odd solutions. It follows from our previous deficiency index computations that these solutions are not normalizable, nonetheless, they obey an analog of plane wave normalizability by virtue of the closure relation for Bessel functions (valid for ν > −1/2 and hence for any values of the positive indexes ν ± in Eq. (23)) Indeed, if we define Bose and Fermi scattering states by then E, − | E, + = 0 and In addition to particle scattering states, it is interesting to look for the osp(1|2) analog of the sp(2) spherical vector. Indeed recall that the spherical vector for the metaplectic representation of Sp(2, R) is the state with minimal eigenvalue of the generator H + K of the maximal compact subgroup SO (2). Indeed, this is none other than the harmonic oscillator ground state ψ 0 = exp(− 1 2 x 2 ). When E 0 = 1 2 , this state is annihilated by S + iQ. For the osp(1|2) algebra, we therefore search for states in the kernel of S + iQ. For bosonic (even) states, the only solution is again which is in the Hilbert space H so long as E 0 > 0. For fermionic (odd) states, we must solve and find The above state is in H whenever E 0 < 1. Note that strictly speaking, for values of the parameter E 0 with 0 < E 0 < 1 a detailed analysis of the self-adjoint extensions of Q is required to decide which combination(s) of the above two states is actually in the kernel of S + iQ. The above states will play the rôle of highest weights in the next section.

XI. PARTICLE ORTHOSYMPLECTIC REPRESENTATION
It remains to identify the orthosymplectic representations realized by the particle solutions to the deformation equations.
First we compute the Casimir operator for the sp(2) subalgebra (H, K, D), which reads Our deficiency index analysis shows that this holds for all E 0 , modulo the choice of self-adjoint extension for 0 < E 0 < 1.
To analyze the osp(1|2) content of the model, we can consider an oscillator-like basis for the generators with the reality condition (B6) by employing the map (B7). Indeed, calling we have (using (25)) and S + = A † , S − = A obey the osp(1|2) algebra (2). (Note that this is a different solution to that given in Eq. (11).) At this point the operators A and A † obey the same algebra as analyzed for the oscillator models in Section VII, so we can inherit that analysis, however some care is required when 0 < E 0 < 1. Firstly when |E 0 − 1 2 | ≥ 1/2 the self-adjoint extension problem gives a unique answer and indeed there is a unique highest weight state The descendants of |E 0 , 0 (generated by acting with A † ) then span the irreducible representation S(E 0 ).
When 0 < E 0 < 1 there are potentially two highest weight states ψ B 0 and ψ F 0 , however, we conjecture that only one combination of these is a zero mode of A for a given choice of self-adjoint extension of Q.
As an example consider the undeformed models with E 0 = 1 2 and Q = d dx . Here, the Hilbert space is H = L 2 (R + ) ⊕ L 2 (R + ). There are of course no selfadjoint extensions of d dx on the half-line, but it is easy to find one for d dx defined on two copies of R + , namely by viewing pairs of wavefunctions there as the even and odd parts of wavefunctions in L 2 (R), on which d dx is essentially self-adjoint. In that case Aψ B 0 = 0 because ψ B 0 is the usual harmonic oscillator ground state, while Aψ F 0 (x) = 2δ(x) = 0. The descendants of ψ B 0 then give the unitary irreducible orthosymplectic representation S( 1 2 ).
We have summarized the orthosymplectic representations realized by deformations of superconformal quantum mechanics in the diagram below:

XII. SUMMARY AND CONCLUSIONS
Although supersymmetric quantum mechanics has a long history [42], its presence in even the simplest of quantum mechanical models is often underappreciated-both the free particle and harmonic oscillator enjoy a hidden osp(1|2) superconformal symmetry realized by employing wave-function parity for the Bose-Fermi Z 2grading. Given a particle/oscillator Hilbert space, we studied the natural question whether other sets of operators realize this algebra. In higher dimensions the moduli space of such operators has a particularly interesting geometric structure: For example, on any (pseudo)-Riemannian manifold whose metric g µν is the gradient of a covector g µν = ∇ µ ξ ν , the triplet of operators {ξ µ ξ µ , ξ µ ∇ µ , ∇ µ ∇ µ } generate the algebra sp (2). Including spinors and the Dirac operator, this algebra can be extended to the osp(1|2) superalgebra studied here and indeed our study is the special case when the underlying manifold is one-dimensional. The fact that we were able to give a detailed classification of this space of operators in a one-dimensional setting suggests that similar general results ought be obtainable in higher dimensions. This is exciting because of its relevance to interacting higher spin and quantum gravity models [2-4, 6, 7]. The one dimensional solutions to the osp(1|2) operator question are parameterized by a one (complex) parameter moduli space. It would be interesting to try and mimic these results for higher hidden quantum mechanical SUSY algebras, the results of [24] indicates that this ought be possible [43]. Here, once one studies Hilbert spaces for mechanics in higher dimensions, one expects a moduli space of solutions with more constraining geometric structures than conformal geometries.
One might wonder whether our results contravene the Stone-Von Neumann theorem on unitary equivalence of Heisenberg representations. This is not the case because the Plyushchay-type models generate osp(1|2) representations with differing values of E 0 and inner product by modifying the commutation relation [a, a † ] = 1 to [S, S † ] = 1 − (2E 0 − 1)(2F − 1), where E 0 = 1/2 gives the standard harmonic oscillator model. It interesting to note that that this deformation is important for deformations higher spin algebras leading to interactions [26,27].
The E 0 = 1/2 orthosymplectic representation is a sum of two discrete series sl(2, R) representations analogous to the double cover half integer spin representations in the theory of angular momentum. Indeed, this is the socalled metaplectic representation of Sl(2, R). It would be interesting to exponentiate these realizations of osp(1|2) representations to give analogs of the metaplectic representation.
The Lie algebra (B1) is extended to the Z 2 graded algebra osp(1|2) ∼ = sp(2) + ✞ ✝ R 2 by adding odd generators s and q that obey In the notation of the introduction, s = 1 2 S + , q = 1 2 S − so the remaining commutation relations may be read off the second line of (2) and read Given the reality conditions (B2), there are two inequivalent reality conditions for the odd generators [14] s † = ±q , q † = ±s . (B6) The first choice above is realized by the harmonic oscillator with s = 1 2 a † and q = 1 2 a. The real linear map induces the map (B3) through the relations (B5) and preserves the osp(1|2) algebra. It gives again the free particle-type reality conditions (B4) and reality conditions The first case corresponds to a free particle on the line with s = 1 2 x and q = i 2 Q = 1 2 ∂ ∂x .
The right hand side above is certainly positive whenever the "ground state energy" E 0 ∈ R >0 . Indeed the Hilbert space D(E 0 ) for real positive E 0 is the unitary irreducible (positive) discrete series representation of sl(2, R). It has quadratic Casimir c sp(2) = 1 4 h 2 + 1 2 (ef + f e) given by Hence the representations D(E 0 ) and D(2 − E 0 ) have the same Casimir. In particular, the harmonic oscillator The unitary irreducible representations of osp(1|2) generalize the harmonic oscillator example and are given by the Z 2 -graded vector space [16] S(E 0 ) = span |E 0 , 2k = e k |E 0 , 0 , |E 0 + 1, 2k = e k s|E 0 , 0 k ∈ Z ≥0 , h|E 0 , 0 = On the orthosymplectic discrete series it takes the value Observe that this is minimized by E 0 = 1 2 which corresponds to the harmonic oscillator.