The Hamiltonian H=xp and classification of osp(1|2) representations
aa r X i v : . [ m a t h - ph ] J a n The Hamiltonian H = xp and classification of osp (1 | representations G. Regniers and J. Van der JeugtDepartment of Applied Mathematics and Computer Science, Ghent University,Krijgslaan 281-S9, B-9000 Gent, [email protected], [email protected]
Abstract
The quantization of the simple one-dimensional Hamiltonian H = xp is of interest for its mathematical properties rather than for its physicalrelevance. In fact, the Berry-Keating conjecture speculates that a properquantization of H = xp could yield a relation with the Riemann hypoth-esis. Motivated by this, we study the so-called Wigner quantization of H = xp , which relates the problem to representations of the Lie superalge-bra osp (1 | . In order to know how the relevant operators act in representa-tion spaces of osp (1 | , we study all unitary, irreducible ∗ -representationsof this Lie superalgebra. Such a classification has already been made by J.W. B. Hughes, but we reexamine this classification using elementary argu-ments. The suggestion that the zeros of the Riemann zeta function might be related tothe spectrum of a self-adjoint operator H goes back to Hilbert and P´olya in theearly th century. It was not until the works of Selberg [1] and Montgomery [2]that this conjecture gained much credibility. Due to papers by Connes [3] andBerry and Keating [4, 5] in the late 1990s, it appears that the Hilbert-P´olya con-jecture might be related to the classical one-dimensional Hamiltonian H = xp .More precisely, Berry and Keating suggest that some sort of quantization of thisHamiltonian might result in a spectrum consisting of the values t n , where the t n are the heights of the non-trivial Riemann zeros + it n . A proper quantizationrevealing such a correspondence is, however, not known.These interesting observations stimulated us to perform a different quantizationof the Hamiltonian H = xp . In Wigner quantization one abandons the canonicalcommutation relations and instead imposes compatibility between Hamilton’sequations and the Heisenberg equations as operator equations. The result is aset of compatibility conditions that are weaker than the canonical commutationrelations. This was applied for the first time in a famous paper by Wigner [6].Wigner’s approach has been applied to many different Hamiltonians, leading tovarious connections with Lie superalgebras [7–9]. In the present text, Wignerquantization will lead to the Lie superalgebra osp (1 | . Since it is our interest todetermine the spectrum of the operators ˆ H and ˆ x , one needs the action of theseoperators in representation spaces of osp (1 | . We present a classification of allirreducible ∗ -representations of this Lie superalgebra, thus reconstructing andimproving some results by Hughes [10].1 Wigner quantization of H = xp The simplest Hermitian operator that corresponds to our Hamiltonian is givenby ˆ H = 12 (ˆ x ˆ p + ˆ p ˆ x ) . (1)Without the assumption of any commutation relations between the position andmomentum operators ˆ x and ˆ p , one can still compute Hamilton’s equations ˙ˆ x = ∂ ˆ H∂p = ˆ x, ˙ˆ p = − ∂ ˆ H∂x = − ˆ p and the equations of Heisenberg ˙ˆ x = i ~ [ ˆ H, ˆ x ] , ˙ˆ p = i ~ [ ˆ H, ˆ p ] and impose that they are equivalent. The resulting compatibility conditions (wechoose ~ = 1 ) [ { ˆ x, ˆ p } , ˆ x ] = − i ˆ x, [ { ˆ x, ˆ p } , ˆ p ] = 2 i ˆ p (2)are weaker than the usual canonical commutation relations [ˆ x, ˆ p ] = i . We wishto find self-adjoint operators ˆ x and ˆ p such that the compatibility conditions (2)are satisfied. For that purpose we define new operators b + and b − , satisfying ( b ± ) † = b ∓ , as b ± = ˆ x ∓ i ˆ p √ . One can rewrite the Hamiltonian ˆ H in terms of the b ± as follows: ˆ H = i b + ) − ( b − ) ) . Evidently the operators ˆ x and ˆ p can be expressed as linear combinations of the b ± . Even the compatibility conditions can be reformulated. They are equivalentto [ ˆ H, b ± ] = − ib ∓ , which in turn can be written as (cid:2) { b − , b + } , b ± (cid:3) = ± b ± . (3)These equations are recognized to be the defining relations of the Lie superalge-bra osp (1 | , generated by the elements b + and b − . So we have found expres-sions of all relevant operators in terms of Lie superalgebra generators.A question one might ask is to find the spectrum of ˆ H and ˆ x in an osp (1 | representation space, which is only possible once these representation spacesare known. The spectral problem will be tackled in a subsequent paper. Rightnow, we wish to present a straightforward way of classifying the irreducible ∗ -representations of osp (1 | . 2 Classification of irreducible ∗ -representations of osp (1 | Although we are aware of the classification by Hughes in [10], we think it ispossible to achieve his results in a more accessible way, based on [11]. In addi-tion we will be able to identify some equivalent representation classes. Beforegiving the details of our classification, we provide the readers with the neces-sary definitions and a general outline of how we will construct all irreducible ∗ -representations of osp (1 | . We will be dealing with the Lie superalgebra osp (1 | , generated by two opera-tors b + and b − that are subject to the relations (3). The generating operators b + and b − are the odd elements of the algebra, while the even elements are h = 12 { b − , b + } , e = 14 { b + , b + } , f = − { b − , b − } . Among others, the following commutation relations can now be computed fromthe defining relations (3): [ h, e ] = 2 e, [ h, f ] = − f, [ e, f ] = h. One can define a ∗ -structure on osp (1 | , which is an anti-linear anti-multiplicative involution X X ∗ . For X, Y ∈ osp (1 | and a, b ∈ C wehave that ( aX + bY ) ∗ = ¯ aX ∗ + ¯ bY ∗ and ( XY ) ∗ = Y ∗ X ∗ . Our ∗ -structureis provided by the dagger operation X X † , so we have (cid:0) b ± (cid:1) ∗ = b ∓ andtherefore h ∗ = h, e ∗ = − f and f ∗ = − e . Once we have constructed such a ∗ -algebra, we need to define representations. Definition 1
Let A be a ∗ -algebra, let H be a Hilbert space and let D be adense subspace of H . A ∗ -representation of A on D is a map π from A into thelinear operators on D such that π is a representation of A regarded as a normalalgebra, together with the condition h π ( X ) v, w i = h v, π ( X ∗ ) w i (4) for all X ∈ A and v, w ∈ D . The representation space D , together with therepresentation π , is called an A -module. A submodule of D is a subspace thatis closed under the action of A . The representation π is said to be irreducible ifthe A -module D has no non-trivial submodules. The even operators h , e and f , together with the previously defined ∗ -structure,form the Lie algebra su (1 , . Both su (1 , and osp (1 | possess a Casimiroperator, denoted by Ω and C respectively: Ω = −
14 (4 f e + h + 2 h ) , C = −
4Ω + 12 ( b − b + − b + b − ) . Ω commutes with every element of su (1 , and similary for C . More-over, we have Ω ∗ = Ω and C ∗ = C .We will construct all possible irreducible ∗ -representations of osp (1 | start-ing from one assumption: h has at least one eigenvector in the representationspace with eigenvalue µ , or π ( h ) v = 2 µ v . (5)Starting from this one vector, we will build other basis vectors of the representa-tion space V by letting operators of osp (1 | act on it. After having determinedthe actions of all osp (1 | operators on all basis vectors of V , we will extend therepresentation π to a ∗ -representation. This is done by defining a sesquilinearform h ., . i : V → C , which is to be an inner product that satisfies (4).The stipulation that h ., . i should be an inner product will be crucial in limitingthe possible representation spaces. However, we will postpone the details of thisdiscussion to the point where we have enough arguments for this end. So let usstart with the actual construction of the representation space V . In this section, the ∗ -structure is of no importance. We will construct an ordi-nary osp (1 | representation space that we will extend to a ∗ -representation inthe next section.The embedding of su (1 , in osp (1 | implies that any irreducible representa-tion of osp (1 | is a representation of su (1 , , the latter being not necessarilyirreducible. V can therefore be written as a direct sum of irreducible representa-tion spaces of su (1 , , or V = M i W i . Without loss of generality, we can regard v as an element of W . Since W is arepresentation space of su (1 , , we know that v k = π ( e ) k v and v − k = π ( f ) k v must be elements of W . All these vectors span the space W , which is generatedby a single vector v .The action of b + on any vector of W must be a vector outside W , providedthat this action differs from zero. Let us define v = π ( b + ) v . We can say that v is an element of W . Similarly, we can look at the action of b − on v : v − = π ( b − ) v . Since b − b + is a diagonal operator (apparent from the definition of the Casimiroperator C ), π ( b − ) v is a certain multiple of v . At this point however, we can-not be sure that π ( b − ) v is different from zero. Likewise, it is impossible to4ell whether π ( b + ) v − = 0 . Since we can neither say that π ( f ) v is a nonzeromultiple of v − , nor that π ( e ) v − is a multiple of v , we must regard v − asan element of a different subspace W − . Note that W and W − are the samespaces when either π ( b − ) v or π ( b + ) v − differs from zero. These actions arezero simultaneously only when µ = 0 .We denote the generating vectors of W − as v − k − = π ( f ) k v − and the gen-erating vectors of W as v k +1 = π ( e ) k v . Lemma 2
The vectors of W , W − and W are connected by the actions of b + and b − in the following manner v k +1 = π ( b + ) v k and v − k − = π ( b − ) v − k , (6) for every positive integer value of k . Proof:
Applying π ( b + ) to the vector v results in a vector of W because π ( b + ) v = 2 π ( e ) v . Thus we find π ( b + ) v = π ( e ) v = v . It is clear that thiscan be generalized to the stated formula for v k +1 . The result for v − k − canbe found analogously. (cid:3) Figure 1 helps to visualize how the representation space is constructed. Weemphasize that the relationship between v and v − is not yet determined. W − W W q q q q v − v v v q q q q v − v − v v ❙❙❙❙♦ ❙❙❙❙♦ b − b − ❙❙❙❙✇ ❙❙❙❙✇ b + b + ✓✓✓✓✼ ✓✓✓✓✼ b + b + ✓✓✓✓✴ b − ✲✲ ee ✛ ✛ f f Figure 1: The representation V = W − ⊕ W ⊕ W The action of h on the entire representation space V can already be deter-mined. Lemma 3
The action of h on V is given by π ( h ) v k = (2 µ + k ) v k , (7) for all k ∈ Z . Proof:
For even values of k , this follows just from the relations [ h, e k ] = 2 ke k , [ h, f k ] = − kf k . k = 1 , these are the commutation relations [ h, e ] = 2 e and [ h, f ] = − f ,and the required identities follow by induction. We then obtain π ( h ) v k = π ( h ) π ( e ) k v = (2 µ + 2 k ) v k . For the odd values of k , we need [ h, b ± ] = ± b ± , which is an instant consequenceof equation (3). From this, we obtain π ( h ) v k +1 = π ( h ) π ( b + ) v k = (2 µ + 2 k + 1) v k +1 , and similarly for v − k − . (cid:3) We would like to determine the actions of b + and b − on every vector of W , W − and W . Our method involves defining the action of the Casimir operatorson the representation space. We write the respective diagonal actions as π ( C ) v = λv ( ∀ v ∈ V ) ,π (Ω) v k = − δ ( δ + 1) v k ( ∀ k ∈ Z ) . We will argue that the choice of λ is not independent of δ . It is a nice exerciceto show with the help of equation (3) that ( b − b + − b + b − ) = 4( b − b + − b + b − ) − . This can be used to show that C = (1 − C + 4Ω) . If we let both sidesof this equation act on a vector v k , we get a quadratic equation in λ . The twopossible solutions are λ = 2 δ (2 δ + 1) and λ = 2( δ + 1)(2 δ + 1) . We choose λ = λ and remark that the results for the choice λ = λ can bereproduced with the transformation δ → − δ − .In order to be able to determine the actions of b + and b − on every vector of V ,we still need the action of the su (1 , Casimir operator Ω on W − and W . Lemma 4
The Casimir operator Ω acts on W − and W as given by π (Ω) v k +1 = − ( δ −
12 )( δ + 12 ) v k +1 , ( k ∈ Z ) . (8) As desired, the su (1 , -Casimir operator is constant on the subspaces W − and W as well. Moreover, the actions on both subspaces are the same. Proof:
To prove equation (8), we will calculate π (Ω) v k +1 as π (Ω b + ) v k . From(3) we can immediately derive that [ b − , b + ] b + = 2 b + − b + [ b − , b + ] . Using this and twice the definition of the Casimir element C , we obtain b + = b + (1 − C − . b + into b − in both sides of the equa-tion. All of the operators on the right hand side can be applied to vectors of W . So now π (Ω b + ) v k can be easily calculated, with equation (8) as a result. (cid:3) It has now become straightforward to find the actions of b + and b − on all thevectors of V . Proposition 5
The actions of the operators b + and b − on the vectors of V aregiven by π ( b − ) v k = ( µ + k + δ ) v k − ,π ( b − ) v k +1 = 2( µ + k − δ ) v k ,π ( b + ) v − k = − ( µ − k − δ ) v − k +1 ,π ( b + ) v − k − = 2( µ − k + δ ) v − k . (9) After the choice λ = λ one would find these actions by means of the transfor-mation δ → − δ − . Since the actions of h , e and f follow directly from these relations, we havenow constructed all representations of osp (1 | generated by a weight vector v .It remains to investigate irreducibility and the ∗ -condition. ∗ -representations Recall that V is the space spanned by all the vectors v k , k ∈ Z . We introduce asesquilinear form h ., . i : V → C such that h π ( X ) v, w i = h v, π ( X ∗ ) w i for all X ∈ osp (1 | and for all v, w ∈ V . We see that h ∗ = h implies that h v k , v l i = 0 for k = l . This means that the set S = { v k | k ∈ Z , v k = 0 } formsan orthogonal basis for V . We denote by I the index set such that v k ∈ S for all k ∈ I .The form h ., . i is defined by putting h v k , v l i = a k δ kl , k, l ∈ I , with a k to be determined and a = 1 . The definition of a ∗ -representationrequires that the representation space is a Hilbert space, so our sesquilinear formneeds to be an inner product. Hence, we want a k > for k ∈ I . From the actionof h and from h ∗ = h we obtain µ = h π ( h ) v , v i = h v , π ( h ) v i = 2¯ µ, so µ must be a real number. Similar calculations for the actions of Ω and C reveal that both δ ( δ + 1) and δ (2 δ + 1) are real. These two conditions togetherimply that δ must be real.From the actions of b + and b − and from ( b ± ) ∗ = b ∓ , we derive a k +1 = (cid:10) v k +1 , π ( b + ) v k (cid:11) = (cid:10) π ( b − ) v k +1 , v k (cid:11) = 2( µ + k − δ ) a k .
7n the same way we find a k = 12 ( µ + k + δ ) a k − . Some readers might care for a closed expression for the a k . This is given by a k = 12 (3 − ( − k ) ( µ − δ ) ⌈ k/ ⌉ ( µ + δ + 1) ⌊ k/ ⌋ , where ( x ) k = x ( x + 1) · · · ( x + k − is the classical Pochhammer symbol.We wish to determine under which conditions h ., . i is an inner product. Alter-natively put, for which parameter values is a k > for all k ∈ I ? Starting from a = 1 this can be derived inductively using the two previous equations. Wefind that all a k can be positive only if µ − δ > and µ + δ + 1 > .A similar reasoning should yield a positivity condition for the a k for negative k . However, the resulting conditions µ ± δ + k > can never be satisfied forall negative values of k . Hence, the representation π must have a lowest weightvector, because otherwise it would not be possible to define an inner product onthe entire representation space. In this case, the restriction of π to an su (1 , subspace is known as a positive discrete series representation.There are two choices for δ to obtain a lowest weight representation. Onechoice is to have v as a lowest weight vector, which will arise when δ = − µ asone sees from the actions (9). For δ = µ − we obtain π ( b + ) v − = 0 , in whichcase v − is the lowest weight vector. After one of these choices Proposition 5must obviously be rewritten. Before we do this, let us make use of the innerproduct h ., . i to construct an orthonormal basis { e k } : e k = v k k v k k ( k ≥ , e k = ( − k v k k v k k ( k < , and e k +1 = v k +1 k v k +1 k ( k ≥ , e k +1 = ( − k − v k +1 k v k +1 k ( k < , for k ∈ I . We can now investigate all irreducible ∗ -representations of osp (1 | . Proposition 6
The only class of irreducible ∗ -representations of osp (1 | isa direct sum of two positive discrete series representations of su (1 , , deter-mined by a parameter µ . For < µ ≤ , there is only one irreducible ∗ -representation of osp (1 | . The actions of the generators on the basis vectors { e k | k = 0 , , , . . . } of the representation space are determined by π ( b + ) e k = p µ + k ) e k +1 ,π ( b − ) e k = √ k e k − ,π ( b + ) e k +1 = p k + 1) e k +2 ,π ( b − ) e k +1 = p µ + k ) e k . (10)8 or µ > , this representation can occur alongside another one, for which theactions of the generators on the basis vectors { e k | k = − , , , , . . . } are givenby π ( b + ) e k = p k + 1) e k +1 ,π ( b − ) e k = p µ + k − e k − ,π ( b + ) e k +1 = p µ + k ) e k +2 ,π ( b − ) e k +1 = p k + 1) e k . (11) The actions of the other generators follow immediately from these relations andare left for the reader to calculate.
Proof:
For δ = − µ , we get the first representation, which is a lowest weightrepresentation since π ( b − ) e = 0 . It is clear that µ must be strictly positive sothat all the given actions are well defined. The case µ = 0 is excluded to be surethat π ( b + ) e k differs from zero.In the case of the second representation, for δ = µ − , we must add thecondition µ > to guarantee that π ( b + ) e − is well defined and different fromzero. We end up with the desired classification. (cid:3) Note that if we were to choose λ = λ in the discussion preceding Lemma4, we would find exactly the same class of irreducible ∗ -representations. Indeed,these two representations would pop up for the choices − δ − − µ or − δ − µ − . It immediately follows that the other actions remain the same in this case.Finally, we notice an equivalence between both representation classes inProposition 6. Thus, we end up with only one class of irreducible represen-tations of osp (1 | . Theorem 7
The only class of irreducible ∗ -representations of osp (1 | is a di-rect sum of two positive discrete series representations of su (1 , , determinedby a parameter µ > . The actions of the generators on the basis vectors { e k | k = 0 , , , . . . } of the representation space are determined by (10) . Proof:
For µ > , define ¯ e k = ¯ e k − for k = 0 , , , . . . . Then the actions (11)prove to be equivalent to (10) for ¯ µ = µ − . Hence, both representations areequivalent. (cid:3) In this text we have obtained a classification of all irreducible ∗ -representationsof osp (1 | . The latter Lie superalgebra showed up naturally in the Wignerquantization of the considered Hamiltonian H = xp . Our main concern how-ever, was to investigate the spectrum of the operators ˆ H and ˆ x . Since theseoperators are written in terms of generators of osp (1 | we felt the need to ex-plore representations of this Lie superalgebra. They provide us with a suitableframework in which we know how the crucial operators act.9esults about the spectrum of ˆ H and ˆ x have already been found and thedetails will be published in a subsequent paper, but it is interesting to summarizethe results here.In order to find all eigenvalues of one of the operators, one defines a formaleigenvector for a specific eigenvalue t , v ( t ) = ∞ X n =0 α n ( t ) e n , where the e n are the eigenvectors of the osp (1 | representation space V and the α n ( t ) are unknown coefficients depending on the eigenvalue t . Demanding that v ( t ) is an eigenvector of the operator in question will gives us a three term recur-rence relation for the coefficients α n ( t ) . These coefficients are then identifiedwith the orthogonal polynomials that comply with the same recurrence relation.The spectrum of the operator is then equal to the support of the weight functionof this type of orthogonal polynomials.Concretely we have that the spectrum of ˆ H is related to Meixner-Pollaczek poly-nomials and is equal to R with multiplicity two. Generalized Hermite polyno-mials are connected with the spectrum of ˆ x , which is simply R .Recall that Wigner quantization is a somewhat more general approach thancanonical quantization. This means that one should be able to recover the canon-ical case from the results after Wigner quantization. Indeed, our results prove tobe compatible with the well-known canonical case for the representation param-eter µ = . Acknowledgments
G. Regniers was supported by project P6/02 of the Interuniversity AttractionPoles Programme (Belgian State — Belgian Science Policy)
References [1] A. Selberg,