We discuss the dynamical quantum systems which turn out to be bi-unitary with
respect to the same alternative Hermitian structures in a infinite-dimensional
complex Hilbert space. We give a necessary and sufficient condition so that the
Hermitian structures are in generic position. Finally the transformations of
the bi-unitary group are explicitly obtained.Comment: Note di Matematica vol 23, 173 (2004

Marmo, G.

Scolarici, G.

Simoni, A.

Ventriglia, F.

English

arXiv

MARMO, GIUSEPPE

SIMONI, ALBERTO

VENTRIGLIA, FRANCO

SCOLARICI G

Archivio della ricerca - Università degli studi di Napoli Federico II

Quantum bi-Hamiltonian systems, alternative Hermitian structures and bi-unitary transformations

We discuss the dynamical quantum systems which turn out to be bi-unitary with respect to the same alternative Hermitian structures in a infinite-dimensional complex Hilbert space.We give a necessary and sufficient condition so that the Hermitian structures are in generic position. 
Finally the transformations of the bi-unitary group are explicitly obtained

Università del Salento: ESE - Salento University Publishing

Note di Matematica 23, n. 2, 2004/2005, 173–181.Quantum Bi-Hamiltonian systems,alternative Hermitian structuresand Bi-Unitary transformationsG. MarmoDipartimento di Scienze Fisiche, Universita’ di Napoli Federico II,Complesso Universitario di Monte S Angelo, Napoli, Italy;I.N.F.N., Sezione di Napoli, Italygiuseppe.marmo@na.infn.itG. ScolariciDipartimento di Fisica, Universita’ di Lecce, 73100, Lecce;I.N.F.N., Sezione di Lecce, 73100 Lecce, Italygiuseppe.scolarici@le.infn.itA. SimoniDipartimento di Scienze Fisiche, Universita’ di Napoli Federico II,Complesso Universitario di Monte S Angelo, Napoli, Italy;I.N.F.N., Sezione di Napoli, Italyalberto.simoni@na.infn.itF. VentrigliaDipartimento di Scienze Fisiche, Universita’ di Napoli Federico II,Complesso Universitario di Monte S Angelo, Napoli, Italy;I.N.F.M and I.N.F.N., Sezione di Napoli, Italyfranco.ventriglia@na.infn.itReceived: 30/5/2004; accepted: 30/5/2004.Abstract. We discuss the dynamical quantum systems which turn out to be bi-unitary withrespect to the same alternative Hermitian structures in a infinite-dimensional complex Hilbertspace. We give a necessary and sufficient condition so that the Hermitian structures are ingeneric position. Finally the transformations of the bi-unitary group are explicitly obtained.Keywords: functional analysis, quantum mechanics, symplectic geometry, spectral theory.MSC 2000 classification: 37K05 (primary), 58D19.1 IntroductionIn the study of bi-Hamiltonian systems (see ref. [1] for quantum systems) onestarts with a given dynamics and looks for alternative Hamiltonian descriptions.In this paper, we deal with the “inverse problem” recently considered by someof us in different combinations [2]. We start with two Hermitian structures oncomplex Hilbert spaces arising from two admissible triples in the corresponding174 G. Marmo, G. Scolarici, A. Simoni, F. Ventrigliareal Hilbert space and look for all dynamical quantum systems which turn outto be bi-unitary with respect to them. However, we limit ourselves to considerhere the case of admissible triples sharing the same complex structure. Thisstudy is a preliminary step to generalize some results in Ref. [3], to the infinite-dimensional case.This paper is organized as follows. In sec. 2 we show how to construct anadmissible Hermitian structure (admissible triple) on a real Hilbert space start-ing from a metric tensor and a complex structure [3]. Moreover we show howto recover a Hermitian scalar product after the complexification of the Hilbertspace. In sec. 3, we briefly discuss the bi-unitary group in the case of finite di-mensional spaces and relate the genericity condition of the Hermitian structureswith the cyclicity of the corresponding connecting Hermitian operator. In sec.4, we discuss bi-unitary groups in the infinite-dimensional case, showing howthe direct integral decomposition of the Hilbert space with respect to a normedring is a suitable theoretical tool to deal with such a problem [4]. In particularwe show how this decomposition with respect to the bicommutant of the op-erator connecting two Hermitian structures can be constructed and used for afull comparison of the two Hermitian structures. In sec. 5, we prove that thecomponent spaces in the decomposition are unidimensional if and only if theHermitian structures are in generic position, what allows to conclude that allthe generic bi-unitary quantum systems commute. This construction resemblesthat of the algebra one obtains for completely integrable systems by means ofthe recursion operator [5]. Moreover, the bi-unitary group is explicitly exhib-ited both in the generic and non generic case. Finally, in the last section, someconcluding remarks are posed.2 Admissible Hermitian structures on the HilbertspaceLet us show that given a metric tensor g and a complex structure J ina infinite-dimensional real Hilbert space HR, one can construct an admissibleHermitian structure h on HR which becomes an Hermitian scalar product in acorresponding complex Hilbert space H.Given a couple g and J on HR, we can always construct out of them anadmissible couple gs, J where gs is defined in the following waygs =:12{g(J., J.) + g(., .)}, (1)which will be positive and nondegenerate if g is positive and nondegenerate. Themetric gs and J are admissible in the sense that J is an anti-Hermitian operatorQuantum Bi-Hamiltonian systems 175with respect to gs. Hereafter we drop s in gs. Such an admissible couple may becompleted defining a symplectic structure ω on HR asω = g ◦ J.Then the triple (g, J, ω) is admissible.It is possible to obtain an admissible triple even starting from g and ω. Infact by Riesz’s theorem a nonsingular linear operator B exists such thatω(x, y) = g(Bx, y) (2)and the antisymmetry of ω together with symmetry of g imply thatg(Bx, y) = −g(x,By) (3)so B is skew-Hermitian and −B2 > 0. Then a (symmetric) non-negative squareroot of B, R,will be injective and densely defined. So R−1 is well defined [6] andputting J =: BR−1 andgω(., .) =: g(R., .) (4)we recoverω(x, y) = g(Bx, y) = gω(Jx, y) (5)and in conclusion the triple (gω, J, ω) is admissible. In other words J is or-thogonal and symplectic and also infinitesimally orthogonal and infinitesimallysymplectic, i.e. it is unitary and skew-Hermitian: J † = −J, J2 = −1.Now, from a metric tensor g and an admissible symplectic form ω, an Her-mitian structure on HR can be obtained. This is a map h : HR × HR → R2defined as followsh : (x, y) 7−→ (g(x, y), ω(x, y)). (6)Equivalently (and having in mind a quantum system) one can complexify HRby defining, for any complex number z = α+ iβ and any vector x:z.x = (α+ iβ)x =: αx+ Jβx. (7)Then h becomes an Hermitian scalar product, linear in the second factor,on this complex Hilbert space H, and we geth(x, y) = g(x, y) + ig(Jx, y). (8)176 G. Marmo, G. Scolarici, A. Simoni, F. Ventriglia3 Bi-unitary descriptions in HIn quantum mechanics the Hilbert space H is given as a complex vectorspace. Now we consider on H two different Hermitian structures, coming fromtwo admissible triples (g1, J1, ω1) and (g2, J2, ω2) on HR with the same complexstructure: J1 = J2 = J . The assumption J1 = J2 implies that the correspondingHermitian structures on H are compatible, in the sense that the group of bi-unitary transformations is non void. [3]Denoting with h1(., .) and h2(., .) the Hermitian structures given on H (bothlinear, for instance, in the second factor), we search for the group that leavesboth h1 and h2 invariant, that is the bi-unitary group.Again by Riesz’s theorem a bounded, positive operator G may be defined,which is self-adjoint both with respect to h1 and h2, ash2(x, y) = h1(Gx, y) ∀x, y ∈ H. (9)Moreover, any bi-unitary operator U must commute with G. Indeedh2(x, U†GUy) = h2(Ux,GUy) = h1(Ux,Uy) =h1(x, y) = h2(Gx, y) = h2(x,Gy)and from thisU †GU = G, [G,U ] = 0. (10)Therefore the group of bi-unitary operators belongs to the commutant G′ of theoperator G.Let us discuss the bi-unitary group when H is finite-dimensional. In this caseG is diagonalizable and the two Hermitian structures result proportional in eacheigenspace of G via the eigenvalue. Then the group of bi-unitary transformationsisU(n1)× U(n2)× · · · × U(nk) n1 + n2 + · · ·+ nk = n = dimH,where nl denotes the degeneracy of the l-th eigenvalue of G.In finite-dimensional complex Hilbert spaces the following definition can bestated [3]:1 Definition. Two Hermitian forms are in generic position iff the eigenval-ues of G are nondegenerate.Quantum Bi-Hamiltonian systems 177Then, if h1 and h2 are in generic position the group of bi-unitary transfor-mations becomesU(1)× U(1)× · · · × U(1)︸ ︷︷ ︸ .n factorsIn other words this means that G generates a complete set of observables.Now we show that:Two Hermitian forms are in generic position if and only if their connectingoperator G is cyclic.In fact the non singular operator G has a discrete spectrum and is diagonal-izable, so when h1 and h2 are in generic position G admits n distinct eigenvaluesλk. Let now {ek} be the eigenvector basis of G and {µk} an n-ple of nonzerocomplex numbers. Then the vectorx0 = µkek (11)is a cyclic vector for G, that is the vectors x0, Gx0, . . . , Gn−1x0 are n linearlyindependent vectors. In fact one obtainsGmx0 = µkλmk ek, (12)and the coefficient determinant is given by(∏kµk)V (λ1, . . . , λn), (13)where V denotes the Vandermonde determinant which is different from zerowhen all the eigenvalues λk are distinct. The converse is also true.4 Infinite-dimensional caseNow we deal with the infinite-dimensional case, when the connecting op-erator G may have a point part and a continuum part in its spectrum. As tothe point part, the bi-unitary group is U(n1) × U(n2) × · · · where now nl canalso be ∞. When G admits a continuum spectrum, the characterization of thebi-unitary group is more involved and suitable mathematical tools are needed,such as spectral theory of operators and theory of rings of operators in Hilbertspaces. In particular in the infinite-dimensional case definition 1 of generic po-sition of two Hermitian forms is no more valid and has to be generalized in thefollowing way [3]:178 G. Marmo, G. Scolarici, A. Simoni, F. Ventriglia2 Definition. Two Hermitian forms are in generic position iff G′′ = G′,that is when the bicommutant G′′ coincides with the commutant G′ of G.First of all, observe that the commutant G′ and the bicommutant G′′ ofthe operator G are weakly closed rings of bounded operators in the ring of allbounded operators B(H) on H and G′′ ⊂ G′. Let E0 be the principal identity ofa set S ⊂ B(H): by definition E0 is the projection operator on the orthogonalcomplement of the set KerS ∩KerS† and satisfies for all A ∈ S the followingrelation:E0A = AE0 = A. (14)It can be proved [4] that the minimal weakly closed ring R(S) containing Sis the strongest closure of the ring Ra∗(S), which is the minimal ring containingS ∪ S†. R(S) contains only those elements A ∈ S ′′ which satisfy the abovecondition (14).Now, by the spectral theorem we express the selfadjoint operator G in termsof its spectral family {P (λ)}:G =∫∆λdP (λ) (15)where ∆ = [a, b] is a closed interval containing the entire spectrum of G. More-over, the positiveness of G ensures that KerG = 0. This implies that 1 ∈ R(G)and hence R(G) ≡ G′′. Therefore G′′ is commutative (in fact R(G) is the w-closure of the symmetric commutative subring G in the ring of B(H)).We recall that each commutative weakly closed ring of operators C in theHilbert space corresponds to a direct integral of Hilbert spaces.The following theorems hold [4]:3 Theorem. To each direct integral of Hilbert spacesH =∫∆Hλdσ(λ),with respect to a measure σ there corresponds a commutative weakly closed ringC = L∞σ (∆) where to each ϕ ∈ L∞σ (∆) there corresponds the operator Lϕ :(Lϕξ) = ϕ(λ)ξλ where ξ ∈ H, ξλ ∈ Hλ and ||Lϕ|| = ||ϕ||.Vice versa:4 Theorem. To each commutative weakly closed ring C of operators in aHilbert space H there corresponds a decomposition of H into a direct integral,for which C is the set of operators of the form Lϕ, ϕ ∈ L∞.Quantum Bi-Hamiltonian systems 179From the previous theorems we get that the weakly closed commutative ringR(G) corresponds to a decomposition of the Hilbert space H into the directintegralH =∫∆Hλdσ(λ), (16)where ∆ = [a, b] is the entire spectrum of the positive selfadjoint operator G.The measure σ(λ) is obtained by the spectral family {P (λ)} and cyclicvectors in the usual way. Now every operator A belonging to the commutant G′is representable in the form of a direct integral of operatorsA =∫∆A(λ) . dσ(λ), (17)where A(λ) is, for almost all λ, a bounded operator in Hλ.Every operator B of the bicommutant G′′ = R(G) is then a multiplicationby a number b(λ) on Hλ for almost all λ.Thus the bi-unitary transformations are in general a direct integral of unitaryoperators U(λ) acting on Hλ.5 Bi-unitary group transformationsMore insight can be gained by a more specific analysis of the direct integraldecomposition.Let G′(λ) be the totality of all the operators A(λ) corresponding to G′,for fixed λ. Then as R(G) is a maximal commutative ring in itself, we mayconclude [7] that the family G′(λ) is irreducible for almost all λ, and the directintegral of H with respect to R(G) can be written as followsH =∫∆Hλdσ(λ) =∑k⊕Hk =∑k⊕∫∆kHλdσ(λ), (18)where now the spectrum ∆ decomposes into a sum of finite or countable numberof measurable sets ∆k, such that for λ ∈ ∆k the spaces Hλ have constantdimension k (k can be ∞). Then any operator A belonging to the commutantG′ is representable as follows:A =∑k⊕∫∆kA(λ) . dσ(λ). (19)As a consequence of Eq. (18), remembering that every operator B of G′′ = R(G)is a multiplication by a number b(λ) on Hλ, we get the following result:180 G. Marmo, G. Scolarici, A. Simoni, F. Ventriglia5 Proposition. Let two Hermitian structures h1 and h2 be given on theHilbert space H. Then there exists a decomposition of H into a direct integral ofHilbert spaces Hλ, of dimension k, such that in each space Hλ, h1 and h2 areproportional: h2 = λh1.Proposition 5 partially generalizes the Lemma in Ref. [3] to the case ofinfinite-dimensional complex Hilbert spaces H. This result belongs to the con-stellation of propositions connected with the so called “Quadratic Hamiltoniantheorems”. [8]As a consequence of Proposition 5, the elements U of the bi-unitary groupcan be written as follows (see Eq. (19)):U =∑k⊕∫∆kUk(λ) . dσ(λ), (20)where Uk : ∆k → U(Hλ), that is Uk(λ) is an element of the unitary groupU(Hλ) on the spaces Hλ of dimension k.Moreover, when the Hermitian forms are in generic position the followingstatement holds:6 Proposition. The component spaces Hλ of the decomposition of H intoa direct integral with respect to R(G) are unidimensional if and only if the twoHermitian forms h1 and h2 are in generic position.Proof. Let us suppose that two Hermitian forms are given in generic po-sition, then by definition R(G) = G′′ = G′, therefore G′ must be commutative.Hence the totality of all irreducibles operators A(λ) corresponding to G′ areunidimensional, then dimHλ = 1 for all λ ∈ ∆.In order to prove the converse, observe that if R(G) = G′′ 6= G′, there exists anon zero subset ∆0 of ∆ such that for λ ∈ ∆0 the totality of irreducible operatorsA(λ) corresponding to G′ is not commutative. Hence dimHλ 6= 1. QEDProposition 6 fully generalizes the Proposition in Ref. [3] to the case ofinfinite-dimensional complex Hilbert space, so that we can say that all thegeneric different admissible quantum dynamical systems are pairwise commut-ing.Finally, Proposition 6 implies that the unitary operators Uk(λ) in Eq. (20),reduce to a multiplication by a phase factor eiϕ(λ) on Hλ for almost all λ, sothat the elements of the bi-unitary group readU =∫∆eiϕ(λ) . dσ(λ). (21)Quantum Bi-Hamiltonian systems 1816 Concluding remarksIn this paper we have shown how to extend to the more realistic case ofinfinite dimensions our results of a previous paper dealing mainly with finitelevel quantum systems.Our approach shows how to deal with “pencils of compatible Poisson Brack-ets” [9] in the framework of quantum systems.Our formulation is already in a form suitable to deal with quantum electro-dynamics.We hope to be able to extend these results to the evolutionary equations forclassical and quantum field theories.References[1] F. Magri: A simple model of integrable Hamiltonian equation, J. Math. Phys. 19 (1978),1156–1162; F. Magri: A geometrical approach to the nonlinear solvable equations, Lect.Notes in Phys. 120 (1980), 233–263; B. Fuchssteiner: The Lie algebra structure anddegenerate Hamiltonian and bi-Hamiltonian systems, Progr. Theor. Phys. 68 (1982), 1082–1104.[2] F. Ventriglia: Alternative Hamiltonian descriptions for quantum systems and non-Hermitian operators with real spectrum, Mod. Phys. Lett. A 17 (2002), 1589–1599; G.Marmo, A. Simoni, F. Ventriglia: Quantum systems: real spectra and nonhermitian(Hamiltonian) operators, Rep. Math. Phys., 51 (2003), 275–285; G. Marmo, A. Simoni,F. Ventriglia: Quantum systems and alternative unitary descriptions, Int. J. Mod. Phys.A, 19 (2004), 2561–2578.[3] G. Marmo, G. Morandi, A. Simoni, F. Ventriglia: Alternative structures and bi-Hamiltonian systems, J. Phys. A: Math. Gen., 35 (2002), 8393–8406.[4] M. A. Naimark: Normed Rings, Wolters-Noordhoff Publishing Groningen, 1970.[5] S. De Filippo, G. Marmo, M. Salerno, G. Vilasi: A new characterization of completelyintegrable systems, Nuovo Cimento B 83 (1984), 97–112.[6] R. Abraham, J. E. Marsden: Foundation of Mechanics, Benjamin-Cummings, New York,1978.[7] See Ref. [4] p. 503.[8] G. Marmo, E. J. Saletan, R. Schmid, A. Simoni: Bi-Hamiltonian Dynamical Systemsand the Quadratic-Hamiltonian Theorem, Nuovo Cimento B 100 (1987), 297–317.[9] I. M. Gel’fand, I Zakharovich: On Local Geometry of Bihamiltonian Structure, Gel’fandMathematical Seminar vol. 1, ed. L. Corwin, I. M. Gel’fand and J. Lepowski, Birkauser,Basel 1993; A. Ibort, F. Magri, G. Marmo, Bihamiltonian structures and Stackel sep-arability, J. Geom. Phys. 33 (2000), 210–228.

Quantum Bi-Hamiltonian systems, alternative Hermitian structuresand Bi-Unitary transformations

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Quantum Bi-Hamiltonian systems, alternative Hermitian structures and
  Bi-Unitary transformations

Quantum Bi-Hamiltonian systems, alternative Hermitian structures and Bi-Unitary transformations

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