Finite dimensional systems of free Fermions and diffusion processes on Spin groups
aa r X i v : . [ m a t h - ph ] F e b Finite dimensional systems of free Fermions anddiffusion processes on
Spin groups
Luigi M. BorasiHausdorff Center of Mathematics &Institute of Applied MathematicsUniversity of Bonn, Germany [email protected]
February 2, 2021
Abstract
In this article we are concerned with “finite dimensional Fermions”, bywhich we mean vectors in a finite dimensional complex space embedded inthe exterior algebra over itself. These Fermions are spinless but possess thecharacterizing anticommutativity property. We associate invariant com-plex vector fields on the Lie group
Spin(2 n + 1) to the Fermionic creationand annihilation operators. These vector fields are elements of the com-plexification of the regular representation of the Lie algebra so (2 n + 1) .As such, they do not satisfy the canonical anticommutation relations,however, once they have been projected onto an appropriate subspace of L (Spin(2 n + 1)) , these relations are satisfied. We define a free time evo-lution of this system of Fermions in terms of a symmetric positive-definitequadratic form in the creation-annihilation operators. The realization ofFermionic creation and annihilation operators brought by the (invariant)vector fields allows us to interpret this time evolution in terms of a pos-itive selfadjoint operator which is the sum of a second order operator,which generates a stochastic diffusion process, and a first order complexoperator, which strongly commutes with the second order operator. Aprobabilistic interpretation is given in terms of a Feynman-Kac like for-mula with respect to the diffusion process associated with the second orderoperator. Probabilistic methods in Quantum Field Theory have proved to be particu-larly fruitful (cf. e.g. [25, 11, 15]). These methods have been almost exclusivelyrestricted to
Bosonic
Field Theories. Some ideas of the Bosonic probabilisticmethods carry over, to an extent, to the Fermionic case using the beautiful alge-braic technique of Berezin integration [3]. However, the Berezin integral, beingdefined in terms of Grassmann variables, does not lend itself easily to an inter-pretation in the context of probability theory or measure theory (nevertheless,1t least for the case of a discrete number of variables, a probabilistic interpre-tation of the Berezin formalism is possible, albeit somewhat cumbersome: cf.e.g. [5, 6]).The aim of this work is to study a finite dimensional system of free Fermioniccreation-annihilation operators in a way which parallels, in the sense explainedbelow, the treatment of the corresponding Bosonic case of a finite dimensionalquantum harmonic oscillator. It is well known that the Hamiltonian of an n -dimensional quantum harmonic oscillator can be interpreted, in Euclideantimes, as the second order differential operator which generates an Ornstein-Uhlenbeck process on R n (cf. e.g. [26, p. 35]). Following this Bosonic parallel westudy a model for a finite dimensional system of Fermions where the FermionicHamiltonian is replaced, in a quite natural way, by a second order differentialoperator. Moreover we study the possibility of interpreting the time evolutiongenerated by such Hamiltonian in terms of stochastic processes.The results in this work are inspired in part by the work of Schulman [24](cf. also [23, Chapters 22-24]) who gives a description of a single -spin particlein terms of the Feynman path integral. The precedents for Schulman’s ideacan be found in early work on Quantum Mechanics connecting the Pauli -spinformalism with the quantum spinning top [4, 21] (cf. also the more recent work[2]). The works which, to the knowledge of the author, are the closest in spirit tothe analysis given here are [8] and [7]. The situation studied here is neverthelessquite different. We are motivated by the parallel between the (Bosonic) quantumharmonic oscillator and the Ornstein-Uhlenbeck process. For this reason weassociate to the Hamiltonian of a finite system of free Fermions a second orderoperator whereas in the works cited above the Hamiltonian was associated to afirst order operator. The analysis that follows is therefore completely different.Moreover, because of our motivation, we pay particular attention to the rigorous,functional analytic details of our description.We start our analysis by considering Fermionic creation-annihilation opera-tors a † j , a j , j = 1 , . . . , n , n ∈ N . We associate to these Fermionic operators firstorder differential operators. We achieve this purpose, similarly to the workscited above, by first exploiting the standard fact that the Fermionic creation-annihilation operators give rise to a faithful, irreducible representation of thecomplex Lie algebra so (2 n +1 , C ) (obtained by complexifying the real Lie algebra so (2 n + 1) of antisymmetric real (2 n + 1) × (2 n + 1) -matrices). This represen-tation is usually called the spin representation of so (2 n + 1 , C ) . We thereforeassociate the Fermionic creation-annihilation operators with abstract elementsin the Lie algebra so (2 n + 1 , C ) . We then use the standard fact that this Lie al-gebra can be realized in terms of (left-invariant) differential operators acting onsmooth functions from the (real) Lie group Spin(2 n + 1) to C . Here Spin(2 n + 1) is the simply-connected Lie group obtained as universal cover of the Lie group SO(2 n + 1) of rotations in n + 1 dimensions. We need to pass from SO(2 n + 1) to Spin(2 n + 1) because the spin-representation of so (2 n + 1 , C ) does not appearinside the representation of so (2 n + 1 , C ) given by left-invariant vector fieldsacting on C ∞ (SO(2 n + 1)) (indeed the spinor representation of SO(2 n + 1) isonly a projective representation of SO(2 n + 1) and not an actual representa-tion). Having associated the Fermionic creation-annihilation operators a † j , a j to differential operators D + j , D − j we consider the free Fermionic Hamiltonian H = P nj =1 E j a † j a j (for positive constants E j ) and we lift it to an element ˜ H of2he universal enveloping algebra of so (2 n +1 , C ) which we look upon as the alge-bra of differential operators on C ∞ (Spin(2 n + 1) generated by the left-invariantvector fields together with the identity. The lift H ˜ H is by its very naturenon-canonical. We choose to define ˜ H = P nj =1 D + j D − j by formally replacing thecreation-annihilation operators in H by their associated first order differentialoperators. This choice differs from the one made by the references cited above.The motivation for our choice is that we want to study a situation parallel tothe Bosonic case, where the free Hamiltonian of a quantum harmonic oscillatorcorresponds to a second order differential operator. The main results we obtainabout the operator ˜ H are contained in theorem 4.7, which contains functionalanalytic properties regarding the operator ˜ H , and in theorem 5.4, where wegive a Feynman-Kac like formula describing the evolution in Euclidean timegenerated by − ˜ H .The layout of the article is as follows. In section 2 we give some basicdefinitions and describe the standard relation between n Fermionic creation-annihilation operators and the spin representation of the Lie algebra so (2 n +1 , C ) , in a way which is well suited for our needs. In section 3 we briefly describethe standard connection between Lie algebra and left-invariant differential op-erators. We then specialize this general relation to our setting and describe howto recover, from this global picture, the spin representation of so (2 n + 1 , C ) .In section 4, after some remarks regarding selfadjointness and the universal en-veloping algebra of a compact Lie group, we define the operator ˜ H and describesome of its most salient functional analytic properties (theorem 4.7). Finallyin section 5 we introduce some standard facts regarding stochastic processes onLie groups and we apply the general theory to our case. The main result is thatthe operator ˜ H splits into two parts, a strictly second order part which is hy-poelliptic and generates a diffusion on Spin(2 n + 1) and a first order part which,as explained in section 5, does not contribute to a diffusion on Spin(2 n + 1) .On the other hand, since it strongly commutes with the second order part, weare able to write a simple Feynman-Kac like formula for the operator ˜ H wherewe average over the process generated by the strictly second order part of − ˜ H .We note that our Feynman-Kac like formula resembles the Feynman-Kac for-mula for the Schrödinger operator of a particle in a magnetic field (cf. e.g. [26,Chapter V, Section 15.]). so (2 n + 1 , C ) In this section, after some notational preliminaries, we describe the relationbetween the Fermionic creation-annihilation operators and the / -spin repre-sentations of the Lie algebra so C . Since this relation is not very broadly knownwe provide some details.Let us denote by C ℓ ( N ) , N ∈ N , the complex Clifford algebra over C N ,that is the unital associative algebra obtained as the quotient of the full tensoralgebra T ( C N ) by the following relations, { v, w } = − h v, w i , v, w ∈ C N , where h· , ·i denotes the standard symmetric bilinear form on C n , and { v, w } def = vw + wv where we have denoted the product of v, w in C ℓ ( N ) simply by vw .3et V be a real or complex , finite dimensional, vector space. We denoteby V V the exterior algebra over V , that is the algebra obtained by quotientingthe full tensor algebra T ( V ) over V by the two sided ideal generated by ele-ments of the form v ⊗ v , v ∈ V . The finite dimensional Fermionic Fock space Γ ∧ C n def = L nk =0 ( C n ) ∧ k , n ∈ N , is defined as the Hilbert space realized by takingthe exterior algebra V C n just as vector space and equipping it with the Hermi-tian scalar product ( · , · ) Γ C n which satisfies ( v ∧ · · · ∧ v n , w ∧ · · · ∧ w n ) Γ ∧ C n def =det jk ( v j , w k ) C n for v j , w k ∈ C n , j, k = 1 , . . . , n , where ( · , · ) C n denotes the stan-dard Hermitian scalar product on C n (antilinear in the left component).We call the element ∈ C ֒ → V C n the vacuum vector . For v ∈ C n , wedenote the Fermionic creation , annihilation operators on Γ ∧ C n respectively by c † ( v ) , c ( v ) . Explicitly, for any v ∈ C n , c ( v ) and c † ( v ) are defined as Hilbertadjoint of each other in Γ ∧ C n with c † ( v ) ψ = v ∧ ψ where ψ ∈ Γ ∧ C n . Notethat by this definition c † ( v ) is (complex) linear in v whereas c ( v ) is (complex)anti-linear. If e j , j ∈ { , . . . , n } , denotes the standard basis of C n , then wedenote c ( e j ) , respectively c ( e j ) † , by c j , respectively c † j . They satisfy the usualcanonical anticommutation relations: { c j , c † k } = δ jk , { c j , c k } = { c † j , c † k } = 0 (where { A, B } def = AB + BA for any A, B in an associative algebra).Let us fix a decomposition C n = C n ⊕ C n orthogonal with respect to thestandard symmetric bilinear form h· , ·i of C n , and let P , respectively P , be theprojection of C n onto the first, respectively the second, copy of C n . Moreoverlet us denote by ¯ v the vector obtained from v ∈ C n by complex conjugatingeach component. Then the standard Hermitian scalar product ( · , · ) on C n satisfies ( v, w ) = h ¯ v, w i , v, w ∈ C n . Let us define the algebra isomorphism γ : C ℓ (2 n ) → End( V C n ) by extending to the whole of C ℓ (2 n ) the relations γ ( v ) = c † ( P v ) − c ( P v ) − i( c † (( P v ) + c ( P v )) , v ∈ C n . (1)The map γ is a representation of C ℓ (2 n ) on the Fock space Γ ∧ C n usually calledthe Fock space representation of C ℓ (2 n ) .Let e j , j = 1 , . . . , n , be the standard basis of C n . We also denote by e ℓ , ℓ = 1 , . . . , n , a basis for each of the copies of C n in the decomposition C n = C n ⊕ C n . To be concrete, in the following we will take P , to be theprojection which sends e j − to e j and e j to zero. Then P sends e j into e j and e j − to zero.Let γ j def = γ ( e j ) , j = 1 , . . . , n . Then, with this choice of P , P we obtain,from (1), the following relations c † j = ( γ j − + i γ j ) c j = ( − γ j − + i γ j ) , j = 1 , . . . n. (2)Note that, under these definitions, the operators γ j = γ ( e j ) , are anti-Hermitianas operators on the finite dimensional Hilbert space Γ ∧ C n .Let so ( N ) , N ∈ N , denote the complex Lie algebra of antisymmetric N × N real matrices. and let so ( N, C ) = so ( N ) ⊗ R C be its complexification.We now define the standard / -spin representation of so (2 n + 1 , C ) on V C n . We give the definition in a form which differs slightly from the standardpresentations (cf. e.g. [9, 12]) therefore we provide some details.4onsider an embedding ι : so (2 n ) ֒ → so (2 n + 1) and the relative vectorspace decomposition so (2 n + 1 , C ) = V n ⊕ ι ( so (2 n, C )) , where V n = so (2 n +1 , C ) /ι ( so (2 n, C )) is a n -dimensional vector space which generates, via the Liebrackets, all of so (2 n + 1 , C ) .Let C ℓ ( V n ) be the complex Clifford algebra generated by the identity andby the symbols κ ( X ) , X ∈ V n , which satisfy { κ ( X ) , κ ( Y ) } = tr( XY ) , X, Y ∈ V n , where tr denotes the trace in the defining representation of so (2 n + 1 , C ) and XY denotes the product of X and Y as (2 n + 1) × (2 n + 1) matrices. Let usidentify V n with C n and denote one such isomorphism by φ . Then φ extendsto an isomorphism of C ℓ ( V n ) with C ℓ (2 n ) and the composition γ ◦ φ : C ℓ ( V n ) → End ( ^ C n ) , of the isomorphism φ with a Fock representation γ of C ℓ (2 n ) , defines a Fockrepresentation of C ℓ ( V n ) . The following proposition shows how the Cliffordalgebra C ℓ ( V n ) with a Fock representation γ ◦ φ gives rise to an irreduciblerepresentation π / of so (2 n + 1 , C ) which is unique up to isomorphism andcoincides with the standard / -spin representation of so (2 n + 1) . Proposition 2.1.
The map κ extends to a Lie algebra homomorphism so (2 n +1 , C ) → C ℓ ( V n ) , still denoted by κ , which sends the Lie brackets of so (2 n +1 , C ) into the commutator [ A, B ] = AB − BA , for A, B ∈ C ℓ ( V n ) . The composition π / = γ ◦ φ ◦ κ : so (2 n + 1 , C ) → End ( ^ C n ) , of the homomorphism κ with the Clifford algebra isomorphism φ and with theClifford algebra representation γ , defines a representation π / of so (2 n + 1 , C ) .This representation is isomorphic to the standard / -spin representation of so (2 n + 1 , C ) , that is, the irreducible representation of so (2 n + 1 , C ) on V C n .Proof. Without loss of generality let us fix { X jk } ≤ j The representation π / restricts to an irreducible represen-tation, also called / -spin representation, of the real Lie algebra so (2 n + 1) .Proof. Indeed, with the same notation as in the proof of the proposition above,we have π / ( X ℓ, n +1 ) = γ ℓ ,π / ( X jℓ ) = γ j γ ℓ , ≤ j < ℓ ≤ n. Being γ ( v ) complex linear in v ∈ C n we have that π / is an irreducible ana-lytic representation of so (2 n + 1 , C ) and it naturally restricts to a well definedrepresentation of the real Lie algebra so (2 n + 1) . By the “Weyl unitary trick”(cf. [1, Theorem 3, ß1, Chapter 8, pp. 202-203]) the restricted representationis indeed irreducible. 6 emark 2.2. We note that, under the same conventions as the proof of theproposition above, the vector ∈ Γ ∧ C n can be seen as a lowest weight vectorwith relative weight ( − , · · · , − ) ∈ C n .To show this let us employ the same notation as in the proof above and letus fix a Cartan subalgebra h of so (2 n + 1 , C ) generated by the elements H j = [ E j E − j ] = i X j − , j , j = 1 , . . . , n. (5)These generators are normalized in such a way that, if we identify h with C n bysending H j into e j and we equip C n with its standard symmetric bilinear form h· , ·i , then the dual space h ∗ is itself isomorphic to C n and the dual of an element H j ∼ = e j is the element H j ∼ = e j itself. Now, H j − c j c † j − . Hence thevector ∈ Γ ∧ C n is associated to the weight λ ∈ h ∗ such that λ ( H j ) = − , forall j = 1 , . . . , n . Under the identifications given above, of h ∗ ∼ = C n ∼ = h , where C n is identified with its dual via the standard symmetric bilinear form on C n ,the weight λ corresponds to the vector (cid:0) − , . . . , − (cid:1) ∈ C n . Under the natural order of R this weight is the lowest weight of the represen-tation. Hence we have shown that, under our conventions, ∈ Γ ∧ C n is the(normalized) lowest weight vector.Similarly, under the same conventions, one can show that e ∧· · ·∧ e n ∈ Γ ∧ C n is the (normalized) highest weight vector corresponding to the highest weight (cid:0) , . . . , (cid:1) ∈ C n .By repeated use of the creation annihilation operators one shows that ageneral weight is of the form ( ± , . . . , ± | {z } n times ) with a given number of “plus signs”and the complementary number of “minus signs”. From a physical perspective,the plus signs in the weight λ denote “filled states”, that is to every “plus sign”there corresponds a Fermionic particle in the respective state.We conclude by stating the following fact which will be regularly used in thefollowing sections. Proposition 2.3. If we let U ( so (2 n + 1 , C )) be the universal enveloping algebraof so (2 n + 1 , C ) then π (1 / extends naturally to a representation of the universalenveloping algebra and we have π (1 / ( U ( so (2 n + 1 , C )) ∼ = C ℓ (2 n ) .Proof. This fact follows at once from the universal property of universal en-veloping algebras (cf. e.g. [13, Theorem 9.7, p. 247]). C ∞ (Spin(2 n + 1)) The proposition 2.3 expresses the complex Clifford algebra C ℓ (2 n ) as representa-tion of the universal enveloping algebra of the complex Lie algebra so (2 n +1 , C ) .In this section, after a short description of the standard representation of theuniversal enveloping algebra in terms of certain differential operators acting ona common domain of functions, we describe how to recover from this (infinitedimensional) representation the / -spin representation π (1 / of so (2 n + 1 , C ) defined in section 2. 7onsider a connected, simply connected, compact Lie group G with Liealgebra g . Let C ∞ ( G ) be the space of complex valued functions on G . Wedenote by L : G → End ( C ∞ ( G )) , L : g L g , the action of G on C ∞ ( G ) bythe left translation L g , g ∈ G , where L g f ( x ) def = f ( g − x ) , g, x ∈ G . Similarly wedenote by R the action of the G on C ∞ ( G ) by the right translation R g , g ∈ G ,where R g f ( x ) def = f ( xg ) , g, x ∈ G .Let us denote by D ( G ) the algebra of differential operators on C ∞ ( G ) gen-erated by the identity and the left-invariant vector fields on G , i.e. the vectorfields which commute with the left translation.We have the following important fact (Cf. [14, Ch. II, Proposition . andits proof, p. 108]): the universal enveloping algebra U ( g ) is isomorphic (asa an algebra) to D ( G ) . Moreover, by this isomorphism, the Lie algebra g isrepresented on C ∞ ( G ) by the representation dR : g → D ( G ) which associatesto each element X ∈ g the corresponding left-invariant vector field dR ( X ) ,where the linear operator dR ( X ) : C ∞ ( G ) → C ∞ ( G ) is defined by dR ( X ) f = ddt R ( e tX ) f | t =0 , f ∈ C ∞ ( G ) . Remark 3.1. The fact that the universal enveloping algebra is isomorphic(as an algebra) to D ( G ) means that the invariant vector fields dR ( X ) , . . . ,dR ( X n ) associated with the generators of the Lie algebra g satisfy the Lie al-gebra commutation relations, that is [ dR ( X ) , dR ( Y )] f = dR ([ XY ]) f , f ∈ C ∞ (Spin(2 n + 1)) , X, Y ∈ g .Let π / = γ ◦ φ ◦ κ be the / -spin representation of so (2 n + 1 , C ) as givenin section 2. With the same notation as in that section we have that π / ( X j, n +1 ) = γ j , j = 1 , . . . , n. Note that γ , . . . γ n satisfy the Lie algebra commutation relations of so (2 n + 1) and the anticommutation relations of the Clifford algebra. Now, we can liftany generator γ j , j ∈ , . . . , n , of the Clifford algebra C ℓ (2 n ) to an invariant(complex) vector field as a differential operator in D ( G ) . These vector fieldswill satisfy the commutation relations of the Lie algebra so (2 n + 1 , C ) but not the Clifford anticommutation relations of the original elements γ , . . . γ n . Torecover the Clifford anticommutation relations we will need to project onto asubspace isomorphic to the Fermionic Fock space Γ ∧ C n . We now turn to thedescription of this procedure.Let L ( G ) denote the space of functions from G to C which are squareintegrable with respect to the normalized Haar measure d g on G . By slightabuse of notation we will still denote by R the extension of the representation,of G on C ∞ ( G ) by right translation, to a representation representation of G on L ( G ) . Note that this extension gives a unitary representation. We now embedthe Fermionic Fock space V C n into L (Spin(2 n + 1)) . Lemma 3.2. Let π (1 / ( g ) denote the / -spin representation of an element g ∈ Spin(2 n + 1) . Then the map F / : Γ ∧ C n ֒ → L (Spin(2 n + 1)) , F / : ψ (cid:0) , π (1 / ( g ) ψ (cid:1) V C n . We denote by [ XY ] (no comma) the Lie brackets of the Lie algebra g and by [ A, B ] = AB − BA (with comma) the comutator in an associative algebra e.g. D ( G ) or U ( g ) . efines an embedding of the Fermionic Fock space Γ ∧ C n into L (Spin(2 n + 1)) .Let Ψ = F / (1) = (cid:0) , π (1 / ( · )1 (cid:1) V C n , F Ψ def = Range( F / ) , (6) where Range( F / ) denotes the image of F / .The restriction of the right regular representation R of Spin(2 n + 1) to F Ψ defines a representation which coincides with the / -spin representation of Spin(2 n + 1) . Moreover, F Ψ ⊂ C ∞ (Spin(2 n + 1)) and the restriction of dR to F Ψ defines a representation of the Lie algebra so (2 n + 1) which coincideswith the / -spin representation of so (2 n + 1) .Proof. Let Y α ( i ) j ( x ) def = √ d α D αij ( x ) , i, j = 1 , . . . , d α x ∈ G , where α labels an irreducible unitary representation of Spin(2 n + 1) , d α denotesthe dimension of such a representation, and D αij ( g ) the i, j -matrix element of g ∈ Spin(2 n +1) in such representation. By Peter-Weyl theorem (cf. [1, Chapter7 §2, Theorem 1 p.172 and Theorem 2 p.174]), for α , i = 1 , . . . , d α fixed, the setof functions ( Y α ( i ) j ) j =1 ,...,d α spans a subspace of dimension d α which is invariantand irreducible for the right regular representation. Now take α = 1 / , and let f j , j = 1 , . . . , d / = 2 n be an orthonormal basis of Γ ∧ C n with f = 1 . Then Y / i ) j = 2 − n/ ( f i , π (1 / ( g ) f j ) Γ ∧ C n . If we pick i = 1 then by the Peter-Weyltheorem, as described above, the set ( Y / j ) j =1 ,...,d / spans a subspace H / which is isomorphic to Γ ∧ C n . And the isomorphism is indeed the F / in thestatement of the theorem. It is also clear that the right regular representationon L (Spin(2 n + 1)) restricts on H / to a representation isomorphic to the / -spin representation of Spin(2 n + 1) .To prove the last part of the statement first note that any Y α ( i ) j , as definedabove, is smooth, that is Y α ( i ) j ∈ C ∞ (Spin(2 n + 1)) (for a sketch of the argumentcf. e.g. [10, Part I, Chapter 2, Appendix to section 2.]). Hence dL is well definedon H / which is by definition the image of F / . By definition of dL it is alsoclear that dL , restricted to H / , realizes a representation of the Lie algebra so (2 n + 1) isomorphic to the / -spin representation. The proof is thereforecomplete. Corollary 3.2.1. The representation dR of so (2 n + 1) extends to a representa-tion dR C of U ( so (2 n +1 , C )) on C ∞ (Spin(2 n +1)) , where, as before, so (2 n +1 , C ) denotes the complexification of the Lie algebra so (2 n + 1) .Proof. The representation dR of U ( so (2 n + 1)) associates to every element X ∈ U ( so (2 n + 1)) a differential operator acting on the complex space C ∞ (Spin(2 n +1)) . Hence the complex-linear extension of dR is well defined on C ∞ (Spin(2 n +1)) and gives a representation dR C of U ( so (2 n +1 , C )) isomorphic to the / -spinrepresentation. For quantum mechanical applications it is not enough to consider an algebraof differential operators on C ∞ (Spin(2 n + 1)) . For example, to discuss the9ime evolution of the system, it is also necessary to consider the operators asunbounded operators in the Hilbert space L ( G ) . In particular the naturalquestion is whether an operator initially defined on C ∞ ( G ) defines a uniqueunbounded operator on L ( G ) . The main objective of this section is to showthat we have a well defined notion of “quasi-Hamiltonian”, which lifts the notionof the Hamiltonian for a system of Fermions, to an unbounded, essentially self-adjoint, positive operator on L (Spin(2 n + 1)) with domain C ∞ (Spin(2 n + 1)) .We begin with some general considerations.Let g be a real semisimple Lie algebra. Let θ : g → g be one of the equivalentCartan involutions on g . In the case where g is the Lie algebra of a compactsemisimple Lie group we take θ to be the identity. Let X ∗ = − θ ( X ) , X ∈ g . (7)We extend this involution to an antilinear involution on U ( g C ) , where g C is thecomplexification of g . This operation makes U ( g C ) into a ∗ -algebra. An element X ∈ U ( g C ) is said to be Hermitian (as an element of the universal envelopingalgebra) when X = X ∗ .Consider the algebra D ( G ) of left-invariant smooth differential operators in L ( G ) with common invariant domain C ∞ ( G ) and let D C ( G ) def = D ( G ) ⊗ R C denote its complexification. On D C ( G ) we have an antilinear involution, whichwe also denote by ∗ , which sends the unbounded operator D ∈ D C G ) to itsHilbert-adjoint D ∗ with respect to the scalar product in L ( G ) . This involutionmakes D ( G ) into a ∗ -algebra. Consider the representation dR of the universalenveloping algebra U ( g C ) . On C ∞ ( G ) we have indeed that dR ( X ) ∗ = − dR ( X ) , X ∈ g . But if we consider dR ( X ) as unbounded operator on L ( G ) withdomain C ∞ ( G ) then the domain of dR ( X ) ∗ will in general be larger than thedomain of dR ( X ) , that is, for X ∈ g , the operator dR (i X ) = i dR ( X ) is ingeneral Hermitian but not selfadjoint . Therefore we cannot say that dR (i X ) ∗ = dR (i X ) holds when we picture dR (i X ) as unbounded operators on L ( G ) withdomain C ∞ ( G ) . One could try to extend the operator dR (i X ) to a selfadjointoperator by enlarging its domain. This might be possible for one operator dR ( X ) for a fixed X ∈ U ( g ) . But, since different X, Y ∈ U ( g ) are elements of an algebra of operators, we need to have a common invariant domain of definitionfor both dR ( X ) and dR ( Y ) . Hence, in general one cannot expect to find anextension of dR which sends Hermitian elements of g C to self-adjoint operatorsin L ( G ) with common domain of selfadjointness. One could argue that thisrequirement is too strong and not necessarily the most natural. Perhaps a morenatural situation, which is obtained in the context of compact semisimple Liegroups, is the following (cf. e.g. [22, Corollary 10.2.10, p.270]). Let G be a compact Lie group with Lie algebra g . Then dR ( X ∗ ) = dR ( X ) ∗ , X ∈ U ( g C ) , (8) In the context of unbounded operators in a Hilbert space, an operator T with domain Dom( T ) is Hermitian when it satisfies Dom( T ) ⊂ Dom( T ∗ ) and T | Dom( T ) = T ∗ Dom( T ) . Theoperator T is selfadjoint when in addition the stronger condition Dom( T ) = Dom( T ∗ ) holds.In the algebraic context of universal enveloping algebras, an element X ∈ U ( g C ) is said to beHermitian when X = X ∗ , where X ∗ is in the sense of (7). These two, in general different,concepts for an object to be Hermitian coincides when we identify the universal envelopingalgebra U ( g C ) with the algebra D ( G ) of smooth right-invariant vector fields acting on C ∞ ( G ) . D C ∈ D ( G ) is automatically essentially selfadjoint .We now turn to the notion of commuting unbounded operators. There aretwo natural notions of commuting unbounded operators, weakly commuting andstrongly commuting. We give the precise definitions.Given two unbounded operators A, B with common domain D in a Hilbertspace H , we say that A, B weakly commute on D when ABv = BAv for all v ∈ D . Given two selfadjoint unbounded operators A, B we say that A, B strongly commute when e i tA e i sB = e i sB e i tA for all s, t ∈ R , where e i tC denotesthe unitary one-parameter group generated by a selfadjoint operator C (cf. [19,Theorem VIII.13] for a justification of this definition).Regarding the relation between strong and weak commutativity of operatorson a Hilbert space we have the following result due to Nelson. Lemma 4.1 ([18, Corollary 9.2]) . Let A, B be two Hermitian unbounded opera-tors on a Hilbert space H and let Q be a dense linear subspace of H such that Q is contained in the domain of A , B , A , AB , BA , and B , and such that A, B weakly commute on Q . If the restriction of A + B to Q is essentially selfad-joint then A and B are essentially selfadjoint and their closures A , B stronglycommute. A direct consequence of this lemma are the following facts, which will beused in this section and the following. Proposition 4.2. Let G be a compact Lie group with Lie algebra g . Let g C be the complexified Lie algebra of g , and U ( g C ) its universal enveloping algebra.Let X, Y ∈ U ( g ) be two commuting operators (in the algebraic sense of elementsin the universal enveloping algebra). Then1. the closed operators dR ( X ) , dR ( Y ) ∈ D ( G ) strongly commute;2. if dR ( X ) is positive (semi-)definite, and dR ( Y ) is Hermitian, then exp( − dR ( X )) exp(i dR ( Y )) = exp(i dR ( Y )) exp( − dR ( X )) , where we recall that dR ( X ) and dR ( Y ) are the unique closed extensionsof dR ( x ) , respectively dR ( Y ) , and dR ( X ) > .Proof. The statement of point follows from (8) and Nelson’s Lemma 4.1. In-deed, if X, Y commute in the universal enveloping algebra then dR ( X ) and dR ( Y ) weakly commute on C ∞ ( G ) because dR is a representation of U ( g ) with domain C ∞ ( G ) . Now, for G a compact group, equation (8) tells usthat any Hermitian element in the algebra D ( G ) is essentially self adjointon C ∞ ( G ) ⊂ L ( G ) . Therefore in particular, for any X, Y ∈ U ( g C ) , wehave that the operators dR ( X ) , dR ( X ) = dR ( X ) , dR ( X ) dR ( Y ) = dR ( XY ) , dR ( X ) + dR ( Y ) = dR ( X + Y ) have the same domain C ∞ ( G ) , and are essen-tially selfadjoint there. Hence the hypothesis of the Lemma 4.1 are satisfiedwith A = dR ( X ) and B = dR ( Y ) and statement follows. The statementof point is a straightforward application of spectral calculus (cf [19, SectionVIII.5]). That is, it admits a unique extension to a selfadjoint operator. emark 4.3. Because of the above proposition we only need to check whethertwo operators commute as elements of the universal enveloping algebra. Fromthe Proposition 4.2 it then follows automatically that their closures are selfad-joint and strongly commuting .With this proposition we have completed the considerations from the generaltheory. We can now turn to the application that we have in mind. Definition 4.4 (Quasi-Fermionic vector fields) . Let X ij , i, j = 1 , . . . , n + 1 ,be the standard basis (cf. (3)) of the Lie algebra so (2 n + 1) of the Lie group Spin(2 n + 1) . Let us denote by D ij def = dR ( X ij ) the corresponding left-invariantvector fields on Spin(2 n + 1) . We define the following operators (cf. (4)) D + k def = D k − , n +1 + i D k, n +1 ,D − k def = − D k − , n +1 + i D k, n +1 , k = 1 , . . . , n , as linear operators on C ∞ (Spin(2 n + 1) , C ) ⊂ L (Spin(2 n + 1)) . Definition 4.5 (Quasi-Hamiltonian operator) . Let us fix n strictly positivenumbers E , . . . , E n , with < E ≤ · · · ≤ E n . Using for D ± k the notation ofthe previous paragraph we call a quasi - Hamiltonian the operator ˜ H = n X k =1 E k D + k D − k , acting on C ∞ (Spin(2 n + 1)) . Remark 4.6. The operators D ± k restricted to the finite dimensional subspace F Ψ ⊂ C ∞ (Spin(2 n + 1)) , given in (6) of 3.2, satisfy the canonical anticom-mutation relations. For this reason we call these operators “quasi-Fermionic”.Similarly we named the operator ˜ H “quasi-Hamiltonian” because, restricted tothe subspace F Ψ ∼ = V C n , it coincides with the free Fermionic Hamiltonianoperator H def = P k E k c † k c k , where the creation-annihilation operators c † k , c k , k = 1 , . . . , n , were defined in section 2. Theorem 4.7. The unbounded operator ˜ H with domain C ∞ (Spin(2 n + 1)) in L (Spin(2 n + 1)) is a positive, essentially selfadjoint operator. Moreover thequasi-Hamiltonian can be decomposed on C ∞ (Spin(2 n + 1)) as ˜ H = P + i B , P = − n X k =1 E k L k , B = n X k =1 E k T k , with T k def = D k − , k , L k def = D k − , n +1 + D k, n +1 , k = 1 , . . . , n , and thefollowing properties are satisfied:1. The operator P and i B , with domains Dom( P ) , Dom(i B ) both equal to C ∞ (Spin(2 n + 1)) , are essentially selfadjoint in L (Spin(2 n + 1)) . More-over P is positive definite. In particular − P and B are closable andtheir closures − P , B are selfadjoint operators which generate respectivelya semigroup and a unitary group (we consider − P because generators ofsemigroups are usually taken to be negative definite). . The operators P and i B strongly commute. The operators T k , L k , k =1 , . . . , n , are essentially selfadjoint. The unique selfadjoint closures T k , k = 1 , . . . , n define a family of strongly commuting unbounded operators.Moreover each T k , k = 1 , . . . , n strongly commutes with each L ℓ , ℓ =1 , . . . , n . In particular i B strongly commutes with P .Proof. First note that ˜ H is well defined on C ∞ (Spin(2 n +1)) , since D ± are linearcombinations of smooth vector fields, in particular D − maps C ∞ (Spin(2 n + 1)) into C ∞ (Spin(2 n + 1)) (indeed D (Spin(2 n + 1) is an algebra). Using the abovedefinition of the operators D + k , D − k in terms of the operators D ij in Definition4.4 we have, on C ∞ (Spin(2 n + 1)) , ˜ H = − n X k =1 E k ( D k − , n +1 + i D k, n +1 )( D k − , n +1 − i D k, n +1 )= − n X k =1 E k ( D k − , n +1 ) − n X k =1 E k ( D k, n +1 ) − i n X k =1 E k [ D k − , n +1 , D k, n +1 ]= − n X k =1 E k (cid:0) ( D k − , n +1 ) + ( D k, n +1 ) (cid:1) − i n X k =1 E k D k − , k . Therefore we obtain ˜ H = P + i B (9)where, P and i B are defined in the statement of the theorem. Note that by(8) all Hermitian operators we are handling are essentially selfadjoint. More-over the operators D + k D − k , k = 1 , . . . , n , are positive definite since D + k is theformal adjoint of D − k . This implies that the quasi-Hamiltonian ˜ H is essentiallyselfadjoint (by (8)) and is positive definite (since is the sum of positive definiteoperators). This concludes the proof of the first part of the theorem.Property is proved by a similar argument. The fact that P is closableand its closure defines a semigroup follows from the fact that P is essentiallyselfadjoint (therefore closable) and positive definite (hence defines a semigroup).Similarly B is closable because i B is essentially selfadjoint and therefore de-fines a unitary one-parameter group.We now turn to the proof of property . By (8) we obtain that i T k =i D k − , k and L k are self-adjoint. Note that, by remark 4.3 if two elements X, Y in the universal enveloping algebra U ( so (2 n + 1)) commute, then theirrepresentation dR ( X ) , dR ( X ) admit closures dR ( X ) , dR ( Y ) which stronglycommute. Hence to prove the commutation properties of point , it is enoughto perform the computation on the universal enveloping algebra.Now, from the fact that the elements X k − , k , k = 1 , . . . , n generate a max-imal commutative subalgebra (Cartan subalgebra) of the Lie algebra so (2 n +1) we obtain that the operators T k = D k − , k = dR ( X k − , k ) , k = 1 , . . . , n forma commuting family of operators.As above, by remark 4.3, to show that L ℓ commutes with X k − , k , for all ℓ, k ∈ { , . . . , n } , it is enough to prove that the corresponding elements of theuniversal enveloping algebra commute. Consider L U ℓ def = ( X ℓ − , n +1 ) + ( X ℓ, n +1 ) , ℓ = 1 , . . . , n, Cf. remark 2.2 13e the element, associated to L k , in the universal enveloping algebra of so (2 n +1) . It is enough to prove that [ L U ℓ , X k − , k ] = 0 , for all ℓ, k = 1 , . . . , n. (10)This follows from the following straightforward computations. Using the identity [ X , Y ] = X [ X, Y ] + [ X, Y ] X for any X, Y ∈ U ( so (2 n + 1) C ) we get [ L U ℓ , X k − , k ] == X ℓ − , n +1 [ X ℓ − , n +1 , X k − , k ] + [ X ℓ − , n +1 , X k − , k ] X ℓ − , n +1 ++ X ℓ, n +1 [ X ℓ, n +1 , X k − , k ] + [ X ℓ, n +1 , X k − , k ] X ℓ, n +1 . (11)Using in this expression the commutation relations (4) of the standard basis of so (2 n + 1) , we obtain for ℓ, k = 1 , . . . , n [ L U ℓ , X k − , k ] == − X ℓ − , n − δ ℓ − , k − X n +1 , k − δ ℓ − , k − X n +1 , k X ℓ − , n +1 ++ X ℓ, n +1 δ ℓ, k X n +1 , k − + δ ℓ, k X n +1 , k − X ℓ, n +1 . Now, using in this expression the fact that X ij = − X ji for all ≤ i < j ≤ n + 1 , and collecting the Kronecker deltas into a unique Kronecker delta whichmultiplies everything, we get [ L U ℓ , X k − , k ] = δ k,ℓ ( X ℓ − , n +1 X k, n +1 + X k, n +1 X ℓ − , n +1 − X ℓ, n +1 X k − , n +1 − X k − , n +1 X ℓ, n +1 ) . Finally, using the identity δ ij f ( i, j ) = δ ij f ( i, i ) where f ( i, j ) is any function of i, j ∈ N , we get [ L U ℓ , X k − , k ] = δ k,ℓ ( X k − , n +1 X k, n +1 + X k, n +1 X k − , n +1 − X k, n +1 X k − , n +1 − X k − , n +1 X k, n +1 )= 0 . This proves (10). As a consequence it is now clear that P commutes with B which concludes the proof of property and of the theorem. From theorem 4.7 we have that the quasi-Hamiltonian is ˜ H = P + i B , with P = P nk =1 E k L k and B = P nk =1 E k T k , where all the operators aredefined on C ∞ (Spin(2 n + 1)) .Since the operator B appears in ˜ H multiplied by the imaginary unit i we cannot associate directly to the closure ˜ H a (real) stochastic process on Spin(2 n + 1) . For this reason we consider, together with P , B , and ˜ H above,the following operator P def = P + B , Dom( P ) def = C ∞ (Spin(2 n + 1)) . (12)14t turns out that it is possible to associate a stochastic diffusion processeson Spin(2 n + 1) to both closures P and P in L (Spin(2 n + 1) . First we see thatboth − P and − P generate probability semigroups in the following sense. Lemma 5.1. The operators − P , respectively − P are essentially selfadjoint on C ∞ (Spin(2 n + 1)) and their closures − P , − P are infinitesimal generators ofstrongly continuous semigroups which act on L (Spin(2 n + 1)) as convolutionsemigroups of probability measures with support on Spin(2 n + 1) .Proof. The statement follows from [17, Theorem 3.1].Now we characterize the stochastic processes generated by − P and − P in terms of the SDEs these processes satisfy. Before doing so let us brieflyintroduce the notions of stochastic differential equation (SDE) on a manifoldand of generator of a diffusion process (cf. e.g. [16]).Let M be a connected smooth manifold of dimension d . Moreover for con-venience let us assume M to be compact. This assumption simplifies somewhatthe discussion and is sufficient for our purposes because we will in the sequelonly deal with manifolds associated to compact Lie groups. In particular if M is a compact manifold, then every C ∞ -vector field on it is complete, that is,the flow associated to the given vector field can be extended to all times. Thisallows us to define a stochastic process globally on the manifold M (withoutthe need of the introduction of an explosion time).Let us denote by X ( M ) the set of C ∞ -vector fields on M . Let us consider r ∈ N vector fields A , A , . . . , A r ∈ X ( M ) on M .Let (Ω , ( F t ) ≤ t< ∞ , P ) be a filtered probability space; we denote by ( W ( t )) =( W ( t ) , . . . , W r ( t )) an r -dimensional F t -adapted Brownian motion starting atzero, W (0) = 0 . Finally, let ξ be an F -measurable M -valued random variable.Consider now an F t -adapted stochastic process X = X ( t ) on M , that is an F t -adapted random variable X = ( X ( t )) with values in the continuous functions C ([0 , ∞ ); M ) . Contrary to the previous sections, in this section the letter X will be reserved to denote a random variable.Suppose that for every f ∈ C ∞ ( M ) the stochastic process X = ( X ( t )) satisfies P -almost surely the following integral equation f ( X ( t )) − f ( ξ ) = Z t r X k =1 ( A k f )( X ( s )) ◦ dW k ( s ) + Z t ( A f )( X ( s )) d s, (13)for all t ∈ [0 , ∞ ) , where ◦ dB denotes integration in the Stratonovich sense (see,e.g. [16]). Then we will say that the M -valued stochastic process X = ( X ( t )) is a solution to (13).Let us spend few words on the notion of strong solution regardless whetherwe are on a manifold M or just in R d . Given a notion of solution it is natural toask whether it satisfies some given initial condition which we take here to be apoint x ∈ M (withoug any randomness). A solution to (13) with (non random)initial conditions ξ = x , is then a stochastic process X x starting at time at x .More preciselly, we are asking for a function F : M × C ([0 , ∞ ); R r ) → C ([0 , ∞ ); M ) which maps the initial condition x ∈ M and the given realization By saying that the equality holds P -almost surely, for all t , we are saying that the righthand side and the left hand side define indistinguishable processes. 15f the Brownian motion W = ( W ( t )) into a realization of a process X x = ( X x ( t )) on the manifold M . Moreover F is such that X x = F ( x, W ) is a solution to(13) with initial condition ξ = x with probability one and with given Brownianmotion W = ( W ( t )) . Since at some point we would like to integrate X x bothwith respect to x ∈ M and with respect to P it is natural to ask that F bejointly measurable in x and W = ( W ( t )) . It turns out that this is not alwayspossible. When it is, X x = F ( x, W ) is called a strong solution to (13) withinitial condition ξ = x ∈ M with probability one (cf. the discussion in [20,Section V.10] and [16, Chapter IV, section 1, esp. pp.162-163]).In the context of smooth manifolds the situation is particularly good becausewe are considering SDE with smooth coefficients. Indeed one has a result (cf.[16, Chapter V, Section 1., Theorem 1.1, p.249]) which states that given aninitial condition x ∈ M and an r -dimensional Brownian motion W = ( W ( t )) ,then a strong solution to (13) always exists and is unique .Once this important detail about how the initial condition is understoodwe can give meaning to the following shorthand, which we shall refer to as a Stratonovich SDE on the (compact) manifold M : ( dX ( t ) = P rk =1 A k ( X ( t )) ◦ dW k ( t ) + A ( X ( t )) dt ,X (0) = x , x ∈ M . (14)The meaning associated to (14) is that we consider a strong solution X of (13)(with initial conditions ξ = x with probability one) and then define a solutionto (14) to be the random variable X x = F ( x, W ) , where F is the map whichdefines our strong solution X .We now define the notion of stochastic diffusion process and of its generator.First consider a more general case. For x ∈ M , let X x be a continuousstochastic process adapted to a filtration F t in the probability space (Ω , F , P ) .For simplicity we consider a stochastic process defined for all t ∈ [0 , ∞ ) andwith values in the space of continuous maps [0 , ∞ ) → M (where M is alwaysassumed to be compact) such that X (0) = x (where equality means P -a.s.).Let P x be the probability law associated to the random variable ( X x ( t )) .This means that P x is the image measure under the measurable mapping X x =( X x ( t )) of the probability measure P . Assume that x P x is universallymeasurable and that P x is uniquely determined by x ∈ M . Moreover assumethat there exists a linear operator L with domain Dom( L ) in C ( M ) , such thatfor every f ∈ Dom( L ) , X f ( t ) def = f ( X ( t )) − f ( X (0)) − Z t ( L f )( X ( s )) d s is a martingale with continuous sample paths and adapted to the filtration F t associated to X x ( t ) (cf. [16, Chapter IV, Theorem 5.2, p.207]). Then the family The idea behind this result is that the manifold M is locally diffeomorphic to R d where d is the dimension of the manifold M . This means that locally the SDE (14) (and hence (13))can be written in coordinates as a standard SDE on R d . One can apply standard results aboutexistence and uniqueness of solutions of SDEs to these local realizations. Finally one needsto patch together different local solutions into a global solution. Details can be found in theabove mentioned [16]. Cf., e.g. [16, p.1]. These conditions are actually automatically satisfied when X x is the strong solution to(14). 16f probability measures ( P x ) x ∈M is called a diffusion generated by the operator L . When, for every x ∈ M , X x is the stochastic diffusion process on the mani-fold M which is the strong solution to (14) with initial condition X (0) = x , thenwe have the following representation [16, Chapter V, Theorem 1.2, p.253]. Thefamily of probability laws ( P x ) x ∈M , associated with the strong solutions X x to(14) with initial conditions x ∈ M , is a diffusion generated by the operator L def = 12 r X j =1 A k ( A k f ) + A f, f ∈ C ∞ ( M ) , (where, as before, the manifold M is assumed to be compact) and the vec-tor fields A , A , . . . , A r ∈ X ( M ) are interpreted as differential operators withcommon domain C ∞ ( M ) .We now go back to our setting where the manifold M = Spin(2 n + 1) andcollect the specialized version of the results recalled above. Doing so we givethe characterization of the generators − P and − P (given by (12)) in terms ofstochastic processes on Spin(2 n + 1) . Remark 5.2 (Notation) . In this section we do not distinguish between anelement X ij in the Lie algebra and the corresponding differential operator D ij = dR ( X ij ) (cf. 4.4). In particular, depending on the context, we identify A k , k =1 , . . . , n , with either D n +1 ,k or X n +1 ,k . Similarly, the differential operator B in theorem 4.7 will be considered also as a vector field without changingnotation. Lemma 5.3 (Stochastic processes associated to P and P ) . The following state-ments hold.1. The Stratonovich SDEs on Spin(2 n + 1)( P ) ( dY ( t ) = P nk =1 p E ′ k A k ( Y ( t )) ◦ dW k ( t ) − B ( Y ( t )) dt ,Y (0) = x , x ∈ Spin(2 n + 1)( P ) ( dX ( t ) = P nk =1 p E ′ k A k ( X ( t )) ◦ dW k ( t ) X (0) = x , x ∈ Spin(2 n + 1) , with E ′ k +1 def = E ′ k def = E k , k = 1 , . . . , n , and ( W k ( t ) , k = 1 , . . . , n ) , astandard Brownian motion in R n , are well defined and admit a uniquestrong solution.2. The operators − P and − P (acting on L (Spin(2 n + 1)) are the genera-tors of the diffusion processes given by the strong solutions of ( P ) , ( P ) respectively.Proof. For the first statement see [16, Chapter 5, Theorem 1.1 p.249]. Thesecond statement follows from [16, Theorem 1.2, p.253].The following result relates the time evolution semigroup generated by thequasi-Hamiltonian ˜ H in L (Spin(2 n + 1)) with a stochastic diffusion process on Spin(2 n + 1) generated by the second order part in − ˜ H .17 heorem 5.4. We have the following representations of the semigroup gener-ated by the closure − ˜ H of − ˜ H ( f, e − t ˜ H g ) L (Spin(2 n +1)) = E X h f (0) (cid:16) e − i t B g (cid:17) (cid:0) X ( t ) (cid:1)i , t ≥ , (15) where E X denotes the expectation with respect to the process generated by − P , B , respectively ˜ H , denotes the closure (which exists by theorem 4.7) of theoperator B , respectively ˜ H ; f (0) denotes complex conjugation, and f, g ∈ C (Spin(2 n + 1)) ⊂ L (Spin(2 n + 1)) .Proof. First note that e − t ˜ H is a bounded operator for all t ∈ R + . Hence f, g can be taken in L (Spin(2 n + 1)) . The equality follows directly from the repre-sentation of the Hamiltonian as ˜ H = P + i B , the fact that P and B stronglycommute, and the Markov property of the semigroup generated by − P (whichis a consequence of point of lemma 5.3): ( f, e − t ˜ H g ) L (Spin(2 n +1)) = ( f, e − t ( P +i B ) g ) L (Spin(2 n +1)) = ( f, e − t P e − i t B g ) L (Spin(2 n +1)) = E X h f (0) (cid:16) e − i t B g (cid:17) (cid:0) X ( t ) (cid:1)i . Acknowledgments I wish to thank Prof. Disertori for her support during this project which con-stitutes part of the research carried out during my Ph.D. I would also liketo thank Prof. Albeverio for inspiring conversations concerning topics relatedto this project. Part of this research was founded by DFG via the grant AL214/50-1 “Invariant measures for SPDEs and Asymptotics”. References [1] A. Barut and R. Raczka. Theory of Group Representations and Applica-tions . World Scientific Publishing Co Inc, November 1986.[2] A.O. Barut, M. Božić, and Z. Marić. 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