C*-Non-Linear Second Quantization

C ∗–non–linear second quantization arXiv:1401.5500v2 [math.OA] 12 Sep 2014 Luigi Accardi, Ameur Dhahri Volterra Center, University of Roma Tor Vergata Via Columbia 2, 00133 Roma, Italy e-mail:[email protected] [email protected] Abstract We construct an inductive system of C ∗ –algebras each of which is isomorphic to a finite tensor product of copies of the one mode n–th degree polynomial extension of the usual Weyl algebra constructed in our previous paper [10]. We prove that the inductive limit C ∗ -algebra is factorizable and has a natural localization given by a family of C ∗ -sub-algebras each of which is localized on a bounded Borel subset of R. Finally we prove that the corresponding families of Fock states, defined on the inductive family of C ∗ -algebras, is projective if and only if n = 1. This is a weak form of the no–go theorems which emerge in the study of representations of current algebras over Lie algebras. Contents 1 Introduction: the C ∗ –non–linear quantization program 2 2 The 1–mode n–th degree Heisenberg ∗–Lie algebra heisR (1, n) 7 3 The Schroedinger representation and the polynomial Heisenberg group Heis(1, n) 8 4 The free group–C ∗ –algebra of Heis(1, n) 9 5 The current algebra of heisR (1, n) over R 11 1 6 Isomorphisms between the current algebra heisR (1, n, RχI ) and heisR (1, n) 12 7 The group Heis(1, n, RχI ) and its C ∗ –algebra 15 ∗ 7.1 C –algebras isomorphism . . . . . . . . . . . . . . . . . . . . 16 8 The inductive limit 19 8.1 Factorizable families of C ∗ –algebras . . . . . . . . . . . . . . 21 9 Existence of factorizable states on W1,n;R 24 1 Introduction: the C ∗–non–linear quantiza- tion program The present paper is a contribution to the program of constructing a theory of renormalized higher powers of quantum white noise (RPWN) or equiva- lently of non relativistic free Boson fields. This program has an old history, but the approach discussed here started in 1999 with the construction of the Fock representation for the renormal- ized square of Boson white noise [2]. This result motivated a large number of papers extending it in different directions and exhibiting connections with almost all fields of mathematics, see for example [18] for the case of free white noise, [1] for the connection with infinite divisibility and for the identifica- tion of the vacuum distributions of the generalized fields with the three non standard Meixner classes, [3] and [17] for finite temperature representations, [9] for the construction of the Fock functor, the survey [5] and the paper [6] for the connections with conformal field theory and with the Virasoro– Zamolodchikov hierarchy, [7] for the connections between renormalization and central extensions. The problem is the following. One starts with the Schroedinger represen- tation of the Heisenberg real ∗–Lie algebra with skew–adjoint generators iq (imaginary unit times position), −ip (−i times momentum), E := i1 (i times central element) and relations [iq, −ip] = i1. The universal enveloping algebra of this Lie algebra, called for brevity the full oscillator algebra (FOA), can be identified with the algebra of differential operators in one real variable with complex polynomial coefficients. The continuous analogue of the Heisenberg Lie algebra is the non relativistic 2 free boson field algebra, also called the current algebra over R of the Heisen- berg algebra, whose only non zero commutation relations are, in the sense of operator valued distributions: [qs , pt ] = δ(s − t)1 ; s, t ∈ R The notion of current algebra has been generalized from the Heisenberg alge- bra to more general ∗–Lie algebras (see Araki’s paper [12] for a mathematical treatment and additional references): in this case the self–adjoint generators of the Cartan sub–algebras are called generalized fields. Notice that the definition of current algebra of a given Lie algebra is inde- pendent of any representation of this algebra, i.e. it does not require to fix a priori a class of states on this algebra. One can speak of ∗–Lie algebra second quantization to denote the transition from the construction of unitary representations of a ∗–Lie algebra to the construction of unitary representations of its current algebra over a measur- able space (typically R with its Borel structure). Contrarily to the discrete case, the universal enveloping algebra of the cur- rent algebra over R of the Heisenberg algebra is ill defined because of the emergence of higher powers of the δ–function. This is the mathematical counterpart of the old problem of defining powers of local quantum fields. Any rule that, giving a meaning to these powers, defines a ∗–Lie algebra structure, is called a renormalization procedure. The survey [5] describes two inequivalent renormalization procedures and the more recent paper [7] shows the connection between them. The second step of the program, after renormalization, is the construction of unitary representations of the resulting ∗–Lie algebra. This step, which is the most difficult one because of the no–go theorems (see discussion below), is usually done by fixing a state and considering the associated cyclic repre- sentation. At the moment, even in the first order case, i.e. for usual fields, the explicitly constructed representations are not many, they are essentially reduced to gaussian (quasi–free) representations. Moreover any gaussian rep- resentation can be obtained, by means of a standard construction, from the Fock representation which is characterized by the property that the cyclic vector, called vacuum, is in the kernel of the annihilation operators. This property has been taken as an heuristic principle to define the notion of Fock state also in the higher order situations (see [5] for a precise definition). It can be proved that, for all renormalization procedures considered up to now, the Fock representation and the Fock state are factorizable, in the sense 3 of Araki–Woods [11]. This property poses an obstruction to the existence of such representation, namely that the restriction of the Fock state on any factorizable Cartan sub–algebra must give rise to a classical infinitely divis- ible process. If this is not the case then no Fock representation, and more generally no cyclic representation associated to a factorizable state, can exist. When this is the case we say that a no–go theorem holds. Nowadays several instances of no–go theorems are available. The simplest, and probably most illuminating one, concerns the Schroedinger algebra, which is the Lie algebra generated by the powers ≤ 2 of p and q (see [18], also [1] and [8] for stronger results). This result implies that there is no natural analogue of the Fock representation for the current algebra over R (for any d ∈ N) of the FOA. On the other side we know (see [1], [2] and the above discussion) that for some sub–algebras of current algebras of the FOA such a representation exist. This naturally rises the problem to characterize these sub–algebras. Since a full characterization at the moment is not available, a natural interme- diate step towards such a characterization is to produce nontrivial examples. To this goal a family of natural candidates is provided by the ∗–Lie– sub–algebras of the FOA consisting of the real linear combinations of the derivation operator and the polynomials of degree less or equal than a fixed natural integer n. Thus the generic element of such an algebra has the form up + P (q) ; u∈R where P is a polynomial of degree n with real coefficients. For n = 1 one finds the Heisenberg algebra; for n = 2 the Galilei algebra and, for n > 2, some nilpotent Lie algebras well studied in mathematics [13], [14] [15], [16] but up to now, with the notable exception of the Galilei algebra (n = 2), not considered in physics. These ∗–Lie–algebras enjoy two very special properties: (i) no renormalization is required in the definition of the associated current algebra over R; (ii) in the Schroedinger representation of the FOA the skew–adjoint ele- ments of these sub–algebras can be explicitly exponentiated giving rise to a nonlinear generalization of the Weyl relations and of the corre- sponding Heisenberg group. This was done in the paper [10]. Property (i) supports the hope of the existence of the Fock representation for the above mentioned current algebra. A direct proof of this fact could be 4 obtained by proving the infinite divisibility of all the vacuum characteristic functions of the generalized fields. Unfortunately even in the case n = 2, in which this function can be explicitly calculated, a direct proof of infinite divisibility can be obtained only for a subset of the parameters which define the generalized fields, but not for all, and this problem is challenging the experts of infinite divisibility since several years. In the present paper we exploit property (ii) and the following heuristic considerations are aimed at making a bridge between the mathematical con- struction below and its potential physical interpretation. Our goal is to consruct a C ∗ –algebra whose generators can be naturally iden- tified with the following formal expressions that we call the non–linear Weyl operators: P ei(p(f0 )+q (fn )+...+q (f1 )) = ei j∈{0,1,...,n+1} Lj (fj ) n 1 (1) The formal generators of the non–linear Weyl operators (called non–linear fields) are heuristically expressed as powers of the standard quantum white noise (or free Boson field), i.e. the pair of operator valued distributions qt , pt with commutation relations [qs , pt ] = iδ(t − s) in the following way: Z Z Z Ln+1 (f0 ) := p(f0 ) = f0 (t)pt dt ; L0 (f ) := fk (t)qt0 dt := 1· f (s)ds R R Z k Lk (f ) := q (fk ) = fk (t)qtk dt ; k ∈ {0, 1, . . . , n} (2) R When n = 1, the expressions (2) are well understood and various forms of second quantization are known. For example one can prove the unitarity of the Fock representation the exponentiability, inside it, of the generators (2) and the commutation relations satisfied by them. A different example, for n = 1, is provided by Weyl second quantization: in it, by heuristic calcula- tions, one guesses the commutation relations that should be satisfied by any representation of the exponentials (1) and then one proves the existence of a C ∗ –algebra which realizes these commutation relations. In the present paper we apply this approach to give a meaning to the expo- nentials (1) and in this sense we speak of C ∗ –second quantization. To this goal we exploit the fact that, if π is a finite Borel partition of a 5 bounded Borel subset of R, then there is a natural way to give a meaning to the generalized Weyl algebra with test functions constant on the sets of π. This is based on the identification of this algebra with the tensor product of |π| (cardinality of π) rescaled copies of the one mode generalized Weyl algebra (see section 8). This identification strongly depends on the specific structure of the Lie algebra considered (see section 6 below). Using this we construct an inductive system of C ∗ –algebras each of which is isomorphic to a finite tensor product of copies of the one mode generalized Weyl algebra but the embeddings defining the inductive system are not the usual tensor product embeddings. The C ∗ –algebra, obtained as inductive limit from the above construction, is naturally intepreted as a C ∗ –quantization, over R, of the initial ∗–Lie alge- bra. This C ∗ –algebra has a localization given by a family of C ∗ –sub–algebras, each of which has a natural localization on bounded Borel subset of R. Moreover this system of local algebras is factorizable in the sense of Definition 8 below. With this construction the problem of constructing unitary representations of the current algebra over R of the initial ∗–Lie algebra is reduced to the problem of finding representations of this C ∗ –algebra: the advantages of this transition from unbounded to bounded case are well known in the case of standard, first order, quantization. In the last section of the paper it is shown that, although the Fock state is defined on each of the C ∗ –algebras of the inductive family, the corresponding family of states is projective if and only if n = 1 (i.e. for the usual Weyl al- gebra). This result can be considered as a C ∗ –version of the no–go theorems proved in [18], [1], [8] for different algebras. The basic construction of the present paper can be extended to more general classes of ∗–Lie algebras (for example the C ∗ –algebras associated to the renormalized square of white noise (RSWN)) and more general spaces (i.e. Rd instead of R). 6 2 The 1–mode n–th degree Heisenberg ∗–Lie algebra heisR (1, n) Definition 1 For n ∈ N∗ the 1–mode n–th degree Heisenberg algebra, denoted heisR (1, n), is the pair {Vn+2 , (Lj )n+1 j=0 } where: - Vn+2 is a (n + 2)–dimensional real ∗–Lie algebra; - (Lj )n+1 j=0 is a skew–adjoint linear basis of Vn+2 ; - the Lie brackets among the generators are given by [Li , Lj ] = 0 ; ∀i, j ∈ {0, 1, · · · , n} [Ln+1 , Lk ] = kLk−1 ; ∀k ∈ {1, · · · , n} , L−1 := 0 Remark 1 1) Multiplying each of the generators (Lj )n+1 j=0 by a strictly ′ n+1 positive number, one obtains a new basis (Lj )j=0 of Vn+2 satisfying the new commutation relations [L′n+1 , L′k ] = kdk L′k−1 ; ∀k ∈ {0, · · · , n} , L′−1 := 0 In this case we speak of a re–scaled copy of the 1–mode n–th degree Heisenberg algebra. 2) Denoting Rn [X] the the vector space of polynomials in one indetermi- nate with real coefficients and degree less or equal than n, the assign- ment of the basis (Lj )n+1 j=0 uniquely defines the parametrization (u, (ak )k∈{0,1,...,n} ) ∈ R × Rn [X] ≡ Rn+2 7→ ℓ0 (u, P ) := X := uLn+1 + ak Lk =: uLn+1 + P (L) ∈ heisR (1, n) (3) k∈{0,1,...,n} of heisR (1, n) by elements of Rn [X]. When no confusion is possible we will use the identification ℓ0 (u, P ) ≡ (u, (ak )k∈{0,1,...,n} ) ∈ Rn+2 (4) 7 3 The Schroedinger representation and the polynomial Heisenberg group Heis(1, n) Let p, q, 1 be the usual momentum, position and identity operators acting on the one mode boson Fock space H1 = Γ(C) = L2 (R) (5) The maximal algebraic domain Dmax (see [4]), consisting of the linear com- binations of vectors of the form q n pk ψz ; k, n ∈ N , z ∈ C where ψz is the exponential vector associated to z ∈ C, is a dense subspace of Γ(C) invariant under the action of p and q hence of all the polynomials in the two non commuting variables p and q. In particular, for each n ∈ N, the real linear span of the set {i1, ip, iq, . . . , iq n }, denoted heisR (F, 1, n), leaves invariant the maximal algebraic domain Dmax . Hence the commutators of elements of this space are well defined on this domain and one easily verifies that they define a structure of ∗–Lie algebra on heisR (F, 1, n). Lemma 1 In the above notations the map Ln+1 7→ ip , L0 7→ i1 , Lk 7→ iq k ; k ∈ {1, . . . , n} (6) admits a unique linear extension from heisR (1, n) onto heisR (F, 1, n) which is a ∗–Lie algebra isomorphism called the Schroedinger representation of the n–th degree Heisenberg algebra heisR (1, n). Proof. The linear space isomorphism property follows from the linear inde- pendence of the set {1, p, q, . . . , q n }. The ∗–Lie algebra isomorphism property follows from direct computation.  In [10] (Theorem 1) it is proved that the unitary operators W (u, P ) := ei(up+P (q)) ∈ Un(L2 (R)) ; (u, P ) ∈ R × Rn [X] (7) satisfy the following polynomial extension of the Weyl relations: W (u, P )W (v, Q) = W ((u, P ) ◦ (v, Q)) ; ∀(u, P ), (v, Q) ∈ R × Rn [X] (8) 8 where −1 (u, P ) ◦ (v, Q) := (u + v, Tu+v (Tu P + Tv Su Q)) (9) and for any u, w ∈ R, the linear operators Tw , Su : Rn [X] → Rn [X] are defined by the following prescriptions: Tw 1 = 1 k−1 k X k! Tw (X ) = w k−h X h + X k ; ∀k ∈ {1, . . . , n} (k + 1 − h)!h! h=0 (Su P )(X) := P (X + u) translation operator on Rn [X] Denote WF,1,n := norm closure in B(Γ(C)) of the linear span of the operators (7). Identity (8) implies that WF,1,n is a C ∗ –algebra. In [10] it is proved that the composition law (9) is a Lie group law on R × Rn [X] whose Lie algebra is heisR (1, n). Since the elements of heisR (1, n) are parametrized by the pairs (u, P ) ∈ R × Rn [X] it is natural to introduce the following notation. Definition 2 (see [10]) The 1–mode n–th degree Heisenberg group is the set Heis(1, n) := eℓ0 (u,P ) : (u, P ) ∈ R × Rn [X]  (10) with composition law −1 eℓ0 (u,P ) ◦ eℓ0 (v,Q) := eℓ0 ((u+v,Tu+v (Tu P +Tv Su Q))) The name Heis(1, n) is motivated by the fact that, for n = 1, Heis(1, n) reduces to the usual the 1–mode Heisenberg group. 4 The free group–C ∗–algebra of Heis(1, n) Definition 3 Let G be a group. The free complex vector space generated by the set {Wg : g ∈ G} 9 has a unique structure of unital ∗–algebra defined by the prescription that the map g 7→ Wg defines a unitary representation of G, equivalently: Wg Wh := Wgh ; g, h ∈ G (Wg )∗ := Wg−1 ; g∈G (11) 1 := We The completion of W 0 (G) under the (minimal) C ∗ –norm kxk := sup{kπ(x)k : π ∈ {∗–representations of G} } ; x ∈ W 0 (G) will be called the free group–C ∗ –algebra of G and denoted W(G). Remark 2 Because of (11) a ∗–representation of W(G) maps the generators Wg (g ∈ G), into unitary operators. Remark 3 If G, G′ are groups, then any group homorphism (resp. isomor- phism) α : G → G′ , extends uniquely to a C ∗ –algebra homorphism (resp. isomorphism) α̃ : W(G) → W(G′ ) characterized by the condition α̃(Wg ) := Wαg ; g∈G Definition 4 If G = Heis(1, n), its free group–C ∗ –algebra is called the 1–mode n–th degree Weyl algebra and denoted 0 W1,n := W 0 (Heis(1, n)) (12) For its generators, called the 1–mode n–th degree Weyl operators, we will use the notation W 0 (u, P ) := Weℓ0 (u,P ) ; (u, P ) ∈ R × Rn [X] (13) By construction the map uF : W 0 (u, P ) ∈ W1,n 0 7→ W (u, P ) ∈ WF,1,n (14) where the operators W (u, P ) are those defined in (7), is a group isomorphism. Hence the definition of free group–C ∗ –algebra implies that it can be extended 0 to a surjective ∗–representation called the Fock representation of W1,n . We will use the same symbol uF for this extension. We conjecture that, in analogy with the case n = 1, the ∗–homomorphism 0 of W1,n onto WF,1,n is in fact an isomorphism and that there is a unique C ∗ – 0 norm on W1,n . 10 5 The current algebra of heisR(1, n) over R Denote \ H0 (R) := L1R (R) ∩ L∞ R (R) = LpR (R) 1≤p≤∞ H0 (R) has a natural structure of real pre–Hilbert algebra with the pointwise operations and the L2 –scalar product. Lemma 2 For any ∗–sub–algebra T of H0 (R) and n ∈ N, there exists a unique real ∗–Lie algebra with skew–adjoint generators {L0 , Lk (f ) : k ∈ {1, . . . , n + 1} ; f ∈ T } where, with the notation Z L0 (f ) := L0 f (t)dt ; L−1 (f ) = 0 ; ∀f ∈ T (15) R the maps f → 7 Lk (f ) (k ∈ {0, 1, . . . , n}) are real linear on T and the Lie brackets are given, for all f, g ∈ T , by [Li (f ), Lj (g)] = 0 ; i, j ∈ {0, 1, . . . , n} (16) [Ln+1 (f ), Lk (g)] = kLk−1 (f g) ; k ∈ {0, 1, 2, . . . , n} , L−1 (f ) = 0 (17) Proof. By definition the Lie brakets of two generators defined by (16), (17) is a multiple of the generators. In order to verify that the Jacobi identity is satisfied notice that, for any i, j, k ∈ {0, 1, . . . , n} [Li (f1 ), [Lj (f2 ), Lk (f3 )]] = 0 unless exactly 2 among the indices i, j, k are equal to n + 1. Moreover, up to change of sign one can assume that i = j = n + 1. In this case one verifies that [Ln+1 (f1 ), [Ln+1 (f2 ), Lk (f3 )]] = k(k − 1)Lk−2 (f1 f2 f3 ) [Ln+1 (f2 ), [Lk (f3 ), Ln+1 (f1 )]] = −k(k − 1)Lk−2 (f1 f2 f3 ) [Lk (f3 ), [Ln+1 (f1 ), Ln+1 (f2 )]] = 0 and adding these identities side by side the Jacobi identity follows.  11 Definition 5 The real ∗–Lie algebra defined in Lemma 2 will be denoted heisR (1, n, T ). If I ⊂ R is a bounded Borel subsets we denote TI := the sub–algebra of T of functions with support in I (18) In analogy with the notation (3) we write the generic element of heisR (1, n, T ) in the form n ˜ X ℓ(f ) := Ln+1 (fn+1 ) + Lk (fk ) ; f0 , . . . , fn+1 ∈ T (19) k=0 where, here and in the following, if (f0 , . . . , fn+1 ) is an ordered (n + 2)–uple of elements of T , we will use the notation f˜ := (f0 , . . . , fn+1 ) (20) 6 Isomorphisms between the current algebra heisR(1, n, RχI ) and heisR(1, n) In the notations of the previous section and of Definition 5, for a bounded Borel subset I of R, we denote 1 if x ∈ J n χJ (x) := 0 if x ∈ /J RχI := {the real algebra of multiples of χI } Thus heisR (1, n, RχI ) ⊂ heisR (1, n, H0 (R)) is the ∗–Lie sub–algebra of heisR (1, n, H0 (R)) with linear skew–adjoint gen- erators {Lk (χI ) : k ∈ {0, 1, . . . , n}} and brackets [Ln+1 (χI ), Lk (χI )] = kLk−1 (χI ) ; k ∈ {0} ∪ {2, . . . , n} (21) for k ∈ {2, . . . , n} and the other commutators vanish. Recalling the notation (15) one must have L0 (χI ) = |I|L0 12 Lemma 3 In the notations of section 3 a real linear map ŝI : heisR (1, n, RχI ) → heisR (F, 1, n) satisfying for some constants aI , bI , ck,I ∈ R∗ := R \ {0} and for each k ∈ {1, . . . , n} ŝI (L0 ) = aI i1 (22) ŝI (Ln+1 (χI )) = bI ip (23) ŝI (Lk (χI )) = ck,I iq k ; ∀k ∈ {1, . . . , n} (24) is a real ∗–Lie algebra isomorphism if and only if ck,I = b−k I |I|aI ; ∀k ∈ {1, . . . , n} (25) The additional condition c1,I = bI (26) implies that aI must be > 0 and: k 1− k2 ck,I = |I|1− 2 aI ; ∀k ∈ {1, . . . , n} (27) Remark 4 In the above statement heisR (F, 1, n) can be replaced by heisR (1, n) because of the real ∗–Lie algebra isomorphism between the two. Proof. By definition ŝI maps a basis of heisR (1, n, RχI ) into a basis of heisR (F, 1, n) because the constants bI , ck,I are non zero hence it defines a unique vector space isomorphism which is a ∗–map because the constants are real. Moreover (22), (23), and (24) imply that [ŝI (Ln+1 (χI ), ŝI (L1 (χI ))] = [bI ip, c1,I iq] = bI c1,I [ip, iq] = bI c1,I i1 while (21) and (24) imply that ŝI ([Ln+1 (χI ), L1 (χI )]) = ŝI (|I|L0 ) = |I|ŝI (L0 ) = |I|aI i1 The isomorphism condition then implies that bI c1,I = |I|aI (28) The same argument, using (21), shows that for all k ∈ {2, . . . , n} [ŝI (Ln+1 (χI )), ŝI (Lk (χI ))] = [bI ip, ck,I iq k ] = bI ck,I [ip, iq k ] = bI ck,I kiq k−1 13  ŝI ([Ln+1 (χI ), Lk (χI )]) = ŝI (kLk−1 (χI )) = kŝI (Lk−1 (χI )) = kck−1,I iq k−1 and the isomorphism condition implies that −(k−1) bI ck,I = ck−1,I ⇔ ck,I = b−1 −2 I ck−1,I = bI ck−2,I = . . . = bI c1,I = b−k I |I|aI which is (25). Finally, if (26) holds, then (28) becomes b2I = |I|aI 1/2 Thus aI must be > 0 and bI = |I|1/2 aI which implies (27).  Remark 5 In the following we fix condition (26) and put aI = 1 (29) for all I so that the real ∗–Lie algebras isomorphism ŝI is given by (22) and (27). Therefore its inverse ŝ−1 I is given, on the generators, by: ŝ−1 I (i1) = L0 1 ŝ−1 I (ip) = |I| −2 Ln+1 (χI ) ŝ−1 k I (iq ) = |I| k/2−1 Lk (χI ) ; ∀k ∈ {1, . . . , n} The reason why the additional conditions (25) and (29) are necessary will be explained in Remark 7 at the end of section 9. Remark 6 Lemma 3 and condition (29) mean that, for any bounded Borel set I ⊂ R, heisR (1, n, RχI ) can be identified to a copy of heisR (1, n) with the rescaled basis 1 k {i|I|L0 , i|I| 2 Ln+1 , i|I|1− 2 Lk , k = 1, . . . , n} (30) In analogy with (3), we parametrize the elements of heisR (1, n, RχI ), with elements of R × Rn [X], and we write ℓI (u, P ) := uLn+1 (χI ) + P (L(χI )) ; u∈R (31) where P := nj=0 aj X j is a polynomial in one indeterminate and we use the P convention n X n X P (L(χI )) := aj Lj (χI ) := a0 |I|L0 + aj Lj (χI ) (32) j=0 j=1 14 The image of such an element under the isomorphism ŝI is 1 ŝI (ℓI (u, P )) = i(u|I| 2 p + PI (q)) (33) where by definition: n n 1− 2j j X X j PI (X) := aj |I| X = a0 |I|1 + aj |I|1− 2 X j (34) j=0 j=1 Introducing the linear change of coordinates in R × Rn [X] defined by 1  1 j  k̂I (u, P ) := (u|I| 2 , PI ) ≡ u|I| 2 , (aj |I|1− 2 ) (35) where PI is defined by (34) we see that, in the notations (3) and (31) one has ŝI ◦ ℓI = ℓ0 ◦ k̂I (36) 7 The group Heis(1, n, RχI ) and its C ∗–algebra In the notations and assumptions of section 6 we have seen that heisR (1, n, RχI ) is isomorphic to heisR (1, n). Since Rn+2 is connected and simply connected, the Lie group of heisR (1, n, RχI ), denoted Heis(1, n, RχI ) is isomorphic to Heis(1, n). In analogy with the notation (10), the generic element of Heis(1, n, RχI ) will be denoted eℓI (u,P ) ; (u, P ) ∈ R × Rn [X] (37) Definition 6 For any bounded Borel set I ⊂ R we denote 0 W1,n;I := W(Heis(1, n, RχI )) the free group–C ∗ –algebra of the group Heis(1, n, RχI )). In analogy with (13), its generators will be called the one mode n–th degree Weyl operators localized on I and denoted 0 WI0 (u, P ) := WeℓI (u,P ) ∈ W1,n;I (38) 15 Remark 7 Since the groups Heis(1, n, RχI )) and Heis(1, n) are isomorphic, the same is true for the corresponding free group–C ∗ –algebras. In the following section we show that, in these C ∗ –algebra isomorphisms, 0 the group generators of W1,n;I are mapped into a set of group generators of 0 W1,n which depends on I and we introduce a construction that allows to get rid of this dependence. 7.1 C ∗–algebras isomorphism In the notations (3) and (37) the map eℓI (u,P ) ∈ Heis(1, n, RχI ) 7→ eŝI (ℓI (u,P )) ∈ Heis(1, n) where ŝI the isomorphism defined in Lemma 3, is a Lie group isomorphism, hence it can be extended to a C ∗ –isomorphism of the corresponding free group–C ∗ –algebras. This extension will be denoted with the symbol: s0I : W1,n;I 0 0 → W1,n In view of the identity (36), and in the notations (13) and (38), the explicit form of s0I on the generators is given by s0I (WI0 (u, P )) = W 0 (k̂I (u, P )) (39) where k̂I is the linear map defined by (35) and (u, P ) ∈ R × Rn [X]. It is clear from (35) and (39) that, as a vector space, s0I (W1,n;I 0 ) coincides 0 with W1,n . In this section we will prove that the map W 0 (u, P ) ∈ W1,n 0 7→ W 0 (k̂I (u, P )) ∈ W1,n 0 (40) induces a C ∗ –algebra automorphism denoted kI . To this goal we use W 0 (k̂I (u, P ))W 0(k̂I (v, Q)) = W 0 (k̂I (u, P ) ◦ k̂I (v, Q)) and the following result. 16 Lemma 4 For all u ∈ R and P ∈ Rn [X], let k̂I be the linear map defined by (35). Then, denoting with the same symbol k̂I its restriction on Rn [X], one has: k̂I ◦ Tu (P ) = Tu|I| 21 ◦ k̂I (P ) k̂I−1 ◦ Tu−1 (P ) = T −1 − 1 ◦ k̂I−1 (P ) u|I| 2 k̂I ◦ Tu−1 (P ) = T −1 1 ◦ k̂I (P ) u|I| 2 k̂I−1 ◦ Tu (P ) = Tu|I|− 12 ◦ k̂I−1 (P ) Proof. Since both Tu and k̂I are linear maps, it is sufficient to prove the lemma for P (X) = X k (k ∈ {0, . . . , n}). For k = 0 all the identities in the lemma are obviously true. Let k ∈ {1, . . . , n}. Then from the identity (10) one has k Tu|I| 21 ◦ k̂I (X k ) = Tu|I| 21 (|I|1− 2 X k ) k = |I|1− 2 Tu|I| 21 (X k ) k−1 1− k2 hX k! k−h i = |I| uk−h |I| 2 X h + X k (k + 1 − h)!h! h=0 k−1 X k! h k = uk−h |I|1− 2 X h + |I|1− 2 X k h=0 (k + 1 − h)!h! = k̂I ◦ Tu (X k ) (41) (41) is equivalent to Tu|I| 21 ◦ k̂I = k̂I ◦ Tu ⇔ k̂I−1 ◦ T −1 1 = Tu−1 ◦ k̂I−1 u|I| 2 1 Replacing u by u|I|− 2 , this yields k̂I−1 ◦ Tu−1 = T −1 − 1 ◦ k̂I−1 (42) u|I| 2 ¿From identities (41) and (42), one gets Tu|I| 21 ◦ k̂I ◦ Tu−1 = k̂I Tu|I|− 21 ◦ k̂I−1 ◦ Tu−1 = k̂I−1 17 or equivalently k̂I ◦ Tu−1 = T −1 1 ◦ k̂I u|I| 2 k̂I−1 ◦ Tu = Tu|I|− 21 ◦ k̂I−1  Proposition 1 k̂I is a group automorphism for the composition law (9). Proof. We have to prove that for all (u, P ), (v, Q) ∈ R × Rn [X], one has   −1 (k̂I (u, P ) ◦ k̂I (v, Q)) = k̂I (u + v); T(u+v) (Tu P + Tv Su Q) We know that 1 1 k̂I (u, P ) ◦ k̂I (v, Q) = (u|I| 2 , PI ) ◦ (v|I| 2 , QI ) 1 1 where PI (X) = P (|I|− 2 X) and QI (X) = Q(|I|− 2 X). But from (11) we know that 1 1  1  −1 (u|I| , PI ) ◦ (v|I| , QI ) = (u + v)|I| , T 2 2 2 1 (T 1 P + Tv|I| 21 Su QI ) u|I| 2 I (u+v)|I| 2  1  = (u + v)|I| 2 , T −1 1 (T 1 k̂ u|I| 2 I (P ) + T 1 S v|I| 2 u I k̂ (Q)) (u+v)|I| 2 Furthermore, from Lemma 4, we know that T −1 1 −1 Tu|I| 21 k̂I (P ) = k̂I T(u+v) Tu (P ) (u+v)|I| 2 Moreover, using Su k̂I (Q) = k̂I Su (Q) We also have T −1 1 Tv|I| 21 Su k̂I (Q) = T −1 1 Tv|I| 12 k̂I Su (Q) (u+v)|I| 2 (u+v)|I| 2 = T −1 1 k̂I Tv Su (Q) (u+v)|I| 2 −1 = k̂I T(u+v) Tv Su (Q) 18 Hence, one gets 1 1 1 −1 (u|I| 2 , PI ) ◦ (v|I| 2 , QI ) = ((u + v)|I| 2 , k̂I T(u+v) (Tu (P ) + Tv Su (Q)))   −1 = k̂I (u + v) , T(u+v) (Tu (P ) + Tv Su (Q)) = k̂I ((u, P ) ◦ (v, Q)) and this proves the statement.  Corollary 1 The map: sI := kI−1 ◦ s0I : W1,n;I 0 0 → W1,n is a C ∗ –algebra isomorphism characterized by the condition sI (WI0 (u, P )) = W 0 (u, P ) ; ∀(u, P ) ∈ Rn+2 (43) Proof. (43) is clear from (39) and the definition (40) of kI . We know that s0I is a C ∗ –algebra isomorphism. ¿From Proposition 1 we know that k̂I is a group automorphism for the composition law defined by (9). Because of the linear independence of the free group algebra genera- tors kI extends to a C ∗ –algebra automorphism. Thus sI is composed of an isomorphism with an automorphism and the thesis follows.  8 The inductive limit In the following, when speaking of tensor products of C ∗ –algebras, it will be understood that a choice of a cross norm has been fixed and that all tensor products are referred to the same choice. 0 For a bounded Borel subset I of R, let W1,n;I be the C ∗ –algebra in Def- 0 0 inition 6 and let the isomorphisms sI : W1,n;I → W1,n defined by (43). For ∗ π = (Ij )j∈F ∈ P artf in (I) define the C –algebra O 0 0 W1,n;I;π := W1,n;I j (44) j∈F the injective C ∗ –homomorphism (C ∗ –embedding)   (diag) O O zI,π :=  s−1 Ij 0  ◦ sI : W1,n;I → 0 W1,n;I j 0 = W1,n;I;π j∈F j∈F 19 Then, for any π ≺ π ′ ∈ P artf in (I), the map   (diag) O O O O zI;π,π′ :=  s−1 I ′  ◦ s Ij : W 0 1,n;I;π → 0 W1,n;I 0 ′ = W1,n;I;π ′(45) j∈F Ij ⊇I ′ ∈π ′ j∈F Ij ⊇I ′ ∈π ′ is a C ∗ –embedding. Moreover, by Pnconstruction and in the notations of Defi- n nition 6, for all u ∈ R and P = j=0 aj X ∈ Rn [X], one has O zI;π,π′ zI,π (WI0 (u, P )) := WI0′ (u, P ) = zI,π′ (WI0 (u, P )) ∈ W1,n;I;π 0 ′ I ′ ∈π ′ Lemma 5 The family  0 (W1,n;I;π )π∈P artf in (I) , (zI;π,π′ )π≺π′ ∈P artf in (I) (46) is an inductive system of C ∗ –algebras, i.e. for all π ≺ π ′ , zI;π,π′ is a morphism and if π ≺ π ′ ≺ π ′′ ∈ P artf in (I) one has zI;π′ ,π′′ zI;π,π′ = zI;π,π′′ (47) Proof. We have already proved that the zI,π,π′ are C ∗ –embeddings. There- fore it remains to prove (47). To this goal, for π, π ′ , π ′′ as in the statement, using the identity O O O = I ′ ∈π ′ I∈π π ′ ∋I ′ ⊆I one finds        (diag) O O O (diag) O zI;π′′ ,π′ zI;π,π′ =   s−1 I ′′  ◦ sI ′    s−1 I′  ◦ sI  I ′ ∈π ′ π ′′ ∋I ′′ ⊆I ′ I∈π π ′ ∋I ′ ⊆I        (diag) O O O O (diag) O =   s−1 I ′′  ◦ sI ′    s−1 I′  ◦ sI  I∈π π ′ ∋I ′ ⊆I π ′′ ∋I ′′ ⊆I ′ I∈π π ′ ∋I ′ ⊆I        (diag) (diag) O O O O =   s−1 I ′′  ◦ sI ′    s−1 I′  ◦ sI  I∈π π ′ ∋I ′ ⊆I π ′′ ∋I ′′ ⊆I ′ π ′ ∋I ′ ⊆I 20     (diag) O O O s−1  ◦ sI ′ s−1
=  I ′′ I′ ◦ sI  I∈π π ′ ∋I ′ ⊆I π ′′ ∋I ′′ ⊆I ′    (diag) O O O =  s−1 I ′′  ◦ sI  I∈π π ′ ∋I ′ ⊆I π ′′ ∋I ′′ ⊆I ′    (diag) O O =  s−1 I ′′  ◦ sI  = zI;π′′ ,π I∈π π ′′ ∋I ′′ ⊆I  Definition 7 For any bounded Borel subset I of R, we denote  W1,n;I , (z̃I;π )π∈P artf in (I) the inductive limit of the family (46) i.e., W1,n;I is a C ∗ –algebra and for any π ∈ P artf in (I) and in the notation (44), 0 z̃I;π : W1,n;I;π → W1,n;I is an embedding satisfying z̃I;π′ zI;π,π′ = z̃I;π ; ∀π ≺ π ′ ∈ P artf in (I) Remark Intuitively one can think of the elements of W1,n;I as a realization of the non–linear Weyl operators: (1) with finitely valued, compact support, test functions. 8.1 Factorizable families of C ∗–algebras Definition 8 A family of C ∗ –algebras {WI }, indexed by the bounded Borel subsets of R, is called factorizable if, for every bounded Borel I ⊂ R and every Borel partition π of I, there is an isomorphism O uI,π : WIj → WI Ij ∈π If this is the case, an operator wI ∈ WI is called factorizable if there exist operators wIj ∈ WIj (Ij ∈ π) such that O u−1 I,π (w I ) = wIj (48) Ij ∈π 21  Remark 8 In the following, for a given bounded Borel set I, when π ≡ {I} is the partition of I, consisting of the only set I, we will use the notation 0 z̃I := z̃I;{I} : W1,n;I → W1,n;I We want to prove that: (i) the family of C ∗ –algebras {W1,n;I : I–bounded Borel subset of R} (49) where the algebras W1,n;I are those introduced in Definition 7, is fac- torizable in the sense of Definition 8; (ii) for any bounded Borel set I, the operators WI (u, P ) := z̃I (WI0 (u, P )) ∈ W1,n;I ; WI0 (u, P ) ∈ W1,n;I 0 (50) are factorizable in the sense of (48). To this goal let us remark that, if I, J are disjoint bounded Borel sets in R, then the map (πI , πJ ) ∈ P artf in (I) × P artf in (J) 7→ πI∪J := {πI ∪ πJ } ∈ P artf in (I ∪ J) (51) defines a canonical bijection between P artf in (I) × P artf in (J) and P artf in (I ∪ J) such that, if πI ≺ πI′ ∈ P artf in (I) and πJ ≺ πJ′ ∈ P artf in (J), ′ then πI∪J ≺ πI∪J ∈ P artf in (I ∪ J). Lemma 6 Let I, J be disjoint bounded Borel sets in R. Then the inductive system of C ∗ –algebras  0 (W1,n;I∪J;πI∪J )πI∪J ∈P artf in (I∪J) , (zI∪J;πI∪J ,πI∪J ′ )πI∪J ≺πI∪J ′ ∈P artf in (I∪J) (52) is isomorphic to the inductive system of C ∗ –algebras  0 0 (W1,n;I;πI ⊗ W1,n;J;π ) J (πI ,πJ )∈P artf in (I)×P artf in (J) , (zI;πI ,πI′ ⊗ zJ;πJ ,πJ′ )πI ≺πI′ ∈P artf in (I),πJ ≺πJ′ ∈P artf in (J) (53) 22 in the sense that, for each πI ∈ P artf in (I) and πJ ∈ P artf in (J), then there exists a C ∗ –algebra isomorphism 0 0 0 uI,J,πI ,πJ : W1,n;I;π I ⊗ W1,n;J;π J → W1,n;I∪J;π I∪J such that, for each πI ≺ πI′ ∈ P artf in (I) and πJ ≺ πJ′ ∈ P artf in (J), one has in the notation (51 ) uI,J,πI ,πJ ◦ (zI;πI ,πI′ ⊗ zJ;πJ ,πJ′ ) = zI∪J;πI∪J ,πI∪J ′ (54) Proof. With the notations above, from (44) one deduces that O O 0 0 0 0 W1,n;I;π I ⊗ W1,n;J;π J := ( W 1,n;I ′) ⊗ ( W1,n;J ′) I ′ ∈πI J ′ ∈πJ O 0 0 ≡ W1,n;K = W1,n;I∪J;π I∪J (55) K∈πI∪J Denote 0 0 0 uI⊗J,I∪J : W1,n;I;π I ⊗ W1,n;J;π J → W1,n;I∪J;π I∪J the isomorphism defined by (55). If πI ≺ πI′ ∈ P artf in (I) and πJ ≺ πJ′ ∈ ′ P artf in (J), then clearly πI∪J ≺ πI∪J ∈ P artf in (I ∪ J) and from (45) we see that uI,J,πI ,πJ ◦ (zI;πI ,πI′ ⊗ zJ;πJ ,πJ′ )         (diag) (diag) O O O O = uI,J,πI ,πJ ◦   s−1 I′  ◦ sI j  ⊗   s−1 J′  ◦ sJh  Ij ∈πI Ij ⊇I ′ ∈πI′ Jh ∈πJ Jh ⊇J ′ ∈πJ′   (diag) O O =  s−1 K  ◦ sHl = zI∪J;πI∪J ,π′ I∪J Hl ∈πI∪J ′ Hl ⊇K∈πI∪J which proves (54).  Theorem 1 (i) The family of C ∗ –algebras defined by (49) is factorizable. (ii) The operators defined by (50) are factorizable. Proof. We apply Definition 8 to the case in which the family F is the family of bounded Borel sets in R. By induction it will be sufficient to prove that, 23 if I, J are disjoint bounded Borel sets in R, then there exists a C ∗ –algebra isomorphism uI,J : W1,n;I ⊗ W1,n;J → W1,n;I∪J Since W1,n;I ⊗ W1,n;J is the inductive limit of the system (53) and W1,n;I∪J is the inductive limit of the system (52), the statement follows from Lemma 6 because isomorphic inductive systems have isomorphic inductive limits. The factorizability of the operators (50) follows from the identity (49).  ¿From Theorem 1 it follows that, if I ⊂ J are bounded Borel sets in R, then the map jI;J : wI ∈ W1,n;I → wI ⊗ 1J\I ∈ W1,n;J (56) is a C ∗ –algebra isomorphism. Since clearly, for I ⊂ J ⊂ K bounded Borel sets in R, 1J\I ⊗ 1K\J ≡ 1K\I , it follows that {(W1,n;I ) , (jI;J ) , I ⊂ J ∈ bounded Borel sets in R} (57) is an inductive system of C ∗ –algebras. Definition 9 The inductive limit of the system (57) will be denoted {W1,n;R , (jI ) , I ∈ bounded Borel sets in R} Since the jI : W1,n;I → W1,n;R are injective embeddings, the family (jI (W1,n;I )) is factorizable and one can introduce the more intuitive notation: jI (W1,n;I ) ≡ W1,n;I ⊗ 1I c 9 Existence of factorizable states on W1,n;R In the notation (38) and with the operators WI (u, P ) defined by (50), using factorizability of the family (W1,n;I ) and of the corresponding generators, for any I ⊂ R bounded Borel and any finite partition π of I, we will use the identifications W1,n;I ≡ jI (W1,n;I ) ≡ W1,n;I ⊗ 1I c ⊂ W1,n;R O WI (u, P ) ≡ WI0 (u, P ) ; ∀(u, P ) ∈ R × Rn [X] I0 ∈π O WI (u, P ) ≡ WI0 (u, P ) ; ∀(u, P ) ∈ R × Rn [X] (58) I0 ∈π 24 omitting from the notations the isomorphisms implementing these identifi- cations. Definition 10 A state ϕ on W1,n;R is called factorizable if for every I ⊂ R bounded Borel, for every finite partition π = (Ij )j∈F of I and for every WI (u, P ) as in (58), one has: Y ϕ(WI (u, P )) = ϕ(WIj (u, P )) ; ∀(u, P ) ∈ R × Rn [X] (59) j∈F The map (14) can be used to lift the Fock state ϕF on WF,1,n to a state, 0 denoted ϕ0 , on W1,n through the prescription ϕ0 (W 0 (u, P )) := ϕF (W (u, P )) (60) (W 0 (u, P ) ∈ W1,n 0 , W (u, P ) ∈ Un(Γ(C))). Then, using the maps z̃I defined by (50), for each bounded Borel set I ⊂ R, one can define the state ϕI 0 on z̃I (W1,n;I ) ⊂ W1,n;I through the prescription that, for each WI0 (u, P ) ∈ 0 W1,n;I , one has ϕI (WI (u, P )) = ϕI (z̃I (WI0 (u, P ))) := ϕF (W (k̂I (u, P ))) (61) Theorem 2 Under the assumption (29), if n = 1 then there exists a factor- izable state ϕ on W1,n;R such that, for each bounded Borel set I ⊂ R, one has ϕ(WI (u, P )) = ϕF (W (k̂I (u, P ))) ; ∀(u, P ) ∈ R × Rn [X] (62) If n ≥ 2, no such state exists. Proof. Let I be a fixed bounded Borel set in R and let π be a finite partition of I. ¿From Definition 10 we know that ϕ is factorizable if and only if for every I ⊂ R bounded Borel set, for every finite partition π of I and for every WI (u, P ) as in (58), (59) holds. If condition (62) is satisfied, the identity (59) becomes equivalent to: Y ϕF (W (k̂I (u, P ))) = ϕF (W (k̂Ij (u, P ))) ; ∀(u, P ) ∈ R × Rn [X] (63) Ij ∈π Thus the statement of the theorem is equivalent to say that, for n = 1 the identity (63) is satisfied and, for n ≥ 2, not. 25 - Case n = 1. For P = a0 + a1 X and u ∈ R, recalling the definition (35) of k̂I , one knows that 1 1 W (k̂I (u, P )) = ei(u|I| 2 p+a0 |I|1+a1 |I| 2 q) whose Fock expectation is known to be 2 +a2 )/4 ϕF (W (k̂I (u, P ))) = e−|I|(u 1 eia0 |I| = (ϕF (W (u, P )))|I| (64) It follows that Y Y ϕF (W (k̂Ij (u, P ))) = (ϕF (W (u, P )))|Ij | Ij ∈π Ij ∈π Therefore, if aI = 1, then ϕF (W (k̂I (u, P ))) = (ϕF (W (u, P )))|I| = ϕF (W (u, P )) - Case n ≥ 2. Since, for n ≥ 2, the 1–mode n–th degree Heisenberg ∗–Lie algebra heisR (1, n) contains a copy of heis(1, 2) (see Definition 1), the algebra W1,n;R contains a copy of W1,2;R . Therefore the non existence of a factorizable state on W1,2;R , satisfying (62), will imply the same conclusion for W1,n;R . In the case n = 2, let P = a0 + a1 X + a2 X 2 and u ∈ R. Then, using again aI = 1, (7) and (30) one has 1 1 2) W (k̂I (u, P )) = ei(|I| 2 up+a0 |I|1+a1 |I| 2 q+a2 q and from [10] (Theorem 2), one knows that 1 4C 2 (A2 +2iA)−3|M |2 |I| ϕF (W (k̂I (u, P ))) = (1 − 2iA)− 2 eia0 |I| e 6(1−2iA) 4C (A +2iA)−3|M |2 2 2 1  |I| = (1 − 2iA)− 2 eia0 e 6(1−2iA) where A = √a22 , B = √a12 , C = √u2 and M = B + iC. On the other hand, if π ∈ P artf in (I) with |π| > 1, then Y ϕF (W (k̂Ij (u, P )) Ij ∈π 4C 2 (A2 +2iA)−3|M |2 Y 1  |Ij |  = (1 − 2iA)− 2 eia0 e 6(1−2iA) Ij ∈π 4C 2 (A2 +2iA)−3|M |2  |I| − |π| ia0 = (1 − 2iA) e e 2 6(1−2iA) 4C 2 (A2 +2iA)−3|M |2  |I| − 12 ia0 6 (1 − 2iA) = e e 6(1−2iA) = ϕF (W (k̂I (u, P ))) 26   Lemma 7 In the case n = 1, the choice of the isomorphism ŝI (see Lemma 3) given by ŝI (L0 ((χI ))) = aI i|I|1 1 ŝI (L2 (χI )) = aI i|I| 2 p 1 ŝI (L1 (χI )) = aI i|I| 2 q gives rise to a factorizable state satisfying (62) if and only if the map I ⊂ R 7→ aI has the form 1 Z aI := p(s)ds |I| I for all Borel subsets I ⊆ R where p( · ) is a locally integrable almost every- where strictly positive function on R. In this case the factorizable state will be translation invariant if and only if p( · ) is a strictly positive constant. Proof. In the case n = 1, if aI 6= 1, then the expression for W (k̂I (u, P )) becomes 1 1/2 1 1/2 W (k̂I (u, P )) = ei(u|I| 2 aI p+a0 |I|aI 1+a1 |I| 2 aI q) consequently its Fock expectation is 2 +a2 )/4 ϕF (W (k̂I (u, P ))) = e−|I|aI (u 1 eia0 aI |I| = (ϕF (W (u, P )))aI |I| (65) Therefore the factorizability condition (63) can hold if and only if the map I ⊂ R 7→ aI |I| is a finitely additive measure. In this case, by construction it will be absolutely continuous with respect to the Lebesgue measure hence there will exist a locally integrable almost everywhere positive function p( · ) satisfying Z aI |I| := p(s)ds ; ∀ Borel I ⊆ R I p( · ) must be almost everywhere strictly positive because, by Lemma 3, aI > 0 for any Borel set I ⊆ R. This proves the first statement of the lemma. The second one follows because the Lebesgue measure is the unique translation invariant positive measure on R.  27 References [1] L. Accardi, U. Franz, M. 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