On the Relation of the Square of White Noise and the Finite Difference Algebra

Transcription

1 On the Relation of the Square of White Noise and the Finite Difference Algebra Luigi Accardi Centro Vito Volterra Università degli Studi di Roma Tor Vergata Via di Tor Vergata, s.n.c Roma, Italia Homepage: Michael Skeide Lehrstuhl für Wahrscheinlichkeitstheorie und Statistik Brandenburgische Technische Universität Cottbus Postfach , D Cottbus, Germany skeide@math.tu-cottbus.de Homepage: skeide.html September 1999 Abstract The algebra of square of white noise [ALV99] contains a subalgebra generated by elements fulfilling the relations of Feinsilver s finite difference algebra [Fei87]. Moreover, Boukas representation space [Bou88] is the same as the representation space of the algebra of square of white noise discovered in [AS99]. In other words, Boukas representation extends to a representation of the algebra of square of white noise. MS is supported by Deutsche Forschungsgemeinschaft. 1

2 For a measurable function on R define Trf = f(t) dt. The algebra of square of white noise (SWN) introduced by Accardi, Volovich and Lu [ALV99] is the unital algebra generated by the following relations. [B f, B + g ] = c Tr(fg) + 4N fg f, g L (R) L (R) (1a) [N a, B + f ] = B+ af a L (R), f L (R) L (R) (1b) and [B + f, B+ g ] = [N a, N a ] = 0. The finite difference algebra (FD algebra) introduced by Feinsilver [Fei87] is the unital *-algebra generated by the relations [P f, Q g ] = [T f, Q g ] = [P f, T g ] = T fg () where, however, f, g are only real-valued step functions on R +. Boukas [Bou88, Bou91a] has realized these relations by operators on a Hilbert space which is spanned by a kind of exponential vectors ψ(h) (h a step function on R + bounded by 1) with inner product ψ(h), ψ(h ) = e R ln(1 h(s)h (s)) ds. (3) Parthasarathy and Sinha [PS91] show that this Hilbert space is isomorphic to the symmetric Fock space Γ(L (R +, l )) where, indeed, the vectors ψ(h) correspond to certain exponential vectors. By Relations (1a,1b) and by multiple polarization, the SWN algebra is spanned linearly by all monomials in the generators B + f, B f, N a which are in normally ordered form B + f n f B n g g N n a a. A Fock representation is a representation with a cyclic vacuum vector Φ for which B f Φ = N a Φ = 0. It is clear that there is at most one Fock representation up to unitary equivalence and that the vectors B + n f Φ form a total subset of the representation space. In [ALV99] existence of the Fock representation was established, by showing that the scalar product of two such vectors (determined uniquely by Relations (1a,1b), if it exists) is, indeed a scalar product. The proof was simplified in [Sni99] where the scalar product is written down in way such that positivity is immediate. In [AS99] we constructed the representation explicitely with the help of Hilbert modules. Our goal in this note is to show that Boukas representation space of the FD algebra and our representation space of the SWN algebra coincide. We do this by identifying in our representation space those exponential vectors which have the same inner product as in (3). Furthermore, we show that the FD algebra has a homomorphic image in the SWN algebra. Therefore, each representation of the SWN algebra gives rise to a representation of the finte difference algebra.

3 Good canditates for exponential vectors are B + n ρχ ψ ρ (t) = t Φ = n! ρ n n! B+ χ t n Φ where ρ C and χ t := χ [0,t]. (If we replace B + by creators on the usual symmetric Fock space we obtain precisely the usual exponential vector ψ(ρχ t ).) Whenever ψ ρ0 (t) exists, then it is an analytic vector-valued function of ρ with ρ < ρ 0. Notice that we have a natural time shift in our representation space. As ψ ρ (t) is analytic in ρ, we may differentiate. It follows that B + χ t n Φ = dn dρ n ρ=0 ψ ρ (t) is in the closed linear span of ψ ρ (t) ( ρ < ρ 0 ). Therefore, if for each t > 0 there exists ρ 0 > 0 such that ψ ρ0 (t) exists, then the vectors ψ ρ (t) and their time shifts form a total subset. Theorem 1 ψ ρ (t) exists, whenever ρ < 1. Moreover, we have ψ ρ (t), ψ σ (t) = e ct ln(1 4ρσ). (4) Proof First, we show that the left-hand side of (4) exists in the simpler case σ = ρ showing, thus, existence of ψ ρ (t). Set f = ρχ t. Then (ff)f = ρ f. This yields the commutation relation N ff B + f = B + f N ff + ρ B + f. Moreover, c Tr(ff) = c ρ t. We find B f B + n f =B + f B fb + f =B + f B fb + f + (c ρ t + 4N ff )B + f + B + f (c ρ t + 8 ρ (n 1)) + B + f 4N ff =B + n f B f 4N ff + nb + f + B + f ρ ( (ct + 4(n 1)) + (ct + 4(n )) (ct + 0) ) =B + n f B f + nb + f 4N ff + B + f n ρ (ct + (n 1)). If we apply this to the vacuum Φ, then the first two summands dissappear. We find the recursion formula B + f It is clear that n Φ, B + n f Φ B + f (n!) ( = 4 ρ ct n + n 1 ) B + f Φ, B + f Φ. n ((n 1)!) n Φ,B + n f Φ converges, if and only if 4 ρ < 1 or ρ < 1. (n!) For fixed ρ U 1 (0) the function ψ ρ (t), ψ ρ (t) is the uniform limit of entire functions on t and, therefore, itself an entire function on t. It is not difficult to check that whenever ψ ρ (s) and ψ ρ (t) exist, then also ψ ρ (s + t) exists and that ψ ρ (s + t), ψ ρ (s + t) = ψ ρ (s), ψ ρ (s) ψ ρ (t), ψ ρ (t) ; see [AS99] for a detailed argument. Thus, there must exist a number κ R (actually, in R +, because ψ ρ (t), ψ ρ (t) 1) such that ψ ρ (t), ψ ρ (t) = e κt. We find κ by differentiating at t = 0. The only contribution in d ( dt 4 ρ ct t=0 n + n 1 ) (... 4 ρ ct ) n + 0 3

4 comes by the Leibniz rule, if we differentiate the last factor and put t = 0 in the remaining ones. We find d dt ψ ρ (t), ψ ρ (t) = t=0 (4 ρ ) n 1 c n = c ln(1 4 ρ ). n=1 The remaining statement follows essentially by the same computations, replacing ρ with ρσ. Corollary Put c =. Then the mapping ψ(ρχ t ) ψ ρ (t) defines a unique time shift invariant isometry from Boukas representation space of the FD algebra into our representation space of the SWN algebra. The range of this isometry is spanned by vectors of the form B + n f Φ where f is different from 0 only on R +. Motivated by this discovery it is natural to ask how the two algebras are related. We find the answer in the following easy to check theorem. Theorem 3 Let f L (R) L (R) and set q f = 1 (B+ f + N f) p f = 1 (B f + N f) t f = 1Trf + p f + q f. Then the elements q, p f, t f (f L (R) L (R)) fulfill Relations (). In other words, there exists a algebra homomorphism from the FD algebra into the SWN algebra. This homomorphism is, however, not injective, unless the relation T f = 1Trf + P f + Q f follows from Relations (). And it is sureley not surjective, because T f is determined by P f, Q f and from these elements we can only gain back B + f B f and B + f + B f + N f. Therefore, is is justified to say that we enlarged Boukas algebra of operators which is generated by certain linear combinations of creators, annihilators and number operators to the full algebra generated by such. A calculus based on SWN operators should generalize Boukas calculus in [Bou91b] to a calculus of creation, annihilation, and number operators. Moreover, the results in [PS91] and our extension indicate that such a calculus should be interpreteable in terms of a calculus on a usual symmetric Fock space. Of course, it would be interesting to know, whether our representation of the SWN algebra is faithful. References [ALV99] L. Accardi, Y.G. Lu, and I.V. Volovich. White noise approach to classical and quantum stochastic calculi. Preprint, Rome, To appear in the lecture notes of the Volterra International School of the same title, held in Trento. 4

5 [AS99] [Bou88] L. Accardi and M. Skeide. Hilbert Module Realization of the Square of White Noise and the Finite Difference Algebra. Preprint, Rome, A. Boukas. Quantum stochastic analysis: a non-brownian case. PhD thesis, Southern Illinois University, [Bou91a] A. Boukas. An example of quantum exponential process. Mh. Math., 11:09 15, [Bou91b] A. Boukas. Stochastic calculus on the finite-difference Fock space. In L. Accardi, editor, Quantum Probability & Related Topics VI. World Scientific, [Fei87] [PS91] [Sni99] P.J. Feinsilver. Discrete analogues of the Heisenberg-Weyl algebra. Mh. Math., 104:89 108, K.R. Parthasarathy and K.B. Sinha. Unification of quantum noise processes in Fock spaces. In L. Accardi, editor, Quantum Probability & Related Topics VI, pages World Scientific, P. Sniady. Quadratic bosonic and free white noise, Preprint, in preparation. 5