A Wavelet Construction for Quantum Brownian Motion and Quantum Brownian Bridges

Transcription

1 A Wavelet Construction for Quantum Brownian Motion and Quantum Brownian Bridges David Applebaum Probability and Statistics Department, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield, England, S3 7RH Dedicated to Robin Hudson on his 65th birthday. Friday 21st October 25, University of Lancaster 1. Probabilistic and Stochastic Hilbertian Structures 2. Wavelet Construction of Quantum Brownian Motion. 3. The Quantum Brownian Bridge Two key references: A.M.Cockroft, R.L.Hudson, Quantum mechanical Wiener processes, J. Mult. Anal. 7, (1977) C.D.Cushen, R.L.Hudson, A quantum-mechanical central limit theorem, J. Appl. Prob. 8, (1971) 1

2 1 Probabilistic and Stochastic Hilbertian Structures Motivation: A quantum random variable in the sense of Accardi, Lewis and Frigerio (1982) is a -homomorphism j between involutive algebras A and B, where B is equipped with a state to determine expectations. A quantum random variable in the sense of Cockcroft and Hudson (1977) is (implicitly) a pair of self-adjoint operators acting in a Hilbert space. Axiomatising the Cockcroft/Hudson approach gives a coherent formalism for quantum Brownian motions/bridges. 1.1 Daggered Spaces Let H be a complex separable Hilbert space and D be a dense linear subspace in H. We will be interested in linear operators T defined on H which have the following properties. (i) D Dom(T ) and the restriction of T to D is closable. (ii) D Dom(T ). (iii) T D D. (iv) Ran(T ) Dom(T ), where T denotes the restriction of T to D. In the sequel, we will often employ the notation T # to mean T or T. Let T 1,, T n be linear operators satisfying (i) to (iv) above. We denote as L n (D) the complex 2n-dimensional linear space generated by {T 1,, T n, T 1,..., T n}. We call L n (D) a daggered space of order n. 2

3 Slightly abusing terminology, {T 1,..., T n } are called the generators of this space. Such a space is said to be symmetric if n j=1 (α jt j α jt j ) is essentially skew-adjoint on D for all α = (α 1,..., α n ) C n. We define {( n ) c } U(α) := exp (α j T j α j T j ) j=1 to be the associated unitary operator in H, (where c denotes closure). 3

4 1.2 Probabilistic and Stochastic Hilbertian Structures Throughout this section, we fix a dense linear subspace D of a complex separable Hilbert space H. A probabilistic Hilbertian structure of order n or PHS(n) is a pair (L n (D), ψ ) where L n (D) is a symmetric daggered space, ψ is a fixed unit vector in H. The characteristic element of a PHS(n) is the mapping φ : C n C given by φ(α) = ψ, U(α)ψ. A PHS(n) is said to be Gaussian with if there exists m C n and an n n positive definite symmetric matrix C such that { φ(α) = exp 1 } 2 (α m)t C(α m). If C is a multiple of the identity, we say that the PHS is i.i.d. Gaussian. The probabilistic motivation for this is that in this case φ is the product of n copies of the characteristic element of a fixed Gaussian PHS(1). Examples include classical multivariate Gaussian vectors, annihilation/creation operators associated to n independent quantum harmonic oscillators. 4

5 Let I be an index set and {X(t), t I} be a family of linear operators in H. We call the pair ({X(t), t I}, ψ ) a stochastic Hilbertian structure or SHS if for each n N and for each t 1,..., t n I, {X(t 1 ),..., X(t n )} generate a symmetric daggered space with respect to D of order n. This latter space is denoted L t1,...,t n (D). The collection of all (L t1,...,t n (D), ψ )s are called the finite-dimensional distributions of the SHS. A SHS is said to be Gaussian if if all of its finite-dimensional distributions are Gaussian. Examples include (classical) Gaussian probability spaces (which themselves include classical and abstract Wiener spaces, and white noise spaces), quantum Wiener integrals, more generally annihilation/creation processes in any boson Fock space. 5

6 Fix I = R +. Following Cockroft and Hudson, we define a (standard) quantum Brownian motion to be a SHS for which (i) X() =. (ii) For all s, t R +, on D, [X(s), X(t)] = [X(s), X(t) ] =, [X(s), X(t) ] = (s t)i. (iii) For all n N, T > and all partitions P = { t < < t n = T } the operators {b j,t,p, 1 j n} generate an i.i.d Gaussian PHS(n) with covariance I, where b j,t,p := 1 tj t j 1 (X(t j ) X(t j 1 )). Boson Fock Brownian motions are obtained by taking H = L 2 (R + ), each X(t) = a(1 [,t) ) where a(f) is the annihilation operator associated to f H, ψ is the vacuum vector and D is the linear span of the exponential vectors. Theorem 1.1 (Cockcroft-Hudson) Any quantum Brownian motion is equivalent to the boson Fock Brownian motion. 6

7 2 The Wavelet Construction We construct a quantum Brownian motion with index set I = [, 1]. The Haar system is constructed as follows. The mother wavelet is 1 if t < 1 2 H(t) := 1 if 1 2 t 1 otherwise. The daughter wavelets are constructed by scaling and translation, H n (t) := 2 j 2 H(2 j t k), n = 2 j + k, j, k < 2 j. If we define H (t) := 1, then (H n, n Z + ) is a complete orthonormal basis for L 2 ([, 1]). The Schauder system is defined as follows. The mother wavelet is t (t) := 2 H(u)du = 2t if t < 1 2 2(1 t) if 1 2 t 1 otherwise. and the daughter wavelets are n (t) := (2 j t k) for n = 2 j + k as above. If we define (t) := t, then ( n, n Z + ) is a Schauder basis for the Banach space C [, 1] of continuous functions on [, 1] which vanish at the origin, equipped with the usual supremum norm. Note in particular that sup n Z + Furthermore, for each n Z +, sup n (t) = 1. (2.1) t [,1] n (t) = 1 t H n (u)du, (2.2) λ n 7,

8 where λ n := 2 j 2 1 for n = 2 j + k as above, and λ := 1. We work in Γ(l 2 (Z + )). For each n Z +, let e n = (,...,, (n) 1,,..., ), so (e n, n Z + ) is an orthonormal basis for l 2 (Z + ). Hence for each g l 2 (Z + ), g = g n e n, n= where g n := g, e n. We define a n := a(e n ) for each n Z +, then we have the canonical commutation relations:- [a m, a n ] = [a n, a m] =, [a n, a m] = δ mn, for each m, n Z +. For all f l 2 (Z + ), ψ(f) will denote the corresponding exponential vector. From now on we fix H = Γ(l 2 (Z + )), D to be the linear span of the exponential vectors and ψ to be the vacuum vector. The following is well-known Fock-law. Proposition ({a n, n Z + }, ψ ) is a Gaussian SHS. In particular all the finite dimensional distributions are i.i.d. Gaussian for all g l 2 (Z + ), n Z +. for all g l 2 (Z + ), n Z +. a n ψ(g) = g n e g 2 2, a nψ(g) = (1 + g n 2 ) 1 2 e g 2 2, Indeed (2) follows directly from the eigenrelation a(f)ψ(g) = f, g, and (3) also follows from this via the commutation relations. 8

9 Theorem 2.1 The series Y (t)ψ := λ n n (t)a n ψ and Y (t) ψ := n= λ n n (t)a nψ (2.3) converge uniformly for each ψ D. The linear operators Y (t) and Y (t) which are so defined are closable with each Y (t) Y (t). Furthermore the maps from [, 1] to Γ(l 2 (Z + )) given by t Y (t)ψ and t Y (t) ψ are continuous. Proof. It is sufficient to take ψ = ψ(g) for some g l 2 (Z + ). For any given t 1, we have n (t) = except for one n in each interval of the form [2 j, 2 j+1 ). We write each such n in the form 2 j + k n where k n < 2 j. Using proposition 2.1 (1), (2.1) and the Cauchy-Schwarz inequality, we have for sufficiently large M 2 J (say) n= λ n n (t)a n ψ n=m λ n n (t) a n ψ = n=m λ n n (t) g n e 1 2 g 2 n=m = 1 2 e 2 j g 2 2 j 2 2j +k(t) g 2j +k k= = 1 2 e 1 2 g 2 2 j 2 g2j +k n 1 2 e 1 2 g 2 2 j 1 2 g 2j +k n

10 1 2 g e 1 2 g 2 as J 2 j 1 2 Using similar arguments and proposition 2.1 (2), we obtain λ n n (t)a nψ n=m 1 2 e 1 2 g 2 2 j 2 (1 + g2j +k n 2 ) e 1 2 g 2 2 j 2 (1 + g2j +k n ) 1 2 e 1 2 g 2 2 j 2 + g as J 2 j 1 2 The closure and mutual adjointness follow from the easily verified fact that for each t 1, φ 1, φ 2 D, Y (t) φ 1, φ 2 = φ 1, Y (t)φ 2, and the continuity follows by the uniform convergence of each series on [, 1]. 1

11 Theorem 2.2 ({Y (t), t }, ψ ) is a quantum Brownian motion. Proof. For each s, t 1 on the domain D, using (2.2) and Parseval s identity, we have [Y (s), Y (t) ] = = = = = m,n= m,n= λ m λ n m (s) n (t)[a m, a n] λ m λ n m (s) n (t)δ mn I λ n n (s).λ n n (t)i n= ( s n= ) ( t ) H n (u)du. H n (u)du I 1 [,s], H n 1 [,t], H n I n= = 1 [,s], 1 [,t] I = (s t)i, as required. The other two commutation relations are immediate. To establish Gaussianity, let P be a partition of [, 1] containing n+1 points and denote the associated operators b j,1,p simply as b j (1 j n). It follows from the commutation relations that [b j, b k ] = [b j, b k] =, [b j, b k] = δ jk, for each 1 j, k n. To see that these are independent Gaussians, for all α j C(1 j n) {( n ) c } ψ ψ, exp j=1 (α j b j ᾱ j b j ) 11

12 { } { = ψ, exp 1 n n } { n } α j 2 exp α j b j exp ᾱ j b j ψ 2 j=1 j=1 j=1 { } = exp 1 n α j 2. 2 j=1 We have used wavelets to construct a quantum Brownian motion on [, 1]. To extend the index set to the whole of R + we work in the countable (KRP) or partial (RFS) or incomplete (JvN) tensor product of an infinite number of copies of Γ(l 2 (N) with respect to the stabilising sequence comprising the vacuum vector in each space. We construct a countable number of i.i.d copies of quantum Brownian motion on [, 1] via the prescription Y (n) (t) = I I (n) Y (t) I with domain the ampliation of D. We then define quantum Brownian motion on R + by as follows. We take ψ to be the infinite tensor product of vacuum vectors, the domain to be the incomplete algebraic tensor product of an infinite number of copies of D and the required operators (A(t), t ) are given by n A(t) = Y (k) (1) + Y (n+1) (t n), k=1 whenever t (n, n + 1]. Note The construction extends to non-fock, fermion, free, Bozejko-Speicher and monotone Brownian motions. For the classical version, see J.M.Steele, Stochastic Calculus and Financial Applications, Springer-Verlag New York Inc. (21) 12

13 3 Quantum Brownian Bridges We return to boson probability. We define a quantum Brownian bridge to be a Gaussian SHS (U(t), t [, 1]), ψ ) for which (i) U() = U(1) =. (ii) For all s, t [, 1], on D, [U(s), U(t)] = [U(s), U(t) ] =, [U(s), U(t) ] = (s t)[1 (s t)]i. To construct a quantum Brownian bridge, let (X(t), t [, 1]), ψ ) be a quantum Brownian motion. Define U(t) := X(t) tx(1), for each t [, 1]. This is a quantum Brownian bridge. Indeed (i) and (ii) are both trivial and Gaussianity follows from writing each U(t) = (1 t)x(t) t(x(1) X(t)) and observing that {X(u), u t), ψ } and {(X(1) X(u), t u 1), ψ } are independent Gaussian SHSs. From the Cockroft-Hudson theorem 1.1, it follows that all quantum Brownian bridges are unitarily equivalent to that obtained from the Fock Brownian motion in this manner. Using the Wiener-Segal duality transformation, we see that for each θ [, 2π), e iθ U(t) + e iθ U(t) is unitarily equivalent to a classical Brownian bridge in Wiener space. The cases θ = and θ = π 2 are naturally associated to canonical position and momentum field operators. The following results are easily established analogues of wellknown classical results. 1. If ({U(t), t [, 1]}, ψ ) is a quantum Brownian bridge then so is ({U(1 t), t [, 1]}, ψ ). 13

14 2. There is a one-to one correspondence between quantum Brownian motions ({A(t), t [, 1]}, ψ ) and quantum Brownian bridges ({U(t), t [, 1]}, ψ ) on the same space given by ( ) t A(t) (t + 1)U, for all t R +, t + 1 ( t U(t) (1 t)a 1 t ), for all t [, 1). Returning to the wavelet expansions of theorems 2.1 and 2.2, it follows just as in the classical case that ({V (t), t [, 1]}, ψ ) is a quantum Brownian bridge, where each V (t) := λ n n (t)a n. n=1 To see this, observe that since each n (1) = for each n N, Y # (1) = a #, hence V (t) # = Y (t) # (t)a # = Y (t)# ty (1) #. The following is a quantum version of a well-known classical example of a Brownian bridge, however the method of the proof is completely different. We work in Γ(L 2 (R + )). 14

15 Theorem 3.1 ({U(t), t [, 1]}, ψ ) is a quantum Brownian bridge where for each t < 1 U(t) := (1 t) t da(u) 1 u. Proof. Gaussianity follows from the properties of quantum Wiener integrals and it is sufficient to verify the non-trivial commutation relation. For s t < 1, we use U(t) = (1 t) s da (u) 1 u + (1 t) t Then using the canonical commutation relations: [ s da(u) s [U(s), U(t) ] = (1 s)(1 t) 1 u, = (1 s)(1 t) s = s(1 t)i. s du (1 u) 2I da (u) 1 u. ] da (u) 1 u A straightforward application of the quantum Itô formula applied to the result of theorem 3.1 yields the quantum stochastic differential equation representation of a quantum Brownian bridge: du(t) = da(t) 1 1 t U(t)dt, for all t < 1. 15

16 To finish, here s a beautiful application of classical Brownian bridges: Let (b(t), t [, 1]) be a Brownian bridge and define { } 2 R := sup b(t) inf π b(t), t [,1] t [,1] then for all z C define ξ(z) := E(R z ). ξ is an entire analytic function. equation It satisfies the functional More spectacularly, ξ(z) = ξ(1 z). ξ(z) = 1 2 z(1 z)π z 2 Γ ( 1 2 z ) ζ(z), where ζ is the Riemann zeta function. Some references for this: D.Williams, Brownian motion and the Riemann zeta-function, Disorder in Physical Systems ed. G.R.Grimmett and D.J.A. Welsh, pp , Clarendon Press, Oxford (199) P.Biane, J.Pitman, M.Yor, Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions, Bull. Amer. Math. Soc., 38, (21) Challenge: Find a link between quantum Brownian bridges and the Riemann zeta function. Also - classical Brownian bridges can be used to prove the Gauss-Bonnet-Chern theorem. Diffusion bridges are applied to 16

17 prove the Atiyah-Singer Index theorem. What can you do with quantum versions? 17