CUBO A Mathematical Journal Vol.14, No¯ 02, (91–109). June 2012 Higher order terms for the quantum evolution of a Wick observable within the Hepp method Sébastien Breteaux IRMAR, UMR-CNRS 6625, Université de Rennes 1, campus de Beaulieu, 35042 Rennes Cedex, France. ENS de Cachan, Antenne de Bretagne, Campus de Ker Lann, Av. R. Schuman, 35170 Bruz, France. email: sebastien.breteaux@ens-cachan.org ABSTRACT The Hepp method is the coherent state approach to the mean field dynamics for bosons or to the semiclassical propagation. A key point is the asymptotic evolution of Wick observables under the evolution given by a time-dependent quadratic Hamiltonian. This article provides a complete expansion with respect to the small parameter ε > 0 which makes sense within the infinite-dimensional setting and fits with finite-dimensional formulae. RESUMEN El método de Hepp describe en términos de estados coherentes la dinámica en campo medio de bosones o la propagación semiclásica. Un punto clave es la evolución asintótica de observables de Wick bajo la evolución dada por un Hamiltoniano cuadrático depen- diente del tiempo. Este art́ıculo proporciona una expansión completa con respecto al parámetro pequeño ε > 0 válido en dimensión infinita y que corresponde a fórmulas en dimensión finita conocidas. Keywords and Phrases: Mean field limit, semiclassical limit, coherent states, squeezed states. 2000 AMS Mathematics Subject Classification: 81R30, 35Q40, 81S10, 81S30. 78 Sébastien Breteaux CUBO 14, 2 (2012) 1 Introduction In this article we derive two expansions with respect to a small parameter ε of quantum evolved Wick observables under a time-dependent quadratic Hamiltonian. The Hepp method was introduced in [20] and then extended in [12, 13] in order to study the mean field dynamics of many bosons systems via a (squeezed) coherent states approach. The asymptotic analysis in the mean field limit is done with respect to a small parameter ε, where the number of particles is of order 1 ε . Remember that the mean field dynamics is obtained as a classical Hamiltonian dynamics which governs the evolution of the center z(t) of the Gaussian state (squeezed coherent state). Meanwhile the covariance of this Gaussian as well as the control of the remainder term is determined by the evolution of a quadratic approximate Hamiltonian around z(t). A key point in this method is the asymptotic analysis of the evolution of a Wick quantized observable according to this quantum time-dependent quadratic Hamiltonian. Only a few results are clearly written about the remainder terms and some possible expansions in powers of ε, see the works of Ginibre and Velo [14, 15]. In the finite-dimensional case, entering into the semiclassical theory, accurate results have been given by Combescure, Ralston and Robert in [6, 7, 8], Hagedorn and Joye in [17, 18, 19]. Another viewpoint is used (in finite dimension) by Paul and Uribe in [25] to get approximate eigenvectors of semiclassical operators in terms of linear superpositions of coherent states. For the mean field infinite-dimensional setting some results have been proved in [16, 11, 28] with a different approach. We stick here with the Hepp method with the presentation of [1] which puts the stress on the similarities and differences between the infinite-dimensional bosonic mean field problem and the finite-dimensional semiclassical analysis. Nevertheless, in [1] the authors only considered the main order term although some of their formulae make possible complete expansions. In this article we derive two expansions of the quantum evolved Wick observables which are equal term by term. Two difficulties have to be solved : (1) Unlike the time-independent finite-dimensional case, no Mehler type explicit formula (see for example [22] or [9]) is available. A general time-dependent Hamiltonian has no explicit dynamics. (2) In the infinite-dimensional framework the quantization of a linear symplectic transforma- tion (a Bogoliubov transformation) requires some care. Useful references on this subject are [3] and [2]. Its realization in the Fock space relies on a Hilbert-Schmidt condition on the antilinear part connected with the Shale theorem (see [29] and [26, 8, 5]). These things are well known but have to be considered accurately while writing complete expan- sions. CUBO 14, 2 (2012) Higher order terms for the quantum evolution of a Wick ... 79 Two different methods, with apparently two different final formulae, will be used. A first one relies on a Dyson expansion approach and provides the successive terms as time-dependent integrals. The second one uses the exact formulae for the finite-dimensional Weyl quantization and after having made explicit the relationship between Wick and Weyl quantizations like in [4] or [1], the proper limit process with respect to the dimension is carried out. The outline of this article is the following. In Section 2 we recall some facts and definitions about the Fock space and Wick quantization. We then present our main results in Section 3 in Theorems 3.1 and 3.2 and illustrate them by a simple example. Section 4 and Section 5 are devoted to the construction and properties of the classical and quantum evolution associated with a symmetric quadratic Hamiltonian. Section 7 and Section 8 contain the proofs of our two expansion formulae. For the convenience of the reader we recall some facts about real-linear symplectomorphisms and symplectic Fourier transform in the appendices. 2 Wick calculus with polynomial observables 2.1 Definitions We recall some definitions and results about Wick quantization. More details can be found in [1]. In this paper (Z,〈·, ·〉) denotes a separable Hilbert space over C, the field of complex numbers. It is also a symplectic space with respect to the symplectic form σ(z1,z2) = ℑ 〈z1,z2〉. We use the physicists convention that all the scalar products over Hilbert spaces are linear with respect to the right variable and antilinear with respect to the left variable. We denote by Sm the symmetrization operator on ⊗m Z (the completion for the natural Hilbert scalar product of the algebraic tensor product ⊗m, alg Z) defined by Sm (z1 ⊗ · · · ⊗ zm) = 1 m! ∑ σ∈Sm zσ1 ⊗ · · · ⊗ zσm , where the zj are vectors in Z and Sm denotes the set of the permutations of {1, . . . ,m}. We will use the notation z1 ∨ · · · ∨zm for Sm (z1 ⊗ · · · ⊗ zm), and z∨m for z∨ · · · ∨z when the m terms of this product are equal to z. We call monomial of order (p,q) ∈ N2 a complex-valued application defined on Z of the form b(z) = 〈 z∨q, b̃z∨p 〉 , with b̃ ∈ L (∨p Z, ∨q Z ) where ∨n Z (or Z∨n) denotes the Hilbert completion of the n-fold symmetric tensor product, and for two Banach spaces E and F, the space of continuous linear applications from E to F is denoted by L (E,F). We then write b ∈ Pp,q (Z). The total order of b is the integer m = p + q. The finite linear combinations of monomials are called polynomials. The set of all polynomials of this type is denoted by P (Z). Subsets of particular interest of P (Z) are Pm (Z) and P≤m (Z), the finite linear combinations of monomials of total order equal to m and not greater than m. 80 Sébastien Breteaux CUBO 14, 2 (2012) The Hilbert space H := ⊕ n∈N n∨ Z is called the symmetric Fock space associated with Z, where tensor products and sum completions are made with respect to the natural Hilbert scalar products inherited from Z. We also consider the dense subspace Hfin of H of states with a finite number of particles Hfin := alg⊕ n∈N n∨ Z , where the tensor products are completed but the sum is algebraic. The Wick quantization of a monomial b ∈ Pp,q (Z) is the operator defined on Hfin by its action on ∨n Z as an element of L( ∨n Z, ∨n+q−p Z), bWick ∣∣∨ n Z = 1[p,+∞) (n) √ n! (n + q − p) ! (n − p) ! ε p+q 2 ( b̃ ∨ I∨n−p Z ) , where IX denotes the identity map on the space X and for Aj ∈ L ( Z∨pj,Z∨qj ) , A1 ∨ A2 = Sq1+q2A1 ⊗ A2Sp1+p2. The Wick quantization is extended by linearity to polynomials. We have a notion of derivative of a polynomial, first defined on the monomials and then extended by linearity. For b ∈ Pp,q (Z) and for any given z ∈ Z, the operator ∂ j z̄∂ k zb(z) := p! (p − k) ! q! (q − j) ! (〈 z∨(q−j) ∣∣∣ ∨ I∨j Z ) b̃ ( z∨(p−k) ∨ I∨k Z ) (2.1) is an element of L (∨k Z, ∨j Z ) . We use the “bra” and “ket” notations of the physicists for vectors and forms in Hilbert spaces. Then we can define the Poisson bracket of order k of two polynomials b1, b2, by {b1,b2} (k) = ∂kzb1.∂ k z̄b2 − ∂ k zb2.∂ k z̄b1 since, for any polynomial b, ∂kzb(z) is a k-form (on Z) and ∂kz̄b(z) is a k-vector. Remark 1. The product denoted by a dot in the definition of the Poisson bracket is a C-bilinear duality-product between k-forms and k-vectors. As an example consider the polynomials b1 (z) = 〈 z∨3,ξ∨31 〉〈 η∨21 ,z ∨2 〉 and b2 (z) = 〈 z∨3,ξ∨32 〉 〈η2,z〉 . The Poisson bracket of order 2 of b1 and b2 is {b1,b2} (2) (z) = 2 × 6 × 〈 z∨3,ξ∨31 〉〈 η∨21 ∨ z,ξ ∨3 2 〉 〈η2,z〉 − 0. 2.2 Some examples of Wick quantizations Here is a quick review of the notations used for some useful examples of Wick quantization. A vector of Z is denoted by ξ, A is a bounded operator and z is the variable of the polynomials. In CUBO 14, 2 (2012) Higher order terms for the quantum evolution of a Wick ... 81 the next table, the first column describes the polynomial and the second the corresponding Wick quantization (as an operator on Hfin). 〈z,Az〉 ↔ dΓ (A) |z| 2 ↔ N 〈z,ξ〉 ↔ a∗ (ξ) 〈ξ,z〉 ↔ a(ξ) √ 2ℜ 〈z,ξ〉 ↔ Φ(ξ) The operator dΓ (A) is the usual second quantization of an operator restricted to Hfin multiplied by a factor ε. If A = IZ we obtain N the usual number operator multiplied by a factor ε. The operators a, a∗ and Φ are the usual annihilation, creation and field operators of quantum field theory with an additional √ ε factor. The real and imaginary parts of a complex number ζ are denoted by ℜζ and ℑζ. The field operators Φ(ξ) are essentially self-adjoint and this enables us to define the (ε-dependent) Weyl operators W (ξ) = eiΦ(ξ) . 2.3 Calculus Here are some calculation rules for Wick quantization of polynomials in P (Z). The proofs can be found in [1]. Proposition 2.1. For every polynomial b ∈ P (Z), • bWick1 bWick2 = (∑min{p1,q2} k=0 εk k! ∂kzb1.∂ k z̄b2 )Wick in Hfin for any bi ∈ Ppi,qi (Z), • bWick is closable and the domain of the closure contains H0 = Vect {W (z)ϕ,ϕ ∈ Hfin, z ∈ Z} , (we still denote by bWick the closure of bWick), • ( bWick )∗ = b̄Wick on Hfin (where the bar denotes the usual conjugation on complex num- bers), • for any z0 in Z, W (√ 2 iε z0 )∗ bWickW (√ 2 iε z0 ) = (b(z0 + z)) Wick holds on H0 where b(z0 + ·) ∈ P (Z). 3 Main results and a simple example Our two hypotheses are: 82 Sébastien Breteaux CUBO 14, 2 (2012) H1 Let (αt)t∈R be a one parameter family of self-adjoint operators on Z defining a strongly continuous dynamical system uα(t,s). H1’ Assume H1 and additionally that the dynamical system preserves a dense set D such that, for any ψ ∈ D, uα (·, ·)ψ belongs to C1 ( R2,Z ) ∩ C0 ( R2,D ) . H2 Let β be in C0 ( R; Z∨2 ) , (βt defines a C-antilinear Hilbert-Schmidt operator by z 7→ (IZ ∨ 〈z|)βt). With H1’ and H2, the classical flow associated with a family Qt (z) = 〈z,αtz〉 + ℑ 〈 βt,z ∨2 〉 of quadratic polynomials is the solution ϕ(t,s) to the equation { i∂tϕ(t,0) [z] = ∂z̄Qt (ϕ(t,0) [z]) ϕ(0,0) = IZ (3.1) where ∂z̄Qt (z) = αz + i(IZ ∨ 〈z|β), written in a weak sense. Although things are better visualized by writing a differential equation, the hypotheses H1 and H2 suffice to define the dynamical system ϕ(t,s). Details about this point are given in Section 4. Actually ϕ(t,s) is a family of symplectomorphisms of (Z,σ) which are naturally decomposed into their C-linear and C-antilinear parts: ϕ = L + A, L ∈ L (Z) , AA∗ ∈ L1 (Z) . See Appendix A for more details about symplectomorphisms and this decomposition. Similarly, the quantum flow associated with Qt is the solution U(t,s) to { iε∂tU(t,0) = Q Wick t U(t,0) U(0,0) = IH . (3.2) The precise meaning of the solutions to this equation is specified in Section 5. We are ready to state our two main results dealing with the evolution of a Wick observ- able bWick, b ∈ P (Z), under the quantum flow, that is to say the quantity U(0,t)bWickU(t,0). (We use the usual notation 〈N〉 = √ N2 + 1.) Theorem 3.1. Assume H1 and H2. Let b ∈ P≤m (Z) be a polynomial. Then, for any time t ≥ 0, the formula U(0,t)bWickU(t,0) = ( b(0),t )Wick + ⌊m/2⌋∑ k=1 ( ε 2 )k ˆ ∆kt ( b(k)t,s̄ k )Wick ds̄k (3.3) holds as an equality of continuous operators from D ( 〈N〉m/2 ) to H, where • s̄k = (s1, . . . ,sk) ∈ Rk+ and ∆kt = { s̄k ∈ Rk+, ∑k j=1 sj ≤ t } , CUBO 14, 2 (2012) Higher order terms for the quantum evolution of a Wick ... 83 • the polynomials b(k)t,s̄k are defined recursively by { b(0)t (z) = b(ϕ(t,0)z) b(k+1)t,s̄ k+1 = λsk+1b(k)t,s̄ k , with λsc = −i {c ◦ ϕ(0,s) ,Qs}(2) ◦ ϕ(s,0) for any polynomial c. Theorem 3.2. Assume H1 and H2. Let m ≥ 2 and b ∈ P≤m (Z) a polynomial. Then introducing • the vector vt ∈ ⊗2 Z such that for all z1, z2 ∈ Z, 〈z1 ⊗ z2,vt〉 = 〈z1,L∗ (t,0)A(t,0)z2〉 , • the operator on P (Z) Λtc(z) = Tr [−2A∗ (t,0)A(t,0)∂z̄∂zc(z)] + 〈vt| .∂2z̄c(z) + ∂2zc(z) . |vt〉 , the formula U(0,t)bWickU(t,0) = ( e ε 2 Λt (b ◦ ϕ(t,0)) )Wick (3.4) holds as an equality of continuous operators from D ( 〈N〉m/2 ) to H. Remark 2. The derivative ∂z̄∂zc(z) is in L (Z) and Tr denotes the trace on the subset of trace class operators of L (Z). Remark 3. For m ≥ 2 the operators λt and Λt send Pm (Z) into Pm−2 (Z). Remark 4. The exponential is intended in the sense e ε 2 Λtb = ⌊m/2⌋∑ k=0 1 k! ( ε 2 Λt )k b for a polynomial b in P≤m (Z). Example 1. To give an idea of the behavior of these formulae we apply them in the simplest (non trivial) possible situation, with Z = C and Qt (z) = ℑ ( z2 ) . As Qt is time-independent the classical evolution equation is autonomous and thus we can write ϕ(t,s) = ϕ(t − s) and i∂tϕ(t)z = ∂z̄Q(ϕ(t)z) = iϕ(t)z. The solution is ϕ(t)z = zcosht + z̄sinht. We can then compute both ˆ t 0 b(1)t,sds and Λt (b ◦ ϕ(t)) . The first one is easily computed as ∂2zQ(z) = −i, ∂ 2 z̄Q(z) = i and, with c = b ◦ ϕ(t), −i {c ◦ ϕ(−s) ,Q(z)}(2) = ( ∂2z + ∂ 2 z̄ ) (c ◦ ϕ(−s)) = [ cosh (−2s) ( ∂2z + ∂ 2 z̄ ) c +2sinh (−2s)∂z̄∂zc] ◦ ϕ(−s) 84 Sébastien Breteaux CUBO 14, 2 (2012) and thus ˆ t 0 b(1)t,sds = ˆ t 0 ( cosh (−2s) ( ∂2z + ∂ 2 z̄ ) + 2sinh (−2s)∂z̄∂z ) ds(b ◦ ϕ(t)) = ( 1 2 sinh (2t) ( ∂2z + ∂ 2 z̄ ) + (1 − cosh (2t))∂z̄∂z ) (b ◦ ϕ(t)) . Now we compute the second one. Since L(t,0)z = L∗ (t,0)z = zcosht and A(t,0)z = A∗ (t,0)z = z̄sinht, we get vt = coshtsinht and then obtain directly Λt = (1 − cosh (2t))∂z̄∂z + 1 2 sinh (2t) ( ∂2z + ∂ 2 z̄ ) . We thus obtain the same result with the two computations for the term of order 1 in ε. Then we can show that ˆ ∆k t b(k)t,s̄ k ds̄k = 1 k! ( Λt )k (b ◦ ϕ(t)) since ˆ ∆kt k∏ j=1 ( 2sinh (−2sj)∂z̄∂z + cosh (−2sj) ( ∂2z + ∂ 2 z̄ )) ds̄k = 1 k! ( (1 − cosh (−2t))∂z̄∂z − 1 2 sinh (−2t) ( ∂2z + ∂ 2 z̄ ))k because d ds [ (1 − cosh (−2s))∂z̄∂z − 1 2 sinh (−2s) ( ∂2z + ∂ 2 z̄ )] = 2sinh (−2s)∂z̄∂z + cosh (−2s) ( ∂2z + ∂ 2 z̄ ) . Remark 5. Since these two formulae will be proven independently and the identification of each term of order k in ε in the expansion of the symbol is clear, we carry out a computation only on the formal level for the convenience of the reader to show the link between the two formulae in the general case. We show (formally) that d ds Λs = λs . Then it is simple to show that ˆ s̄k∈∆kt λskλsk−1 · · ·λs1ds̄k = 1 k! ( Λt )k as operators on P (Z) once the case k = 2 is understood: 2 ˆ s̄2∈∆2t λs2λs1ds̄2 = ˆ t 0 ˆ s1 0 λs2λs1ds2ds1 + ˆ t 0 ˆ s2 0 λs2λs1ds1ds2 = ˆ t 0 Λs1λs1ds1 + ˆ t 0 λs2Λs2ds2 = ( Λt )2 . CUBO 14, 2 (2012) Higher order terms for the quantum evolution of a Wick ... 85 In this computation we have used that Λ0 = 0 as A(0,0) = 0. We first give λs in a more explicit way. As ∂2z̄Q = i |β〉 and ∂2zQ = −i〈β| we first get λc = [ ∂2z ( c ◦ ϕ−1 ) . |β〉 + 〈β| .∂2z̄ ( c ◦ ϕ−1 )] ◦ ϕ with ϕ = ϕ(t,0) and omitting the time dependence everywhere. Then with ϕ = L + A (and thus ϕ−1 = L∗ − A∗) and 〈z1,Az2〉 = 〈z1 ⊗ z2,wA〉 we obtain λc(z) = ∂2zc(z) . ∣∣(L∗∨2 + A∗∨2 ) β 〉 + 〈( L∗∨2 + A∗∨2 ) β ∣∣ .∂2z̄c(z) −2 (〈( IZ ⊗ ∂z̄∂zc(z)∗ L∗ ) β,wA 〉 + 〈wA,(IZ ⊗ ∂z̄∂zc(z)L∗)β〉 ) . Then we compute d ds Λs in several steps. The linear and antilinear parts of the equation i∂sϕ(s,0)z = ∂z̄Qs (ϕ(s,0)z) give ∂sLz = −iαLz + (〈Az| ∨ IZ) |β〉 ∂sAz = −iαAz + (〈Lz| ∨ IZ) |β〉 . We now show that ∂svs = ∣∣(L∗∨2 + A∗∨2 ) β 〉 , ∂s 〈z1 ⊗ z2,vs〉 = ∂s 〈Lz1,Az2〉 = 〈−iαLz1,Az2〉 + 〈β,Az2 ∨ Az1〉 + 〈Lz1,−iαAz2〉 + (〈Lz2| ∨ 〈Lz1|) |β〉 = 〈β,(A ∨ A) (z1 ∨ z2)〉 + 〈(L ∨ L) (z1 ∨ z2) ,β〉 = 〈 z1 ∨ z2, ( L∗∨ 2 + A∗∨ 2 ) β 〉 . And thus ∂s ( ∂2z. |v〉 + 〈v| .∂2z̄ ) = ∂2z. ∣∣(L∗∨2 + A∗∨2 ) β 〉 + 〈( L∗∨2 + A∗∨2 ) β ∣∣ .∂2z̄. We then show that ∂sTr [A ∗A∂z̄∂zc(z)] = 〈β,(IZ ⊗ L∂z̄∂zc(z))wA〉 + 〈wA,(IZ ⊗ ∂z̄∂zc(z)L∗)β〉 . We first observe that Tr [A∗A∂z̄∂zc(z)] = 〈wA,(IZ ⊗ ∂z̄∂zc(z))wA〉. A simple calculation us- ing ∂sAz = −iαAz + (〈Lz| ∨ IZ) |β〉 shows that ∂swA = (−iα ⊗ IZ)wA + (IZ ⊗ L∗)β and this immediately gives the result. 4 Classical evolution of a Wick polynomial under a quadratic evolution The adjoint of a C-antilinear operator is defined in Appendix A. Definition 4.1. A C-antilinear operator A on Z is said of Hilbert-Schmidt class if ‖A‖La 2 (Z) := ‖AA∗‖1/2L1(Z) is finite, where ‖·‖L1(Z) is the usual trace norm for C-linear operators. The set of Hilbert-Schmidt antilinear operators is denoted by La2 (Z). 86 Sébastien Breteaux CUBO 14, 2 (2012) Let X (Z) = L (Z) + La2 (Z) with norm ‖T‖X (Z) = ‖L‖L(Z) + ‖A‖La 2 (Z) for T = L + A, where L and A are respectively C-linear and C-antilinear. The space X (Z) is a Banach algebra. Remark 6. The norm ‖T‖X (Z) is well defined as the decomposition T = L + A is unique (L = 1 2 (T − iTi) and A = 1 2 (T + iTi)). 4.1 Construction of the classical flow without the α term Let β ∈ C0 ( R; Z∨2 ) and Qt = ℑ 〈 βt,z ∨2 〉 . Observe that ∂z̄Q(t) (z) = i(IZ ∨ 〈z|)βt and so (∂z̄Qt)t is a continuous one parameter family of X (Z), so that the theory of ordinary dif- ferential equations in Banach algebras (see for example [21]) asserts that there exists a unique two parameters family ϕ(t2,t1) of elements of X (Z) such that { i∂tϕ(t,0) = ∂z̄Qt ϕ(t,0) ϕ(0,0) = IZ , with ϕ of C1 class in both parameters such that for all r, s and t, ϕ(t,s)ϕ(s,r) = ϕ(t,r) . The classical flow ϕ(t,s) is a symplectomorphism with respect to the symplectic form σ(z1,z2) = ℑ 〈z1,z2〉. It can be checked deriving σ(ϕ(t,s)z1,ϕ(t,s)z2) with respect to t. 4.2 The strongly continuous dynamical system associated with (αt) We first state a proposition which is a direct consequence of Theorem X.70 in [27] in the unitary case. This proposition provides a set of assumptions ensuring the existence of a strongly continuous dynamical system associated with a family (αt)t of self-adjoint operators. Other more general situations can be considered as in [23, 24] for example. Proposition 4.2. Let (αt)t∈R be a family of self-adjoint operators on the Hilbert space Z satisfying the following conditions. (1) The αt have a common domain D (from which it follows by the closed graph theorem that c(t,s) = (αt − i) (αs − i) −1 is bounded). CUBO 14, 2 (2012) Higher order terms for the quantum evolution of a Wick ... 87 (2) For each z ∈ Z, (t − s)−1 c(t,s)z is uniformly strongly continuous and uniformly bounded in s and t for t 6= s lying in any fixed compact interval. (3) For each z ∈ Z, c(t)z = limsրt (t − s)−1 c(t,s)z exists uniformly for t in each compact interval and c(t) is bounded and strongly continuous in t. The approximate propagator uk is defined by uk (t,s) = exp(− (t − s)iαj−1 k ) if j−1 k ≤ s ≤ t ≤ j k and uk (t,r) = uk (t,s)uk (s,r). Then for all s, t in a compact interval and any z ∈ Z, u(t,s)z = lim k→+∞ uk (t,s)z exists uniformly in s and t. Further, if z ∈ D, then u(t,s)z is in D for all s, t and satisfies { i d dt u(t,s)z = αtu(t,s)z u(s,s)z = z . 4.3 Construction of the classical flow with the α term Assume H1 and H2. Let ϕ̂ be the solution of { i∂tϕ̂(t,0) = ∂z̄Q̂t ϕ̂(t,0) ϕ̂(0,0) = IZ , with Q̂t (z) = ℑ 〈 β̂t,z ∨2 〉 , β̂t = uα (t,0) ∗∨2 βt. What we call here the solution of { i∂tϕ(t,0) = ∂z̄Qt ϕ(t,0) ϕ(0,0) = IZ , (4.1) with Qt = 〈z,αtz〉 + ℑ 〈 βt,z ∨2 〉 is ϕ(t,0) = uα (t,0) ◦ ϕ̂(t,0) . Depending on the assumptions on (αt) it will be possible to precise if ϕ solves Equation (4.1) in a usual sense (strongly, weakly, on some dense subset...). With the particular set of assumptions of Theorem 4.2 we get that for all z1 ∈ D and z2 ∈ Z, { i∂t 〈z1,ϕ(t,0)z2〉 = 〈αz1,ϕ(t,0)z2〉 + i〈z1 ∨ ϕ(t,0)z2,β〉 ϕ(0,0) = IZ . 88 Sébastien Breteaux CUBO 14, 2 (2012) 4.4 Composition of a Wick polynomial with the classical evolution The composition of a polynomial with the classical flow defines a time-dependent polynomial. Definition 4.3. We define a norm on P (Z) by ‖b‖P(Z) = ∑ p, q ‖bp,q‖q←p where b = ∑ p, q bp,q is a polynomial with bp,q ∈ Pp,q (Z) and ‖bp,q‖q←p is a shorthand for ‖b̃p,q‖L(∨p Z,∨q Z). For a polynomial b in Pm (Z), we will sometimes write ‖b‖Pm(Z). Proposition 4.4. Let b ∈ Pm (Z) be a polynomial, and ϕ ∈ X (Z). Then b ◦ ϕ ∈ Pm (Z) and we have the estimate ‖b ◦ ϕ‖Pm(Z) ≤ ‖ϕ‖ m X (Z) ‖b‖Pm(Z) . Proof. The proof is essentially the same as in Proposition 2.12 of [1]. 5 Quantum evolution of a Wick polynomial 5.1 Without the α term Definition 5.1. Let β ∈ C0 ( R; Z∨2 ) and Qt (z) = ℑ 〈 βt,z ∨2 〉 . A family U(t,s) of unitary opera- tors on H defined for s, t real is a solution of { i∂tU(t,0) = QWickt ε U(t,0) U(0,0) = IH (5.1) if (1) U(t,s) is strongly continuous in H with respect to s, t with U(s,s) = I, (2) U(t,r) = U(t,s)U(s,r), r ≤ s ≤ t, (3) i d dt U(t,s)y exists for almost every t (depending on s) and is equal to QWickt U(t,s)y, (4) iε d ds U(t,s)y = −U(t,s)QWicks y, y ∈ D (N + 1), 0 ≤ s ≤ t. This definition is made to fit the general framework of Theorems 4.1 and 5.1 of [23]. More precisely we may check the following theorem. Theorem 5.2. Let β ∈ C0 ( R; Z∨2 ) and Qt (z) = ℑ 〈 βt,z ∨2 〉 . Then the quantum flow equation (5.1) associated to the family 1 ε Qt has a unique solution. This solution preserves the sets D(〈N〉k/2) for k ≥ 2. To establish this theorem we will use the following estimates. CUBO 14, 2 (2012) Higher order terms for the quantum evolution of a Wick ... 89 Lemma 5.1. Let β ∈ Z∨2 and Q(z) = ℑ 〈 β,z∨2 〉 . Then, on Hfin, and for k ≥ 1, QWick satisfies the estimates ∥∥QWick/εΨ ∥∥ ≤ 3 2 ‖β‖Z∨2 ‖(N/ε + 1)Ψ‖ (5.2) and ± i [ QWick/ε,(N/ε + 1) k ] ≤ 3k √ 2‖β‖Z∨2 (N/ε + 1) k . (5.3) The second estimate is in the sense of quadratic forms, for all Ψ ∈ Hfin, ± i (〈 1 ε QWickΨ,(N/ε + 1) k Ψ 〉 − 〈 (N/ε + 1) k Ψ, 1 ε QWickΨ 〉) ≤ 3 k √ 2 ‖β‖Z∨2 〈 Ψ,(N/ε + 1) k Ψ 〉 . Proof. The first estimate is a consequence of n + 2 ≤ 2(n + 1) associated to 2i ε QWick ∣∣ Z∨n = √ n(n − 1) 〈β| ∨ I∨n−2 Z − √ (n + 2) (n + 1) |β〉 ∨ I∨n Z . For the second estimate, consider 2i ε 〈 Ψ, [ (1 + N/ε) k ,QWick ] Ψ 〉 . The first term of this commutator is ∑ n (n + 1) k (√ (n + 2) (n + 1) 〈 Ψ(n) ∨ 〈β| ,Ψ(n+2) 〉 − √ n(n − 1) 〈 Ψ(n), |β〉 ∨ Ψ(n−2) 〉) . Then we deduce easily the second term and a reindexation gives the following form for the whole commutator: ∑ n [ (n + 1) k − ((n + 2) + 1) k ]√ (n + 2) (n + 1) × (〈 Ψ(n) ∨ 〈β| ,Ψ(n+2) 〉 + 〈 Ψ(n+2), |β〉 ∨ Ψ(n) 〉) . Newton’s binomial formula and the inequalities ∑k−1 l=0 ( k l ) 2k−l ≤ 3k and (n + 1)l ≤ (n + 1)k−1 yield (n + 1) k − ((n + 2) + 1) k ≤ 3k (n + 1)k−1 . Using also n + 2 ≤ 2(n + 1) to control √ (n + 2) (n + 1) we obtain ±i 〈 Ψ, [ (1 + N/ε) k , QWick ε ] Ψ 〉 ≤ 1 2 ∑ n 3k (n + 1) k−1 √ 2(n + 1) ∥∥∥Ψ(n) ∥∥∥‖β‖Z∨2 ∥∥∥Ψ(n+2) ∥∥∥ . Cauchy-Schwarz’s inequality gives the claimed estimate. Lemma 5.2. Let β ∈ Z∨2 and Q(z) = ℑ 〈 β,z∨2 〉 . Then QWick is essentially self-adjoint on Hfin and its closure is essentially self-adjoint on any other core for N/ε+1. Inequalities (5.2) and (5.3) still hold on D (N/ε + 1). 90 Sébastien Breteaux CUBO 14, 2 (2012) We still denote by QWick this self-adjoint extension. Proof. We apply the commutators Theorem X.37 of [27] with the estimates of Lemma 5.1 for k = 1. Lemma 5.3. If a solution of the quantum flow equation (5.1) exists then it leaves Q((N/ε + 1)k) = D((N/ε + 1)k/2) invariant for any integer k ≥ 2. In the time-independent case the estimate ‖U(t,0)‖L(D((N/ε+1)k/2)) ≤ exp ( 3k √ 2‖β‖ |t| ) holds. Proof. From Lemma 5.2, for any k ≥ 2, D((N/ε + 1)k/2) ⊂ D(QWick). We can adapt the proof of Theorem 2 of [10] to the case of the quantization of a continuous one parameter family of quadratic polynomials with the estimates of Lemma 5.1. Proof of theorem 5.2. We use Theorems 4.1 and 5.1 of [23] with the family of operators iQ(t) Wick /ε (here we directly consider the self-adjoint extension of QWickt /ε). We set Y = D((N/ε + 1) k/2 ). (1) This family is stable in the sense that ‖ ∏k j=1 e −isjQ(tj) Wick/ε‖L(H) ≤ 1 (we actually have an equality here). (2) The space Y is admissible for this family in the sense that for each t, (iQWickt /ε + λ) −1 leaves Y invariant and ∥∥∥ ( iQWickt /ε + λ )−1∥∥∥ L(Y) ≤ ( λ − 3k √ 2‖β‖ )−1 for ℜλ > 3k √ 2‖β‖. This is true because, as we have seen in Lemma 5.3, (e−isQ Wick t /ε)s∈R leaves Y invariant and, thanks to the estimate of the same lemma, we can apply the resolvent formula ( iQWickt /ε + λ )−1 = ˆ +∞ 0 e−λse−isQ Wick t /εds and obtain the desired estimate. (3) Y ⊂ D ( QWickt /ε ) so that QWickt /ε ∈ L (Y,H) for each t, and the map t → QWickt /ε ∈ L (Y,H) is continuous. (4) Y = D((N/ε + 1)k/2) is reflexive. Theorems 4.1 and 5.1 of [23] thus apply and give the existence of an evolution operator. The preservation of the set D((N/ε + 1)k/2) comes from the application of Lemma 5.3 to the solution of the time-dependent problem. To conclude it is then enough to observe that the domains D(〈N〉k/2) and D((N/ε + 1)k/2) are the same and have equivalent norms. CUBO 14, 2 (2012) Higher order terms for the quantum evolution of a Wick ... 91 5.2 With the α term Assume H1 and H2. Let Û be the solution of { i∂tÛ(t,0) = Q̂Wickt ε Û(t,0) Û(0,0) = IH (5.4) with Q̂t (z) = ℑ 〈 β̂t,z ∨2 〉 , β̂t = uα (t,0) ∗∨2 βt. What we call here the solution of { i∂tU(t,0) = QWickt ε U(t,0) U(0,0) = IH (5.5) with Qt = 〈z,αtz〉 + ℑ 〈 βt,z ∨2 〉 is U(t,0) = Γ (uα (t,0)) ◦ Û(t,0) . 6 Removal of the α part Proposition 6.1. Assume H1 and H2. Suppose Theorems 3.1 and 3.2 hold with a null one param- eter family of self-adjoint operators on Z, and β̂t = uα (t,0)∗∨2 βt. We denote with a hat the quantities associated with this solution. Then Theorems 3.1 and 3.2 hold. Proof. For Equation 3.3, we forget during the proof the (t,0) dependency in our notations and write ˆ ∆0t b(0)t,s̄ 0 ds̄0 instead of b(0),t. Then U∗bWickU = Û∗Γ (u∗α)b WickΓ (u)Û = Û∗ (b ◦ uα)Wick Û = ⌊ m2 ⌋∑ k=0 ( ε 2 )k ˆ ∆k t ( b̂ ◦ uα (k)t,s̄k )Wick ds̄k where the b̂(k)t,s̄ k are defined recursively by { b̂(0)t (z) = b ◦ ϕ̂ b̂(k+1)t,s̄ k+1 = λ̂sk+1b̂(k)t,s̄ k with λ̂sc = −i { c ◦ ϕ̂(0,s) ,Q̂s }(2) ◦ ϕ̂(s,0) for any polynomial c. Thus it suffices to prove that b̂ ◦ uα (k)t,s̄k = b(k)t,s̄ k . 92 Sébastien Breteaux CUBO 14, 2 (2012) This is clear for k = 0 as uα ◦ ϕ̂ = ϕ. Then we observe that λ̂sc = −i { c ◦ ϕ̂−1,Q̂ }(2) ◦ ϕ̂ = −i { c ◦ ϕ−1 ◦ uα,Q̂ }(2) ◦ u−1α ◦ ϕ = −i { c ◦ ϕ−1,Q }(2) ◦ ϕ where we used that ∂2z 〈z,αz〉 = 0, ∂2z̄ 〈z,αz〉 = 0 and βt = uα (t,0) ∨2 β̂t. We can thus restrict our proof to the case of a polynomial Qt of the form Qt (z) = ℑ 〈 βt,z ∨2 〉 with βt ∈ C0 ( R; Z∨2 ) and no (αt) term. 7 A Dyson type expansion formula for the Wick symbol of the evolved quantum observable In this section we prove Theorem 3.1. Proof. We first prove that the formula, for c ∈ P≤m (Z), U(0,s) (c ◦ ϕ(0,s))Wick U(s,0) = cWick − iε 2 ˆ s 0 U(0,σ) {c ◦ ϕ(0,σ) ,Qσ}(2)Wick U(σ,0)dσ holds as an equality of continuous operators from D(〈N〉m/2) to H, with 〈N〉 = (N2 + 1)1/2. This is a consequence of the fact that the derivative of the left hand term as a function of s is −iε 2 U(0,s) {c ◦ ϕ(0,s) ,Qs}(2)Wick U(s,0) as it can be seen from the relation i∂σ (c ◦ ϕ(0,σ)) = −∂z (c ◦ ϕ(0,σ)) .∂z̄Qσ + ∂zQσ.∂z̄ (c ◦ ϕ(0,σ)) and Proposition 2.1. Applying the previous formula with c = b(K)t,s̄ K we get recursively U(0,t)bWickU(t,0) = K−1∑ k=0 ( ε 2 )k ˆ s̄k∈∆kt ( b(k)t,s̄ k )Wick ds̄k + ( ε 2 )K ˆ s̄K∈∆K t U(0,sK) ( b(K)t,s̄ K ◦ ϕ(0,sK) )Wick U(sK,0)ds̄ K . This process gives a null remainder as soon as K > m/2 as for K ≤ ⌊m/2⌋, since the polyno- mial b(K)s̄ K is of total order m − 2K. 8 An exponential type expansion formula for the Wick sym- bol of the evolved observable In this section we prove Theorem 3.2. CUBO 14, 2 (2012) Higher order terms for the quantum evolution of a Wick ... 93 8.1 Quantum evolution as a Bogoliubov implementation Some basic facts about symplectomorphisms are recalled in Appendix A. Definition 8.1. A symplectomorphism T is called implementable if and only if there exists a unitary operator U on H , called a Bogoliubov implementer of T, such that ∀ξ ∈ Z, U∗W (ξ)U = W (Tξ) . Proposition 8.2. Assume αt ≡ 0 and H2. Let Qt = ℑ 〈 βt,z ∨2 〉 , ϕ(t,s) the associated classical evolution (see Section 4) and U(t,s) the associated quantum evolution (see Section 5). Then for all t in R, U(t,0) is a Bogoliubov implementer of −iϕ(0,t)i. Remark 7. Note that the symplectomorphism ϕ(0,t) is only R-linear and not C-linear in general, and thus −iϕ(0,t)i 6= ϕ(0,t). Proof. We begin with a formal computation which will be justified further. It suffices to show that iε∂t [U(0,t)W (−iϕ(t,0)iξ)U(t,0)] = 0. Computing this derivative and omitting the time and −iϕ(t,0)iξ dependencies in our notations, we get with U(t,0) = U U∗W { −W∗QWickW + QWick + W∗iε∂tW } U. Then from Proposition 2.10 (iii) in [1], the differential formula of Weyl operators recalled in Propo- sition 8.3 below and with ft = −iϕ(t,0)iξ it suffices to show that Q ( z + iε√ 2 ft ) = Q(z) + iε ( iε 2 ℑ 〈ft,∂tft〉 + i √ 2ℜ 〈∂tft,z〉 ) to get the result. This equality results from the expansion of Q(z) = ℑ 〈 β,z∨2 〉 , recalling that i∂tϕ(t,0)ξ = ∂z̄Q(ϕ(t,0)ξ), and observing that ∂z̄Q(z) = i(〈z| ∨ IZ) |β〉. We now need to clarify the meaning of this computation. It suffices to show that the quantity 〈Φ,U(0,t)W (−iϕ(t,0)iξ)U(t,0)Ψ〉 is constant for Ψ, Φ in D (N + 1). Since this domain is preserved by the operators U(t,s), the Weyl operators are weakly derivable on this domain (see next proposition), and U(t,s) is derivable on this domain, then we get the justification of the previous formal computation. Proposition 8.3. Let z, h be vectors in Z, t be a real parameter and ϕ, ψ be in the domain of Φ(h). Then lim t→0 1 t (〈ϕ, [W (z + th) − W (z)]ψ〉) = 〈 ϕ,W (z) [ iΦ(h) + iε 2 ℑ 〈z,h〉 + ] ψ 〉 = 〈 ϕ, [ iΦ(h) − iε 2 ℑ 〈z,h〉 ] W (z)ψ 〉 . 94 Sébastien Breteaux CUBO 14, 2 (2012) Proof. For the first equality. The Weyl commutation relations give 1 t 〈ϕ, [W(z + th) − W(z)]ψ〉 = 1 t 〈 W(−z)ϕ, [ e iε 2 ℑ〈z,th〉W(th) − IZ ] ψ 〉 = 〈 W(−z)ϕ,e iε 2 ℑ〈z,th〉 1 t (W(th) − IZ)ψ 〉 + 1 t ( e iε 2 ℑ〈z,th〉 − 1 ) 〈W(−z)ϕ,ψ〉 → t→0 〈 ϕ,W(z) [ iΦ(h) + iε 2 ℑ 〈z,h〉 ] ψ 〉 . The convergence of the first term is due to the continuous one parameter group structure of W (th). The other equality is obtained in the same way. 8.2 Action of Bogoliubov transformations on Wick symbols A theorem due to Shale (see [29]) characterizes implementable symplectomorphisms. We quote here a version of this theorem fitting our needs. Theorem 8.4 (Shale, 1962). A symplectomorphism T is implementable if and only if the C-linear part of T∗T − Id is trace class. We can now quote the main result of this part. Theorem 8.5. Let T = L + A with L C-linear and A C-antilinear, be an implementable symplec- tomorphism with a Bogoliubov implementer U preserving D(〈N〉k/2) for any integer k≥ 2, then for any polynomial b in P≤m (Z) with m ≥ 2, U∗bWickU = ( e ε 2 Λ[T] [b(T∗·)] )Wick (8.1) as an equality of continuous operators from D(〈N〉m/2) to H, with 〈N〉 = (N2 + 1)1/2, where • the exponential is a finite expansion whose rank depends on the degree of the polynomial b, • the operator Λ [T] is defined on any polynomial c by Λ [T]c(z) = Tr [−2AA∗∂z̄∂zc(z)] + 〈v| .∂2z̄c(z) + ∂2zc(z) . |v〉 with v ∈ ⊗2 Z the vector such that for all z1, z2 ∈ Z, 〈z1 ⊗ z2,v〉 = 〈z1,LA∗z2〉. In order to prove this result, we use intermediate steps. (1) We prove that U∗bWeylU = b(T∗·)Weyl in finite dimension. CUBO 14, 2 (2012) Higher order terms for the quantum evolution of a Wick ... 95 (2) We use the Fourier transform and the formula bWeyl = 1 (πε/2) d ( b ∗ e− |z|2 ε/2 )Wick to get the result in finite dimension. (3) We extend the result to infinite dimension. 8.3 Action of Bogoliubov transformations on Weyl quantizations of poly- nomials in finite dimension Definition 8.6. In a finite-dimensional Hilbert space Z identified with Cd, the symplectic Fourier transform is defined by Fσ [f] (z) = ˆ Z e 2πiσ(z,z′)f(z′)L(dz′) where L denotes the Lebesgue measure, and f is any Schwartz tempered distribution. We associate with each polynomial b ∈ Pp,q (Z) a Weyl observable by bWeyl = ˆ Z Fσ [b] (z)W ( −i √ 2πz ) L(dz) . (8.2) This formula has a meaning as an equality of quadratic forms on S (Z) since for any Φ, Ψ in S (Z), z 7→ 〈 Φ,W(−i √ 2πz)Ψ 〉 and its derivative are continuous bounded functions and Fσ [b] is made of derivatives of the delta function. Proposition 8.7. Let b ∈ P≤m (Z) with m ≥ 2 be a polynomial on a finite-dimensional Hilbert space Z. Let T be an implementable symplectomorphism with implementation U preserving the domain D(〈N〉m/2). Then U∗bWeylU = b(T∗·)Weyl as a continuous operator from D(〈N〉m/2) to H. Proof. We compute, in the sense of quadratic forms on S (Z), U∗bWeylU = ˆ Fσ [b] (z)W ( − √ 2πTiz ) L(dz) = ˆ Fσ [b] (T∗z)W ( −i √ 2πz ) L(dz) = ˆ Fσ [b(T∗·)] (z)W ( −i √ 2πz ) L(dz) = b(T∗·)Weyl where we made use of the relation Ti = i(T∗) −1 , the volume preservation of T∗ in Z seen as a R-vector space and the property of composition of a symplectic Fourier transform by a symplec- tomorphism (see Appendix C). The boundedness from D(〈N〉m/2) to H is deduced from the facts 96 Sébastien Breteaux CUBO 14, 2 (2012) that the Fourier transform of b involves only derivatives of the delta function of order smaller or equal to m and that a derivation of the Weyl operator gives at worse a field factor which is controlled by 〈N〉1/2. 8.4 Action of Bogoliubov transformations on Wick quantization of poly- nomials in finite dimension Proposition 8.8. Let b ∈ P≤m (Z) with m ≥ 2 be a polynomial on a finite-dimensional Hilbert space Z. Let T be an implementable symplectomorphism with implementation Upreserving the domain D(〈N〉m/2). Then U∗bWickU = ( e ε 2 Λ[T] [b(T∗·)] )Wick , (8.3) as a continuous operator from D(〈N〉m/2) to H, where Λ [T] is defined as in Theorem 8.5. Proof. We search the polynomial c such that U∗bWickU = cWick. In finite dimension for polyno- mials we can use the well known deconvolution formula cWick = ( c ∗ 1 (πε/2) d e |z|2 ε/2 )Weyl . By Proposition 8.7 we boil down to search for a polynomial c such that ( b ∗ 1 (πε/2) d e |z|2 ε/2 ) (T∗·) = c ∗ 1 (πε/2) d e |z|2 ε/2 . Using symplectic Fourier transform (see appendix C) and its properties with respect to convolution, composition with symplectomorphisms and Gaussians, we get Fσc = [Fσb(T∗·)] ×  Fσ   e |z|2 ε/2 (πε/2) d  (T∗·)   ×  Fσ   e − |z|2 ε/2 (πε/2) d     = e π2ε(|T ∗·|2−|·|2) 2 × Fσb(T∗·) . Writing T = L + A with L the C-linear and A the C-antilinear part of T we obtain |T∗z| 2 − |z| 2 = 〈L∗z,L∗z〉 + 〈A∗z,A∗z〉 + 〈L∗z,A∗z〉 + 〈A∗z,L∗z〉 − 〈z,z〉 = 〈z,LL∗z〉 + 〈z,AA∗z〉 + 〈LA∗z,z〉 + 〈z,LA∗z〉 − 〈z,z〉 = 〈z,2AA∗z〉 + 〈 v,z∨ 2 〉 + 〈 z2,v 〉 with v ∈ ⊗2 Z the vector such that for all z1, z2 ∈ Z, 〈z1 ⊗ z2,v〉 = 〈z1,LA∗z2〉. By Fourier transforming again, we get π2Fσ [( |T∗·|2 − |·|2 ) × · ] Fσc = Tr [−2AA∗∂z̄∂zc(z)] + 〈v|∂2z̄c(z) + ∂2zc(z) |v〉 as the C-linear and C-antilinear parts behave differently under Fourier transform (the C-linear part has a minus sign added, see appendix C). We then obtain the claimed result. CUBO 14, 2 (2012) Higher order terms for the quantum evolution of a Wick ... 97 8.5 Extension to infinite dimension on a “cylindrical” class of polyno- mials Theorem 8.9. Let T̂ be symplectomorphism of the form T̂ = ecρ, with c a conjugation and ρ a positive, self-adjoint, Hilbert-Schmidt operator commuting with c. Let (ξj)j∈N a Hilbert basis in which ρ is diagonal. Let πK be the orthogonal projection on the finite-dimensional space ZK = Vect({ξj}j≤K). Then for any polynomial b in Pm (Z) with m ≥ 2 and any integer K Û∗bWickK Û = ( e ε 2 Λ[T̂] [ bK ( T̂∗· )])Wick as continuous operators from D(〈N〉 m 2 ) to H where bK (z) = b(πKz). Proof. We first remark that, with Q(z) = ℑ 〈cρz,z〉, e−iQWick/ε is a Bogoliubov implementer of T̂ as it can be seen using Proposition 8.2 and the Hilbert-Schmidt property of ρ. We define ρL = ρπL, T̂L = T̂πL and the operator QL (z) Wick = ℑ 〈cρLz,z〉Wick. We use the identification H = Γs (ZL)⊗ Γs(Z⊥L ) and observe that on Γs (ZL) ⊗ {ΩZ ⊥ L }, e−iQ Wick/ε = e−iQ Wick L /ε. For K ≤ L we obtain on Γs (ZL) ⊗ {ΩZ ⊥ L } Û∗Lb Wick K ÛL = ( e ε 2 Λ[T̂L] [ bK ( T̂∗L· )])Wick by Proposition 8.8, with ÛL = e −iQWickL /ε. But on this domain it is the same as Û∗bWickK Û = ( e ε 2 Λ[T̂] [ bK ( T̂∗· )])Wick with Û = e−iQ Wick/ε. We thus get an equality on ∪LΓs (ZL), and by continuity of the involved operators from D(〈N〉 m 2 ) to H we get the expected result. We will first show that Formula (8.1) apply in particular to a well chosen class of cylindrical polynomials, and then extend it by density to every polynomial. 8.6 Extension to general polynomials We split the proof of Formula (8.1) for general polynomials into several lemmata and propositions. Lemma 8.1. Let (ξj)j∈N be a Hilbert basis of Z, πm be the orthogonal projector on Zm = Vect({ξj}j≤m). Let b be a polynomial in Pp,q (Z) and define bK = b(πK·). Then (b̃K)K∈N is bounded and b̃ = w − lim j→∞ b̃K . To formulate more clearly some convergence results we need some extra definitions. 98 Sébastien Breteaux CUBO 14, 2 (2012) Definition 8.10. We define the spaces L∨p,q (Z) = L ( Z∨p,Z∨q ) , L∨m = ⊕ p+q=m L∨p,q and L∨≤m = ⊕ m ′≤m L∨m ′ corresponding to Pp,q (Z), Pm (Z) and P≤m (Z). Let b = ∑ p,q bp,q be a polynomial, with bp,q ∈ P (Z). We note b̃ = (b̃p,q) ∈ ⊕ p,q L∨p,q (Z). The norm of b̃ = (b̃p,q) ∈ L∨≤m (Z) is ‖b̃‖L∨ ≤m (Z) = ∑ p,q ‖b̃p,q‖L(∨p Z,∨q Z) . A sequence (b̃K)K∈N of elements of L≤m (Z) converges weakly to b̃ in L≤m (Z) if b̃Kp,q converges weakly to b̃p,q for every p and q as K → +∞. Lemma 8.2. Let T be an operator in X (Z), (bK)K∈N and b be polynomials in Pm (Z) such that (b̃K)K∈N converges weakly to b̃. Then bK (T·) and b(T·) are in Pm (Z) and b̃K (T·) converges weakly to b̃(T·). Lemma 8.3. Let T be an operator in X (Z), (bK)K∈N and b be polynomials in Pm (Z) such that (b̃K)K∈N is bounded and converges weakly to b̃. Then ( ˜e ε 2 Λ[T]bK)K∈N converges weakly to ˜e ε 2 Λ[T]b. Proof. It is enough to show that weak convergence is preserved by the action of Λ [T]. But, for any polynomial b, Λ̃ [T]b = Tr 1 [ (−2A∗A ⊗ IZ∨q−1) b̃ ] + (〈v| ∨ IZ∨q−2) b̃ + b̃(|v〉 ∨ IZ∨p−2) , where Tr1 is the partial trace on the first Z subspace on the left and any direction on the right (so that if b̃ ∈ L∨p,q (Z), then Tr1[(−2A∗A ⊗ IZ∨q−1)b̃] is in L∨p−1,q−1 (Z)). With this formula the preservation of the weak convergence is clear. Proposition 8.11. Let b and (bK)K∈N be Wick polynomials in Pp,q (Z) such that w − lim b̃K = b̃. Then w − lim K (bK − b) Wick 〈N〉− p+q 2 = 0. Proposition 8.12. Let b and (bK)K∈N be Wick polynomials in Pp,q (Z) such that w − lim b̃K = b̃. Let U be a unitary operator on the Fock space H such that, for all k ≥ 2, 〈N〉 k 2 U〈N〉− k 2 is a bounded operator. Then w − lim K U∗ (bK − b) Wick U〈N〉− m ′ 2 = 0 with m′ = max (m,2), m = p + q. Proposition 8.13. Let T be an implementable symplectomorphism with Bogoliubov implementer U. Then for any polynomial b in P≤m (Z), m ≥ 2, U∗bWickU = ( e ε 2 Λ[T] [b(T∗·)] )Wick as continuous operators from D(〈N〉 m 2 ) to H. CUBO 14, 2 (2012) Higher order terms for the quantum evolution of a Wick ... 99 Proof. From the results 8.1 to 8.12 we deduce that Proposition 8.13 holds for symplectomorphisms of the form T̂ = ecρ, with c a conjugation and ρ a positive, self-adjoint, Hilbert-Schmidt operator commuting with c. By Theorem A.8 this assumption on the form of T̂ is not restrictive. Indeed, if T = uT̂ with u unitary and Û is a Bogoliubov implementer for T̂, then U = ÛΓ (u∗) is a Bogoliubov implementer for T and Γ (u) ( e ε 2 Λ[T̂] [ b ( T̂∗· )])Wick Γ (u∗) = ( e ε 2 Λ[T] [b(T∗·)] )Wick . Indeed for any polynomial c, and operator ϕ in X (Z), Γ (ϕ)cWickΓ (ϕ∗) = c(ϕ∗·)Wick and Λ [ T̂ ]k [ b ( T̂∗· )] (u∗·) = Λ [ uT̂ ]k [ b ( T̂∗u∗· )] as can be checked using that L = uL̂ and A = u (L, L̂ and A,  denote respectively the C-linear and C-antilinear parts of T and T̂). This achieves the proof. 8.7 An evolution formula for the Wick symbol We can now prove Theorem 3.2. Proof. We only need to apply propositions 8.2 and 8.13 with T = −iϕ(0,t)i = L∗ (t,0) +A∗ (t,0) (with ϕ(t,0) = L(t,0) + A(t,0)). We remark that for any symplectomorphism T, (−iTi) ∗ = T−1 so that (−iϕ(0,t)i) ∗ = ϕ(t,0) and thus we get the result. 8.8 Estimates We now give estimates for the different terms of the expansion of the symbol. Proposition 8.14. Let T = L + A be an implementable symplectomorphism with L C-linear and A C-antilinear. Then the operator Λ [T] defined on P (Z) by Λ [T]c(z) = Tr [−2AA∗∂z̄∂zc] + 〈v|∂2z̄c(z) + ∂2zc(z) |v〉 , with v ∈ ⊗2 Z the vector such that for all z1, z2 ∈ Z, 〈z1 ⊗ z2,v〉 = 〈z1,LA∗z2〉 is such that, for c in Pm (Z) ‖Λ [T]c‖Pm−2(Z) ≤ 2‖T‖X (Z) ‖A‖La2 (Z) ‖c‖Pm(Z) . Proof. We only have to remark that for any polynomial c in Pp,q (Z) the following estimates hold ‖Tr [B∂z̄∂zc(z)]‖q−1←p−1 ≤ ‖B‖L1(Z) ‖c‖q←p for any trace class operator B, and ∥∥〈v|∂2z̄c(z) ∥∥ q−2←p ≤ ‖v‖∨2 Z ‖c‖q←p and that ‖v‖∨2 Z = ‖LA∗‖La 2 (Z) ≤ ‖L‖L(Z) ‖A‖La 2 (Z). The same estimate holds for ∂ 2 zc(z) |v〉. 100 Sébastien Breteaux CUBO 14, 2 (2012) We apply this result to the expression given in the theorem 3.2. Proposition 8.15. Let (Qt)t be a continuous one parameter family of quadratic polynomials, ϕ the classical flow associated to (Qt)t, and Λ t the operator defined in theorem 3.2. Then, for b in P≤m (Z) ∥∥∥e ε 2 Λt (b ◦ ϕ(t,0)) ∥∥∥ P(Z) ≤ ‖b‖P(Z) ‖ϕ(t,0)‖ m X (Z) m∑ k=0 1 k! ( ε‖ϕ(t,0)‖X (Z) ‖A(t,0)‖La 2 (Z) )k where A is the C-antilinear part of ϕ. Proof. It is enough to combine the propositions 4.4 and 8.14. Remark 8. The norm ‖ϕ(t,0)‖X (Z) is bigger than 1 as for any symplectic transformation T = L + A with L C-linear and A C-antilinear, L∗L = IZ + A ∗A ≥ IZ (see proposition A.4) and thus ‖T‖X (Z) ≥ ‖L‖L(Z) ≥ 1. Appendices A R-linear symplectic transformations In this part we adapt and recall some results of [26] to fit our needs. Let (Z,〈·, ·〉) be a separable Hilbert space over the complex numbers field C. The scalar products is linear with respect to the right variable and antilinear with respect to the left variable. We note AutR (Z) the group of R-linear continuous automorphisms on Z. We define a symplectic form σ on Z by σ(z1,z2) := ℑ 〈z1,z2〉 . Definition A.1. A R-linear automorphism T is a symplectomorphism if it preserves the symplectic form, i.e. if ∀z1,z2 ∈ Z, σ(Tz1,Tz2) = σ(z1,z2) . We note Sp R (Z) the set of symplectic transformations over the Hilbert space Z. It is a subgroup of AutR (Z). Proposition A.2. A R-linear application T : Z → Z can be written as a sum of two applications respectively C-linear and C-antilinear in a unique way : T = T − iTi 2 + T + iTi 2 . Definition A.3. Let A be a (bounded) C-antilinear operator on the Hilbert space Z. We define its adjoint A∗ as the only antilinear operator such that ∀z1,z2 ∈ Z, 〈z1,Az2〉 = 〈z2,A∗z1〉 . CUBO 14, 2 (2012) Higher order terms for the quantum evolution of a Wick ... 101 Let T = L + A = Z → Z be a R-linear application with L C-linear and A C-antilinear. The adjoint T∗ of T is defined by T∗ = L∗ + A∗. Proposition A.4. Let T = L + A be a R-linear automorphism with L C-linear and A C-antilinear, then the following conditions are equivalent. (1) L + A is a symplectomorphism. (2) (L∗ − A∗) (L + A) = IZ. (3) (L∗ + A∗) (L − A) = IZ. (4) L∗L − A∗A = IZ and L ∗A = A∗L. (5) L∗ − A∗ is a symplectomorphism. (6) L − A is a symplectomorphism. (7) LL∗ − AA∗ = IZ and A ∗L = L∗A. Proof. (1) ⇔ (2) Let T = L + A a symplectomorphism, for all z1,z2 ∈ Z, σ(z1,z2) = ℑ 〈z1,z2〉 = ℑ 〈(L + A)z1,Tz2〉 = ℑ ( 〈z1,L∗Tz2〉 + 〈z1,A∗Tz2〉 ) = ℑ 〈z1,(L∗ − A∗)Tz2〉 . Replacing z1 by iz1 we get the same relation with a real part instead of an imaginary part and finally 〈z1, [(L∗ − A∗) (L + A) − IZ]z2〉 = 0 and this in turn implies (L∗ − A∗) (L + A) = IZ. We can reverse the order of these calculations in order to obtain the first equivalence. (2) ⇔ (3) The C-linearity and antilinearity properties of L and A give (L∗ − A∗) (L + A)i = i(L∗ + A∗) (L − A) so that we get the equivalent condition (3). ((2) and (3)) ⇔ (4) The sum and the difference of the equations of (2) and (3) give (4) and the sum and difference of the equations in (4) give (2) and (3). (1) ⇔ (5) From (1) and (3) we know that the inverse of a symplectomorphism T = L + A is T−1 = L∗−A∗ which is necessarily a symplectomorphism too, and thus (1) ⇒ (5). We get (5) ⇒ (1) exchanging T and T−1. (1) ⇔ (6) ⇔ (7) is easily deduced from the previous equivalences. 102 Sébastien Breteaux CUBO 14, 2 (2012) Proposition A.5. Let T = L + A be a symplectomorphism with L C-linear and A C-antilinear, then L is invertible. Proof. From Proposition A.4 we get L∗L = IZ + A ∗A ≥ IZ and LL∗ = IZ + AA∗ ≥ IZ and thus L and L∗ are one to one. As L∗ is one to one we get RanL = (KerL∗) ⊥ = {0} ⊥ = Z. It is now enough to show that the range of L is closed. Pick a vector y ∈ Z, there is a sequence (xn) ∈ ZN such that Lxn → y. The relation L∗L ≥ IZ gives |Lxm − Lxn| ≥ |xn − xm|. The left hand part of the inequality goes to 0 for m,n → ∞, so that (xn) is a Cauchy sequence and thus converges to a limit x. By continuity of L, Lx = y and L is indeed one to one. Definition A.6. An application c from Z to Z is a conjugation if and only if it satisfies the following conditions. (1) c is R-linear. (2) c2 = IZ. (3) For all z1, z2 in Z, 〈cz1,z2〉 = 〈cz2,z1〉. Remark 9. It follows from the third condition in this definition that a conjugation is antilinear. One may define different conjugations on the same Hilbert space over C (even for a one dimensional Hilbert space). As an example one can consider a Hilbert basis (ej) and define the application c : ∑ j αjej 7→ ∑ j αjej. Definition A.7. Let c be a conjugation on the Hilbert space Z. The real and imaginary parts of a vector z ∈ Z (with respect to the conjugation c) are defined as ℜz := z + cz 2 and ℑz := z − cz 2i . They verify z = ℜz + iℑz. The space Ec R := ℜZ = ℑZ is a subspace of Z as R-vector space, 〈·, ·〉 restricted to Ec R is a real scalar product and E = Ec R ⊕ iEc R . Let f be a R-linear application on Z, then we can define the applications from Ec R to itself α : z 7→ ℜf(z) , γ : z 7→ ℜf(iz) , β : z 7→ ℑf(z) , δ : z 7→ ℑf(iz) . Then, if a, b ∈ Ec R , then f(a + ib) = α(a) + iβ(a) + γ(ib) + iδ(ib), and f can be represented as an application on Ec R × Ec R by the matrix ( α γ β δ ) . The following relations hold with the above sign if f is C-linear and with the below sign if f is C-antilinear: β = ∓γ and α = ±δ and f∗ is represented by the matrix ( αT ∓βT βT ±αT ) . CUBO 14, 2 (2012) Higher order terms for the quantum evolution of a Wick ... 103 We want to show a reduction result for the symplectomorphisms in the spirit of the polar decomposition, in the case of an implementable symplectomorphism (see Definition 8.1 and The- orem 8.4). Theorem A.8. Let T be an implementable symplectomorphism. Then T = uecρ where • u is a unitary operator, • c is a conjugation, • ρ is a Hilbert-Schmidt, self-adjoint, non-negative operator commuting with c. Remark 10. The operator u is the unitary operator of the polar decomposition L = u |L| of the C-linear part of T. The conjugation c is a specific conjugation associated with L and will be constructed during the proof and ρ = arg cos |L|. Proof. Let us write T = L + A with L C-linear and A C-antilinear. With L = u |L| the polar decomposition of L we get T = u(|L| + u∗A) so that it is enough to show the two next lemmas. Lemma A.1. Let (E,〈·, ·〉) be a finite-dimensional Hilbert space over C. Let f : E → E be a C-antilinear application such that ff∗ = IE and f = f ∗ . Then there exists an orthonormal basis (uj) of E such that ∀j, f(uj) = uj . Proof. Let us consider an arbitrary conjugation c0 on E and the ( α β β −α ) “matrix” of f (as a R-linear operator) on E = E c0 R ⊕ iEc0 R identified with E c0 R × Ec0 R . The matrix associated to f∗ is ( αT βT βT −αT ) so that the relation f = f∗ gives α = αT and β = βT . From ff∗ = IE we deduce α 2 + β2 = Id and αβ = βα. We can thus diagonalize simultaneously α and β, and so in a convenient basis of E c0 R the matrix of f is of the form   ... 0 ... 0 λαj λ β j 0 ... 0 ... ... 0 ... 0 λ β j −λ α j 0 ... 0 ...   . 104 Sébastien Breteaux CUBO 14, 2 (2012) We can thus confine ourself to the case of a space E of complex dimension 1 and of f with a matrix of the form ( α β β −α ) with α and β real numbers. We search a normalized vector z = ( cos θ sin θ ) = ( x y ) and a real λ such that f(z) = λz, i.e. λ( x y ) = ( αx+βy −αy+βx ) = ( x y −y x ) ( α β ) = √ α2 + β2 ( cos θ sin θ − sin θ cos θ )( cos φ sin φ ) = √ α2 + β2 ( cos(φ−θ) sin(φ−θ) ) so that if we choose θ such that φ − θ = θ we get the desired result with λ = √ α2 + β2. Finally, from ff∗ = IE we deduce that λ = 1 and the result follows. Lemma A.2. Let T = L+A be an implementable symplectomorphism with L C-linear self-adjoint and positive, A C-antilinear. Then L and A commute, there exist a conjugation c commuting with L and A such that Ac is self-adjoint and non-negative and T = ecρ with ρ = arg coshL = arg sinh (Ac) a Hilbert-Schmidt, non-negative and self-adjoint operator com- muting with c. Proof. As AA∗ ∈ L1 (Z), AA∗ = ∑ j λ 2 j |ej〉 〈ej|, with λj ∈ R and ∑ j λ 2 j < ∞, from L 2 = IZ +AA ∗ we deduce L2 = ∑ j µ 2 j |ej〉 〈ej| with µj = √ 1 + λ2j and thus L = ∑ j µj |ej〉 〈ej|. From the equivalent characterizations of a symplectomorphism we get L2 − AA∗ = IZ and L 2 − A∗A = IZ multiplying the first equality on the right and the second on the left by A and computing the difference we get [ L2,A ] = 0. As L is self-adjoint and positive one can use the functional calculus and L = √ L2 to obtain [L,A] = 0. From [L,A] = 0, L = L∗ and the characterizations of a symplectomorphism, we also get AL = LA = L∗A = A∗L so that (A − A∗)L = 0 and from the invertibility of L one deduces A = A∗. The proper subspaces associated with L and ker (L − µIZ), are thus stable by the action of A (and finite-dimensional). We also remark that on ker (L − µIZ), AA ∗ = L2 − IZ = (µ 2 − 1)IZ, so that two cases are possible: µ = 1, thenA = 0 or µ > 1, then 1√ µ2 − 1 A 1√ µ2 − 1 A∗ = IZ . We apply Lemma A.1 to the C-antilinear applications induced by the applications A/ √ µ2 − 1 on the Hilbert spaces ker (L − µIZ). This provides us with a Hilbert basis (ej) of Z which diago- nalizes both L and A. We can also define a conjugation c (∑ j αjej ) = ∑ j αjej. This conjugation commutes with L and A, and Ac is clearly a non-negative self-adjoint operator and so is necessar- ily √ AA∗. We finally get for every vector ej of the basis the relations Lej = µjej and Aej = λjej CUBO 14, 2 (2012) Higher order terms for the quantum evolution of a Wick ... 105 with µ2j − λ 2 j = 1, and thus one can define ρj = arg coshµj (ρj = arg sinhλj as λj ≥ 0) and so we can define ρ = arg coshL = arg sinhAc so that T = ecρ. B Relations between Weyl and Wick symbols in finite di- mension We want to use the relation between the Weyl and Wick symbols associated to a same Wick polynomial in finite dimension, working with Z = Cr we have b = 1 (πε/2) r b̆ ∗ e − |z|2 ε/2 where b is the Wick symbol and b̆ is the Weyl symbol and bWick = b̆Weyl. We want to get rid of the convolution and for this we use the Fourier transform Ff(x′) = 1 (2π) r ˆ R2r e−ix.x ′ f(x)dx where x,x′ ∈ R2r ∼= Cr. The inverse Fourier transform is then F−1f(x) = 1 (2π) r ˆ R2r eix.x ′ f(x′)dx′ . We can use the formulae F (f ∗ g) = (2π)r Ff.Fg F [ e−α |x|2 2 ] (x′) = 1 αr e− |x ′| 2 2α F−1 (x × ·) F = Dx . We then obtain with m = 2n Fb(z′) = (2π) r (πε/2) r F [ e − |z|2 ε/2 ] Fb̆(z′) = ( 4 ε )r ( ε 4 )r e − ε 8 |z ′| 2 Fb̆(z′) = e − ε 8 |z ′| 2 Fb̆(z′) and b = F−1e− ε 8 |z ′| 2 Fb̆ = e − ε 8 F−1|z′| 2F b̆ = e ε 2 ∂z.∂z̄b̆ 106 Sébastien Breteaux CUBO 14, 2 (2012) using the fact that F−1 |z′|2 F = D2(x,ξ) = −4 × 1 2 (∂x − i∂ξ) . 1 2 (∂x + i∂ξ) = −4∂z.∂z̄ . It is clear that if b̆ is a polynomial in P≤m (Z), then b is in this class of polynomials, as we can see deriving the convolution product. We want to show that the map P≤m (Z) → P≤m (Z) b̆ 7→ b = 1 (πε/2) n b̆ ∗ e − |z|2 ε/2 is bijective. As the dimension of Z is finite, the dimension of P≤m (Z) is finite and it is enough to show that this map is one to one. For this we want to justify that on the part of main degree this application is the identity. This is obvious from the following facts: • ∂qz̄∂ p zb = 1 (πε/2)r ∂ q z̄∂ p zb̆ ∗ e− |z|2 ε/2 • this application is the identity on the constants. Thus we can also consider the reverse application that we will improperly note b̆ = e− ε 2 ∂z.∂z̄b. C Symplectic Fourier transform Let us then consider the symplectic Fourier transform on L2 ( Cd; C ) ≡ L2 ( R2d ) with z = x + iy, defined by Fσ (f) (z) = ˆ e i2πσ(z,z′)f(z′)L(dz′) with σ(z,z′) = ℑ 〈z,z′〉 = ℑ [〈x,x′〉 + 〈y,y′〉 + i〈x,y′〉 − i〈y,x′〉] and L denotes the Lebesgue measure. We list here some properties of the symplectic Fourier transform. (1) Inverse. (Fσ)−1 = Fσ (2) Convolution. Fσ (f ∗ g) = Fσf.Fσg (3) Composition with a symplectic transformation. Let T be a symplectomorphism, then Fσ [f(T·)] (z) = Fσ [f] (Tz) . CUBO 14, 2 (2012) Higher order terms for the quantum evolution of a Wick ... 107 (4) Gaussians. For a > 0, Fσ [ e−a|·| 2 ] (z) = ( π a )d e−π 2 |z|2/a . (5) Derivation. We consider the derivations ∂z = 1 2 (∂x − i∂y) and ∂z̄ = 1 2 (∂x + i∂y) then − 1 π ∂z.z0 = Fσ (z̄.z0×) Fσ and 1 π z̄0.∂z̄ = Fσ (z̄0.z×) Fσ . 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Introduction Wick calculus with polynomial observables Definitions Some examples of Wick quantizations Calculus Main results and a simple example Classical evolution of a Wick polynomial under a quadratic evolution Construction of the classical flow without the term The strongly continuous dynamical system associated with (t) Construction of the classical flow with the term Composition of a Wick polynomial with the classical evolution Quantum evolution of a Wick polynomial Without the term With the term Removal of the part A Dyson type expansion formula for the Wick symbol of the evolved quantum observable An exponential type expansion formula for the Wick symbol of the evolved observable Quantum evolution as a Bogoliubov implementation Action of Bogoliubov transformations on Wick symbols Action of Bogoliubov transformations on Weyl quantizations of polynomials in finite dimension Action of Bogoliubov transformations on Wick quantization of polynomials in finite dimension Extension to infinite dimension on a ``cylindrical'' class of polynomials Extension to general polynomials An evolution formula for the Wick symbol Estimates R-linear symplectic transformations Relations between Weyl and Wick symbols in finite dimension Symplectic Fourier transform