Higher order terms for the quantum evolution of a Wick observable within the Hepp method
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 which makes sense within the infinite-dimensional setting and fits with finite-dimensional formulae.
Mathematics subject classification (2000): 81R30, 35Q40, 81S10, 81S30.
Keywords: mean field limit, semiclassical limit, coherent states, squeezed states.
In this article we derive two expansions with respect to a small parameter of quantum evolved Wick observables under a time-dependent quadratic Hamiltonian.
The Hepp method was introduced in  and then extended in [11, 12] 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 , where the number of particles is of order .
Remember that the mean field dynamics is obtained as a classical Hamiltonian dynamics which governs the evolution of the center 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 .
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 , see the works of Ginibre and Velo [13, 14]. In the finite-dimensional case, entering into the semiclassical theory, accurate results have been given by Combescure, Ralston and Robert in . For the mean field infinite-dimensional setting some results have been proved in [15, 10, 23] with a different approach.
We stick here with the Hepp method with the presentation of  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  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 :
In the infinite-dimensional framework the quantization of a linear symplectic transformation (a Bogoliubov transformation) requires some care. Useful references on this subject are  and . Its realization in the Fock space relies on a Hilbert-Schmidt condition on the antilinear part connected with the Shale theorem (see  and [21, 7, 5]).
These things are well known but have to be considered accurately while writing complete expansions.
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  or , 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
We recall some definitions and results about Wick quantization. More details can be found in .
In this paper denotes a separable Hilbert space over , the field of complex numbers. It is also a symplectic space with respect to the symplectic form . 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 the symmetrization operator on (the completion for the natural Hilbert scalar product of the algebraic tensor product ) defined by
where the are vectors in and denotes the set of the permutations of . We will use the notation for , and for when the terms of this product are equal to . We call monomial of order a complex-valued application defined on of the form
with where (or ) denotes the Hilbert completion of the -fold symmetric tensor product, and for two Banach spaces and , the space of continuous linear applications from to is denoted by . We then write . The total order of is the integer . The finite linear combinations of monomials are called polynomials. The set of all polynomials of this type is denoted by . Subsets of particular interest of are and , the finite linear combinations of monomials of total order equal to and not greater than .
The Hilbert space
is called the symmetric Fock space associated with , where tensor products and sum completions are made with respect to the natural Hilbert scalar products inherited from . We also consider the dense subspace of of states with a finite number of particles
where the tensor products are completed but the sum is algebraic.
The Wick quantization of a monomial is the operator defined on by its action on as an element of ,
where denotes the identity map on the space and for , . 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 and for any given , the operator
is an element of . 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 of two polynomials , , by
since, for any polynomial , is a -form (on ) and is a -vector.
The product denoted by a dot in the definition of the Poisson bracket is a -bilinear duality-product between -forms and -vectors. As an example consider the polynomials
The Poisson bracket of order of and is
Some examples of Wick quantizations
Here is a quick review of the notations used for some useful examples of Wick quantization. A vector of is denoted by , is a bounded operator and is the variable of the polynomials. In the next table, the first column describes the polynomial and the second the corresponding Wick quantization (as an operator on ).
The operator is the usual second quantization of an operator restricted to multiplied by a factor . If we obtain the usual number operator multiplied by a factor . The operators , and are the usual annihilation, creation and field operators of quantum field theory with an additional factor. The real and imaginary parts of a complex number are denoted by and . The field operators are essentially self-adjoint and this enables us to define the (-dependent) Weyl operators
Here are some calculation rules for Wick quantizations of polynomials in . The proofs can be found in .
For every polynomial ,
in for any ,
is closable and the domain of the closure contains
(we still denote by the closure of ),
on (where the bar denotes the usual conjugation on complex numbers),
for any in , holds on where .
3 Main results and a simple example
Our two hypotheses are:
Let be a one parameter family of self-adjoint operators on defining a strongly continuous dynamical system .
Assume H1 and additionally that the dynamical system preserves a dense set such that, for any , belongs to .
Let be in , ( defines a -antilinear Hilbert-Schmidt operator by ).
With H1’ and H2, the classical flow associated with of quadratic polynomials is the solution to the equation
where , 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 . Details about this point are given in Section 4. Actually is a family of symplectomorphisms of which are naturally decomposed into their -linear and -antilinear parts:
See Appendix A for more details about symplectomorphisms and this decomposition.
Similarly, the quantum flow associated with is the solution of
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 observable , , under the quantum flow, that is to say the quantity . (We use the usual notation .)
Assume H1 and H2. Let be a polynomial. Then, for any time , the formula
holds as an equality of continuous operators from to , where
the polynomials are defined recursively by
with for any polynomial c.
Assume H1 and H2. Let and a polynomial. Then introducing
the vector such that for all ,
the operator on
holds as an equality of continuous operators from to .
The derivative is in and Tr denotes the trace on the subset of trace class operators of .
For the operators and send into .
The exponential is intended in the sense
for a polynomial in .
To give an idea of the behavior of these formulae we apply them in the simplest (non trivial) possible situation, with and . As is time-independent the classical evolution equation is autonomous and thus we can write and . The solution is . We can then compute both
The first one is easily computed as , and, with ,
Now we compute the second one. Since and , we get and then obtain directly
We thus obtain the same result with the two computations for the term of order 1 in .
Then we can show that
Since these two formulae will be proven independently and the identification of each term of order in 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
Then it is simple to show that
as operators on once the case is understood:
In this computation we have used that as .
We first give in a more explicit way. As and we first get
with and omitting the time dependence everywhere. Then with (and thus ) and we obtain
Then we compute in several parts. The linear and antilinear parts of the equation give
We now show that ,
And thus .
We then show that
We first observe that . A simple calculation using shows that and this immediately gives the result.
4 Classical evolution of a Wick polynomial under a quadratic evolution
The adjoint of a -antilinear operator is defined in Appendix A.
A -antilinear operator on is said of Hilbert-Schmidt class if is finite, where is the usual trace norm for -linear operators. The set of Hilbert-Schmidt antilinear operators is denoted by .
Let with norm
for , where and are respectively -linear and -antilinear. The space is a Banach algebra.
The norm is well defined as the decomposition is unique ( and ).
Construction of the classical flow without the term
Let and . Observe that and so is a continous one parameter family of , so that the theory of ordinary differential equations in Banach algebras (see for example ) asserts that there exists a unique two parameters family of elements of such that
with of class in both parameters such that for all , and ,
The classical flow is a symplectomorphism with respect to the symplectic form . It can be checked deriving
with respect to .
The strongly continuous dynamical system associated with
We first state a proposition which is a direct consequence of Theorem X.70 in  in the unitary case. This proposition provides a set of assumptions ensuring the existence of a strongly continuous dynamical system associated with a family of self-adjoint operators. Other more general situations can be considered as in [19, 20] for example.
Let be a family of self-adjoint operators on the Hilbert space satisfying the following conditions.
The have a common domain (from which it follows by the closed graph theorem that is bounded).
For each , is uniformly strongly continuous and uniformly bounded in and for lying in any fixed compact interval.
For each , exists uniformly for in each compact interval and is bounded and strongly continuous in .
The approximate propagator is defined by if and .
Then for all , in a compact interval and any ,
exists uniformly in and . Further, if , then is in for all and satisfies
Construction of the classical flow with the term
Assume H1 and H2. Let be the solution of
with , . What we call here the solution of
Depending on the assumptions on it will be possible to precise if sloves Equation (4.1) in a usual sense (strongly, weakly, on some dense subset…).
With the particular set of assumptions of Theorem 4.3 we get that for all and ,
Composition of a Wick polynomial with the classical evolution
The composition of a polynomial with the classical flow defines a time-dependent polynomial.
We define a norm on by
where is a polynomial with and is a shorthand for . For a polynomial in , we will sometimes write .
Let be a polynomial, and . Then and we have the estimate
The proof is essentially the same as in Proposition 2.12 of . ∎
5 Quantum evolution of a Wick polynomial
Without the term
Let and . A family of unitary operators on defined for real is a solution of
is strongly continuous in with respect to , with ,
exists for almost every (depending on ) and is equal to ,
, , .
This definition is made to fit the general framework of Theorems 4.1 and 5.1 of . More precisely we may check the following theorem.
Let and .
Then the quantum flow equation (5.1) associated to the family has a unique solution. This solution preserves the sets for .
To establish this theorem we will use the following estimates.
Let and . Then, on , and for , satisfies the estimates
The second estimate is in the sense of quadratic forms, for all ,
The first estimate is a consequence of associated to
For the second estimate, consider . The first term of this commutator is
Then we deduce easily the second term and a reindexation gives the following form for the whole commutator:
Newton’s binomial formula and the inequalities and yield
Using also to control we obtain
Cauchy-Schwarz’s inequality gives the claimed estimate. ∎
We still denote by this self-adjoint extension.
If a solution of the quantum flow equation (5.1) exists then it leaves invariant for any integer .
In the time-independent case the estimate
Proof of theorem 5.2.
We use Theorems 4.1 and 5.1 of  with the family of operators (here we directly consider the self-adjoint extension of ). We set .
This family is stable in the sense that (we actually have an equality here).
The space is admissible for this family in the sense that for each , leaves invariant and
This is true because, as we have seen in Lemma 5.5, leaves invariant and, thanks to the estimate of the same lemma, we can apply the resolvent formula