*2.3. Open System*

We shall discuss how to describe the dynamics of open systems. In the context of quantum statistical mechanics, open systems are a subject that has been discussed for a long time. Open systems are also fundamental in quantum field theory, and are closely related to scattering theory. In particular, it is a necessary description of the dynamics in the paper concerning the DP as a typical example of off-shell quantum fields. This is because the DP phenomena are known to involve the process of generation by incident light and annihilation that changes to scattered light. On the other hand, it is essential that the quantum field considered here is a quantum system with an infinite degree of freedom system, and we should pay attention to the description of its dynamics (see Section 4 for details). In the following, we introduce the mathematical concepts necessary to describe the dynamics of open systems.

The discussion below is based on the understanding that closed systems are a special case of open systems. We consider a quantum system **S** described by a C∗-algebra X . Every time evolution of **S** as a closed system is described by an automorphism of X . Furthermore, when the time *t* is parametrized by R, the time evolution of **S** as a closed system is described by a strongly continuous automorphism group *α* : R *t* → *αt* ∈ *Aut*(X ) satisfying *α*0 = id X , *αs* ◦ *αt* = *αs*+*t* and *α*−*t* = *α*<sup>−</sup><sup>1</sup> *t* for all *s*, *t* ∈ R. In contrast to a closed system, the time evolution of an open system is described by a completely positive map *T* : X→X . The complete positivity of maps between C∗-algebras is defined as follows:

**Definition 2** (Complete positivity [24–27])**.** *Let* C *and* D *be C*∗*-algebras. A linear map T* : C → D *is said to be completely positive (CP) if*

$$\sum\_{i,j=1}^{n} D\_i^\* T(\mathbf{C}\_i^\* \mathbf{C}\_j) D\_j \ge 0 \tag{5}$$

*for all n* ∈ N*, C*1, ··· , *Cn* ∈ C *and D*1, ··· , *Dn* ∈ D*.*

It is known that a CP map is positive, but the converse is not true. Every homomorphism of a C∗-algebra C into a C∗-algebra D is CP. In particular, all automorphisms of a C∗-algebra C are CP. For every C∗-algebra C and *n* ∈ N, *Mn*(C) denotes the C<sup>∗</sup>- algebra of square matrices of order *n* whose entries are elements of C. For every linear map *T* : C→D and *n* ∈ N, a linear map *T*(*n*) : *Mn*(C) → *Mn*(D) is defined by *T*(*n*)(*C*)=( *T*(*Cij*)) for all *C* = ( *Cij*) ∈ *Mn*(C). A linear map *T* : C→D is said to be *n*-positive if *T*(*n*) : *Mn*(C) → *Mn*(D) is positive. A linear map *T* : C→D is CP if and only if it is *n*-positive for all *n* ∈ N. The dual map *T*∗ : D∗ → C∗ of *T* : C→D is defined by

$$(T^\*\!\!\!\!\!\!\/)(\mathbb{C}) = \mathcal{q}(T(\mathbb{C})) \tag{6}$$

for all *ϕ* ∈ D∗ and *C* ∈ C. *T* is CP if and only if the linear map D∗ *ϕ* → ∑*n i*,*j*=1 *CiT*<sup>∗</sup>(*DiϕD*<sup>∗</sup> *j* ) *C*∗ *j* ∈ C∗ is positive for all *n* ∈ N, *C*1, ··· , *Cn* ∈ C and *D*1, ··· , *Dn* ∈ D. Here, for every *A*, *B* ∈ D and *ϕ* ∈ D∗, *Aϕ*, *ϕB*, *AϕB* ∈ D∗ are defined by

$$(A\varphi)(D) = \varphi(DA),\tag{7}$$

$$(\varphi B)(E) = \varphi(BE),\tag{8}$$

$$(A\!\!\!\!\!\!\/)(F) = \!\!\!\!\!\/(BFA),$$

respectively, for all *D*, *E*, *F* ∈ D. The following structure theorem for normal CP maps defined on *B*(H) is well-known.

**Theorem 1.** *Let* H *be a separable Hilbert space. Let T be a normal CP map on <sup>B</sup>*(H)*.* (1) *There exist a separable Hilbert space* K*, an element ξ of* K*, a positive operator R on* K*, and a unitary operator U on* H⊗K *such that*

*T*(*X*) = TrK[*U*<sup>∗</sup>(*<sup>X</sup>* ⊗ *R*)*U*(1 ⊗ |*ξξ*|)] (10)

*for all X* ∈ *<sup>B</sup>*(H)*.*

(2) *There exists a family* {*Ki*}<sup>∞</sup>*i*=<sup>1</sup> *of bounded operators on* H *such that*

$$T(X) = \sum\_{i=1}^{\infty} K\_i^\* X K\_i \tag{11}$$

*for all X* ∈ *<sup>B</sup>*(H)*.*

The proof of this theorem is given in Appendix B. The dynamics of open systems in the Heisenberg picture are described by a quantum stochastic process in the sense of Accardi– Frigerio–Lewis [28,29]. Following their study, measurement theory in the Heisenberg picture is formulated in [20].
