EXTENDED WEAK COUPLING LIMIT Jan Derezi´ nski Based on joint work with Wojciech De Roeck

2. Various levels of description used in physics • More exact fundamental description; • More approximate effective description. One of the aims of theoretical and mathematical physics is to justify effective models as limiting cases of more fundamental theories.

3. Small quantum system weakly interacting with a large reservoir. We are interested in a class of dynamics generated by a Hamiltonian (self-adjoint operator) H λ of the form Hamiltonian of the small system + Hamiltonian of the large reservoir + λ times interaction. There are a number of varieties of such Hamiltonians used in quantum physics and they go under various names. We use the name Pauli-Fierz Hamiltonians.

4. Reduced weak coupling limit (Pauli, van Hove,...,Davies) • Reduce the dynamics to the small system. • Consider weak coupling λ → 0 . t • Rescale time as λ 2 . • Subtract the dynamics of the small system. In the limit one obtains a dynamics given by a completely positive Markov semigroup. It is an irre- versible non-Hamiltonian dynamics.

5. Extended weak coupling limit (Accardi-Frigerio-Lu, D.-De-Roeck) Known also as stochastic limit. • Consider weak coupling λ → 0 . t • Rescale time as λ 2 . • Rescale the reservoir energy by the factor of λ 2 around the Bohr frequencies. • Subtract the dynamics of the small system. In the limit one obtains a (reversible) quantum Langevin dynamics, which gives a dilation of the completely pos- itive semigroup obtained in the reduced weak coupling limit.

PLAN OF THE MINICOURSE 1. DILATIONS OF CONTRACTIVE SEMIGROUPS 2. WEAK COUPLING LIMIT FOR FRIEDRICHS OPERATORS 3. COMPLETELY POSITIVE MAPS 4. COMPLETELY POSITIVE SEMIGROUPS 5. PAULI FIERZ OPERATORS 6. LANGEVIN DYNAMICS OF MARKOV SEMIGROUPS 7. WEAK COUPLING LIMIT FOR PAULI-FIERZ OPERATORS 8. CANONICAL COMMUTATION RELATIONS 9. REPRESENTATIONS OF THE CCR IN FOCK SPACES 10. SMALL SYSTEM IN CONTACT WITH BOSE GAS

7. DILATIONS OF CONTRACTIVE SEMIGROUPS

8. Dilations of contractive semigroups Let K be a Hilbert space and e − i t Υ a contractive semi- group. This implies that iΥ is dissipative: − iΥ + iΥ ∗ ≤ 0 . Let Z be a Hilbert space containing K , I K the embed- ding of K in Z and e − i tZ a unitary group on Z . We say that ( Z , I K , e − i tZ ) is a dilation of e − i t Υ iff K e − i tZ I K = e − i t Υ , I ∗ t ≥ 0 . This clearly implies K e − i tZ I K = e − i t Υ ∗ , I ∗ t ≤ 0 . We say that the dilation is minimal if { e − i tZ K : t ∈ R } is total in Z .

9. Standard construction of a dilation I We define the vector space ˜ F of functions f from R to K , such that { s ∈ R | f ( s ) � = 0 } is a finite set. We equip ˜ F with a bilinear form ( f ( s ) | e − iΥ | t − s | f ′ ( t )) K + ( f ( s ) | e iΥ ∗ | t − s | f ′ ( t )) K ( f | f ′ ) := � � t ≥ s t<s One checks that the form ( ·|· ) is positive definite. Let N denote the subspace of f , for which ( f | f ) = 0 . Let F denote the completion of the pre-Hilbert space ˜ F � N .

10. Standard construction of a dilation II For u ∈ K define ˜ Pu ( s ) := δ s, 0 u , where δ s, 0 is Kronecker’s delta. Then Pu := [ Pu ] ∈ F defines an isometric em- bedding of P : K → F . Define now ˜ W t f ( s ) = f ( s − t ) . W t is a one-parameter group on ˜ ˜ F that preserves the form ( ·|· ) . Therefore, it defines a one-parameter uni- tary group W t on F . W t dilates the semigroup e − i t Υ : PW t P = e − i t Υ . In fact, it is a minimal dilation of e − i t Υ .

11. Construction of a dilation Let h be an auxiliary space and ν : K → h satisfy 1 2i(Υ − Υ ∗ ) = − πν ∗ ν. Let (1 | be a linear functional with domain L 2 ( R ) ∩ L 1 ( R ) : � (1 | f = f ( x )d x. Let Z R be the operator of multiplication on L 2 ( R ) by the variable x . Define Z := K ⊕ h ⊗ L 2 ( R ) . Introduce the singular Friedrichs operator given by the following formal expression: (2 π ) − 1 � 2 ν ∗ ⊗ (1 | � 1 2 (Υ + Υ ∗ ) Z := (2 π ) − 1 2 ν ⊗ | 1) Z R Then ( Z , I K , e − i tZ ) is a dilation of e − i t Υ .

12. Construction of a dilation – the unitary group K e − i t Υ I K U t = I ∗ R e − i tZ R I R + I ∗ � t − i(2 π ) − 1 d u e − i( t − u )Υ ν ∗ ⊗ (1 | e − i uZ R I R 2 I ∗ K 0 � t − (2 π ) − 1 d u e − i( t − u ) Z R ν ⊗ | 1) e − i u Υ I K 2 i I ∗ R 0 � d u 1 d u 2 e − i u 2 Z R ν ⊗ | 1) e − i( t − u 2 − u 1 )Υ ν ∗ ⊗ (1 | e − i u 1 Z R I R . − (2 π ) − 1 I ∗ R 0 ≤ u 1 ,u 2 , u 1 + u 2 ≤ t We check that U t is a strongly continuous unitary group. Therefore, we can define Z as its unitary gener- ator: U t = e − i tZ . (Here I R is the embedding of h ⊗ L 2 ( R ) in Z ). .

13. Construction of a dilation – resolvent of the generator For z ∈ C + , we define R ( z ) := I ∗ R ( z − Z R ) − 1 I R + I ∗ K ( z − Υ) − 1 I K +(2 π ) − 1 K ( z − Υ) − 1 ν ∗ ⊗ (1 | ( z − Z R ) − 1 I R 2 I ∗ +(2 π ) − 1 2 I ∗ R ( z − Z R ) − 1 ν ⊗ | ν )( z − Υ) − 1 I K R ( z − Z R ) − 1 ν ⊗ | 1)( z − Υ) − 1 ν ∗ ⊗ ( ν | ( z − Z R ) − 1 I R ; +(2 π ) − 1 I ∗ R ( z ) := R ( z ) ∗ . We can check that R ( z 1 ) − R ( z 2 ) = ( z 2 − z 1 ) R ( z 1 ) R ( z 2 ) , Ker R ( z ) = { 0 } . Therefore, we can define Z as the self- adjoint operator Z satisfying R ( z ) = ( z − Z ) − 1 .

14. Construction of a dilation – removing a cutoff Z is the norm resolvent limit for r → ∞ of the following regularized operators: (2 π ) − 1 2 ν ∗ ⊗ (1 | 1 [ − r,r ] ( Z R ) 2 (Υ + Υ ∗ ) 1 Z r := (2 π ) − 1 2 ν ⊗ 1 [ − r,r ] ( Z R ) | 1) 1 [ − r,r ] ( Z R ) Z R (Note that it is important to remove the cut-off in a symmetric way).

15. False quadratic form of the generator of dilations On D := K ⊕ h ⊗ ( L 2 ( R ) ∩ L 1 ( R )) we can define the (non-self-adjoint) quadratic form (2 π ) − 1 � 2 ν ∗ ⊗ (1 | � Υ Z + := (2 π ) − 1 2 ν ⊗ | 1) Z R One can say that it is a “false form” of Z . In fact, for ψ, ψ ′ ∈ D , the function R ∋ t �→ ( ψ | e − i tZ ψ ′ ) is differen- d t ( ψ | e − i tZ ψ ′ ) tiable away from t = 0 , its derivative t �→ d is continuous away from 0 and at t = 0 it has the right limit equal to ψ | (e − i tZ − 1) ψ ′ � t ↓ 0 t − 1 � − i( ψ | Z + ψ ′ ) = lim .

16. Scaling invariance For λ ∈ R , introduce the following unitary operator on Z j λ g ( y ) := λ − 1 g ( λ − 2 y ) , u ∈ K ; g ∈ Z R . j λ u = u, Note that j ∗ λ Z R j λ = λ 2 Z R , j ∗ λ | 1) = λ | 1) . Therefore, the operator Z is invariant with respect to the following scaling: λ (2 π ) − 1 � 2 ν ∗ ⊗ (1 | � λ 21 2 (Υ + Υ ∗ ) Z = λ − 2 j ∗ j λ . λ λ (2 π ) − 1 2 ν ⊗ | 1) Z R

17. WEAK COUPLING LIMIT FOR FRIEDRICHS OPERATORS

18. Friedrichs operators Let H := K ⊕ H R be a Hilbert space, where K is finite dimensional. Let I K be the embedding of K in H . Let K be a self-adjoint operator on K and H R be a self- adjoint operator on H R . Let V : K → H R . Define the Friedrichs Hamiltonian � K λV ∗ � H λ := . λV H R

19. Reduced weak coupling limit for Friedrichs operators � V ∗ e − i tH R V � d t < ∞ . Define the � Assume that Level Shift Operator � ∞ 1 k ( K ) V ∗ e − i t ( H R − k ) V 1 k ( K )d t. � Υ := 0 k Note that Υ K = K Υ . Theorem. λ → 0 e i tK/λ 2 I ∗ K e − i tH λ /λ 2 I K = e − i t Υ . lim

20. Continuity of spectrum Assumption. We suppose that for any k ∈ sp K there exists an open I k ⊂ R such that k ∈ I k , Ran1 I k ( H R ) ≃ h k ⊗ L 2 ( I k , d x ) , 1 I k ( H R ) H R is the multiplication operator by the vari- able x ∈ I k and � ⊕ 1 I k ( H R ) V ≃ v ( x )d x. I k We assume that I k are disjoint for distinct k and x �→ v ( x ) ∈ B ( K , h k ) is continuous at k .

21. Asymptotic space h k , Z := K ⊕ h ⊗ L 2 ( R ) . Let ν : K → h be Define h := ⊕ k defined as 1 2 ⊕ ν := (2 π ) v ( k )1 k ( K ) . k Note that it satisfies ν ∗ ν = 1 i (Υ − Υ ∗ ) . As before, we set Z R to be the multiplication by x on L 2 ( R ) and (2 π ) − 1 � 2 ν ∗ ⊗ (1 | � 1 2 (Υ + Υ ∗ ) Z := , (2 π ) − 1 2 ν ⊗ | 1) Z R so that ( Z , I K , e − i tZ ) is a dilation of e − i t Υ .

22. Scaling For λ > 0 , we define the family of partial isometries J λ,k : h k ⊗ L 2 ( R ) → h k ⊗ L 2 ( I k ) : � λ g k ( y − k 1 λ 2 ) , if y ∈ I k ; ( J λ,k g k )( y ) = 0 , if y ∈ R \ I k . We set J λ : Z → H , defined for g = ( g k ) ∈ Z R by � J λ g := J λ,k g k , k and on K equal to the identity. Note that J λ are partial isometries and λ ց 0 J ∗ s − lim λ J λ = 1 .

23. Extended weak coupling limit for Friedrichs operators h k ⊗ L 2 ( R ) . we define the renormalizing On Z = K ⊕ � k Hamiltonian Z ren := K ⊕ ⊕ k. k Theorem. λ ց 0 e i λ − 2 tZ ren J ∗ λ e − i λ − 2 tH λ J λ = e − i tZ . s ∗ − lim Here we used the strong* limit: s ∗ − lim λ ց 0 A λ = A means that for any vector ψ λ ց 0 A λ ψ = Aψ, lim λ ց 0 A ∗ λ ψ = A ∗ ψ. lim

24. COMPLETELY POSITIVE MAPS

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