SCATTERING THEORY IN NONRELATIVISTIC QUANTUM FIELD THEORY Jan - PowerPoint PPT Presentation

Cross-sections for long-range potentials In many quantum mechanics textbooks one approximates long-range potentials by a sequence of short-range potentials, e.g., the Coulomb potential by the Yukawa potentials V µ = z e − µ | x | | x | − 1 . For short-range potentials one can construct Møller and scattering operators, which leads to scattering cross-sections σ µ ( λ, ˆ ξ 1 , ˆ ξ 2 ) . Then one shows that there exist µ ց 0 σ µ ( λ, ˆ ξ 1 , ˆ lim ξ 2 ) , which is interpreted as the scattering cross-section for V . 26

Modified Møller operators There exist better approaches to long-range scattering. One can define modified Møller operators for long-range potentials. For instance, for an appropriate S ( t, ξ ) = tξ 2 2 + corrections there exists t →±∞ e i tH e − i S ( t,D ) . S ± lr := s − lim It is isometric, S ± lr H 0 = HS ± lr and Ran S ± lr = Ran1 c ( H ). 27

Freedom of definition However, in general there is no canonical choice of S ± lr . If we have two modified Møller operator S ± lr , 1 and S ± lr , 2 , then there exists a phase ψ ± such that lr , 2 e i ψ ± ( D ) . S ± lr , 1 = S ± This arbitrariness disappears in scattering cross-sections, which are canonically defined. 28

Asymptotic momenta For long-range potentials, there exists a self-adjoint operator D ± such that, for any g ∈ C c ( R d ), t →∞ e i tH g ( D ) e − i tH 1 c ( H ) . g ( D ± ) = s − lim Unlike modified Møller operators, asymptotic momenta are canonically defined. Modified Møller operators can be introduced as isometric operators satisfying g ( D ± ) = S ± lr g ( D ) S ±∗ lr . 29

SECOND QUANTIZATION 1-particle Hilbert space: Z . Symmetrization/antisymmetrization projections 1 � Θ s := Θ( σ ) , n ! σ ∈ S n 1 � Θ a := sgn σ Θ( σ ) . n ! σ ∈ S n s / a Z := Θ s / a ⊗ n Z . n -particle bosonic/fermionic space: ⊗ n ∞ n =0 ⊗ n Bosonic/fermionic Fock space: Γ s / a ( Z ) := ⊕ s / a Z . Vacuum vector: Ω = 1 ∈ ⊗ 0 s / a Z = C . 30

Creation and annihilation operators For z ∈ Z we define the creation operator √ a ∗ ( z )Ψ := Ψ ∈ ⊗ n n + 1 z ⊗ s / a Ψ , s / a Z , and the annihilation operator a ( z ) := ( a ∗ ( z )) ∗ . Traditional notation: identify Z with L 2 (Ξ) for some measure space (Ξ , d ξ ). If z equals a function Ξ ∋ ξ �→ z ( ξ ), then � � a ∗ ( z ) = z ( ξ ) a ∗ ξ d ξ, a ( z ) = z ( ξ ) a ξ d ξ. 31

Field and Weyl operators For f ∈ Z we introduce field operators φ ( f ) := 1 2( a ∗ ( f ) + a ( f )) , √ and Weyl operators W ( f ) := e i φ ( f ) . For later reference note that (Ω | W ( f )Ω) = e −� f � 2 / 4 . 32

Wick quantization � � ⊗ n s / a Z , ⊗ m Let b ∈ B s / a Z with the integral kernel b ( ξ 1 , · · · ξ m , ξ ′ n , · · · , ξ ′ 1 ). The Wick quantization of the polynomial b is the operator � b ( ξ 1 , · · · ξ m , ξ ′ n , · · · , ξ ′ B = 1 ) a ∗ ( ξ 1 ) · · · a ∗ ( ξ m ) a ( ξ ′ n ) · · · a ( ξ ′ 1 )d ξ 1 , · · · ξ n d ξ ′ 1 · · · d ξ ′ m . For Φ ∈ ⊗ k + m s / a Z , Ψ ∈ ⊗ k + n s / a Z , it is defined by � ( n + k )!( m + k )! (Φ | b ⊗ 1 ⊗ k (Φ | B Ψ) = Z Ψ) . k ! 33

Second quantization For an operator q on Z we define the operator Γ( q ) on Γ s / a ( Z ) by � Γ( q ) s / a Z = q ⊗ · · · ⊗ q. � � ⊗ n Similarly, for an operator h we define the operator dΓ( h ) by � s / a Z = h ⊗ 1 ( n − 1) ⊗ + · · · 1 ( n − 1) ⊗ ⊗ h. dΓ( h ) � � ⊗ n Traditional notation: If h is the multiplication operator � h ( ξ ) a ∗ by h ( ξ ), then dΓ( h ) = ξ a ξ d ξ. Note the identity Γ(e i th ) = e i t dΓ( h ) . 34

NONRELATIVISTIC QED Free photons Let L 2 tr ( R 3 , R 3 ) describe divergenceless (transversal) square integrable vector fields on R 3 . Free photons are described by the Hilbert space H ph := Γ s ( L 2 tr ( R 3 , R 3 )) and the Hamiltonian � � a ∗ H ph = s ( ξ ) | ξ | a s ( ξ )d ξ. s where e s ( ξ ) · ξ = 0, e s ( ξ ) · e s ′ ( ξ ) = δ s,s ′ are two polarization vectors. 35

Vector potential The vector potential is the operator given by s ( ξ ) e i xξ � � (2 π ) − 3 e s ( ξ ) a ∗ A ( x ) = d ξ + hc � 2 | ξ | s Actually, we will need to replace it by the smeared vector potential s ( ξ ) e i xξ � � (2 π ) − 3 ρ ( ξ ) e s ( ξ ) a ∗ A ρ ( x ) = d ξ + hc � 2 | ξ | s where ρ ∈ C c ( R 3 ) is a cutoff equal 1 for | ξ | < Λ. (In what follows we drop the subscript ρ ). 36

N -body matter Hamiltonians n identical particles of charge z , mass m , Bose/Fermi statistics, in a vector potential A ( x ) and in a scalar potential V ( x ), are described by the Hilbert space Γ n s / a ( L 2 ( R d )) and the Hamiltonian � 1 n � 2 m ( D i − zA ( x i )) 2 + zV ( x i ) � H n = i =1 z 2 � 4 π | x i − x j | − 1 . + 1 ≤ i<j ≤ n 37

2nd-quantized matter Hamiltonians Matter can be described in the 2nd quantized formalism, with the Hilbert space H = Γ s / a ( L 2 ( R d )) and the Hamiltonian ∞ n =0 H n H = ⊕ � 1 � � 2( D − zA ( x )) 2 + zV ( x ) b ∗ ( x ) = b ( x )d x z 2 +1 � � b ∗ ( x ) b ∗ ( y ) | x − y | − 1 b ( y ) b ( x )d x d y. 2 4 π 38

Matter interacting with photons Suppose we have a number of species of particles. The total Hilbert space is ⊗ j H j ⊗ H ph and the total Hamiltonian is � 1 � � ( D − z j A ( x )) 2 + z j V ( x ) � b ∗ H = j ( x ) b j ( x )d x 2 m j j +1 z j z k � � � b ∗ j ( x ) b ∗ k ( y ) | x − y | − 1 b k ( y ) b j ( x )d x d y 2 4 π j,k � � a ∗ + s ( ξ ) | ξ | a s ( ξ )d ξ. s 39

Single particle interacting with photons A single particle interacting with radiation is described by a Hilbert space L 2 ( R d ) ⊗ H ph and the Hamiltonian (sometimes called the Pauli-Fierz Hamiltonian) 1 ( D − zA ( x )) 2 + zV ( x ) � � H = 2 m � � a ∗ + s ( ξ ) | ξ | a s ( ξ )d ξ. s It is an example of a Hamiltonian where a small system (a particle) interacts with a large quantum environment (photons). 40

SCATTERING FOR HAMILTONIANS OF QUANTUM FIELD THEORY Typical Hamiltonians of QFT have (at least formally) the form � h ( ξ ) a ∗ ( ξ ) a ( ξ )d ξ H λ := � � v n,m ( ξ 1 , · · · ξ m , ξ ′ n , · · · , ξ ′ + λ 1 ) n,m a ∗ ( ξ 1 ) · · · a ∗ ( ξ m ) a ( ξ ′ n ) · · · a ( ξ ′ 1 )d ξ 1 , · · · ξ m d ξ ′ 1 · · · d ξ ′ n ξ 2 + m 2 describes the 1-particle � where e.g. h ( ξ ) = energy. The polynomials should be even in fermionic variables. 41

Localized interactions Assume that v n,m ( ξ 1 , · · · ξ m , ξ ′ n , · · · , ξ ′ 1 ) are smooth and decay fast in all directions. This is a simplifying assumption, which is not satisfied in most interesting theories. Nevertheless, there are physically relevant examples, where this assumption is fulfilled, besides we can use it as an introductory step before studying more relevant translation invariant systems. 42

Self-adjointness We do not worry too much about the self-adjointness of H λ . If we do not know how to do otherwise, we work with formal power series. In fact, in the case of fermions there is no problem, since the perturbation is bounded. In the case of bosons, the self-adjointness is OK if the perturbation is of degree 1 or 2 but small enough. Otherwise it can be proven only under special assumptions (e.g. for P ( φ ) 2 interactions). 43

Literature Scattering operator was used in QFT from the very beginning. It was present already in the work of Schwinger, Tomonaga, Feynman and Dyson Much of mathematical literature about scattering in QFT is old and often not very satisfactory. Let us mention 1. K.O. Friedrichs: “Perturbations of spectra in Hilbert spaces” 1965 2. K. Hepp: “La theorie de la renormalisation” 1969 3. A. S. Xvarc: “Matematiqeskie osnovy kvantovo i teorii pol� ” 1975 44

Interactions that do not polarize the vacuum Suppose that v n, 0 = v 0 ,n = 0. Then Ω is an eigenvector of both H 0 and H . Then standard wave operators exist, at least formally. Unfortunately, physically realistic Hamiltonians often polarize the vacuum. 45

Ground states for localized interactions One can show, at least formally, that H λ possesses a ground state H λ Ω λ = E λ Ω λ , ∞ � λ n Ω n , Ω λ = n =0 ∞ � λ n E n . E λ = n =0 46

Møller operators for localized interactions Unrenormalized Møller operators exist, at least as formal power series � ∞ e − 2 ǫt e ± i tH e ∓ i t ( H 0 − E ) d t S ± = s − lim ǫ ց 0 2 ǫ ur 0 ∞ � λ n S ± = ur ,n . n =0 Z = S −∗ ur S − ur = S + ∗ ur S + ur is proportional to identity and equals Z = | (Ω λ | Ω) | 2 . The renormalized Møller operators ur Z − 1 / 2 are formally unitary and so is the S ± rn := S ± renormalized scattering operator S rn := S + ∗ rn S − rn . 47

Asymptotic fields for localized interactions Lehman-Symanzik-Zimmermann introduce an alternative approach based on asymptotic fields t →±∞ e i tH a (e − i th f ) e − i tH , a ± λ ( f ) := lim t →±∞ e i tH a ∗ (e − i th f ) e − i tH , a ∗± λ ( f ) := lim (at least as formal power series). They satisfy the usual CCR. Asymptotic annihilation operators kill the perturbed ground state a ± λ ( f )Ω λ = 0 48

Møller operator from asymptotic fields The (renormalized) Møller operators can be defined with help of asymptotic fields S ± rn ,λ a ∗ ( f 1 ) · · · a ∗ ( f n )Ω a ∗± λ ( f 1 ) · · · a ∗± = λ ( f n )Ω λ They are formally unitary and intertwine the CCR: S ± rn ,λ a ∗ ( f ) a ∗± λ ( f ) S ± = rn ,λ , S ± a ± λ ( f ) S ± rn ,λ a ( f ) = rn ,λ . Note that there is no need for renormalization. 49

Scattering operator from asymptotic fields The (renormalized) scattering operators can be defined with help of asymptotic fields, even skipping Møller operators, as the unique (up to a phase factor) unitary operators satisfying ˜ λ ( f ) ˜ S rn ,λ a ∗− a ∗ + λ ( f ) = S rn ,λ , ˜ λ ( f ) ˜ S rn ,λ a − a + λ ( f ) = S rn ,λ . 50

Translation-invariant interactions Basic Hamiltonians of QFT have a translation-invariant interaction, and their scattering theory (even just formal) is more complicated. On the level of the interactions this is expressed by a delta function: v n,m ( ξ 1 , · · · ξ m , ξ ′ n , · · · , ξ ′ 1 ) v n,m ( ξ 1 , · · · ξ m , ξ ′ n , · · · , ξ ′ = ˜ 1 ) δ ( ξ 1 + · · · + ξ m − ξ ′ n − · · · − ξ ′ 1 ) , 51

SCATTERING THEORY OF VAN HOVE HAMILTONIANS Let ξ �→ h ( ξ ) ∈ [0 , ∞ [ describe the energy and ξ �→ z ( ξ ) the interaction. Van Hove Hamiltonian is a self-adjoint operator formally defined as � � � h ( ξ ) a ∗ z ( ξ ) a ∗ H = ξ a ξ d ξ + z ( ξ ) a ξ d ξ + ξ d ξ. To avoid the ultraviolet problem we will always assume � | z ( ξ ) | 2 d ξ < ∞ . h ≥ 1 52

Van Hove Hamiltonian Infrared case A Let | z ( ξ ) | 2 � h ( ξ ) 2 d ξ < ∞ . h< 1 Introduce the dressing operator − a ∗ ( z h ) + a ( z � � U := exp h ) . and the ground state energy � | z ( ξ ) | 2 E := − h ( ξ ) d ξ. 53

Van Hove Hamiltonian Infrared case A continued Let � h ( ξ ) a ∗ H 0 = ξ a ξ d ξ. In Case A, the operator H is well defined and, up to a constant, is unitarily equivalent to H 0 : H − E = UH 0 U ∗ Therefore H has the spectrum [ E, ∞ [ and � | z ( ξ ) | 2 � � �� a ∗ ( ξ ) z ( ξ ) � Ψ = exp − 2 h ( ξ ) 2 d ξ exp h ( ξ )d ξ Ω . is its unique ground state. 54

Van Hove Hamiltonian Infrared case B Let | z ( ξ ) | 2 � h ( ξ ) d ξ < ∞ ; h< 1 | z ( ξ ) | 2 � h ( ξ ) 2 d ξ = ∞ . h< 1 Then H is well defined, has the spectrum [ E, ∞ [, but has no bound states. 55

Van Hove Hamiltonian Infrared case C Let � | z ( ξ ) | 2 d ξ < ∞ ; h< 1 | z ( ξ ) | 2 � h ( ξ ) d ξ = ∞ . h< 1 Then H is well defined, but sp H =] − ∞ , ∞ [. 56

Unrenormalized Møller operators for Van Hove Hamiltonians Assume that h has an absolutely continuous spectrum (as an operator on L 2 (Ξ)) and Case A or B: � | z ( ξ ) | 2 h ( ξ ) d ξ < ∞ . Then there exists � ∞ e − ǫt e i tH e − i t ( H 0 + E ) d ξ. S ± := s − lim ǫ ց 0 ǫ ur 0 We have S ± ur = UZ , where � | z ( ξ ) | 2 � � Z = exp − h 2 ( ξ ) d ξ . 57

Renormalized Møller and scattering operators for Van Hove Hamiltonians In Case A, the vacuum renormalization constant is nonzero and we can renormalize S ± ur , obtaining the dressing operator: ur Z − 1 / 2 = U. S ± rn := S ± The scattering operator is (unfortunately) trivial: S = S + ∗ rn S − rn = 1 . 58

Asymptotic fields for Van Hove Hamiltonians It is easy to see that in Case A, B and C, for f ∈ Dom h − 1 , there exist asymptotic fields: t →±∞ e i tH a (e − i th f ) e − i tH = a ( f ) + ( f | h − 1 z ) , a ± ( f ) := lim t →±∞ e i tH a ∗ (e − i th f ) e − i tH = a ∗ ( f ) + ( z | h − 1 f ) . a ∗± ( f ) := lim This allows us to compute that the scattering operator ( ˜ S = 1) even in Case B and C. In Case A the asymptotic representation of the CCR is Fock but in Case B and C it is not. 59

SPECTRAL PROPERTIES OF PAULI-FIERZ HAMILTONIANS Let K be a Hilbert space with a self-adjoint operator K describing the small system. Typical example of K is a Schr¨ odinger operator. Usually, we will assume that K has discrete eigenvalues, which is the case if K = − ∆ + V ( x ) with lim | x |→∞ V ( x ) = ∞ . The full Hilbert space will be H := K ⊗ Γ s ( L 2 ( R d )). 60

Generalized spin-boson or Pauli-Fierz Hamiltonians We will discuss at length a class of Hamiltonians, which is often used in physics and mathematics literature to ilustrate basic properties of a small system interacting with bosonic fields. Let ξ �→ v ( ξ ) ∈ B ( K ). ξ 2 + m 2 , m ≥ 0. � We take, e.g. h ( ξ ) := Set H := H 0 + V where � h ( ξ ) a ∗ ( ξ ) a ( ξ )d ξ, H 0 = K ⊗ 1 + 1 ⊗ � v ( ξ ) ⊗ a ∗ ( ξ )d ξ + hc . V = 61

Spectrum of Pauli-Fierz Hamiltonians erard Assume that ( K + i) − 1 is compact Theorem D.-G´ and � (1 + h ( ξ ) − 1 ) � v ( ξ ) � 2 d ξ < ∞ . Then H is self-adjoint and bounded from below. If E := inf sp H , then sp ess H = [ E + m, ∞ [. 62

Ground state of Pauli-Fierz Hamiltonians Theorem Bach-Fr¨ ohlich-Sigal, Arai-Hirokawa, G´ erard. If in addition � (1 + h ( ξ ) − 2 ) � v ( ξ ) � 2 d ξ < ∞ , then H has a ground state (the infimum of its spectrum is an eigenvalue). 63

Embedded point spectrum of Pauli-Fierz Hamiltonians One does not expect that H has point spectrum embedded in its continuous spectrum. In fact, one can often prove for a small nonzero coupling constant that the spectrum of H λ := H + λV in ] E + m, ∞ [ is purely absolutely continuous, e.g. Bach-Fr¨ ohlich-Sigal-Soffer. In particular, if m = 0, this means that the only eigenvalue of H λ is at the bottom of its spectrum. It often can be proven to be nondegenerate. 64

SCATTERING THEORY OF PAULI-FIERZ HAMILTONIANS I In the case of Pauli-Fierz Hamiltonians the usual formalism of scattering in QFT does not apply, because of the presence of the small system. It is convenient to use a version of the LSZ formalism and start with asymptotic fields. I will follow the formalism of D-Gerard. Fr¨ ohlich-Griesemer-Schlein use a slightly different setup. Set Z 1 := Dom h − 1 / 2 ⊂ L 2 ( R d ). 65

Basic theorem Theorem D.-G´ erard. Let for f from a dense subspace � ∞ � � � e i th ( ξ ) f ( ξ ) v ( ξ )d ξ + hc � � � d t < ∞ . � � � 0 1. for f ∈ Z 1 there exists t →±∞ e i tH 1 ⊗ W (e − i th f ) e − i tH ; W ± ( f ) := s − lim 2. W ± ( f 1 ) W ± ( f 2 ) = e − iIm( f 1 | f 2 ) W ± ( f 1 + f 2 ), f 1 , f 2 ∈ Z 1 ; 3. R ∋ t �→ W ± ( tf ) is strongly continuous; 4. e i tH W ± ( f ) e − i tH = W ± (e i th f ); 5. if H Ψ = E Ψ, then (Ψ | W ± ( f )Ψ) = e −� f � 2 / 4 � Ψ � 2 . 66

Asymptotic fields for Pauli-Fierz Hamiltonians We introduce asymptotic fileds φ ± ( f ) := d � id tW ± ( tf ) � � t =0 and asymptotic creation/annihilation operators 1 a ∗± ( f ) √ := 2( φ ( f ) + i φ (i f )) , 1 a ± ( f ) √ := 2( φ ( f ) − i φ (i f )) . 67

Asymptotic vacua for Pauli-Fierz Hamiltonians Two equivalent definitions: � Ψ : (Ψ | W ± ( f )Ψ) = e −� f � 2 / 4 � Ψ � 2 � K ± := 0 Ψ : a ± ( f )Ψ = 0 � � = . The last item of the previous theorem can be reformulated as H p ( H ) ⊂ K ± 0 , where H p ( H ) denotes the span of eigenvectors of H . 68

Asymptotic Fock representation Define H ± [0] := Span cl � W ± ( f )Ψ : Ψ ∈ K ± � 0 , f ∈ Z 1 . Then H ± [0] is the smallest space containing the asymptotic vacua and invariant wrt asymptotic creation operators. 69

Asymptotic completeness for massive Pauli-Fierz Hamiltonians Theorem Assume that m > 0. Then erard. H ± 1. Hoegh-Kroehn, D.-G´ [0] = H , in other words, the asymptotic representations of the CCR are Fock. erard. K ± 2. D.-G´ 0 = H p ( H ), in other words, all the asymptotic vacua are linear combinations of eigenvectors. 70

Conjectures about asymptotic completeness for massless Pauli-Fierz Hamiltonians Conjectures. D.-G´ erard. Assume that h ( ξ ) = | ξ | and � (1 + h ( ξ ) − 2 ) � v ( ξ ) � 2 d ξ < ∞ . Then 1. H ± [0] = H , 2. K ± 0 = H p ( H ). Conjecture is true if dim K = 1 (i.e. for van Hove Hamiltonians). It is also true if v ( ξ ) = 0 for | ξ | < ǫ , ǫ > 0, (as remarked by Fr¨ ohlich-Griesemer-Schlein). 71

Asymptotic Hamiltonian for the asymptotic Fock sector � The operator K ± 0 := H describes the energies of � � K ± 0 asymptotic vacua (bound state energies, if asymptotic completeness is true). Define the asymptotic space H ± as := K ± 0 ⊗ Γ s ( L 2 ( R d )) 0 and the asymptotic Hamiltonian � H ± as K ± h ( ξ ) a ∗ ( ξ ) a ( ξ )d ξ. := 0 ⊗ 1 + 1 ⊗ 0 72

Møller operators for the asymptotic Fock sector There exists a unitary operator S ± 0 : H ± as → H ± [0] ⊂ H 0 called the Møller operator (for the asymptotic Fock sector) such that S ± 0 Ψ ⊗ a ∗ ( f 1 ) · · · a ∗ ( f n ) Ω a ∗± ( f 1 ) · · · a ∗± ( f n ) Ψ , Ψ ∈ K ± = 0 . 73

Intertwining properties of Møller operators We have S ± 0 1 ⊗ a ∗ ( f ) a ∗± ( f ) S ± = 0 , S ± a ± ( f ) S ± 0 1 ⊗ a ( f ) = 0 , S ± 0 H ± as HS ± = 0 . 0 74

Scattering operators for the asymptotic Fock sector Define S 00 = S + ∗ 0 S − 0 . It satisfies S 00 H − as = H +as S 00 . 0 0 If H + [0] = H − [0] , then S 00 is unitary on H +as = H − as . 0 0 75

Relaxation to the ground state I In practice, one often expects (and sometimes one can prove) that H only absolutely continuous spectrum except for a unique ground state Ψ gr . Thus | t |→∞ e i tH = | Ψ gr )(Ψ gr | . w − lim If in addition asymptotic completeness holds, then the asymptotic space is H ± as = Γ s ( Z ). 0 76

Relaxation to the ground state II Introduce the C ∗ -algebra A := B ( K ) ⊗ CCR( Z ) where CCR( Z ) = Span cl { W ( f ) : f ∈ Z} . Theorem Assume asymptotic completeness and the absence of bound states except for a unique ground state. Let A ∈ A . Then | t |→∞ e i tH A e − i tH = | Ψ gr )(Ψ gr | (Ψ gr | A Ψ gr ) . w − lim 77

REPRESENTATIONS OF THE CCR Let Y be a real vector space equipped with an antisymmetric form ω . (Usually we assume that ω is symplectic, i.e. is nondegenerate). Let U ( H ) denote the set of unitary operators on a Hilbert space H . We say that Y ∋ y �→ W π ( y ) ∈ U ( H ) is a representation of the CCR over Y in H if 2 y 1 ωy 2 W π ( y 1 + y 2 ) , W π ( y 1 ) W π ( y 2 ) = e − i y 1 , y 2 ∈ Y . 78

Regular representations of the CCR Let Y ∋ y �→ W π ( y ) be a representation of the CCR. Clearly, R ∋ t �→ W π ( ty ) ∈ U ( H ) is a 1-parameter group. We say that a representation of the CCR ) is regular if this group is strongly continuous for each y ∈ Y . 79

Field operators Assume that y �→ W π ( y ) is a regular representation of the CCR. φ π ( y ) := − i d � d tW π ( ty ) t =0 . � � φ π ( y ) will be called the field operator corresponding to y ∈ Y . We have Heisenberg canonical commutation relation [ φ π ( y 1 ) , φ π ( y 2 )] = i y 1 ωy 2 . 80

Creation/annihilation operators Let Z be a complex vector space with a scalar product ( ·|· ). It has a symplectic form Im( ·|· ) Suppose that Z ∋ f �→ W π ( f ) ∈ U ( H ) is a regular representation of the CCR. For f ∈ Z we introduce creation/annihilation operators a π ∗ ( f ) := 1 a π ( f ) := 1 2( φ π ( f ) + i φ π (i f )) , 2( φ π ( f ) − i φ π (i f )) . √ √ They satisfy the usual relations [ a π ( f 1 ) , a π ( f 2 )] = 0 , [ a π ∗ ( f 1 ) , a π ∗ ( f 2 )] = 0 , [ a π ( f 1 ) , a π ∗ ( f 2 )] = ( f 1 | f 2 ) . 81

Fock representation of the CCR Consider the creation/annihilation operators acting on the Fock space Γ s ( Z cpl ). Then φ ( f ) := 1 2 ( a ∗ ( f ) + a ( f )) √ are self-adjoint and Z ∋ f �→ exp i φ ( f ) is a regular representation of the CCR called the Fock representation. The vacuum Ω is characterized by either of the following equivalent equations: a ( f )Ω = 0 , f ∈ Z ; (Ω | e i φ ( f ) Ω) e − 1 4 ( f | f ) , = f ∈ Z . 82

Coherent representation of the CCR I Let g be an antilinear functional on Z (not necessarily bounded), that is g ∈ Z ∗ . Then Z ∋ f �→ W g ( f ) := W ( f ) e iRe( g | f ) ∈ U (Γ s ( Z cpl )) is a regular representation of the CCR called the [ g ]-coherent representation. The corresponding creation/annihilation operators are a ( f ) + 1 √ a g ( f ) = 2( f | g ) , a ∗ ( f ) + 1 a ∗ √ g ( f ) = 2( g | f ) . 83

Coherent representation of the CCR II The vector Ω is characterized by either of the following equations: 1 √ a g ( f )Ω = 2( f | g )Ω , 4 ( f | f )+iRe( f | g ) . e − 1 (Ω | W g ( f )Ω) = The representation f �→ W g ( f ) is unitarily equivalent to the Fock representation iff g is a bounded functional g ∈ Z cpl . More generally, W g 1 is equivalent to W g 2 iff g 1 − g 2 ∈ Z cpl . 84

Coherent sectors of a CCR representation I Suppose that Z ∋ f �→ W π ( f ) ∈ U ( H ) is a representation of the CCR (e.g. obtained by asymptotic limits, so that π = ± ). Let g be be an antilinear functional on Z . How can we find all subrepresentations of W π equivalent to a multiple of the [ g ]-coherent representation? 85

Coherent sectors of a CCR representation II Define √ K π { Ψ ∈ H : a π ( f )Ψ = := 2( g | f )Ψ } g { Ψ ∈ H : (Ψ | W π ( f )Ψ) = � Ψ � 2 e − 1 4 ( f | f )+iRe( f | g ) } , = H π Span cl � a π ∗ ( f 1 ) · · · a π ∗ ( f 1 )Ψ : Ψ ∈ K π � := g , f i ∈ Z [ g ] Span cl � W π ( f )Ψ : Ψ ∈ K π � = g , f ∈ Z . 86

Coherent sectors of a CCR representation III We define an isometric operator S π g : K π g ⊗ Γ s ( Z cpl ) → H by S π g Ψ ⊗ a ∗ g ( f 1 ) · · · a ∗ g ( f n )Ω a π ∗ ( f 1 ) · · · a π ∗ ( f n )Ψ , = S π g Ψ ⊗ W g ( f )Ω W π ( f )Ψ . = 87

Coherent sectors of a CCR representation IV Theorem. 1. H π [ g ] is an invariant subspace for W π . 2. S π g : K π g ⊗ Γ s ( Z cpl ) → H π [ g ] is unitary. 3. S π g 1 ⊗ W g ( f ) = W π ( f ) S π g . 4. If U is unitary such that U 1 ⊗ W g ( f ) = W π ( f ) U, then Ran U ⊂ H π [ g ] . Thus on [ g ] ∈Z ∗ / Z cpl H π ⊕ [ g ] ⊂ H the representation W π is well understood – it is of the coherent type. 88

Covariant CCR representations Let h be a self-adjoint operator on Z cpl and H a self-adjoint operator on H . We say that ( W π , h, H ) is a covariant representation of the CCR iff e i tH W π ( f ) e − i tH = W π (e i th f ) , f ∈ Z . Example. Fock representation, ( W, h, dΓ( h )): e i t dΓ( h ) W ( f ) e − i t dΓ( h ) = W (e i th f ) . 89

Covariant coherent CCR representations 1 Let g ∈ h − 1 Z cpl . Set z = 2 hg . Introduce the van Hove √ Hamiltonian dΓ g ( h ) := dΓ( h ) + a ∗ ( z ) + a ( z ) + ( z | h − 1 z ) . Then ( W g , h, dΓ g ( h )) is covariant: e i t dΓ g ( h ) W g ( f ) e − i t dΓ g ( h ) = W g (e i th f ) . This is obvious for g ∈ Z cpl , because then dΓ g ( h ) = W (i g )dΓ( h ) W ( − i g ) , W g ( f ) = W (i g ) W ( f ) W ( − i g ) . 90

Restricting covariant representation to a Fock sector Suppose that Z ∋ f �→ W π ( f ) ∈ U ( H ) is a representation of the CCR covariant for h, H : e i tH W π ( f ) e − i tH = W π (e i th f ) . It is easy to restrict it to the Fock sector: Theorem. K π 0 and H π [0] are e i tH -invariant. Let � K π and on K π 0 ⊗ Γ s ( Z cpl ) set 0 := H � � K π 0 H π 0 = K π 0 ⊗ 1 + 1 ⊗ dΓ( h ) . Then HS π 0 = S π 0 H π 0 . 91

Restricting covariant representation to a coherent sector I Theorem. Let g ∈ h − 1 / 2 Z . Then H π [ g ] is e i tH -invariant and there exists a unique operator K π g on K π g such that if on K π g ⊗ Γ s ( Z cpl ) we set H π g := K π g ⊗ 1 + 1 ⊗ dΓ g ( h ) , then HS π g = S π g H π g . 92

Restricting covariant representation to a coherent sector II Thus restricted to H π [ g ] , the covariant representation ( W π , h, H ) is unitarily equivalent to 1 ⊗ W g , 1 ⊗ h, K π � � g ⊗ 1 + 1 ⊗ dΓ g ( h ) . In particular, if g �∈ Z cpl , then the Hamiltonian does not have a ground state inside this sector. Nevertheless, inside this sector, we have good control on the dynamics! 93

SCATTERING THEORY OF PAULI-FIERZ HAMILTONIANS II Below we reformulate the basic theorem. Theorem Under the same assumptions as before 1. for f ∈ Z 1 there exists t →±∞ e i tH 1 ⊗ W (e − i th f ) e − i tH ; W ± ( f ) := s − lim 2. Z 1 ∋ f �→ W ± ( f ) are representations of the CCR. 3. These representations are regular. 4. ( W ± , h, H ) are covariant. 5. The Fock sector of W ± contains all eigenvectors of H . 94

Asymptotic g -coherent subspace Let g ∈ Z ∗ . Then one can define { Ψ ∈ H : (Ψ | W ± ( f )Ψ) = � Ψ � 2 e − 1 K ± 4 ( f | f )+iRe( f | g ) } , := g H ± Span cl � W ± ( f )Ψ : Ψ ∈ K π � := g , f ∈ Z , [ g ] as well as the asymptotic Hilbert spaces H ± as := K ± g ⊗ Γ s ( Z cpl ) g asymptotic Hamiltonians H ± as := K ± g ⊗ 1 + 1 ⊗ dΓ( h ) . g 95

g -coherent Møller operators I → H ± The Møller operators S ± g : H ± as [ g ] ⊂ H intertwine g field operators and the Hamiltonians: S ± g 1 ⊗ a ∗ a ∗± ( f ) S ± g ( f ) = g , S ± a ± ( f ) S ± g 1 ⊗ a g ( f ) = g , S ± g H ± as HS ± = g . g One can define scattering operator between sectors g 1 and g 2 : S g 2 ,g 1 := S + ∗ g 2 S − g 1 . 96

g -coherent Møller operators II Define the g -coherent identifier J ± g : H ± as → H by g J ± g Ψ ⊗ W g ( f )Ω = 1 ⊗ W ( f ) Ψ . Then we can introduce Møller operators using this identifier: t →±∞ e i tH J ± g e − i tH ± as S ± g = s − lim . g 97

Incoming/outgoing coherent subspaces In the physical space we can distinguish the space where asymptotic CCR are coherent: H ± g ∈Z ∗ / Z cpl H ± := ⊕ [ g ] ⊂ H . [coh] We also introduce the corresponding asymptotic spaces H ± as g ∈Z ∗ / Z cpl H ± as := ⊕ . g coh 98

Coherent Møller and scattering operators We have the Møller operators S ± coh : H ± as coh → H ± [coh] S ± g ∈Z ∗ / Z cpl S ± := ⊕ g . coh intertwining the asymptotic and the physical Hamiltonian S ± coh H ± as coh = HS ± coh . Finally, we have an object that is perhaps the most interesting physically: the coherent scattering operator S coh : H − as coh → H +as coh S coh := S + ∗ coh S − coh . 99

Soft bosons I Assume that all asymptotic fields are g -coherent for some unbounded g . Typically one can expect that all the unboundedness of g is concentrated at the zero energy, that is for any ǫ > 0, � 1 [ ǫ, ∞ [ ( h ) g � < ∞ . By modifying g we can assume that 1 [ ǫ, ∞ [ ( h ) g = 0. The one-particle space can be split as Z = Z ≤ ǫ ⊕ Z >ǫ , where Z ≤ ǫ := 1 [0 ,ǫ ] ( h ) Z , Z >ǫ := 1 ] ǫ, ∞ [ ( h ) Z . Then the Fock space splits as Γ s ( Z ) ≃ Γ s ( Z ≤ ǫ ) ⊗ Γ s ( Z >ǫ ) , and the vacuum splits as Ω = Ω ≤ ǫ ⊗ Ω >ǫ . 100