SCATTERING THEORY IN NONRELATIVISTIC QFT Jan Derezi nski 1 - PowerPoint PPT Presentation

SCATTERING THEORY IN NONRELATIVISTIC QFT Jan Derezi´ nski 1

SECOND QUANTIZATION 1-particle Hilbert space: Z . ∞ n =0 ⊗ n Bosonic/fermionic Fock space: Γ s / a ( Z ) := ⊕ s / a Z . Vacuum vector: Ω = 1 ∈ ⊗ 0 s / a Z = C . 2

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. 3

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

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 ∗ = ξ a ξ d ξ + z ( ξ ) a ξ d ξ + ξ d ξ. H To avoid the ultraviolet problem we will always assume � | z ( ξ ) | 2 d ξ < ∞ . h ≥ 1 5

Van Hove Hamiltonian Infrared case A I 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. 6

Van Hove Hamiltonian Infrared case A II Let � h ( ξ ) a ∗ = ξ a ξ d ξ. H 0 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 ) d ξ exp h ( ξ )d ξ Ω . is its unique ground state. 7

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. 8

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 =] − ∞ , ∞ [. 9

Asymptotic fields for Van Hove Hamiltonians Assume that h has an absolutely continuous spectrum (as an operator on L 2 (Ξ)). It is easy to see that in Case A, B and C 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 10

Wave and scattering operators for Van Hove Hamiltonians In the case A we have Ua ( f ) U ∗ a ± ( f ) , = Ua ∗ ( f ) U ∗ a ∗± ( f ) . = Thus we can interpret U as the wave operator (both incoming and outgoing). Since a − ( f ) and a −∗ ( f ) coincide with a + ( f ) and a + ∗ ( f ), the scattering operator is identity, also in case B and C. 11

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 )). 12

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, = K ⊗ 1 + 1 ⊗ H 0 � v ( ξ ) ⊗ a ∗ ( ξ )d ξ + hc . = V 13

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, ∞ [. 14

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 ). 15

Basic theorem about scattering for Pauli-Fierz Hamiltonians 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 . 16

Asymptotic fields for Pauli-Fierz Hamiltonians We introduce asymptotic fields 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 )) . 17

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 . 18

Asymptotic Fock representation Define [0] := Span cl � H ± 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. 19

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. 20

Conjectures about asymptotic completeness for massless Pauli-Fierz Hamiltonians Conjectures. D.-G´ erard. Assume that h ( ξ ) = | ξ | , ( K + i) − 1 is compact 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). 21

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 � K ± ⊗ 1 + 1 ⊗ H ± as h ( ξ ) a ∗ ( ξ ) a ( ξ )d ξ. := 22

Møller operators for the asymptotic Fock sector There exists a unitary operator → H ± S ± 0 : H ± as [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 . We have 0 H ± as = HS ± S ± 0 . 23

Scattering operator for the asymptotic Fock sector The scattering operator is defined as S 0 := S + ∗ 0 S − 0 . It is an operator H − as → H +as satisfying 0 0 S 0 H − as = H +as S 0 . Its matrix elements are called scattering amplitudes and can be used to compute scattering cross sections of various processes. 24