Scattering in a Euclidean formulation of relativistic quantum - PowerPoint PPT Presentation

Scattering in a Euclidean formulation of relativistic quantum mechanics W. N. Polyzou The University of Iowa

Contributors (students/former students) Victor Wessels: Few-Body Systems, 35(2004)51 Philip Kopp: Phys. Rev. D85,016004(2012) Gordon Aiello: Phys. Rev. D93,056003(2016) Gohin Samad Useful discussions with colleagues at Iowa F. Coester P. Jørgensen

Motivation and Observations • Constructing relativistic quantum mechanical models satisfying cluster properties is complicated. • Locality is logically independent of the rest of the axioms of Euclidean field theory → Euclidean formulation of relativistic quantum theory satisfying cluster properties. • Reconstruction theorem: The physical Hilbert space and a unitary representation of the Poincar´ e group can be directly formulated in the Euclidean representation. Analytic continuation is not necessary. • Given these elements it should be possible to formulate a relativistic treatment of scattering in a Euclidean representation using standard quantum mechanical methods.

Elements of relativistic quantum mechanics � ψ | φ � Hilbert space U (Λ , a ) ↔ { P µ , J µν } Relativity P 0 = H Dynamics P 0 = H ≥ 0 Spectral condition → stability ( x i − x j ) 2 → ∞ [ U (Λ , a ) − ⊗ U i (Λ , a )] | ψ � → 0 Cluster properties: scattering asymptotic conditions

Osterwalder-Schrader (Euclidean) reconstruction Input: { G En (x 1 , · · · , x n ) } Relevant properties • Euclidean covariant (invariant) • Cluster property • Reflection positivity

Construction of the physical Hilbert space: H M Vectors (dense set) ψ ( x ) := ( ψ 1 (x 11 ) , ψ 2 (x 21 , x 22 ) , · · · ) 0 < x 0 n 1 < x 0 n 2 < · · · < x 0 ψ n (x n 1 , x n 2 , · · · , x nn ) = 0 unless nn . θ x := θ ( τ, x ) = ( − τ, x ) Euclidean time reflection Physical Hilbert space inner product � � d 4 k xd 4 n y ψ ∗ � ψ | φ � M = ( θψ, G E φ ) E = n ( θ x n 1 , θ x n 2 , · · · , θ x nn ) × kn G E , n + k (x nn , · · · , x 1 n ; y 1 k , · · · , y kk ) φ k (y k 1 , y k 2 , · · · , y kk ) All variables are Euclidean - no analytic continuation.

Reflection positivity - property of { G En } � ψ | ψ � M = ( ψ, Π + > Θ G E Π + > ψ ) E ≥ 0 ⇓ Gives the physical Hilbert space and spectral condition.

Illustration Two-point Green function: Euclidean → Minkowski � φ ∗ ( − τ x , x ) d 4 p ρ ( m ) dm e ip · ( x − y ) p 2 + m 2 ψ ( τ y , y ) d 4 xd 4 y � φ | ψ � M = (2 π ) 4 � m ( p ) d p ρ ( m ) dm ξ ∗ = χ m ( p ) 2 e m ( p ) Euclidean wave function → Minkowski wave function � d 4 y (2 π ) 3 / 2 e − e m ( p ) τ y − i p · y ψ ( τ y , y ) χ m ( p ) := � d 4 x (2 π ) 3 / 2 e − e m ( p ) τ x − i p · x φ ( τ x , x ) ξ m ( p ) := m 2 ψ ( τ x , x ) = ∇ 2 4 ψ ( τ x , x )

Euclidean invariance → Poincar´ e invariance Relativity and SL (2 , C ) × SL (2 , C ) � t + z � i τ + z � � x − iy x − iy X m := X e := x + iy t − z x + iy i τ − z det ( X M ) = t 2 − x 2 det ( X E ) = − ( τ 2 + x 2 ) X → X ′ = AXB t det ( A ) = det ( B ) = 1 Preserves both t 2 − x 2 and τ 2 + x 2 Complex Lorentz group = complex orthogonal group Real orthogonal group = subgroup of complex Lorentz group Real Lorentz ( A , B ) = ( A , A ∗ ) , A ∈ SL (2 , C ) ; Real orthogonal ( A , B ) ∈ SU (2) × SU (2)

Relation between Euclidean and Poincar´ e generators • Euclidean time translations → contractive Hermitian semigroup on H M : H E = P 0 E = − iH M = − iP 0 M • Euclidean space-time rotations → local symmetric semigroup on H M : J 0 j E = iJ 0 j M = iK j • Euclidean space rotations → unitary one parameter groups on H M : J ij E = J ij M • Euclidean space translations → unitary one parameter groups on H m : P i E = P i M { P µ M , J µν M } = 10 self-adjoint generators satisfying the Poincar´ e commutation relations on H M

Spinless case n ∂ � H ψ n (x n 1 , x n 2 , · · · , x nn ) = ψ n (x n 1 , x n 2 , · · · , x nn ) ∂ x 0 nk k =1 n ∂ � P ψ n (x n 1 , x n 2 , · · · , x nn ) = − i ψ n (x n 1 , x n 2 , · · · , x nn ) ∂ x nk k =1 n ∂ � J ψ n (x n 1 , x n 2 , · · · , x nn ) = − i x nk × ψ n (x n 1 , x n 2 , · · · , x nn ) ∂ x nk k =1 n ∂ ∂ � − x 0 K ψ n (x n 1 , x n 2 , · · · , x nn ) = ( x nk ) ψ n (x n 1 , x n 2 , · · · , x nn ) . nk ∂ x 0 ∂ x nk nk k =1 All integration variables are Euclidean; Minkowski time is a parameter .

Cluster properties G E , n + m → G E , n G E , m ⇓ Generators become additive in asymptotically separated subsystems Used to formulate scattering asymptotic conditions.

Multichannel scattering theory Scattering probability = | S fi | 2 = |� ψ + | ψ − �| 2 | ψ ± � = Ω ± | ψ 0 ± � � e iHt � e − ie n t Ω ± | ψ 0 ± � = lim | φ n , p n , µ n � f n ( p n , µ n ) d p n t →±∞ � �� � � �� � n e − iH 0 t | ψ 0 ±� � �� � J t →±∞ e iHt Je − iH 0 t | ψ 0 ±� = lim Elements: Cluster properties, subsystem bound states: | φ n � , wave packets: f n , dynamics: H , strong limits.

Field theoretic implementation: Haag-Ruelle scattering (Minkowski case) Φ( x ) = interpolating field � 1 ˜ e − ip · x Φ( x ) d 4 x Φ( p ) = (2 π ) 2 Φ( p ) , h ( − m 2 ) = 1 , h ( p 2 ) = 0 , − p 2 �∈ ( m 2 − ǫ, m 2 + ǫ ) Φ m ( p ) = h ( p 2 )˜ ˜ � 1 e ip · x ˜ Φ m ( p ) d 4 p Φ m ( x ) = (2 π ) 2 e − i √ � i p 2 + m 2 t + i p · x ˜ f m ( x ) = f ( p ) d p (2 π ) 3 / 2 � ∂ Φ m ( t , x ) � � f m ( t , x ) − Φ m ( t , x ) ∂ f m ( t , x ) a † m ( f m , t ) = − i d x − ∂ t ∂ t t →±∞ Π i a † Ω ± | ψ 0 ± � = s − lim m i ( f m i , t ) | 0 �

Euclidean formulation of HR scattering - technical issues • ( M 2 = ∇ 2 ) One-body solutions must satisfy the time support condition: support ( h ( ∇ 2 ) � x | ψ � ) = support ( � x | ψ � ) • Products of one-body solutions must satisfy the relative time support condition ( n = 2 , no spin) . J : � x 1 | φ 1 , p 1 �� x 2 | φ 2 , p 2 � = 1 h 1 ( ∇ 2 1 ) δ (x 0 1 − τ 1 ) h 2 ( ∇ 2 2 ) δ (x 0 (2 π ) 3 e i p 1 · x 1 + i p 2 · x 2 2 − τ 2 ) τ 2 > τ 1

• Delta functions in Euclidean time × f ( x ) are square integrable in H M ! • A sufficient condition for h i ( ∇ 2 ) to preserve the support condition is for polynomials in ∇ 2 to be complete with respect to the inner product on H M h i ( ∇ 2 ) ≈ P ( ∇ 2 ) • The J defined on the previous slide can be used to satisfy the time-support conditions.

Completeness of P n ( ∇ 2 ) sufficient to construct h ( m 2 ) without violating positive Euclidean time-support condition. Proving completeness - Stieltjes moment problem G E 2 moments e − √ � ∞ m 2 + p 2 τ m 2 + p 2 ρ ( m ) m 2 n dm γ n := � 2 0 where τ = τ 1 + τ 2 > 0. Carleman’s condition ∞ � | γ n | − 1 2 n > ∞ n =0 Satisfied for ρ ( m 2 ) a tempered distribution ⇒ P ( ∇ 2 ) complete. 1 | γ n | − 1 2 n ∼ n + c

Existence - sufficient condition (Cook) � ∞ � ( HJ − JH 0 ) U 0 ( ± t ) | ψ 0 �� M dt < ∞ a � ( HJ − JH 0 )Φ U 0 ( ± t ) | ψ 0 �� 2 M = ( ψ 0 U 0 ( ∓ t )( J † H − H 0 J † ) θ G E ( HJ − JH 0 ) U 0 ( ± t ) | ψ 0 ) E The effect of using one-body solutions for 2-2 scattering is that the contribution from the disconnected part of G E to the above is zero. This fails for LSZ scattering. The connected part is expected to behave like ct − 3 for large t , satisfying the Cook condition.

Computational tricks for scattering Invariance principle: t →±∞ e iHt Je − iH 0 t | ψ � = t →±∞ e if ( H ) t Je − if ( H 0 ) t | ψ � lim lim f ( x ) = − e − β x n →∞ e ∓ ine − β H Je i ± ne − β H 0 | ψ � t →±∞ e iHt Je − iH 0 t | ψ � = lim lim σ ( e − β H ) ∈ [0 , 1] → | e inx − P ( x ) | < ǫ x ∈ [0 , 1] |� e ine − β H − P ( e − β H ) |� < ǫ Matrix elements of e − n β H are easy to calculate: � τ, x | e − n β H | ψ � = � τ − n β, x | ψ �

Model tests (of computational methods) H = k 2 / m − λ | g �� g | ( M 2 = 4 k 2 + 4 m 2 − 4 m λ | g �� g | ) 1 � k | g � = k 2 + m 2 π Attractive - one pion exchange range, bound state with deuteron mass. e − 2 ine − β H ≈ P ( e − β H ) � k f | T ( E + i 0) | k i � ≈ � ψ f | ( I − e − ine − β M 0 P ( e − β H ) e − ine − β H 0 ) | ψ i � 2 π i � ψ f | δ ( E − H 0 ) | ψ i �

• Choose sufficiently narrow initial and final wave packets. • Choose sufficiently large n . • Replace e 2 ine − β H by a polynomial approximation. • Calculations formally independent of β , adjust β for faster convergence. • Model allows independent tests of each approximation. • Approximations must be done in the proper order.

Results • Converges to exact sharp momentum transition matrix elements. • Tests converge for . 050 − 2 GeV. • Biggest source of error is the wave packet width.