Multi-Time Formalism in Quantum Field Theory Sascha Lill - PowerPoint PPT Presentation

Multi-Time wave functions The consistency condition Quantum Multi-time models Proof of existence and uniqueness Open questions Multi-Time Formalism in Quantum Field Theory Sascha Lill sascha.lill@online.de University T¨ ubingen October 27, 2019 Sascha Lill University T¨ ubingen October 27, 2019 1 / 23

Multi-Time wave functions The consistency condition Quantum Multi-time models Proof of existence and uniqueness Open questions Outline Multi-Time wave functions The consistency condition Quantum Multi-time models Interacting potentials An (almost)-consistent QFT model QFT in Multi-Time The Initial Value Problem Proof of existence and uniqueness Tools and ideas Solution construction Final proof: Existence and Uniqueness Open questions Sascha Lill University T¨ ubingen October 27, 2019 2 / 23

Multi-Time wave functions The consistency condition Quantum Multi-time models Proof of existence and uniqueness Open questions Multi-Time wave functions ◮ state in Schr¨ odinger picture: t | Ψ t � = Ψ( t, x 1 , ..., x N ) ◮ perform Lorentz boost ◮ Ψ ′ ( t, x ′ 1 , ..., x ′ N ) is unclear! t 3 ◮ introduce separate time for each t 2 particle: t 1 x φ ( q ) = φ ( t 1 , x 1 , ..., t N , x N ) x 2 x 1 x 3 ◮ ”Multi-Time wave function” (Dirac, 1932) Sascha Lill University T¨ ubingen October 27, 2019 3 / 23

Multi-Time wave functions The consistency condition Quantum Multi-time models Proof of existence and uniqueness Open questions Multi-Time wave functions ◮ φ ( q ) = φ ( x 1 , ..., x N ) = φ ( t 1 , x 1 , ..., t N , x N ) ◮ recovery of single-time wave function: Ψ t ( x 1 , ..., x N ) = φ ( t, x 1 , ..., t, x N ) ◮ usually only defined for space-like separated particles: � x j − x j ′ � > | t j − t j ′ | ⇔ : q ∈ S ◮ equations of motion: i∂ t 1 φ ( q ) = H 1 φ ( q ) i∂ t Ψ = H Ψ → ... i∂ t N φ ( q ) = H N φ ( q ) ◮ Hamiltonian has to be split : H = � N j =1 H j � � ◮ consistency condition: H j − i∂ t j , H j ′ − i∂ t j ′ = 0 Sascha Lill University T¨ ubingen October 27, 2019 4 / 23

Multi-Time wave functions The consistency condition Quantum Multi-time models Proof of existence and uniqueness Open questions What time dynamics should look like: t q 3 Σ q 1 q 2 Ω x 2 x 1 U Σ ' q' 2 q' 3 x q' 1 0 ◮ We would like to make sense of: � ∞ � � � � dq ′ d 4 n x T ( H ( x 1 ) · · · H ( x n )) φ ( q ′ ) φ ( q ) = Q x k ∈ Ω n =0 � �� � U ( q,q ′ ) Sascha Lill University T¨ ubingen October 27, 2019 5 / 23

Multi-Time wave functions The consistency condition Quantum Multi-time models Proof of existence and uniqueness Open questions The consistency condition t 2 ◮ first, consider ∂H i ∂t j = 0 : H 1 t 2 ◮ unitary time evolution U depends H 2 H 2 on order of time increase H 1 U 12 = e − iH 2 t 2 e − iH 1 t 1 t 0 t 1 U 21 = e − iH 1 t 1 e − iH 2 t 2 t 0 t 1 � e − iH 1 t 1 , e − iH 2 t 2 � ! ⇒ U 21 − U 12 = = 0 [ H 1 , H 2 ] = 0 Sascha Lill University T¨ ubingen October 27, 2019 6 / 23

Multi-Time wave functions The consistency condition Quantum Multi-time models Proof of existence and uniqueness Open questions Mathematical proof (bounded H i ) ◮ Take arbitrary paths ( ∂H i ∂t j � = 0 ): t 2 � � � t 2 γ j U 1 = T exp − i γ 1 H j ( s ) · ˙ 1 ( s ) ds A � � � γ j t 0 U 2 = T exp − i γ 2 H j ( s ) · ˙ 2 ( s ) ds t 1 t 0 t 1 ◮ and set them equal: � � � 1 ! = U 1 γ j ( s ) ds γ = γ 1 ⋄ γ − 1 U 2 = T exp − i γ H j ( s ) · ˙ 2 � � [ H 1 ,H 2 ] � � � + ∂H 1 ∂t 2 − ∂H 2 ⇔ 1 = T exp − i dA A i ∂t 1 ◮ consistency condition : [ H 1 , H 2 ] + i∂H 1 − i∂H 2 = 0 ∂t 2 ∂t 1 Sascha Lill University T¨ ubingen October 27, 2019 7 / 23

Multi-Time wave functions Interacting potentials The consistency condition An (almost)-consistent QFT model Quantum Multi-time models QFT in Multi-Time Proof of existence and uniqueness The Initial Value Problem Open questions Interacting potentials ◮ M particles with interaction potential: M M � � H free H = + V ( x j − x k ) j j =1 k,j =1 k � = j � � � − ∆ j H free 2 m, − iα a ∂ a e.g. ∈ j + mβ, − ∆ j + m 2 , |∇ j | j 1 V ( x j − x k ) = 2 � x j − x k � + � M ◮ splitting is simple: H j = H free V ( x j − x k ) k =1 j k � = j Sascha Lill University T¨ ubingen October 27, 2019 8 / 23

Multi-Time wave functions Interacting potentials The consistency condition An (almost)-consistent QFT model Quantum Multi-time models QFT in Multi-Time Proof of existence and uniqueness The Initial Value Problem Open questions Interacting potentials ◮ Hamiltonians with interacting potentials violate consistency : � H j = − ∆ j 1 2 m + 2 � x j − x k � k � = j ◮ consistency is: = [ H j , H k ] = ( x j − x k ) · ( ∇ j + ∇ k ) 0 ! � = 0 � 2 m � x j − x k � 3 ◮ Happens with all Lorentz-Invariant potentials! [Petrat, Tumulka (2014)], [Nickel, Deckert (2016)] Sascha Lill University T¨ ubingen October 27, 2019 9 / 23

Multi-Time wave functions Interacting potentials The consistency condition An (almost)-consistent QFT model Quantum Multi-time models QFT in Multi-Time Proof of existence and uniqueness The Initial Value Problem Open questions An (almost)-consistent QFT model ◮ M spin- 1 / 2 fermions ( x k ) and N ∈ N 0 spin- 1 / 2 bosons ( y l ) ◮ configuration space with spin: � ( R 3 ) 4 � M × � ∞ � ( R 3 ) 4 � N Q = N =0 N=0 N=1 N=2 ◮ wave function Ψ t : Q → C Ψ t ( q ) = Ψ ( N ) r 1 ,...,r M ,s 1 ,...,s N ( x 1 , ..., x M , y 1 , ..., y N ) ◮ free Dirac evolutions: � � − i � 3 H free a =1 ( α a ) r k r ′ x k Ψ r k = k ∂ x a k + m x ( β ) r k r ′ Ψ r ′ � � k k − i � 3 H free a =1 ( α a ) s l s ′ Ψ s l = l ∂ y a l + m y ( β ) s l s ′ Ψ s ′ y l l l Sascha Lill University T¨ ubingen October 27, 2019 10 / 23

Multi-Time wave functions Interacting potentials The consistency condition An (almost)-consistent QFT model Quantum Multi-time models QFT in Multi-Time Proof of existence and uniqueness The Initial Value Problem Open questions ◮ boson annihilation by x k : use cutoff with supp ( ϕ δ ) ⊂ B δ (0) √ � a s ( x op � ( N ) ( q ) = � d 3 ˜ y − x k )Ψ ( N +1) k )Ψ N + 1 y ϕ δ ( ˜ s N +1 = s ( q , ˜ y ) ◮ boson creation by x k : � � ( N ) ( q ) = � N s ( x op l =1 δ ss l ϕ δ ( y l − x k )Ψ ( N − 1) 1 a † k )Ψ ( q \ y l ) √ � N s l ◮ interaction = creation + annihilation: x k = � 4 s =1 ( g s,k a s ( x op s ( x op H int k ) + g ∗ s,k a † k )) ◮ full Hamiltonian: � M � ( N ) � � N � � ( H Ψ) ( N ) = H free + H int H free Ψ + Ψ x k x k y l k =1 l =1 ◮ Would be consistent without cutoff [Petrat, Tumulka (2014)] ◮ Cutoff allows for rigorous construction of a unique solution to Multi-time equations of motion [Lill (2018)] Sascha Lill University T¨ ubingen October 27, 2019 11 / 23

Multi-Time wave functions Interacting potentials The consistency condition An (almost)-consistent QFT model Quantum Multi-time models QFT in Multi-Time Proof of existence and uniqueness The Initial Value Problem Open questions Multi-Time x 0 j -x 0 k ◮ challenge: define admissible wave functions x j - x k 0 ◮ only space-like configurations q ∈ S δ ◮ particles close together are forced to equal times t 1 , ..., t J ◮ admissible wave functions: 1. partial derivatives ∂ x a k , ∂ y a l , ∂ t j to arbitrary order are continuous � k 2 + m 2 ) , N = � 2. define H f = d Γ( k ( − ∆ k ) + H f + 1 Now, Ψ t ∈ dom ( N n ) ∀ n ∈ N ⇒ sector sum � ∂ α Ψ t � 2 = � ∞ N =0 � ∂ α Ψ ( N ) � k, 2 < ∞ is finite t ⇒ finite Sobolev norms 3. 3D-support R 3 ⊃ supp 3 Ψ t is compact ◮ we write: φ ∈ C ∞ P,c and Ψ t ∈ H ∞ c Sascha Lill University T¨ ubingen October 27, 2019 12 / 23

Multi-Time wave functions Interacting potentials The consistency condition An (almost)-consistent QFT model Quantum Multi-time models QFT in Multi-Time Proof of existence and uniqueness The Initial Value Problem Open questions The Initial Value Problem ◮ IVP to be solved: φ (0 , x 1 , ..., 0 , y N ) = φ 0 ( x 1 , ..., y N ) ∈ H ∞ c    � �  Ψ( q ) i∂ t 1 φ ( q ) = H 1 φ ( q ) = H x k + H y l x k ∈ P 1 y l ∈ P 1 ...    � �  φ ( q ) i∂ t J φ ( q ) = H J φ ( q ) = H x k + H y l x k ∈ P J y l ∈ P J ◮ Theorem 1: A unique solution φ ∈ C ∞ P,c exists ∀ q ∈ S δ Sascha Lill University T¨ ubingen October 27, 2019 13 / 23

Multi-Time wave functions The consistency condition Tools and ideas Quantum Multi-time models Solution construction Proof of existence and uniqueness Final proof: Existence and Uniqueness Open questions Assembling time evolutions ◮ solution: assemble single-time evolutions t ◮ start with Ψ( t 0 , x 1 , ..., x M ) , sort x 0 1 ≤ ... ≤ x 0 M ◮ evolve with H = � M j =1 H j first up to x 0 1 ◮ evolve only with H = � M j =2 H j t 2 up to x 0 H 2 H 3 2 t 1 ◮ proceed with decreasing sums x until x 0 M is reached ◮ works if � M x 2 x 1 x 3 j = k H j , k ∈ { 1 , ..., N } are ess. self-adjoint Sascha Lill University T¨ ubingen October 27, 2019 14 / 23