Momentum relation and classical limit in the future-not-included - PowerPoint PPT Presentation

Momentum relation and classical limit in the future-not-included complex action theory † Keiichi Nagao † Ibaraki Univ. Aug. 7, 2013 @ YITP Based on the work with H.B.Nielsen PTEP(2013) 073A03 (+PTP126 (2011)102, IJMPA27 (2012)1250076, PTEP(2013) 023B04)

Introduction Complex action theory (CAT) • coupling parameters are complex • dynamical variables such as q and p are fundamentally real but can be complex at the saddle points (asymptotic values are real). Possible extension of quantum theory Expected to give falsifiable predictions Intensively studied by H. B. Nielsen and M. Ninomiya

Complex coordinate formalism KN, H.B.Nielsen, PTP126 (2011)102 Non-Hermitian operators ˆ q new and ˆ p new : q † ˆ new | q ⟩ new = q | q ⟩ new for complex q , p † ˆ new | p ⟩ new = p | p ⟩ new for complex p , [ˆ q new , ˆ p new ] = i � . Our proposal is to replace the usual Hermitian operators ˆ q , ˆ p , and their eigenstates | q ⟩ and | p ⟩ , q | q ⟩ = q | q ⟩ , ˆ p | p ⟩ = p | p ⟩ , and [ˆ which obey ˆ q , ˆ p ] = i � q † p † new , | q ⟩ new and | p ⟩ new . for real q and p , with ˆ new , ˆ

1 1 ( ˆ q ) , p + i ϵ ′ ˆ q new ≡ ˆ √ 1 − ϵϵ ′ (ˆ q − i ϵ ˆ p ) , ˆ p new ≡ √ 1 − ϵϵ ′ ) 1 √ ( 1 − ϵϵ ′ 1 − ϵϵ ′ 4 e − 1 4 � ϵ (1 − ϵϵ ′ ) q 2 | | q ⟩ new ≡ 2 � ϵ q ⟩ coh , 4 π � ϵ ) 1 √ ( 1 − ϵϵ ′ 1 − ϵϵ ′ 4 4 � ϵ ′ (1 − ϵϵ ′ ) p 2 | i 1 e − | p ⟩ new ≡ 2 � ϵ ′ p ⟩ coh ′ . 4 π � ϵ ′ | λ ⟩ coh ≡ e λ a † | 0 ⟩ satisfies a | λ ⟩ coh = λ | λ ⟩ coh , where √ 1 a = 2 � ϵ (ˆ q + i ϵ ˆ p ) . √ | λ ⟩ coh ′ ≡ e λ a ′† | 0 ⟩ , where a ′† = ϵ ′ ( q − i ˆ p ) ˆ , is ϵ ′ 2 � another coherent state defined similarly.

Modified complex conjugate ∗ {} : ex.) for f ( q , p ) = aq 2 + bp 2 , f ( q , p ) ∗ q , p = f ∗ ( q , p ) = a ∗ q 2 + b ∗ p 2 , Modified bra m ⟨ | , {} ⟨ | : Modified hermitian conjugate † m , † {} : m ⟨ λ | = ⟨ λ ∗ | = ( | λ ⟩ ) † m . ( | ⟩ ) † {} = {} ⟨ | . For example, a wave function : ψ ( q ) = ⟨ q | ψ ⟩ → ψ ( q ) = m ⟨ new q | ψ ⟩

We decompose some function f as f = Re {} f + i Im {} f , where Re {} f and Im {} f are the “ {} -real” and “ {} -imaginary” parts of f defined by Re {} f ≡ f + f ∗{} and Im {} f ≡ f − f ∗{} 2 i . 2 ex) for f = kq 2 , Re q f = Re ( k ) q 2 , Im q f = Im ( k ) q 2 . If f satisfies f ∗ {} = f , we say f is {} -real, while if f obeys f ∗ {} = − f , we call f purely {} -imaginary.

Theorem on matrix elements m ⟨ new q ′ or p ′ |O (ˆ new ) | q ′′ or p ′′ ⟩ new , q † p † q new , ˆ new , ˆ p new , ˆ where O is a Taylor-expandable function, can be evaluated as if inside O we had the hermiticity q † p † conditions ˆ q new ≃ ˆ new ≃ ˆ q and ˆ p new ≃ ˆ new ≃ ˆ p for q ′ , q ′′ , p ′ , p ′′ such that the resulting quantities are well defined in the sense of distribution. → We do not have to worry about the q † p † anti-Hermitian terms in ˆ q new , ˆ new , ˆ p new and ˆ new , provided that we are satisfied with the result in the distribution sense.

Deriving the momentum relation via FPI KN, H.B.Nielsen, IJMPA27 (2012)1250076 Lagrangian in a system with a single d.o.f.: q ( t )) = 1 q 2 − V ( q ) , L ( q ( t ) , ˙ 2 m ˙ V ( q ) = ∑ ∞ n = 2 b n q n , V = V R + iV I , L = L R + iL I , where n = 2 Re b n q n , V R ≡ Re q ( V ) = ∑ ∞ n = 2 Im b n q n , V I ≡ Im q ( V ) = ∑ ∞ q 2 − V R ( q ) , L R ≡ Re q ( L ) = 1 2 m R ˙ q 2 − V I ( q ) . m = m R + im I . L I ≡ Im q ( L ) = 1 2 m I ˙

∫ i � ∆ tL ( q , ˙ q ) m ⟨ new q t + dt | ψ ( t + dt ) ⟩ = m ⟨ new q t | ψ ( t ) ⟩ dq t . e C We consider m ⟨ new q t | ξ ⟩ which obeys p new | ξ ⟩ = � ∂ m ⟨ new q t | ˆ m ⟨ new q t | ξ ⟩ ∂ q t i = ∂ L q t , ξ − q t ( ) m ⟨ new q t | ξ ⟩ . ∂ ˙ q dt Introducing a dual basis m ⟨ anti ξ | , we have ∫ m ⟨ new q t | ψ ( t ) ⟩ ≃ d ξ m ⟨ new q t | ξ ⟩ m ⟨ anti ξ | ψ ( t ) ⟩ C ∫ d ξ m ⟨ new q t | ψ ( t ) ⟩| ξ . = C

Then, we obtain m ⟨ new q t + dt | ψ ( t + dt ) ⟩| ξ [ im √ 2 π � dt ] 2 � dt ( q 2 t + dt − ξ 2 ) = m ⟨ anti ξ | ψ ( t ) ⟩ exp m × { δ c ( ξ − q t + dt ) ) n  ( � dt ∂ n δ c ( ξ − q t + dt ) ( − i ) n idt  ∑  − � b n  .  ∂ξ n m   n = 2 → Only the component with ξ = q t + dt contributes to m ⟨ new q t + dt | ψ ( t + dt ) ⟩ . Thus, we have obtained the momentum relation : p = ∂ L q = m ˙ q . ∂ ˙

Properties of the future-included theory KN, H.B.Nielsen, PTEP(2013) 023B04 Nielsen and Ninomiya, Proc. Bled 2006, p87. ∫ i � S TA = −∞ to t D path , ⟨ q | A ( t ) ⟩ = e path ( t ) = q ∫ i � S t to TB = ∞ D path , ⟨ B ( t ) | q ⟩ ≡ e path ( t ) = q | A ( t ) ⟩ and | B ( t ) ⟩ time-develop according to dt | A ( t ) ⟩ = ˆ dt | B ( t ) ⟩ = ˆ i � d H | A ( t ) ⟩ , i � d H B | B ( t ) ⟩ , where H B = ˆ ˆ H † . ⟨O⟩ BA ≡ ⟨ B ( t ) |O| A ( t ) ⟩ ⟨ B ( t ) | A ( t ) ⟩

dt ⟨O⟩ BA = ⟨ i Utilizing d � [ ˆ H , O ] ⟩ BA , we obtain • Heisenberg equation • Ehrenfest’s theorem: d q new ⟩ BA = 1 p new ⟩ BA , dt ⟨ ˆ m ⟨ ˆ d p new ⟩ BA = −⟨ V ′ (ˆ q new ) ⟩ BA . dt ⟨ ˆ * momentum relation p = m ˙ q KN, H.B.Nielsen, IJMPA27 (2012)1250076 • Conserved probability current density

Properties of the future-not-included theory KN, H.B.Nielsen, PTEP(2013) 073A03 { ˆ i � d O⟩ AA = ⟨ [ ˆ H h ] ⟩ AA + } dt ⟨ ˆ O , ˆ O − ⟨ ˆ O⟩ AA , ˆ H a , ≃ ⟨ [ ˆ O , ˆ H h ] ⟩ A ( t ) A ( t ) , O⟩ AA ≡ ⟨ A ( t ) |O| A ( t ) ⟩ where ⟨ ˆ ⟨ A ( t ) | A ( t ) ⟩ . Thus, we obtain d 1 q new ⟩ AA ≃ p new ⟩ AA , dt ⟨ ˆ ⟨ ˆ m eff d p new ⟩ AA ≃ −⟨ V ′ q new ) ⟩ AA , dt ⟨ ˆ R (ˆ m 2 where m eff ≡ m R + m R . → p = m eff ˙ q . I We show that the method works also in FNIT.

They give Ehrenfest’s theorem: d 2 q new ⟩ AA ≃ −⟨ V ′ q new ) ⟩ AA . m eff dt 2 ⟨ ˆ R (ˆ This suggests that the classical theory of FNIT is described not by a full action S , but S eff : ∫ t S eff ≡ dtL eff , T A q , q ) ≡ 1 q 2 − V R ( q ) � L R . L eff (˙ 2 m eff ˙ Thus, we claim that in FNIT the classical theory is q = ∂ L eff described by δ S eff = 0 , and p = m eff ˙ q . ∂ ˙ This is quite in contrast to the classical theory of FIT, which would be described by δ S = 0 , where ∫ T B S = T A dtL , and p = m ˙ q .

Table: Comparison between FIT and FNIT FIT FNIT ∫ T B ∫ t action S = T A dtL S = T A dtL O⟩ BA = ⟨ B ( t ) | ˆ O⟩ AA = ⟨ A ( t ) | ˆ ⟨ ˆ O| A ( t ) ⟩ ⟨ ˆ O| A ( t ) ⟩ “exp. value” ⟨ B ( t ) | A ( t ) ⟩ ⟨ A ( t ) | A ( t ) ⟩ dt ⟨ ˆ dt ⟨ ˆ i � d O⟩ BA i � d O⟩ AA time development = ⟨ [ ˆ ≃ ⟨ [ ˆ O , ˆ H ] ⟩ BA O , ˆ H h ] ⟩ AA classical δ S = 0 δ S eff = 0 , S eff = ∫ t theory T A dtL eff momentum p = m ˙ q p = m eff ˙ q relation

Reconsideration of the method in FNIT In the method we looked at a transition amplitude from t i to t f , which is similar to that in FIT: ⟨ B ( t ) | A ( t ) ⟩ = ⟨ B ( T B ) | e − i � ˆ H ( T B − T A ) | A ( T A ) ⟩ . In FNIT : I ≡ ⟨ A ( t ) | A ( t ) ⟩ � ˆ i H † ( t − T A ) e − i � ˆ H ( t − T A ) | A ( T A ) ⟩ = ⟨ A ( T A ) | e ∫ ∫ C ′ D q ′ e − i � S TA to t ( q ) ∗ q e i � S TA to t ( q ′ ) D q = C × ψ A ( q T A , T A ) ∗ qTA ψ A ( q ′ T A , T A ) . → a path from T A to t , and that from t to T A .

We formally rewrite ⟨ A ( t ) | A ( t ) ⟩ into another expression similar to ⟨ B ( t ) | A ( t ) ⟩ by inverting the time direction of the transition amplitude from T A to t , and introduce L formal . S T A to t ( q ) ∗ q ∫ t dt ′ L ( q ( t ′ ) , ˙ q ( t ′ )) ∗ q = T A ∫ − T A + 2 t dt ′′ L ( q formal ( t ′′ , t ) , − ∂ t ′′ q formal ( t ′′ , t )) ∗ q formal , = t where t ′′ = − t ′ + 2 t , q formal ( t ′′ , t ) ≡ q ( − t ′′ + 2 t ) = q ( t ′ ) .

Then I is written as ∫ t ∫ ∫ i TA dt ′ L ( q ′ ( t ′ ) , ˙ q ′ ( t ′ )) C ′ D q ′ I = C ′′ D q formal e � ∫ TB dt ′′ L ( q formal ( t ′′ , t ) , − ∂ t ′′ q formal ( t ′′ , t )) ∗ q formal J ψ A ( q ′ × e − i T A , T A ) , � t where C ′′ is a contour of q formal ( t ′′ , t ) , and ∫ − TA + 2 t ∗ q ′ ∫ formal e − i dt ′′ L ( q ′ formal ( t ′′ , t ) , − ∂ t ′′ q ′ formal ( t ′′ , t )) formal C ′′′ D q ′ J = � TB formal ( − T A + 2 t , t ) , T A ) ∗ q ′ × ψ A ( q ′ formal = ⟨ A (2 t − T B ) | q ′ formal ( T B , t ) ⟩ formal ( T B , t ) , 2 t − T B ) ∗ q ′ = ψ A ( q ′ formal .