Momentum relation and classical limit in the future-not-included complex action theory

Transcription

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

2 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

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

4 ˆq new 1 1 ϵϵ (ˆq iϵ ˆp), ˆp new ( 1 ϵϵ q new 4π ϵ ( 1 ϵϵ p new 4π ϵ ) 1 4 e 1 4 ϵ (1 ϵϵ )q 1 ϵϵ 2 2 ϵ ) 1 4 e 1 4 ϵ (1 ϵϵ )p 2 i 1 ( ˆp + iϵ ˆq ), 1 ϵϵ q coh, 1 ϵϵ 2 ϵ p coh. λ coh e λa 0 satisfies a λ coh = λ λ coh, where 1 a = 2 ϵ (ˆq + iϵ ˆp). ) λ coh e λa 0, where a ϵ = 2 (ˆq i ˆp ϵ, is another coherent state defined similarly.

5 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

6 We decompose some function f as f = Re {} f + iim {} f, where Re {} f and Im {} f are the {}-real and {}-imaginary parts of f defined by Re {} f f + f {} 2 and Im {} f f f {} 2i. 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.

7 Theorem on matrix elements m new q or p O(ˆq new, ˆq new, ˆp new, ˆp new) q or p new, where O is a Taylor-expandable function, can be evaluated as if inside O we had the hermiticity conditions ˆq new ˆq new ˆq and ˆp new ˆp new ˆp 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 anti-hermitian terms in ˆq new, ˆq new, ˆp new and ˆp new, provided that we are satisfied with the result in the distribution sense.

8 Deriving the momentum relation via FPI KN, H.B.Nielsen, IJMPA27 (2012) Lagrangian in a system with a single d.o.f.: L(q(t), q(t)) = 1 2 m q2 V(q), V(q) = n=2 b n q n, V = V R + iv I, L = L R + il I, where V R Re q (V) = n=2 Reb n q n, V I Im q (V) = n=2 Imb n q n, L R Re q (L) = 1 2 m R q 2 V R (q), L I Im q (L) = 1 2 m I q 2 V I (q). m = m R + im I.

9 m new q t+dt ψ(t + dt) = e i tl(q, q) m new q t ψ(t) dq t. C We consider m new q t ξ which obeys m new q t ˆp new ξ = m new q t ξ i q t = 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) dξ m new q t ξ m anti ξ ψ(t) C = dξ m new q t ψ(t) ξ. C

10 Then, we obtain = m new q t+dt ψ(t + dt) ξ 2π dt m m anti ξ ψ(t) exp {δ c (ξ q t+dt ) ( ) n dt ( i) nidt m b n n=2 [ im ] 2 dt (q2 t+dt ξ2 ) n δ c (ξ q t+dt ) ξ n. 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.

11 Properties of the future-included theory KN, H.B.Nielsen, PTEP(2013) 023B04 Nielsen and Ninomiya, Proc. Bled 2006, p87. q A(t) = e i S T A = to t Dpath, path(t)=q B(t) q e i S t to T B = Dpath, path(t)=q A(t) and B(t) time-develop according to i d dt A(t) = Ĥ A(t), i d dt B(t) = Ĥ B B(t), where Ĥ B = Ĥ. O BA B(t) O A(t) B(t) A(t)

12 Utilizing d dt O BA = i [Ĥ, O] BA, we obtain Heisenberg equation Ehrenfest s theorem: d dt ˆq new BA = 1 m ˆp new BA, d dt ˆp new BA = V (ˆq new ) BA. * momentum relation p = m q KN, H.B.Nielsen, IJMPA27 (2012) Conserved probability current density

13 Properties of the future-not-included theory KN, H.B.Nielsen, PTEP(2013) 073A03 i d dt Ô AA = [Ô, Ĥ h ] AA + { Ô Ô AA, Ĥ a }, [Ô, Ĥ h ] A(t)A(t), where Ô AA A(t) O A(t) A(t) A(t). Thus, we obtain d dt ˆq new AA 1 ˆp new AA, m eff d dt ˆp new AA V R (ˆq new) AA, where m eff m R + m2 I m R. p = m eff q. We show that the method works also in FNIT.

14 They give Ehrenfest s theorem: d 2 m eff dt 2 ˆq new AA V R (ˆq new) AA. This suggests that the classical theory of FNIT is described not by a full action S, but S eff : S eff t T A dtl eff, L eff ( q, q) 1 2 m eff q 2 V R (q) L R. Thus, we claim that in FNIT the classical theory is described by δs eff = 0, and p = m eff q = L eff q. This is quite in contrast to the classical theory of FIT, which would be described by δs = 0, where S = T B dtl, and p = m q. T A

15 Table: Comparison between FIT and FNIT FIT FNIT action S = T B dtl S = t dtl T A T A exp. value Ô BA = B(t) Ô A(t) B(t) A(t) Ô AA = A(t) Ô A(t) A(t) A(t) time i d dt Ô BA i d dt Ô AA development = [Ô, Ĥ] BA classical theory momentum relation [Ô, Ĥ h ] AA δs = 0 δs eff = 0, S eff = t dtl T eff A p = m q p = m eff q

16 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 Ĥ(T B T A ) A(T A ). In FNIT : I A(t) A(t) = A(T A ) e i Ĥ (t T A ) e i Ĥ(t TA) A(T A ) = Dq Dq e i S T A to t(q) qe i S T A to t(q ) C C ψ A (q TA, T A ) q TA ψ A (q T A, T A ). a path from T A to t, and that from t to T A.

17 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 TA to t(q) q t T A dt L(q(t ), q(t )) q TA +2t t dt L(q formal (t, t), t q formal (t, t)) q formal, where t = t + 2t, q formal (t, t) q( t + 2t) = q(t ).

18 Then I is written as I = Dq C e i TB t C Dq formal e i t T A dt L(q (t ), q (t )) dt L(q formal (t,t), t q formal (t,t)) q formal JψA (q T A, T A ), where C is a contour of q formal (t, t), and TA +2t T J = B C Dq formal e i ψ A (q formal ( T A + 2t, t), T A ) q formal = A(2t T B ) q formal (T B, t) = ψ A (q formal (T B, t), 2t T B ) q formal. dt L(q formal (t,t), t q formal (t,t)) q formal

19 Expressing q (t ) for T A t t as q formal (t, t), we can rewrite I as I Dq formal e i TB T A dt { ϵ(t t)}l formal (q formal (t,t), t q formal (t,t),t t) ψ A (q formal (T B, t), 2t T B ) q formalψa (q formal (T A, t), T A ), where ϵ(t) is 1 for t > 0 and 1 for t < 0, and L formal (q formal (t, t), t q formal (t, t), t t) = 1 2 m formal(t t) ( t q formal (t, t)) 2 V formal (q formal (t, t), t t), m formal (t t) m R iϵ(t t)m I, V formal (q formal (t, t), t t) V R (q formal (t, t)) iϵ(t t)v I (q formal (t, t)).

20 Replacing L with L formal in the method, we obtain p formal (t, t) = L formal(q formal (t, t), t q formal (t, t), t t) ( t q formal (t, t)) = m formal (t t) t q formal (t, t). We take the time average of t q formal around t = t. { } d dt q(t) t q formal(t, t) t =t 1 2 t = 1 2 t t+ t t t t+ t t t dt t q formal (t, t) dt p formal (t, t) m formal (t t) 1 m eff p(t), where p(t) p formal (t, t). Thus, we have reproduced p = m eff q.

21 Summary In our previous paper we derived the momentum relation p = m q by considering a transition amplitude from some initial time to final time, which is similar to that in FIT. In this paper we provided a way to properly apply the method to FNIT by rewriting the transition amplitude in FNIT into another expression similar to that in FIT, and by introducing L formal. We explicitly derived the momentum relation p = m eff q in FNIT via this method.

22 In FNIT classical physics is described not by a full action S but a certain real action S eff ( S R ): S eff = t L eff, where L eff = 1 2 m eff q 2 V R (q). momentum relation is given by ˆp new AA d = m eff dt ˆq new AA, p = m eff q, where m eff = m R + m2 I m R. quite different from those in FIT. In FIT classical theory is described by a full action S. momentum relation is given by ˆp new BA = m d dt ˆq new BA, p = m q.

23 Outlook It is interesting to see the dynamics of the CAT in a simple model such as a harmonic oscillator. The potential of the slow roll inflation is extremely flat. The imaginary part might help us to have more natural potential.