Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Scattering with partial information Daniel Carney University of British Columbia, Canada Gordon Semenoff Laurent Chaurette arxiv:1606.xxxxx How can we understand (relativistic) scattering in the language of quantum measurement theory? Measurement theory: |0iA ⊗ X i U ci |iiS − → X i ci |iiA ⊗ |iiS Measurement theory: |0iA ⊗ X i U ci |iiS − → Scattering: A = φA = apparatus field(s) S = φS = system field(s) X i ci |iiA ⊗ |iiS Measurement theory: |0iA ⊗ X i U ci |iiS − → Scattering: A = φA = apparatus field(s) S = φS = system field(s) U = S : H− → H+ = S-matrix X i ci |iiA ⊗ |iiS Measurement theory: |0iA ⊗ X i Scattering: A = φA = apparatus field(s) S = φS = system field(s) U = S : H− → H+ = S-matrix ± H ± = HA ⊗ HS± U ci |iiS − → X i ci |iiA ⊗ |iiS Questions 1. Given some property of S we want to measure, what do we observe with A? 2. How much A − S entanglement does scattering generate? 1. Ex: verifying spatial superpositions p 1/2 x xL S xL |Li |Ri p p S =− Z d 4x p q 1 1 1 1 λ (∂µ φS )2 + (∂µ φA )2 + mS2 φ2S + mA2 φ2A + φ2S φ2A 2 2 2 2 4 q − − ρ = |p i hp |A ⊗ 1 2 α Convex family 0 ≤ α ≤ 1. α → 1: coherent superposition of |Li, |Ri α → 0: classical ensemble |Li, |Ri α 1 2 S − − ρ = |p i hp |A ⊗ 1 2 α α 1 2 S Convex family 0 ≤ α ≤ 1. α → 1: coherent superposition of |Li, |Ri α → 0: classical ensemble |Li, |Ri Question: How do we measure α (in perturbation theory in λ?) Answer: position space interference in A. Answer: position space interference in A. P(p) ∼ 1 − |M|2 (1 + α) ∼ 1 − λ2 (1 + α) Answer: position space interference in A. P(p) ∼ 1 − |M|2 (1 + α) ∼ 1 − λ2 (1 + α) P(x, t) = A(x, t) sin (φLR (x, t)) A ∼ αM ∼ αλ p x Answer: position spacexLinterference in A. 1/2 S P(p) ∼ 1 − |M|2 (1 + α) ∼ 1 − λ2 (1 + α) xL |Li |Ri P(x, t) = A(x, t) sin (φLR (x, t)) A ∼ αM ∼p αλ p p p q q p q p p p q q + p q q (1) p 2. Entanglement entropy generated by scattering ↵ p p q1 q2 q3 q1 q4 q2 ↵ − |ψi = |pi− A ⊗ |α = q1 , q2 , . . .iS Assume precisely one apparatus particle in initial and final state. Then easy calculation: Assume precisely one apparatus particle in initial and final state. Then easy calculation: ρ+ A = trH+ ρ S Assume precisely one apparatus particle in initial and final state. Then easy calculation: ρ+ A = trH+ ρ S 1 + I0 + F (p) F (p1 ) ρ+ = A F (p2 ) .. . I0 = −2T ImMpαpα 2 X F (p) = 2πT Mpαpα δp+pα ,p+pα δ(EpA + EαS − EpA − EαS ) α Assume precisely one apparatus particle in initial and final state. Then easy calculation: ρ+ A = trH+ ρ S 1 + I0 + F (p) F (p1 ) ρ+ = A F (p2 ) .. . I0 = −2T ImMpαpα 2 X F (p) = 2πT Mpαpα δp+pα ,p+pα δ(EpA + EαS − EpA − EαS ) α F (p): “conditional total cross-section” So, entanglement entropy: SA = −(1 + I0 + F (p)) ln(1 + I0 + F (p)) − X p6=p F (p) ln F (p) So, entanglement entropy: SA = −(1 + I0 + F (p)) ln(1 + I0 + F (p)) − X F (p) ln F (p) p6=p Exact, non-perturbative. Due entirely to symmetry and assumption of single A particle. So, entanglement entropy: SA = −(1 + I0 + F (p)) ln(1 + I0 + F (p)) − X F (p) ln F (p) p6=p Exact, non-perturbative. Due entirely to symmetry and assumption of single A particle. Assuming perturbation theory holds, X F (p) ln F (p) SA ≈ − p p p Ex: 2 → 2 scattering in λφ2A φ2S theory at lowest order. (cf. Seki-Park-Sin 1412.7894, Peschanski-Seki 1602.00720) q p q iMpqpq = q + = p q iλ q (2π)3 16EpA EqS EpA EqS (1) p p Ex: 2 → 2 scattering in λφ2A φ2S theory at lowest order. (cf. Seki-Park-Sin 1412.7894, Peschanski-Seki 1602.00720) q p q iMpqpq = q + = p q iλ q (2π)3 16EpA EqS EpA EqS (1) 2 T λ2 pcm (E A + E S ) T λ2 SA = − ln V 16π (E A E S )2 V 2 16(E A E S )2 Comments Comments 1. Can use interference measurements to detect things more sensitively than usually considered (i.e. λ vs. λ2 ). Comments 1. Can use interference measurements to detect things more sensitively than usually considered (i.e. λ vs. λ2 ). 2. Entropy: wave packets? Inelastic regime for apparatus? RT? Comments 1. Can use interference measurements to detect things more sensitively than usually considered (i.e. λ vs. λ2 ). 2. Entropy: wave packets? Inelastic regime for apparatus? RT? 3. Black holes: entropy in soft modes? String scattering?