Download Scattering with partial information

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?