Measurement schemes for quantum field theory
in curved spacetimes
CJ Fewster
University of YorkFirst Virtual LQP/AQFTUKJune 2020
arXiv:1810.06512 - with Rainer Verch; short summary in arXiv:1904.06944
arXiv:2003.04660 with Henning Bostelmann and Maximilian Ruep
CJ Fewster University of York

Measurement schemes for QFT in CST

1 / 24

A gap in the literature
Measurement theory in quantum mechanics has a long and controversial history.
I Simple rules are taught to students
I Measurement chain analysed in quantum measurement theory

CJ Fewster University of York

Measurement schemes for QFT in CST

2 / 24

A gap in the literature
Measurement theory in quantum mechanics has a long and controversial history.
I Simple rules are taught to students
I Measurement chain analysed in quantum measurement theory
Much less is said in quantum field theory
I Lecture courses and texts are silent
I QMT rarely discussed for QFT; still less in curved spacetimes.
I Algebraic QFT is founded on the idea of local observables,
but little discussion of how they are actually measured.

CJ Fewster University of York

Measurement schemes for QFT in CST

2 / 24

In this talk...

I Analyse the measurement chain in QFT
I Provide a general operational framework for measurement
I Covariant; applies in curved as well as flat spacetime
I Passes consistency tests
I Can be used for calculation

CJ Fewster University of York

Measurement schemes for QFT in CST

3 / 24

Relativity, quantum theory and measurement – it’s complicatedc<∞measurements occupy bounded
spacetime regions

ReportingFunding

CJ Fewster University of York

Measurement schemes for QFT in CST

4 / 24

Relativity, quantum theory and measurement – it’s complicatedc<∞measurements occupy bounded
spacetime regions~>0...but are not performed at points

CJ Fewster University of York∆E ∆t & ~Measurement schemes for QFT in CST

4 / 24

Relativity, quantum theory and measurement – it’s complicatedc<∞measurements occupy bounded
spacetime regions~>0...but are not performed at points
Reduced state

Lorentz invarianceno preferred frame
no instantaneous collapse at constant t
Original state
Hellwig & Kraus

CJ Fewster University of York

Measurement schemes for QFT in CST

4 / 24

Relativity, quantum theory and measurement – it’s complicatedc<∞measurements occupy bounded
spacetime regions~>0...but are not performed at points

Lorentz invarianceno preferred frame
no instantaneous collapse at constant t

Relativity ofsimultaneity

no preferred order for
spacelike separated measurements

CJ Fewster University of York

Measurement schemes for QFT in CST

4 / 24

Relativity, quantum theory and measurement – it’s complicatedc<∞measurements occupy bounded
spacetime regions~>0...but are not performed at points

Lorentz invarianceno preferred frame
no instantaneous collapse at constant t

Relativity ofsimultaneity

no preferred order for
spacelike separated measurements

Curved spacetime

lack of symmetry, nontrivial topology...

CJ Fewster University of York

Measurement schemes for QFT in CST

Ligo

4 / 24

Relativity, quantum theory and measurement – it’s complicatedc<∞measurements occupy bounded
spacetime regions~>0...but are not performed at points

Lorentz invarianceno preferred frame
no instantaneous collapse at constant t

Relativity ofsimultaneity

no preferred order for
spacelike separated measurements

Curved spacetime

lack of symmetry, nontrivial topology...

Ligo

Extension of QM measurement rules to QFT is nontrivial and risks pathology
CJ Fewster University of York

Measurement schemes for QFT in CST

4 / 24

Impossible measurements Sorkin 1993
‘By hand’ extension of QM rules to QFT

C

B

A

Claim: nonselective measurement of a typical observable B allows C to determine
whether A has conducted a measurement – superluminal communication.
Presumably, therefore, B represents an impossible measurement.

CJ Fewster University of York

Measurement schemes for QFT in CST

5 / 24

Impossible measurements Sorkin 1993
‘By hand’ extension of QM rules to QFT

C

B

A

Spacetime extension of B is critical.

CJ Fewster University of York

Measurement schemes for QFT in CST

5 / 24

Impossible measurements Sorkin 1993
‘By hand’ extension of QM rules to QFT

C

B

A

“[I]t becomes a priori unclear, for quantum field theory, which observables can bemeasured consistently with causality and which can’t.

CJ Fewster University of York

Measurement schemes for QFT in CST

5 / 24

Impossible measurements Sorkin 1993
‘By hand’ extension of QM rules to QFT

C

B

A

“[I]t becomes a priori unclear, for quantum field theory, which observables can bemeasured consistently with causality and which can’t.
This would seem to deprive [QFT] of any definite measurement theory, leaving theissue of what can actually be measured to (at best) a case-by-case analysis”
See e.g., Borsten, Jubb, Kells (2019) for such an analysis.
CJ Fewster University of York

Measurement schemes for QFT in CST

5 / 24

Operational approach CJF & Verch, 2018
Instead of constructing rules for QFT de novo, apply a systematic approach bymodelling the measurement process.

CJ Fewster University of York

Measurement schemes for QFT in CST

6 / 24

Operational approach CJF & Verch, 2018
A QFT (system) is coupled to another QFT (probe) in a compact spacetime region K(a proxy for the experimental design). The probe is measured elsewhere.
Measure probe

K

Prepare system and probe
CJ Fewster University of York

Measurement schemes for QFT in CST

6 / 24

Operational approach CJF & Verch, 2018
Measurements are performed on the coupled system–probe set-up,
A QFT (system) is coupled to another QFT (probe) in a compact spacetime region K
but are described in the language of a fictitious uncoupled system.
(a proxy for the experimental design). The probe is measured elsewhere.
Updated state

Measure probe

?

≈

K

Prepare system and probe
CJ Fewster University of York

Hypothetical
local systemmeasurement

Prepare system
Measurement schemes for QFT in CST

6 / 24

Outline of the idea
Describe the system and probe by QFTs A, B on spacetime M (globally hyperbolic).
A(M) is the algebra of system observables on M. We compare
I the uncoupled combination U of A and B
I a coupled combination C with compact coupling region K .

CJ Fewster University of York

Measurement schemes for QFT in CST

7 / 24

Outline of the idea
Describe the system and probe by QFTs A, B on spacetime M (globally hyperbolic).
A(M) is the algebra of system observables on M. We compare
I the uncoupled combination U of A and B
I a coupled combination C with compact coupling region K .
Define geometrical ‘in’/‘out’ regions M −/+ on which U and C agree.

M+K

M ± = M \ J ∓ (K )M−

CJ Fewster University of York

Measurement schemes for QFT in CSTK

7 / 24

Outline of the idea
Describe the system and probe by QFTs A, B on spacetime M (globally hyperbolic).
A(M) is the algebra of system observables on M. We compare
I the uncoupled combination U of A and B
I a coupled combination C with compact coupling region K .
Define geometrical ‘in’/‘out’ regions M −/+ on which U and C agree.
M ± contain Cauchy surfaces =⇒ ∃ identifications τ ± : U(M) → C(M).
M+K

M ± = M \ J ∓ (K )M−

CJ Fewster University of York

Measurement schemes for QFT in CSTK

7 / 24

Outline of the idea
Describe the system and probe by QFTs A, B on spacetime M (globally hyperbolic).
A(M) is the algebra of system observables on M. We compare
I the uncoupled combination U of A and B
I a coupled combination C with compact coupling region K .
Define geometrical ‘in’/‘out’ regions M −/+ on which U and C agree.
M ± contain Cauchy surfaces =⇒ ∃ identifications τ ± : U(M) → C(M).
τ ± translate fictitious uncoupled language to the physical coupled system.
I ‘Prepare system and probe states separately at early times’
I ‘Measure a probe observable at late times’
Measurements on the probe are interpreted as measurements of system observables.

CJ Fewster University of York

Measurement schemes for QFT in CST

7 / 24

Algebraic QFT
Describe a QFT on M in terms of a ∗-algebra A(M) with unit and subalgebrasA(M; N) for suitable open regions N ⊂ M.
Minimal conditions
Isotony N1 ⊂ N2 =⇒ A(M; N1 ) ⊂ A(M; N2 )Timeslice A(M; N) = A(M) if N contains a Cauchy surface of M
Einstein [A(M; N1 ), A(M; N2 )] = 0 if N1,2 are causally disjoint

CJ Fewster University of York

Measurement schemes for QFT in CST

8 / 24

Algebraic QFT
Describe a QFT on M in terms of a ∗-algebra A(M) with unit and subalgebrasA(M; N) for suitable open regions N ⊂ M.
Minimal conditions
Isotony N1 ⊂ N2 =⇒ A(M; N1 ) ⊂ A(M; N2 )Timeslice A(M; N) = A(M) if N contains a Cauchy surface of M
Einstein [A(M; N1 ), A(M; N2 )] = 0 if N1,2 are causally disjoint
A = A∗ ∈ A(M; N) is interpreted by fiat as an observable localisable in N
NB An observable may be localisable in many distinct regions.

CJ Fewster University of York

Measurement schemes for QFT in CST

8 / 24

Algebraic QFT
Describe a QFT on M in terms of a ∗-algebra A(M) with unit and subalgebrasA(M; N) for suitable open regions N ⊂ M.
Minimal conditions
Isotony N1 ⊂ N2 =⇒ A(M; N1 ) ⊂ A(M; N2 )Timeslice A(M; N) = A(M) if N contains a Cauchy surface of M
Einstein [A(M; N1 ), A(M; N2 )] = 0 if N1,2 are causally disjoint
A = A∗ ∈ A(M; N) is interpreted by fiat as an observable localisable in N
NB An observable may be localisable in many distinct regions.

A state is a positive, normalised linear functional ω : A(M) → C,assigning an expectation value ω(A) to A ∈ A(M).

CJ Fewster University of York

Measurement schemes for QFT in CST

8 / 24

Coupled combinations and scattering
Describe both the system and the probe by AQFTs A, B on M.
Their uncoupled combination is U = A ⊗ B.
Theory C is a coupled combination of A and B with compact coupling region K .
ch (K ) = J + (K ) ∩ J − (K )Minimal abstract definition:∀L outside the causal hull ch (K )∃ an isomorphismch K

U(M; L) → C(M; L)

Lcompatible with isotony.

CJ Fewster University of York

Measurement schemes for QFT in CST

9 / 24

Coupled combinations and scattering
Describe both the system and the probe by AQFTs A, B on M.
Their uncoupled combination is U = A ⊗ B.
Theory C is a coupled combination of A and B with compact coupling region K .
M+

M ± = M \ J ∓ (K ) contain Cauchy surfaces for M
Applying timeslice, ∃ isomorphismsKτ ± : U(M) → C(M)

τ ± : U(M) = U(M; M ± ) −→ C(M; M ± ) = C(M)CJ Fewster University of York

Measurement schemes for QFT in CST

9 / 24

Coupled combinations and scattering
Describe both the system and the probe by AQFTs A, B on M.
Their uncoupled combination is U = A ⊗ B.
Theory C is a coupled combination of A and B with compact coupling region K .
Upshot: covariantly described advanced/retarded response maps∼=

τ −/+ : U(M) −→ C(M)are identifications of the uncoupled and coupled combinations at early/late times.
The scattering map isΘ = (τ − )−1 ◦ τ + ∈ Aut(U(M))
Locality: Θ  U(M; N) = id, if N ⊂ K ⊥ .

CJ Fewster University of York

Measurement schemes for QFT in CST

9 / 24

Measurement scheme: prepare early, measure late
Describe measurements of C(M) in uncoupled language.
Fixing a probe preparation state σ and system state ω, the stateω σ = ((τ − )−1 )∗ (ω ⊗ σ)

ω σ (X ) = (ω ⊗ σ)((τ − )−1 X )eeof C(M) is uncorrelated at early times.

CJ Fewster University of York

Measurement schemes for QFT in CST

10 / 24

Measurement scheme: prepare early, measure late
Describe measurements of C(M) in uncoupled language.
Fixing a probe preparation state σ and system state ω, the stateω σ = ((τ − )−1 )∗ (ω ⊗ σ)

ω σ (X ) = (ω ⊗ σ)((τ − )−1 X )eeof C(M) is uncorrelated at early times.

e := τ + (1 ⊗ B) tests probe degrees of freedom at late times.
An observable B

CJ Fewster University of York

Measurement schemes for QFT in CST

10 / 24

Measurement scheme: prepare early, measure late
Describe measurements of C(M) in uncoupled language.
Fixing a probe preparation state σ and system state ω, the stateω σ = ((τ − )−1 )∗ (ω ⊗ σ)

ω σ (X ) = (ω ⊗ σ)((τ − )−1 X )eeof C(M) is uncorrelated at early times.

e := τ + (1 ⊗ B) tests probe degrees of freedom at late times.
An observable B
e in state ω σ ; interpret as a measurement of a system observable A in state ωMeasure Bechosen so thate = (ω ⊗ σ)(Θ(1 ⊗ B))
ω(A) = ω σ (B)for all ω.

e

CJ Fewster University of York

Measurement schemes for QFT in CST

10 / 24

Measurement scheme - ctd
Problem: find A so that ω(A) = (ω ⊗ σ)(Θ(1 ⊗ B)) for all ω
Observation:(ω ⊗ σ)(P ⊗ Q) = ω(P)σ(Q) = ω(σ(Q)P) = ω(ησ (P ⊗ Q))
where ησ : A(M) ⊗ B(M) → A(M) linearly extends P ⊗ Q 7→ σ(Q)P.
Solution:def

A = εσ (B) = ησ (Θ(1 ⊗ B))

CJ Fewster University of York

Measurement schemes for QFT in CST

11 / 24

Measurement scheme - ctd
Problem: find A so that ω(A) = (ω ⊗ σ)(Θ(1 ⊗ B)) for all ω
Observation:(ω ⊗ σ)(P ⊗ Q) = ω(P)σ(Q) = ω(σ(Q)P) = ω(ησ (P ⊗ Q))
where ησ : A(M) ⊗ B(M) → A(M) linearly extends P ⊗ Q 7→ σ(Q)P.
Solution:def

A = εσ (B) = ησ (Θ(1 ⊗ B))
εσ (B) = ησ (Θ(1 ⊗ B)) is called the induced system observablecorresponding to probe observable B.
In QMT language, (C, τ ± , σ, B) is a measurement scheme for system observable εσ (B).
CJ Fewster University of York

Measurement schemes for QFT in CST

11 / 24

Induced system observables – localisation
Recall: Θ acts trivially on U(M; L) if L ⊂ K ⊥ .
Theorem (a) If B ∈ B(M; L) with L ⊂ K ⊥ then εσ (B) = σ(B)1.

CJ Fewster University of York

Measurement schemes for QFT in CST

L

K

12 / 24

Induced system observables – localisation
Recall: Θ acts trivially on U(M; L) if L ⊂ K ⊥ .
Theorem (a) If B ∈ B(M; L) with L ⊂ K ⊥ then εσ (B) = σ(B)1.

L

K(b) If A ∈ A(M; L) with L ⊂ K ⊥ then [A, εσ (B)] = 0 for all B.

CJ Fewster University of York

Measurement schemes for QFT in CST

12 / 24

Induced system observables – localisation
Recall: Θ acts trivially on U(M; L) if L ⊂ K ⊥ .
Theorem (a) If B ∈ B(M; L) with L ⊂ K ⊥ then εσ (B) = σ(B)1.

L

K(b) If A ∈ A(M; L) with L ⊂ K ⊥ then [A, εσ (B)] = 0 for all B.
Corollary If A obeys a Haag property, thenεσ (B) ∈ A(M; N)

for all B ∈ B(M),where N is any open connected causally convex set containing K .
NB N must contain ch K . The localisation of B is irrelevant.
Consistent with the idea that A(M; N) consists of observables
that are measurable in N.
CJ Fewster University of York

Measurement schemes for QFT in CST

ch K

12 / 24

Induced system observables – fluctuations
True and hypothetical expectation values agree, by constructione = ω(εσ (B))
ω σ (B)

for all B ∈ B(M).

eεσ : B(M) → A(M) is linear, completely positive, and obeys
εσ (1) = 1,εσ (B ∗ ) = εσ (B)∗ ,εσ (B)∗ εσ (B) ≤ εσ (B ∗ B).

Consequently, the true measurement displays greater variance than the hypotheticalone due to detector fluctuationse ω σ ) ≥ Var(εσ (B); ω).
Var(B;eCJ Fewster University of York

Measurement schemes for QFT in CST

13 / 24Effects and effect-valued measures
An effect is an observable s.t. B and 1 − B are positive,corresponding to a true/false measurement
Prob(B | ω) = ω(B),

Prob(¬B | ω) = ω(1 − B).

Unsharp unless B is a projection.
Because εσ is completely positive, but not a homomorphism in general:
I probe effects induce system effects
I even sharp probe effects typically induce unsharp system effects.

CJ Fewster University of York

Measurement schemes for QFT in CST

14 / 24

Post-selection and pre-instruments
Suppose a probe-effect B is tested when the system state is ω.
The post-selected system state, conditioned on the effect being observed, should
correctly predict the probability of any system effect being observed, given that B was.

CJ Fewster University of York

Measurement schemes for QFT in CST

15 / 24

Post-selection and pre-instruments
Probability of a joint successful measurement of system effect A and probe effect B isdef

Prob(A&B) = (ω ⊗ σ)(Θ(A ⊗ B)) = (Iσ (B)(ω))(A)so

Prob(A|B) =Prob(A&B)(Iσ (B)(ω))(A)=,
Prob(B)(Iσ (B)(ω))(1)

Call Iσ (B) : A(M)∗+ → A(M)∗+ a pre-instrument.

CJ Fewster University of York

Measurement schemes for QFT in CST

15 / 24

Post-selection and pre-instruments
Probability of a joint successful measurement of system effect A and probe effect B isdef

Prob(A&B) = (ω ⊗ σ)(Θ(A ⊗ B)) = (Iσ (B)(ω))(A)so

Prob(A|B) =Prob(A&B)(Iσ (B)(ω))(A)=,
Prob(B)(Iσ (B)(ω))(1)

Call Iσ (B) : A(M)∗+ → A(M)∗+ a pre-instrument.
The post-selected state, conditioned on B, isω0 =Iσ (B)(ω)(Iσ (B)(ω))(1)(if defined).

Non-selective measurement results in ωB = Iσ (B)(ω) + Iσ (1 − B)(ω) = Iσ (1)(ω)
independent of B!

CJ Fewster University of York

Measurement schemes for QFT in CST

15 / 24

Locality and post-selection
Theorem For A localisable in K ⊥ ,ω 0 (A) =ω(Aεσ (B))
ω(εσ (B))

Corollary ω 0 (A) = ω(A) iff A is uncorrelated with εσ (B) in ω.
ω 0 (A) = ω(A) for nonselective measurement of B

CJ Fewster University of York

Measurement schemes for QFT in CST

16 / 24

Locality and post-selection
Theorem For A localisable in K ⊥ ,ω 0 (A) =ω(Aεσ (B))
ω(εσ (B))

Corollary ω 0 (A) = ω(A) iff A is uncorrelated with εσ (B) in ω.
ω 0 (A) = ω(A) for nonselective measurement of B
If ω has a Reeh–Schlieder property, (e.g., Minkowski vacuum state)ω 0 (A) = ω(A) ⇐⇒ εσ (B) = ω(εσ (B))1for observables A localisable in K ⊥ .
Post-selecting on nontrivial effects alters expectation values in K ⊥ due to correlation.

CJ Fewster University of York

Measurement schemes for QFT in CST

16 / 24

Example – correlated observables
Commuting effects A, A0 are perfectly correlated in state ω if
Prob(A&A0 ) + Prob(¬A&¬A0 ) = 1⇐⇒

ω(A(1 − A0 )) = 0 = ω((1 − A)A0 )Consider a measurement scheme in which A = εσ (B) for probe effect B.
If ω 0 is the updated state, conditioned on successful measurement of B, thenProb(A0 | ω 0 ) = ω 0 (A0 ) = 1 − E

CJ Fewster University of York

Measurement schemes for QFT in CST

17 / 24

Example – correlated observables
Commuting effects A, A0 are perfectly correlated in state ω if
Prob(A&A0 ) + Prob(¬A&¬A0 ) = 1

ω(A(1 − A0 )) = 0 = ω((1 − A)A0 )⇐⇒Consider a measurement scheme in which A = εσ (B) for probe effect B.
If ω 0 is the updated state, conditioned on successful measurement of B, thenProb(A0 | ω 0 ) = ω 0 (A0 ) = 1 − E
where E ≥ 0 is bounded by!

e ωσ )
Var(B;E ≤ (ω ⊗ σ)(∆ ) 1 +
,
e
ω(A)2
2

2

∆ = (id − Θ)(A0 ⊗ 1).

Both factors can be reduced by experimental design.
CJ Fewster University of York

Measurement schemes for QFT in CST

17 / 24

Example – correlated observables
Commuting effects A, A0 are perfectly correlated in state ω if
Prob(A&A0 ) + Prob(¬A&¬A0 ) = 1

ω(A(1 − A0 )) = 0 = ω((1 − A)A0 )⇐⇒Consider a measurement scheme in which A = εσ (B) for probe effect B.
If ω 0 is the updated state, conditioned on successful measurement of B, thenProb(A0 | ω 0 ) = ω 0 (A0 ) = 1 − E
where E ≥ 0 is bounded by!

e ωσ )
Var(B;E ≤ (ω ⊗ σ)(∆ ) 1 +
,
e
ω(A)2
2

2

Prob(A0 | ω 0 ) → 1∆ = (id − Θ)(A0 ⊗ 1).

in the limit of ideal experimentation

CJ Fewster University of York

Measurement schemes for QFT in CST

17 / 24

What versus how?
The updated state depends on Θ and B; not just the system observable A = εσ (B).
Depends on how the measurement was made, not just what was measured.

CJ Fewster University of York

Measurement schemes for QFT in CST

18 / 24

What versus how?
The updated state depends on Θ and B; not just the system observable A = εσ (B).
Depends on how the measurement was made, not just what was measured.
E := max

However,ω 0 (A0 ) −ω(A0 εσ (B))
ω(εσ (B)A0 ), ω 0 (A0 ) −ω(εσ (B))ω(εσ (B))



,

!

obeys

e ωσ )
Var(B;,E ≤ (ω ⊗ σ)(∆ ) 1 +eω(εσ (B))2

ω 0 (A0 ) ≈ω({A0 , εσ (B)})
for those A0 only slightly affected by the interaction.
2ω(εσ (B))

2

2

CJ Fewster University of York∆ = (id − Θ)(A0 ⊗ 1).

Measurement schemes for QFT in CST

18 / 24

Successive measurement of two probes
For i = 1, 2 consider Bi with coupling regions Ki and scattering morphisms Θi .
Combined probe B1 ⊗ B2 has coupling region K1 ∪ K2 and morphism Θ̂.

CJ Fewster University of York

Measurement schemes for QFT in CST

19 / 24

Successive measurement of two probes
For i = 1, 2 consider Bi with coupling regions Ki and scattering morphisms Θi .
Combined probe B1 ⊗ B2 has coupling region K1 ∪ K2 and morphism Θ̂.
Suppose K2 ∩ J − (K1 ) = ∅, so K2 is later than K1 according
to some observers and assume causal factorisation, i.e.,

K2
K1Θ̂ = Θ̂1 ◦ Θ̂2 ,where Θ̂1 = Θ1 ⊗3 idand Θ̂2 = Θ2 ⊗2 id

CJ Fewster University of York

Measurement schemes for QFT in CST

19 / 24

Successive measurement of two probes
For i = 1, 2 consider Bi with coupling regions Ki and scattering morphisms Θi .
Combined probe B1 ⊗ B2 has coupling region K1 ∪ K2 and morphism Θ̂.
Suppose K2 ∩ J − (K1 ) = ∅, so K2 is later than K1 according
to some observers and assume causal factorisation, i.e.,

K2
K1Θ̂ = Θ̂1 ◦ Θ̂2 ,where Θ̂1 = Θ1 ⊗3 idand Θ̂2 = Θ2 ⊗2 id

Theorem Coherence of successive measurement
Iσ2 (B2 ) ◦ Iσ1 (B1 ) = Iσ1 ⊗σ2 (B1 ⊗ B2 )Post-selection on B1 and then B2 agrees with post-selection on B1 ⊗ B2 .

CJ Fewster University of York

Measurement schemes for QFT in CST

19 / 24

Successive measurement of two probes
For i = 1, 2 consider Bi with coupling regions Ki and scattering morphisms Θi .
Combined probe B1 ⊗ B2 has coupling region K1 ∪ K2 and morphism Θ̂.
Suppose K2 ∩ J − (K1 ) = ∅, so K2 is later than K1 according
to some observers and assume causal factorisation, i.e.,Θ̂ = Θ̂1 ◦ Θ̂2 ,where Θ̂1 = Θ1 ⊗3 idand Θ̂2 = Θ2 ⊗2 id

Corollary If K1 and K2 are causally disjoint,Iσ2 (B2 ) ◦ Iσ1 (B1 ) = Iσ1 ⊗σ2 (B1 ⊗ B2 ) = Iσ1 (B1 ) ◦ Iσ2 (B2 )CJ Fewster University of York

Measurement schemes for QFT in CST

K1

K2

19 / 24

Impossible measurements? Bostelmann, CJF & Ruep arXiv:2003.04660
CB
A

I Alice chooses whether to make a nonselective measurement
I Bob certainly makes a nonselective measurement
I Can Charlie determine whether Alice performed the measurement?
?

ωAB (C ) 6= ωB (C )CJ Fewster University of York

Measurement schemes for QFT in CST

20 / 24

Impossible measurements? Bostelmann, CJF & Ruep arXiv:2003.04660
Model A and B measurements using probes

C

B
A

N

More detailed investigation of scattering map locality properties givesΘ̂2 C ⊗ 1 ⊗ 1 ∈ U(M; N)

for region N ⊂ KA⊥ ∩ MB−

Consequently, Charlie cannot determine whether Alice has measured:ωAB (C ) = (ω ⊗ σ1 ⊗ σ2 )(Θ̂1 Θ̂2 C ⊗ 1 ⊗ 1) = (ω ⊗ σ1 ⊗ σ2 )(Θ̂2 C ⊗ 1 ⊗ 1) = ωB (C )CJ Fewster University of York

Measurement schemes for QFT in CST

20 / 24

Impossible measurements? Bostelmann, CJF & Ruep arXiv:2003.04660
Model A and B measurements using probes

C

B
A

N

The analysis shows that the measurement scheme is free of Sorkin-type pathologies.
Key assumption – the probes and couplings are described by physics respecting locality.
Impossible measurements can only be performed using impossible apparatus.

CJ Fewster University of York

Measurement schemes for QFT in CST

20 / 24

A specific probe modelTwo free scalar fields: Φ (system) and Ψ (probe) coupled via an interaction term
Lint = −ρΦΨ,ρ ∈ C0∞ (M),K = supp ρ.

Linear equations: standard quantisation applies at least for sufficiently weak coupling.
As formal power series in h ∈ C0∞ (M + ),hΘ(1 ⊗ e iΨ(h) ) = e iΦ(f−)−⊗ e iΨ(h )where f − and h− − h are supported insupp ρ ∩ J − (supp h).

CJ Fewster University of Yorkρ

Measurement schemes for QFT in CST

21 / 24

A specific probe modelTwo free scalar fields: Φ (system) and Ψ (probe) coupled via an interaction term
Lint = −ρΦΨ,ρ ∈ C0∞ (M),K = supp ρ.

Linear equations: standard quantisation applies at least for sufficiently weak coupling.
As formal power series in h ∈ C0∞ (M + ),hΘ(1 ⊗ e iΨ(h) ) = e iΦ(f−)−⊗ e iΨ(h )ρwhere f − and h− − h are supported insupp ρ ∩ J − (supp h).
−εσ (e iΨ(h) ) = σ e iΨ(h ) e iΦ(f

−)

−

−

= e −S(h ,h )/2 e iΦ(f−)

if σ is quasifree with two-point function S.
CJ Fewster University of York

Measurement schemes for QFT in CST

21 / 24

Examples of induced observables−−

εσ (e iΨ(h) ) = e −S(h ,h )/2 e iΦ(f−)εσ (Ψ(h)) = Φ(f − )εσ (Ψ(h)2 ) = Φ(f − )2 + S(h− , h− )1Consequently,] ω σ ) = ω(Φ(f − ))E(Ψ(h);e] ω σ ) = Var(Φ(f − ); ω) + S(h− , h− )Var(Ψ(h);eIncreased variance in true measurement from detector fluctuations.
CJ Fewster University of York

Measurement schemes for QFT in CST

22 / 24

Deformed product on the probe system
εσ : B(M) → A(M) is not a homomorphism. BUT ∃ a deformed product on B(M),00e iΨ(h) ? e iΨ(h ) =σ(e iΨ(h) )σ(e iΨ(h ) ) −iEP (f − ,f 0− )/2 iΨ(h+h0 )ee
σ(e iΨ(h+h0 ) )in which εσ is a homomorphism (though not injective).
Consequence: the induced observables form a subalgebra of A(M)Im εσ ∼= B(M)/ ker εσ .
so the system is partially represented in the probe algebra.
Example: Ψ(h)’s do not necessarily ?-commute at spacelike separation,[Ψ(h), Ψ(h0 )]? = iEP (f − , f 0− ),allowing for the creation of long-range correlations.
CJ Fewster University of York

Measurement schemes for QFT in CST

23 / 24

Summary
I Operational framework of QMT adapted to AQFT
I covariant, formulated for curved as well as flat spacetimes
I framework derived from minimal assumptions

I Probe observables induce local system observables,
I localisable in the causal hull of coupling region

I Post-selected states
I updated state derived from required properties rather than posited
I reproduces idealised correlations in a limit of idealised measurement
I coherence under successive measurements
I invariant under re-ordering of causally disjoint measurements

I Framework is free of impossible measurements
I Computation of induced observables for specific model

CJ Fewster University of York

Measurement schemes for QFT in CST

24 / 24

Local modification of couplings in QFT

An interaction term
ψ1 ψ2 ϕ1 ϕ2
provides a tunable coupling
between ϕ1 and ϕ2 .

ψ1

ψ2

CJ Fewster University of York

Measurement schemes for QFT in CST

25 / 24

Localisation of induced observables
εσ (Ψ(h)n ) may be localised in any open causally convex nhd of
supp f − ⊂ supp ρ ∩ J − (supp h)

D

W

Localisation region for finite-time coupling is a diamond D.
Localisation region for eternal coupling is a wedge W (can’t do better).
CJ Fewster University of York

Measurement schemes for QFT in CST

26 / 24

Back to Unruh

T

Localisation region for finite-time coupling is a diamond D
Measurements may be taken along future of curve beyond D.
Localisation region for eternal coupling is a wedge. HIghly nonlocal.

CJ Fewster University of York

Measurement schemes for QFT in CST

27 / 24